Riemannian Manifolds and Homogeneous Geodesics [1 ed.] 9783030566579, 9783030566586

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Springer Monographs in Mathematics

Valerii Berestovskii Yurii Nikonorov

Riemannian Manifolds and Homogeneous Geodesics

Springer Monographs in Mathematics Editors-in-Chief Isabelle Gallagher, Département de Mathématiques et Applications, Ecole Normale Supérieure, Paris, France Minhyong Kim, School of Mathematics, Korea Institute for Advanced Study, Seoul, South Korea; Mathematical Institute, University of Warwick, Coventry, UK Series Editors Sheldon Axler, Department of Mathematics, San Francisco State University, San Francisco, CA, USA Mark Braverman, Department of Mathematics, Princeton University, Princeton, NJ, USA Maria Chudnovsky, Department of Mathematics, Princeton University, Princeton, NJ, USA Tadahisa Funaki, Department of Mathematics, University of Tokyo, Tokyo, Japan Sinan Güntürk, Department of Mathematics, Courant Institute of Mathematical Sciences, New York University, New York, NY, USA Claude Le Bris, CERMICS, Ecole des Ponts ParisTech, Marne la Vallée, France Pascal Massart, Département de Mathématiques, Université de Paris-Sud, Orsay, France Alberto A. Pinto, Department of Mathematics, University of Porto, Porto, Portugal Gabriella Pinzari, Department of Mathematics, University of Padova, Padova, Italy Ken Ribet, Department of Mathematics, University of California, Berkeley, CA, USA René Schilling, Institut für Mathematische Stochastik, Technische Universität Dresden, Dresden, Germany Panagiotis Souganidis, Department of Mathematics, University of Chicago, Chicago, IL, USA Endre Süli, Mathematical Institute, University of Oxford, Oxford, UK Shmuel Weinberger, Department of Mathematics, University of Chicago, Chicago, IL, USA Boris Zilber, Mathematical Institute, University of Oxford, Oxford, UK

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Valerii Berestovskii Yurii Nikonorov •

Riemannian Manifolds and Homogeneous Geodesics

123

Valerii Berestovskii Sobolev Institute of Mathematics Siberian Branch of the Russian Academy of Sciences Novosibirsk, Russia

Yurii Nikonorov Southern Mathematical Institute Scientific Centre of the Russian Academy of Sciences Vladikavkaz, Russia

ISSN 1439-7382 ISSN 2196-9922 (electronic) Springer Monographs in Mathematics ISBN 978-3-030-56657-9 ISBN 978-3-030-56658-6 (eBook) https://doi.org/10.1007/978-3-030-56658-6 Mathematics Subject Classification: 53C30, 53C22, 53C20, 53B21, 22F30, 22E60, 22E15 © Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Introduction

The main purpose of this book is to give an exposition of recent results (obtained with the essential participation of the authors) on geodesic orbit Riemannian manifolds and their special classes of generalized normal homogeneous and Clifford– Wolf homogeneous Riemannian manifolds. All of the included results of the authors on Killing vector fields (of constant length, in particular), on one-parameter isometry groups on Riemannian manifolds, as well as on the Chebyshev norms on the Lie algebras of the isometry groups of compact homogeneous Finsler (Riemannian, in particular) manifolds, are also closely related to this subject. The book also contains new results of the authors on special classes of geodesic orbit Riemannian manifolds, restrictively Clifford–Wolf homogeneous Riemannian manifolds, and some useful formulas expressing the curvature tensor and the sectional curvature of homogeneous Riemannian manifolds through Killing vector fields. For the purpose of independence of the exposition, we consider useful results not only on geodesic orbit manifolds, but also on smooth and Riemannian manifolds, Lie groups and Lie algebras, homogeneous Riemannian manifolds, and compact homogeneous spaces of positive Euler characteristic. The proofs are given in most cases, otherwise their sketches or all necessary references are provided. The main priorities for proofs are their brevity and simplicity. On the other hand, the proofs of the main results of this book are complete but (as a rule) are not so simple that they may be connected with the use of various ideas and methods. In this Introduction, we give a quite detailed review of the book’s contents and the logical connections between the considered results. The results on Riemannian manifolds, Lie groups and Lie algebras in Chapters 1 and 2 are mainly standard. The main but not unique sources are the book [291] by S. Kobayashi and K. Nomizu and the book [461] by F. Warner. In Chapter 1, we provide the necessary background to study in the following chapters. We introduce the main notions, and prove, discuss or simply quote the basic results on smooth manifolds, manifolds with covariant derivatives, and Riemannian manifolds. In particular, we consider the Lie algebra of smooth vector fields on smooth manifolds, the Levi-Civita connection on Riemannian manifolds, their curv

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vature tensors, sectional and other curvatures, geodesics, the intrinsic metric, and shortest curves. As a rule, we prove the main results. In Section 1.5, we discuss some classical results describing global properties of Riemannian manifolds, for instance, Rauch’s comparison theorem 1.5.25, the Hadamard–Cartan theorem 1.5.78, and Synge’s theorem 1.5.80. Let us especially note the proofs of the Hopf–Rinow theorem 1.5.1 and the S. Myers theorem 1.5.79, A quite simple proof of the S. Myers theorem 1.5.79 on compactness of a complete Riemannian manifold with the Ricci curvature bounded from below by a positive constant and finiteness of its fundamental group is based on the original paper [349] and uses Proposition 1.4.10 (which states that the first variation of a given geodesic generates a Jacobi field), but unlike the original proof, it does not use coordinates. We also give or indicate some geometric applications of the mentioned results. The O’Neill formulas for Riemannian submersions are derived in Section 1.6. These formulas are applied to prove that the Riemannian submersion of connected complete Riemannian manifolds is a submetry (Corollary 1.6.12) and locally trivial fiber bundle (Theorem 1.6.14). Submetries are needed in the study of generalized normal homogeneous Riemannian manifolds in Chapter 6. From Theorem 1.6.14 and Proposition 4.9.2, the following important fact from the theory of homogeneous manifolds follows: The canonical projection of a given Lie group to the quotient space of this group by its compact subgroup is a locally trivial fiber bundle. In Section 1.7 some results about Riemannian manifolds with restrictions on curvature (allowing simple statements) are considered. In Chapter 2, besides the usual results about Lie groups and Lie algebras, we select the results described below, supplied with short proofs. We give a sketch of the proof of S. Lie’s third theorem 2.3.2 on the existence of a Lie group with a given Lie algebra, based on Ado’s theorem 2.3.1 on the existence of an exact finitedimensional representation for any finite-dimensional Lie algebra and G. Frobenius’ Theorem 1.1.41 on full integrability; the short proof of Ado’s theorem was obtained by Yu. A. Neretin in [353]. The H. Weyl theorem 2.4.15 on finiteness of the fundamental group and compactness for a Lie group with negative definite Killing form follows from the positivity of the Ricci curvature of the corresponding bi-invariant Riemannian metric on this ´ Cartan group and the S. Myers theorem 1.5.79; this group is semisimple due to the E. criterion. In Theorem 2.5.11, a generalization of the H. Weyl theorem is obtained, and in Theorem 2.5.12, it is proved that a Lie algebra is compact and semisimple if and only if its Killing form is negative definite. In Theorem 2.5.13, it is proved that each derivation of a compact semisimple Lie algebra is inner. Any simply connected Lie group with a compact Lie algebra is isomorphic to a direct product of a vector group and a compact simply connected semisimple Lie group (Theorem 2.5.14). Each Cartan subalgebra of the Lie algebra g of a compact Lie group G coincides with the Lie subalgebra t of some of its maximal tori T (Theorem 2.5.22), and maximal tori are pairwise adjoint with respect to inner automorphisms of the group G (Theorem 2.5.23). On the basis of Theorems 2.5.14 and 2.5.23, it is proved in Proposition 2.5.33 that the Weyl group of a connected compact Lie group G is isomorphic to a finite quotient group NG (T )/T , where NG (T ) is a normalizer of the

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torus T in G. On the basis of Theorem 2.7.3 by B. Kostant [296], we give a new proof of the E. B. Vinberg theorem [448] in Theorems 2.7.2 and 2.7.5: Under the same conditions, each invariant with respect to the Weyl group norm (in particular, induced by a scalar product) on t is the restriction of some Ad(G)-invariant norm on g (induced by a scalar product). Necessary information on the Weyl groups, the root systems, and relations between real (in particular, compact) Lie algebras and complex Lie algebras are provided in Subsections 2.5.3, 2.5.4, and Section 2.6 without proofs; the root systems and Dynkin diagrams of simple Lie algebras are presented in Tables 2.1 and 2.2. For a more detailed exposition on Lie algebras and Lie groups we refer to the books [6], [257], [266], [271], [473]. The majority of results in Chapter 3 are devoted to isometric flows, i.e. to 1parameter isometry groups on Riemannian manifolds; they were established in the authors’ papers [75, 77, 74] for the first time. Each such flow is generated by a Killing vector field X; the main focus is on Killing vector fields X of constant length. Let us specify only some of the main results. In Theorem 3.2.3, we consider a new formula expressing the value of the covariant curvature 4-tensors on four Killing vector fields through the metric tensor and the Lie commutators of fields. This formula and its Corollary 3.2.4 allow us to calculate the curvature tensor and the sectional curvature of homogeneous Riemannian manifolds. At the same time, a well known formula for the sectional curvature is usually obtained by using the O’Neill formulas. In Theorem 3.5.1, along with other assertions, we prove the following: If the length of a Killing vector field X reaches a positive maximum at some point x of a Riemannian manifold, then the integral curve of the field X through x is a geodesic, and the sectional curvature of all two-dimensional sections containing tangent vectors of this geodesic are non-negative. There are corollaries of this theorem: Theorem 3.5.6 (the Ricci curvature in the direction of a unit Killing vector field is non-negative; it is identically equal to zero exactly when the field is parallel), the M. Berger theorem 3.5.10 [96] (any Killing vector field on a compact evendimensional Riemannian manifold of positive sectional curvature vanishes at some point), Corollary 3.5.16 (the S. Bochner theorem [111]), Sh. Kobayashi’s Corollary 3.5.17 (a Killing vector field on a compact Riemannian manifold of non-positive Ricci curvature is parallel); if the field is non-trivial, then the manifold has an infinite fundamental group (Corollary 3.5.18), and other results. The Killing vector field has constant length if and only if all its integral curves are geodesics (Proposition 3.3.4). An isometry of a metric space moving all points the same distance is called a Clifford–Wolf translation, briefly, a CW-translation (Definition 3.4.1). Any one-parameter group of CW-translations of a Riemannian manifold is generated by a Killing vector field X of constant length (Proposition 3.4.5). For symmetric spaces, the converse is true (Theorem 4.8.1); generally, including homogeneous manifolds, it is true only partially (Proposition 3.4.8). A Killing field of constant length with closed trajectories is called regular if all of them have the same length, otherwise it is called quasiregular. A Killing vector field of constant length is regular only in the case when it generates a free isometric action of the group S1 . Under the conditions of Proposition 3.4.5, a field X is either regular or all its trajec-

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tories are not closed. The construction of regular and quasiregular vector fields is based on Theorem 3.7.5: A smooth vector field X with closed non-point trajectories on a smooth manifold M is generated by some smooth action η : S1 × M → M if and only if X is a unit Killing vector field for some smooth Riemannian metric g on M. Using this theorem, we construct Riemannian metrics with quasiregular unit Killing vector fields on all spheres S2n+1 , n ≥ 1, (Corollary 3.7.7), including real-analytic co-homogeneity one metrics with unusual properties (Theorem 3.7.8), on all homotopic 7-spheres (Corollary3.7.6), and on the bundle of tangent unit vectors with the Sasaki metric over some Riemannian manifolds with closed geodesics (Subsection 3.7.3). In Section 3.8, we survey important results related to discrete subgroups of Lie groups, for instance, the Bieberbach groups and the Margulis lemma. In many cases these subgroups are fundamental groups of complete Riemannian manifolds with non-positive or negative sectional curvature, especially when these manifolds are compact or of finite volume. We used the books by W. Thurston [436], W. Ballmann–M. Gromov–V. Schroeder [43], J. A. Wolf [483], and the paper [338] by J. Milnor. The main focus is on Bieberbach’s solution of Hilbert’s problem 18 and refinements of this solution. Let us mention particularly the paper [133] by P. Buser, where the author essentially simplifies Bieberbach’s proofs using Gromov’s ideas and results on almost flat manifolds [242]. Unfortunately, we could not pay the same attention to the especially interesting case of hyperbolic Riemannian manifolds which are compact or have finite volume. Chapter 4 is devoted to homogeneous spaces and homogeneous Riemannian manifolds. We discuss transitive isometry groups for a given homogeneous Riemannian manifold and topological properties of homogeneous spaces. We consider the infinitesimal structure of homogeneous Riemannian manifolds and the structure of the set of G-invariant Riemannian metrics on a homogeneous space G/H. Moreover, we derive useful formulas for the sectional curvature, the Ricci curvature, and the scalar curvature of a given homogeneous Riemannian space. Finally, we discuss various topological and algebraical restrictions for homogeneous Riemannian spaces with positive or negative sectional curvature, as well as positive or negative Ricci curvature, and the structure of compact homogeneous Riemannian spaces with Killing vector fields of constant length. Due to Theorem 3.1.4 and the Myers–Steenrod theorem 4.1.1, the set of Killing vector fields on Riemannian manifold (M, g) constitutes the Lie algebra of the Lie group Isom(M, g) of all isometries of the space (M, g) (Theorem 4.1.4). If G ⊂ Isom(M, g) is transitive on (M, g) with the stabilizer H at some point x ∈ M, then (M = G/H, g) is reductive and G admits a left-invariant and H-right-invariant Riemannian metric g0 such that the natural projection π : (G, g0 ) → (G/H, g) is a Riemannian submersion with totally geodesic fibers (Proposition 4.9.2). If the metric g0 is bi-invariant, then the space (M, g) is called normal homogeneous. Special attention is paid to Killing vector fields of constant length and the corresponding isometric flows on symmetric spaces that constitute a very important subclass of homogeneous Riemannian manifolds. It is proved that such a flow on any symmetric space is free or induced by a free isometric action of the circle and

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consists of Clifford–Wolf translations, see Theorem 4.8.1. In Theorem 4.8.16, a criterion for the existence of quasiregular (or having both closed and non-closed trajectories) unit Killing vector fields on locally Euclidean space is obtained. Theorem 4.8.17 is proved: A homogeneous Riemannian manifold of constant sectional curvature admits quasiregular unit Killing fields if and only if it is not diffeomorphic to a sphere or to a real projective space. All such manifolds are classified in the book [483]. If the Lie groups G and H ⊂ G are connected and compact, then the Euler characteristic χ(G/H) is non-negative, and is positive if and only if the ranks of the groups G and H (dimensions of their maximal tori) coincide; in the latter case χ(G/H) is equal to the quotient of the orders of the Weyl groups of G and H (the Hopf–Samelson theorem 4.3.1). Due to Theorems 4.1.1, 3.5.10, 4.3.1, and 4.12.1, each even-dimensional homogeneous Riemannian manifold of positive sectional curvature has positive Euler characteristic (Proposition 4.12.7); it is indecomposable in a direct metric product. Let us notice that each compact triangulated odddimensional manifold has zero Euler characteristic due to Poincar´e duality. A compact connected and simply connected almost effective homogeneous Riemannian manifold (G/H, g) of positive Euler characteristic is indecomposable in a direct metric product exactly in the case when the Lie group G is simple (the B. Kostant theorem 4.12.3 [294]). A classification of the corresponding homogeneous spaces G/H was obtained by A. Borel and J. de Siebenthal in [117] (see also the book [483] by J. A. Wolf). In Theorem 4.10.2, the B. Kostant criterion for the natural reductivity of invariant Riemannian metrics on homogeneous space G/H is formulated. Compact homogeneous naturally reductive Riemannian manifold of positive Euler characteristic are normal homogeneous relative to some semisimple isometry Lie group (Theorem 4.12.13 and [79]). In Theorems 4.14.1, 4.14.2, and 4.14.5, the Corollary 3.2.4 mentioned earlier is adapted for algebraic calculations of the sectional curvature, the Ricci curvature, and the scalar curvature of homogeneous Riemannian manifolds; simplified formulas for naturally reductive metrics are obtained in Theorem 4.14.6. In Theorem 4.15.1, some criteria on the scalar curvature of invariant Riemannian metrics on a homogeneous space G/H with compact and connected Lie groups G and H ⊂ G are stated. Section 4.15 is devoted to the study of properties of homogeneous Riemannian spaces under some restriction to their curvatures. In Subsections 4.15.1, 4.15.3, and 4.15.4, we provide without proofs known information on homogeneous Riemannian spaces of positive sectional curvature, non-positive sectional curvature and non-positive Ricci curvature respectively. Using the S. Myers theorem 1.5.79 and Theorem 4.14.6, we prove Theorem 4.15.6: A homogeneous effective space M = G/H with connected G and compact H admits invariant Riemannian metrics of positive Ricci curvature if and only if M is compact and its fundamental group is finite or G is compact and its Levi subgroup (i.e. a maximal connected semisimple subgroup in G) acts transitively on M; for this it suffices to take any normal homogeneous metric for G/H. In Section 4.16 we discuss known results on the signature of the Ricci curvature on homogeneous Riemannian spaces. In Section 4.17, some results on the struc-

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Introduction

ture of the set of Killing vector fields of constant length on compact homogeneous Riemannian manifolds are considered. Chapter 5 is devoted to geodesic orbit spaces and contains both classical results and some recent results from [18], [360], and [366]. A Riemannian homogeneous space (M = G/H, g), where H is a compact subgroup of the Lie group G and g is a G-invariant Riemannian metric, is called a geodesic orbit space, briefly GOspace, if each of its geodesics is an orbit of some one-parameter subgroup of the group G; for G = Isom(M, g) it gives a concept of Riemannian GO-manifold. Let us identify the Lie algebra g of the Lie group of G with a suitable subalgebra of the Lie algebra of Killing vector fields on the Riemannian manifold (M, g). Then an element X ∈ g is called a geodesic vector if the integral curve of the field of X through eH ∈ G/H is geodesic in (G/H, g). Lemma 5.2.5 and Theorems 5.1.2, 5.7.1 give a characterization of geodesic vectors, Theorems 5.2.11 and 5.2.12 explain their properties, and Theorems 5.2.6 and 5.7.1 give a characterization of GO-spaces. A Riemannian manifold is called weakly symmetric if for any pair of its points there is an isometry interchanging these points. Each weakly symmetric space is a geodesic orbit space according to the paper [100] by J. Berndt, O. Kowalski, and L. Vanhecke. We give a new proof of this important result in Theorem 5.3.6. In Section 5.4, important results on the Lie algebra g of the isometry group G of an arbitrary geodesic orbit space (G/H, ρ) are discussed. Some structure results on the radical and the nilradical (the maximal nilpotent ideal) of g are considered in Theorems 5.4.1, 5.4.3, 5.4.5, and 5.4.7. If (M, g) is a Riemannian GO-manifold, g is the Lie algebra of its Killing fields, a is a commutative ideal in g, then any field X ∈ a has constant length on (M, g) (Theorem 5.5.1). In Theorem 5.6.1, it is proved that each geodesic orbit Riemannian manifold of non-positive Ricci curvature is a symmetric space. This theorem was first proved (with some gaps) by C. Gordon in [230]; a complete and essentially different proof was given in [360]. The classification of all geodesic orbit Riemannian metrics on spheres is considered in Section 5.9. The main result of Chapter 5 is Theorem 5.10.4 [18] which can be restated as follows: “Every simply connected indecomposable non-normal homogeneous GOmanifold of positive Euler characteristic is isometric either to (G/H = SO(2n + 1)/U(n), g), n ≥2, or to (G/H = Sp(n)/U(1) · Sp(n − 1), g), n ≥2. Moreover, up to a homothety, g may be any G-invariant Riemannian metric, excepting two metrics, one of which is G-normal homogeneous, and the second one is symmetric. Besides, all these manifolds are weakly symmetric flag manifolds.” Theorem 4.12.3 and Proposition 5.8.11 (“Let (M = G/H, g) be a geodesic orbit Riemannian space. Then any connected Lie subgroup K ⊂ G containing H has a totally geodesic orbit P = K(eH) = K/H which is a GO-space (with the induced metric); moreover, the projection π : G/H → G/K is a Riemannian submersion with totally geodesic fibers where both fibers and the base are GO-spaces.”) play an important role in the proof of Theorem 5.10.4. At first on the basis of [16], Theorem 5.10.4 is proved for the case of rank 2 compact (simple) Lie groups G. Then by using Lemma 5.10.5 and Proposition 5.8.11, it is established that in the general case the root system of a compact (simple) Lie group G (taking into account its

Introduction

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subgroups H) has to allow a so-called special decomposition with properties from Corollary 5.10.8. The study of the root systems of simple Lie algebras implies that special decompositions exist only for the pairs of Lie algebras h ⊂ g from Theorem 5.10.4. The property of being weakly symmetric for all invariant Riemannian metrics on homogeneous spaces from Theorem 5.10.4 is proved in paper [503] by W. Ziller. New methods to study geodesic orbit spaces, based on the properties of principal orbit types for linear actions of compact Lie groups, are discussed in Section 5.11. These methods are applied in Subsection 5.12.1 for the classification of compact geodesic orbit Riemannian spaces (G/H, ρ) such that the isotropy representation has exactly two irreducible components. The final section of Chapter 5 contains some other recent remarkable results related to homogeneous geodesics and geodesic orbit spaces. The main results of Chapter 6 are published in papers [72, 78, 79, 93, 73]. An isometry f of a metric space M to itself is called a δ (x)-translation for a point x ∈ M if x is a point of maximal displacement for f . A metric space M is called (G)-δ -homogeneous (respectively, (G)-Clifford–Wolf homogeneous) if for any pair of points x, y ∈ M there is a δ (x)-translation (respectively, a CW-translation) of the space M (from the isometry group G of the space M), moving x to y. It is clear that each CW-translation of the space M is a δ (x)-translation for all x ∈ M, and any (G)Clifford–Wolf homogeneous space is (G)-δ -homogeneous. A locally compact space with intrinsic metric or a Riemannian homogeneous manifold (M = G/H, ρ) with closed transitive connected group of isometries G and the stabilizer H at (some) point x ∈ M is called G-normal in the generalized (respectively, usual) sense if G admits a bi-invariant (respectively, a Riemannian bi-invariant) inner metric r such that the natural projection (G, r) → (G/H, ρ) is a submetry. A compact homogeneous Riemannian manifold (M, µ) is (G)-δ -homogeneous if and only if it is (G)normal homogeneous in the generalized sense; moreover, it suffices to consider the bi-invariant metric p r on G that is induced by the Ad(G)-invariant “Chebyshev norm” kXk = maxy∈M µ(X(y), X(y)) on g (Corollary 6.6). We prove Theorem 6.2.9: “For a connected complete Riemannian manifold (M, ρ) and an isometry group G of M the following conditions are equivalent: 1) (M, ρ) is G-δ -homogeneous, 2) every geodesic in (M, ρ) with the origin at some point x ∈ M is an integral curve of some Killing vector field on the manifold (M, ρ) from the Lie algebra of the Lie group of G that reaches the maximum value of length at the point x, 3) for each point x ∈ M and each vector v ∈ Mx there is a Killing vector field X on the manifold (M, ρ) from the Lie algebra of the group G with condition X(x) = v that reaches the maximum value of length at the point x.” The vector field X with condition 3) of Theorem 6.2.9 is called a δ -vector. From item 2) of Theorem 6.2.9, Corollary 6.2.10 follows: Every (G)-δ -homogeneous Riemannian manifold is a geodesic orbit Riemannian space (with the isometry group G). On the basis of item 3) of Theorem 6.2.9 and Theorem 3.5.1, we obtain Corollary 6.3.1: Every δ -homogeneous Riemannian manifold (M, µ) with the intrinsic metric ρ has non-negative sectional curvature. Each closed totally geodesic submanifold of a δ homogeneous (geodesic orbit) Riemannian manifold is itself δ -homogeneous (re-

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Introduction

spectively, geodesic orbit) (Theorem 6.6.2). From Theorem 6.6.2 and the Toponogov theorem 1.7.10 [438], we get that every (non-)compact δ -homogeneous Riemannian manifold is isometric to a direct metric product of (a Euclidean space and) compact indecomposable δ -homogeneous Riemannian manifolds (Theorem 6.6.5). It is clear that the converse is true and a direct product is simply connected if and only if all factors are simply connected. Every locally isometric (in particular, universal) covering space of a δ -homogeneous Riemannian manifold is a δ homogeneous Riemannian manifold (Theorem 6.3.5). From Corollary 6.2.10, Proposition 5.8.11, and Theorem 6.6.2, we get Proposition 6.6.6: “Let (G/H, µ) be a G-δ -homogeneous manifold and K be a connected Lie subgroup of the Lie group G such that H ⊂ K ⊂ G. Then the orbit of the group K in G/H through the point x = eH is a totally geodesic submanifold in (G/H, µ). In particular, K/H with the metric induced by the tensor µ is a δ -homogeneous space.” On the grounds of Corollary 6.2.10 and Theorem 5.10.4, compact simply connected indecomposable generalized normal homogeneous Riemannian manifolds of positive Euler characteristic, that are not normal, are among the manifolds (G/H, µ) from Theorem 5.10.4. One a priori condition for the required Riemannian metrics µ on such spaces G/H is found in Proposition 6.9.1. In the case of spaces Sp(l)/(U(1) · Sp(l − 1)), l ≥ 2, this condition is equivalent to positivity of the sectional curvature with the δ -pinching in the interval (1/16, 1/4) [452]; further in Theorem 6.9.2, it is proved that in fact all required Riemannian manifolds are of such kind. First, it is proved that the Riemannian manifold (SO(7)/U(3), µ) with conditions from Proposition 6.9.1 is not generalized normal homogeneous (Proposition 6.9.9). Then using Proposition 6.6.6, it is proved that the homogeneous Riemannian manifolds (SO(7)/U(3), µ) and (Sp(2)/(U1 · Sp(1)), µ1 ) admit totally geodesic isometric embeddings into the homogeneous spaces SO(2l + 1)/U(l) and Sp(l)/(U(1) · Sp(l − 1)), l ≥ 3, with the corresponding invariant Riemannian metrics; all metrics satisfy the conditions of Proposition 6.9.1 (Lemma 6.9.15 and Proposition 6.9.19). The proof of Theorem 6.9.2 is completed with some quite difficult considerations concerning δ -vectors and characteristic polynomials of matrices. In Section 6.12, general results on the Chebyshev norm on the Lie algebra of the isometry group of a compact homogeneous Finsler manifold are obtained. In Subsections 6.13.2, 6.13.3, and 6.13.4, we find the Chebyshev norms on the Lie algebras (and their Cartan subalgebras) of complete connected isometry groups of Euclidean spheres, Berger’s spheres, invariant Riemannian metrics for SO(5)/U(2) = Sp(2)/(U1 · Sp(1)) as well as the Loewner–John ellipsoids for unit spheres of these norms. In Section 6.14, we study almost normal homogeneous Riemannian manifolds. Chapter 7 (whose main results are obtained in the authors’ paper [80]) is devoted to the study of Clifford–Wolf homogeneous Riemannian manifolds and their natural generalizations. The main text of this chapter is divided into 11 sections. In the first section, some examples of Clifford–Wolf homogeneous Riemannian manifolds are discussed. Theorem 7.2.1 on the nullity of the covariant derivative of the curvature tensor on a Riemannian manifold when one uses three times one and the same Killing vector field of constant length, which is the main technical tool for a fur-

Introduction

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ther study, is given in the second section. In the next section we get the main result of this chapter, Theorem 7.0.1, which implies the complete classification of simply connected Clifford–Wolf homogeneous Riemannian manifolds. First of all, on the basis of Theorem 7.2.1 and one result from [105], we immediately obtain that every simply connected Clifford–Wolf homogeneous (briefly, CW-homogeneous) Riemannian manifold is a symmetric space. After that, a careful but not so long study of the CW-homogeneity condition in the symmetric case allows us to finish the proof. In the fourth section, we study Clifford–Killing spaces, i.e. real vector spaces of Killing vector fields of constant length. The manifolds in the title of the fifth section were defined and studied in the paper [168] by D’Atri and Nickerson. These are exactly Riemannian manifolds that locally admit Clifford-Killing spaces of dimension equal to the dimension of the manifold. We prove in Theorem 7.5.6 that in the case of simply connected complete manifolds these manifolds are classified similarly to the CW-homogeneous spaces in Theorem 7.0.1, but from odd-dimensional spheres it is necessary to leave only the seven-dimensional one. It is very interesting that the well-known classical results by Radon and Hurwitz from [393] and [267], discussed in the sixth section, could be interpreted exactly as the building of Clifford–Killing spaces of maximal dimension on round odd-dimensional spheres, and the well-known Radon–Hurwitz function gives their dimensions. In the next section, we establish a close connection of Clifford–Killing spaces on round odd-dimensional spheres with Clifford algebras and modules. It allows us to classify all vector Clifford–Killing spaces on round spheres in Theorems 7.7.4 and 7.7.6 up to isometric transformations of the spheres (preserving orientation). Radon noticed in his paper [393] that his study of the Hurwitz problem (Problem 7.6.1 in Section 7.6) is closely related to some (topological) spheres in Lie groups O(2n), n ≥2. In Section 7.8, we prove that each such Radon sphere is a totally geodesic sphere in the group O(2n) supplied with a bi-invariant Riemannian metric µ. In the next section, we consider a more general question on totally geodesic spheres in (SO(2n), µ), connected with triple Lie systems in the Lie algebra of the Lie group SO(2n) and Clifford–Killing spaces on round spheres S2n−1 . In particular, besides the results similar to results of the previous sections, it is proved that the Radon spheres coincide with the totally geodesic Helgason spheres (of constant sectional curvature k equal to the maximal sectional curvature in (SO(2n), µ)) from article [255] only in the case n = 2. In Proposition 7.9.5, the curvature k is calculated. In Section 7.10, Lie algebras in Clifford–Killing spaces on round odd-dimensional spheres are studied. Section 7.11 is devoted to the study of the sets of Killing fields of constant length on round spheres. In particular, we give a description of all transitive groups G on a given round sphere such that this sphere is G-Clifford–Wolf homogeneous. The final section is devoted to the study of restrictively Clifford–Wolf homogeneous manifolds. The classification of such manifolds is given in Theorem 7.12.6. Two examples of restrictively Clifford–Wolf homogeneous but not Clifford–Wolf homogeneous Riemannian manifolds are given.

xiv

Introduction

The notation in this book is standard in general. We list some of most important notations below. The symbols N and Z as usual mean the set of natural numbers and the ring of integers. The fields of real numbers, complex numbers, quaternions, and the Cayley algebra of octonions are denoted by R, C, H, and Ca respectively. All manifolds and (in particular) Lie groups are denoted by capital letters. For a smooth manifold M, we use the following notations. Mp TM C∞ (M) F ∞ (M) dim(M) π1 (M) χ(M)

the tangent space to M at a point p ∈ M the tangent bundle of M the ring of smooth real-valued functions on M the C∞ -module and the Lie algebra of smooth vector fields on M the dimension of M the fundamental group of M the Euler characteristic of a compact manifold M

The following notations are quite standard. ∇ ∇X A T R [X,Y ] LX AX Y Cl

the covariant derivative the covariant derivative of a tensor A along a smooth vector field X the torsion tensor of a covariant derivative the curvature tensor of a covariant derivative the Lie bracket of smooth vector fields X and Y the Lie derivative along a vector field X the O’Neill tensor the Clifford algebra

For a Riemannian manifold (M, g) with a metric tensor g, we use the following notations. K(σ ) Ric ric sc Exp Expx Radinj Isom(M, g) Isom0 (M, g) Ψ Su M d(·, ·) B(x, r) U(x, r) S(x, r) diam(M)

the sectional curvature of (M, g) in the direction of a 2-subspace σ the Ricci tensor of (M, g) the Ricci curvature of (M, g) the scalar curvature of (M, g) the exponential mapping of (M, g) the exponential mapping of (M, g) at a point x the injectivity radius of (M, g) the full isometry group of (M, g) the full connected isometry group of (M, g) the geodesic flow of (M, g) the spherical bundle of vectors of length u on (M, g) the intrinsic metric of (M, g) the closed ball with center x and radius r in (M, d) the open ball with center x and radius r in (M, d) the sphere with center x and radius r in (M, d) the diameter of (M, d)

As a rule, the Lie algebra of a Lie group G is denoted by (g, [·, ·]) or simply g (the corresponding fraktur letter). Moreover, we use the following additional notations.

Introduction

lg rg LG RG exp Ad ad B = Bg t = tg rk(g) rk(G) W = W (t)

xv

the left shift on the Lie group G by element g the right shift on the Lie group G by element g the Lie algebra of left invariant vector fields on the Lie group G the Lie algebra of right invariant vector fields on the Lie group G the exponential map from g to G the adjoint representation of the Lie group G on g the adjoint representation of the Lie algebra g on g the Killing form of a Lie algebra g the Cartan subalgebra of the Lie algebra g the rank of the Lie algebra g the rank of the Lie group G the Weyl group of a Cartan subalgebra t

By G/H we denote the homogeneous space of a Lie group G by its closed subgroup H (almost everywhere in the book we are dealing with a compact subgroup H). We use also the following special notations for some important homogeneous spaces. Sn RP n CP n HP n CaP 2 Hn CH n HH n CaH 2 F m / diag(F)

the n-dimensional sphere the real projective space the complex projective space the quaternionic projective space the Cayley projective plane the hyperbolic (Lobachevsky) space the complex analogue of the hyperbolic space the quaternionic analogue of the hyperbolic space the octave analogue of the hyperbolic plane the Ledger–Obata space defined by a compact simple Lie group F

All notations that are not generally accepted are explained in the relevant parts of the book and could also be clarified by the subject index.

The first edition of this book [82] was published (in Russian) in 2012. In this edition, we include many recent results obtained in the last few years as well as more detailed expositions of some classical results related to the main topics of the book. The authors hope that the present book will be interesting and useful for experts in the field of analysis and geometry, for researchers, and for teachers in higher education institutions working in adjacent topics of mathematics, as well as to students and graduate students of corresponding mathematical specialties. We will be grateful to all readers who will inform us of any inaccuracies in the presentation or of any gaps in our citations.

V. N. Berestovskii & Yu. G. Nikonorov

Contents

1

Riemannian Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Fundamentals of the Theory of Smooth Manifolds . . . . . . . . . . . . . . . 1.1.1 Smooth Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 The Tangent Bundle over a Smooth Manifold . . . . . . . . . . . . . 1.1.3 Smooth Vector Fields on M as Derivations of the Ring C∞ (M) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.4 The Lie Commutator of Two Smooth Vector Fields . . . . . . . . 1.1.5 Tensor Fields on Smooth Manifolds . . . . . . . . . . . . . . . . . . . . . 1.2 Manifolds with Connections and Riemannian Manifolds . . . . . . . . . . 1.2.1 Covariant Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Riemannian Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 The Levi-Civita Connection on a Riemannian Manifold . . . . 1.3 Curvature of Manifolds with Connection and Riemannian Manifolds 1.3.1 The Curvature Tensor of a Manifold with a Covariant Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 The Curvature Tensor of a Riemannian Manifold . . . . . . . . . . 1.3.3 The Ricci Tensor of a Manifold with a Covariant Derivative 1.3.4 The Ricci Tensor of a Riemannian Manifold . . . . . . . . . . . . . . 1.3.5 Curvatures of Riemannian Manifolds . . . . . . . . . . . . . . . . . . . . 1.4 Fundamentals of the Geometry of Riemannian Manifolds . . . . . . . . . 1.4.1 Parametrization of a Curve by the Arc Length . . . . . . . . . . . . 1.4.2 The First Variation of the Length of a Curve . . . . . . . . . . . . . . 1.4.3 Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.4 The Geodesic Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.5 The Exponential Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.6 The Shortest Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.7 The Intrinsic Metric of a Riemannian Manifold . . . . . . . . . . . 1.5 Some Global Properties of Riemannian Manifolds . . . . . . . . . . . . . . . 1.6 Riemannian Submersions and the O’Neill Formulas . . . . . . . . . . . . . . 1.7 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 1 3 9 10 14 16 16 18 19 21 21 21 24 24 25 26 26 27 30 31 33 35 36 36 66 71 xvii

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2

Lie Groups and Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 2.1 Main Properties of Lie Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 2.2 Smooth Actions of Lie Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 2.3 Main Properties of Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 2.3.1 Connections with Matrix Lie Algebras . . . . . . . . . . . . . . . . . . 87 2.3.2 Important Classes of Lie Algebras and Structural Results . . . 87 2.3.3 Automorphisms and Derivations . . . . . . . . . . . . . . . . . . . . . . . . 90 2.4 Lie Groups with Left-invariant Riemannian Metrics . . . . . . . . . . . . . . 91 2.4.1 The Levi-Civita Connection on a Lie Group with a Left-invariant Riemannian Metric . . . . . . . . . . . . . . . . . . . . . . 91 2.4.2 Curvatures of a Lie Group with a Left-invariant Riemannian Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 2.4.3 Curvature of Nilpotent Group-manifolds . . . . . . . . . . . . . . . . . 93 2.4.4 Curvatures of Bi-invariant Riemannian Metrics on Lie Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 2.5 Compact Lie Algebras and Lie Groups . . . . . . . . . . . . . . . . . . . . . . . . . 97 2.5.1 Compact Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 2.5.2 Cartan Subalgebras of Compact Lie Algebras . . . . . . . . . . . . . 103 2.5.3 Root Systems and the Weyl Groups . . . . . . . . . . . . . . . . . . . . . 105 2.5.4 Abstract Root Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 2.6 Connections with Complex Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . 113 2.7 Invariant Norms and Inner Products on Lie Algebras . . . . . . . . . . . . . 114

3

Isometric Flows and Killing Vector Fields on Riemannian Manifolds . 117 3.1 Isometries and Killing Vector Fields on Riemannian Manifolds . . . . 118 3.2 The Curvature Tensor and Killing Vector Fields . . . . . . . . . . . . . . . . . 123 3.3 Killing Vector Fields of Constant Length . . . . . . . . . . . . . . . . . . . . . . . 125 3.4 Killing Vector Fields of Constant Length and Clifford–Wolf Translations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 3.5 Killing Vector Fields and Curvatures . . . . . . . . . . . . . . . . . . . . . . . . . . 132 3.6 Isometric Flows and Points with Finite Period . . . . . . . . . . . . . . . . . . . 137 3.7 Regular and Quasiregular Killing Vector Fields . . . . . . . . . . . . . . . . . . 144 3.7.1 Orbits of Isometric S1 -actions on Riemannian Manifolds . . . 144 3.7.2 Flows on Simply Connected Manifolds . . . . . . . . . . . . . . . . . . 149 3.7.3 Geodesic Flows and the Sasaki Metric on Tangent Bundles of Riemannian Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 3.8 Discrete Subgroups of Lie Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 3.9 Open Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

4

Homogeneous Riemannian Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 4.1 The Isometry Group of a Riemannian Manifold . . . . . . . . . . . . . . . . . 167 4.2 Homogeneous Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 4.3 On the Topology of Compact Homogeneous Spaces . . . . . . . . . . . . . . 177 4.4 General Results on Reductive Decompositions . . . . . . . . . . . . . . . . . . 178 4.5 Invariant Affine Connections on Reductive Homogeneous Spaces . . 184

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4.6 4.7 4.8

Main Properties of Homogeneous Riemannian Manifolds . . . . . . . . . 186 Symmetric and Locally Symmetric Spaces . . . . . . . . . . . . . . . . . . . . . 188 Killing Vector Fields of Constant Length on Locally Symmetric Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 4.8.1 Symmetric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 4.8.2 Non-simply Connected Manifolds . . . . . . . . . . . . . . . . . . . . . . 194 4.8.3 Locally Euclidean Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 4.8.4 Homogeneous Spherical Space Forms . . . . . . . . . . . . . . . . . . . 200 Infinitesimal Structure of a Homogeneous Riemannian Space . . . . . . 204 Special Invariant Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 The Structure of the Set of Invariant Metrics . . . . . . . . . . . . . . . . . . . . 209 Compact Homogeneous Riemannian Manifolds of Positive Euler Characteristic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 Some Special Compact Homogeneous Spaces . . . . . . . . . . . . . . . . . . . 219 4.13.1 Aloff–Wallach Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 4.13.2 Generalized Wallach Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 4.13.3 Ledger–Obata Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 Curvatures of Homogeneous Riemannian Spaces . . . . . . . . . . . . . . . . 228 Homogeneous Riemannian Manifolds with Restrictions on Curvatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 4.15.1 Homogeneous Riemannian Spaces of Positive Sectional Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 4.15.2 Homogeneous Riemannian Spaces of Positive Ricci Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238 4.15.3 Homogeneous Riemannian Spaces of Non-positive Sectional Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242 4.15.4 Homogeneous Riemannian Spaces of Non-positive Ricci Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 Signature of the Ricci Curvature on Homogeneous Spaces . . . . . . . . 245 4.16.1 Signature of the Ricci Operator on Lie Groups . . . . . . . . . . . . 246 4.16.2 On Two Negative Eigenvalues of the Ricci Operator . . . . . . . 246 4.16.3 Lie Groups with Metrics of Negative Ricci Curvature . . . . . . 248 Homogeneous Riemannian Spaces with Killing Vector Fields of Constant Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 4.17.1 Examples of Killing Vector Fields of Constant Length . . . . . 252 4.17.2 Additional Properties of Killing Vector Fields of Constant Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 4.17.3 Unsolved Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

4.9 4.10 4.11 4.12 4.13

4.14 4.15

4.16

4.17

5

Manifolds With Homogeneous Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . 257 5.1 Homogeneous Geodesics on Riemannian Manifolds . . . . . . . . . . . . . 257 5.1.1 Homogeneity Criterion for a Geodesic . . . . . . . . . . . . . . . . . . 259 5.1.2 On the Closure of a Homogeneous Geodesic . . . . . . . . . . . . . 263 5.1.3 Examples of Homogeneous Geodesics . . . . . . . . . . . . . . . . . . 266 5.1.4 A Special Quadratic Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . 269

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5.2 5.3 5.4

Geodesic Orbit Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270 Important Subclasses of Geodesic Orbit Spaces . . . . . . . . . . . . . . . . . 276 On the Radical and the Nilradical of the Lie Algebra of the Isometry Group of a Geodesic Orbit Space . . . . . . . . . . . . . . . . . . . . . 281 5.5 Killing Vector Fields of Constant Length on GO-spaces . . . . . . . . . . 288 5.6 Geodesic Orbit Manifolds of Non-positive Ricci Curvature . . . . . . . . 288 5.7 Compact Geodesic Orbit Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291 5.8 Applications of the Totally Geodesic Property . . . . . . . . . . . . . . . . . . . 294 5.9 Geodesic Orbit Riemannian Metrics on Spheres . . . . . . . . . . . . . . . . . 299 5.9.1 On Invariant Metrics on S4n+3 for Transitive Actions of the Groups Sp(n + 1), Sp(n + 1)U(1), and Sp(n + 1)Sp(1) . 301 5.9.2 Sp(n + 1)-invariant Geodesic Orbit Metrics on the Sphere S4n+3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 5.10 Compact GO-spaces of Positive Euler Characteristic . . . . . . . . . . . . . 305 5.10.1 Basic Facts on Compact Homogeneous Manifolds of Positive Euler Characteristic . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 5.10.2 The Classification Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306 5.10.3 Proof of the Classification Theorem . . . . . . . . . . . . . . . . . . . . . 307 5.11 GO-spaces and Representations with Non-trivial Principal Isotropy Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313 5.12 Miscellaneous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316 5.12.1 Compact GO-spaces with Two Isotropy Summands . . . . . . . . 316 5.12.2 Geodesic Orbit Metrics on Ledger–Obata Spaces . . . . . . . . . . 318 5.12.3 GO-spaces Fibered over Irreducible Symmetric Spaces . . . . 320 5.12.4 Geodesic Orbit Riemannian Structures on Rn . . . . . . . . . . . . . 321 5.12.5 On Left-invariant Einstein Riemannian Metrics that are not Geodesic Orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322 5.12.6 Various Results on Homogeneous Geodesics and GO-spaces 324 5.12.7 Open Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328 6

Generalized Normal Homogeneous Manifolds With Intrinsic Metrics 329 6.1 δ -homogeneous and Clifford–Wolf Homogeneous Spaces . . . . . . . . 329 6.2 δ -homogeneous and Restrictively Clifford–Wolf Homogeneous Riemannian Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330 6.3 Some Structural Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334 6.4 Generalized Normal Homogeneous and δ -homogeneous Riemannian Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336 6.5 Additional Symmetries of δ -homogeneous Metrics . . . . . . . . . . . . . . 341 6.6 Totally Geodesic Submanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 6.7 Properties of δ -vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345 6.7.1 The Case of Positive Euler Characteristic . . . . . . . . . . . . . . . . 348 6.8 Generalized Normal Homogeneous Manifolds of One Special Type 350 6.9 Classification of Generalized Normal Homogeneous Riemannian Manifolds of Positive Euler Characteristic . . . . . . . . . . . . . . . . . . . . . . 352 6.9.1 General Ideas and Constructions . . . . . . . . . . . . . . . . . . . . . . . 355

Contents

6.10 6.11

6.12 6.13

6.14 7

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6.9.2 The Spaces SO(2l + 1)/U(l), l ≥ 3 . . . . . . . . . . . . . . . . . . . . . 357 6.9.3 The Spaces Sp(l)/U(1) · Sp(l − 1), l ≥ 2 . . . . . . . . . . . . . . . . 363 Generalized Normal Homogeneous Metrics and 2-means in the Sense of W. J. Firey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364 Generalized Normal Homogeneous Riemannian Metrics on Spheres and Projective Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 6.11.1 Homogeneous Riemannian Manifolds Being Investigated . . 369 6.11.2 G-generalized Normal Homogeneous Metrics on Spheres . . 371 6.11.3 Spin(9)-generalized Normal Homogeneous Metrics on the Sphere S15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378 The Chebyshev Norm on the Lie Algebra of the Isometry Group of a Compact Homogeneous Finsler Manifold . . . . . . . . . . . . . . . . . . . 384 Calculations of the Chebyshev Norms for Some Manifolds . . . . . . . . 391 6.13.1 On the L¨owner–John Ellipsoids . . . . . . . . . . . . . . . . . . . . . . . . 392 6.13.2 The Chebyshev Norm for Euclidean Spheres . . . . . . . . . . . . . 393 6.13.3 The Chebyshev Norm for the Berger Spheres . . . . . . . . . . . . . 396 6.13.4 The Chebyshev Norms for Invariant Metrics on SO(5)/U(2) 401 Almost Normal Homogeneous Riemannian Manifolds . . . . . . . . . . . . 408

Clifford–Wolf Homogeneous Riemannian Manifolds . . . . . . . . . . . . . . . 413 7.1 Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414 7.2 Once More about Killing Vector Fields of Constant Length . . . . . . . . 415 7.3 The Proof of the Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416 7.4 Clifford–Killing Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419 7.5 Riemannian Manifolds with the Killing Property . . . . . . . . . . . . . . . . 420 7.6 Results by A. Hurwitz and J. Radon . . . . . . . . . . . . . . . . . . . . . . . . . . . 422 7.7 Clifford–Killing Spaces on Spheres and Clifford Algebras and Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424 7.8 Clifford–Killing Spaces for S2n−1 and Unit Radon Spheres in O(2n) 427 7.9 Triple Lie Systems in so(2n) and Totally Geodesic Spheres in SO(2n) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429 7.10 Lie Algebras in Clifford–Killing Spaces for S2n−1 . . . . . . . . . . . . . . . 434 7.11 Killing Vector Fields of Constant Length and G-Clifford–Wolf Homogeneous Metrics on Round Spheres . . . . . . . . . . . . . . . . . . . . . . 436 7.11.1 Descriptions of Killing Vector Fields of Constant Length and G-Clifford–Wolf Homogeneous Metrics . . . . . . . . . . . . . 436 7.11.2 The Spaces of Unit Killing Fields on Spheres . . . . . . . . . . . . . 439 7.11.3 Unit Killing Fields on the Sphere S15 = Spin(9)/Spin(7) . . . 441 7.12 Restrictively Clifford–Wolf Homogeneous Riemannian Manifolds . 446

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453 List of tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473 Symbol Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475

xxii

Contents

Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477

Chapter 1

Riemannian Manifolds

Abstract The necessary foundations for the study of the following chapters are provided. We introduce the main notions, and prove, discuss or simply mention the basic results on smooth manifolds, manifolds with covariant derivatives, and Riemannian manifolds that will be needed later. In particular, we consider the Lie algebra of smooth vector fields on smooth manifolds, the Levi-Civita connection on Riemannian manifolds, their curvature tensors, sectional and other curvatures, geodesics, the intrinsic metric, and shortest curves. As a rule, we prove the main results. In particular, we prove the most important results of this chapter, namely the Hopf–Rinow theorem, the Myers theorem on Riemannian manifolds with strictly positive Ricci curvature, the Synge theorem, the Rauch comparison theorem, the Hadamard–Cartan theorem, and the O’Neill formulas for Riemannian submersions. We also give or indicate some geometric applications of the main results.

1.1 Fundamentals of the Theory of Smooth Manifolds 1.1.1 Smooth Manifolds We give here the main definitions for smooth manifolds and their simplest examples. A Hausdorff topological space M n is called a topological n-manifold (without boundary) if for any point x ∈ M n there are a neighborhood U of the point x and a homeomorphism φ : U ≈ V onto an open subset V ⊂ Rn . Usually, we assume that M n is connected and satisfies the second countability axiom. Let M n be a topological n-manifold (without boundary). A triple (U, φ ,V ) as above is called a map or chart on M n . A family of maps (Uα , φα ,Vα ), α ∈ A, on M n is called an atlas if {Uα , α ∈ A} is a cover of M n . An atlas is smooth if for any pair α, β ∈ A of indices the (transition) mapping φαβ = φα ◦ φβ−1 : φβ (Uα ∩Uβ ) → φα (Uα ∩Uβ ) © Springer Nature Switzerland AG 2020 V. Berestovskii and Y. Nikonorov, Riemannian Manifolds and Homogeneous Geodesics, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-56658-6_1

(1.1) 1

2

1 Riemannian Manifolds

is a C∞ -mapping, i.e. its coordinate functions have partial derivatives of any order. This requirement makes sense because φβ (Uα ∩ Uβ ) and φα (Uα ∩ Uβ ) are open sets of Euclidean space. A maximal smooth atlas D on M n is called a smooth or differential structure on M n . A differentiable or smooth manifold is a manifold together with some differential structure D on it. Any smooth atlas on an n-manifold M defines a unique differential structure containing it. Therefore it is enough to introduce a smooth atlas on M. A differentiable manifold is orientable if there exists some atlas (Uα , φα ,Vα ), α ∈ A, from its smooth structure such that every map (1.1) has a positive Jacobian; any maximal atlas with these properties is called an orientation on M n . Otherwise, a (smooth) manifold is non-orientable. Every smooth orientable manifold admits exactly two orientations. A smooth manifold together with an orientation on it is called an oriented manifold. A continuous map f : (M n1 , D1 ) → (M n2 , D2 ) of two smooth manifolds is called smooth at a point x ∈ M n1 if for some charts (U1 , φ1 ,V1 ) ∈ D1 and (U2 , φ2 ,V2 ) ∈ D2 such that x ∈ U1 and f (x) ∈ U2 , the map φ2 ◦ f ◦ φ1−1 is a C∞ -map. This requirement makes sense because the domain of the map φ2 ◦ f ◦ φ1−1 , φ1 ( f −1 (U2 ) ∩ U1 ), is an open subset in Rn1 and φ2 ◦ f ◦ φ1−1 maps it into an open subset φ2 (U2 ) in Rn2 . This map f is smooth if it is smooth at any point in M n1 . Such a map f is called a diffeomorphism if it is bijective and both maps f and f −1 are smooth. Two smooth manifolds are diffeomorphic if there exists a diffeomorphism between them. In differential topology two diffeomorphic manifolds are identified. Example 1.1.1. Euclidean space En = Rn admits a natural smooth structure defined by one map (En , idEn , En ). Example 1.1.2. If U is an open subset in a smooth n-manifold M n with a smooth atlas (Uα , φα ,Vα ), α ∈ A, then the family (U ∩ Uα , φα |U∩Uα , φα (U ∩Uα )), α ∈ A, is a smooth atlas on U. The set U together with the corresponding smooth structure is called a smooth open submanifold in M n . Example 1.1.3. Let Sn ⊂ En+1 be the standard unit sphere with the center at the origin. Take UN = Sn − {(0, . . . , 0, 1)}, US = Sn − {(0, . . . , 0, −1)}, and define homeomorphisms (stereographic projections) φN : UN → En and φS : US → En by the formulas φN (x1 , . . . , xn , xn+1 ) =

1 (x1 , . . . , xn ) 1 − xn+1

(1.2)

φS (x1 , . . . , xn , xn+1 ) =

1 (x1 , . . . , xn ). 1 + xn+1

(1.3)

and

It is clear that UN ∪US = Sn . One can easily check that φN ◦ φS−1 = φS ◦ φN−1 = invSn−1 : En − {0} → En − {0}

1.1 Fundamentals of the Theory of Smooth Manifolds

3

is an inversion map invSn−1 with respect to Sn−1 ⊂ En , i.e. invSn−1 (y1 , . . . , yn ) = invSn−1 (y) =

y . |y|2

Hence we have defined a smooth (even real-analytic) structure on Sn . Example 1.1.4. If M m and N n are smooth manifolds with smooth atlases (Uα , φα ,Vα ), α ∈ A, and (Wβ , ψβ ,Yβ ), β ∈ B, then the space Pm+n := M m × N n is a smooth (n + m)-manifold with the smooth atlas (Uα × Wβ , φα × ψβ ,Vα × Yβ ), (α, β ) ∈ A × B. The manifold Pm+n with such an atlas is called the product of smooth manifolds M m and N n .

1.1.2 The Tangent Bundle over a Smooth Manifold Definition 1.1.5. Let M be a smooth manifold. A smooth map f : M → R is called a smooth real-valued function on M. Proposition 1.1.6. The set C∞ (M) of all smooth real-valued functions on M is a commutative associative ring relative to the operations ( f + g)(p) = f (p) + g(p), ( f g)(p) = f (p)g(p);

f , g ∈ C∞ (M), p ∈ M.

The function 1M identically equal to the real number 1 is the unit of this ring. Definition 1.1.7. An arbitrary map v : C∞ (M) → R with the properties 1) v(c f ) = c(v f ), 2) v( f + g) = v f + vg, 3) v( f g) = f (p)vg + g(p)v f for all f , g ∈ C∞ (M), c ∈ R, is called a tangent vector to the smooth manifold M at the point p ∈ M. The set M p of all such maps is called a tangent vector space to the smooth manifold M at the point p ∈ M. The following theorem justifies the usage of these words. Theorem 1.1.8. Let v, w ∈ M p , c ∈ R. Then the maps v + w and cv, defined by formulas (v + w) f = v f + w f , (cv) f = c(v f ) for all f ∈ C∞ (M), c ∈ R, are elements of M p . With respect to these operations, the set M p is a real vector space of dimension dim(M). All assertions of this theorem, besides the statement on the dimension, are evident. The proof of the latter assertion is divided into several lemmas. First of all, we need the following (Lemma 1.10 from [461]).

4

1 Riemannian Manifolds

Lemma 1.1.9. For every real number r > 0, there exists a smooth real function f = fr , 0 ≤ f ≤ 1, on Rn , which is equal to 1 on the closed cube [−r, r]n , and vanishes on the complement to the open cube (−2r, 2r)n . Proof. First we shall consider the case r = 1. It is enough to define f as a product f = (h ◦ pr1 ) · · · (h ◦ prn ),

(1.4)

where pri : Rn → R are the canonical projections (coordinate functions) on Rn and h is a smooth function h : R → R, equal to 1 on [−1, 1] and to 0 outside of (−2, 2). To construct such a function h we start with a smooth function φ : R → R such that φ (t) = e−1/t , if t > 0 and φ (t) = 0 otherwise. Then the function g(t) =

φ (t) φ (t) + φ (1 − t)

is smooth, non-negative, attains the value 1 for t ≥ 1, and vanishes for t ≤ 0. We obtain the required function h by setting h(t) = g(t + 2)g(2 − t). Now, for any number r > 0, a suitable function could be defined by the formula fr (x) = f ((1/r)x),

x ∈ Rn . t u

The lemma is proved. Mn,

Lemma 1.1.10. Let U be an open submanifold of a smooth manifold p0 ∈ U, and φ : U → R be a smooth function. Then there exist some open neighborhood U0 of the point p0 and a function ψ ∈ C∞ (M n ) such that U0 ⊂ U, ψ(p) = φ (p) for all p ∈ U0 and ψ(q) = 0 for all q ∈ M n −U. Proof. There is a smooth chart (U1 , φ1 ,V1 ) of the manifold M n such that p0 ∈ U1 ⊂ U and φ1 (p0 ) = (0, . . . , 0) ∈ Rn . Then there exists a number r > 0 such that [−2r, 2r]n ⊂ V1 . Define a function ψ : M n → R by conditions ψ(p) = fr (φ1 (p))φ (p), where fr is the function from Lemma 1.1.9 if p ∈ U1 , and ψ(q) = 0 if q ∈ M n −U1 . Thus ψ is a desired function for U0 = φ1−1 ((−r, r)n ). t u Lemma 1.1.11. Let V0 be an open convex subset in Rn with the canonical coordinate functions x1 , . . . , xn , and g ∈ C∞ (V0 ). Then for any points x0 , x1 ∈ V0 , n

g(x1 ) = g(x0 ) + ∑ (xi (x1 ) − xi (x0 )) i=1

n

+

∑ (xi (x1 ) − xi (x0 ))(x j (x1 ) − x j (x0 ))

i, j=1

Z 1 0

(1 − t)

∂g (x0 ) ∂ xi

∂ 2g (x0 + t(x1 − x0 ))dt. ∂ xi ∂ x j

Moreover, for a fixed point x0 ∈ V0 , all the summands from the right-hand side as well as their factors are smooth functions on x1 ∈ V0 .

1.1 Fundamentals of the Theory of Smooth Manifolds

5

Proof. Due to the convexity of the set V0 , the following smooth function of one variable is well-defined: θ (t) := g(x0 + t(x1 − x0 )),

t ∈ [0, 1].

Then, applying the Newton–Leibnitz formula and integration by parts in the definite integral, we get θ (1) =

θ (0) + θ (t)|10

Z 1

= θ (0) +

= θ (0) + (t − 1)θ 0 (t)|10 + = θ (0) + θ 0 (0) +

Z 1

Z

0 1

θ 0 (t)dt

(1 − t)θ 00 (t)dt

0

(1 − t)θ 00 (t)dt.

0

Thus we get the formula θ (1) = θ (0) + θ 0 (0) +

Z 1

(1 − t)θ 00 (t)dt.

(1.5)

0

It is easy to calculate that θ (0) = g(x0 ), n

θ (1) = g(x1 ),

θ 0 (0) = ∑ (xi (x1 ) − xi (x0 )) i=1

θ 00 (t) =

n

∂g (x0 ), ∂ xi ∂ 2g

∑ (xi (x1 ) − xi (x0 ))(x j (x1 ) − x j (x0 )) ∂ xi ∂ x j (x0 + t(x1 − x0 )).

(1.6) (1.7) (1.8)

i, j=1

The equalities (1.5), (1.6), (1.7), and (1.8) imply the formula from Lemma 1.1.11. The last assertion of Lemma 1.1.11 is well known. t u

Lemma 1.1.12. Suppose that v ∈ M pn0 , f , g ∈ C∞ (M n ), and f (p) = g(p) for all points p from some open subset U in M n such that p0 ∈ U. Then v f = vg.

Proof. Let φ : U → R be defined by the equality φ (p) = 1 for all p ∈ U. Hence φ ∈ C∞ (U). Let ψ ∈ C∞ (M n ) be the function constructed in Lemma 1.1.10 for the chosen function φ . Then ψ( f − g) ≡ 0 and, consequently, by “the Leibnitz rule”, we have 0 = v(ψ( f − g)) = ( f − g)(p0 )vψ + ψ(p0 )v( f − g) = v f − vg. The lemma is proved.

t u

6

1 Riemannian Manifolds

Lemma 1.1.13. Let v ∈ M pn0 and U be an open submanifold in M n , containing the point p0 . Then there exists a unique vector v|U ∈ U p0 such that vg = v|U g|U

for every function

g ∈ C∞ (M n ),

(1.9)

where g|U denotes the restriction of the function g to U. Moreover, the correspondence v ∈ M pn0 → v|U ∈ U p0 is a linear automorphism of vector spaces. Proof. By Lemma 1.1.10, for any function φ ∈ C∞ (U), there exist a neighborhood U0 of the point p0 and a function ψ ∈ C∞ (M n ) such that U0 ⊂ U and ψ(p) = φ (p) for all p ∈ U0 . Define v|U φ = vψ. Lemma 1.1.12 implies the correctness of the definition of the map vU : C∞ (U) → R and the inclusion v|U ∈ U p0 . It is obvious that the condition (1.9) is fulfilled. Now, both assertions of the lemma are corollaries of Lemma 1.1.12 and the condition (1.9). t u Proof (of Theorem 1.1.8). Assume that dim(M) = n, p = p0 ∈ M n , v ∈ M pn0 , and consider a function g ∈ C∞ (M n ). There exists a smooth chart (U, φ ,V ) on M n such that p0 ∈ U and φ (p0 ) = (0, . . . , 0). One can find a number r > 0 such that [−r, r]n ⊂ V . Put V0 = (−r, r)n and U0 = φ −1 (V0 ). So we get a new chart (U0 , φ0 ,V0 ), where φ0 = φ |U0 , such that p0 ∈ U0 , and the set V0 is convex. On the grounds of Lemma 1.1.13, the condition (1.9) is fulfilled if we replace U by U0 . Writing for brevity g instead of g|U0 ◦ φ0−1 , we get by Lemma 1.1.11 and the equalities x0 = φ0 (p0 ) = (0, . . . , 0), φ0i = xi ◦ φ0 , that vg = v|U0 g|U0 , where n

g|U0 = g(x0 ) + ∑

i=1

φ0i

n ∂g (x ) + φ0i φ0j 0 ∑ ∂ xi i, j=1

Z 1

(1 − t)

0

∂ 2g (t(x1 , . . . , xn ))dt. ∂ xi ∂ x j

Now, applying properties of the tangent vector v|U0 and the equality φ0 (p0 ) = 0, we get that n ∂g v(g) = v|U0 g|U0 = ∑ (v|U0 φ0i ) i (φ0 (p0 )). ∂x i=1 It is easy to see from the above arguments that this formula is fulfilled if we remove all lower indices 0 and consider a vector v ∈ M pn , where p ∈ U is an arbitrary point. Since the function g from C∞ (M n ) was arbitrary, this means that n

v(·) = ∑ ζ i i=1

∂ (· ◦ φ −1 ) (φ (p)), ∂ xi

(1.10)

where ζ i = ζ i (v) = (v|U )(xi ◦ φ ). By Lemma 1.1.13 and the formulas (1.11), (1.10), the correspondence v ∈ M pn → (ζ 1 (v), . . . , ζ n (v))

(1.11)

1.1 Fundamentals of the Theory of Smooth Manifolds

7

is linear. It is clear that an arbitrary collection of numbers (ζ 1 , . . . , ζ n ) ∈ Rn defines a unique vector v ∈ M pn by the formula (1.10). Thus, the vector spaces M pn and Rn are isomorphic and the theorem is proved. t u Definition 1.1.14. The numbers ζ 1 , . . . , ζ n from the formula (1.11) are called the components of the vector v ∈ M pn in the chart (U, φ ,V ) (where p ∈ U ⊂ M n ). Let M n be a smooth n-manifold. Define the tangent manifold T M n to M n as the set of all pairs (p, v), where p ∈ M n , v ∈ M pn . Here p is called the origin of the vector v. Notice that if p 6= q then formally M pn ∩ Mqn is not empty and consists of the (joint) zero vector; therefore, together with a vector its origin is indicated. The map π : T M n → M n , defined by the formula π(p, v) = p, is called the projection. If (U, φ ,V ) is a chart on M n , we define a chart (π −1 (U), T φ ,V × Rn ) on T M n by the formula T φ (p, v) = (φ (p), ζφ (v)), (1.12) where ζφ (v) is an n-vector, consisting of the components of v in the chart (U, φ ,V ) (Definition 1.1.14). Theorem 1.1.15. Let (U, φ ,V ) and (U1 , φ1 ,V1 ) be two charts on a smooth manifold M n , and let (π −1 (U), T φ ,V × Rn ) and (π −1 (U1 ), T φ1 ,V1 × Rn ) be the corresponding charts on T M n . Then the transition function T φ1 ◦ (T φ )−1 : φ (U ∩U1 ) × Rn → φ1 (U ∩U1 ) × Rn for the last two charts has the form T φ1 ◦ (T φ )−1 = (φ1 ◦ φ −1 ) × D(φ1 ◦ φ −1 ), where D(φ1 ◦ φ −1 ) is the usual Fr´echet differential of the smooth diffeomorphism φ1 ◦ φ −1 of open sets of Euclidean space. Consequently, the requirement that the map T φ for a chart (U, φ ,V ) on M n is homeomorphism correctly defines a topology on T M n and T M n with charts (π −1 (U), T φ ,V × Rn ) becomes a smooth manifold of dimension 2n. Proof. It suffices to check the mentioned dependence of components ζ1j ; j = 1, . . . , n, of a vector v ∈ M p , p ∈ U ∩ U1 , in the chart (U1 , φ1 ,V1 ) on its components ζ i ; i = 1, . . . , n, in the chart (U, φ ,V ). We shall denote coordinates of points in V1 by x11 , . . . , x1n . It follows from the formulas (1.11) and (1.10), that n

ζ1j = v|U∩U1 (x1j ◦φ1 ) = v|U∩U1 (x1j ◦(φ1 ◦φ −1 )◦φ ) = ∑ ζ i i=1

∂ (x1j ◦ (φ1 ◦ φ −1 )) (φ (p)), ∂ xi

as required.

t u

Corollary 1.1.16. The projection π : T M → M is a smooth mapping. Definition 1.1.17. The smooth tangent manifold T M together with the smooth projection π : T M → M is called the tangent bundle over n-manifold M n . A smooth section X to the projection π, i. e. a smooth map X : M → T M such that π ◦ X = idM , is called a (smooth) vector field on M n .

8

1 Riemannian Manifolds

Proposition 1.1.18. The set F ∞ (M) of all smooth vector fields on M is a module over the ring C∞ (M) with respect to the operations (X +Y )(p) = X(p) +Y (p), f X(p) = f (p)X(p),

f ∈ C∞ (M),

p ∈ M,

X,Y ∈ F ∞ (M).

Definition 1.1.19. Let M and N be smooth manifolds and let φ : M → N be a smooth map. The differential T φ : T M → T N of the map φ is defined by the requirement that for any elements v ∈ T M, f ∈ C∞ (N), (T φ (v)) f = v( f ◦ φ ).

(1.13)

Similar to the proof of Theorem 1.1.15, it is established that T φ is a smooth map and we get the following commutative diagram: Tφ

T M −→ T N πM ↓

↓ πN φ

M −→

N

If φ : M → N and ψ : N → P are smooth maps, then ψ ◦ φ : M → P is also a smooth map and we get the equality T (ψ ◦ φ ) = T ψ ◦ T φ for T (ψ ◦ φ ) : T M → T P. Moreover, T idM = idT M . These two properties mean that the correspondences T : M → T M and T : {φ : M → N} → {T f : T M → T N} define a covariant functor from the category Smooth of smooth manifolds and their smooth maps to itself. Due to the above diagram, this could be considered also as a covariant functor from Smooth to the category Vect of smooth vector bundles over smooth manifolds. Definition 1.1.20. We shall call a smooth map c : I → M n of an arbitrary interval I of the real line, containing more than one point, a (smooth) path in M n . Definition 1.1.19 implies immediately Proposition 1.1.21. Let c : (−ε, ε) → M n be a smooth path in M n , c(0) = p, c 0 (0) := T c(d/dx(0)) = w ∈ M pn , f ∈ C∞ (M n ). Then w f = ( f ◦ c) 0 (0), where in the right-hand side, the usual derivative of smooth real functions on (−ε, ε) is applied. Definition 1.1.22. Let ψ : N → M be a smooth map. 1) The map ψ is called an immersion if its differential Tx ψ := T ψ|Nx is nondegenerate for every point x ∈ N. 2) A pair (N, ψ) is called a (smooth) virtual submanifold of a manifold M if ψ is an injective immersion.

1.1 Fundamentals of the Theory of Smooth Manifolds

9

3) A pair (N, ψ) is called a (smooth) submanifold of a manifold M if ψ is an injective immersion, which is also a topological embedding, i.e. a homeomorphism onto its image. (Virtual) submanifolds are often defined by stating that N ⊂ M, ψ is an inclusion map and the subset N is supplied with some smooth structure D. Using the inverse map theorem, it is not difficult to prove Proposition 1.1.23. A smooth structure D of a smooth submanifold N ⊂ M is uniquely defined. Definition 1.1.24. Let M and B be smooth manifolds. A smooth map p : M → B is called a submersion if for any point x ∈ M, the differential Tx p : Mx → B p(x) is a surjective map. Preimages-submanifolds p−1 (b), b ∈ p(M), (of dimension dim M − dim B) are called fibres, a vector field on M (respectively, a vector v ∈ T M) is called vertical, if X( f ◦ p) = 0 (respectively, v( f ◦ p) = 0) for all f ∈ C∞ (B). The vector subspace of all vertical vectors in Mx will be denoted by Vx .

1.1.3 Smooth Vector Fields on M as Derivations of the Ring C∞ (M) If X ∈ F ∞ (M), f ∈ C∞ (M), then it is possible to define a new real function X f on M by the formula (X f )(p) = X(p) f , p ∈ M. (1.14) It follows from (1.10) and (1.11) that X f ∈ C∞ (M). The following proposition is a direct corollary of (1.14) and Definition 1.1.7. Proposition 1.1.25. Every smooth vector field X on a smooth manifold M defines a derivation D = DX : C∞ (M) → C∞ (M) of the ring C∞ (M) by the formula DX ( f ) = X f . This means that for all functions f , g ∈ C∞ (M), 1) X(c f ) = cX f , 2) X( f + g) = X f + Xg, 3) X( f g) = gX f + f Xg, or, in the other notation, 1) D(c f ) = cDX f , 2) D( f + g) = DX f + DX g, 3) D( f g) = gDX f + f DX g. We also have the following converse to Proposition 1.1.25 (cf. [461]): Theorem 1.1.26. For any derivation D : C∞ (M n ) → C∞ (M n ) of the ring C∞ (M n ), there exists a unique smooth vector field X on M n such that D = DX .

10

1 Riemannian Manifolds

Proof. Comparing the properties 1), 2), 3) for D from Proposition 1.1.25 and for v from Definition 1.1.7, we see that for any point p ∈ M, the correspondence f ∈ C∞ (M) → (D f )(p) is given by the vector X(p) ∈ M p . If (U, φ ,V ) is a chart on M n such that p ∈ U, then by Lemma 1.1.13 and (1.10), n

(D f )|U = ∑ (X|U )(xi ◦ φ ) i=1

∂ ( f ◦ φ −1 ) ∂ xi

for any function f ∈ C∞ (M). Then (X|U )(xi ◦φ ) ∈ C∞ (U), X ∈ F ∞ (M), and D = DX . t u Proposition 1.1.25 and Theorem 1.1.26 characterize smooth vector fields on a smooth manifold M as derivations of the ring C∞ (M). Note also that for f , g ∈ C∞ (M) and X,Y ∈ F ∞ (M), we get (X +Y ) f = X f +Y f ,

(1.15)

( f X)g = f (Xg).

(1.16)

Definition 1.1.27. If f ∈ C∞ (M), v ∈ T M, then by the definition, d f (v) = v f . The smooth function d f : T M → R, defined in this way, is also called the differential of the function f . This immediately implies that d f ◦X = X f,

where

f ∈ C∞ (M), X ∈ F ∞ (M).

(1.17)

Definitions (1.1.27), (1.1.7), and (1.10) imply the following proposition. Proposition 1.1.28. Let f , g ∈ C∞ (M), and let c be a constant function on M. Then we have 0) dc = 0, 1) d(c f ) = cd f , 2) d( f + g) = d f + dg, 3) d( f g) = gd f + f dg. The last assertion of this proposition is the famous Leibnitz rule. If the last two properties hold then property 1) is equivalent to property 0).

1.1.4 The Lie Commutator of Two Smooth Vector Fields Routine calculations lead to the following statement.

1.1 Fundamentals of the Theory of Smooth Manifolds

11

Proposition 1.1.29. Let X,Y, Z be smooth vector fields on a smooth manifold M. Then the formula [X,Y ] f = X(Y f ) −Y (X f ),

f ∈ C∞ (M),

gives a derivation of the ring C∞ (M) and therefore defines by Theorem 1.1.26 a unique smooth vector field [X,Y ] on M. Moreover, 1) the operation [X,Y ] is bilinear over R, 2) [X,Y ] = −[Y, X] (antisymmetry), 3) [X, [Y, Z]] + [Y, [Z, X]] + [Z, [X,Y ]] = 0 (the Jacobi identity), 4) [X, fY ] = f [X,Y ] + (X f )Y . Definition 1.1.30. For vector fields X and Y , the vector field [X,Y ] is called the Lie bracket or the commutator of X and Y . Any vector space L over R together with a bilinear operation [·, ·], satisfying conditions 1), 2), and 3) from Proposition 1.1.29, is called a Lie algebra over the field R. Now, suppose that ξ i , η j : U → R are the components of vector fields X,Y in a chart (U, φ ,V ). Let us calculate the components ζ i : U → R of the vector field [X,Y ] in the same chart. On the basis of (1.10) and (1.11), ζ i = [X,Y ]φ i = X(Y φ i ) −Y (Xφ i ) = Xη i −Y ξ i n ∂ (η i ◦ φ −1 ) 1 ∂ (ξ i ◦ φ −1 ) 1 (φ , . . . , φ n )ξ j − ∑ (φ , . . . , φ n )η j . j j ∂ x ∂ x j=1 j=1 n

=

∑

If we denote η i ◦ φ −1 by η i and ξ i ◦ φ −1 by ξ i for brevity, then we get ζi =

∂ ηi j ∂ ξ i j ξ − jη . ∂xj ∂x

(1.18)

We used here the Einstein summation rule in order to remove the sum sign Σ . This formula can be considered as a different definition of the Lie bracket [X,Y ]. Definition 1.1.31. Let M and N be smooth manifolds, X and Y smooth vector fields on M and N respectively, and let f : M → N be a smooth mapping. We say that the vector field X is f -connected with the vector field Y if T f (X(x)) = Y ( f (x)) for any point x ∈ M. As a corollary of Definitions 1.1.31 and 1.1.19 we get Corollary 1.1.32. Let f : M → N be a smooth mapping of two smooth manifolds. Then a smooth vector field X on the manifold M is f -connected with a smooth vector field Y on N if and only if for any smooth real function φ on N, Y φ ◦ f = X(φ ◦ f ). It is easy to check that Corollary 1.1.32, Definitions 1.1.30 and 1.1.19 immediately imply

12

1 Riemannian Manifolds

Theorem 1.1.33. Let M and N be smooth manifolds and f : M → N a smooth mapping. Assume that smooth vector fields X and Y on M are f -connected respectively with smooth vector fields X1 and Y1 on N. Then the smooth vector field [X,Y ] on M is f -connected with the smooth vector field [X1 ,Y1 ] on N. Definition 1.1.34. A smooth path c(p,t), t ∈ I, where p is a fixed point in M n and I is an open (bounded or unbounded) interval of the real line, containing the number 0, is called an integral curve of a smooth vector field X with origin at p, if c(p, 0) = p,

c 0 (p,t) = X(c(p,t)),

t ∈ I.

(1.19)

If I is a maximal domain of definition of such a path, then the path c(p,t), t ∈ I, is called the maximal integral curve. If c(p,t), t ∈ I0 ⊂ I, is a part of such an integral curve with condition c(p, I0 ) ∈ U, where (U, φ ,V ) is some map in M n , then the differential equation in (1.19) for this curve is written in the form of a first order system of ODEs for xi (t) := xi (φ (c(p,t))): (xi )0 (t) = ζ i (φ −1 (x1 (t), . . . , xn (t))), t ∈ I0 , where ζ i , i = 1, . . . , n, are the components of the vector field X in the chart (U, φ ,V ). The theory of first order systems of ODEs with a smooth right-hand part implies the existence of a unique smooth mapping Ψ : W → M with the following properties: 1) W is an open neighborhood of the set M × {0} in M × R; 2) If p ∈ M then W ∩ ({p} × R) = {p} × I p , where I p is an open connected neighborhood of the point {0} in R; 3) If p ∈ M then Ψp (t) := Ψ (p,t) = c(p,t), t ∈ I p , is an integral curve of the vector field X; 4) W is the maximal domain of definition for the mappings Ψ with properties 1), 2), 3). Notice that Condition 4) is equivalent to the statement that c(p,t), t ∈ I p , from Condition 3) is a maximal integral curve of the field X. As a corollary of these properties, α) Ψ (·, 0) = idM ; β ) Ψ (p,t) and Ψ (Ψ (p,t), s) are defined for s,t ∈ R if and only if Ψ (p,t + s) is defined; moreover, Ψ (Ψ (p,t), s) = Ψ (p,t + s). Definition 1.1.35. The above mapping Ψ : W → M, defined by a smooth vector field X on M, is called the flow (on M) of the vector field X. This flow and the vector field X are called complete if W = M × R. The property β ) can be expressed as follows: Ψt := Ψ (·,t) is a local 1-parameter group of local diffeomorphisms of the manifold M n . If the flow Ψ is complete, then the word “local” could be omitted. Proposition 1.1.36. A smooth vector field X on a smooth manifold M is complete if and only if the domain of definition of the flow Ψ , corresponding to the vector field X, includes the set M × (−ε, ε) for some number ε > 0.

1.1 Fundamentals of the Theory of Smooth Manifolds

13

Proof. The necessity is obvious. Now, let us fix s ∈ R, then there exist numbers t ∈ (−ε, ε) and m ∈ N such that s = mt. By the condition, Ψt := Ψ (·,t) as well as Ψ (·, s) = (Ψt )m are defined on M. t u Proposition 1.1.37. Every smooth vector field X on a compact smooth manifold M is complete. Proof. For any point p ∈ M, there exist its open neighborhood U p in M and a number ε p > 0 such that the domain of the flow Ψ , corresponding to the field X, includes U p × (−ε p , ε p ). The system {U p , p ∈ M} constitutes an open cover of the compact M. Therefore, there is a finite subcover {U pk , k = 1, . . . , m} of M. If we consider ε = min{ε pk | k = 1, . . . , m}, then the domain of the flow Ψ includes the set M × (−ε, ε). By Proposition 1.1.36, the field X is complete. t u The following proposition (cf. Proposition 1.9 of Chapter 1 in [291]) gives a geometric interpretation of the Lie bracket [X,Y ] for X,Y ∈ F ∞ (M). Proposition 1.1.38. Let X and Y be smooth vector fields on M and let Ψt be the local 1-parameter group of local diffeomorphisms of the field X on M. Then 1 [X,Y ] = lim [Y − (TΨt )Y ]. t→0 t More exactly, 1 [X,Y ](p) = lim [Y (p) − ((TΨt )Y )(p)], t→0 t

p ∈ M.

Proof. Choose an arbitrary open subset V := U × (−ε, ε) of the domain of Ψ . For any function f ∈ C∞ (M), consider f (p,t) := f (Ψt (p)) − f (p), (p,t) ∈ V . Then f (p,t) = f (p,t) − f (p, 0) = t

Z 1

f 0 (p,ts)ds := tg(p,t).

0

Thus, f ◦Ψt = f +tg(·,t) on U for all t ∈ (−ε, ε). In consequence of these equalities, the definition of Ψ , and Proposition 1.1.21, 1 1 X f (p) = lim [ f (Ψt (p)) − f (p)] = lim f (p,t) = lim g(p,t) = g(p, 0). t→0 t t→0 t t→0 Put p(t) = Ψ−t (p). Hence, ((TΨt )Y )(p) f = (Y ( f ◦Ψt ))(p(t)) = [(Y f ) + t(Y g(·,t))](p(t)); 1 1 lim [Y − (TΨt )Y ](p) f = lim [(Y f )(p) − (Y f )(p(t))] − lim(Y g(·,t))(p(t)) t→0 t t→0 t t→0 = X(p)(Y f ) −Y (p)g(·, 0) = X(p)(Y f ) −Y (p)(X f ) = [X,Y ](p) f . The proposition is proved.

t u

14

1 Riemannian Manifolds

Definition 1.1.39. We say that a k-dimensional distribution D on a smooth manifold M n is defined if, for every point x ∈ M n , a k-dimensional vector subspace D(x) ⊂ Mx , 1 ≤ k < n, is chosen. The distribution D is called smooth if for any point x ∈ M n , there exist its open neighborhood U and smooth vector fields X1 , . . . , Xk on U such that the linear span of the vectors X1 (y), . . . , Xk (y) coincides with D(y) for every point y ∈ U. We say that the vector field X ∈ F ∞ (M) belongs to the distribution D if X(x) ∈ D(x) for any point x ∈ M. A smooth distribution D is called involutive if [X,Y ] ∈ D for any vector fields X,Y ∈ F ∞ (M) belonging to D. Definition 1.1.40. A (virtual) submanifold (N, ψ) of a smooth manifold M is called an integral manifold of a smooth distribution D on M if T ψ(Nx ) = D(ψ(x)) for every point x ∈ N. A smooth distribution D on a smooth manifold M is called completely integrable if, for any point of the manifold M, there is an integral manifold of the distribution D passing through this point. Maximal connected integral manifolds of a smooth completely integrable distribution D on a smooth manifold M are called leaves, and the set of all leaves F = F (D) is called the foliation (defined by the distribution D). Theorem 1.1.41. A smooth distribution D on a smooth manifold M is completely integrable if and only if it is involutive, in other words, the C∞ (M)-module F ∞ (D) of smooth vector fields, belonging to D, is a Lie algebra. This theorem, due to G. Frobenius, is a corollary of Proposition 1.59 and Theorem 1.60 in [461]. Furthermore, in Theorem 1.60 in [461], a statement is proved which implies that the notion of foliation from Definition 1.1.40 is equivalent to a more usual notion of foliation. In [282], simultaneously with a derivation of the main O’Neill equations for Riemannian submersions [374], a short proof is given of the Frobenius theorem. On the grounds of Theorem 1.1.41 and its proof, it is not difficult to prove the following theorem. Theorem 1.1.42. A leaf L ∈ F (D) of a smooth k-dimensional distribution D on a smooth manifold M n is a smooth k-dimensional submanifold L ⊂ M n if and only if L is closed in M. In the general case, L is a smooth k-dimensional manifold with respect to the leaf topology, which has as a base connected components of sets U ∩L, where U are open subsets in M.

1.1.5 Tensor Fields on Smooth Manifolds A smooth vector field on a smooth manifold is a tensor field of degree (1, 0). Let M n be a smooth manifold and m be a natural number. An m-linear map F : ∞ (F (M))m → C∞ (M) (respectively F : (F ∞ (M))m → F ∞ (M)) is called a smooth tensor (or smooth tensor field) of degree (0, m) (respectively (1, m)) on M. This means that for any i = 1, . . . , m, smooth vector fields X1 , . . . , Xi ,Yi , . . . , Xm , and any smooth real function f on M, the following relations are satisfied:

1.1 Fundamentals of the Theory of Smooth Manifolds

15

1) F(X1 , . . . , Xi +Yi , . . . , Xm ) = F(X1 , . . . , Xi , . . . , Xm ) + F(X1 , . . . ,Yi , . . . , Xm ), 2) F(X1 , . . . , f Xi , . . . , Xm ) = f F(X1 , . . . , Xi , . . . , Xm ). Theorem 1.1.43. Let F, X1 , . . . , Xm , and x ∈ M be respectively a tensor of degree (0, m) or (1, m), smooth vector fields, and a point on a smooth n-manifold M. Then the value F(X1 , . . . , Xm )(x) depends only on the values X1 (x), . . . , Xm (x). Moreover, for any point x ∈ M, F correctly defines some m-linear function F(x) : (Mx )m → R or F(x) : (Mx )m → Mx . Proof. By Lemma 1.1.10, there exist a chart (U, φ ,V ) such that x ∈ U and a smooth real function g on M that is equal to 1 in some neighborhood of the point x and vanishes outside of U. Since g(x) = 1, then by the above property 2) of the tensor F, we obtain F(X1 , . . . , Xm )(x) = F(ggX1 , . . . , ggXm )(x). (1.20) Denote by ξi j , where i = 1, . . . , m, j = 1, . . . , n; the components of the restrictions of vector fields Xi , i = 1, . . . , m, to U in the chart (U, φ ,V ) and by el , l = 1, . . . , n, smooth vector fields on U with the components ξ j ≡ δlj in the chart (U, φ ,V ). Here δlj is the so-called Kronecker symbol which is equal to 1 for j = l and to 0 otherwise. It is clear that we can consider gξi j and gel respectively as smooth functions and vector fields on M and ggXi = gξi j ge j (1.21) in the Einstein notation. It follows from properties 1) and 2) of the tensor F and formulas (1.20) and (1.21), that F(X1 , . . . , Xm )(x) = F(gξ1j1 ge j1 , . . . , gξmjm ge jm )(x) = gξ1j1 (x) · · · gξmjm (x)F(ge j1 , . . . , ge jm )(x) = ξ1j1 (x) · · · ξmjm (x)F(ge j1 , . . . , ge jm )(x). Therefore, F(X1 , . . . , Xm )(x) = ξ1j1 (x) · · · ξmjm (x)F(ge j1 , . . . , ge jm )(x).

(1.22)

Now, note that if h is another real function on M, equal to 1 at the point x and to 0 outside of U, then by property 1) of the tensor F, we get F(ge j1 , . . . , ge jm )(x) = F(ghe j1 , . . . , ghe jm )(x) = F(he j1 , . . . , he jm )(x). The first assertion is proved. Now let w1 , . . . , wm ∈ Mx be vectors at the point x ∈ M. Take an arbitrary chart (U, φ ,V ) with x ∈ U. If ξi j are the components of the vectors wi , i = 1, . . . , m, in the chart (U, φ ,V ), then define a smooth vector field Xi by the formula Xi = ξi j ge j , (so Xi (x) = wi ) and then define F(x)(w1 , . . . , wm ) = F(X1 , . . . , Xm )(x).

(1.23)

16

1 Riemannian Manifolds

If Y1 , . . . ,Ym are some other smooth vector fields on M such that Yi (x) = wi for i = 1, . . . , m, then by the first part of the theorem, F(X1 , . . . , Xm )(x) = F(Y1 , . . . ,Ym )(x), t u

so the formula (1.23) correctly defines F(x).

Let F be a smooth tensor field of degree (0, m) (or (1, m)) on a smooth n-manifold M and (U, φ ,V ) be a chart on M. Then as a consequence of the last assertion of Theorem 1.1.43, it is possible to define smooth real functions (or vector fields) Fj1 ,..., jm = F(e j1 , . . . , e jm ) = F(x)(e j1 (x), . . . , e jm (x)),

x ∈ U,

on U. In the vector case, we get the vector fields with the components ξ j = Fjj1 ,..., jm

in the chart (U, φ ,V ). The real functions Fj1 ,..., jm (respectively, Fjj1 ,..., jm ) are called the components of the tensor F in the chart (U, φ ,V ). Now, let ξkj , j = 1, . . . , n, be the components of some vector fields Xk , k = 1, . . . , m, in the chart (U, φ ,V ). Then the function (or the vector field) F(X1 , . . . , Xm ) has a restriction to U (respectively, the components in the chart (U, φ ,V )), equal to Fj1 ,..., jm ξ1j1 . . . ξmjm (respectively, Fjj1 ,..., jm ξ1j1 . . . ξmjm ). Definition 1.1.44. The Lie derivative LX A of an (s, r)-tensor field A along a vector field X is defined by its values on vector fields X1 , X2 , . . . , Xr by the formula r

LX A(X1 , . . . , Xr ) = X · A(X1 , . . . , Xr ) − ∑ A(X1 , . . . , [X, Xi ], . . . , Xr ), i=1

if A is a (0, r)-tensor, and by the formula r

LX A(X1 , . . . , Xr ) = [X, A(X1 , . . . , Xr )] − ∑ A(X1 , . . . , [X, Xi ], . . . , Xr ), i=1

if A is a (1, r)-tensor. Notice that the operator LX is linear and it is a derivation with respect to the tensor product. In particular, the action of LX on functions gives the derivative in the direction of the vector field X; if Y is a vector field then LX Y = [X,Y ].

1.2 Manifolds with Connections and Riemannian Manifolds 1.2.1 Covariant Derivative Definition 1.2.1. A linear connection (or a covariant derivative) on a smooth manifold M is a mapping

1.2 Manifolds with Connections and Riemannian Manifolds

∇ : F ∞ (M) × F ∞ (M) → F ∞ (M),

17

∇(X,Y ) := ∇X Y ∈ F ∞ (M)

such that for all Z ∈ F ∞ (M) and f ∈ C∞ (M), the following relations are satisfied: 1) ∇X+Y Z = ∇X Z + ∇Y Z; 2) ∇ f X Y = f ∇X Y ; 3) ∇X (Y + Z) = ∇X Y + ∇X Z; 4) ∇X fY = f ∇X Y + (X f )Y . As an immediate consequence of Definition 1.2.1 we get Proposition 1.2.2. The difference of any two covariant derivatives is a (1, 2)-tensor. Theorem 1.2.3. Let ∇ be a linear connection on a smooth manifold M. Then the mapping T : F ∞ (M) × F ∞ (M) → F ∞ (M),

T (X,Y ) = ∇X Y − ∇Y X − [X,Y ],

defines a skew-symmetric (1, 2)-tensor on M. Proof. It follows from the definition of T and Properties 1) and 3) in Definition 1.2.1 that T is an additive function with respect to X and Y and T (X,Y ) = −T (Y, X). Therefore, one needs only to show that T ( f X,Y ) = f T (X,Y ),

f ∈ C∞ (M).

In consequence of the definition of T , Properties 2) and 4) in Definition 1.2.1, and properties of [·, ·], we get T ( f X,Y ) = ∇ f X Y − ∇Y f X − [ f X,Y ] = f (∇X Y − ∇Y X − [X,Y ]) − (Y f )X + (Y f )X = f T (X,Y ).u t Definition 1.2.4. The tensor T from the statement of Theorem 1.2.3 is called the torsion tensor of the linear connection ∇. An analogue to Definition 1.1.44 of the Lie derivative of tensor fields in the direction of a given vector field is the extension of the covariant derivative of vector fields along a given vector field to tensor fields. It is uniquely defined by natural requirements that the derivative commutes with the convolution of tensors and satisfies the Leibnitz rule. Definition 1.2.5. The covariant derivative ∇X A of an (s, r)-tensor field A along a vector field X is defined by its values on vector fields X1 , . . . , Xr by the formula r

(∇X A)(X1 , . . . , Xr ) = X · A(X1 , . . . , Xr ) − ∑ A(X1 , . . . , ∇X Xl , . . . , Xr ) l=1

if A is a (0, r)-tensor and by the formula

18

1 Riemannian Manifolds r

(∇X A)(X1 , . . . , Xr ) = ∇X A(X1 , . . . , Xr ) − ∑ A(X1 , . . . , ∇X Xl , . . . , Xr ) l=1

if A is a (1, r)-tensor.

1.2.2 Riemannian Metrics A smooth metric tensor or a Riemannian metric on a smooth manifold M n is a smooth symmetric positively defined tensor g : F ∞ (M) × F ∞ (M) → C∞ (M) of degree (0, 2) on M such that for all smooth vector fields X,Y on M, the following relations are satisfied: 1) g(X,Y ) = g(Y, X), 2) g(X, X) ≥ 0, and 3) g(X, X)(x) > 0, if X(x) 6= 0 and x ∈ M. Definition 1.2.6. A pair (M, g), where M is a smooth (connected) manifold and g is a (smooth) metric tensor on M, is called a Riemannian manifold. If (U, φ ,V ) is a map on M and g is a metric tensor on M, then g has some components gi j : U → R, i, j = 1, . . . , n, in the map (U, φ ,V ). Usually these components are identified with functions gi j ◦ φ −1 : V → R. In terms of these components, Condition 1) above is equivalent to the equalities gi j = g ji , while Conditions 2) and 3) together are equivalent to the statement that for any point x ∈ U the (n × n)-matrix G(x) := (gi j (x)) is positively defined. If, according to the previous section, smooth vector fields X and Y respectively have components ξ i and η j in a map (U, φ ,V ), then g(X,Y )|U = gi j ξ i η j . In Exercise 25 in [461, p. 52]), the following important theorem is stated. Theorem 1.2.7. Any smooth (connected) manifold admits a smooth Riemannian metric. The proof of this theorem uses the statement that any Hausdorff locally compact topological space M with the second countability axiom (all these conditions are satisfied for smooth manifolds M in our sense) is paracompact, i.e. every open cover of the space M admits some open locally finite refinement (Lemma 1.9 in [461]) and Theorem 1.11 in [461] on the existence of a (smooth) partition of the unit on a smooth manifold. The proof of these two assertions in [461] takes up only four pages. Using these results, one can prove Theorem 1.2.7 as an easy exercise. The details are omitted.

1.2 Manifolds with Connections and Riemannian Manifolds

19

1.2.3 The Levi-Civita Connection on a Riemannian Manifold Definition 1.2.8. A linear connection ∇ on a Riemannian manifold (M, g) is called a Levi-Civita connection if for all smooth vector fields X,Y, Z on M, the following conditions are satisfied: 1) Xg(Y, Z) = g(∇X Y, Z) + g(X, ∇X Z); 2) The torsion tensor T of the connection ∇ is equal to zero. The following theorem demonstrates the importance of the introduced notion. Theorem 1.2.9. On any Riemannian manifold (M, g), there exists a unique LeviCivita connection ∇. Proof. Obviously, T = 0 ⇐⇒ ∇X Y − ∇Y X = [X,Y ].

(1.24)

For brevity we shall write hX,Y i instead of g(X,Y ). Let us make the cyclic permutations in the equality 1) from Definition 1.2.8: XhY, Zi = h∇X Y, Zi + hY, ∇X Zi, Y hZ, Xi = h∇Y Z, Xi + hZ, ∇Y Xi, ZhX,Y i = h∇Z X,Y i + hX, ∇Z Y i. Then in consequence of (1.24), XhY, Zi +Y hZ, Xi − ZhX,Y i = h∇X Z − ∇Z X,Y i + h∇Y Z − ∇Z Y, Xi + h∇X Y, Zi + h∇Y X, Zi = h[X, Z],Y i + h[Y, Z], Xi + h∇X Y, Zi + h∇X Y, Zi − h[X,Y ], Zi . We get from here that 1 h∇X Y, Zi = [XhY, Zi +Y hZ, Xi − ZhX,Y i 2 +h[X,Y ], Zi + h[Z, X],Y i − h[Y, Z], Xi].

(1.25)

Now, if we define h∇X Y, Zi by the right-hand side of (1.25), then it is easy to check that for fixed vector fields X and Y this gives a (0, 1)-tensor function F(Z) on Z ∈ F ∞ (M). Consequently, by the easily proved Lemma 1.2.10 below, there exists a unique vector field W = W (X,Y ) on M such that F(Z) = hW, Zi. This means that it is possible to define ∇X Y as the smooth vector field W (X,Y ). Finally, using formula (1.25), one can easily check that the above defined mapping ∇ has all the properties from Definitions 1.2.1 and 1.2.8. t u Lemma 1.2.10. For any smooth (0, 1)-tensor F on a Riemannian manifold (M, g), there exists a unique smooth vector field W on M such that for every smooth vector field Z on M the equality F(Z) = hW, Zi holds.

20

1 Riemannian Manifolds

The definition of a covariant derivative ∇ on a smooth manifold M n , Lemma 1.1.10, and the formulas (1.10), (1.11) imply the following: if smooth vector fields X and Y coincide on an open subset U ⊂ M with smooth vector fields X 0 and Y 0 respectively, then ∇X Y = ∇X 0 Y 0 on U. Therefore, for any map (U, φ ,V ) on M n and the corresponding smooth vector fields e1 , . . . , en on U it is possible to define the smooth vector fields ∇ei e j = Γi kj ek , (1.26) where Γi kj are some smooth real functions on U that are called the Christoffel symbols (in the map (U, φ ,V )). Now, if X = ξ l el and Y = η k ek then by Definition 1.2.1,  k  ∂η k j ∇X Y = ξ l e + η Γ e k lk j . ∂ xl

(1.27)

Let e1 , . . . , en be the basis of local 1-forms ((0, 1)-tensors) dual to e1 , . . . , en , i.e. ei (e j ) = δ ji and A = ηk ek (local) 1-form. Then by Definition 1.2.5 and (1.27), ∇X A(ei ) = ξ

l



   ∂ ηk k j k l ∂ ηi k e (ei ) − ηk e (Γli e j ) = ξ − ηk Γli . ∂ xl ∂ xl

Therefore, ∇X A = ξ

l



 ∂ ηi k − ηk Γli ei . ∂ xl

(1.28)

We want to calculate the Christoffel symbols for the Levi-Civita connection ∇ on a Riemannian manifold (M, g) as functions of the components gi j and their partial derivatives of the metric tensor g in the same map (U, φ ,V ). Using the fact that [ei , e j ] = 0 for all i, j = 1, . . . , n, we get from (1.25) and (1.26): h∇ei e j , el i = hΓi kj ek , el i = Γi kj hek , el i = Γi kj gkl = glk Γi kj 1 = (ei he j , el i + e j hel , ei i − el hei , el i) 2   1 ∂ g jl ∂ gli ∂ gi j 1 = (ei g jl + e j gli − el gi j ) = + − l . 2 2 ∂ xi ∂xj ∂x Here we used the brief notation

∂ g jl ∂ xi

instead of the exact expression

∂ g jl ◦φ −1 ∂ xi

◦ φ , etc.

The condition 3) for the metric tensor g, as was said before, is equivalent to the statement that for any point x ∈ U the matrix G(x) = (gi j (x)), i, j = 1, . . . , n, is positively defined. In particular, there exists an inverse matrix G(x)−1 = (grs (x)) with smooth real elements grs , r, s = 1, . . . , n, on U and we have the condition grs gsk ≡ δkr . Using these facts, we deduce from the expressions in the previous paragraph that 1 ml ml k Γi m j = g glk Γi j = g 2



∂ g jl ∂ gli ∂ gi j + − l ∂ xi ∂xj ∂x

 .

(1.29)

1.3 Curvature of Manifolds with Connection and Riemannian Manifolds

21

m In particular, this formula implies Γi m j = Γji .

1.3 Curvature of Manifolds with Connection and Riemannian Manifolds 1.3.1 The Curvature Tensor of a Manifold with a Covariant Derivative Theorem 1.3.1. Let M be a smooth manifold with a covariant derivative ∇. Then the function R : (F ∞ (M))3 → F ∞ (M), defined by the formula R(X,Y, Z) = R(X,Y )Z = ∇X ∇Y Z − ∇Y ∇X Z − ∇[X,Y ] Z,

(1.30)

is a (3, 1)-tensor on M (the curvature tensor of the linear connection ∇). Moreover, R(X,Y )Z = −R(Y, X)Z .

(1.31)

Proof. It is clear that the function R is additive in any of its arguments. The last formula is evident. Let f ∈ C∞ (M). Using properties of the connection ∇, we get R( f X,Y )Z = ∇ f X ∇Y Z − ∇Y ∇ f X Z − ∇[ f X,Y ] Z = f (∇X ∇Y Z − ∇Y ∇X Z − ∇[X,Y ] Z) − (Y f )∇X Z − ∇−(Y f )X Z = f R(X,Y )Z. Now the same property for Y follows from (1.31). Using the properties of ∇, we get R(X,Y )( f Z) = ∇X ∇Y ( f Z) − ∇Y ∇X ( f Z) − ∇[X,Y ] ( f Z) = f (∇X ∇Y Z − ∇Y ∇X Z − ∇[X,Y ] Z) + ∇X (Y f )Z +(X f )∇Y Z − ∇Y (X f )Z − (Y f )∇X Z − ([X,Y ] f )Z = f R(X,Y )Z + (Y f )∇X Z + (X f )∇Y Z − (X f )∇Y Z −(Y f )∇X Z + [X(Y f ) −Y (X f )]Z − ([X,Y ] f )Z = f R(X,Y )Z. The theorem is proved.

t u

1.3.2 The Curvature Tensor of a Riemannian Manifold The curvature tensor R of the Levi-Civita connection ∇ on any Riemannian manifold (M, g) has additional symmetry properties. Theorem 1.3.2. Let (M, g) be a smooth Riemannian manifold and ∇ the corresponding Levi-Civita connection on (M, g). Then for any smooth vector fields X, Y , Z, and W on M, we have

22

1 Riemannian Manifolds

R(X,Y )Z + R(Y, Z)X + R(Z, X)Y = 0,

(1.32)

hR(X,Y )Z,W i = −hR(X,Y )W, Zi,

(1.33)

hR(X,Y )Z,W i = hR(Z,W )X,Y i.

(1.34)

Proof. 1) Using formulas (1.24), (1.30), and the Jacobi identity for the Lie bracket, we get R(X,Y )Z + R(Y, Z)X + R(Z, X)Y = ∇X ∇Y Z − ∇Y ∇X Z − ∇[X,Y ] Z +∇Y ∇Z X − ∇Z ∇Y X − ∇[Y,Z] X + ∇Z ∇X Y − ∇X ∇Z Y − ∇[Z,X]Y = ∇X [Y, Z] + ∇Y [Z, X] + ∇Z [X,Y ] − (∇[X,Y ] Z + ∇[Y,Z] X + ∇[Z,X]Y ) = [X, [Y, Z]] + [Y, [Z, X]] + [Z, [X,Y ]] = 0. 2) By property 1) of the connection ∇ from Definition 1.2.1 in Subsection 1.2.3, hR(X,Y )Z,W i = h∇X ∇Y Z − ∇Y ∇X Z − ∇[X,Y ] Z,W i = Xh∇Y Z,W i − h∇Y Z, ∇X W i −Y h∇X Z,W i + h∇X Z, ∇Y W i −[X,Y ]hZ,W i + hZ, ∇[X,Y ]W i = −h∇Y Z, ∇X W i + h∇X Z, ∇Y W i +hZ, ∇[X,Y ]W i + X(Y hZ,W i) − XhZ, ∇Y W i −Y (XhZ,W i) +Y hZ, ∇X W i − [X,Y ]hZ,W i = −h∇Y Z, ∇X W i + h∇X Z, ∇Y W i +hZ, ∇[X,Y ]W i − XhZ, ∇Y W i +Y hZ, ∇X W i. Doing only a part of the second step above, we get hR(X,Y )W, Zi = h∇X ∇Y W − ∇Y ∇X W − ∇[X,Y ]W, Zi = Xh∇Y W, Zi − h∇Y W, ∇X Zi −Y h∇X W, Zi +h∇X W, ∇Y Zi − h∇[X,Y ]W, Zi. Now the required formula is obtained by comparing the two expressions above and the symmetry property of the tensor g. 3) The property (1.34) is a corollary of all previous properties (1.31), (1.32), and (1.33). In view of (1.33), the equality (1.34) is equivalent to the equality hR(X,Y )Z,W i + hR(Z,W )Y, Xi = 0 .

(1.35)

In consequence of (1.32) and (1.33), the left-hand part in (1.35) is equal to −hR(Y, Z)X,W i − hR(Z, X)Y,W i + hR(Z,W )Y, Xi = hR(Y, Z)W, Xi − hR(Z, X)Y,W i + hR(Z,W )Y, Xi = −hR(W,Y )Z, Xi − hR(Z, X)Y,W i = hR(Z, X)W,Y i + hR(W,Y )X, Zi. Comparing the last expression with the left-hand part in (1.35) (especially their second terms), we see that it can be obtained from (1.35) by the cyclic permutation of Z → W → Y → X → Z. Applying this permutation once more and reversing the

1.3 Curvature of Manifolds with Connection and Riemannian Manifolds

23

order of summands, we see that the last expression, consequently the left-hand part in (1.35), is equal to hR(Y, X)Z,W i + hR(W, Z)Y, Xi. In view of (1.31), the corresponding summands in this expression and in the lefthand part in (1.35) are negative to each other. This proves the equality (1.35) and the theorem. t u Definition 1.3.3. The tensor hR(X,Y )Z,W i is called the covariant curvature tensor of the Riemannian manifold (M, g). Now we will calculate the components Rli jk (respectively, Ri jkl ) of the (covariant) curvature tensor of a Riemannian manifold (M, g) in a chart (U, φ ,V ) on M in terms of the Christoffel symbols Γi m j and their derivatives. From the definition of the curvature tensor and the preceding section, m Ri jkl = hR(ei , e j )ek , el i = hRm i jk em , el i = glm Ri jk

= h∇ei ∇e j ek − ∇e j ∇ei ek − ∇[ei ,e j ] ek , el i * =

= h∇ei Γjkm em − ∇e j Γikm em , el i ! + ∂Γjks ∂Γiks m s m s Γjk Γim + − Γik Γjm − es , el ∂ xi ∂xj ! s s ∂Γ ∂Γ jk s m = gls Γims Γjkm + − Γjm Γik − ikj . ∂ xi ∂x

Therefore, Rsijk = Γims Γjkm +

∂Γjks

s m − Γjm Γik −

∂ xi Ri jkl = gls Rsijk .

∂Γiks , ∂xj

(1.36) (1.37)

In terms of the components, the symmetry properties (1.31), (1.32), (1.33), and (1.34) can be written respectively in the form of the equalities Rsijk = −Rsjik ,

(1.38)

Rsijk + Rsjki + Rski j = 0,

(1.39)

Ri jkl = −Ri jlk ,

(1.40)

Ri jkl = Rkli j .

(1.41)

Furthermore, we get from (1.37) and (1.38) (respectively, (1.39)) Ri jkl = −R jikl ,

(1.42)

Ri jkl + R jkil + Rki jl = 0.

(1.43)

24

1 Riemannian Manifolds

1.3.3 The Ricci Tensor of a Manifold with a Covariant Derivative Recall that the trace trace(l) of a linear mapping l : V → V of a finite-dimensional vector space V is the sum of all diagonal elements of the matrix (l) of the mapping l in any basis e of the space V . This definition does not depend on the choice of e. Now, assume that R is the curvature tensor of a linear connection ∇ on a smooth manifold M and X,Y are smooth vector fields on M. Then the function R(·, X)Y : Z ∈ F ∞ (M) → R(Z, X)Y ∈ F ∞ (M) is a (1, 1)-tensor on M. Consequently by Theorem 1.1.43, for any point x ∈ M, this tensor defines an R-linear mapping R(·, X(x))Y (x) : w ∈ Mx → R(w, X(x))Y (x) ∈ Mx . Then it is easy to see that the formula Ric(X,Y )(x) = trace R(·, X(x))Y (x)

(1.44)

defines a smooth (0, 2)-tensor on M. Definition 1.3.4. The tensor Ric is called the Ricci tensor of the linear connection ∇ on M. In terms of the components, if Rli jk are the components of the curvature tensor R in a chart (U, φ ,V ), then the components of the Ricci tensor Ric in the chart (U, φ ,V ) are equal to Ric jk = Rll jk . (1.45)

1.3.4 The Ricci Tensor of a Riemannian Manifold Theorem 1.3.5. The Ricci tensor of the Levi-Civita connection for a smooth Riemannian manifold (M, g) is symmetric, i.e. Ric(X,Y ) = Ric(Y, X) for any smooth vector fields X and Y on M. Proof. Take an arbitrary point x ∈ M n and some orthonormal basis {v1 , . . . , vn } in Euclidean space (Mx , gx ). The matrix of the linear map R(·, X(x))Y (x) : Mx → Mx in the basis {v1 , . . . , vn } is equal to (ai j ) = (hR(v j , X(x))Y (x), vi i). Then in consequence of properties (1.34), (1.31), and (1.33), n

n

Ric(X,Y )(x) = ∑ aii = ∑ hR(vi , X(x))Y (x), vi i i=1

i=1

1.3 Curvature of Manifolds with Connection and Riemannian Manifolds n

25

n

= ∑ hR(Y (x), vi )vi , X(x)i = ∑ hR(vi ,Y (x))X(x), vi i = Ric(Y, X)(x). i=1

t u

i=1

Definition 1.3.6. A Riemannian manifold (M, g) is called an Einstein manifold if Ric = λ g for some number λ ∈ R.

1.3.5 Curvatures of Riemannian Manifolds Definition 1.3.7. Let σ be a 2-plane in the tangent vector space Mx to a Riemannian manifold (M n , g), n ≥ 2, and u, v be two linear independent vectors in σ . The sectional curvature K(σ ) of the manifold (M n , g) in the direction of σ is the number K(σ ) = hR(u, v)v, ui/|u ∧ v|,

where

|u ∧ v| = hu, uihv, vi − hu, vi2 . Using the properties of the curvature tensor R, it is easy to check that K(σ ) does not depend on the choice of the basis {u, v} in σ . In particular, if {u, v} is an orthonormal basis in σ , then K(σ ) = hR(u, v)v, ui.

(1.46)

Definition 1.3.8. Let (M n , g), n ≥ 2, be a Riemannian manifold. The Ricci curvature in the direction of a vector v ∈ Mx is defined as ric(v) = Ric(v, v). Usually ric(v) is calculated in the direction of a unit vector v ∈ (Mx , gx ), x ∈ M. In this case it is possible to include v in some orthonormal basis {v1 = v, v2 , . . . , vn } of the space (Mx , gx ). Then, using the formula from the proof of Theorem 1.3.5, we get n

n

ric(v) = Ric(v, v) = ∑ hR(vi , v)v, vi i = ∑ K(σi ), i=1

(1.47)

i=2

where σi is the linear span of vectors v and vi , i ≥ 2. Definition 1.3.9. Let (M, g) be a Riemannian manifold, dim(M) = n ≥ 2. The scalar curvature of the manifold (M, g) at the point x ∈ M is the real number n

sc(x) = ∑ ric(vi ) = i=1

n

∑ hR(vi , v j )v j , vi i,

(1.48)

i, j=1

where {v1 , . . . , vn } is any orthonormal basis in (Mx , gx ). Remark 1.3.10. This definition does not depend on the choice of orthonormal basis {v1 , . . . , vn } in (Mx , gx ).

26

1 Riemannian Manifolds

We have the following classical theorem. Theorem 1.3.11 (F. Schur). Let (M, g) be a Riemannian manifold of dimension n ≥ 3. If for any point x in M, all sectional curvatures of 2-planes at the point x are equal (to a number which a priori may depend on x), then (M, g) has constant sectional curvature. I. G. Nikolaev obtained important stability results related to Schur’s theorem in the paper [355]. Analogously to Schur’s theorem (cf. Remark (v) to Section 3.6 in [241]), we have the following. Theorem 1.3.12. A Riemannian manifold (M, g) of dimension n ≥ 3 is Einstein if and only if Ric = λ g, where λ is some smooth real function on M.

1.4 Fundamentals of the Geometry of Riemannian Manifolds 1.4.1 Parametrization of a Curve by the Arc Length We will use a general agreement that a map f : S → M of a subset S in a smooth manifold N is called smooth if there exist an open submanifold U ⊂ N, where S ⊂ U, and a smooth extension F : U → N of the map f . Definition 1.4.1. A smooth path in a smooth manifold M is a smooth mapping c : [a, b] → M, where [a, b], a < b, is a finite segment in R. A smooth path c is regular if c0 (t) := T c(t, 1) is a non-zero vector in Mc(t) for every number t ∈ [a, b]. We shall say that a smooth path c in M is equivalent to a smooth path c1 : [a1 , b1 ] → M (c ∼ c1 ) if there is a smooth mapping φ of the segment [a, b] onto the segment [a1 , b1 ] such that φ 0 (t) > 0, a ≤ t ≤ b, and c = c1 ◦ φ . It is clear that c is regular if and only if c1 is regular. The equivalence class of a smooth path with respect to the equivalence relation ∼ is called a smooth oriented curve in M (regular, if it has a regular representative). Any representative of a smooth curve in M is called its parametrization. Definition 1.4.2. By definition, the length l = l(c) of a smooth path c : [a, b] → M in a smooth Riemannian manifold (M, g) is the number Z b

l = l(c) = a

|c0 (t)|dt =

Z bp

g(c0 (t), c0 (t))dt.

(1.49)

a

Proposition 1.4.3. The length of a smooth curve, defined by some path c in a Riemannian manifold (M, g), does not depend on its parametrization, i.e. if c ∼ c1 , then l(c) = l(c1 ). Proof. Let c : [a, b] → M, c1 : [a1 , b1 ] → M and c = c1 ◦ φ . Then c0 (t) = T c(t, 1) = T (c1 ◦ φ )(t, 1)

1.4 Fundamentals of the Geometry of Riemannian Manifolds

27

= T c1 (T φ (t, 1)) = T c1 ((φ (t), φ 0 (t))) = φ 0 (t)c01 (φ (t)), and we get that |c0 (t)| =

p

g(c0 (t), c0 (t)) =

q

g(φ 0 (t)c01 (φ (t)), φ 0 (t)c01 (φ (t)))

q = φ 0 (t) g(c01 (t), c01 (t)) = φ 0 (t)|c01 (φ (t))| due to the metric tensor properties. Now, using the change of variables τ = φ (t) in the definite integral, we get Z b

l(c) = a

|c0 (t)|dt =

Z b a

|c01 (φ (t))|φ 0 (t)dt =

Z b1 a1

|c01 (τ)|dτ = l(c1 ).

t u

It is convenient to use the special parametrization of a regular smooth curve, containing a smooth path c : [a, b] → (M, g), by the arc length of the path c: Z t

s(t) =

|c0 (τ)|dτ,

a ≤ t ≤ b.

(1.50)

a

Clearly, s(a) = 0, s(b) = l(c) = l, and s0 (t) = |c0 (t)| > 0. Then by the inverse map theorem for one variable, there exists a smooth diffeomorphism φ (s) = t(s) of the segment [0, l] onto [a, b] such that φ (s(t)) = t and a ≤ t ≤ b. Now, it is possible to define a smooth regular path c1 : [0, l] → M by the formula c1 (s) = c(t(s)),

s ∈ [0, l].

(1.51)

It is clear that c(t) = c1 (s(t)), s0 (t) > 0, a ≤ t ≤ b, so c ∼ c1 . The path c1 = c1 (s), defined by the formula (1.51), is called a parametrization of the (corresponding) curve by the arc length. Sometimes we will use the parameter σ instead of s, where a ≤ σ ≤ b, s = Kσ + const, K = const > 0, for the corresponding a < b, and will say that the path c2 (σ ) = c1 (Kσ + const), a ≤ σ ≤ b, is parameterized proportionally to the arc length. Proposition 1.4.4. A smooth path c = c(t) : [a, b] → (M, g) is parameterized proportionally to the arc length if and only if |c0 (t)| = K, a ≤ t ≤ b, where K is some positive number. In addition, l(c) = K(b − a) and c is parameterized by the arc length if and only if K = 1 and a = 0.

1.4.2 The First Variation of the Length of a Curve Let f : N → M be a smooth map and M be supplied with a linear connection ∇. A vector field along f is defined as a smooth map W : N → T M such that for every point x ∈ N, W (x) ∈ M f (x) . In other words, the relation πM ◦ W = f should be fulfilled. Such a vector field W is tangent to f if there exists a smooth vector field

28

1 Riemannian Manifolds

e : N → T N on N such that W = T f ◦ W e . The set F ∞ ( f ) (respectively, Ft∞ ( f )) of W all smooth (tangent) vector fields along f is a C∞ (N)-module with respect to the point-wise multiplication (φW )(x) = φ (x)W (x), x ∈ N. From the properties 2) and 4) of the connection ∇ in Section 1.2.3 and the formulas (1.10), (1.11), one can easily deduce the following: for any smooth vector fields X,Y on M and a point y ∈ M, ∇X Y (y) depends only on X(y) and the values of Y in an arbitrary neighborhood U of the point y. In fact it is not difficult to show that ∇X Y (y) is defined by X(y) and the restriction of the field Y to an arbitrary path with the tangent vector X(y). This permits us to define a smooth vector field ∇V W along f for any smooth vector fields V and W along f , if V is tangent to f , with the properties, analogous to the previous one: 1) 2) 3) 4)

∇V1 +V2 W = ∇V1 W + ∇V2 W , ∇φV W = φ ∇V W, φ ∈ C∞ N, ∇V (W1 +W2 ) = ∇V W1 + ∇V W2 , ∇V (φW ) = φ ∇V W + (Ve φ )W, where V = T f ◦ Ve .

We are not going to discuss it in detail, but we will give the coordinate descripe is a chart on M, {e1 , . . . , en } and Γ k tion. Let x ∈ N, f (x) ∈ U, where (U, φ , U) ij are the corresponding smooth vector fields and the Christoffel symbols for ∇ in the e Then f −1 (U) is an open neighborhood of the point x in N and the chart (U, φ , U). restrictions of vector fields V and W to f −1 (U) can be presented in the form V = ξ i (ei ◦ f ),

W = η j (e j ◦ f ),

where ξ i and η j are smooth real functions on f −1 (U). Then the restriction of ∇V W to f −1 (U) can be defined as ∇V W = (ξ i η j (Γi kj ◦ f ) + Ve η k )(ek ◦ f ).

(1.52)

Now, if V,W, Z are smooth vector fields along f and V,W are tangent to f , then it is possible to define the curvature tensor R(V,W )Z = ∇V ∇W Z − ∇W ∇V Z − ∇T f ◦[Ve ,We ] Z.

(1.53)

Using the component expressions of the objects in the case of the charts on N and M, given earlier, it is not difficult (even somewhat troublesome and tedious) to check the following statements. For any point x ∈ N, R(V,W )Z(x) = R(V (x),W (x))Z(x),

(1.54)

where the right-hand side is the curvature tensor of a space with connection (M, ∇), so we also obtain a tensor in the left-hand side. If (M, g) is a smooth Riemannian manifold and ∇ is its Levi-Civita connection, then for any vector fields V,W, Z along f , where V is tangent to f , we have the usual properties Ve g(W, Z) = g(∇V W, Z) + g(W, ∇V Z);

(1.55)

1.4 Fundamentals of the Geometry of Riemannian Manifolds

29

if W is also tangent to f , then e ]. ∇V W − ∇W V = T f ◦ [Ve , W

(1.56)

Assume that a smooth path c = c(t) : [a, b] → (M, g) is parameterized proportionally to the arc length and |c0 (t)| = K, a ≤ t ≤ b. A smooth mapping C = C(t, s), (t, s) ∈ [a, b] × (−ε, ε) → M, is called the variation of the path c if C(t, 0) = c(t) for all t ∈ [a, b]. This is the variation with fixed end points if C(a, s) = c(a) and C(b, s) = c(b) for all s ∈ (−ε, ε). Let us consider such a variation. Let X = TC ◦ ∂ /∂t and Y = TC ◦∂ /∂ s be tangent vector fields along C. For any fixed s ∈ (ε, ε), the mapping cs (t) = C(t, s), t ∈ [a, b], is a smooth path in (M, g) and in consequence of (1.49), Z b

l(s) := l(cs ) =

|X(t, s)| dt =

a

Z bp

g (X(t, s), X(t, s))dt.

(1.57)

a

Since |c0 (t)| = K > 0 and [a, b] is compact, then by continuity we may suppose that the integrand in formula (1.57) is positive (otherwise it is possible to take a smaller positive ε). Then the function l(s), −ε < s < ε, is smooth. Let us calculate its derivative l 0 (0) with respect to s at s = 0. By formulas (1.55) and (1.56),  Z  1 b ∂ l 0 (0) = g (X(t, s), X(t, s)) dt 2K a ∂ s s=0 =

1 2K

Z b

1 b g (∇Y X, X) (t, 0)dt K a Z 1 b = g (∇X Y, X) (t, 0)dt K a Z

(Y g(X, X)) (t, 0)dt = a

=

1 K

=−

Z b a

[Xg(Y, X)(t, 0) − g(Y, ∇X X)(t, 0)] dt

Z 1 b

K

a

g(Y, ∇X X)(t, 0)dt +

1 [g(Y, X)]ba . K

Preserving the notations X, Y for the restrictions of these vector fields to c(·) = C(·, 0), it is possible to write down the obtained result in the form l 0 (0) = −

1 K

Z b a


b 1 g (∇c0 c0 )(t),Y (t) dt + g(Y (·), c0 (·)) a . K

(1.58)

If C is the variation with fixed ends, then l 0 (0) = −

1 K

Z b a


g (∇c0 c0 )(t),Y (t) dt.

(1.59)

The formulas (1.58) and (1.59) are called the formulas of the first variation of arc length.

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1 Riemannian Manifolds

1.4.3 Geodesics Definition 1.4.5. A smooth curve c : [a, b] → (M, g), parameterized proportionally to the arc length, is called a geodesic if for any its variations C : [a, b] × (−ε, ε) → (M, g) with fixed ends, l 0 (0) = 0. Definition 1.4.6. A smooth vector field V along a smooth curve c : [a, b] → (M, g) is called parallel (along c) if ∇c0 V = 0. Theorem 1.4.7. A smooth path c : [a, b] → (M, g) is a geodesic if and only if its vector field of tangents c0 is parallel along c. Proof. Sufficiency. First of all we get that d g(c0 (t), c0 (t)) = c0 (t)g(c0 , c0 ) = 2g((∇c0 c0 )(t), c0 (t)) = 0, dt i.e. |c0 (t)| ≡ K for some number K, which can be supposed positive. Therefore, the formula (1.59) is applicable to the path c and it implies that l 0 (0) = 0. Thus, c is a geodesic. Necessity. Let c : [a, b] → (M, g) be a geodesic. It follows from the definition that any part c : [a1 , b1 ] → (M, g), [a1 , b1 ] ⊂ [a, b], of the path c is a geodesic. Assume that c([a1 , b1 ]) ⊂ U, where (U, φ ,V ) is a chart in M. Preserving the same notation [a, b] for [a1 , b1 ], we may apply formula (1.59). By Lemma 1.1.9, there exists a smooth function χ : [a, b] → R which is positive inside [a, b] and equal to zero at a and b. Let x m (t) = (x m ◦ φ ◦ c)(t) and ξ m (t) be correspondingly the coordinates of the path c and the components of vector field (∇c 0 c 0 )(t) in the chart (U, φ ,V ), t ∈ [a, b], m = 1, . . . , n. Then the formula (x m ◦ φ ◦C)(t, s) = x m (t) + sχ(t)ξ m (t),

(t, s) ∈ [a, b] × (−ε, ε),

defines a variation C of the path c with fixed ends for small ε > 0. Moreover, Y (t) = (TC ◦ ∂ /∂ s)(t, 0) = χ(t)(∇c0 c0 )(t). In this case formula (1.59) gives us l 0 (0) = −

1 K

Z b a


χ(t)g (∇c 0 c 0 )(t), (∇c0 c0 )(t) dt.

(1.60)

This number is negative if there is a point t ∈ (a, b) such that (∇c 0 c 0 )(t) 6= 0, which is impossible by definition of a geodesic. Therefore, (∇c 0 c 0 )(t) = 0 for all t ∈ [a1 , b1 ], hence, for all t ∈ [a, b]. t u Definition 1.4.8. A smooth vector field Y along a geodesic c = c(t), a ≤ t ≤ b, is called the Jacobi field (along c) if for the tangent vector field X = c0 , ∇X ∇X Y + R(Y, X)X = 0.

(1.61)

Definition 1.4.9. A smooth map C(t, s), (t, s) ∈ [a, b] × (−ε, ε), where ε > 0, is called a geodesic variation if for any fixed number s ∈ (−ε, ε) the path cs (t) = C(t, s), t ∈ [a, b], is a geodesic.

1.4 Fundamentals of the Geometry of Riemannian Manifolds

31

Proposition 1.4.10. If C(t, s), (t, s) ∈ [a, b] × (−ε, ε), is a geodesic variation then for any number s ∈ (−ε, ε) the field Ys (t) = TC(t, s)(∂ /∂ s), t ∈ [a, b], is the Jacobi field along cs . Proof. Let X = TC ◦ (∂ /∂t) and W be an arbitrary smooth vector field along C. Then Y (g(X,W )) = g(∇Y X,W ) + g(X, ∇Y W ), X(Y (g(X,W ))) = g(∇X ∇Y X,W ) + g(∇Y X, ∇X W ) +X(g(X, ∇Y W )) = g(∇X ∇Y X − ∇Y ∇X X,W ) −g(∇X ∇Y X,W ) + X(g(∇Y X,W )) + X(g(X, ∇Y W )) = g(R(X,Y )X,W ) − g(∇X ∇X Y,W ) + X(Y (g(X,W ))). Since the vector field W is arbitrary, this observation and properties of the tensors g and R imply the identity (1.61). t u

1.4.4 The Geodesic Flow Theorem 1.4.7 states that any smooth path c : [a, b] → (M, g) in a Riemannian manifold (M, g) is geodesic if and only if its tangent vector field c0 = T c ◦ d/dx is parallel along c, i.e. ∇c 0 c 0 = 0. (1.62) Then, as proved earlier, |c 0 (t)| = K for all t ∈ [a, b] and some constant K ≥ 0, so c is parameterized proportionally to the arc length. If c([a, b]) ⊂ U, where (U, φ ,V ) is a map in M, then according to properties of the covariant derivative from Definition 1.2.1 and (1.26), the equation (1.62) is equivalent to the following system of ODEs of second order: dx i dx j d2x m 1 n (t) + Γ m (t) (t) = 0, i j (x (t), . . . , x (t)) 2 dt dt dt

t ∈ [a, b],

(1.63)

where x i (t) = x i (φ ◦ c(t)),

m −1 Γm ij = Γ ij ◦φ ,

and x 1 , . . . , x n are the standard coordinate functions in the open set V ⊂ Rn . In turn, the system (1.63) is equivalent to the following system of ordinary differential equations of the first order dx i (t) = ξ i (t), dt

dξ m 1 n i j (t) = −Γ m i j (x (t), . . . , x (t))ξ (t)ξ (t). dt

Applying these equations (maybe to a part of the curve), we get easily

(1.64)

32

1 Riemannian Manifolds

Proposition 1.4.11. If l : [c, d] → [a, b] is an affine surjective map with l 0 (τ) ≡ k, τ ∈ [c, d], and c : [a, b] → (M, g) is a geodesic, then σ (τ) = (c ◦ l)(τ), τ ∈ [c, d], is also a geodesic. The chart (U, φ ,V ) on M n defines the chart (π −1 (U), ψ,V × Rn ) on T M with coordinate functions x1 , . . . , xn ; ξ 1 , . . . , ξ n , where π : T M → M is the canonical projection. It is possible to consider the tangent vector field c0 (t) along the smooth path c(t), t ∈ [a, b], in M as a smooth path in T M with the property π ◦ c0 = c. If c([a, b]) ⊂ U then (ψ ◦ c0 )(t) = (x 1 (t), . . . , x n (t);

ξ 1 (t) =

dx 1 dx n (t), . . . , ξ n (t) = (t)). dt dt

(1.65)

Comparing the equations (1.65) and (1.64) we see that the path c(t), a ≤ t ≤ b, is geodesic if and only if c0 (t), a ≤ t ≤ b, is an integral curve of a smooth vector field F on T M with the components ξ 1, . . . , ξ n;

−Γ 1i j (x 1 , . . . , x n )ξ i ξ j , . . . , −Γ nij (x 1 , . . . , x n )ξ i ξ j

in the chart (π −1 (U), ψ,V × Rn ). It is clear that the vector field F does not depend on the choice of chart and F defines a smooth vector field on T M. Definition 1.4.12. The vector field F on T M, defined as above, is called the geodesic vector field. The equations (1.64) and the existence and uniqueness theorems for the solution of systems of first order differential equations with a given initial data imply that the converse to the previous assertion is also true, thus we get Theorem 1.4.13. If ψ(t), a ≤ t ≤ b, is an integral curve of the vector field F (on T M) then c(t) := (π ◦ ψ)(t), a ≤ t ≤ b, is geodesic in (M, g) and c0 (t) = ψ(t),

a ≤ t ≤ b.

Every geodesic in (M, g) has such a form. Definition 1.4.14. The flow Ψ , defined by the geodesic vector field F on T M for a Riemannian manifold (M, g), is called the geodesic flow of the manifold (M, g). Besides general properties of flows, the geodesic flow has some special properties. Theorem 1.4.15. If Ψ is the geodesic flow of the geodesic vector field F on T M for a Riemannian manifold (M, g), then Ψ (sv,t) = sΨ (v, st) for all

s,t ∈ R, v ∈ T M.

(1.66)

1.4 Fundamentals of the Geometry of Riemannian Manifolds

33

Moreover, T π ◦ F = idT M = πT M ◦ F.

(1.67)

Proof. We can reformulate the statement of Theorem 1.4.13 as follows: cv (t) := (π ◦Ψ )(v,t), c 0v (t) = Ψ (v,t),

(v,t) ∈ W ;

(v,t) ∈ W, c 0v (0) = Ψ (v, 0) = v,

(1.68) (1.69)

where cv (t), t ∈ Iv is a geodesic curve in (M, g) with the initial data cv (0) = π(v),

c 0v (t) = v.

Then by Proposition 1.4.11, for every number s ∈ R, the curve σ (t) = cv (st),

st ∈ Iv ,

is a geodesic in (M, g) with the tangent vectors σ 0 (t) = sc 0v (st) and the initial data σ (0) = cv (0) = π(v) = π(sv),

σ 0 (0) = sc 0v (0) = sv.

By Theorem 1.4.13 and the definition of Ψ , this is equivalent to the equation (1.66). Now the second equality in (1.67) follows from the general definition of vector fields. Differentiating the equation (1.68) and using the equation (1.69), we get
v = c 0v (0) = T π(Ψv0 (0)) = T π F(Ψv (0)) = T π(F(v)) for any vector v ∈ T M. We have proved the first equality in (1.67).

t u

1.4.5 The Exponential Map In the notation of the previous section, let us define the following set e := {v ∈ T M | (v, 1) ∈ W }. W Since W is an open neighborhood of the set T M × {0}, we obtain, as a corollary of e is an open neighborhood of the set S0 M of all zero vectors the equation (1.66), that W e → M of a in T M; S0 M is naturally identified with M. The exponential map Exp : W Riemannian manifold (M, g) is defined by the formula Exp(v) := π(Ψ (v, 1)),

e. v∈W

(1.70)

e := If x is a point in M, then the restriction of the map Exp to the intersection Mx ∩ W ex is called the exponential map at the point x and is denoted by Expx . Since W e is W e an open neighborhood of the set S0 M, then the equation (1.66) implies that Wx is an

34

1 Riemannian Manifolds

open star-shaped neighborhood of the zero vector 0x in Mx . This means that for any ex and any number s ∈ [0, 1], sw ∈ W ex too. Notice the following vector w ∈ W Proposition 1.4.16. For any Riemannian manifold (M, g), the following assertions are equivalent: 1) The geodesic flow Ψ of the space (M, g) is complete. 2) The domain of the exponential map Exp of the space (M, g) is equal to T M. 3) There is a point x ∈ M with the domain of the exponential map Expx coinciding with Mx . Hint of the proof. The equivalence of the assertions 1) and 2) is evident. Obviously, 2) implies 3). The statement 3) =⇒ 2) will be discussed later. Theorem 1.4.17. For every point x ∈ (M, g), there exists a maximal number 0 < rx ≤ +∞ such that Expx is defined on Vx = U(0x , rx ) ⊂ Mx , where U(0x , rx ) is the open ball in Mx with the center at zero 0x and radius rx , and is a diffeomorphism of this set onto an open neighborhood Ux = Expx (Vx ) of the point x in M. Proof. Since Ψ (0x ,t) = 0x for all t ∈ R, then Expx (0x ) := π(Ψ (0x , 1)) = π(0x ) = x. The statement of the theorem will be a corollary of the inverse map theorem if we prove that T (Expx )(0x ) = idMx (1.71) ((Mx )0x is naturally identified with Mx ). If v is an arbitrary vector in Mx and t ∈ R is close enough to 0, then according to the definitions and (1.66), (1.68), Expx (tv) = π(Ψ (tv, 1)) = π(Ψ (v,t)) = cv (t).

(1.72)

Now, it follows from (1.72) and (1.69) that T (Expx )(0x )(v) = c0v (0) = v. We have proved the equality (1.71) and the theorem.

t u

Definition 1.4.18. The map (Ux , φx = (Expx )−1 ,Vx ), constructed in Theorem 1.4.17, is called the Riemannian normal coordinate system at the point x (if we also use some rectangular Euclidean coordinates in Vx with the origin at 0x ). If the geodesic flow Ψ of the space (M, g) is complete, then the number rx from Theorem 1.4.17 is called the injectivity radius of the space (M, g) at the point x and is denoted by Radinj(x). Proposition 1.4.19. Let (M, g) be a Riemannian manifold and x ∈ M. The tangent map to the exponential map Expx at the point x ∈ M can be described as follows. Let V,W be elements of Mx and let J be the Jacobi field along the geodesic s 7→ Expx (sV ) with initial conditions J(0) = 0, J 0 (0) = W , then TV Expx (W ) = J(1). Proof. This is clear since TsV Expx (W ) is simply dtd Expx (s(V +tW ))|0 , which is the Jacobi field J defined by a 1-parameter family of geodesics through x with initial conditions J(0) = 0, J 0 (0) = W (cf. Proposition 1.4.10). t u

1.4 Fundamentals of the Geometry of Riemannian Manifolds

35

1.4.6 The Shortest Curves Theorem 1.4.20. Let y be any point in Ux − {x} and c = c(t) = Expx (tv),

0 ≤ t ≤ 1, |v| = r0

be a geodesic in (M, g) with the end points x and y. Then the length e l of an arbitrary piecewise smooth path ce(s), a ≤ s ≤ b, with the end points x and y is not less than the length l = r0 of the geodesic c. Furthermore, e l = l if and only if the path ce is obtained from c by a parameter change.

Proof. Assume at first that ce([a, b]) ⊂ Ux . We can suppose that ce(s) 6= x for s > a. Define a geodesic variation by the formula C = C(t, s) = Expx (t(Exp−1 c(s)))), x (e

(t, s) ∈ [0, 1] × [a, b].

For any s > a the path cs (t) = C(t, s), t ∈ [0, 1], is a geodesic parameterized proportionally to the arc length, with the length l(s) > 0, and cb = c. As a consequence of this, Theorem 1.4.7 and (1.58),   1 1  c 0s (1) 0 0 0 l (s) = g(Y (·, s), c s (·)) 0 = g ce (s), 0 ≤ |e c 0 (s)|. (1.73) l(s) |c s (1)| Integrating this inequality, we get l = r0 = l(c) = l(b) = l(b) − l(a) =

Z b a

l 0 (s)ds ≤

Z b a

|e c 0 (s)|ds = e l.

It is clear from the obtained evaluations that e l = l if and only if for all s ∈ (a, b] the inequality in (1.73) is an equality, i.e. ce 0 (s) = |e c 0 (s)|(cs 0 (1)/|cs 0 (1)|); this means that the path ce is obtained from the geodesic c by a parameter change. If ce([a, b]) does not lie in Ux , then by continuity there exists a number a < d < b such that ce([a, d]) ⊂ Ux and r0 < r(d) = r1 < rx , where r(d) = |(Expx |Vx )−1 (c(d))|. ˜ Then according to the obtained estimates,
e l ≥ l ce|[a,d] ≥ r1 > r0 = l. The theorem is proved.

t u

Corollary 1.4.21. If c = c(t), a ≤ t ≤ b, is a shortest piecewise smooth path in (M, g) with given ends, parameterized proportionally to the arc length, then c is a (smooth) geodesic in (M, g).

36

1 Riemannian Manifolds

1.4.7 The Intrinsic Metric of a Riemannian Manifold We will denote by xcy the set of piecewise paths c joining two points x and y (on a given manifold). Definition 1.4.22. Let (M, g) be a connected smooth Riemannian manifold. The distance d(x, y) between the points x and y in M is defined by the formula d(x, y) = inf l(c), xcy

(1.74)

i.e. it is equal to the greatest lower bound of the lengths of all piecewise smooth paths c in (M, g) joining the points x and y. Proposition 1.4.23. If (M, g) is a connected Riemannian manifold then the pair (M, d), where d : M × M → R, is a metric space. Moreover, the metric topology on (M, d) coincides with the initial topology on the smooth manifold M. Proof. Since M is a connected topological manifold with the second countability axiom, it is arc-wise connected. Using an approximation of paths by piecewise smooth paths, we see that every point x in M can be joined with any other point y in M by some piecewise smooth path c (with end points x and y). Then 0 ≤ d(x, y) ≤ l(c) < +∞ for any two points x, y ∈ M. Changing the orientation of a path to its converse does not change its length. Therefore d(x, y) = d(y, x). The triangle inequality immediately follows from the definitions. The inequality d(x, y) > 0 for x 6= y, as well as the coincidence of the metric and the initial topologies on M, are deduced from Theorem 1.4.20. t u Definition 1.4.24. The metric d on a (connected) smooth Riemannian manifold (M, g) is called the intrinsic metric.

1.5 Some Global Properties of Riemannian Manifolds In this section, we discuss some important classical results in global Riemannian geometry. The following theorem is an essential addition to Propositions 1.4.16 and 1.4.23. Theorem 1.5.1 (Hopf–Rinow, [259]). For any (connected smooth) Riemannian manifold (M, g), the following assertions are equivalent: 1) The metric space (M, d) is complete; 2) The domain of the exponential map Exp of the manifold (M, g) is T M; 3) There is a point x ∈ M such that the domain of the exponential map Expx of (M, g) at the point x is equal to Mx ; 4) There is a point x ∈ M such that for every point y ∈ M there exists a geodesic of length d(x, y), joining the points x and y.

1.5 Some Global Properties of Riemannian Manifolds

37

5) There is a point x ∈ M such that every closed ball B(x, r), 0 ≤ r < +∞, in (M, d) is compact. Proof. It is clear that 2) ⇒ 3) and 5) ⇒ 1). Now, let us prove that 1) ⇒ 2). Take any point x ∈ M and an arbitrary unit vector v ∈ Mx . Let T be the set of all real numbers τ, for which Expx (τv) = Exp(τv) is defined. Recall that Exp(τv) = Ψ (1, τv) = Ψ (τ, v), where Ψ is the geodesic flow of the Riemannian manifold (M, g). The second general property of flows defined by smooth vector fields implies that T = Iv is an open connected neighborhood of zero in R. Assume that 0 < τ0 = sup T < +∞. By definition, for every number τ ∈ (0, τ0 ), Expx (tv) = cv (t), 0 ≤ t ≤ τ, is a segment of a geodesic in (M, g) with the initial tangent vector c0v (0) = v. Since v is a unit vector, every such segment is parameterized by the arc length. Therefore, d(Expx (tv), Expx (sv)) ≤ |t − s| for all t, s ∈ [0, τ0 ). Then by the completeness of the space (M, d), there exists a limit limt→τ0 Expx (tv) = Expx (τ0 v) and τ0 ∈ T . We get a contradiction. Since the point x ∈ M and the unit vector v ∈ Mx are arbitrary, it is proved that the map Exp is defined on T M. Let us prove 3) ⇒ 4). Assume that assertion 3) is fulfilled. Theorem 1.4.20 implies that assertion 4) is true for the same point x and any point y such that d(x, y) < rx , where rx is the injectivity radius of the manifold (M, g) at the point x. Put s0 := d(x, y) ≥ rx and let s be some number such that 0 < s < rx . It follows from the definition of the intrinsic metric d that for any natural number k, there exists a point zk such that d(x, zk ) = s and d(x, zk ) + d(zk , y) ≤ s0 + (1/k). By Theorem 1.4.20, all points zk lie in the compact ball B(x, s), the image of a closed ball of radius s with center at the origin in Euclidean space (Mx , gx ) under Expx . Then it is possible to choose a subsequence of the sequence zk converging to some point z. By construction, d(x, z) + d(z, y) = s0 = d(x, y) and d(x, z) = s. Then by Theorem 1.4.20, there exists an arc length-parameterized shortest curve geodesic γ(t) = Expx (tv), 0 ≤ t ≤ s, kvk = 1, joining points x and z. By the condition, c(t) := Expx (tv) is defined for all t ∈ [0, +∞). If we can show that the set S of all numbers t ∈ [0, s0 ] such that d(x, c(t)) + d(c(t), y) = d(x, y) coincides with the segment [0, s0 ], then c(s0 ) = Expx (s0 ) = y and the assertion will be proved. By construction, [0, s] ⊂ S. It follows from the triangle inequality for d that if 0 ≤ t1 ≤ t2 ≤ s0 and t2 ∈ S, then t1 ∈ S. It is clear that the set S is closed, so s1 := sup S ∈ S. Assume that s1 < s0 . Then, repeating the above argument to get the point z, with the point z1 = c(s1 ) instead of x, we can find a point z2 such that d(z1 , z2 ) + d(z2 , y) = d(z1 , y), z1 6= z2 6= y, and points z1 , z2 are joined by some shortest geodesic c1 (t), s1 ≤ t ≤ s2 < s0 of length d(z1 , z2 ), parameterized by the arc length. By the triangle inequality and Corollary 1.4.21, the union of the shortest geodesics c(t), 0 ≤ t ≤ s1 , and c1 will give some shortest geodesic, parameterized by the arc length and defined on the segment [0, s2 ]. Then, by the uniqueness of solutions for a smooth ODE system with given initial data, we have the identity c1 (t) = c(t), s1 ≤ t ≤ s2 . Therefore, [0, s2 ] ⊂ S, a contradiction.

38

1 Riemannian Manifolds

Now, if assertion 4) holds, then B(x, r) = Expx (B(0, r)) (where B(0, r) ⊂ (Mx , gx )) for any real number r ≥ 0, and the set B(x, r) is compact. Therefore, we get 4) ⇒ 5). t u Theorem 1.5.1, especially its point 5), implies the following corollary. Corollary 1.5.2. The metric space (M, d), defined by a given connected Riemannian manifold, is complete if and only if every closed ball B(x, r), 0 ≤ r < +∞, in M is compact. Theorem 1.5.1 and Corollary 1.4.21 also imply Corollary 1.5.3. The metric space (M, d), defined by any connected smooth Riemannian manifold (M, g), is complete if and only if any two points x, y in M can be joined by a (not necessary unique) shortest geodesic in (M, g) of length d(x, y), (parameterized by arc length). Definition 1.5.4 ([12]). For any metric space (M, ρ), one can define an intrinsic metric ρin on M (possibly with both finite and infinite values): ρin (x, y) is defined as the infimum of lengths of paths in (M, ρ) joining the points x and y. It is clear that ρ ≤ ρin . Then (M, ρ) is called a space with intrinsic metric if ρin = ρ. Theorem 1.5.5 (S. Cohn-Vossen, [164]). A locally compact space with intrinsic metric (X, ρ) is complete if and only if every closed ball in X is compact. Under these conditions, any points x, y ∈ X can be joined by a shortest arc, i.e. a path of length equal to d(x, y). Notice that the Hopf–Rinow theorem (Theorem 1.5.1) can be easily deduced from this Cohn-Vossen theorem and Theorem 1.4.20. From now on, we assume that all considered Riemannian manifolds are complete, unless otherwise noted. Lemma 1.5.6. At the origin of the Riemannian normal coordinate system, all its Christoffel symbols are zero. Proof. Let e1 , . . . , en be an orthonormal basis in (Mx , gx ), adapted to a Riemannian normal coordinate system (Ux , φ ,Vx ). For any pair of indexes i, j, consider a geodesic γ(t) = Expx (t(ei + e j )) if j 6= i and a geodesic γ(t) = Expx (t(ei )) if j = i. Put x(t) = φ (γ(t)) and Γi kj (x) := Γi kj (φ −1 (x)). By definition of the Riemannian normal coordinate system, x m (t) = t for m = i, j and zero otherwise. Then x˙ m (t) = 1 for m = i, j and zero otherwise. In any case, x¨ m (t) = 0. Then 0 = x¨ k (t) = Γi kj (x(t)) · 1 · 1 by the equation (1.63). Thus, Γi kj (γ(t)) ≡ 0 for any k = 1, . . . , n. This implies the assertion of the lemma. Theorem 1.5.7 (J. H. C. Whitehead, [465]). For any point x ∈ (M, g) there exists a number sx , where 0 < sx < r2x , such that every shortest curve γ(t), 0 ≤ t ≤ a, joining some points y, z ∈ U(x, sx ), satisfies the condition d(x, γ(t)) < max{d(x, y), d(x, z)} for all t ∈ (0, a).

1.5 Some Global Properties of Riemannian Manifolds

39

Proof. Note that by the triangle inequality, every shortest curve γ(t), 0 ≤ t ≤ a, joining any points y, z ∈ U(x, r), where r = rx /2, lies in U(x, rx ). First let us consider such γ and its coordinate presentation x(t) = φ (γ(t)) in a Riemannian normal coordinate system (U(x, rx ), φ ,Vx ). Then f (t) = ∑ni=1 (x i (t))2 = d(x, γ(t)), f 0 (t) = 2 ∑ni=1 x i (t)x˙ i (t), and by equation (1.63) we get n

n

f 00 (t) = 2 ∑ [x˙ i (t)2 + x i (t)x¨ i (t)] = 2 ∑ [x˙ i (t)2 − Γjki (x(t))x˙ j (t)x˙ k (t)x i (t)] i=1

i=1

"

i

#

= 2 δ jk − ∑ Γjki (x(t)xi (t)) x˙ j (t)x˙ k (t). i=1

In consequence of Lemma 1.5.6, the last expression is positive if γ(t) ∈ U(x, r1 ) for a sufficiently small positive r1 ≤ r. In this case, f 00 (t) > 0 and the well-known result of Calculus implies that f (t) < max( f (0), f (a)) for 0 < t < a. Hence the assertion of the theorem is true for sx = r1 /2. t u Proposition 1.5.8. There exists a positive number σx ≤ sx such that for any point y ∈ U(x, σx ), the ball U(x, σx ) is a diffeomorphic image relative to Expy of a starshaped neighborhood Wy of zero in My . Proof. Suppose for simplicity that (M, g) is complete. Define the mapping Φ : T M → M × M by the formula Φ(v) = (p(v), Exp(v)), where v ∈ T M and p : T M → M is the canonical projection. It is clear that its differential T Φ is non-singular at the zero vector 0x ∈ Mx and Φ(0x ) = (x, x) ∈ M × M. Then by the inverse mapping theorem, there exists an open neighborhood of the point 0x in T M of the form W = ∪y∈U 0 U(0y , δ ) where U 0 is an open neighborhood of x in M and δ > 0, such that Φ is a diffeomorphism of W onto some neighborhood Φ(W ) of the point (x, x) in M × M. Therefore, there exists a positive number σx ≤ sx , where sx is from the statement of Theorem 1.5.7, such that (x, x) ∈ U(x, σx ) ×U(x, σx ) ⊂ Φ(W ). It is clear that σx satisfies the desired condition. t u Proposition 1.5.9. Let γi (s) = Expx (svi ), s ∈ R, vi ∈ Mx , i = 1, 2, be two different geodesics in (M, g) parameterized
by arc length and 0 < si < σx , i = 1, 2. Then the function f (s) := d γ1 (s1 ), γ2 (s) , −s2 < s < s2 , is smooth and f 0 (0) = −hv1 , v2 i. Proof. By the construction of σx , Expγ1 (s1 ) is a diffeomorphism of a star-shaped region in Mγ1 (s1 ) with center at the origin onto the open ball U(x, σx ). Therefore, the vector-function v(s) := Exp−1 (γ (s)), −s2 < s < s2 , is smooth. Then the formula γ (s ) 2 1

1


C(t, s) = Expγ1 (s1 ) (t/s1 ) · v(s) ,

0 ≤ t ≤ s1 ,

−s2 < s < s2 ,

40

1 Riemannian Manifolds

defines a smooth geodesic variation in U(x, σx ) such that cs (t) = C(t, s) is parameterized proportionally to arc length and c0 is parameterized by arc length. Let l(s) be the length of the path cs . Then f (s) = l(s) and by (1.58), there exists the derivative f 0 (0) = l 0 (0) = −hv1 , v2 i. t u Definition 1.5.10. Let γ = γ(s) = Expx (sv), v ∈ Mx , be a geodesic in a Riemannian manifold (M, g) with the initial point x = γ(0) parameterized by the arc length. A finite number t > 0 is called the conjugate value of γ if the differential (T Expx )tv is degenerate. Then the point γ(t) is called a conjugate point of γ. The set of all conjugate points of γ is denoted by Conj(γ). It is clear that if Conj(γ) is not empty then there exists the minimal conjugate value t1 of γ. Definition 1.5.11. The number t1 and γ(t1 ) are called respectively the first conjugate value and the first conjugate point of γ. Lemma 1.5.12. Let x ∈ (M, g), v, w ∈ Mx , hv, vi = 1, C(t, s) := Expx (t(v + sw)),

t ∈ [0, T ],

s ∈ (−ε, ε),

(1.75)

be a nontrivial geodesic variation in (M, g), and X = TC(∂ /∂t), Y = TC(∂ /∂ s) be the corresponding tangent vector fields along C. Then hX(t, 0),Y (t, 0)i = thv, wi,

∇X Y (0, 0) = w,

where h·, ·i = gx (·, ·). Proof. Let l(t, s) be the length of the geodesic Expx (τ(v + sw)), 0 ≤ τ ≤ t ≤ T , for a fixed s ∈ (−ε, ε). It is clear that p p l(t, s) = ht(v + sw),t(v + sw)i = t hv + sw, v + swi. Then by the formula of the first variation, hX(t, 0),Y (t, 0)i =

∂ l(t, s) (t, 0) = thv, wi. ∂s

Let e1 , . . . , en be an orthonormal basis in (Mx , h·, ·i) defining the normal coordinates y 1 , . . . , y n with the origin x. We can assume also that e1 = v and ei = ∂∂y i , i = 1, . . . , n, are corresponding vector fields. Then it is clear that Y (t, 0) = ∑ni=1 tw i ei , where w 1 , . . . , w n are the components of the vector w in the basis e1 , . . . , en . Hence ! ! n

∇X Y (0, 0) =

∑ ∇e1 tw i ei

i=1

n

(x) =

∑ [w i ei + 0 · ∇e1 ei ]

i=1

From Proposition 1.4.10 and Lemma 1.5.12 we easily get

(x) = w.

t u

1.5 Some Global Properties of Riemannian Manifolds

41

Proposition 1.5.13. Let γ : R → M be a geodesic in a Riemannian manifold (M, g) parameterized by arc length, and fix a point a ∈ R . Then there exists a basis Y1 , . . . ,Yn−1 , where n = dim(M), in the space of Jacobi vector fields along γ, orthogonal to γ 0 and equal to zero at t = a. Lemma 1.5.14 (the Gauss lemma). Let x ∈ (M, g), v, w ∈ Mx , hv, vi = 1, γv (t) = Expx (tv), t ∈ R. Then for any t ∈ R, we have h(T Expx )tv (v), (T Expx )tv (w)i = hγ 0v (t), (T Expx )tv (w)i = hv, wi. Proof. It is clear that in the notation of Lemma 1.5.12, (T Expx )tv (v) = X(t, 0) = γ 0v (t),

Y (t, 0) = t(T Expx )tv (w).

Therefore, the required equality follows from Lemma 1.5.12.

t u

Proposition 1.5.15. The following three conditions are equivalent for any geodesic γ(t) = Expx (tv), v ∈ Mx , hv, vi = 1, in any complete Riemannian manifold (M, g): 1) T > 0 is a conjugate value of a geodesic γ = γ(t). 2) There exists a nontrivial Jacobi vector field Y (t), 0 ≤ t ≤ T , along the geodesic γ = γ(t) such that Y (0) = Y (T ) = 0. 3) There exists a geodesic variation C(t, s), 0 ≤ t ≤ T , −ε < s < ε, such that C(t, 0) = γ(t), TC(∂ /∂ s)(t, 0) is not identically zero, and TC(∂ /∂ s)(0, 0) = TC(∂ /∂ s)(T, 0) = 0. Moreover, hY (t), γ 0 (t)i ≡ 0 in 2) and hTC(∂ /∂ s)(t, 0), γ 0 (t)i ≡ 0 in 3) for all 0 ≤ t ≤ T. Proof. Assume that condition 1) is fulfilled. Then by Definition 1.5.10 there exists a nonzero vector w ∈ Mx such that T (Expx )T v (w) = 0. Define the geodesic variation C(t, s) by the formula (1.75). Then by Lemma 1.5.12, the corresponding vector field Y (t, 0) = TC(t,0) ( ∂∂s ) is the Jacobi vector field such that Y (0) = 0, ∇vY = w, and hY (t), γ 0 (t)i ≡ hv, wi. We have proved that 1) implies 2) and 3). Proposition 1.4.10 and condition 3) implies 2). Assume that condition 2) is fulfilled for the Jacobi vector field Y (t), 0 ≤ t ≤ T . The vector field Y is a solution of the Jacobi equation, which is the second order (vector) ODE. Since Y is nontrivial and Y (0) = 0, then ∇γ 0 (0)Y = w 6= 0. Otherwise, the vector field Y has the zero initial condition and must be identically zero itself. Define C(t, s), 0 ≤ t ≤ T , −ε < s < ε, by the formula (1.75). Then Proposition 1.4.10 and Lemma 1.5.12, and the uniqueness of solution of ODEs with given initial data imply that TC(t, 0)(∂ /∂ s) is exactly the given Jacobi vector field Y (t) for 0 ≤ t ≤ T . Therefore, (Expx )T v (w) = 0. We have proved that 2) implies 1) and 3). Let us suppose now that conditions 2) and 3) are fulfilled. Then Y (T ) = (T Expx )T v (w) = 0. Therefore, Lemma 1.5.12 implies hv, wi = 0 and hY (t), γ 0 (t)i ≡ 0, which completes the proof of the proposition. t u

42

1 Riemannian Manifolds

Proposition 1.5.16. Let γ = γ(t) = Expx (tv), hv, vi = 1, be a geodesic in (M, g) with the initial point x = γ(0) and t1 > 0 be the first conjugate value of γ. Then there exists a Jacobi vector field Y (t), 0 ≤ t ≤ t1 , along the geodesic γ(t) such that Y (0) = Y (t1 ) = 0, Y (t) 6= 0 for 0 < t < t1 , ∇vY = w 6= 0, hw, vi = 0, and hY (t), γ 0 (t)i = 0 for all t ∈ [0,t1 ]. Proof. The proposition follows from Proposition 1.5.15, Lemma 1.5.12, and the fact that t1 is the first conjugate value of γ. t u Proposition 1.5.17. Let Y (t) be a Jacobi vector field along a part of a geodesic γ(t), 0 ≤ t ≤ t1 , in (M, g) such that Y (0) = 0, Y (t) 6= 0, and hY (t), γ 0 (t)i = 0 for all t ∈ (0,t1 ). Then
f 00 (t) + K γ 0 (t),Y (t)/kY (t)k f (t) ≥ 0 for all t ∈ (0,t1 ), (1.76) p where f (t) = hY (t),Y (t)i. Moreover, we have an equality in (1.76) for a given t only if ∇γ 0 (t) (Y /kY k) = 0 . Proof. We have h∇γ 0 (t) ∇γ 0 (t)Y,Y (t)i = h∇γ 0 (t)Y,Y (t)i0 − h∇γ 0 (t)Y, ∇γ 0 (t)Y i 1 = hY (t),Y (t)i00 − h∇γ 0 (t)Y, ∇γ 0 (t)Y i = ( f (t) f 0 (t))0 − h∇γ 0 (t)Y, ∇γ 0 (t)Y i 2 = ( f 0 (t))2 + f (t) f 00 (t) − h∇γ 0 (t)Y, ∇γ 0 (t)Y i. Put Y (t) = f (t)V (t), so kV (t)k ≡ 1. Since h∇γ 0 (t)V,V i = (1/2)γ 0 (t)hV,V i = 0, then h∇γ 0 (t)Y, ∇γ 0 (t)Y i = h f 0 (t)V (t) + f (t)∇γ 0 (t)V, f 0 (t)V (t) + f (t)∇γ 0 (t)V i = ( f 0 (t))2 + f (t)2 h∇γ 0 (t)V, ∇γ 0 (t)V i . Since Y = Y (t) is a Jacobi vector field along γ = γ(t), we get h∇γ 0 (t) ∇γ 0 (t)Y,Y (t)i = − f (t)2 hR(V (t), γ 0 (t))γ 0 (t),V (t)i ,
f (t) f 00 (t) + f (t)2 hR(V (t), γ 0 (t))γ 0 (t),V (t)i − h∇γ 0 (t)V, ∇γ 0 (t)V i = 0 , which implies the proposition.

t u

Theorem 1.5.18. No geodesic in a complete Riemannian manifold with non-positive sectional curvature has a conjugate point. If all sectional curvatures of a complete Riemannian manifold do not exceed a positive number K then for any of its geodesics the first conjugate value is not less than √πK . Moreover, the first conjugate value for any geodesic is equal to the last number if and only if the space has constant sectional curvature K. In the last case, in the notation from Propositions 1.5.16 and 1.5.17, ∇γ 0 (t)Y is always collinear to Y (t) if Y (t) 6= 0.

1.5 Some Global Properties of Riemannian Manifolds

43

Proof. The well-known Sturm comparison theorem says that if we have solutions of two ODEs u 00 + K(t) u = 0, v 00 + L(t) v = 0 where L(t) ≤ K(t), v(t) has m zeros in the interval 0 ≤ t ≤ T and u(0) = v(0), u0 (0) = v0 (0), then u(t) has at least m zeros in the same interval. √ If K(t) ≡ K√> 0 then a solution of the first equation is of the form u(t) = C1 sin Kt + C2 cos Kt. Therefore, the distance between any√two consecutive zeros of a nontrivial solution of the second equation is at least π/ K. These facts together with Propositions 1.5.16 and 1.5.17 imply the theorem. t u Using arguments from the proof of Theorem 1.5.18, it is easy to get the following Proposition 1.5.19. For any point x ∈ (M, g) there exists a number s > 0 such that every pair of points in B(x, s) is joined by a unique shortest geodesic segment and these points are not conjugate to each other for this segment. A similar argument proves Theorem 1.5.20. Under the conditions and using the notation of Proposition 1.5.17, f 0 (t) > 0 for all t ≥ 0 if a complete Riemannian manifold has non-positive sectional curvature. If all sectional curvatures of a complete Riemannian h manifold  are less π 0 than or equal to a positive number K, then f (t) > 0 for all t ∈ 0, 2√K . Remark 1.5.21. The conjugate values of a geodesic in a complete Riemannian manifold M are the same as for the corresponding geodesic in any locally isometric covering M˜ of M. In any Euclidean sphere Sn , n ≥ 3, any two-dimensional great sphere is a totally geodesic submanifold of Sn . These two facts imply the last assertion in Theorem 1.5.18. Below we will consider variations of curves C(t, s), a ≤ t ≤ b, −ε < s < ε, in a Riemannian manifold (M, g) such that: 1) C(t, 0) = γ(t), a ≤ t ≤ b is a geodesic parameterized by arc length. 2) C(t, s) = Expγ(t) (sY (t)), where Y = Y (t), a ≤ t ≤ b, is a smooth vector field along γ such that hY (t), γ 0 (t)i ≡ 0. p Using the notation |v| = hv, vi for v ∈ T M, set X(t, s) =

∂C(t, s) , ∂t

Y (t, s) =

∂C(t, s) , ∂s

Z b

l (s) =

|X(t, s)|dt.

a

Since |X(t, 0)| ≡ 1 we can and shall suppose that |X(t, s)| > 0 everywhere. We need to calculate l 0 (s) and l 00 (0). In many places below we shall omit t and s. We have   Z b 1 ∂ l 0 (s) = hX(t, s), X(t, s)i dt a 2|X(t, s)| ∂ s Z b Z b 1 = h∇Y X, Xidt = h∇X Y, X/|X|idt. a |X| a

44

1 Riemannian Manifolds

This implies that l 0 (0) =

Z b a

[XhY (t, 0), γ 0 (t)i − hY (t, 0), ∇γ 0 (t) γ 0 (t)i]dt = 0, l 00 (s) =

Z b ∂ a

Z b

= a

∂s

(1.77)

h∇X Y, X/|X|idt

[h∇Y ∇X Y, X/|X|i + h∇X Y, ∇Y X/|X|i + h∇X Y,Y (1/|X|)Xi]dt.

Then conditions 1) and 2) and the symmetry properties of the curvature tensor imply that Z b a

[hR(Y, γ 0 )Y, γ 0 i + h∇X Y, ∇X Y i]dt

= l 00 (0) =

Z b a

[h∇γ 0 Y, ∇γ 0 Y i − hR(Y, γ 0 )γ 0 ,Y i]dt.

(1.78)

Corollary 1.5.22. If Y is a Jacobi vector field along γ then l 00 (0) = [h∇γ 0 Y,Y i]ba . In particular, if Y (γ(0)) = Y (γ(b)) = 0 then l 00 (0) = 0. Proof. It follows from (1.78) that l 00 (0) =

Z b a

Z b

= a

[γ 0 (t)h∇γ 0 Y,Y i − h∇γ 0 (t) ∇γ 0 Y,Y (t)i − hR(Y (t), γ 0 (t))γ 0 (t),Y (t)i]dt [γ 0 (t)h∇γ 0 Y,Y i]dt = [h∇γ 0 Y,Y i]ba , t u

which proves the corollary.

Lemma 1.5.23. Let Y1 ,Y2 be two Jacobi vector fields along a geodesic γ = γ(t), a ≤ t ≤ b, parameterized by arc length. Then  d  h∇γ 0 (t)Y1 ,Y2 (t)i − hY1 (t), ∇γ 0 (t)Y2 i = 0. dt

(1.79)

Proof. The expression on the left of (1.79) is equal to h∇γ 0 ∇γ 0 Y1 ,Y2 i + h∇γ 0 Y1 , ∇γ 0 Y2 i − h∇γ 0 Y1 , ∇γ 0 Y2 i − hY1 , ∇γ 0 ∇γ 0 Y2 i = −hR(Y1 , γ 0 )γ 0 ,Y2 i + hR(Y2 , γ 0 )γ 0 ,Y1 i = 0 by the symmetry properties of R.

t u

Lemma 1.5.24. Assume that we are given the following objects: a) γ = γ(t), a ≤ t ≤ b, is a geodesic in (M, g) parameterized by arc length, which has no conjugate point for γ(a); b) Z = Z(t) is a (possibly piecewise continuous) smooth vector field and Y = Y (t) is a Jacobi vector field along γ, both orthogonal to γ 0 = γ 0 (t), such that Z(a) = Y (a) = 0, Z(b) = Y (b) 6= 0.

1.5 Some Global Properties of Riemannian Manifolds

45

c) C1 (t, s), C2 (t, s) are variations of curves with the above conditions 1), 2), and l1 (s), l2 (s) are the lengths, corresponding to Z and Y . Then l 001 (0) ≥ l 002 (0) with equality only in the case when Z(t) ≡ Y (t). This statement is also true when Z(b) = Y (b) = 0, Z(a) = Y (a) 6= 0 in condition b). Proof. By Proposition 1.5.13, there exists a basis Y1 , . . . ,Yn−1 , where n = dim(M), for Jacobi vector fields along γ, orthogonal to γ 0 and equal to zero at t = a. Then Z(t) = f j (t)Y j (t) for some (possibly continuous piecewise) smooth functions f j (t), a ≤ t ≤ b, j = 1, . . . , n − 1. It follows from Lemma 1.5.23 and equalities Y j (a) = 0, j = 1, . . . , n − 1, that h∇γ 0 (t)Yi ,Y j (t)i − hYi (t), ∇γ 0 (t)Y j i = 0. Similarly to this, d h∇ 0 Yi ,Y j i − h∇γ 0 ∇γ 0 Yi ,Y j i dt γ d = h∇γ 0 Yi ,Y j i + hR(Yi , γ 0 )γ 0 ,Y j i. dt

h∇γ 0 Yi , ∇γ 0 Y j i =

Using these two relations, we get h∇γ 0 Z, ∇γ 0 Zi = h∇γ 0 ( f iYi ), ∇γ 0 ( f jY j )i, h( f i )0Yi , ( f j )0Y j i + f i f j h∇γ 0 Yi , ∇γ 0 Y j i + [( f i )0 f j + f i ( f j )0 ]h∇γ 0 Yi ,Y j i d = h( f i )0Yi , ( f j ) 0Y j i + [ f i f j h∇γ 0 Yi , ∇γ 0 Y j i] + f i f j hR(Yi , γ 0 )γ 0 ,Y j i. dt Now these computations and formula (1.78) imply l 001 (0) =

Z b a

Z b

= a

Z b

= a

[h( f i )0Yi , ( f j )0Y j i +

d i j [ f f h∇γ 0 Yi , ∇γ 0 Y j i]]dt dt

h( f i )0Yi , ( f j )0Y j idt + f i (b) f j (b)h(∇γ 0 Yi )(b), (∇γ 0 Y j )(b)i h( f i )0Yi , ( f j )0Y j idt + l 002 (0)

because of the equality Z(b) = Y (b) and Corollary 1.5.22. Thus, l 001 (0) ≥ l 002 (0) with equality only when ( f i )0 ≡ 0 for i = 1, . . . , n − 1 i.e. when f i (t) ≡ f i (b) for i = 1, . . . , n − 1, that is, Z(t) ≡ Y (t). t u Theorem 1.5.25 (Rauch’s comparison theorem, [395]). Let M and N be Riemannian manifolds of the same dimension n. Let g = g(t) (respectively h = h(t)) be a geodesic of M (respectively, of N) parameterized by arc length and X = X(t) (respectively, Y = Y (t)) a Jacobi vector field along g (respectively, h). Assume that the following conditions are fulfilled:

46

1 Riemannian Manifolds

1) X(t), Y (t) are perpendicular to g 0 (t), h0 (t) and X(0) = 0, Y (0) = 0 respectively; 2) X 0 := ∇g 0 X, Y 0 := ∇h 0 Y have the same length at t = 0; 3) neither X nor Y vanishes in the interval (0, b); 4) for each t ∈ (0, b), KM (P) ≥ KN (Q), where KM (P) (respectively, KN (Q)) is the sectional curvature by an arbitrary tangent plane P (respectively, Q) at g(t) tangent to g (respectively, at h(t) tangent to h). Then Y (b) is longer than or has the same length as X(b). Proof. Let u(t) = |X(t)|, v(t) = |Y (t)|. We need to show that u(b) ≤ v(b). It suffices to prove that u(t) lim = 1; (1.80) t→0 v(t)   u(t) 0 ≤ 0, (1.81) v(t) that is, u(t)/v(t) decreases. Since (u2 )0 = 2hX 0 , Xi, (u2 )00 = 2hX 0 , X 0 i + 2hX 00 , Xi; (v2 )0 = 2hY 0 ,Y i, (v2 )00 = 2hY 0 ,Y 0 i + 2hY 00 ,Y i, we get (1.80) by l’Hospital’s rule applied twice to u2 (t)/v2 (t) and the second assumption. Proving (1.81) is equivalent to proving u 0 (t)/u(t) ≤ v 0 (t)/v(t). We fix a parameter value s, 0 < s < b. By the third assumption, u(s) 6= 0 and v(s) 6= 0. We may thus set U = X/u(s) and V = Y /v(s). Since X and Y are Jacobi fields, so are U and V . Therefore, we have u 0 (s) 1 = hU,Ui0 (s) = u(s) 2

Z s

hU,Ui00 (t)dt =

0

Z s 0

[hU 0 ,U 0 i − KM (P)hU,Ui]dt,

where KM (P) denotes the sectional curvature of the planes P spanned by U and g 0 . Similarly, v 0 (s) 1 = hV,V i0 (s) = v(s) 2

Z s

hV,V i00 (t)dt =

0

Z s 0

[hV 0 ,V 0 i − KN (Q)hV,V i]dt.

There exists a vector field U along g such that hU,Ui(t) = hV,V i(t),

hU 0 ,U 0 i(t) = hV 0 ,V 0 i(t).

(1.82)

For this, let fs be an isometric linear isomorphism of h0 (s)⊥ onto g 0 (s)⊥ such that fs (V (s)) = U(s). Let σts (respectively τts ) be the parallel translation along g from g0 (s)⊥ into g 0 (t)⊥ (respectively, the parallel translation along h from h0 (s)⊥ into h0 (t)⊥ ). Define ft : h0 (t)⊥ → g 0 (t)⊥ by ft = σts ◦ fs ◦ (τts )−1 . If we set U(t) = ft (V (t)), then (1.82) holds. Now we shall show that u 0 (s) ≤ u(s)

Z s 0

[hU 0 ,U 0 i − KM (P)hU,Ui]dt ≤

v 0 (s) . v(s)

1.5 Some Global Properties of Riemannian Manifolds

47

The second inequality follows from the fourth assumption, (1.82), and the preceding integral expression for v 0 (s)/v(s). The first inequality is a consequence of the preceding integral expression for u 0 (s)/u(s), (1.78), and Lemma 1.5.24. Since s ∈ (0, b) was arbitrary, we get (1.81), as well as the required inequality u(b) ≤ v(b). t u Using Rauch’s theorem 1.5.25 and Sturm’s comparison theorem, mentioned earlier in the proof of Theorem 1.5.18, we get the following refinement of Theorem 1.5.18 for positive sectional curvature. Theorem 1.5.26. Let γ be a geodesic in M and K(P) the sectional curvature of a plane section P tangent to γ. If 0 < L ≤ K(P) ≤ H for all such plane sections P, then the distance d along γ between any two consecutive conjugate points satisfies the following inequalities: π π √ ≤d≤ √ . H L Lemma 1.5.27. Let γ = γ(t) = Expx (tv), 0 ≤ t ≤ T , be a unit speed geodesic in a Riemannian manifold (M, g) with x ∈ M having no conjugate point to x along itself. Assume that Expx : Mx → M is not a diffeomorphism. Then for any t ∈ [0, T
] there exists a maximal number m(t) > 0 with a unique smooth map ft : U γ(t), m(t) → M such that ft (γ(t)) = tv and ft is a right inverse to Expx . Moreover, |m(t1 ) − m(t2 )| ≤ |t1 − t2 | for all t1 ,t2 ∈ [0, T ].

(1.83)


V = (t, u) | 0 ≤ t ≤ T, u ∈ U γ(t), m(t)

(1.84)

If and f : V → M is defined by the formula f (t, u) = ft (u), then f is smooth. Proof. By the condition, for any t ∈ [0, T ], the differential T (Expx )tv is nondegenerate. Then the existence of a maximal number m(t) > 0 with mentioned map ft follows from the inverse map theorem. The inequality (1.83) follows from the definitions, inclusion U(z, r −d(y, z)) ⊂ U(y, r) and inequality d(γ(t1 ), γ(t2 )) ≤ |t1 −t2 |. It follows from (1.83) that (t, u) ∈ V and |s − t| < m(t) − d(γ(t), u) for s ∈ [0, T ] imply (s, u) ∈ V, so ft (u) = fs (u). This and the smoothness of maps ft , t ∈ [0, T ], imply the smoothness of f . t u Proposition 1.5.28. Let γ = γ(t), 0 ≤ t ≤ T , be a unit speed geodesic in a Riemannian manifold (M, g). Suppose that there is no point conjugate to x := γ(0) along γ. Then for any piecewise smooth curve c : [a, b] → M with c(a) = x, c(b) = γ(T ), and a continuous function t(τ) : τ ∈ [a, b] → t(τ) ∈ [0, T ] such that t(a) = 0, t(b) = T ,
d γ(t(τ)), c(τ) < m(t(τ)), τ ∈ [a, b], we have l(c) ≥ l(γ) = T with equality if and only if c is a reparametrization of γ. Proof. If Expx is a diffeomorphism of Mx onto M, then any curve c : [a,
b] → M with c(a) = x, c(b) = γ(T ) can be presented as c(τ) = Expx Exp−1 x (c(τ)) , and it is enough to apply the argument from the proof of Theorem 1.4.20.

48

1 Riemannian Manifolds

Assume that Expx : Mx → M is not diffeomorphism. Any curve c as above can be
presented as c(τ) = Expx f (t(τ), c(τ)) , a ≤ τ ≤ b, where f (t(τ), c(τ)), a ≤ τ ≤ b, is a piecewise smooth curve in Mx by the last assertion of Lemma 1.5.27 (even if t(τ) is continuous only), so we can apply the argument from the proof of Theorem 1.4.20. t u Proposition 1.5.29. Any part of a geodesic between its origin and a point after its first conjugate point is not a shortest arc. Proof. Let γ = γ(t), a ≤ t ≤ b, be a geodesic parameterized by arc length, γ(b) the first conjugate point of γ(a) for γ, and let c > b be any number. By Proposition 1.5.15, there exists a Jacobi vector field Y = Y (t), t ∈ R along γ such that Y (a) = Y (b) = 0, Y (t) 6= 0 for a < t < b, and Y is orthogonal to γ 0 . By Proposition 1.5.19, there exists a δ > 0 such that 0 < δ < b − a and no two points on γ(t), b − δ ≤ t ≤ b + δ , are conjugate to each other for this segment. Let us take any c such that b < c ≤ b + δ . Then there exists a Jacobi vector field W along γ such that W (b − δ ) = Y (b − δ ) and W (c) = 0. Then W is orthogonal to γ 0 by Lemma 1.5.12. Now introduce two continuous piecewise smooth vector fields Ye (t) and Z(t), a ≤ t ≤ c, along γ such that Ye (t) = Y (t) if a ≤ t ≤ b, Ye (t) = 0 if b ≤ t ≤ c, and Z(t) = Y (t) if a ≤ t ≤ b − δ , Z(t) = W (t) if b − δ ≤ t ≤ c. Define the variations of curves C1 (t, s), C2 (t, s), a ≤ t ≤ c, −ε < s < ε, corresponding to vector fields Ye , Z respectively according to conditions 1), 2) above. Let l1 (s), l2 (s), −ε < s < ε, be the corresponding lengths of curves. By Corollary 1.5.22, l 001 (0) = 0, and l 01 (0) = l 02 (0) = 0 by (1.77). The second derivatives l 00i (0), i = 1, 2, are presented as sums of integrals over the segments [a, b − δ ] and [b − δ , c]. The integrals over [a, b − δ ] are equal, while the integral over [b − δ , c] for l 001 (0) is greater than that for l 002 (0) by Lemma 1.5.24. Thus, l 02 (0) = 0, l 002 (0) < 0. Therefore, the geodesic γ = γ(t), a ≤ t ≤ c, is not a shortest arc. In the above argument, c > b was arbitrarily close to b. Therefore, the statement of the theorem is valid for all c > b. t u Definition 1.5.30. Let (M, g) be a complete Riemannian manifold, γ = γ(t), t ≥ 0, a geodesic ray in (M, g) parameterized by arc length, γ(0) = x. A point γ(b) = y is said to be a cut point of x on γ if 1) d(x, y) = b; 2) for every c > b, the geodesic γ(t), 0 ≤ t ≤ c, is not a shortest arc. A point y ∈ M is called a cut point of x if y is a cut point of x on some shortest geodesic segment joining x and y. The set Cx of all cut points of x is called the cut locus of x. Proposition 1.5.31. A point y = γ(b) is a cut point of x = γ(0) for a ray γ(t), t ≥ 0, if and only if a) y is a first conjugate point on γ for x or b) there is another shortest geodesic segment joining x and y. Proof. Sufficiency. The statement that y is a cut point of x on γ follows from condition a) by Corollary 1.4.21 and from condition b) by Proposition 1.5.29. Necessity. Let us suppose that y = γ(b) is a cut point of x on γ. Then for every c > b, γ(t), 0 ≤ t ≤ c, is not a shortest arc. Therefore, by the Hopf–Rinow theorem 1.5.1,

1.5 Some Global Properties of Riemannian Manifolds

49

there is a shortest geodesic segment γc (t), 0 ≤ t ≤ s(c) < c, such that γc (0) = x, γc (s(c)) = γ(c). There are only two possibilities: 1) There exists a sequence of unit vectors vk ∈ (Mx , gx ) such that (a) γvk (t) = Expx (tvk ) 0 ≤ t ≤ s(ck ) = γ(ck ), ck > b, ck → b, and (b) vk → w 6= v. 2) For any sequence of unit vectors vk ∈ (Mx , gx ) as in 1), (a), and (c) vk → v. It follows from the continuity of the intrinsic distance d : M × M → R that in case 1) there is another shortest geodesic segment γw (t), 0 ≤ t ≤ b, joining the point x and y. For the same reason, (T Expx )bv is degenerate in case 2), i.e. γ(b) is the first conjugate point for γ(0) on γ. Otherwise, we get a contradiction with the inverse mapping theorem. t u Proposition 1.5.32. 1) If y is a cut point of x on a geodesic γ, then x is a cut point of y on the same geodesic with the opposite orientation. 2) If y is a cut point of x then x is a cut point of y. Proof. Obviously, the second statement is a corollary of the first one. Assume that condition 1) is satisfied for x = γ(0) and y = γ(b). Then statement a) or b) from Proposition 1.5.31 holds. Evidently, statement 1) is true in case b). Condition a) and Proposition 1.5.15 imply that there is a Jacobi vector field Y along γ such that Y (0) = Y (b) = 0 and Y (t) 6= 0 for t ∈ (0, b). Obviously, Y is a Jacobi vector field for γ − , i.e. γ is supplied with the opposite orientation. Then by Propositions 1.5.15 and 1.5.29, x is a cut point for y on γ − . t u Definition 1.5.33. A mapping f : X → Y , where X is a Hausdorff topological space and Y is a locally compact topological space, is said to be proper if the inverse image f −1 (K) of every compact subset K ⊂ Y is a compact subset in X. There is a useful characterization of continuous proper mappings in terms of closed mappings. Proposition 1.5.34 (Proposition 7 on p. 104 in [123]). Let f be a continuous mapping of a Hausdorff topological space X into a locally compact space Y . Then f is proper if and only if the mapping f × iZ : X × Z → Y × Z is closed for every topological space Z. Lemma 1.5.35. Let π : T M → M be thetangent bundle. Then the
restriction of the mapping Φ := π × Exp to the set B = u ∈ Mx : |u| = d Φ(u) is a continuous proper map. Proof. The lemma is a consequence of the assertion that π is a locally trivial bundle. In turn, one can easily get this assertion using local coordinates (x, ξ ) for T M, see Definition 1.1.14. t u

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Definition 1.5.36. Let (M, g) be a complete Riemannian manifold. By Conjx for x ∈ M we shall denote the conjugate set of x, i.e. the set of all points y ∈ M such that y is a first conjugate point of x for some geodesic, joining x and y. Then the set Conj = {(x, y) ∈ M × M : y ∈ Conjx } (respectively, C = {(x, y) ∈ M × M : y ∈ Cx }) is called the conjugate (respectively the cut) set of (M, g). In what follows, we will use the notation S1 M for the unit spherical bundle in T M. Lemma 1.5.37. The conjugate set Conj is closed. Proof. It is clear that the set ND = {v ∈ T M : T (Expπ(v) )v is nondegenerate} is open in T M. Then its complement D = T M −ND is closed. Evidently, (x, y) ∈ Conj if and only if (x, y) = Φ(D ∩ d(x, y)S1 M). The above statements and Lemma 1.5.35 imply that the set Conj is closed (see also Proposition 1.5.34). t u Lemma 1.5.38. The function s : S1 M → R defined by 
s(u) = sup t : d Φ(tu) = t , where d is the intrinsic distance and Φ = π × Exp, is upper semicontinuous. 
Proof. It is clear that the set w ∈ T M : d Φ(w) − |w| = 0 is closed in T M. This implies the assertion. t u Proposition 1.5.39. The function s : S1 M → R is continuous. Proof. By Lemma 1.5.38, it is enough to prove that s is lower semicontinuous, i.e. for every u0 ∈ S1 M, s(u0 ) ≤ lim inf s(u) when u ∈ S1 M. Take a unit vector u0 ∈ S1 M. u→u0

It follows from Proposition 1.5.8 that s(u0 ) > s0 and lim inf s(u) > s0 for some s0 > u→u0

0. If lim inf s(u) = +∞ we are done. Assume that lim inf s(u) = σ < +∞. u→u0

u→u0

Suppose that there is a neighborhood U of u0 such that for every u ∈ U, γu (s(u)) is the first conjugate point to π(u) on the geodesic γu (t) = Exp(tu), t ∈ R. Then Lemma 1.5.37 implies that s(u0 ) = lim s(u). u→u0

In the opposite case, there is a sequence uk → u0 such that γuk (s(uk )) is not a first conjugate point, but the cut point, to π(uk ) on γuk (t) = Exp(tuk ), t ∈ R. Assume that such a sequence can be chosen so that s(uk ) → σ < s(u0 ). Let x0 = π(u0 ) and w = σ u0 . Then T (Expx0 )w is nondegenerate and x0 is joined with y0 := Exp(w) by the unique shortest arc γu0 (t), 0 ≤ t ≤ σ . Thus, there is a neighborhood V of w such that Φ maps V diffeomorphically onto a neighborhood V 0 of the point Φ(w) = (x0 , y0 ) in M × M. The above assumptions and statements together with Proposition 1.5.31 imply that there are vectors vk ∈ S1 M such that vk 6= uk , π(vk ) = π(uk ) := xk , Exp(s(vk )vk ) = Exp(s(uk )uk ) := yk . The last two equalities mean that γuk (t) and γvk (t), where 0 ≤ t ≤ s(vk ) = s(uk )
are two different
shortest arcs joining the same points xk and yk , while Φ s(vk )vk = Φ s(uk )uk = (x0 , y0 ) for s(vk )vk 6= s(uk )uk . Since

1.5 Some Global Properties of Riemannian Manifolds

51



(x0 , y0 ) = Φ(σ u0 ) = lim Φ s(uk )uk = lim Φ s(vk )vk n→∞

n→∞

and x0 , y0 are joined by the unique shortest arc γu0 (t), 0 ≤ t ≤ σ , we must have vk → u0 . Therefore, for sufficiently large k, uk , vk ∈ U, uk 6= vk while Φ(uk ) = Φ(vk ). This contradicts the fact that Φ is injective on U. t u Proposition 1.5.40. The cut set C is closed for any complete Riemannian manifold (M, g). Proof. It follows from Proposition 1.5.39 that the set Ce = ∪x∈M Cex = ∪x∈M {s(u)u : u ∈ Mx } (1.85) 
is the joint boundary of a closed subset B := w∈ T M : d Φ(w) −
|w| = 0 in T M, and its complement in T M, an open subset w ∈ T M : d Φ(w) − |w| < 0 e The above statements and Lemma in T M. Then Ce is closed in T M and C = Φ(C). 1.5.35 imply that the set C is closed (see also Proposition 1.5.34). t u Proposition 1.5.41. For every point x in a complete Riemannian manifold, 1) The cut locus Cx is closed. 2) The intrinsic distance between the point x and the set Cx is calculated by the formula d(x,Cx ) = Radinj(x) = min{s(u), u ∈ S1 M ∩ Mx }.

(1.86)

Proof. 1) It is clear that Cex = π −1 ({x}) ∩ Ce is closed and Cx = Expx (Cex ). It is easy to see that Expx | B ∩Mx is a continuous proper mapping, and Cx is closed (see also Proposition 1.5.34). 2) Let r = Radinj(x). Since Expx is a diffeomorphism on the open ball U = U(0, r) ⊂ (Mx , gx ) (and Expx (U) = U(x, r)), it is injective on U and T Expx is not degenerate on U. Therefore, by Proposition 1.5.31, d(x,Cx ) ≥ r, and s(u) ≥ r for every u ∈ S1 M ∩ Mx . On the other hand, by definition of r = Radinj(x), for every t > r, either Expx is non-injective on U(0,t) ⊂ Mx or T Expx is degenerate at some point in U(0,t). So, by Proposition 1.5.31, for every t > r, there are some points in Cx ∩ (B(x,t) −U(x, r)). Then statement 1) and the compactness of every closed ball in (M, g) imply that there is a point y ∈ Cx ∩ S(x, r), and d(x,Cx ) = r. It is clear now that r = min{s(u), u ∈ S1 M ∩ Mx }. t u Theorem 1.5.42. The injectivity radius Radinj is continuous for any complete Riemannian manifold. Proof. The theorem follows immediately from Proposition 1.5.39 and the last equality in (1.86). t u Definition 1.5.43. A smooth function c : [a, b] ⊂ R → M, where M is a smooth manifold, such that c(a) = c(b) is called a loop. If a loop c is injective on [a, b), then it is called a simple loop. A (simple) loop c such that c 0 (a) = c 0 (b) is called a (simple) closed curve.

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1 Riemannian Manifolds

Note that any (simple) closed geodesic in a Riemannian manifold is periodic. In the proof of the following important theorem of Klingenberg we follow Proposition 4.13 in [405]. Theorem 1.5.44. Let (M, g) be a complete Riemannian manifold, x ∈ M, and s0 (x) = d(x, Conjx ). Denote by l0 (x) the infimum of the lengths of nontrivial simple geodesic loops, originating and ending at x. Then Radinj(x) = min(s0 (x), l0 (x)/2).
Proof. First, we have Radinj(x) ≤ min s0 (x), l0 (x)/2 by Definition 1.5.30 and Propositions 1.5.31, 1.5.41. If Radinj(x) = +∞ we are done. Assume that Radinj(x) < +∞. It follows from Proposition 1.5.41 that there exists an element u ∈ (S1 )x M = S1 M ∩ Mx with s(u) = Radinj(x). If s(u) is equal to the first conjugate value s0 (u) to x along the geodesic γu (t) = Expx (tu), t ≥ 0, then it is clear that s(u) = s0 (x). If s(u) < s0 (u), then by Proposition 1.5.31 there exists a v ∈ (S1 )x M (v 6= u) such that γu (s(u)) = γv (s(u)) := y. When s(u) is the first conjugate value s0 (v) to x along γv , we again get s(u) = s0 (x). Therefore we may assume that s(u) < s0 (v). Then T (Exp p ) is nondegenerate at s(u)u and s(v)v, and we can take disjoint open neighborhoods U,V of s(u)u, s(v)v respectively, so that Expx is diffeomorphic on these neighborhoods. Since 0 0 γu (s(u))  = γv (s(u)) = y, if we can show that γ u (s(u)) = −γ v (s(u)), then γ := γu |[0,s(u)]

S

γu |−1 is a geodesic loop based at x with l(γ) = 2 Radinj(x), [0,s(u)]

which implies l0 (x) ≤ 2 Radinj(x). Suppose that γ 0u (s(u)) 6= −γ 0v (s(u)). Then there is a vector w ∈ Sy M that is different from −γ 0u (s(u)), −γ 0v (s(u)) and makes an acute angle with both of these vectors. Setting p(t) = Expy (tw), we consider the e1 (t) = (Expx |U )−1 p(t), w e2 (t) = (Expx |V )−1 p(t) in Mx that pass through curves w e 01 (0), w e 02 (0) make acute s(u)u, s(u)v respectively. By the Gauss lemma 1.5.14, w angles with −u, −v respectively. Therefore, for sufficiently small t > 0, we have e1 (t) ∈ U ∩U(0x , s(u)), w e2 (t) ∈ V ∩U(0x , s(u)), i.e. |w1 (t)|, |w1 (t)| < s(u). On the w other hand, since

e1 (t)| = p(t) = γ we2 (t)/|we2 (t)| |w e2 (t)| , γ we1 (t)/|we1 (t)| |w e1 (t)| ≥ s(w e1 (t)/|w e1 (t)|), which contradicts the choice of it follows that s(u) > |w u. t u Definition 1.5.45. For a complete Riemannian manifold (M, g), the value Radinj(M) := inf{Radinj(x), x ∈ M} is said to be the injectivity radius of M. Corollary 1.5.46. Let (M, g) be a compact Riemannian manifold. Then the inequality Radinj(M) > 0 holds. Moreover, we have Radinj(M) = min{s0 , L0 /2}, where s0 := min{s0 (x), x ∈ M} and L0 is the infimum of the lengths of nontrivial simple closed geodesics of M. Proof. It is clear that Radinj(M) < +∞. Theorem 1.5.42 and Proposition 1.5.8 imply that there is a point x ∈ M such that Radinj(M) = Radinj(x) > 0.

1.5 Some Global Properties of Riemannian Manifolds

53

By Theorem 1.5.44, Radinj(x) = min(s0 (x), l0 (x)/2). There are two possibilities: 1) s0 (x) ≤ l0 (x0 )/2, 2) s0 (x) > l0 (x)/2. In consequence of Lemma 1.5.37 and Proposition 1.5.39, the function s0 (z), z ∈ M, is continuous and so attains its minimum s0 on compact M. It is clear that in the first case we must have s0 (x) = s0 . In the second case, there is a nontrivial geodesic loop γ = γ(t), 0 ≤ t ≤ l0 (x), with origin and end at x such that γ(t1 ) 6= γ(t2 ) if 0 < t1 < t2 < l0 (x). We state that really γ must be a closed geodesic without intersections. Let assume that this is not true, that is γ 0 (0) 6= −γ 0 (l0 (x)), and y = γ(l0 /2). Condition 2) implies that y is not conjugate to x along γ. Then the proof of Proposition 1.5.32 shows that x is not conjugate to y along the different shortest segments γ(t), l0 (x)/2 ≤ t ≤ l0 (x) and (γ|[0,l0 (x)/2] )−1 . Now the same argument as in the second paragraph of the proof of Theorem 1.5.44 will show that there are shorter different geodesic segments of equal length with joint origin y and end z 6= y. Hence Radinj(y) < Radinj(x), a contradiction. Thus, γ(t), 0 ≤ t ≤ l0 (x), is a (nontrivial) simple closed geodesic. It is clear that in the second case l0 (x) = l0 is the minimum of the lengths of nontrivial simple closed geodesics of M. t u Remark 1.5.47. If (M, g) admits at least one nontrivial closed geodesic without selfintersections, then L0 is the minimum of the lengths of nontrivial simple closed geodesics in (M, g). Unfortunately, the authors do not know whether such geodesics exist on every compact Riemannian manifold. On the other hand, any compact Riemannian manifold has closed geodesics by the Lyusternik–Fet theorem, see [328] or [405, Theorem 3.5 on P. 299]. Some later considerations of similar questions can be found in the book [287] by W. Klingenberg. Corollary 1.5.46 and Theorem 1.5.18 imply Corollary 1.5.48. Let (M, g) be a compact Riemannian manifold with empty conjugate set Conj. Then Radinj(M) = L0 /2 where L0 is the minimum of the lengths of nontrivial closed (simple) geodesics without intersections in (M, g). In particular, this is true for compact Riemannian manifolds with non-positive sectional curvature. Remark 1.5.49. There are many other compact Riemannian manifolds with empty conjugate set and interesting properties. We recall that for two-dimensional surfaces in the three-dimensional Euclidean space, the Gauss curvature coincides with the sectional curvature (see e.g. §4.4.2 in [98] or Example 5.2 in Chapter VII of [291]). Below we prove some important properties of geodesics on ellipsoids of revolution. For this we shall need two facts on a special smooth surface of revolution p z = z(r), r = x2 + y2 (1.87) obtained by rotation around the z-axis of the graph of a strongly monotonic function z = z(x), 0 ≤ x1 ≤ x ≤ x2 , where x1 < x2 .

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1 Riemannian Manifolds

By §2.6.1 in [439], the Gauss curvature of the surface (1.87) is equal to z 0 (r)z 00 (r) K= 
2  2 . r 1 + z 0 (r)

(1.88)

We have the classical Clairaut rule for any geodesic γ = γ(t) on a surface of revolution: r2 |v| ˙ r cos ψ = = const := I. (1.89) ˙ |γ| Here r is the radius of the parallel (of the surface) going through a point of the geodesic and ψ is the angle between the tangent vector of the geodesic and the parallel at the same point, v is the polar angle around the rotation axis of the surface. Formula (1.89) is a restatement of the first part of Proposition 9.3.2 in [391]. Let us consider the following (oblate) ellipsoid of revolution x 2 + y2 +

z2 = 1, a2

0 < a < 1.

(1.90)

Its intersection with the plane y√ = 0 is the ellipse x2 + z2 /a2 = 1, whose upper half is defined by the function z = a 1 − x2 , 0 ≤ x ≤ 1. Then −ax z 0 (x) = √ , 1 − x2

z 00 (x) =

−a , (1 − x2 )3/2

1 + z 0 (x)


2

=

1 − k 2 x2 , 1 − x2

k2 = 1 − a2 ,

1 . a2

(1.91)

which together with (1.88) imply K = K(r) =

a2 (1 − k2 r2 )2

,

a2 = K(0) ≤ K(r) ≤ K(1) =

˙ Corollary 1.5.50. Every geodesic γ = γ(t), t ∈ R, on the ellipsoid with |γ(t)| =1 and 0 < I := r0 ≤ 1, is uniquely defined by r0 up to shift of the parameter t and rotation around the axis, v(t) ˙ 6= 0 and so it is always positive or negative, r0 is the minimal radius of parallels which meets the geodesic. This geodesic coincides with the equator for r0 = 1, but intersects the equator and tangents to both parallels with radius r0 for r0 ∈ (0, 1). Proof. It is not difficult to see that all assertions of this corollary follow from the Clairaut rule (1.89) and the facts that rotations around the axis and the mirror reflection relative to the equator are isometries of the ellipsoid. t u Corollary 1.5.51. For any point on a geodesic of the ellipsoid, the distance along this geodesic to its nearest conjugate (with respect to the geodesic) point is less than π/a and is not less than πa; it is equal to πa only for one geodesic, the equator. The length of any closed geodesic, consisting of two opposite meridians (a double meridian), is less than 2π and greater than 4.

1.5 Some Global Properties of Riemannian Manifolds

55

Proof. It follows from (1.88) that the sectional curvature of the ellipsoid is an increasing positive function on the radius r of parallels, attains its maximal value 1/a2 at the equator and its minimal value a2 at the top. This observation and Corollary 1.5.50 imply both parts of the first sentence. The last statement is a consequence of the fact that any meridian meets the top with r0 = 0, so it is a geodesic by (1.89), the length of a meridian is greater than 2, double the length of its orthogonal projection to the plane of the equator, and one of the doubled meridians is the ellipse x2 + z2 /a2 = 1, a convex plane curve inscribed into the circle of unit radius whose length is equal to 2π. t u Our next goal is to get an estimate (from below) of the lengths of simple geodesic loops distinct from the equator and double meridians. Theorem 1.5.52. Let us consider any geodesic on the ellipsoid (1.90), that is distinct from the equator and double meridians. Then the absolute value of the difference between two consequent values of the polar angle, when this geodesic intersects the equator, is less than π. p √ Proof. Since z = a 1 − x2 − y2 = a 1 − r2 , it suffices to find the orthogonal projection of a geodesic onto the plane z = 0 with polar coordinates (r, ϕ = v), in other words to find a function r = r(ϕ) along a geodesic. It follows from (1.89) that q ± r2 − r02 r0 cos ψ r0 cos ψ = , ϕ˙ = = 2 , u˙ = ± sin ψ = , r r r r where r0 is as in Corollary 1.5.50. Further, omitting the sign, we get (recall that k2 = 1 − a2 ) q √ √ r2 − r02 1 − r2 1 − r2 r˙ = √ · u˙ = √ · , r 1 − k2 r 2 1 − k2 r 2 q q 2 )(r 2 − r 2 ) r (1 − r r (1 − r2 )(r2 − r02 ) dr dr dt 0 √ = = > . dϕ dt dφ r0 r0 1 − k 2 r 2 Since the last term is equal to proved. Lemma 1.5.53. Let f (x) =

dr dϕ

sin x x ,

computed for the unit sphere, Theorem 1.5.52 is t u

0 < x < π2 . Then f 0 (x) < 0 and

2 π

< f (x) < 1.

x Proof. The inequality f 0 (x) = cosx x − sin < 0 is equivalent to the well-known inx2 equality tan x > x. Since lim f (x) = 1 and lim f (x) = π2 , we immediately get the x→0

second assertion.

x→π/2

t u

Theorem 1.5.54. The length of every geodesic loop on the ellipsoid (1.90) is greater than 8/3.

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1 Riemannian Manifolds

Proof. The statement is true for the equator, whose length is equal to 2π > 4 > 8/3, and for any double meridian by Corollary 1.5.51. Let us estimate the length L of any other loop. By Theorem 1.5.52, the length of its part between two consequent points of intersection with the equator is greater than the Euclidean distance between these points, which is equal to 2 sin(v/2), where v < π is the polar angle between these points. Notice that two adjacent such parts are obtained from each other by a combination of mirror reflection with respect to the equator and mirror reflection with respect to a plane of a double meridian. Let n be the minimal natural number such that nv ≥ 2π. Then n > 2, v/2 < π/2 and by Lemma 1.5.53, n−1 n − 1 2π · n · 2 sin(v/2) ≥ · · 2 sin(v/2) n n v n−1 π n−1 sin(v/2) 2 8 = · · 2 sin(v/2) = · 2π · > ·4 = . n v/2 n v/2 3 3

L > (n − 1) · 2 sin(v/2) =

t u

Theorem 1.5.54 and Corollary 1.5.51 imply the following Theorem 1.5.55. If 0 < a ≤ z2 a2

4 3π ,

then the injectivity radius of the ellipsoid x2 + y2 +

= 1 is equal to πa, the conjugate radius at any point of the equator, which in turn is less than half the length of any geodesic loop (in particular, any closed geodesic). It is useful to compare this theorem with Corollary 1.5.48. It seems that the paper [346] by Morse and Hedlund, where they considered a surface, is the first paper on manifolds without conjugate points, which do not necessarily have non-positive curvature. Then E. Hopf proved in [258] that any twodimensional torus with Riemannian metric, which has no conjugate point, is flat. In the first part of this paper he proved that the total scalar curvature of a closed surface without conjugate points is nonpositive and vanishes only if the surface is flat. L. Green generalized the last results as follows: The integral of the scalar curvature of a compact C4 -Riemannian manifold without conjugate points is nonpositive, and it vanishes only if the metric is locally Euclidean [235]. Then L. Green noted that whether an n-dimensional torus without conjugate points is flat is still an open question. This question was solved affirmatively by D. Burago and S. Ivanov only in 1994 [131]. Definition 1.5.56. A Riemannian manifold M has no conjugate (respectively, focal) point if the exponential map at each point (respectively, on the normal bundle of each geodesic) is non-singular. When M is complete, these are equivalent to the following: Any two points in e can be joined by a unique geodesic the universal Riemannian covering manifold M e and, respectively, every closed ball in M is strictly convex. Definition 1.5.57. A subset X in a Riemannian manifold M is (strictly) convex if, given any x, y ∈ X, there exists a (unique) shortest geodesic γ : [a, b] → M, connecting x with y, such that (respectively, and) γ([a, b]) ⊂ X.

1.5 Some Global Properties of Riemannian Manifolds

57

Definition 1.5.58. A subset X in a Riemannian manifold M is said to be strongly convex if X and every closed ball B(x, q) ⊂ X, q ≥ 0, are strictly convex. Remark 1.5.59. Note that there are different definitions of convex, strictly convex, and strongly convex sets in the literature. There are also some other type of convexity for sets in Riemannian manifolds (locally convex, totally convex, etc.). See e.g. p. 252 in [98], p. 417 in [152], p. 74 in [183], Section 5.2 in [241], p. 168 in [405], [411], and [465]. From the definition of strongly convex sets, we get Proposition 1.5.60. Let U be an open strongly convex set in a Riemannian manifold and B(a, r) ⊂ U, 0 < r < +∞. Then for every pair of distinct points x and y in B(a, r), the shortest arc joining them is contained in U(a, r) with the possible exception of the set {x, y} ∩ S(a, r). Definition 1.5.61. The convexity radius of a Riemannian manifold M at a given point x ∈ M is r(x) = sup {R > 0 : U(x, s) is strongly convex for all 0 < s < R}. The convexity radius of M is defined by r(M) = inf{r(x), x ∈ M}.
Remark 1.5.62. The open ball U x, r(x) is strongly convex for every x ∈ M. Lemma 1.5.63. Suppose that 1) y, z ∈ U(x, r), r > 0, and none of the points x, y, z lies between two others; 2) γ = γ(s), 0 ≤ s ≤ s0 , is a geodesic segment in (M, g) parameterized proportionally to arc length such that γ([0, s0 ]) ⊂ U(x, r), y = γ(0), z = γ(s0 ); 3) the sectional curvature of (M, g) is less than or equal to K in all two-dimensional directions at all points in U(x, r); 4) r < Radinj(x); √ 5) r < π/2 K if K > 0; 6) φ (s) = d 2 (x, γ(s))/2. Then φ 00 (s) > 0, 0 ≤ s ≤ s0 . Proof. It follows from conditions 2), 4) that γ(s) = Expx (ω(s)), where ω = ω(s) is a smooth regular curve in Mx . Then φ (s) = 12 g(ω(s), ω(s)). Define the map C : [0, 1] × [0, s0 ] → M,

C(τ, s) = Expx (τω(s))

and the vector fields X(τ, s) = ∂τ C(τ, s), Y (τ, s) = ∂sC(τ, s). For any fixed s ∈ [0, s0 ], the curve cs (τ) = C(τ, s), 0 ≤ τ ≤ 1, is the unique shortest geodesic in (M, g) joining the points x and γ(s) and parameterized proportionally to arc length. The first variation formula (1.58) implies that φ 0 (s) = g(X,Y )(1, s).

(1.92)

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1 Riemannian Manifolds

It follows from this and Lemma 1.5.2 that
g X(τ, s),Y (τ, s) = τ · φ 0 (s).

(1.93)

Now by (1.92), ∂ g(X,Y )(1, s) = Y g(X,Y )(1, s) = g(∇Y X,Y )(1, s) ∂s 1 1 ∂ = g(∇X Y,Y )(1, s) = Xg(Y,Y )(1, s) = g(Y,Y )(1, s), 2 2 ∂τ

φ 00 (s) =

1 ∂ g(Y,Y )(1, s). (1.94) 2 ∂τ We have a decomposition Y (τ, s) = V (τ, s) +W (τ, s) such that g(X,V ) ≡ 0. It is clear that for fixed s ∈ [0, s0 ], both V (τ, s) and W (τ, s) are Jacobi vector fields along the cs (τ), 0 ≤ τ ≤ 1. It follows from (1.93) that p p
τφ 0 (s) = g(X,W )(τ, s) = 2φ (s)g X(τ, s)/ 2φ (s),W (τ, s) ,
2
g(W,W )(τ, s) = τφ 0 (s) / 2φ (s) . φ 00 (s) =

Therefore


2
g(Y,Y )(τ, s) = g(V,V )(τ, s) + τφ 0 (s) / 2φ (s) ;

this and (1.94) implies 1 φ (s) = 2 00




2 ∂ 0 g(V,V )(1, s) + φ (s) /φ (s) . ∂τ

(1.95)

Let t be the natural parameter p for the geodesic segment cs (τ), 0 ≤ τ ≤ 1 and fixed s ∈ [0, s0 ]. It is clear that t = 2φ (s)τ. Hence in the notation of Proposition 1.5.17 (and s) instead of Y (t)), the first summand in (1.95) is equal pfor the Jacobi field V (t, p √ to 2φ (s) f 0 (t) f (t) for t = 2φ (s). Conditions 2) and 5) imply that t < π/2 K. This inequality and condition 3) mean that all conditions of Theorem 1.5.20 are fulfilled. Thus, by Theorem 1.5.20, f (t) > 0, f 0 (t) > 0, hence φ 00 (s) > 0, s ∈ [0, s0 ], because of (1.95). t u Corollary 1.5.64. Under the conditions of Lemma 1.5.63, we have the inequality
d(x, γ(s)) < max d(x, y), d(x, z) for all s ∈ (0, s0 ). Proposition 1.5.65. Let (M, g) be a complete Riemannian manifold and K the maximum of sectional curvatures in all two-dimensional directions at all points in the closed ball B(x, 2r), r > 0. Then =  the open ball√U(x,  r) is convex  if r < Radinj(x)/2 √  l0 (x)/4, K ≤ 0 and r < min Radinj(x), π/ K /2 = min l0 (x)/2, π/ K /2, if K > 0. Proof. Let y, z ∈ U(x, r) and γ(s), 0 ≤ s ≤ s0 be any shortest geodesic segment in (M, g) joining the points y and z (it exists by the Hopf–Rinow theorem 1.5.1).

1.5 Some Global Properties of Riemannian Manifolds

59

The triangle inequality implies that γ([0, s0 ]) ⊂ U(x, 2r), 2r < Radinj(x). Then the proposition follows from Lemma 1.5.63 and Corollary 1.5.64. The equalities in the √ statements follow from Theorem 1.5.44 and the inequality s0 ≥ π/ K for K > 0, which follows from Theorem 1.5.18. t u Let (M, g) be a complete Riemannian manifold. By Theorem 1.5.42, the injectivity radius Radinj(y) is a continuous function of y. For a given r > 0, denote by R and K the minimum of Radinj and maximum of sectional curvatures at points over B(x, r) respectively. √ Theorem 1.5.66. If r < R/2 for K ≤ 0 and r < min(R, π/ K)/2 for K > 0, then U(x, r) is strongly convex. Proof. Assume that d(x, y) = r1 < r and B(y, r2 ) ⊂ U(x, r). Then there exists a point z on the sphere S(y, r2 ) such that d(x, z) = d(x, y) + d(y, z) = r1 + r2 . Hence r1 + r2 < r and r2 < r − r1 < r. We need to prove that B(y, r2 ) is strictly convex. Let z, w be arbitrary points in B(y, r2 ). Then d(z, w) ≥ d(z, y) + d(y, w) ≤ 2r2 < 2r < R, and by definition of R, there is a unique minimal geodesic segment [zw] joining the points z, w. It is clear that [zw] continuously depends on z, w ∈ B(y, r2 ). It remains to show that always [z, w] ⊂ B(y, r2 ). To prove this, we will follow the proof of Theorem 5.14 from the book [150] by Cheeger and Ebin. There exists an ε > 0 such that r2 + ε < r − r1 . Then U(y, r2 + ε) ⊂ B(y, r2 + ε) ⊂ U(x, r). Set Vr2 +ε = {(z, w) ∈ B(y, r2 ) × B(y, r2 ) : [z, w] ⊂ U(y, r2 + ε)}. Using the fact that [z, w] varies continuously with z, w, we see that Vr2 +ε is nonempty and relatively open. Moreover, by Lemma 1.5.63 and Corollary 1.5.64, if (z, w) ∈ Vr2 +ε , then [z, w] − {z, w} ⊂ U(y, r2 ). Since [z, w] varies continuously with z, w, it follows that if (z, w) ∈ Vr2 +ε , then [z, w] ∈ B(y, r2 ) ⊂ U(y, r2 + ε). Therefore Vr2 +ε ⊂ Vr2 +ε , that is Vr2 +ε is closed. Thus Vr2 +ε is nonempty, relatively open, and closed. By connectedness, Vr2 +ε = B(y, r2 ) × B(y, r2 ). Since ε > 0 can be taken arbitrarily small, then [z, w] ⊂ B(y, r2 ) for any z, w ∈ B(y, r2 ). t u Theorem 1.5.66 and Definition 1.5.61 immediately imply the following two theorems. Theorem 1.5.67. The convexity radius r of any complete Riemannian manifold (M, g) is positive and satisfies the inequality |r(x) − r(y)| ≤ d(x, y). Theorem 1.5.68. Let (M, g) be a complete Riemannian manifold with injectivity radius Radin j(M) > 0 and sectional curvatures ≤ K. Then  √  r(M) ≥ min Radinj(M), π/ K /2 if K > 0, and r(M) ≥ Radinj(M)/2 if K ≤ 0.

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Theorem 1.5.68, Corollary 1.5.46, and Proposition 4.6.7 imply Theorem 1.5.69. If (M, g) is a homogeneous or compact Riemannian manifold then   √  √  r(M) ≥ min Radinj(M), π/ K /2 = min L0 (M)/2, π/ K /2 if K > 0, and r(M) ≥ Radinj(M)/2 = L0 (M)/4 if K ≤ 0. Here K is equal to the maximum of the sectional curvatures at x for any point x ∈ M in the first case and to the maximum of the sectional curvatures over M in the second case. To prove the next proposition, we follow the proof of Proposition IX.6.1 in [148]. Proposition 1.5.70 (Berger, [97]). For any complete Riemannian manifold (M, g) we have r(M) ≤ Radinj(M)/2. Proof. Assume that for a complete Riemannian manifold (M, g) this is not true, that is, Radinj(M) < 2r(M). Then there exist points x, y ∈ M such that x and y are cut points one to the other (see Proposition 1.5.32), and d(x, y) < 2r(M). Let γ = γ(t), t ∈ R, be some unit speed geodesic in (M, g) such that γ(0) = x, γ(t0 ) = y, t0 = d(x, y), and z = γ(t0 /2). Then x, y ∈ B(z,t0 /2) and t0 /2 ∈ r(M), which implies that γ is the unique geodesic with the mentioned properties. Therefore, by Proposition 1.5.31, x and y are conjugate along γ. It follows from Proposition 1.5.29 that for small t − t0 > 0, we have t/2 < r(M), d(x, γ(t)) < t, which determines a unique minimizing unit speed geodesic γt from x to γ(t), with γt → γ as t → t0 . Then
γt d(x, γ(t))/2 ∈ U(x,t/2) ∩U(γ(t),t/2), (1.96) γ(t/2) ∈ B(x,t/2) ∩ B(γ(t),t/2).

(1.97)

Since both closed balls in the last inclusion are strictly convex, there exists a unique shortest geodesic segment ω = ω(s), 0 ≤ s ≤ s0 , in (M, g) such that ω(0) = γ(t/2) and ω(s0 ) = γt (d(x, γ(t))/2); then ω([0, s0 ]) ⊂ B(x,t/2) ∩ B(γ(t),t/2). (1.98)

It is clear that the functions φx (s) = d x, ω(s) , φγ(t) = d γ(t), ω(s) , 0 ≤ s ≤ s0 , are smooth. Since any closed ball in M of radius < r(M) is strictly convex, we have φ 00x (s) ≥ 0, φ 00γ(t) (s) ≥ 0 for s ∈ [0, s0 ]. Moreover, the first variation formula (compare with (1.92)) implies that φ 0x (0) + φ 0γ(t) (0) = 0, hence φ 0x (0) ≥ 0
or φ 0γ(t) (0) ≥ 0. In the first case d x, γt (d(x, γ(t))/2) ≥ t/2, in the second case
d γ(t), γt (d(x, γ(t))/2) ≥ t/2. This contradicts the inclusion (1.96). t u
Proposition 1.5.71. Suppose that y, z ∈ U(x, r) where 0 < r ≤ min Radinj(x), r(x) , and [y, z] is a (necessarily unique) geodesic minimizer, joining y and z. Then
d(x, w) < max d(x, y), d(x, z) for every w ∈ [y, z] − {y, z}.

1.5 Some Global Properties of Riemannian Manifolds

61

Proof. Assume that there is a w ∈ [y, z] − {y, z} such that
d(x, w) ≥ max d(x, y), d(x, z) .

Then d(x, w) = max d(x, y), d(x, z) < r ≤ min Radinj(x), r(x) . Since the injectivity and the convexity radiuses are continuous, there exists a point x1 6= x such that
d(x1 , w) = d(x1 , x) + d(x, w) < min Radinj(x1 ), r(x1 ) . Then d(x1 , y) < d(x1 , x) + d(x, y) < d(x1 , w),

d(x1 , z) < d(x1 , x) + d(x, z) < d(x1 , w),

which contradicts the fact that the closed ball B x1 , max d(x1 , y), d(x1 , z) is strictly convex. t u The following characterizations are given by J. O’Sullivan [378] (here, a geodesic ray γ : [0, ∞) → M means a locally isometric mapping). Theorem 1.5.72. Let (M, g) be a complete Riemannian manifold. Then the following hold: a) M has non-positive sectional curvature if and only if for every geodesic ray γ : 2 [0, ∞) → M and every Jacobi field Y along γ, dtd 2 |Y |2 > 0; b) M has no focal points if and only if for every geodesic ray γ : [0, ∞) → M and every Jacobi field Y along γ that satisfies Y (0) = 0, dtd |Y |2 > 0; c) M has no conjugate points if and only if for every geodesic ray γ : [0, ∞) → M, every non-trivial Jacobi field Y along γ that satisfies Y (0) = 0, and every positive time, |Y |2 > 0. It follows that complete Riemannian manifolds with non-positive sectional curvature have no focal points and that those with no focal points have no conjugate points. This refines the Hadamard–Cartan theorem 1.5.78. On the other hand, R. Gulliver constructed in [247] examples of complete Riemannian manifolds with sectional curvature of both signs but without focal points and examples with focal points but without conjugate points. These examples may be used to show that, for r(M) = 0 over the class of compact n-dimensional manifolds M, each n ≥ 2, inf Radinj(M) where r(M) and Radinj(M) are the convexity radius and the injectivity radius of M respectively [179]. Many of the major results on Riemannian manifolds with non-positive sectional curvature generalize to those with no focal points. These include the center theorem [378], the flat torus theorem [379], and the higher rank rigidity theorem [462]. The latter theorem generalizes Theorem 3.8.16. Moreover, it is applied to generalize for manifolds without focal points a result of Ballmann and Eberlein stating that for compact manifolds of nonpositive curvature, the rank is an invariant of the fundamental group. Let us note also the following result from [379]: A compact Riemannian manifold M without focal points is flat if and only if its fundamental group π1 (M) is solvable.

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The following conjecture was mentioned by H. E. Rauch in [395]: If x is any point in a compact, simply connected Riemannian manifold M, then the conjugate locus and cut locus of x in M must have a common point. The following theorem shows that this conjecture is false in general. Theorem 1.5.73 ([463]). Let M be a connected, compact smooth differentiable manifold of dimension ≥ 2 not homeomorphic to S2 . Then there is a Riemannian metric g on M and a point x ∈ M whose conjugate locus and cut locus are disjoint. Theorem 1.5.44 implies that Radinj(x) = l0 (x)/2 for this (M, g). Remark 1.5.74. Recall that there is a compact Riemannian manifold (M, g) such that Radinj(x) < l0 (x)/2 for a suitable point x ∈ M, see Theorem 1.5.55. Detailed information on covering maps, fundamental groups, and their properties can be found in [483]. Here, we restrict ourselves to corresponding definitions in the category of connected smooth manifolds and present the main statements. Definition 1.5.75. A smooth mapping p : M → N of connected smooth manifolds is called a covering mapping (onto N) or covering (of N) if for any point x ∈ N there exists an open connected neighborhood U of x such that the preimage p−1 (U) is a disconnected union of its open connected components in M and the restriction of p to every such component is a diffeomorphism onto U. In addition, M is called a covering manifold for N, the preimages of points from N relative to p are called the fibres of p, and a diffeomorphism f of M onto itself is called a deck transformation if p◦ f = p. A smooth connected manifold M is called simply connected or universal e→M if any covering mapping onto M is a diffeomorphism. A covering mapping M e is called universal if M is simply connected. It is clear that the composition of any two covering mappings is again a covering mapping; for a given covering mapping p, deck transformations preserve all fibres and constitute the so-called group of deck transformations Γ (p) relative to compositions. A universal covering mapping exists for any connected smooth manifold M. Moreover, for any two universal covering maps p1 : M1 → M, p2 : M2 → M there exists a diffeomorphism f : M1 → M2 such that p2 ◦ f = p1 . Then any mapping g : M1 → M2 of the form g = γ2 ◦ f or g = f ◦ γ1 where γ2 ∈ Γ (p2 ) and γ1 ∈ Γ (p1 ) has the same properties as f and any diffeomorphism with these properties can be presented in the first or in the second form. This implies that for the universal coverings p1 and p2 of a given connected smooth manifold M, the groups of deck transformations Γ (p1 ) and Γ (p2 ) are isomorphic. Definition 1.5.76. A covering mapping is called regular if its group of deck transformations acts (simply) transitively on any fibre of the covering. The group of deck e → M is called the fundatransformations Γ = Γ (p) of a universal covering p : M mental group of M and denoted by π1 (M). e → M and a A universal covering is regular. For any universal covering p : M subgroup Γ ⊂ π1 (M) there exists, up to diffeomorphism, a unique smooth connected

1.5 Some Global Properties of Riemannian Manifolds

63

e → N, p2 : N → M such that π1 (N) = Γ and p2 ◦ manifold N and coverings p1 : M p1 = p. In addition, the covering p2 is regular if and only if Γ is a normal subgroup of π1 (M); in this case Γ (p2 ) ∼ = π1 (M)/π1 (N). The following result by Cheeger and Ebin in [150] is very useful. Theorem 1.5.77. Let M and N be Riemannian manifolds and f : M → N a local isometry. If M is complete, then f is a covering mapping. Theorem 1.5.78 (the Hadamard–Cartan theorem). Let (M, g) be a complete Riemannian manifold of non-positive curvature. Then for any x ∈ M, Expx : Mx → M is a universal covering mapping. Proof. Since M is complete, the Hopf–Rinow theorem implies that Expx is defined on the whole Mx and maps Mx onto M. By Theorem 1.5.18, there is no conjugate point in M for x, and T Expx is nonsingular on Mx . Then we can define a new Riemannian metric on Mx by the formula (v, w) = h(T Expx )u (v), (T Expx )u (w)i,

v, w ∈ (Mx )u ,

u ∈ Mx .

The Gauss lemma 1.5.14 implies that the lines through the origin in Mx are geodesics in the new metric. Then (Mx , (·, ·)) is complete by the Hopf–Rinow theorem and Expx : (Mx , (·, ·)) → (M, g) is a local isometry with respect to the corresponding intrinsic metrics δ and d. Assume that y, z ∈ Mx and Expx (y) = Expx (z) = p ∈ M. Let σ p be the number from Proposition 1.5.8. It is clear that U(y, σ p ), U(z, σ p ) possess the same properties as U(p, σ p ) and Expx maps isometrically U(y, σ p ) and U(z, σ p ) onto U(p, σ p ). Therefore, if a ∈ U(y, σ p ) ∩U(z, σ p ) then δ (y, a) = δ (z, a) < σ p and unique shortest arcs in U(y, σ p ) and U(z, σ p ) joining a respectively with y and z are mapped by Expx to the same shortest arc in U(p, σ p ) joining points Expx (a) and p. Since Expx is a local isometry, the above two shortest arcs in U(y, σ p ) and U(z, σ p ) coincide, so y = z and U(y, σ p ) = U(z, σ p ). The above argument shows that Expx : Mx → M is a universal covering mapping. t u Theorem 1.5.79 (the Myers theorem, [349]). If (M n , g) is a complete Riemannian manifold such that its Ricci curvature (for every unit vector tangent to M n ) is no less than (n − 1)k2 , where k > 0, then the manifold M n is compact, its diameter (with respect to the intrinsic metric d) does not exceed π/k, and its fundamental group is finite. Proof. Consider an arbitrary geodesic c = c(t), 0 ≤ t ≤ a, where a > 0, parameterized by the arc length and joining the points x and y. Choose an arbitrary orthonormal basis v1 , . . . , vn in (Mx , gx ) with the condition vn = c0 (0). Extend these vectors to parallel vector fields V1 , . . . ,Vn along c. Then Vn (t) = c0 (t) and V1 (t), . . . ,Vn (t) is an orthonormal basis in (Mc(t) , g(c(t))) for all t ∈ [0, a]. Define the geodesic variation c by the formulas Ci (t, s) = Exp(s f (t)Vi (t)), where

(t, s) ∈ [0, a] × (−ε, ε),

(1.99)

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i = 1, . . . , n − 1;

f (t) = sin

 πt  a

,

0 ≤ t ≤ a.

Then f (0) = f (a) = 0,

f 00 +

f (t) > 0, 0 < t < a;

π2 f = 0. a2

(1.100)

Now calculate the second derivative of the curve length for these variations at s = 0 (the first variations are equal to zero). ∂ Define the vector fields Xi = TCi ◦ ∂t and Yi = TCi ◦ ∂∂s for i = 1, . . . , n − 1. For any fixed t0 ∈ (0, a) the curves Ci (t0 , ·), i = 1, . . . , n − 1, are geodesics. Therefore, according to Proposition 1.4.10, every vector field Xi is the Jacobi field along the geodesic Ci (t0 , ·); moreover, Xi (t, 0) = c 0 (t) are unit vector fields along c and

∇Yi Xi (t, 0) = ∇Xi Yi (t, 0) = f 0 (t)Vi (t),

Yi (t, s) = f (t)Yei (t, s),

(1.101)

where Yei are unit vector fields. This, the smoothness of the curvature tensor, the compactness of the segment [0, a], the continuous dependence of solutions for systems of ODEs on the initial data and independent variable imply that there exists an ε > 0 such that Xi (t, s) 6= 0 for all (t, s) ∈ [0, a] × (−ε, ε). We will assume that this condition is fulfilled. Then li (s) =

Z ap 0

li0 (s) =

g(Xi , Xi )(t, s) dt,

Z a g(∇Yi Xi , Xi )

|Xi |

0

(t, s) dt.

According to the last equality in formula (1.101), g(∇Yi Xi , Xi )(t, 0) = g( f 0 (t)Vi (t), c 0 (t)) = 0. In consequence of what we said before, we get that l 00i (0) = Z a

= 0

Z a 0

[g(∇Yi ∇Yi Xi , Xi ) + g(∇Yi Xi , ∇Yi Xi )](t, 0) dt

[−g(R(Xi ,Yi )Yi , Xi )(t, 0) + g( f 0 (t)Vi (t), f 0 (t)Vi (t))] dt Z a

= 0

[−g(R(c 0 , fVi )( fVi ), c 0 )(t) + f 0 (t)2 ] dt Z a

= 0

[− f 2 K(c 0 ,Vi ) + ( f 0 )2 ](t) dt.

Integrating by parts and using conditions (1.100), we obtain Z a 0

( f 0 )2 (t)dt = −

Z a 0

f f 00 (t)dt =

π2 a2

Z a 0

f 2 (t)dt.

1.5 Some Global Properties of Riemannian Manifolds

65

Finally, we get l 00i (0) = − n−1

∑

l 00i (0) =

i

=−

Z a

−

Z a

n−1

Z a

f

f

0

0

f ( f K(c 0 ,Vi ) + f 00 )(t) dt, !

∑ K(c 0 ,Vi ) + (n − 1) f 00

(t) dt

i

f (t)( f (t) ric(c 0 (t)) + (n − 1) f 00 (t))dt =

0

n−1

∑ l 00i (0), i

n−1

∑ i

l 00i (0) =

Z a

2

f (t) 0



π 2 (n − 1) a2

0



− ric(c (t)) dt.

By the conditions of the theorem, ric(c 0 (t)) ≥ (n − 1)k2 , and if a > π/k, then the integrand is negative for 0 < t < a. Consequently, at least for one i = i0 , l 00i0 (0) < 0 and l 0i0 (0) = 0. Therefore, the geodesic c(t), 0 ≤ t ≤ a, parameterized by the arc length, cannot be a shortest curve joining the points x and y. Since a geodesic joining points x and y could be arbitrary, we get d(x, y) ≤ π/k due to Corollary 1.5.3. Hence, diam(M) ≤ π/k since the points x, y were arbitrary. Therefore, (M, d) is compact by Theorem 1.5.1. All these arguments are valid also for the universal locally isometric e of M. Therefore, M e is compact and the fundamental group π1 (M) is covering M finite. t u Theorem 1.5.80 (Synge’s theorem, [427]). Let M n be a compact Riemannian manifold of positive sectional curvature. 1) If n is an even number and M is orientable, then M is simply connected. 2) If n is an odd number, then M is orientable. Proof. We present here the original proof of this theorem. We will prove both assertions simultaneously. Assume that some of these assertions are not true. Then M is not simply connected and there exists a non-shrinkable loop c1 in M. By compactness of M, the free homotopic class [c1 ] of the loop c1 contains some nontrivial shortest loop. On the basis of Corollary 1.4.21, this shortest loop must be a closed periodic geodesic. Let c = c(t), 0 ≤ t ≤ a, be its parametrization by the arc length and c(0) = x, c 0 (0) = c 0 (a) = u. Define the parallel translation P(t) : Mc(0) → Mc(t) , 0 ≤ t ≤ a, along this geodesic. Then P(a) is a linear orthogonal transformation of Euclidean space (Mx , gx ), moreover, P(a)(u) = u. In the first case P(a) preserves orientation, while in the second case changes it (under the corresponding choice of the loop c1 , possible in consequence of the nonorientability of M). Therefore, P(a) is an orientation preserving (respectively, changing) orthogonal transformation of the orthogonal complement u⊥ = Mx Ru. Then it is known [222] that in both cases there exists a unit vector v ∈ u⊥ such that P(a)(v) = v. Let V (t) := P(t)(v), 0 ≤ t ≤ a and C(t, s) := Expc(t) (sV (t)); (t, s) ∈ [0, a] × (−ε, ε) be a smooth variation of the geodesic c. Notice that this variation is entirely analogous to the variations Ci (t, s) from (1.99), when f (t) ≡ 1. Obviously, for every s ∈ (−ε, ε), the curve cs (t) := C(t, s), 0 ≤ t ≤ a, is a smooth loop in M of some length l(s), which is freely

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homotopic to the loop c. Then l 0 (0) = 0 by(1.58). We get, analogously to Theorem 1.5.79, that l 00 (0) =

Z a 0

[− f 2 K(c 0 ,V ) + ( f 0 )2 ](t)dt = −

Z a

K(c 0 (t),V (t))dt < 0.

0

Consequently, contrary to the assumption, the loop c is not the shortest loop in the class [c1 ]. The theorem is proved. t u Remark 1.5.81. The variation C constructed in this proof of Synge’s theorem is the so-called Synge’s film.

1.6 Riemannian Submersions and the O’Neill Formulas Definition 1.6.1. Let M and B be Riemannian manifolds. A submersion p : M → B (see Definition 1.1.24) is called Riemannian if for any point x ∈ M, the linear map Tx p : Hx (M) → B f (x) , where Hx (M) = Vx (M)⊥ is the orthogonal complement to Vx (M) in Mx , is an isometry. The vectors of the form v ∈ Hx (M), x ∈ M, are called horizontal. Let p : M → B be a Riemannian submersion. Then one can define (smooth orthogonal) projections H : T M → HM and V : T M → V M onto the horizontal and the vertical distributions HM and V M. Define a map A : F ∞ (M) × F ∞ (M) → F ∞ (M) by the formula AX Y = V ∇HX (HY ) + H∇HX (VY ). It is easy to see that A is a (1, 2)-tensor; it is called the O’Neill tensor and has the following properties: 1) A is horizontal, i.e. AX = AHX ; 2) For any vector field X ∈ F ∞ M, AX is skew-symmetric, moreover, AX (HM) ⊂ V M and AX (V M) ⊂ HM; 3) If X,Y ∈ HM then AX Y = −AY X. The first assertion is evident. Also it is not difficult to check the second assertion. The third assertion will be proved in Lemma 1.6.3, which shows that A is the integrability tensor of the horizontal distribution HM. In calculations with tensor expressions, it is possible to choose some special horizontal vector fields. A horizontal vector field X on M is called basic if it is p connected with some vector field X∗ on B. Any vector field X∗ on B has a unique horizontal lift X on M. Therefore, X ←→ X∗ is a one-to-one correspondence between basic vector fields on M and all vector fields on B. The following lemma shows that, to some extent, this correspondence preserves the Lie brackets, inner products, and covariant derivatives. Later we use the same notation h·, ·i for metric tensors on M and B.

1.6 Riemannian Submersions and the O’Neill Formulas

67

Lemma 1.6.2. If X,Y are basic vector fields on M, then 1) hX,Y i = hX∗ ,Y∗ i ◦ p , 2) H[X,Y ] is the basic vector field corresponding to the field [X∗ ,Y∗ ], 3) H∇X Y is the basic vector field corresponding to the field ∇∗X∗ Y∗ . Proof. The first property is a direct corollary of the definitions. According to Theorem 1.1.33, the vector field [X,Y ] is p -connected with the vector field [X∗ ,Y∗ ], which implies the second property. The proof of the third property is based on the formula (1.25). In fact, for a basic vector field Z on M, we get 2h∇X Y, Zi = XhY, Zi +Y hZ, Xi − ZhX,Y i +h[X,Y ], Zi + h[Z, X],Y i − h[Y, Z], Xi

(1.102)

by this formula. On the other hand, for instance, by the first property, XhY, Zi = X{hY∗ , Z∗ i ◦ p } = X∗ hY∗ , Z∗ i ◦ p . The first two properties imply h[X,Y ], Zi = hH[X,Y ], Zi = h[X∗ ,Y∗ ], Z∗ i ◦ p etc. This implies that 2h∇∗X∗ Y∗ , Z∗ i ◦ p is equal to the right-hand side of the equality (1.102). Then ∇X Y is p -connected with ∇∗X∗ Y∗ , which implies the third property. t u Lemma 1.6.3. Let X and Y be horizontal vector fields. Then AX Y = 12 V [X,Y ]. Proof. Since [X,Y ] = ∇X Y − ∇Y X, then V [X,Y ] = AX Y − AY X. Therefore, it suffices to prove the above-mentioned property 3), or, equivalently, to show that AX X = 0. We may assume that the field X is basic. Then 0 = W hX, Xi = 2h∇W X, Xi for any vertical vector field W . But [W, X] = ∇W X − ∇X W is vertical, because W is p -connected with the zero vector field on B. Therefore h∇W X, Xi = h∇X W, Xi = −hW, ∇X Xi = −hW, AX Xi. Since AX X is vertical, we obtain the required result.

t u

Theorem 1.6.4. Let p : M → B be a Riemannian submersion, K and K∗ be the sectional curvatures of the spaces M and B, and u, w be horizontal mutually orthogonal unit vectors at a point z ∈ M, u∗ = T p (u), w∗ = T p (w). Then K(σ (uw)) = K∗ (σ (u∗ w∗ )) − 3|Au w|2 . Proof. There are basic vector fields X,Y on M such that X(z) = u, Y (z) = w. Let R and R∗ be the curvature tensors of the spaces M and B. Then by Lemma 1.6.2, the definition of the tensor A , and the verticality of the vector field [V [X,Y ],Y ] (see the proof of Lemma 1.6.3), we get

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hR(X,Y )Y, Xi = h∇X ∇Y Y − ∇Y ∇X Y − ∇[X,Y ]Y, Xi = hH∇X H∇Y Y − H∇Y H∇X Y − H∇H[X,Y ]Y, Xi +hH∇X V ∇Y Y − H∇Y V ∇X Y − H∇V [X,Y ]Y, Xi = hR∗ (X∗ ,Y∗ )Y∗ , X∗ i ◦ p +hAX (AY Y ), Xi − hAY (AX Y ), Xi − hH∇Y V [X,Y ], Xi.

Let us calculate every of the last three summands. The first summand is zero, because AY Y = 0 according to property 3) of the tensor A . In consequence of properties 2) and 3) of the tensor A , the second summand is equal to hAX Y, AY Xi = −hAX Y, AX Y i. By Lemma 1.6.3 and properties 2) and 3) of A , the third summand is equal to −hAY V [X,Y ], Xi = hV [X,Y ], AY Xi = −2hAX Y, AX Y i. Now the theorem is an immediate corollary of the above calculations.

t u

Lemma 1.6.5. Let p : M → B be a Riemannian submersion and M be complete. Then for any shortest geodesic c∗ = c∗ (t), 0 ≤ t ≤ a, in B, parameterized by the arc length, and a point z ∈ p −1 (c∗ (0)), there exists a unique horizontal lift c(t), 0 ≤ t ≤ a, of the path c∗ . This means that p (c(t)) = c∗ (t) for all t ∈ [0, a] and all tangent vectors c0 (t), 0 ≤ t ≤ a, are horizontal. Moreover, the path c(t), 0 ≤ t ≤ a, is a shortest geodesic in M, joining the fibres p −1 (c∗ (0)), p −1 (c∗ (a)) and parameterized by the arc length. Proof. The conditions imply the existence of a geodesic c∗1 = c∗1 (t), −ε ≤ t ≤ a + ε, where ε > 0, extending c∗ and being a topological embedding. Put C1 := c∗1 (−ε, a + ε). Then the set C = p −1 (C1 ) − p −1 (C1 ) is closed in M (the upper bar denotes closure). Since p is a submersion, M1 := p−1 (C1 ) is a submanifold in M0 = M −C. It is clear that z ∈ M1 . Moreover, the restriction p : M1 → C1 is a Riemannian submersion. Let X be the horizontal lift of the (unit) tangent vector field c0∗1 along c∗1 relative to this Riemannian submersion and c be the maximal integral curve of the vector field X, originated at z. Then c is parameterized by the arc length and is defined on (−ε, a + ε) due to the completeness of M. Moreover, T p ◦ X = c0∗1 ◦ p and p (c(0)) = p (z) = c∗1 (0). Therefore, p (c(t)) = c∗1 (t) for all t ∈ (−ε, a + ε). According to Lemmas 1.6.3 and 1.6.2, ∇X X = H∇X X +V ∇X X = H∇X X + AX X = H∇X X, T p (H∇X X) = ∇∗c0 c0∗1 = 0. ∗1

Then ∇X X = 0 and the curve c is a geodesic in M. Moreover, the path c(t), 0 ≤ t ≤ a, is a shortest geodesic in M, joining the fibres p−1 (c∗ (0)) and p−1 (c∗ (a)). Otherwise there is a shorter path c1 (t), 0 ≤ t ≤ b, in M, joining some points from

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69

fibres p −1 (c(0)) and p −1 (c(a)). Then the path p (c1 (t)), 0 ≤ t ≤ b, joins the points c∗ (0), c∗ (a), and Z bp

h(p ◦c1 ) 0 (t), (p ◦c1 ) 0 (t)i dt Z bq Z bq = hH(c 01 (t)), H(c 01 (t))i dt ≤ hc 01 (t), c 01 (t)i dt l(p ◦ c1 ) =

0

0

0

= l(c1 ) < l(c) = l(c∗ ),

where l denotes the length of the corresponding path. Therefore, the path c∗ (t), 0 ≤ t ≤ a, is not a shortest geodesic in B, a contradiction. t u Theorem 1.6.6. Let p : M → B be a Riemannian submersion, where M is complete and B is connected. Then B is complete and p is surjective. Moreover, if M has sectional curvature ≥ k, then B also has sectional curvature ≥ k. Proof. Put x ∈ p (M). According to Lemma 1.6.5, any geodesic c∗ (t), 0 ≤ t < a, in B with the origin x and parameterized by the arc length has a horizontal lift c(t), 0 ≤ t < a, in M, which is also geodesic and is parameterized by the arc length. Since M is complete, this lift extends onto [0, a] and p (c(a)) = lim p (c(t)) = lim c∗ (t). t%a

t%a

This implies that Expx is defined on all tangent spaces Bx . By the Hopf–Rinow theorem 1.5.1, B is complete. It follows from the connectedness of B and Corollary 1.5.3 that the point x can be joined with any point y ∈ B by a shortest geodesic c∗ (t), 0 ≤ t ≤ a, parameterized by the arc length. Then by Lemma 1.6.5, y ∈ p (M) and p is surjective. The last assertion is an immediate corollary of the proved assertions and Theorem 1.6.4. t u More detailed information on Riemannian submersion and related topics can be found e.g. in [106, 210, 239]. Definition 1.6.7. A submanifold N of a Riemannian manifold M is called totally geodesic if any geodesic in M, tangent at some point to N, lies in N. According to Proposition 8.2 of Chapter VII in [291], the following proposition is fulfilled. Proposition 1.6.8. A smooth (possibly, virtual) submanifold N of a Riemannian manifold (M, g) is totally geodesic if and only if ∇X Y ∈ F ∞ (N) for all vector fields X,Y ∈ F ∞ (N) and the Levi-Civita connection ∇ on (M, g). Corollary 1.6.9. Assume that p : M → B is a Riemannian submersion, M is complete, and B is connected. Then any geodesic in M with horizontal tangent vector at its initial point is horizontal. Moreover, if M is connected and the fibres of the submersion p are totally geodesic, then the fibres are mutually isometric.

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Proof. Let us use the notation from Lemma 1.6.5. Let z1 6= z2 ∈ p −1 (c∗ (0)) and ci (t), 0 ≤ t ≤ a, i = 1, 2, be lifts of c∗ (t), 0 ≤ t ≤ a, such that ci (0) = zi , i = 1, 2. Suppose that d(z1 , z2 ) and t0 ∈ (0, a) are sufficiently small. Then for every t ∈ [0,t0 ] there exists a unique shortest geodesic γt (s), 0 ≤ s ≤ 1, in (M, g), joining c1 (t) and c2 (t). Hence γt (s) ∈ p −1 (c∗ (t)) for all s ∈ [0, 1] and the map f (t, s) = γt (s), (t, s) ∈ [0,t0 ] × [0, 1], is a variation of geodesics γt , t ∈ [0,t0 ]. Let us define vector fields along f by the following formulae Y = T f ◦ ∂ /∂t and X = T f ◦ ∂ /∂ s. By Lemma 1.6.5, g(X(t, s),Y (t, s)) = 0 for s = 0 and s = 1. Thus, interchanging the roles of X and Y in (1.58), we get d(l(γt ))/dt = 0 for all t ∈ [0,t0 ]. This implies that d(c1 (t), c2 (t)) ≡ d(z1 , z2 ) for all t ∈ [0,t0 ] and hence the corollary. t u Definition 1.6.10. A map of metric spaces f : (M, r) → (N, q) is called a submetry if it maps any closed ball B(x, s) ⊂ (M, r) with radius s and center x onto the closed ball B( f (x), s) ⊂ (N, q) with radius s and center f (x) [71]. Definition 1.6.11. A metric fibration in the sense of Efremovich–Gorelik–Vainstein [56, 201, 202, 445] could be defined as a submetry with mutually isometric fibres, i.e. preimages of points in the region of the submetry. Corollary 1.6.12. Assume that p : M → B is a Riemannian submersion, M is complete, and B is connected. Then p is a submetry. Moreover, for any points x∗ , y∗ ∈ B, x ∈ p −1 (x∗ ), ρB (x∗ , y∗ ) = min{ρM (x, y) : y ∈ p −1 (y∗ )}. Remark 1.6.13. In [71], it is proved that any submetry of a complete Riemannian C∞ -manifold M into a Riemannian C∞ -manifold B is a Riemannian submersion of the class C1,1 . In the general case this result cannot be improved even for metric fibrations [201]. Corollaries 1.6.9 and 1.6.12 imply that any Riemannian submersion with complete total space, totally geodesic fibres, and connected base is a metric fibration. Later, the term “the metric fibration” was applied in [209], [239], and other sources to any Riemannian submersion. It is proved in [239] that any nontrivial Riemannian submersion with connected fibres and a Euclidean sphere as a total space is a Hopf fibration. Natural additions to this result are the papers [445] and [209]. The book [239] also contains other important results on Riemannian submersions. Theorem 1.6.14 ([268], [291], [420]). Suppose that p : M → B is a Riemannian submersion, M is complete, and B is connected. Then p is a locally trivial fiber bundle. Proof. In consequence of Theorem 1.6.6, B is complete. Let x be any point in B, r = r(x) be the injectivity radius of the Riemannian manifold B at the point x, U = U(x, r) be an open ball of radius r with center x, and F = p −1 (x). Define a map φ : U × F :→ p −1 (U) by the formula 
 φ (u, f ) = ExpM H T p ( f )−1 ((Expx |U )−1 (u)) . (1.103)

1.7 Miscellaneous

71

On the grounds of Lemma 1.6.5, Corollary 1.6.9, and (1.103) it is not difficult to establish that p(φ (u, f )) = u. In fact, it is possible to prove that φ is a diffeomorphism, but it suffices to prove that φ is homeomorphism. It is clear that φ is continuous. It remains to check that φ is a bijection. If z ∈ p −1 (u), where u ∈ U, then by Lemma 1.6.5, there exists a unique horizontal lift [z, f ] of a unique shortest geodesic segment [u, x] in B, and it is also a shortest geodesic segment in M. This observation, Corollary 1.6.9, and the formula (1.103) imply that φ ( f , u) = z and φ is a bijection. t u

1.7 Miscellaneous Here we consider some additional very important results on Riemannian manifolds. Definition 1.7.1. Let (M, g) be a compact Riemannian manifold of positive secmin K tional curvature K. Its δ -pinching δ (M, g) is defined as max . K It is clear that δ (M, g) ∈ (0, 1]. If δ (M, g) = 1 and M is simply connected, then (M, g) is isometric, up to a homothety, to the standard sphere. Definition 1.7.2. A compact Riemannian manifold (M, g) of positive sectional curvature is said to be weakly δ -pinched for a given δ ∈ (0, 1] if δ (M, g) ≥ δ . If strict inequality holds, we say that (M, g) is strictly δ -pinched. Theorem 1.7.3 (S. Brendle–R. Schoen, [128]). Let (M, g) be a compact strictly 1/4-pinched Riemannian manifold. Then M is diffeomorphic to a spherical space form. In particular, no exotic sphere admits a metric with strictly 1/4-pinched sectional curvature. Note that this result (the Differentiable Sphere Theorem) is also true under changing of the δ (M, g)-pinching by its point-wise analogue [127, 128]. It should be noted that the proof of Theorem 1.7.3 is based on the use of Ricci flow, which allows us to deform the given 1/4-pinched metric to a metric of constant curvature 1. The pinching constant 1/4 is optimal: in fact, any compact symmetric space of rank 1 admits a metric whose sectional curvatures lie in the interval [1, 4]. The list of these spaces includes the following examples: the complex projective space CPm (dimension 2m ≥ 4), the quaternionic projective space HPm (dimension 4m ≥ 8), and the projective plane over the octonions CaP2 (dimension 16), see Chapter 4 for more details. If n = dim(M) is odd then it is possible to change the number 1/4 to some number δ (n) ∈ (0, 1/4) [127]. There is a good exposition of injectivity radius estimates and sphere theorems (up to 1997) in [4]. More recent surveys on sphere theorems are in [127] and [129]. For any natural number n ≥ 2 there exists a real number δ (n) ∈ (0, 1/4) such that any compact simply connected Riemannian manifold of dimension n with condition δ (M, g) > δ (n) is diffeomorphic to the sphere or compact Riemannian symmetric space of rank one [387]. If (M, g) is a compact Riemannian manifold with sectional

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curvature K ≥ 1 and diameter diam(M) ≥ π/2, then M is homeomorphic to the sphere or locally symmetric [237], [467]. In [238], D. Gromoll and W. Meyer constructed an exotic seven-dimensional sphere (Σ 7 , g) of sectional curvature K ≥ 0. It is obtained as the orbit space of some isometric action of the Lie group Sp(1) on the Lie group Sp(2) (supplied with a bi-invariant Riemannian metric) with the induced Riemannian metric, so (Σ 7 , g) is a base of Riemannian submersion. The inequality K ≥ 0 follows from Theorems 2.4.13 and 1.6.6. Recall the following result for non-negatively curved Riemannian manifolds. Theorem 1.7.4 (M. Gromov, [243]). Let (M, g) be a compact Riemannian manifold of dimension n with non-negative sectional curvature. Then the sum of the Betti numbers of M is bounded above by some constant C(n), which depends only on the dimension. Any non-compact connected complete Riemannian manifold M n with K ≥ 0 admits a compact totally geodesic submanifold (the soul) S and a diffeomorphism onto the normal vector fiber bundle νS over S, identical on S (J. Cheeger–D. Gromoll [152], V. A. Sharafutdinov [411]). There exists a metric retraction (the Sharafutdinov map) p : M → S [412], which is the Riemannian submersion of the class C∞ [468]. If also at some point x ∈ M n all sectional curvatures are positive then M n is diffeomorphic to Rn [385]. Any smooth manifold M n , n ≥ 3, admits a complete Riemannian metric gM with Ricci curvature a(n) < ric(gM ) < b(n) < 0, where the numbers a(n) and b(n) depend only on n [325]. Now, we recall some important results from the paper [242] by M. Gromov, see also [135]. Let M be a connected compact n-dimensional Riemannian manifold, d = d(M) the diameter of M. Denote by c+ = c+ (M) and c− = c− (M) respectively, the upper and lower bounds of the sectional curvature of M, and set c = c(M) = max{|c− |, |c+ |}. We say that M is ε-flat, ε > 0, if cd 2 < ε. Note, that every compact flat manifold is ε-flat for any ε > 0. Moreover, every compact nilmanifold (a manifold is called a nilmanifold if it admits a transitive action of a nilpotent Lie group) possesses an ε-flat metric for any ε > 0. The latter example shows that for every n > 3 and every ε > 0, there are infinitely many ε-flat n-dimensional manifolds with different fundamental groups. Let us define inductively ex i (x) = exp(ex i−1 (x)), ex 0 = x, and set b ε (n) = exp(−ex j (n)), where j = 200. Theorem 1.7.5 (M. Gromov, [242]). Let M be a compact b ε (n)-flat manifold, and π its fundamental group. Then: 1) There exists a maximal nilpotent normal divisor N ⊂ π; 2) The order of the group π/N is less that ex 3 (n); 3) The finite covering of M corresponding to N is diffeomorphic to a nilmanifold. It should be noted that the true value of the constants b ε (n) with the above property is unknown. Theorem 1.7.5 implies the following result.

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73

Corollary 1.7.6 ([242]). If M is b ε (n)-flat, then its universal covering is diffeomorphic to Rn . If M is b ε (n)-flat and π is commutative, then V is diffeomorphic to a torus. Let M be a compact connected smooth Riemannian n-manifold with diameter d(M) ≤ D. One says that M has “almost non-positive” curvature if the sectional curvature K(M) < ε for a small ε > 0 depending on D and n. In [219] the authors study the topology of manifolds of almost nonpositive curvature under the condition K(M) ≥ −1. They denote by M (n, D) the family of compact Riemannian n-manifolds M with d(M) ≤ D and K(M) ≥ −1. The following result from [219] is an addition to the Hadamard–Cartan theorem. Theorem 1.7.7. There exists a positive number εn (D) such that the following holds. If M ∈ M (n, D) satisfies K(M) < εn (D) then the universal covering space of M is diffeomorphic to Rn . Corollary 1.7.8. The homotopy type of any smooth complete connected Riemannian manifold M of nonpositive or almost nonpositive curvature is defined by its fundamental group π1 (M), i.e. M is the so-called Eilenberg–MacLane space K(π, 1). Theorem 1.7.7 was conjectured by M. Gromov. One might hope to eliminate the condition K(M) ≥ −1. But, for n = 3, there is a counterexample due to Gromov [242], which has been verified in a paper by Buser and Gromoll [134]. Namely, for given ε > 0, there exists a Riemannian metric gε on S3 such that d(gε ) ≤ ε and K(gε ) ≤ ε. Let us recall Toponogov’s Triangle Theorem, which has various consequences, see e.g. [437, 150, 405]. It is one of a family of theorems that quantify the assertion that a pair of geodesics emanating from a point p spread apart more slowly in a region of high curvature than they would in a region of low curvature. Theorem 1.7.9 (V. A. Toponogov, [437]). Let M be an n-dimensional Riemannian manifold with sectional curvature K satisfying K ≥ δ for some δ ∈ R. Let PQR be a geodesic triangle, i.e. a triangle whose sides are geodesics, in M, such that√the geodesic PQ is minimal and if δ > 0, the length of the side PR is less than π/ δ . Let P0 Q0 R0 be a geodesic triangle in the model space Mδ , i.e. the simply connected space of constant curvature δ , such that the length of sides P0 Q0 and P0 R0 are equal respectively to that of PQ and PR and the angle at P0 is equal to that at P. Then d(Q, R) ≤ d(Q0 , R0 ) for the corresponding distance functions. Interesting discussions on comparison theorems in Riemannian geometry can be found in [281], [150], and [245]. Recall that a geodesic γ : R → M in a Riemannian manifold (M, g) is called a metric line if each of its segments is the shortest path between its endpoints. The following splitting theorem is well known.

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Theorem 1.7.10 (V. A. Toponogov, [438]). Every complete Riemannian manifold (M, g) with non-negative sectional curvature, containing a metric line, is isometric to a direct Riemannian product (N, ν) × R. This theorem has the following generalization. Theorem 1.7.11 (J. Cheeger–D. Gromoll, [151]). Every complete Riemannian manifold (M, g) with non-negative Ricci curvature, containing a metric line, is isometric to a direct Riemannian product (N, ν) × R. More comprehensive expositions on Riemannian geometry can be found e.g. in the books [98], [183], [221], [277], [345], [386], [390], and [405].

Chapter 2

Lie Groups and Lie Algebras

Abstract In this chapter, we recall some well-known results on Lie groups and Lie algebras. In particular, we discuss the third Lie theorem, the Ado theorem, and the Cartan semisimplicity criterion. Some important types of Lie algebras and Lie groups together with their important ideals and normal subgroups are discussed. Much attention is paid to compact Lie groups and compact Lie algebras. We consider deep structure results on compact Lie algebras and Cartan subalgebras, discuss important properties of maximal tori in connected compact Lie groups, as well as their roots, root systems, and the Weyl groups. Moreover, we provide useful formulas for calculation of the Ricci curvature and the scalar curvature of Lie groups, supplied with left-invariant Riemannian metrics. Some important geometric properties of nilpotent and compact Lie groups with left-invariant Riemannian metrics are discussed.

2.1 Main Properties of Lie Groups Definition 2.1.1. A Lie group is a smooth manifold G that is simultaneously a group with the product operation · , equipped with smooth mappings · : (g, h) ∈ G × G → g · h ∈ G, −1

: g ∈ G → g−1 ∈ G.

The unit element of the group G is denoted by e. For simplicity, we write simply gh instead of g · h. Definition 2.1.1 implies that for a given element g in G, the operations lg : h ∈ G → gh ∈ G and rg : h ∈ G → hg ∈ G of left and right multiplications by the element g are smooth. Moreover, lg and rg are diffeomorphisms of G onto itself since lg−1 ◦ lg = idG and rg−1 ◦ rg = idG . The mappings lg and rg are called the left translation and the right translation (by the element g) respectively. © Springer Nature Switzerland AG 2020 V. Berestovskii and Y. Nikonorov, Riemannian Manifolds and Homogeneous Geodesics, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-56658-6_2

75

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Sometimes it is convenient to deal with more general objects than Lie groups. Definition 2.1.2. A topological group is a group G that is also a topological space, such that the group operations · : (g, h) ∈ G × G → g · h ∈ G, where G × G has the product topology, and −1

: g ∈ G → g−1 ∈ G,

are continuous. A topological group G is called locally compact if the underlying topology is locally compact and Hausdorff. It is clear that Lie groups are locally compact topological groups. It should be noted, that locally compact groups are important because many examples of groups that arise in various branches of mathematics are locally compact and such groups have a natural measure called the Haar measure. A detailed exposition can be found e.g. in [343]. Definition 2.1.3. A smooth mapping φ : G → H of Lie groups, that is a group homomorphism, is called a Lie group homomorphism; if φ is a group monomorphism, a group epimorphism, or a group isomorphism, then it is called a Lie group monomorphism, a Lie group epimorphism, or a Lie group isomorphism respectively. Definition 2.1.4. A vector field X on a Lie group G is left-invariant (respectively, right-invariant) if X(gh) = T (lg )(X(h)) (respectively, X(hg) = T (rg )(X(h))) for all elements g, h in G. In other words, a vector field X is lg -related (respectively, rg related) with itself for any element g in G (see Subsection 1.1.4). It is clear that any left-invariant and any right-invariant vector field on a given Lie group is smooth. Proposition 2.1.5. The set LG of all left-invariant vector fields on a Lie group G constitutes a Lie algebra of dimension n = dim G. Proof. The linearity of the differential of a smooth mapping at every point implies that LG is a linear space over R. If X ∈ LG, then it is completely determined by its value at e since X(g) = T (lg )(X(e)) for any g ∈ G and every element v ∈ Ge correctly defines a vector field X ∈ LG via X(g) = T (lg )(v). Therefore, LG has dimension n = dim G. If X,Y ∈ LG, then X and Y are lg -related with themselves for every g ∈ G. Then according to Theorem 1.1.33, their Lie bracket [X,Y ] is lg related with itself for every g ∈ G. Therefore, LG constitutes a n-dimensional Lie subalgebra of the Lie algebra F ∞ (G) of all smooth vector fields on G. t u Remark 2.1.6. Similar arguments show that the set RG of all right-invariant vector fields on the Lie group G constitutes a Lie algebra of dimension n = dim G. Connections between the Lie algebras RG and LG are explained in Theorem 2.1.29.

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77

Definition 2.1.7. The pair (LG, [·, ·]) is called the Lie algebra of the Lie group G. Remark 2.1.8. The proof of Proposition 2.1.5 implies that the Lie algebra LG could be identified with the tangent space Ge , supplied with a corresponding Lie bracket [·, ·]. In what follows, we will use this identification. Theorem 2.1.9. If φ : G → H is a Lie group homomorphism, then T φe : (Ge , [·, ·]) → (He , [·, ·]) is a Lie algebra homomorphism. If φ is a Lie group monomorphism, a Lie group epimorphism, or a Lie group isomorphism, then T φe is a Lie algebra monomorphism, a Lie algebra epimorphism, or a Lie algebra isomorphism respectively. Proof. To prove the first assertion, it suffices to show that any left-invariant vector field X on G is φ -related with a left-invariant vector field Y on H such that T φ (X(e)) = Y (e). Then by Theorem 1.1.33, T φ ([X,Y ](e)) = [T φ (X(e)), T φ (Y (e))] for all X,Y ∈ LG, which means that T φe is a Lie algebra homomorphism. Let g be an arbitrary element in G. Since φ is a Lie group homomorphism, we have φ (gh) = φ (g)φ (h) for all g, h ∈ G, i.e. φ ◦ lg = lφ (g) ◦ φ .

(2.1)

Therefore, Y (φ (g)) = (T lφ (g) )(Y (e)) = (T lφ (g) )(T φ (X(e))) = T (lφ (g) ◦ φ )(X(e)) = T (φ ◦ lg )(X(e)) = T φ (T lg (X(e))) = T φ (X(g)). It follows from the formula (2.1) that T φ (g) ◦ T lg (e) = T lφ (g) (e) ◦ T φ (e). This implies that the rank of the linear mapping T φg does not depend on g ∈ G, since T lg (e), T lφ (g) (e) are linear space isomorphisms. Then φ (G) has zero measure in H if T φe is not surjective (by the A. Sard theorem), but the kernel of the mapping φ is a closed submanifold (and a normal subgroup) in G of dimension at least 1 if T φe is not injective (this is a well-known result on smooth manifolds). These arguments imply the second assertion of the theorem. t u From the second assertion of Theorem 2.1.9, its proof and the inverse mapping theorem, we get Corollary 2.1.10. If φ : G → H is a Lie group isomorphism in the sense of Definition 2.1.3, then the inverse mapping φ −1 : H → G is also a Lie group isomorphism in the sense of the same definition.

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Example 2.1.11. Let us consider the group G = (Rn , +) with standard coordinate functions x1 , . . . , xn . Then the vector fields ∂∂xi , i = 1, . . . , n, constitute a basis of the Lie algebra LG over R. Moreover, for any real smooth function f on G, we have       ∂ ∂ ∂ ∂ ∂ ∂ , f := i f − j f = 0. ∂ xi ∂ x j ∂x ∂xj ∂ x ∂ xi Therefore, [X,Y ] = 0 for all X,Y ∈ LG. Definition 2.1.12. A Lie algebra (L, [·, ·]) with zero brackets is called a commutative or abelian Lie algebra. Example 2.1.13. Let Zn ⊂ Rn be an integer lattice. Then the quotient group T n := Rn /Zn is a Lie group, the natural homomorphism φ : Rn → T n is a Lie group epimorphism, and the differential T φ 0 : Rn → Rn is the identical isomorphism of a commutative Lie algebra. The Lie group T n is called an n-dimensional torus, and the epimorphism φ is called a universal covering epimorphism. Definition 2.1.14. An arbitrary Lie group homomorphism φ : (R, +) → G is called a 1-parameter subgroup of the Lie group G. Proposition 2.1.15. All left-invariant and all right-invariant vector fields on a Lie group are complete. Proof. Let us suppose that X is a left-invariant vector field on a Lie group G. Then there is an integral curve Ψ (e,t), t ∈ (−ε, ε), ε > 0, of the vector field X with the initial point e. Since X is left-invariant then for any element g ∈ G, Ψ (g,t) := gΨ (e,t),

t ∈ (−ε, ε),

(2.2)

is an integral curve of the vector field X with the initial point g. In other words, the domain of the flow Ψ of the vector field X includes the set G × (−ε, ε). By Proposition 1.1.36, the vector field X is complete. The case of a right-invariant vector field could be considered analogously. t u Proposition 2.1.16. Let Ψ be the flow of either a left-invariant or a right-invariant vector field X on a Lie group G. Then φ (t) = Ψ (e,t), t ∈ R, is a 1-parameter subgroup of the Lie group G. Proof. If X is left-invariant then, according to properties of the flow Ψ , φ : R → G is a smooth mapping and φ (t + s) = Ψ (e,t + s) = Ψ (Ψ (e,t), s) = Ψ (e,t)Ψ (e, s) = φ (t)φ (s), i.e. φ : R → G is a homomorphism. The case of a right-invariant vector field could be considered analogously. t u Proposition 2.1.17. Let G be a Lie group. Then for every vector v ∈ Ge , there exists a unique 1-parameter subgroup φv : R → G with condition φv0 (0) = v.

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Proof. By the definition, for all t, s ∈ R and φ = φv , the equalities φ (t + s) = φ (t) · φ (s) and φ 0 (0) = v hold. Differentiating the first equality in s at s = 0, we get φ 0 (t) = T (lφ (t) )(φ 0 (0)) = T (lφ (t) )(v). This means that φ (t), t ∈ R, is an integral curve of a left-invariant vector field X with conditions X(e) = v and φ (0) = e. This observation together with Propositions 2.1.15 and 2.1.16 imply the existence of a suitable 1parameter subgroup φv in G. The uniqueness is obvious. t u Definition 2.1.18. Let G be a Lie group and let g = Ge be its Lie algebra. Then the mapping exp : g → G, defined by the formula exp(v) = φv (1), is called the exponential map of the Lie group G. Proposition 2.1.19. Let us consider Lie groups G and H with the Lie algebras g and h respectively and a Lie group homomorphism f : G → H. Then the following diagram is commutative: g

T fe

→

exp ↓ G

h ↓ exp

f

→

H

Proof. Consider any v ∈ g and φv : R → G, the corresponding 1-parameter subgroup in G. Then ψ := f ◦ φv is a 1-parameter subgroup in H, ψ 0 (0) = ( f ◦ φv )0 (0) = T f (φv (0))(φ 0 (0)) = T fe (v), exp(T fe (v)) = ψ(1) = f (φv (1)) = f (exp(v)). The proposition is proved.

t u

Theorem 2.1.20. The exponential map exp : g → G of a Lie group G is a diffeomorphism on some neighborhood of zero 0 ∈ g. Proof. Let v be any vector in g. Let us consider the straight line c(s) = sv, s ∈ R, in g. Obviously, for a given s ∈ R, ψ(t) := φv (st), t ∈ R, is a 1-parameter subgroup in G with the tangent vector ψ 0 (0) = sv. Hence, (exp ◦c)(s) = exp(sv) = φsv (1) = ψ(1) = φv (s). Differentiating this equality in s at s = 0, we get T exp 0 (c0 (0)) = T exp 0 (v) = v. Therefore, T exp 0 = Idg since v ∈ g is arbitrary. Now, the required assertion follows from the inverse mapping theorem. t u Let us consider G = {X ∈ M(n, R) | det(X) 6= 0}, where M(n, R) is the real vector space of all real (n × n)-matrices. For a matrix X ∈ M(n, R), we assign its elements xi j ; i, j = 1, . . . , n, in the lexicographic order, hence we get a bijection

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φ : M(n, R) → Rn . Then V = φ (G) = φ (M(n, R) \ det−1 (0)) is an open subset 2 in Rn . If we consider φ as a homeomorphism, and the triple (G, φ ,V ) as a chart, we see that the group G is a Lie group. Indeed, the elements of the product of two matrices are polynomials in elements of the factors, and the elements of the inverse matrix X −1 are rational functions in elements of the matrix X ∈ G. The manifold G consists of two connected components: the first component is a connected Lie group GL+ (n, R) of matrices with positive determinants, and the second one consists of matrices with negative determinants and is a coset with respect to the subgroup GL+ (n, R). Theorem 2.1.21. The Lie algebra Ge = gl(n, R) of the Lie group G = GL(n, R) coincides with M(n, R). Moreover, the Lie bracket [A, B] of elements A, B ∈ gl(n, R) is calculated by the formula [A, B] = AB − BA.

(2.3)

Proof. The product XY , for X,Y ∈ M(n, R), is linear with respect to X and to Y . Hence, an element A ∈ Ge determines a corresponding left-invariant vector field A on G by the formula A(X) = XA, X ∈ G. (2.4) For the same reason, if A, B ∈ Ge and A, B are the corresponding left-invariant vector fields on G, then the elements n

α i j (X) =

n

∑ xik ak j , k=1

β lm (X) =

∑ xlr brm

(2.5)

r=1

of the matrices XA and XB are the components of the vector fields A and B at the point X with respect to the mentioned chart for GL(n, R). Now, a direct calculation using formulas (1.18) and (2.5) gives the equality (2.3). t u It is easy to prove the following Proposition 2.1.22. The exponential map exp : gl(n, R) → GL(n, R) (in the sense of Definition 2.1.18) is calculated by the formula +∞

exp(A) =

Ak , k=0 k!

∑

A ∈ gl(n, R).

Let V = V (n, R) be a real vector space of dimension n. The group of all nondegenerate linear transformations of the space V is called the general linear group GL(V ). The group GL(V ) admits a unique Lie group structure such that an isomorphism φ : GL(V ) → GL(n, R) of abstract groups, determined by the choice of a basis in the space V , is a Lie group isomorphism. The Lie algebra gl(V ) of the Lie group GL(V ) is a vector space of all linear endomorphisms A : V → V with bracket (2.3).

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Definition 2.1.23. A Lie group homomorphism f : G → GL(V ) (a Lie algebra homomorphism φ : g → gl(V )) is called a representation of the group G (respectively, of the Lie algebra g) in the space V . The representation f (respectively, φ ) is called faithful if it is a monomorphism, and is called irreducible if it has no non-trivial invariant subspace in V . Definition 2.1.24. A subgroup H ⊂ G of a Lie group G that is a smooth (virtual) submanifold in G is called a (virtual) Lie subgroup (of the Lie group G). The topology of a virtual Lie subgroup can be different from the topology induced by the ambient group. This is the case for a dense winding of the torus T2 which carries the Lie group (in particular, the topology) structure of R but intersects with any nonempty open subset of the torus on an unbounded subset of R. Proposition 2.1.25. Let G be a Lie group, g ∈ G, I(g) : G → G, where I(g)(v) = gvg−1 , v ∈ G. Then the following assertions hold. 1) I(g) : G → G is a Lie group automorphism G (such automorphisms are called inner for the Lie group G); 2) The linear map Ad(g) := T I(g)e : Ge → Ge is an automorphism of the Lie algebra (Ge , [·, ·]) of the Lie group G; 3) The mapping g ∈ G → Ad(g) is a homomorphism of the Lie group G into the Lie group Aut(Ge , [·, ·]) of all automorphisms of the Lie algebra (Ge , [·, ·]) of the group G and, consequently, is a representation of the Lie group G in the vector space Ge . Proof. 1) Obviously, I(g−1 ) = I(g)−1 , and I(g)(v1 v2 ) = g(v1 v2 )g−1 = (gv1 g−1 )(gv2 g−1 ) = I(g)(v1 )I(g)(v2 ) for every v1 , v2 ∈ G, i.e. I(g) is an automorphism of the abstract group G. The definition of a Lie group implies that I(g) is a diffeomorphism. Therefore, I(g) is an automorphism of the Lie group G. 2) This is a partial case of Theorem 2.1.9. 3) At first, note that Aut(Ge , [·, ·]) is a closed subgroup of the Lie group GL(Ge ), hence, according to Definition 2.1.24 and to Theorem 2.1.32 below, it is a Lie subgroup of the Lie group GL(Ge ). It is clear that −1 I(g1 g2 )(v) = (g1 g2 )v(g1 g2 )−1 = g1 (g2 vg−1 2 )g1 = I(g1 )(I(g2 )(v))

for every v, g1 , g2 ∈ G. Then, due to properties of smooth mappings and the identity I(g)(e) = e, we get Ad(g1 g2 ) = T (I(g1 g2 ))e = T (I(g1 )I(g2 ))e = T (I(g1 ))e T (I(g2 ))e = Ad(g1 ) Ad(g2 ), i.e. Ad : G → GL(Ge ) is an abstract group homomorphism. The definition of a Lie group implies that Ad is a smooth mapping, hence, a Lie group homomorphism and a representation of the Lie group G in the vector space Ge . The last assertion of

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Theorem 2.1.9 implies that Ad is a homomorphism from the Lie group G to the Lie group of all automorphisms of the Lie algebra (Ge , [·, ·]). t u Definition 2.1.26. The representation Ad : g ∈ G → Ad(g) is called the adjoint representation of the Lie group G, and its differential ad = (T (Ad))e is called the adjoint representation of the Lie algebra (Ge , [·, ·]) of the Lie group G. Our next goal is to calculate explicitly the mapping ad. Theorem 2.1.27. Identifying left-invariant vector fields X and Y on the Lie group G with their values at e, we get ad(X)(Y ) = [X,Y ].

(2.6)

Proof. Let φ (t), ψ(s), t, s ∈ R, be 1-parameter subgroups in G that are defined by the vector fields X and Y as in Proposition 2.1.16. Then I(φ (t))(ψ(s)) = φ (t)ψ(s)φ (−t); Ad(φ (t))(Y ) = T (I(φ (t)))e (ψ 0 (0)) = (T (rφ (−t) ) ◦ T (lφ (t) ))(Y ) = T (rφ (−t) )(Y ). According to formula (2.2), we get T (rφ (−t) )(Y ) = T (Φ−t )(Y ), where Φ is the flow of the vector field X. Therefore, Ad(φ (t))(Y ) = T (Φ−t )(Y ).

(2.7)

By Proposition 1.1.38, 1 ad(X)(Y ) = (Ad(φ (t))(Y ))0 (0) = lim [T (Φ−t )(Y ) −Y ] t→0 t 1 = − lim [Y − T (Φ−t )(Y )] = −[−X,Y ] = [X,Y ]. t→0 t t u

The theorem is proved. From Propositions 2.1.19, 2.1.22 and Theorem 2.1.27, we get

Corollary 2.1.28. For every element X of the Lie algebra g of the Lie group G, the equality Ad(exp(tX)) = exp(t ad(X)), t ∈ R, holds. Theorem 2.1.29. Let X and Y (respectively, Xe and Ye ) be left-invariant (rightinvariant) vector fields on a Lie group G such that e = X(e), X(e)

Ye (e) = Y (e).

Then e Ye ](e) = −[X,Y ](e). [X,

(2.8)

2.1 Main Properties of Lie Groups

83

Proof. Let φ (t), ψ(s), t, s ∈ R, be 1-parameter subgroups in G that are integral curves of the vector fields X and Y . Due to (2.8) and the equalities φ (t + s) = φ (t)φ (s) = φ (s)φ (t), ψ(t + s) = ψ(t)ψ(s) = ψ(s)ψ(t), we get that φ (t), ψ(s), t, s ∈ R, are also integral curves of the fields Xe and Ye . From the computations of the proof of Theorem 2.1.27 for X,Y and analogous computae Ye , we get the equalities tions for X, e Ye ](e), [X,Y ](e) = ad(X)(Y ) = −[X, t u

which proves the theorem. In what follows, we will need the following proposition.

Proposition 2.1.30. Let g be the Lie algebra of a Lie group G, X,Y ∈ g and [X,Y ] = 0. Then the elements exp(tX), exp(sY ) ∈ G commute for all numbers t, s ∈ R. Proof. By Propositions 2.1.25, 2.1.19, Theorem 2.1.27 and Proposition 2.1.22, we have exp(tX) exp(sY ) exp(−tX) = I(exp(tX))(exp(sY )) = exp(T I(exp(tX))e (sY )) = exp(s Ad(exp(tX)(Y ))) !! +∞ ad(tX)k (Y ) = exp(sY ), = exp(s exp(ad(tX))(Y )) = exp s Y + ∑ k! k=1 which proves the proposition.

t u

From Definitions 1.1.22, 2.1.24 and Theorems 1.1.41, 1.1.42, we get Theorem 2.1.31. Every Lie subalgebra h of the Lie algebra g = Ge of a given Lie group G determines a left-invariant involutive distribution D = D(h) = {T lg (h) | g ∈ G}. The leaf H of the corresponding foliation F (D), containing the unit e ∈ G, is a connected virtual Lie subgroup. Conversely, for every connected virtual Lie subgroup H ⊂ G, the tangent space He = h is a Lie subalgebra h ⊂ g. Moreover, H is a Lie subgroup of the Lie group G if and only if H is closed on G. In the general case, H is a Lie group with respect to the leaf topology . In Proposition 2.2.6 and Theorem 2.27 of [6], there is a relatively short proof of the following theorem of Cartan, which is a natural addition to Theorem 2.1.31. Theorem 2.1.32. Every closed (in the topological sense) subgroup H of the Lie group G is a Lie (sub)group.

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It is clear that a connected virtual Lie subgroup (also called an analytic subgroup) of a given Lie group is naturally arcwise connected. We have the following result. Theorem 2.1.33 (M. Kuranishi–H. Yamabe, [493]). Let G be a Lie group, and let A be an arc-wise connected subgroup of G. Then A is a virtual Lie subgroup. This theorem was first proved by M. Kuranishi and later by H. Yamabe [493] in different ways. The proof of M. Kuranishi has never been published, and that of H. Yamabe is quite short [493]. A detailed proof can be found e.g. in [233]. It should be noted that there are connected (but not arc-wise connected) subgroups of Lie groups that are not virtual Lie subgroups. Example 2.1.34. We construct a special subgroup of the torus T2 = R2 /Z2 . Let us put A = {(x, πx)} ∪ {(x + 1/2, πx)}, x ∈ R, where {u} means the fractional part of a number u ∈ R. It is easy to check the following equalities modulo Z2 : (x, πx) + (y, πy) = (x + y, π(x + y)), (x + 1/2, πx) + (y + 1/2, πy) = (x + y + 1, π(x + y)) = (x + y, π(x + y)), (x, πx) + (y + 1/2, πy) = ((x + y) + 1/2, π(x + y)). It is clear that A contains the identity (0, 0) and for any element in A, there is an inverse element. Indeed, (−x, π(−x)) is an inverse for (x, πx) and (−x + 1/2, π(−x)) is an inverse for (x + 1/2, πx). Hence, A is a subgroup in T2 . Moreover, A is connected (as a subset in T2 ), but is not a virtual Lie subgroup. Note that there is a connected and dense subgroup in R2 such that its complement is also connected and dense in R2 , see [275] and [329]. Using Theorems 2.1.31 and 2.1.32, we get Corollary 2.1.35. The intersection of any family of Lie subgroups of a given Lie group G is also a Lie subgroup of the Lie group G. The subsets SL(n, R) ⊂ GL(n, R) and O(n) ⊂ GL(n, R), consisting of matrices with unit determinants and orthogonal matrices (i.e. X ∈ GL(n, R) with the property XX t = E = e) respectively, are closed subgroups and, hence, Lie subgroups of the Lie group GL(n, R). Moreover, the intersection SO(n) = O(n) ∩ SL(n, R) is a (closed-open) connected component of the unit of the group O(n). A real vector space V (n, R, (·, ·)) of dimension n with an inner (scalar) product (·, ·) is called n-dimensional Euclidean space and is denoted by En . In this case, the compact Lie subgroup O(V ) ⊂ GL(V ) (respectively, SO(V ) ⊂ GL(V )) of all linear (respectively, preserving a given orientation of the space V ) transformations of the space V , preserving the inner product (·, ·), is isomorphic to the Lie group O(n) (respectively, SO(n)). Now we consider one more important property of representations of Lie groups and Lie algebras.

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85

Definition 2.1.36. A representation f : G → GL(V ) of a Lie group G (a representation ϕ : g → gl(V ) of a Lie algebra g) is completely reducible if for any f (G)invariant (ϕ(g)-invariant) submodule V 0 in V , there is an f (G)-invariant (ϕ(g)invariant) submodule V 00 in V such that V = V 0 ⊕V 00 . Proposition 2.1.37. A representation f : G → GL(V ) of a Lie group G (a representation ϕ : g → gl(V ) of a Lie algebra g) is completely reducible if and only if V is a direct sum of f (G)-irreducible (ϕ(g)-irreducible) submodules. The property of being completely reducible is inherited by invariant submodules and is transferred to direct sums of completely reducible representations. Proof. We prove the proposition only for Lie groups (the proof for Lie algebras is similar). Let f : G → GL(V ) be a representation such that any f (G)-invariant submodule V 0 ⊂ V admits a complementary f (G)-invariant submodule V 00 in V . First, we show that this property is inherited by any f (G)-invariant submodule W in V . Indeed, if W 0 is an f (G)-invariant submodule in W ⊂ V , then there is an f (G)invariant submodule V 00 in V such that V = W 0 ⊕V 00 . It is clear that W 00 = W ∩V 00 is f (G)-invariant, W 0 ∩W 00 is trivial, and dim W 00 ≥ dim W + dim V 00 − dim V = dim W − dimW 0 , hence W = W 0 ⊕W 00 , as required. Let f be completely reducible. Now, if f : G → GL(V ) is irreducible, then V is a sum of unique irreducible submodules. Suppose that it is not irreducible. Then it has an f (G)-irreducible submodule that admits a complementary f (G)-invariant submodule, which in turn is again completely reducible. Iterating this procedure for the latter submodules several times, we get a complementary submodule which is irreducible itself, hence, we represent V as a direct sum of f (G)-irreducible submodules. Now, suppose that a representation f : G → GL(V ) is such that V = V1 ⊕ V2 ⊕ · · · ⊕Vs for some f (G)-irreducible submodules Vi , i = 1, ..., s. It is clear that a direct sum of such representations have the same property. Now, let W be any f (G)-invariant submodule in V . We should find a complementary f (G)-invariant submodule for W in V . First, suppose that W is f (G)-irreducible. For any X ∈ W we have a unique decomposition X = X1 + X2 + · · · + Xs , where Xi ∈ Vi , i = 1, ..., s. Hence, we get linear maps ηi : W → Vi such that ηi (X) = Xi . It is clear that all maps ηi are f (G)equivariant. Therefore, ηi (W ) are either trivial or non-trivial f (G)-irreducible, since W is irreducible. In the second case any ηi is an isomorphism of f (G)-modules W and Vi , since Vi is also irreducible. Obviously, at least one ηi , say η1 , is non-trivial. Then V2 ⊕V3 ⊕ · · · ⊕Vs is a complementary f (G)-invariant submodule for W in V . Now, suppose that W is not f (G)-irreducible. Put V 0 = V , W 0 = W and take any f (G)-irreducible submodule W1 in W . Then, as we have seen, there is a complementary f (G)-invariant submodule V 1 to W1 in V = V 0 . In particular, we get an f (G)-invariant submodule W 1 = W 0 ∩V 1 in V 1 . We can continue this procedure as follows: if the submodule W i is not trivial, take any irreducible submodule Wi+1 in

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W i . Then there is a complementary f (G)-invariant submodule V i+1 in V = V i for Wi+1 . This procedure will be finished when W i is trivial for some i = t. This means that W = W1 ⊕W2 ⊕ · · · ⊕Wt−1 and V t is a complementary f (G)-invariant submodule in V . t u Remark 2.1.38. It should be noted that a representation of V as a direct sum of f (G)irreducible (ϕ(g)-irreducible) submodules in Proposition 2.1.37 is not unique in general (see e.g. Example 4.2.14). This effect is related to the existence of isomorphic submodules in V . Detailed expositions of Lie groups and Lie algebras can be found e.g. in the books [6], [257], and [461].

2.2 Smooth Actions of Lie Groups Smooth manifolds and Lie groups are connected with the very important notion of a smooth action. This notion has already been applied earlier and will be applied further in various cases. It is appropriate to consider such cases from a unified point of view. Definition 2.2.1. A smooth (left) action of a Lie group G on a smooth manifold M is a smooth mapping µ : G × M → M, µ : (g, x) ∈ G × M → g(x) ∈ M such that e(x) = x and (g1 · g2 )(x) = g1 (g2 (x)) for all g1 , g2 ∈ G and all x ∈ M. A right action is defined in the same manner. It is clear that a right action coincides with a left action for any commutative Lie group. Flows of complete smooth vector fields on smooth manifolds, in particular, geodesic flows and flows of Killing vector fields on complete Riemannian manifolds are examples of smooth actions of the additive group of real numbers. The multiplication in a Lie group could be considered both as a smooth left and a smooth right action on itself. Definition 2.2.2. Let µ : G × M → M be a smooth left action of a Lie group G. The stabilizer of the point x ∈ M with respect to the action µ is the set Gx = {g ∈ G : g(x) = x}. The action µ is called (almost) free if Gx = {e} (respectively, Gx is discrete) for every point x ∈ M. The action µ is called (almost) effective if GM := ∩x∈M Gx = {e} (respectively, GM is discrete). It is clear that the sets Gx , x ∈ M, and GM are closed subgroups of the Lie group G, hence, Lie subgroups due to Theorem 2.1.32. There are other names of the group Gx : the stationary subgroup of the point x or the isotropy subgroup of the point x; the group T (Gx ) = {T g, g ∈ Gx } is called the linear isotropy group. The group GM is called the ineffective kernel of the action µ. Obviously, GM is a normal Lie subgroup of the Lie group G, every free action is effective, and every almost free action is almost effective. The action µ induces a smooth effective action of the Lie group G/GM on M.

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87

Definition 2.2.3. The orbit of a point x ∈ M with respect to a smooth action µ : G × M → M of a Lie group G is the set Ox = {g(x) : g ∈ G}. The action µ is called (simply) transitive if Ox = M for some point x ∈ M (and the action µ is free). The definitions of smooth transitive and simply transitive actions of Lie groups on smooth manifolds are the bases of the notion of a homogeneous space. Homogeneous spaces are considered in Chapter 4.2.

2.3 Main Properties of Lie Algebras 2.3.1 Connections with Matrix Lie Algebras It should be noted that every Lie algebra is isomorphic to some matrix algebra. More precisely, we have the following Theorem 2.3.1 (Ado’s theorem [228]). Every finite-dimensional Lie algebra admits a faithful finite-dimensional representation. A short proof of Ado’s theorem using universal enveloping algebras (see [181]) can be found in the paper [353] by Yu. A. Neretin. We know that every Lie group G uniquely determines its (finite-dimensional) Lie algebra (LG, [·, ·]). From Theorems 2.3.1 and 2.1.31, it is easy to get the opposite (in some reasonable sense) assertion: Theorem 2.3.2 (Lie’s third theorem). For any finite-dimensional Lie algebra (L, [·, ·]), there is a unique, up to isomorphism, connected and simply connected Lie group G with the Lie algebra (LG, [·, ·]) isomorphic to (L, [·, ·]).

2.3.2 Important Classes of Lie Algebras and Structural Results Among the Lie subalgebras of a given Lie algebra, ideals are very important. Definition 2.3.3. A vector subspace f of a Lie algebra g is called an ideal of the Lie algebra g if [X,Y ] ∈ f for all elements X ∈ g and Y ∈ f. Definition 2.3.4. It is said that a Lie algebra g is semisimple if {0} is its unique commutative ideal. A non-commutative Lie algebra g is called simple if it has no ideals except g and {0}. A Lie group with a simple (respectively, semisimple) Lie algebra is called a simple (respectively, semisimple) Lie group. The center of a Lie algebra g is the set C(g) := {X ∈ g | [X,Y ] = 0 for all Y ∈ g}. It is easy to prove the following proposition.

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Proposition 2.3.5. Every ideal of a Lie algebra g is a Lie subalgebra. The center C(g) ⊂ g is a commutative ideal of the Lie algebra g. Every simple Lie algebra is semisimple. Every non-trivial semisimple Lie algebra has dimension ≥ 3. For every Lie algebra g, there are several important subalgebras and ideals. Definition 2.3.6. Let k be a Lie subalgebra of a Lie algebra g. The following formulas cg (k) = {X ∈ g | [X,Y ] = 0 for all Y ∈ k}, ng (k) = {X ∈ g | [X,Y ] ⊂ k for all Y ∈ k} define respectively the centralizer and the normalizer of the Lie subalgebra k in the Lie algebra g. It is easy to see that cg (k) ⊂ ng (k), and cg (g) is the center of the Lie algebra g. Definition 2.3.6 and the Jacobi identity imply Proposition 2.3.7. For every Lie subalgebra k of a Lie algebra g, the sets cg (k) and ng (k) are Lie subalgebras in g, and k is an ideal in ng (k). Now, let us consider an arbitrary Lie algebra g. For linear subspaces V and W in g, [V,W ] denotes a linear span of all vectors of the type [v, w], where v ∈ V and w ∈ W. The derived algebra of a Lie algebra g is an important ideal in g that is defined by the equality g0 = [g, g]. Moreover, it is natural to define the linear spaces gk for all non-negative integer k as follows: g0 = g,

g1 = [g, g],

...,

gk+1 = [g, gk ],

...

The Jacobi identity implies gk+1 ⊂ gk , k ≥ 0, and every subspace gk is an ideal in the Lie algebra g. Definition 2.3.8. A Lie algebra g is called k-step nilpotent if gk = {0} and gk−1 6= {0}. It is called nilpotent if it is k-step nilpotent for some k. A connected Lie group G is called nilpotent if its Lie algebra is nilpotent. Example 2.3.9. Every commutative Lie algebra is 1-step nilpotent. It should be noted that, for a k-step nilpotent Lie algebra g, the subalgebra gk−1 6= {0} is contained (according to the definition of gk = {0}) in the center g. Consequently, every nilpotent Lie algebra has non-trivial center. Theorem 2.3.10 (Engel’s theorem, [266, 473]). A Lie algebra g is nilpotent if and only if the operator ad X : g → g is nilpotent for all X ∈ g. For a given Lie algebra g, there is a natural sequence of its ideals: g0 = g,

g1 = [g0 , g0 ],

...,

gl+1 = [gl , gl ],

...

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89

Definition 2.3.11. A Lie algebra g is called (k-step) solvable if there are numbers l with the property gl = {0} (and k is the smallest such number). A connected Lie group G is called solvable if its Lie algebra is solvable. Obviously, any nilpotent, in particular, commutative, Lie algebra is solvable. It can be proved that the sum of two solvable (nilpotent) ideals in a given Lie algebra is again solvable (respectively, nilpotent) ideal. Hence, the following definition makes sense. Definition 2.3.12. The radical rad = rad(g) (the nilradical nilrad = nilrad(g)) of a Lie algebra g is the biggest solvable (respectively, nilpotent) ideal in g. Theorem 2.3.13 (the Levi–Mal’tsev theorem, [266, 473]). For every Lie algebra g, there is a semisimple subalgebra s ⊂ g such that the decomposition g = rad(g) ⊕ s (the direct sum of linear spaces) holds. The subalgebra s is unique up to an automorphism of g of the form exp(ad z), where z is an element of the nilradical nilrad(g). Any semisimple subalgebra s of g as in Theorem 2.3.13 is called a Levi subalgebra or a Levi factor of the Lie algebra g. Note that g could be represented as a semidirect sum of its Lie subalgebras: g = rad(g) o s. Levi subalgebras are exactly maximal semisimple subalgebras of g and every semisimple subalgebra of g is situated in some Levi subalgebra. For a solvable Lie algebra g, we get nilrad(g) ⊃ g0 = [g, g], since g0 is a nilpotent ideal in g [266, 473]. Definition 2.3.14. Let g be an arbitrary Lie algebra. The function B(X,Y ) = Bg (X,Y ) = trace(ad X ◦ adY ),

X,Y ∈ g,

where ad X(Y ) = [X,Y ], is called the Killing form of the Lie algebra g. It is easy to prove the following proposition. Proposition 2.3.15. The Killing form of a Lie algebra g is bilinear, symmetric, and invariant with respect to all automorphisms of g. Theorem 2.3.16 (Cartan’s criterion, [228, 473]). A Lie algebra g is solvable if and only if Bg ([X,Y ], Z) = 0 for all X,Y, Z ∈ g (in other words, Bg ([g, g], g) = 0). Theorem 2.3.17 (the semisimplicity criterion, [228, 266, 473]). A Lie algebra g is semisimple if and only if its Killing form Bg is non-degenerate. Definition 2.3.18. A Lie algebra g is called unimodular if, for all X ∈ g, the equality trace(ad(X)) = 0 holds. A Lie group G is called unimodular if it admits a biinvariant Haar measure. It should be noted that for a connected Lie group G the property of being unimodular is equivalent to the same property for its Lie algebra g. In particular, all compact, all semisimple, and all connected nilpotent Lie groups are unimodular [256, Proposition 1.4 in Chapter 10].

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2.3.3 Automorphisms and Derivations As we have already mentioned, the group Aut(g) of all automorphisms of a Lie algebra g is a closed subgroup of the Lie group GL(g) and, therefore, is a Lie group itself. Proposition 2.3.19. 1) The set Der = Der(g) of all derivations of a Lie algebra g, i.e. linear maps D ∈ gl(g) such that D[X,Y ] = [D(X),Y ] + [X, D(Y )]

for all

X,Y ∈ g,

(2.9)

is a subalgebra of the Lie algebra gl(g). 2) For every D ∈ Der and X ∈ g, we have [D, ad X] = ad(D(X)).

(2.10)

The set Dg := {ad(X) : X ∈ g} is an ideal of the Lie algebra Der. 3) The Lie algebra aut(g) of the Lie group Aut(g) coincides with Der. Proof. 1) This is directly verified using the equality [D1 , D2 ] = D1 ◦ D2 − D2 ◦ D1 for D1 , D2 ∈ Der ⊂ gl(g). 2) First, note that the Jacobi identity for the Lie algebra g is equivalent to the fact that all elements of the type ad X, X ∈ g, are derivations of the Lie algebra g. Formula (2.10) is verified by direct calculations. The second assertion is a corollary of this formula. 3) Let γ(t), t ∈ R, be an arbitrary 1-parameter subgroup in GL(g) and γ 0 (0) = D ∈ gl(g). The relation γ(t)([X,Y ]) = [γ(t)(X), γ(t)(Y )]

for all

X,Y ∈ g

for all

X ∈g

(2.11)

can be rewritten as follows: γ(t) ◦ ad X = ad(γ(t)X) ◦ γ(t) or γ(t) ◦ ad(·) ◦ γ(−t) = ad ◦γ(t),

(2.12)

but the equality (2.10) is equivalent to the next one: D ◦ ad(·) + ad(·) ◦ (−D) = ad ◦D.

(2.13)

Now, if D ∈ aut(g), then differentiating (2.11), we get that D([X,Y ]) = [D(X),Y ] + [X, D(Y )], i.e. D ∈ Der. If D ∈ Der and we consider the matrix exponent γ(t) = exp(tD), t ∈ R, then the relation (2.13) holds, hence, (2.12) holds, which is equivalent to (2.11). t u

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Definition 2.3.20. A derivation D of a Lie algebra g is called inner if D = ad X for some X ∈ g. The previous proposition implies that the set of all inner derivations Dg constitutes an ideal in the Lie algebra of all derivations Der(g) of a given Lie algebra g. It should be noted that, for some Lie algebras, all derivations are exhausted by inner derivations (for instance, this is the case for semisimple Lie algebras), while other Lie algebras (for instance, nilpotent) have derivations that are not inner.

2.4 Lie Groups with Left-invariant Riemannian Metrics In this section, we apply the formulas obtained in Section 4.14 for the computation of sectional curvature, Ricci curvature, and scalar curvature of homogeneous Riemannian manifolds for the partial case of left-invariant Riemannian metrics on Lie groups. In particular, we consider nilpotent Lie groups and Lie groups with biinvariant Riemannian metrics. The latter case is especially important for elucidating the structure of compact Lie groups and Lie algebras in Section 2.5.

2.4.1 The Levi-Civita Connection on a Lie Group with a Left-invariant Riemannian Metric Definition 2.4.1. A Riemannian metric (metric tensor) g on a Lie group G is called left-invariant if for any element h ∈ G, the left translation lh is an isometry of the space (G, g). It is clear that such a metric g is determined by an orthonormal basis X1 , . . . , Xn in (Ge , g(e)), or, equivalently, in the Lie algebra LG of all left-invariant vector fields on G. Then any smooth vector field X on G is uniquely represented in the form X = ξ i Xi with smooth component function ξ i . Moreover, X is left-invariant if and only if all functions ξ i are constant. For all left-invariant vector fields X,Y, Z on G, Formula (1.25) implies (we use the notation g(X,Y ) := hX,Y i) h2∇X Y, Zi = XhY, Zi +Y hZ, Xi − ZhX,Y i +h[X,Y ], Zi + h[Z, X],Y i − h[Y, Z], Xi = h[X,Y ], Zi + h[Z, X],Y i − h[Y, Z], Xi,

where

1 h∇X Y, Zi = h[X,Y ], Zi + hU(X,Y ), Zi, 2

(2.14)

1 hU(X,Y ), Zi = (h[Z, X],Y i + hX, [Z,Y ]i) 2

(2.15)

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for any Z ∈ LG. It is clear that U(X,Y ) is well-defined, bilinear and symmetric. Since Z could be an arbitrary left-invariant vector field, then by Lemma 1.2.10, we get 1 ∇X Y = [X,Y ] +U(X,Y ). (2.16) 2

2.4.2 Curvatures of a Lie Group with a Left-invariant Riemannian Metric In Theorems 4.14.1, 4.14.2, and 4.14.5, the formulas for computations of the sectional curvature, the Ricci curvature, and the scalar curvature (respectively) of a homogeneous Riemannian space are derived. By a homogeneous Riemannian space, we mean a homogeneous space M = G/H of a connected Lie group G by its compact subgroup H, supplied with a Riemannian metric g that is G-invariant with respect to the canonical left action of the Lie group G on G/H. These formulas are given in terms of an Ad H-invariant direct sum decomposition Ge = g = h⊕p and an Ad H-invariant inner product (·, ·) on p such that the canonical identification of Euclidean vector spaces (p, (·, ·)) and (MH , g(H)) is an isometry. Moreover, these formulas contain a symmetric bilinear form U : p × p → p with subscripts g, h, and p that indicate the projections to the corresponding spaces. Obviously, in the case of Lie groups, we have H = {e}, G/H = G, h = {0}, p = g, and (·, ·) = h·, ·i. Note that any right-invariant vector field on G is a Killing vector field on (G, h·, ·i). Moreover, the comparison of Formulas (2.15) and (4.31) shows that the meaning of the symbol U coincides in these formulas under the identification of g = Ge with LG. Since the metric h·, ·i is left-invariant, it suffices to compute their curvatures only at the point e ∈ G. The above formulas, (2.15), and Theorem 2.1.29 imply Corollary 2.4.2. The formulas for the sectional curvature, the Ricci curvature, and the scalar curvature of a left-invariant Riemannian metric on a Lie group G at the point e ∈ G are preserved if we replace left-invariant vector fields with rightinvariant vector fields with the same values at the point e. Hence, in order to specify the mentioned formulas, we should remove all terms that contain the subscript h, remove the subscripts g and p, and replace (·, ·) with h·, ·i. As a result, we get the following formulas. hR(X,Y )Y, Xi = hU(X,Y ),U(X,Y )i − hU(Y,Y ),U(X, X)i 3 1 1 − h[X,Y ], [X,Y ]i − h[X, [X,Y ]],Y i − hX, [Y, [Y, X]]i. 4 2 2 If X and Y are two left-invariant orthogonal unit vector fields on G, then the above expression is constant and gives the sectional curvature K(X,Y ). The Ricci curvature Ric in the direction of a left-invariant vector field X is determined by the formula

2.4 Lie Groups with Left-invariant Riemannian Metrics

93

1 1 Ric(X, X) = − B(X, X) − ∑ |[X, Xi ]|2 2 2 i +

1 h[Xi , X j ], Xi2 − h[Z, X], Xi, 4∑ i, j

(2.17)

where B is the Killing form of the Lie algebra g, and Z is a left-invariant vector field on G, defined by the formula Z = ∑ U(Xi , Xi ).

(2.18)

i

Remark 2.4.3. It is clear from (2.15) that hZ, Xi = trace(ad(X)) for all X ∈ g. This equality completely determines the field Z, hence, it does not depend on the choice of basis {Xi }. Therefore, Z = 0 if and only if the Lie algebra g is unimodular. Note also that trace(ad(Z)) = hZ, Zi. The scalar curvature sc is determined by the formula sc = −

1 1 |[Xi , X j ]|2 − ∑ B(Xi , Xi ) − |Z|2 . 4∑ 2 i i, j

(2.19)

2.4.3 Curvature of Nilpotent Group-manifolds Now, we specify the formulas for the Ricci curvature and scalar curvature from the previous subsection for nilpotent Lie groups with left-invariant Riemannian metrics. Proposition 2.4.4. Let (G, g) be a Lie group supplied with a left-invariant Riemannian metric, and suppose its Lie algebra g = LG is k-step nilpotent. Then (g = LG, g(e)) admits the following direct orthogonal decomposition into a sum of vector subspaces: g = V0 ⊕ · · · ⊕Vk−1 ,

where

Vm = gm gm+1 .

(2.20)

Proposition 2.4.5. If (g, [·, ·]) is a finite-dimensional nilpotent Lie algebra, then trace ad X = 0,

trace ad2 X = 0

(2.21)

for every element X ∈ g. Proof. We can choose an orthonormal basis X1 , . . . , Xn , adapted to decomposition (2.20). Now, it is clear that for any element X ∈ g, the matrices of linear maps ad X and ad2 X in this basis are upper triangular with zeros on the main diagonal. Hence, we get (2.21). t u Proposition 2.4.5, Remark 2.4.3, and the equalities (2.17), (2.19) imply the following formulas

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Ric(X, X) = −

1 1 |[X, Xi ]|2 + ∑h[Xi , X j ], Xi2 , 2∑ 4 i, j i

sc = −

1 |[Xi , X j ]|2 . 4 i,∑ j=1

(2.22)

(2.23)

The obtained formulas imply Corollary 2.4.6. Let (G, g) be a (connected) nilpotent non-commutative Lie group with a left-invariant Riemannian metric. Then its scalar curvature sc is non-positive. Moreover, sc = 0 if and only if G is commutative. In the latter case, all sectional curvatures are zero. In turn, this implies Corollary 2.4.7. Let (G, g) be a (connected) nilpotent Lie group with a left-invariant Riemannian metric. Then there are unit orthogonal vectors X,Y ∈ (LG, h·, ·i) such that Ric(X, X) < 0 and K(X,Y ) < 0. It is interesting that the dual assertion is also true. Proposition 2.4.8. Let (G, g) be a connected k-step nilpotent Lie group, k > 1, with a left-invariant Riemannian metric. Then Ric(X, X) > 0 for every unit vector X ∈ (LG)k−1 . As a corollary, there is a unit vector Y ∈ (LG, h·, ·i), orthogonal to X, such that K(X,Y ) > 0. Proof. It is clear that [Xi , X] ∈ LGk = {0} for all vectors Xi , hence, (2.22) takes the form 1 n Ric(X, X) = ∑ h[Xi , X j ], Xi2 . 4 i, j=1 Obviously, this number is positive. Now, the second assertion is also clear.

t u

2.4.4 Curvatures of Bi-invariant Riemannian Metrics on Lie Groups Definition 2.4.9. Let G be a connected Lie group and let K be its connected Lie subgroup with a Lie algebra k ⊂ g = LG. A left-invariant Riemannian metric on the Lie group G is called K-right-invariant if it is invariant under right translations by elements of the subgroup K; in the case K = G such a metric is called bi-invariant. Proposition 2.4.10. A Riemannian metric g = h·, ·i on a connected Lie group G satisfies the conditions of Definition 2.4.9 if and only if hU(X,Y ), ki = 0 for all X,Y ∈ g, i.e. the operator ad Z : g → g is skew-symmetric for all Z ∈ k. At the same time, U(X,Y ) = 0 for all X,Y ∈ k.

2.4 Lie Groups with Left-invariant Riemannian Metrics

95

Proof. It is easy to see that the transformations Ψt , t ∈ R, defined by an arbitrary element Z ∈ k, consist of right translations by the element of the group K. Hence, the metric h·, ·i on the group G is K-right-invariant with respect to a connected Lie subgroup K if and only if every such field Z is a Killing vector field on (G, h·, ·i). Hence, by Theorem 3.1.4, the left-invariant metric h·, ·i on G is a K-right-invariant if and only if 1 1 hU(X,Y ), Zi = (h[Z, X],Y i + hX, [Z,Y ]i) = ZhX,Y i = 0 2 2

(2.24)

for all X,Y ∈ LG, Z ∈ k. Consider g = k ⊕ m, a h·, ·i-orthogonal decomposition into a direct sum of vector spaces. Putting X ∈ m and Y, Z ∈ k in Formula (2.24), we see that [k, m] ⊂ m, since [k, k] ⊂ k. Therefore, putting Z ∈ m and X,Y ∈ k in Formula (2.24), we get hU(X,Y ), Zi = 0, hence, U(X,Y ) = 0. t u From Proposition 2.4.10 and Equality (2.16), we get Corollary 2.4.11. Under the conditions of Definition 2.4.9, 1 ∇X Y = [X,Y ] 2

(2.25)

for all vector fields X,Y ∈ k ⊂ LG. This corollary and the Hopf–Rinow theorem 1.5.1 imply Corollary 2.4.12. For any left-invariant vector field X on a Lie group with a biinvariant Riemannian metric ∇X X = 0. Geodesics of this metric are 1-parameter subgroups and their left and right translations. If the group G is connected, then the map exp(= Expe ) : g → G is surjective. Theorem 2.4.13. For any pair of unit and orthogonal vector fields X,Y ∈ k on a Lie group G with a left-invariant and K-right-invariant Riemannian metric, the sectional curvature is calculated by the formula 1 K(X,Y ) = |[X,Y ]|2 . 4

(2.26)

Theorem 2.4.14. Let X = X1 , . . . , Xn be an orthonormal basis of left-invariant vector fields on a connected Lie group G with a bi-invariant Riemannian metric h·, ·i. Then 1 1 Ric(X, X) = − B(X, X) = ∑h[X, Xi ], [X, Xi ]i (2.27) 4 4 i for every X ∈ g. Moreover, Ric(X, X) = 0 if and only if X is in the center of the Lie algebra (LG, [·, ·]). For the bi-invariant metric h·, ·i, the Lie algebra (LG, [·, ·]) has trivial center if and only if the Ricci curvature of this metric is positive. At the same time, the Lie group G is compact and has a finite fundamental group.

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Proof. The first equality in (2.27) is a direct corollary of Theorem 4.14.6; the second equality follows from the first one, the definition of the Killing form, and Proposition 2.4.10. The obtained results and the Myers theorem 1.5.79 imply all other assertions. t u Theorem 2.4.15 (the H. Weyl theorem). If the Killing form of the Lie algebra of a connected Lie group G is negative definite, then G is compact and has finite center e is and finite fundamental group. As a corollary, its universal covering Lie group G compact. Proof. Obviously, an inner product on Ge extends to a bi-invariant Riemannian metric on the Lie group G if and only it is invariant with respect to all automorphisms Ad(g), g ∈ G, of the Lie algebra (Ge , [·, ·]). Then, according to Propositions 2.3.15, 2.4.10, Theorem 2.1.27, and the conditions of this theorem, the inner product −B extends to a bi-invariant Riemannian metric on the Lie group G. Theorem 2.4.14 implies the first assertion, which, in turn, implies the second one. t u Theorem 2.4.14 directly implies Corollary 2.4.16. The scalar curvature of a bi-invariant Riemannian metric h·, ·i on a connected Lie group G is calculated in an arbitrary orthonormal basis X1 , . . . , Xn of left-invariant vector fields on (G, h·, ·i) as follows: sc =

1 4

n

∑

|[X j , Xk ]|2 .

j,k=1

It is non-negative, and it is zero if and only if the Lie group G is commutative. Let us consider one more corollary from Theorem 2.4.14. Corollary 2.4.17. Let G be a semisimple (not necessarily compact) Lie group, K its compact subgroup, and k ⊂ g the corresponding Lie algebras. Then the restriction to k of the Killing form Bg of the Lie algebra g is negative definite. Proof. Let us take any inner product (·, ·) on g = Ge and extend it to a left-invariant Riemannian metric on G, preserving the notation. Since K is compact, there is a R bi-invariant Haar measure dk on K with the condition K 1dk = 1. It is clear that the formula hu, vi :=

Z K

(Trk (u), Trk (v))dk,

where

u, v ∈ Gx , x ∈ G,

defines a left-invariant and K-right-invariant Riemannian metric on the Lie group G. Let X = X1 , . . . , Xn be a h·, ·i-orthonormal basis in the Lie algebra g. Since the center of the Lie algebra g is trivial, by Proposition 2.4.10 and the definition of the Killing form we get that Bg (X, X) = ∑i h[X, [X, Xi ], Xi i = − ∑i h[X, Xi ], [X, Xi ]i < 0 for all non-trivial X ∈ k. t u

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2.5 Compact Lie Algebras and Lie Groups In this section we recall important results on compact Lie groups and compact Lie algebras. For more details see e.g. [6, 7, 121, 227, 228, 257, 409].

2.5.1 Compact Lie Algebras Definition 2.5.1. A Lie algebra g is called compact if there is a compact Lie group G with the Lie algebra isomorphic to g. Definition 2.5.2. A bilinear symmetric form h·, ·i on a Lie algebra g is called invariant if h[Z, X],Y i + hX, [Z,Y ]i = 0 for all X,Y, Z ∈ g. Further in this subsection, we consider Lie algebras g with a given invariant (positive definite) inner product h·, ·i. For any Lie subalgebra f in g, f⊥ denotes its orthogonal complement in g. Proposition 2.5.3. The orthogonal complement to an ideal in a Lie algebra is an ideal itself. Proof. Let f be an ideal in the Lie algebra g. Then, according to Definition 2.5.2, for all elements X ∈ g, Y ∈ f, and Z ∈ f⊥ , we get h[X, Z],Y i = −hZ, [X,Y ]i = 0, i.e. [X, Z] ∈ f⊥ and f⊥ is an ideal of the Lie algebra g.

t u

Proposition 2.5.4. The center C(g) of a Lie algebra g is its maximal commutative ideal. Proof. By Proposition 2.3.5, C(g) is an ideal of the Lie algebra g. Let f be an arbitrary commutative ideal in g. Then g = f ⊕ f⊥ and, according to Proposition 2.5.3, we have [X, Z] ∈ f ∩ f⊥ = {0} for all X,Y ∈ f and Z ∈ f⊥ . Moreover, [X,Y ] = 0 since the ideal f is commutative. Therefore, X ∈ C(g) and f ⊂ C(g). t u Theorem 2.5.5. Every Lie algebra g, supplied with an (arbitrary) invariant inner product, is represented as a direct orthogonal sum of ideals g = C(g) ⊕ l, where C(g) is the center of the Lie algebra g, and l is a semisimple Lie algebra. Moreover, l is represented as a direct orthogonal sum of ideals, that are simple Lie algebras. Proof. Let us consider l := C(g)⊥ . Then l is an ideal of the Lie algebra g according to Propositions 2.3.5, 2.5.3 and g = C(g) ⊕ l. Moreover, by Proposition 2.5.4, the ideal l is a semisimple Lie algebra. The last assertion is proved by induction on the dimension of the Lie algebra l by Proposition 2.5.3. t u

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Proposition 2.5.6. Every compact Lie algebra admits an invariant inner product. Proof. Let g be a compact Lie algebra. By Definition 2.5.1, there is a compact connected Lie group G with the Lie algebra g. Let us take any inner product (·, ·) on g = Ge and extend it to a left-invariant Riemannian metric on G, preserving the notation. Since G is compact, there is a biinvariant Haar measure dg on G with the R condition G 1dg = 1. It is clear that the formula hu, vi :=

Z G

(Trg (u), Trg (v))dg,

where

u, v ∈ Gx , x ∈ G,

defines a bi-invariant Riemannian metric on G. By Proposition 2.4.10 and (2.15), its restriction to g = Ge is an invariant inner product on g. t u Lemma 2.5.7. For any semisimple Lie algebra g with an invariant inner product h·, ·i, every derivation D of g satisfying the condition hD(X),Y i + hX, D(Y )i = 0

(2.28)

for all X,Y ∈ g is inner. Proof. Since the Lie algebra g is semisimple, the Lie algebra homomorphism ad g → Dg (see item 2) of Proposition 2.3.19) is an isomorphism. Let h·, ·i be an invariant inner product on g. Definition 2.5.2 implies that Dg ⊂ so(g, h·, ·i) automatically. According to Proposition 2.3.19, Dg is an ideal in the Lie algebra aut(g) of the Lie group Aut(g). Consider A = Aut(g) ∩ SO(g, h·, ·i). Then Dg is an ideal in the Lie algebra a of the Lie group A. It is clear that D ∈ a, if D satisfies the conditions of the lemma. Since A is a compact Lie group, then, according to Proposition 2.5.6, there is an invariant inner product (·, ·) on a. Let us suppose that Dg 6= a. Then by Proposition 2.5.3, the (·, ·)-orthogonal complement b := Dg⊥ is non-trivial and also is an ideal in a. Take any non-zero elements X ∈ g and D ∈ b. Then [D, ad X] ∈ Dg ∩ b = {0}. On the other hand, [D, ad X] = ad(D(X)) due to (2.10). Since the kernel of the homomorphism ad on g is trivial, D = 0. t u Theorem 2.5.8. A Lie algebra g is compact if and only if there is an invariant inner product on g. Proof. The necessity is proved in Proposition 2.5.6. For the sufficiency: Let g be a Lie algebra with an invariant inner product h·, ·i. By Theorem 2.5.5, there is an orthogonal decomposition g = C(g) ⊕ l into a direct sum of ideals, where C(g) is the center and l is a semisimple Lie algebra. The proof of Lemma 2.5.7 implies that the Lie algebra l is isomorphic to the Lie algebra of the compact Lie group A = Aut(l) ∩ SO(l, h·, ·il ). The Lie algebra C(g) is isomorphic to the Lie algebra of the torus T m , where m = dim(C(g)). Then, the Lie algebra g is isomorphic to the Lie algebra of the compact Lie group T m × A. t u

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From Theorem 2.5.8 and Definition 2.5.2, we get Corollary 2.5.9. Any Lie subalgebra of a compact Lie algebra is a compact Lie algebra itself. Proposition 2.5.10. Any solvable Lie subalgebra of a compact Lie algebra is commutative. Proof. Let h be a solvable Lie subalgebra of a compact Lie algebra g. By Corollary 2.5.9, h is a compact Lie algebra. Let us suppose that h is not commutative. Then by Theorems 2.5.8 and 2.5.5, h is represented as the direct sum of ideals h = C(g) ⊕ l1 ⊕ · · · ⊕ lk , where C(g) is the center of h and l1 , . . . , lk (k ≥ 1) are simple Lie algebras. Since [g, g] is an ideal of a given Lie algebra g, then by the simplicity of subalgebras l1 , . . . , lk , we get that [h, h] = [l1 , l1 ] ⊕ · · · ⊕ [lk , lk ] = l1 ⊕ · · · ⊕ lk . It follows that h is not solvable Lie algebra, a contradiction.

t u

Theorem 2.5.11. Let G be a connected Lie group. Then the following conditions are equivalent: 1) The Killing form of the Lie algebra g of the Lie group G is negative definite. 2) G is compact and semisimple. 3) G is compact and has finite center. 4) G is compact and has finite fundamental group. Proof. If the Killing form B of the Lie algebra g is negative definite, then (·, ·) = −B(·, ·) is an invariant inner product on g by Proposition 2.3.15. Note that g has trivial center, since B is zero on the center. Then according to Theorem 2.5.5, g is semisimple, hence G is semisimple too. Moreover, G is compact and has finite center and finite fundamental group by the H. Weyl theorem 2.4.15. Hence, 1) implies 2), 3), and 4). The implication 2) ⇒ 1) follows from Corollary 2.4.17, where one needs to take K = G. Now, suppose that G is compact, hence g is also compact and by Theorem 2.5.8, there is an invariant inner product (·, ·) on g. Moreover, by Theorem 2.5.5, there is an orthogonal decomposition g = C(g) ⊕ l into a direct sum of ideals, where C(g) is the center and l is a semisimple Lie algebra. Let X = X1 , . . . , Xn be an (·, ·)-orthonormal basis in g, then (using the fact that (·, ·) is invariant) we get B(X, X) = ∑([X, [X, Xi ], Xi ) = − ∑([X, Xi ], [X, Xi ]) ≤ 0 i

i

for every X ∈ g. Moreover, B(X, X) = 0 if and only if X is in the center of g. If G has finite center, then the center C(g) of g is trivial. Indeed, for any U ∈ C(g), the 1-parameter subgroup exp(tU), t ∈ R, is in the center of G. Therefore, B is negative definite and 3) ⇒ 1).

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If G has finite fundamental group, then the center C(g) of g is trivial. Otherwise, G is covered by the Lie group Rm × L, where m = dim C(g) > 0 and L is a simply connected Lie group with the Lie algebra l. Since G is compact, the fundamental group of G should be infinite, which is impossible. Therefore, C(g) is trivial, B is negative definite, and 4) ⇒ 1). t u Proposition 2.3.15 and Theorems 2.4.15, 2.5.8, 2.5.11 imply Theorem 2.5.12. A Lie algebra g is compact and semisimple if and only if its Killing form is negative definite. Theorem 2.5.13. Every derivation of a compact semisimple Lie algebra is inner. Proof. According to Theorem 2.5.12, the Killing form Bg is negative definite on g. Propositions 2.3.15 and 2.3.19 imply that the inner product h·, ·i = −Bg is invariant on g and every derivation D of the Lie algebra g satisfies (2.28). Now, it suffices to apply Lemma 2.5.7. Theorems 2.5.5, 2.5.8, and 2.5.11 directly imply Theorem 2.5.14. Any compact Lie algebra g is the Lie algebra of a connected and simply connected Lie group of the type G = Rm × G1 × · · · × Gk , where G1 , . . . , Gk are compact connected simply connected simple Lie groups. In the rest of this subsection we provide some important properties of compact Lie algebras. The following result is important in the study of homogeneous spaces. Proposition 2.5.15. Let g be a Lie algebra and let h be its Lie subalgebra such that there is an ad(h)-invariant complement p to h in g. Then the linear subspaces u := {Z ∈ h | [Z, p] = 0} and q := p+[p, p] are ideals in g. If, in addition, g is compact and p is h·, ·i-orthogonal to h in g with respect to some ad(g)-invariant inner product h·, ·i on g, then q and u are complementary ideals in g. Proof. Since [u, h] ⊂ h and p is ad(h)-invariant, we get by the Jacobi identity [p, [u, h]] ⊂ [[p, u], h]] + [u, [p, h]] = 0, hence [u, h] ⊂ u. Since [u, p] = 0, we get that u is an ideal in g. Further, [h, p] ⊂ p ⊂ q and [p, p] ⊂ q, hence, [g, p] ⊂ q. By the Jacobi identity, we get also [h, [p, p]] ⊂ [[h, p], p] ⊂ [p, p] ⊂ q. Hence, it suffices to show that [p, [p, p]] ⊂ q. Take some X,Y, Z ∈ p, then [Y, Z] = [Y, Z]h + [Y, Z]p , where [Y, Z]h ∈ h and [Y, Z]p ∈ p. Further, [X, [Y, Z]p ] ∈ [p, p] ⊂ q and [X, [Y, Z]h ] ∈ [h, p] = p ⊂ q, hence, [X, [Y, Z]] ∈ q. Since X,Y, Z ∈ p can be chosen arbitrary, [p, [p, p]] ⊂ q and q is an ideal in g. Let us prove the second assertion. It is easy to check that for Z ∈ h the condition hZ, [p, p]i = 0 is equivalent to each of the conditions h[Z, p], pi = 0 and [Z, p] = 0. Hence, q and u are complementary ideals in g. t u

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101

Remark 2.5.16. The ideals u and q from Proposition 2.5.15 are not complementary in g for some special ad(h)-invariant complements p. For instance, if g = f ⊕ f for some semisimple compact Lie algebra f, h = diag(f) = {(X, X) ∈ g | X ∈ f}, and p = {(X, 0) ∈ g | X ∈ f}, then u is trivial and q = p. The following proposition is also very useful. Proposition 2.5.17. Let g be a compact Lie algebra, h·, ·i an ad(g)-invariant inner product on g, and h a subalgebra of g. Suppose that the h·, ·i-orthogonal complement p to h in g is of the type p = p1 ⊕ p2 ⊕ · · · ⊕ pk , where every pi is ad(h)-invariant and ad(h)-irreducible, hpi , p j i = 0 and [pi , p j ] = 0
for i 6= j. For i = 1, . . . , k, put hi := pi + [pi , pi ] ∩ h and gi := hi ⊕ pi . Then the following assertions hold: 1) gi = pi + [pi , pi ] for every 1 ≤ i ≤ k; 2) hgi , g j i = 0 for i 6= j; 3) every gi is an ideal in g; 4) every pair (gi , hi ) is effective and isotropy irreducible, i.e. ad(hi ) acts irreducibly on pi ; L 5) u, the h·, ·i-orthogonal complement to i gi in g, is an ideal in g; 6) g = u ⊕ g1 ⊕ g2 ⊕ · · · ⊕ gk , h = u ⊕ h1 ⊕ h2 ⊕ · · · ⊕ hk . Proof. First, we prove that gi = pi + [pi , pi ] for every 1 ≤ i ≤ k. Consider any j 6= i, 1 ≤ j ≤ k. For every X, Z ∈ pi and Y ∈ p j , we have h[Z, X],Y i = −hX, [Z,Y ]i = 0, since [pi , p
j ] = 0 and h·, ·i is ad(g)-invariant. Hence, h[pi , pi ], p j i = 0, which implies prp [pi , pi ] ⊂ pi and, therefore, gi = pi + [pi , pi ]. Since [pi , p j ] = 0, by the Jacobi identity we get [p j , [pi , pi ]] = 0. Therefore, h[p j , [pi , pi ]], gi = 0 and h[pi , pi ], [p j , g]i = 0 by the ad(g)-invariance of h·, ·i. In particular, h[pi , pi ], [p j , p j ]i = 0. This equality together with the equalities h[pi , pi ], p j i = 0, h[p j , p j ], pi i = 0, and hpi , p j i = 0 imply hgi , g j i = 0, since gi = pi + [pi , pi ] and g j = p j + [p j , p j ]. Since [p j , [pi , pi ]] = 0 and [p j , pi ] = 0 for i 6= j, we have [p j , gi ] = 0, 0 = [p j , [gi , p j ]] = [gi , [p j , p j ]], and [gi , g j ] = 0. Moreover, [pi , gi ] = [pi , pi +hi ] = [pi , pi ]+ [pi , hi ] ⊂ [pi , pi ]+pi = gi , [[pi , pi ], gi ] ⊂ gi , and [gi , gi ] ⊂ gi . Hence, every gi is an ideal in g. It is clear that u ⊂ h. Since hu, [pi , pi ]i = 0 for any i, we have h[pi , u], pi i = 0, which means that [u, p] = 0. It is easy to check that u = {Z ∈ h | [Z, p] = 0}. Indeed, if V ∈ h is such that [V, pi ] = 0 for every i = 1, . . . , k, then 0 = h[X,V ], pi ]i = −hV, [X, pi ]i for any X ∈ pi . This means that hV, [pi , pi ]i = 0 and, consequently, hV, gi i = 0. Since i can be arbitrary, we get V ∈ u. By Lemma 2.5.15 we get that u is an ideal in g. Now, all assertions of the lemma are clear. t u Now, we consider a special decomposition of a compact Lie algebra, related to a given inner derivation. Let us consider any compact Lie algebra g with ad(g)invariant inner product h·, ·i. Take any Z ∈ g and consider the operator LZ : g → g,

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LZ (X) = [Z, X]. This operator is skew-symmetric, but LZ2 is a symmetric operator on g with respect to h·, ·i. Put m0 = Ker LZ = {Y ∈ g | [Z,Y ] = 0} and consider all nonzero eigenvalues of the operator LZ2 : −λi2 , i = 1, . . . , s, where 0 < λ1 < λ1 < · · · < λs , and the corresponding eigenspaces mλi = {Y ∈ g | LZ2 (Y ) = [Z, [Z,Y ]] = −λi2Y }. It is clear that m0 is a Lie subalgebra of g and m := mλ1 ⊕ mλ2 ⊕ · · · ⊕ mλs = Im LZ .

(2.29)

Note that this decomposition and the decomposition g = m0 ⊕ m are h·, ·iorthogonal. This follows from the simple observation: If P ⊂ g is an invariant subspace of the operator LZ2 , then its h·, ·i-orthogonal complement P⊥ is also an invariant subspace of LZ2 . We will need the following simple Lemma 2.5.18. For any X ∈ m0 and Y ∈ mα , we have [X,Y ] ∈ mα , i.e. every mα is ad(m0 )-invariant. In particular, [Z, mα ] ⊂ mα . Proof. We have LZ ([X,Y ]) = [X, [Z,Y ]] and LZ2 ([X,Y ]) = [X, [Z, [Z,Y ]]] = −α 2 [X,Y ]. t u Now we define a linear operator σ : m → m as follows: σ (Y ) =

1 [Z,Y ], λi

Y ∈ mλi .

(2.30)

Remark 2.5.19. If G is a compact Lie group with the Lie algebra g, then the operator σ defines a complex structure on the flag manifold G/CG (Z), where CG (Z) is a centralizer of Z in the group G (see e.g. [106, Chapter 8]). For any X,Y ∈ m we also define
1 [X,Y ] − [σ (X), σ (Y )] , 2
1 − [X,Y ] : = [X,Y ] + [σ (X), σ (Y )] . 2 [X,Y ]+ : =

(2.31) (2.32)

Obviously, [X,Y ] = [X,Y ]+ + [X,Y ]− . Proposition 2.5.20. In the above notation, we have σ (mα ) ⊂ mα for all α and σ (σ (Y )) = −Y for all Y ∈ m. If U ∈ mα , V ∈ mβ , then [U,V ]+ ∈ mα+β ,

[U,V ]− ∈ m|α−β | .

Proof. The first assertion follows from lemma 2.5.18, the equality σ 2 = − Id is obvious. By this equality and the definitions we have

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LZ ([U,V ]) = [Z, [U,V ]] = [[Z,U],V ] + [U, [Z,V ]] = α[σ (U),V ] + β [U, σ (V )], LZ ([σ (U), σ (V )]) = [[Z, σ (U)], σ (V )] + [σ (U), [Z, σ (V )]] = −α[U, σ (V )] − β [σ (U),V ], LZ2 ([U,V ]) =

2

[Z, [Z, [U,V ]]] = −(α + β 2 )[U,V ] + 2αβ [σ (U), σ (V )], LZ2 ([σ (U), σ (V )]) = [Z, [Z, [σ (U), σ (V )]]]

= −(α 2 + β 2 )[σ (U), σ (V )] + 2αβ [U,V ].

Hence, we get LZ2 [U,V ] −
[σ (U), σ (V )] = −(α + β )2 [U,V ] −
[σ (U), σ (V )] and LZ2 [U,V ] + [σ (U), σ (V )] = −(α − β )2 [U,V ] + [σ (U), σ (V )] . t u

2.5.2 Cartan Subalgebras of Compact Lie Algebras Definition 2.5.21. A nilpotent Lie subalgebra t of a Lie algebra g is called a Cartan subalgebra if ng (t) = t (i.e. it coincides with its normalizer in g). Theorem 2.5.22. Cartan subalgebras of a compact Lie algebra g are exactly the maximal commutative subalgebras in g. Cartan subalgebras of the Lie algebra g of a compact Lie group G coincide with Lie algebras of maximal tori in G. Moreover, the adjoint representation Ad(G) (of a connected Lie group G) acts transitively on the set of all Cartan subalgebras in g. Proof. According to Definition 2.5.21 and Proposition 2.5.10, every Cartan subalgebra t ⊂ g is commutative. Hence, if h is any commutative subalgebra in g that contains t, then t ⊂ h ⊂ ng (t) = t. Therefore, t is a maximal commutative subalgebra in g. Now, suppose that t is a maximal commutative subalgebra in g, but t 6= ng (t). Then, by Proposition 2.3.7, t is a maximal commutative ideal in ng (t). Hence, for any element X ∈ ng (t) that is not in t, Lin(t ∪ {X}) is solvable, and hence a commutative Lie subalgebra in g by Proposition 2.5.10, which contradicts the maximality of t with respect to this property. The first assertion is proved. Let us prove the second assertion. Let t be a Cartan subalgebra in g, i.e. (according to the proved result) a maximal commutative subalgebra in g. By Proposition 2.1.30, for all numbers t, s ∈ R and all vectors X,Y ∈ t, the elements exp(tX), exp(sY ) ∈ G commute. If T is a minimal closed subgroup in G, containing all elements of such type, then it is connected, compact, and commutative. By Cartan’s theorem 2.1.32, the group T ⊂ G is a Lie group, hence, a torus. Let T 0 ⊃ T be a maximal torus in the group G with the (commutative!) Lie algebra t0 . By the construction, t ⊂ t0 , and t = t0 due to the maximality of t. Conversely, let T be a maximal torus in G with the (commutative) Lie algebra t. Then t is a maximal commutative subalgebra in g. Otherwise, using the just applied arguments, we may find another torus T 0 ⊃ T , where T 0 6= T . Now, let t be a Cartan subalgebra in g. Let us prove that there is a vector V in t with cg (V ) = t, where cg (V ) = {X ∈ g | [X,V ] = 0} is the centralizer of V in g. It is clear that there is a V ∈ t such that the 1-parameter group exp(sV ), s ∈ R, is dense

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in T . For any U ∈ cg (V ) and any s ∈ R, we get ad(sV )(U) = [sV,U] = 0, hence Ad(exp(sV ))(U) = exp(ad(sV ))(U) = U by Corollary 2.1.28. Recall that exp(sV ), s ∈ R, is dense in T , hence, Ad(a)(U) = U for any a ∈ T and, therefore, [t,U] = 0. Since t is a maximal commutative subalgebra in g, we have U ∈ t, hence, cg (V ) = t. Now consider two maximal tori T1 and T2 in the compact Lie group G with the vectors X1 and X2 in their Lie subalgebras (Cartan subalgebras) t1 and t2 , such that cg (X1 ) = t1 and cg (X2 ) = t2 , i.e. ti is the centralizer of Xi in g for i = 1, 2. Without loss of generality, we may suppose that dim t2 ≤ dim t1 (we do not know a priori that these dimensions coincide). Since the Lie group G is compact, its Lie algebra g admits an Ad(G)-invariant (invariant, in other terms) inner product (·, ·) (cf. the proof of Proposition 2.5.6). Now, let us consider a continuous function f (g) = (X1 , Ad(g)(X2 )) on G. Since the group G is compact, this function has a point g0 of absolute minimum. This means that, for any Z ∈ g, the function fe(t) = f (g0 exp(tZ)) attains the minimal value at the point t = 0. Consequently,   0 = fe0 (0) = X1 , Ad(g0 )([Z, X2 ]) −1 = (Ad(g−1 0 )X1 , [Z, X2 ]) = −([Ad(g0 )X1 , X2 ], Z),

hence, [Ad(g−1 0 )(X1 ), X2 ] = [X1 , Ad(g0 )(X2 )] = 0. The obtained equality means that Ad(g0 )(X2 ) ∈ t1 . Since Ad(g0 ) is an automorphism of g, we get t1 ⊂ cg (Ad(g0 )(X2 )) = Ad(g0 )(cg (X2 )) = Ad(g0 )(t2 ). Due to the assumption dim t2 ≤ dim t1 , this implies dim t2 = dim t1 and t1 = cg (Ad(g0 )(X2 )) = Ad(g0 )(t2 ). t u Theorem 2.5.23. Let G be a connected compact Lie group. Then every two maximal tori in G are conjugate by an inner automorphism of the Lie group G and any element of G is contained in a maximal torus of this group. Moreover, any maximal torus in G is a maximal abelian subgroup in G, and the centralizer of any torus T ⊂ G is a union of all maximal tori containing T (in particular, this centralizer is connected). Proof. The first assertion follows from Theorem 2.5.22 and the second one follows from Proposition 2.5.6 and Corollary 2.4.12. The third assertion has been proved in Proposition 4.26 in [6], and the fourth one is its corollary. t u According to Theorems 2.5.22 and 2.5.23 the following definition makes sense. Definition 2.5.24. The dimension of a maximal commutative subalgebra of a compact Lie algebra g (respectively, of a maximal torus of a compact Lie group G) is called the rank of the Lie algebra g (of the compact Lie group G) and is denoted by rk g (rk G). Definition 2.5.25. Let g be a Lie algebra. A vector V ∈ g is called regular if its centralizer cg (V ) = {X ∈ g | [X,V ] = 0} has minimal possible dimension among all vectors in g.

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Proposition 2.5.26. A vector V in a compact Lie algebra g is regular if and only if its centralizer cg (V ) in g is a Cartan subalgebra. The set of all regular vectors in g is open and everywhere dense in g. Proof. Consider any V ∈ g and its centralizer cg (V ) = {X ∈ g | [X,V ] = 0} in g. It is clear that V lies in some maximal commutative Lie algebra (i.e. Cartan subalgebra due to Theorem 2.5.22) t ⊂ g. Obviously, t ⊂ cg (V ), hence, dim cg (V ) ≥ dim t = rk g. By the proof of Theorem 2.5.22, for any Cartan subalgebra t in g, there is a vector V in t with cg (V ) = t. Therefore, the minimal value of dim(cg (V )) is exactly rk g, and it is attained only when cg (V ) is a Cartan subalgebra. Let us consider any basis X1 , X2 , . . . , Xn in g, n = dim g. For a given V ∈ g, the centralizer cg (V ) is determined by the equation [X,V ] = 0. If [Xk , Xi ] = ∑nj=1 Ckij X j for all k and i, then this equation is equivalent to the following equations: ! n

n

i=1

k=1

∑ ∑ Ckij vk

xi = 0,

j = 1, . . . , n,

where the variables xi are taken from the decomposition X = ∑ni=1 xi Xi and the constants vi are taken from V = ∑ni=1 vi Xi . Let us consider an (n × n)-matrix M(V ) = M = (mi j ) with the entries mi j = ∑nk=1 Ckij vk . The characteristic polynoi mial of this matrix has the form χ(M(V )) = λ n + ∑n−1 i=r di (V )λ , where r = rk g, di (V ) are some polynomials in coordinates of the vector V . Note that dr (V ) 6= 0 if and only if the vector V is regular. Since dr (V ) is a polynomial in coordinates of the vector V and dr (V ) 6= 0 for some vectors V , then the set {V ∈ g | dr (V ) 6= 0} is open and everywhere dense in g (a polynomial, vanishing on a open set, is a zero polynomial). t u A detailed description of regular vectors in compact Lie algebras can be obtained in terms of root systems (cf. Subsection 2.5.3 below and Subsection 7.2.1 in [409]). Theorem 2.5.22 directly implies Proposition 2.5.27. Let g = C(g) ⊕ g1 ⊕ · · · ⊕ gk be a representation of a compact Lie algebra g as a direct sum of its center and simple Lie algebras from Theorem 2.5.5. Then every Cartan subalgebra t ⊂ g is represented as a direct sum t = C(g) ⊕ t1 ⊕ · · · ⊕ tk , where ti ⊂ gi , i = 1, . . . k, are Cartan subalgebras in gi .

2.5.3 Root Systems and the Weyl Groups Definition 2.5.28. The integer lattice of a torus T with the Lie algebra t is the set exp−1 (e), where exp : t → T is the exponential map. Let g be the Lie algebra of a compact Lie group G, supplied with a bi-invariant Riemannian metric h·, ·i, and let T be a maximal torus in G. In what follows, we will need the following proposition (cf. Proposition 4.12 in [6]).

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Proposition 2.5.29. A compact Lie algebra (g, h·, ·i) splits into a direct orthogonal sum of Ad(T )-invariant vector subspaces of the form g = t ⊕ ∑m i=1 vi , where Ad(T ) acts trivially on t, dim vi = 2 for i ≥ 1, and the action Ad(T ) on vi (according to a suitable orthonormal basis {ei,1 , ei,2 }) is determined by the matrix   cos(2πθi (t)) − sin(2πθi (t)) . sin(2πθi (t)) cos(2πθi (t)) Here, θi (t) := θi (exp−1 (t)), t ∈ T , is correctly defined by the linear form θi : t → R, which takes integral values on the integer lattice. Moreover (by Theorem 5.5 in [6]), the forms θi and θ j are linearly independent for i 6= j. Definition 2.5.30. The forms ±θi : t → R are called the roots of the Lie group G. Remark 2.5.31. According to Theorem 3.24 in [6] and Theorem 2.5.23, the linear forms ±θi : t → R are uniquely determined by the Lie group G. If the Lie group G is simply connected (hence, semisimple), then these forms are usually called the roots of the Lie algebra g. The roots and the system of all roots are denoted by small Greek letters and by ∆ = ∆ (g) respectively. The restriction of the inner product h·, ·i to t allows us to identify the roots with some elements of t in the standard way. Then α ⊥ Uα := ker α for every root α ∈ ∆ (g). Definition 2.5.32. Let T be a maximal torus of a connected compact Lie group G with the Lie algebra t. The Weyl group W of the Lie group G relative to the given maximal torus T (respectively, of the Lie algebra t) is a group of automorphisms of the torus T (the Lie algebra t) that are (respectively, are induced by) restrictions of inner automorphisms of the group G. Proposition 2.5.33. The Weyl group W of a connected compact Lie group G (relative to the given maximal torus T ) as well as the Weyl group of the Cartan subalgebra t ⊂ g are isomorphic to the finite group NG (T )/T , where NG (T ) is the normalizer of the group T in G. Moreover, the group W , up to automorphism, does not depend on the choice of a compact connected Lie group G with a given compact Lie algebra g. Proof. It suffices to prove the proposition for a Lie group G. Every automorphism in W has the form t → ntn−1 , where n ∈ NG (T ). Therefore, one can correctly define a corresponding epimorphism f : NG (T ) → W with the kernel coinciding with the centralizer ZG (T ). The subgroup NG (T ) is closed in G, hence, is compact. By Theorem 2.5.23, we have ZG (T ) = T , hence, W ∼ = NG (T )/T and W is compact. It is clear that every automorphism of the torus T is covered by a linear transformation of its universal covering vector Lie group t, preserving the integer lattice exp−1 (e). Therefore, Aut(T ) ⊂ GL(t), and, consequently, its subgroup W is discrete. Being compact and discrete, the group W is finite. It is clear that the group W acts trivially on the center Z(G) ⊂ T of the group G. Therefore, the action of the group W on T is completely determined by its induced action on T /Z(G), i.e. by the action of the Weyl group of the semisimple compact connected Lie group G/Z(G) with the

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107

trivial center on its maximal torus T /Z(G) (cf. Theorem 2.5.14 in connection with this). t u Corollary 2.5.34. Let G be a connected compact Lie group, and T be a maximal torus in G. Suppose that for a1 , a2 ∈ T , there is a g ∈ G such that ga2 g−1 = a1 . Then there is a w ∈ W with w(a2 ) = a1 , where W is the Weyl group of G. Proof. Let us consider H = {a ∈ G | aa1 a−1 = a1 }, the centralizer of a1 in G, and its identity component H0 . By Theorem 2.1.32, H and H0 are Lie subgroups in G. Obviously, T ⊂ H0 , and gT g−1 is also a maximal torus. Since a1 ∈ T ∩ gT g−1 , we have gT g−1 ⊂ H and, moreover, gT g−1 ⊂ H0 (T ∪ gT g−1 is a connected set containing the unit e of G). By Theorem 2.5.23, T and gT g−1 are conjugate in H0 , hence, there is a b ∈ H0 such that b(gT g−1 )b−1 = bgT (bg)−1 = T . Therefore, the conjugation by the element c := bg preserves T , moreover, ca2 c−1 = bga2 (bg)−1 = b(a1 )b−1 = a1 , since b ∈ H0 ⊂ H. Hence, we have found a suitable element w = c of the Weyl group. Corollary 2.5.35. Let G be a connected compact Lie group, g its Lie algebra, and t a Cartan subalgebra in g. Suppose that for V1 ,V2 ∈ t, there is a g ∈ G such that Ad(g)(V2 ) = V1 . Then there is a w ∈ W with w(V2 ) = V1 , where W is the Weyl group of t. Proof. Let us consider exp T : t → T . There is a neighborhood U of zero in t such that exp T is a diffeomorphism on U. There is a small number s > 0 such that sV1 , sV2 ∈ U. Then Ad(g)(sV2 ) = sV1 and by Corollary 2.1.28, ga2 g−1 = a1 , where ai = expT (sVi ), i = 1, 2. By Corollary 2.5.34, there exists an element v ∈ NG (T ) such that va2 v−1 = a1 . Hence, by Corollary 2.1.28, Ad(v)(sV2 ) = Ad(v)(sV1 ) and Ad(v)(V2 ) = Ad(v)(V1 ), hence, w(V2 ) = V1 , where w is the element of the Weyl group of t, generated by Ad(v). From Remark 2.5.31, Proposition 4.37 and Theorem 5.13 in [6], we get the following theorem. Theorem 2.5.36. The Weyl group W of a Cartan subalgebra t of a compact Lie algebra g rearranges the roots from ∆ (g) and is generated by (orthogonal with respect to h·, ·it ) reflections in the hyperplanes Uα ⊂ t, α ∈ ∆ (g), from Remark 2.5.31. We provide here some additional information on the root system of a compact simple Lie algebra (g, h·, ·i = −B) with the Killing form B, that can be found e.g. in [256, 483]. Let us fix a Cartan subalgebra t (that is, a maximal abelian subalgebra) of the Lie algebra g. As mentioned above, there is a set ∆ (root system) of (non-zero) real-valued linear forms α ∈ t∗ on the Cartan subalgebra t, which are called roots.

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Given a root system ∆ we can always choose (in many ways) a set of positive roots. This is a subset ∆ + ⊂ ∆ such that • For each root α ∈ ∆ exactly one of the roots α, –α is contained in ∆ + . • For any two distinct α, β ∈ ∆ + such that α + β is a root, α + β ∈ ∆ + . If a set of positive roots ∆ + is chosen, elements of −∆ + are called negative roots. An element of ∆ + is called a simple root if it cannot be written as the sum of two elements of ∆ + . The set of simple roots is also referred to as a base for ∆ . The set Π = {π1 , . . . , πr }, r = rk(g), of simple roots in ∆ + is a basis of t with the following additional property: • For every α ∈ ∆ + , there is a unique decomposition α = ∑ri=1 ai πi with nonnegative integers ai . The set ∆ + determines the following partial ordering ≺ on roots: α ≺ β , α, β ∈ ∆ , if β − α ∈ ∆ + . There is the maximal root β ∈ ∆ + with respect ≺, that is characterized by the fact that β − α is a non-negative linear combination of roots in ∆ + for all other positive roots α. We will denote this root by αmax . It should be noted that for each root system ∆ there are many different choices of the set of positive roots (or, equivalently, of the simple roots), but any two sets of positive roots differ by the action of the Weyl group W = W (t) of the Lie algebra g. Let us consider some positive root system ∆ + ⊂ ∆ . For any α ∈ ∆ , we denote by |α| a unique positive root among the roots ±α. The Lie algebra g admits a direct h·, ·i-orthogonal decomposition (cf. Proposition 2.5.29) g = t⊕

M

vα

(2.33)

α∈∆ +

into vector subspaces, where each subspace vα is 2-dimensional and ad(t)-invariant. We describe this decomposition in more detail. Using the restriction of the (nondegenerate) inner product h·, ·i to t, we will naturally identify α with some vector in t. Note that [vα , vα ] is a one-dimensional subalgebra in t spanned by the root α, and [vα , vα ] ⊕ vα is a Lie algebra isomorphic to su(2). The vector subspaces vα , α ∈ ∆ + , admit bases {Uα ,Vα }, such that hUα ,Uα i = hVα ,Vα i = 1, hUα ,Vα i = 0 and [H,Uα ] = hα, HiVα ,

[H,Vα ] = −hα, HiUα ,

∀ H ∈ t,

[Uα ,Vα ] = α. (2.34)

Note also, that [vα , vβ ] = vα+β + v|α−β | , assuming vγ := {0} for γ ∈ / ∆ +. For a positive root system ∆ + the (closed) Weyl chamber is defined by the equality C = C(∆ + ) := {H ∈ t | hα, Hi ≥ 0 ∀ α ∈ ∆ + }.

(2.35)

Recall some important properties of the Weyl group W = W (t) of the Cartan subalgebra t.

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109

1) For every root α ∈ ∆ ⊂ t the Weyl group W contains the orthogonal reflection ϕα in the plane Pα , which is orthogonal to the root α with respect to h·, ·i. It is easy to see that ϕα (H) = H − 2 hH,αi hα,αi α, H ∈ t. 2) Reflections from 1) generate W . 3) The root system ∆ is invariant under the action of the Weyl group W . 4) W acts irreducibly on t and simply transitively on the set of positive root systems. For any H ∈ t, there is a w ∈ W such that w(H) ∈ C(∆ + ). 5) For any X ∈ g, there is an inner automorphism ψ of g such that ψ(X) ∈ t. For any w ∈ W , there is an inner automorphism η of g such that t is stable under η, and the restriction of η to t coincides with w. 6) The Weyl group W acts transitively on the set of positive roots of fixed length. We will consider more deep properties of root systems in a more general context in Subsection 2.5.4.

2.5.4 Abstract Root Systems An exposition on abstract root systems can be found in various sources (cf. e.g., [121, 409, 473]). In this subsection, Rl means an l-dimensional Euclidean space with an inner product (·, ·). Every non-zero vector α ∈ Rl defines a reflection σα in the mirror (the plane of reflection) Uα = {β ∈ Rl | (β , α) = 0}. It is easy to get an explicit ,α) formula for this reflection: σα (β ) = β − 2(β (α,α) α. Definition 2.5.37. A finite set ∆ ⊂ Rl is called a root system, and its element are called roots, if the linear span of the set ∆ coincides with Rl and the following conditions hold: 1)

2(β ,α) (α,α)

∈ Z for all α, β ∈ ∆ ;

,α) 2) σα (β ) = β − 2(β (α,α) α ∈ ∆ for all α, β ∈ ∆ .

A root system is called reduced, if it satisfies the following additional condition: 3) if α, β ∈ ∆ and β is parallel to α, then β = ±α. Remark 2.5.38. It should be noted that the root system of a compact simple Lie algebra is a root system in this more general sense, see e.g. [256, 483]. The conditions from this definition imply that the angle between any two root vectors can take only one of the values 0, π/6, π/4, π/3, π/2, 2π/3, 3π/4, 5π/6, π; and the ratio of squared lengths of non-orthogonal roots of a reduced root system can be equal only to 1, 2, or 3, depending on the angle between the vectors. Definition 2.5.39. The finite group W , generated by all reflections σα , α ∈ ∆ , is called the Weyl group of the root system ∆ .

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The complement of the union of all mirrors Uα , α ∈ ∆ , in Rl is decomposed into connected components, called open Weyl chambers, and their closures are called simply Weyl chambers. It is easy to see that the Weyl group W is simply transitive on the set of Weyl chambers. We say that a vector α ∈ Rl is regular if it lies in an open Weyl chamber, and singular if it belongs to at least one mirror. A set Π ⊂ ∆ is called a simple root system (a basis) if Π is a basis in Rl and every root β ∈ ∆ can be represented in the form β = ∑α∈Π kα α, where the coefficients kα are all simultaneously non-negative or non-positive. In this case, roots with nonnegative (nonpositive) coefficients are called positive (respectively, negative), while the roots from Π are called simple. Any root system is effectively recovered by a certain system of simple roots, which certainly exists. Moreover, different systems of simple roots are mapped onto each other by means of the Weyl group W . Let α1 , α2 , . . . , αl be simple roots of a root system Π . The Cartan matrix of the 2(α ,α ) system Π is a matrix with the entries ai j = (αii,αi )j . This matrix contains all information on the root system ∆ . There is also a convenient graphical method to express the information contained in the Cartan matrix by using Dynkin diagrams. Consider the graph Γ whose vertices correspond to simple roots (numbered also as simple roots). Two different vertices with numbers i and j are connected with mi, j edges, where mi, j = ai j · a ji . If |ai j | > |a ji |, then we need to add an arrow leading from the i-th vertex to the j-th one (that is, the arrow points to a shorter root, since (αi , α j ) < 0 in this case). The graph Γ equipped with such additional marks is called the Dynkin diagram of the root system Π (see Table 2.2 for examples of such diagrams). Definition 2.5.40. A reduced root system ∆ is called irreducible if it cannot be decomposed into two non-empty subsystems, such that each root of one of the subsystems were orthogonal to each root from another subsystem. We note that for an irreducible root system ∆ , the corresponding Weyl group W acts irreducibly on Rl . In particular, the W -orbit of an arbitrary root generates Rl . It can be shown that in any irreducible root system, there are no more than two root lengths, which leads to the concepts of short and long roots, and the Weyl group W acts transitively on the set of roots of the same length. It is clear that the classification of reduced root systems is reduced to the classification of irreducible root systems, which is obtained from the classification of irreducible systems of simple roots. For each irreducible system of simple roots, the corresponding Dynkin diagram is connected. In the language of Dynkin diagrams, one can also state a well-known classification of irreducible systems of simple roots. For more information about these systems, see Tables 2.1 and 2.2. It should be noted that there is a natural one-to-one correspondence between irreducible systems of simple roots and simple complex Lie algebras, and also with simple compact Lie algebras. In accordance with the notation in paper [228], we give in Table 2.1 the classification of irreducible root systems, which is equivalent to the classification of simple complex Lie algebras or simple compact real Lie algebras (their dimensions dim g are also shown in the table). There exist four infinite series Al , Bl , Cl , and Dl of such

2.5 Compact Lie Algebras and Lie Groups

111

Table 2.1 Simple Lie algebras Type of g

dim g

Al (l ≥ 1)

l 2 + 2l

εi − ε j

Bl (l ≥ 2)

2l 2 + l

±εi ± ε j , ±εi

Cl (l ≥ 2)

2l 2 + l

Dl (l ≥ 3)

2l 2 − l

E6

78

E7

133

E8

248

Roots

±εi ± ε j , ±2εi

±εi ± ε j

Simple roots αi = εi − εi+1 (1 ≤ i ≤ l) αi = εi − εi+1 (1 ≤ i < l), αl = εl αi = εi − εi+1 (1 ≤ i < l), αl = 2εl αi = εi − εi+1 (1 ≤ i < l), αl = εl−1 + εl

εi − ε j , ±ε,

αi = εi − εi+1 (1 ≤ i < 6),

εi + ε j + εk + ε

α6 = ε4 + ε5 + ε6 + ε

εi − ε j ,

αi = εi − εi+1 (1 ≤ i < 7),

εi + ε j + εk + εl

α7 = ε5 + ε6 + ε7 + ε8

εi − ε j ,

αi = εi − εi+1 (1 ≤ i < 8),

±(εi + ε j + εk )

α8 = ε6 + ε7 + ε8 α1 = (ε1 − ε2 − ε3 − ε4 )/2,

F4

52

±εi ± ε j , ±εi ,

α2 = ε4 ,

(±ε1 ± ε2 ± ε3 ± ε4 )/2

α3 = ε3 − ε4 , α4 = ε2 − ε3

G2

14

εi − ε j , ±εi

α1 = −ε2 , α2 = ε2 − ε3

systems, and also five special systems: F4 , E6 , E7 , E8 and G2 . The subscript below indicates the rank of the system (the corresponding Lie algebra). In Table 2.1, the roots for the systems Bl , Cl , Dl , and F4 are expressed in terms of an orthonormal basis (ε1 , . . . , εl ). The roots for the systems Al , E6 , E7 , E8 , and G2 are expressed in terms of vectors ε1 , . . . , εl+1 , connected by the relation ∑i εi = 0. For l 1 these vectors, (εi , εi ) = l+1 , (εi , ε j ) = − l+1 for i 6= j. The roots for E6 are expressed in terms of vectors ε1 , . . . , ε6 , constructed as for A5 , and the vector ε, orthogonal to them, and such that (ε, ε) = 1/2. Indices i, j, . . . in the record of any root are considered distinct. All the roots of any of the Lie algebras Al , Dl , E6 , E7 , E8 have the same length. Simple Lie algebras (and corresponding root systems) with this property are called simply laced. The roots of the remaining simple Lie algebras have two distinct lengths, and we obtain subsystems ∆l ⊂ ∆ and ∆s ⊂ ∆ of all long and all short √roots, respectively. If α ∈ ∆√ (respectively, G2 ), then |α|1 = 2 |β |1 l , β ∈ ∆ s for Bl , Cl , F4 p (respectively, |α|1 = 3 |β |1 ), where |X|1 = hX, Xi. In all cases, roots of the same

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Table 2.2 Properties of simple Lie algebras Type g

Al (l ≥ 1) Bl (l ≥ 2) Cl (l ≥ 2) Dl (l ≥ 3)

E6

E7

E8

F4

Dynkin diagram

α1

α2

αl−1 αl

α1

α2

αl−1

αl

α1

α2

αl−1

αl

α1

α2

αl−2 αl−1

α1

α2

α3

α4

α5

α1

α2

α3

α4

α5

α6

α1

α2

α3

α4

α5

α6

α1

α2

◦−◦−···−◦−◦

◦−◦−···−◦ ⇒ ◦

◦−◦−···−◦ ⇐ ◦ ◦−◦−···− ◦ − ◦ | ◦αl ◦−◦−◦−◦−◦ | ◦α6

◦−◦−◦−◦−◦−◦ | ◦α7

α7

◦−◦−◦−◦−◦−◦−◦ | ◦α8 α3

α4

◦−◦ ⇐ ◦−◦

α1 G2

Order of the Weyl group

α2

◦W◦

Compact Lie algebra

(l + 1)!

su(l + 1)

2l · l!

so(2l + 1)

2l · l!

sp(l)

2l−1 · l!

so(2l)

27 · 34 · 5

e6

210 · 34 · 5 · 7

e7

214 · 35 · 52 · 7

e8

27 · 32

f4

12

g2

length can form angles π3 , π2 , 2π 3 . Roots of different length for Bl , Cl , F4 (respectively, π π 3π G2 ) can form angles 4 , 2 , 4 (respectively, π6 , π2 , 5π 6 ). In Table 2.2 we provide Dynkin diagrams for the corresponding root systems, the orders of the corresponding Weyl groups and the corresponding compact simple Lie algebras.

2.6 Connections with Complex Lie Algebras

113

2.6 Connections with Complex Lie Algebras Lie algebras can be defined over any field F. If the characteristic of the field is distinct from 2, then for such Lie algebras, a deep theory is developed, the fundamentals of which are remarkably exposed in the books [271] by N. Jacobson, [266] by J. Humphreys, and [473] by D. J. Winter. For geometric studies, complex Lie algebras (in addition to real Lie algebras) play an important role. The difference between a complex Lie algebra g and a real one consists only in the fact √ that g is a linear space over the field C. The multiplication by the imaginary unit −1 in g generates a natural complex structure J, i.e. an R-linear endomorphism J of the Lie algebra g such that J 2 = − Id. A complex Lie algebra g can also be considered as a Lie algebra over the field of real numbers (the commutation operation is inherited in this case). The resulting real Lie algebra gR has a complex structure J and is called the realification of the complex Lie algebra g. On the other hand, if g0 is a real Lie algebra, then supplied in a natural √ g0 ⊗ C is √ way with the structure of a Lie algebra: for X + −1Y, U + −1V ∈ g0 ⊗ C we define the commutator by the formula √ √ √ [X + −1Y,U + −1V ] = [X,U] − [Y,V ] + −1([X,V ] + [Y,U]). The complex Lie algebra g = gC 0 obtained in this way is said to be the complexification of the Lie algebra g0 . Definition 2.6.1. Let g be a complex Lie algebra. A real form of the Lie algebra g is a Lie subalgebra g0 of the real Lie algebra gR such that gC to g. In 0 is isomorphic √ this case, every Z ∈ g is uniquely represented in the form Z = X + −1Y , where X,Y ∈ g0 . It should be noted that there exist complex Lie algebras that do not have a real form. On the other hand, every semisimple complex Lie algebra has a real form, that is not unique in general. One can learn more about real forms from Chapter 3 of the book [256] or from [228]. We note that each complex semisimple Lie algebra g generates an abstract root system, and can be uniquely reconstructed from such a system [266, 256]. Using the invariance and the non-degeneracy of the Killing form, g can be decomposed into a direct sum of simple (i.e. without proper ideals) Lie algebras. The root systems of simple Lie algebras are irreducible. The classification of such systems is essentially equivalent to the classification of simple complex Lie algebras. More details about this can be found in [271, 266, 256, 228]. It is also remarkable that every semisimple complex Lie algebra has a compact real form (cf. Theorem 6.3 in [256]). Moreover, any two compact real forms of a semisimple complex Lie algebra are conjugate, i.e. are mapped onto each other by inner automorphisms of the Lie algebra (Theorem 1.2 in [228, Chapter 4]). Therefore, the classification of compact simple Lie algebras can easily be obtained from the classification of simple complex Lie algebras.

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2.7 Invariant Norms and Inner Products on Lie Algebras In this section, following [78], we provide results on the connection between invariant norms and inner products on a Lie algebra g with norms and inner products, defined on some Cartan subalgebra t ⊂ g, that are invariant with respect to the corresponding Weyl group. Let V be a finite-dimensional vector space over the field of real numbers and F be some compact group of linear transformations of the space V . Let us consider a norm | · | : V → R (an inner product (·, ·) : V × V → R) on V . It is said that | · | is invariant with respect to the group F if |X| = | f (X)| for all X ∈ V and all f ∈ F. Analogously, we shall say that the inner product (·, ·) is invariant with respect to the group F if (X,Y ) = ( f (X), f (Y )) for all X,Y ∈ V and f ∈ F. Remark 2.7.1. If the norm | · | is Euclidean, i.e. is defined by some inner product (·, ·), then its invariance with respect to the group F is equivalent to the invariance of (·, ·) with respect to the group F. Indeed, 2( f (X), f (Y )) − 2(X,Y ) = ( f (X +Y ), f (X +Y )) −( f (X), f (X)) − ( f (Y ), f (Y )) − (X +Y, X +Y ) + (X, X) + (Y,Y ). In what follows, we will consider invariant norms | · | and invariant inner products (·, ·) on the Lie algebra g, assuming the invariance with respect to the group of inner automorphisms Int(g), generated by elements exp(ad X), X ∈ g. If G is a compact connected Lie group with the tangent Lie algebra g, then Int(g) = Ad(G) according to Proposition 2.1.25, Theorem 2.1.27, and Corollary 2.1.28, hence, the invariance in the above sense is equivalent to the Ad(G)-invariance. Let G be a compact connected Lie group with the Lie algebra g. Let us fix some Cartan subalgebra t in g. Now, let us consider two subgroups in the group Int(g), connected with the Cartan subalgebra t: e = { f ∈ Int(g) | f (t) = t}, W

e0 = { f ∈ Int(g) | f |t = Id}. W

e0 is a normal subgroup in the group W e , and the group W = W e /W e0 It is clear that W is the Weyl group of the Cartan subalgebra t (cf. Definition 2.5.32 and Proposition 2.5.33). For any Ad(G)-invariant norm | · | (any Ad(G)-invariant inner product (·, ·)) on g, we shall denote by | · |t (respectively, (·, ·)t ) its restriction to the Cartan subalgebra t. It is clear that | · |t and (·, ·)t are invariant with respect to the Weyl group W . ´ B. Vinberg is well known. The following result of E. ´ B. Vinberg, [448]). Let G be a compact connected Lie group Theorem 2.7.2 (E. and t any Cartan subalgebra in the Lie algebra g. Then every norm on t, which is invariant with respect to the Weyl group W , is the restriction of a unique Ad(G)invariant norm on g. Here, we provide the proof of this theorem, based on the following result by B. Kostant.

2.7 Invariant Norms and Inner Products on Lie Algebras

115

Theorem 2.7.3 (B. Kostant, [296]). Let G be a compact connected Lie group, Q an Ad(G)-invariant inner product on the Lie algebra g and t any Cartan subalgebra in g. Let us consider an arbitrary orbit O of the adjoint action of G on g. Then the orthogonal (with respect to Q) projection of O to t coincides with the convex hull of the orbit of an arbitrary element in t ∩ O with respect to the Weyl group W . Proof (of Theorem 2.7.2). Let us consider an arbitrary norm | · |1 on t, which is invariant with respect to the Weyl group W . According to Theorem 2.5.22, for every Y ∈ g, there is an s ∈ G such that Ad(s)(Y ) ∈ t. Now, let us define a map |·| : g → R

(2.36)

such that |Y | = | Ad(s)(Y )|1 , where Ad(s)(Y ) ∈ t. Let us prove that this is well-defined. Suppose that s1 , s2 ∈ G are such that Y1 := Ad(s1 )(Y ) ∈ t and Y2 := Ad(s2 )(Y ) ∈ t. Since Ad(v)(Y1 ) = Y2 , where v = s2 s−1 1 , then, according to Corollary 2.5.35, there is a w in the Weyl group W of the Lie algebra t such that w(Y1 ) = Y2 . Since the norm | · |1 on t is invariant with respect to W , we have |Y1 |1 = |Y2 |1 . Thus, the map (2.36) is correctly defined. It is also obvious that it is continuous and Ad(G)-invariant. Now, let us show that the resulting map | · | : g → R is a norm on g. We need only to prove the inequality |X +Y | ≤ |X| + |Y | for every X,Y ∈ g. Put Z = X +Y . By the Ad(G)-invariance, we can suppose without loss of generality that Z ∈ t. Let us fix on g some Ad(G)-invariant inner product Q, and let Xt and Yt be the Q-orthogonal projections to t of vectors X and Y respectively. Then Z = Xt + Yt . Therefore, |Z| = |Z|1 ≤ |Xt |1 +|Yt |1 . Now, let us consider a vector Xe (respectively, Ye ) in the intersection of t with the orbit of X (respectively, Y ) with respect to the adjoint e action. By Theorem 2.7.3, the vector Xt lies in the convex hull of the W -orbit of X. Since the | · |1 -norms of all elements of this orbit are the same and are equal to e 1 = |X|, we have |Xt |1 = |Xt | ≤ |X|. Analogously we get that |Yt |1 = |Yt | ≤ |Y |. |X| Hence, |X +Y | = |Z| = |Z|1 ≤ |Xt |1 + |Yt |1 ≤ |X| + |Y |, as is required. Therefore, we have found an Ad(G)-invariant norm on g with the restriction to t equal to | · |1 . The uniqueness of such a norm evidently follows from the definition of the norm | · |. t u Now, let us turn to the study of Ad(G)-invariant inner products. It is clear that the restriction of an arbitrary Ad(G)-invariant inner product on the Lie algebra g to the Cartan subalgebra t is invariant with respect to the Weyl group. The following is well known Theorem 2.7.4 ([121]). For a simple compact Lie group G with the Lie algebra g, any inner product on a Cartan subalgebra t of g, invariant with respect to the Weyl group, is proportional to the restriction of the Killing form of the Lie algebra g to t.

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With the help of this theorem it is not difficult to get the following assertion. Theorem 2.7.5. Let G be a compact connected Lie group and t any Cartan subalgebra in the Lie algebra g. Then every W -invariant inner product on t is the restriction of a unique Ad(G)-invariant inner product on g. Proof. Let (·, ·) be an arbitrary Ad(G)-invariant inner product on the Lie algebra g of the Lie group G. According to Theorem 2.5.5, there is a (·, ·)-orthogonal decomposition g = C(g) ⊕ g1 ⊕ · · · ⊕ gl into a direct sum of the center C(g) and simple Lie algebras gi , 1 ≤ i ≤ l. By the definition of the Killing form B on g, this decomposition is also orthogonal with respect to B, moreover, the restrictions of B to gi coincide with the Killing forms Bi of the Lie algebras gi , 1 ≤ i ≤ l. Since the latter Lie algebras are simple (in other words, are irreducible under the action of their inner automorphism groups), an invariant inner product on each of these Lie algebras is defined uniquely, up to a multiplication by a positive number. Moreover, the Killing form B is negative definite on g1 ⊕ · · · ⊕ gl due to Theorem 2.5.11 and is zero on C(g). Therefore, (·, ·) = Q ⊕ x1 · B1 ⊕ · · · ⊕ xl · Bl , where Q is an arbitrary inner product on C(g), xi < 0, 1 ≤ i ≤ l. Now, it suffices to apply Proposition 2.5.27 and Theorem 2.7.4.

t u

The above results are useful in the study of the Chebyshev norms (cf. Sections 6.12 and 6.13), corresponding to homogeneous Riemannian manifolds. In particular, Theorem 2.7.2 allows us to calculate the values of the Chebyshev norm only for vectors from some fixed Cartan subalgebra of the Lie algebra of the isometry group.

Chapter 3

Isometric Flows and Killing Vector Fields on Riemannian Manifolds

Abstract In this chapter we deal with Killing vector fields on Riemannian manifolds and their connections with isometric flows, corresponding periodicity functions, and the sectional and Ricci curvatures. Special attention is paid to Killing vector fields of constant length, the corresponding isometric flows on complete smooth Riemannian manifolds, and their connections with Clifford–Wolf translations. The subjects of this chapter are Killing vector fields on Riemannian manifolds and their connections with isometric flows, corresponding periodicity functions, and the sectional and Ricci curvatures. Tightly connected to the latter case are the wellknown theorems of Berger, Bochner, and Synge, which we prove here. It is shown that an isometric flow is generated by an effective almost free action of the circle group if there are no points of infinite or zero period. In the last case, the set of periods is at most countable and naturally generates an invariant stratification of the set of singular points by closed totally geodesic strata; the union of all regular orbits is a connected open dense subset. Given an effective almost free isometric action of the circle group, there exists another Riemannian metric for which the same action is generated by a nontrivial Killing vector field of constant length. Then we concentrate on these fields, the corresponding isometric flows on complete smooth Riemannian manifolds, and their connections with Clifford–Wolf translations. Various examples of the Killing vector fields of constant length, generated by the isometric effective almost free but not free actions of the circle on the Riemannian manifolds, close in some sense to symmetric spaces, are constructed. The latter manifolds include “almost round” odd-dimensional spheres and unit vector bundles over Riemannian manifolds. It is proved that such a flow on a symmetric space is free or induced by a free isometric action of the circle and consists of Clifford–Wolf translations. Examples of unit Killing vector fields, generated by an almost free but not free action of the circle on locally symmetric Riemannian spaces, are found; there are homogeneous (non-simply connected) Riemannian manifolds of constant positive sectional curvature and locally Euclidean spaces among them. Everywhere in this chapter, unless otherwise stated, all Riemannian manifolds are assumed to be connected and complete. © Springer Nature Switzerland AG 2020 V. Berestovskii and Y. Nikonorov, Riemannian Manifolds and Homogeneous Geodesics, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-56658-6_3

117

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3.1 Isometries and Killing Vector Fields on Riemannian Manifolds Definition 3.1.1. A diffeomorphism φ : (M, g) → (M, g) of a Riemannian manifold (M, g) onto itself is called an isometry or a motion of the Riemannian manifold (M, g) if for every point x ∈ M and all vectors u, v ∈ Mx , g(T φ (u), T φ (v)) = g(u, v) (in the other notation, φ ∗ g = g). Theorem 3.1.2. Let φ : (M, g) → (M, g) be a bijection of a Riemannian manifold (M, g). Then the following statements are equivalent: 1) φ is an isometry of the Riemannian manifold (M, g); 2) φ is an isometry of the metric space (M, d), where d is the intrinsic metric on (M, g) defined by (1.74). Proof. 1) =⇒ 2) follows from the definitions. 2) =⇒ 1). Let x be an arbitrary point in (M, g). Corollary 1.5.3 implies that φ maps any geodesic γ = γ(t), t ∈ R, in (M, g) onto a geodesic φ (γ) and φ (γ(t)) = φ (γ)(t), t ∈ R. This implies that we can take σφ (x) = σx for σx > 0 from Proposition 1.5.8. Then Proposition 1.5.9 implies that for any two geodesics γi (s), s ∈ R, i = 1, 2, parameterized by the arc length, such that γi (0) = x, i = 1, 2, cos ∠(φ (γ1 ), φ (γ2 ))x = gφ (x) (φ (γ1 )0 (0), φ (γ2 )0 (0)) = gx (γ10 (0), γ20 (0)) = cos ∠(γ1 , γ2 )x . The Euclidean vector space (Mx , gx ), x ∈ M, is naturally identified with the set of tangent vectors γ 0 (0) of geodesics γ(t), t ∈ R, in (M, g) with the origin γ(0) = x, where gx (γ10 (0), γ20 (0)) = kγ10 (0)k · kγ20 (0)k · cos ∠(γ1 , γ2 )x . Further, γ10 (0) + γ20 (0) is unique γ30 (0) such that gx (γ30 (0), γ10 (0)) = kγ10 (0)k2 + gx (γ10 (0), γ20 (0)), gx (γ30 (0), γ20 (0)) = kγ20 (0)k2 + gx (γ10 (0), γ20 (0)), and kγ30 (0)k2 = kγ10 (0)k2 + kγ20 (0)k + 2gx (γ10 (0), γ20 (0)). Moreover, α · γ 0 (0) is defined as (αγ)0 (0), where (αγ)(t) = γ(αt), α,t ∈ R. By the previous argument the correspondence L : γ 0 (0) ∈ (Mx , gx ) → (φ ◦ γ)0 (0) ∈ (Mφ (x) , gφ (x) ) is a linear isometric isomorphism of Euclidean vector spaces. Thus, by Theorem 1.5.1 and Corollary 1.5.2, the restriction of φ to U(x, σx ) has the form ∞ Expφ (x) ◦L ◦ Exp−1 x , so φ is a C -diffeomorphism and T φx = L. Since x was chosen arbitrarily, the theorem is proved. t u

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Definition 3.1.3. A smooth vector field X on Riemannian manifold (M, g) is called a Killing vector field if its flow Ψ is complete and the transformations Ψt , t ∈ R, are isometries of the Riemannian manifold (M, g). Theorem 3.1.4. A smooth vector field X on a complete Riemannian manifold (M, g) is a Killing vector field if and only if Xg(Y, Z) = g([X,Y ], Z) + g(Y, [X, Z])

(3.1)

for all smooth vector fields Y, Z on M, or, equivalently, when LX g = 0. Proof. Necessity. Let X be a smooth Killing vector field on (M, g), Ψ its flow and p ∈ M an arbitrary point. Then according to Propositions 1.1.21 and 1.1.38, 1 Xg(Y, Z)(p) = lim [g(Y, Z)(p) − g(Y, Z)(Ψ−t (p))] t→0 t 1 = lim [g(Y, Z)(p) − g((TΨt )Y, (TΨt )Z)(p)] t→0 t 1 = lim [g(Y, Z) − g((TΨt )Y, Z)] t→0 t 1 + lim [g((TΨt )Y, Z) − g((TΨt )Y, (TΨt )Z)](p) t→0 t 1 1 = lim g(Y − (TΨt )Y, Z)(p) + lim g((TΨt )Y, Z − (TΨt )Z))(p) t→0 t t→0 t = g([X,Y ], Z)(p) + g(Y, [X, Z])(p). Sufficiency. Let X be a smooth vector field on a complete Riemannian manifold (M, g), satisfying the relation (3.1), Ψ the flow of the vector field X with the domain W , and Y, Z arbitrary smooth vector fields on M. Let (p,t0 ) ∈ W , where t0 > 0, be any point in W . Define a smooth path c(t) = Ψ (p,t) and smooth real functions f (t) and φ (t) on [0,t0 ] by the formulas f (t) = g(Y, Z)(c(t)) − g(Y, Z)(p), φ (t) = g(Y, Z)(c(t)) − g((TΨt )Y (p), (TΨt )Z(p)).   ∂ It is clear that c0 (t) = T c ∂t = X(c(t)) for all t ∈ [0,t0 ]. Then on the basis of the above calculations, Definition 1.1.31, and Corollary 1.1.32, we get f 0 (t) = Xg(Y, Z)(c(t)),

φ 0 (t) = g([X,Y ], Z)(c(t)) + g(Y, [X, Z])(c(t))

for all t ∈ [0,t0 ]. This observation and the equalities f (0) = 0 = φ (0) and (3.1) imply f (t0 ) =

Z t0 0

f 0 (t)dt =

Z t0 0

φ 0 (t)dt = φ (t0 ).

Therefore, g((TΨt0 )Y (p), (TΨt0 )Z(p)) = g(Y, Z)(p), i.e. Ψt is a local 1-parameter group of local isometric transformations of the Riemannian manifold (M, g). Now,

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if c(t) = Ψ (p0 ,t), (p0 ,t) ∈ W , is a maximal integral curve of the field X with the origin at an arbitrary point p0 , then c0 (t) = (TΨt )(c0 (0)),
g(c0 (t), c0 (t)) = g (TΨt )(c0 (0)), (TΨt )(c0 (0)) = g(c0 (0), c0 (0)), i.e. the tangent vectors c0 (t) have constant length for all t from the domain of c. From here, the completeness of the Riemannian manifold (M, g), and the Hopf–Rinow theorem one can easily deduce that c(t) is defined for all t ∈ R. Consequently, X defines a 1-parameter isometry group Ψt , t ∈ R, of the Riemannian manifold (M, g) and, therefore, is a Killing vector field on (M, g). It remains to note that the equality (3.1) is equivalent to the equality LX g = 0, which immediately follows from Definition 1.1.44. t u Proposition 3.1.5. The classical coordinate Killing equation in [286]: ξi, j + ξ j,i = 0

(3.2)

for covariant components ξi = gil ξ l of the Killing vector field X is exactly the coordinate expression of the equation LX g = 0 for local vector fields Y = ∂ /∂ xi and Z = ∂ /∂ x j in coordinates x1 , . . . , xn . Proof. The equation LX g = 0 is exactly (3.1) with its right-hand side moved to the left. Then with the help of (1.18) we obtain on the left-hand side of the last equation             ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ l l   , −g ξ , i , j −g , ξ , j L l ∂ ξ ∂ xi ∂ x j ∂x ∂x ∂ xi ∂x ∂ xl ∂ xl ∂ xl     l l l ∂ gi j ∂ξ ∂ ∂ ∂ ∂ξ ∂ ∂ξ ∂ξl l ∂ gi j = ξl l +g , + g , = ξ + g + g . jl il ∂ xi ∂ xl ∂ x j ∂ xi ∂ x j ∂ xl ∂ xi ∂xj ∂x ∂ xl Thus we get the equation ξl

∂ gi j ∂ξl ∂ξl + g + g = 0. jl il ∂ xi ∂xj ∂ xl

By (1.28), ξi, j = ∂ ξi /∂ x j − Γi lj ξl , so ξi, j + ξ j,i = ∂ (gil ξ l )/∂ x j − Γi kj gkl ξ l + ∂ (g jl ξ l )/∂ xi − Γjik gkl ξ l   = ∂ gil /∂ x j − 2glk Γi kj + ∂ g jl /∂ xi ξ l + gil ∂ ξ l /∂ x j + g jl ∂ ξ l /∂ xi = ξl

∂ gi j ∂ξl ∂ξl + g jl i + gil j . l ∂x ∂x ∂x

In the last step we interchanged the last two summands and used the equality   1 ∂ g jl ∂ gli ∂ gi j glk Γi kj = , + − 2 ∂ xi ∂xj ∂ xl

(3.3)

3.1 Isometries and Killing Vector Fields on Riemannian Manifolds

121

which follows from (1.29). Thus, we have shown that equations (3.2) and (3.3) are identical and proved the proposition. t u To study properties of a Killing vector field X, the operator AX defined on vector fields by the formula AX V = −∇V X (3.4) is useful. Obviously, AX = LX − ∇X . We have the following statement (see e.g. [291]). Proposition 3.1.6. Let X be a Killing vector field on a Riemannian manifold (M, g). Then the following assertions hold: 1) For all vector fields U and V on M, g(∇U X,V ) + g(U, ∇V X) = 0. In other words, the operator AX is skew-symmetric. 2) For any vector field U on M, R(X,U) = [∇X , ∇U ] − ∇[X,U] = [∇U , AX ], where R is the curvature tensor of the manifold (M, g). 3) For all vector fields U,V,W on the Riemannian manifold (M, g), the following formula holds: −g(R(X,U)V,W ) = g(∇U ∇V X,W ) + g(∇U V, ∇W X). Proof. It is clear that X · g(U,V ) = g(∇X U,V ) + g(U, ∇X V ) and X · g(U,V ) = g([X,U],V ) + g(U, [X,V ]). Therefore, g(∇U X,V ) + g(U, ∇V X) = g(∇X U − [X,U],V ) + g(U, ∇X V − [X,V ]) = 0, which proves the first assertion. We will get the second assertion following ideas in the proof of Lemma 2.2 in [292]. First, note that for any vector field V , the equality ∇V (AX ) = [∇V , AX ] holds (see Proposition 2.10 in [291, Chapter III]), and the operator ∇V (AX ) is skewsymmetric in view of the skew symmetry of the operator AX . Consider the bilinear form ψ(V,U) = ∇V (AX )U − R(X,V )U, where U,V are vector fields and R is the curvature tensor of the manifold under consideration. It is easy to see that ∇V (AX )U = [∇V , AX ] ·U = ∇V AX U − AX ∇V U = −∇V ∇U X − AX ∇V U and ψ(V,U) − ψ(U,V ) = [∇V , AX ] ·U − [∇U , AX ] ·V + R(X,U)V − R(X,V )U = [∇U , ∇V ]X − ∇[U,V ] X − R(U,V )X = 0, because R(X,U)V + R(V, X)U + R(U,V )X = 0 (see the equality (1.32)) and AX ∇U V − AX ∇V U = AX [U,V ] = −∇[U,V ] X. Thus, the bilinear form ψ is symmetric in its two arguments. Now we shall consider the three-linear form ϕ(U,V,W ) =

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g(ψ(V,U),W ). It is clear that it is symmetric in the variables V and U. Note now that the expression ϕ(U,V,W ) = g(∇V (AX )U − R(X,V )U,W ) = g(∇V (AX )U,W ) − g(R(X,V )U,W ) is skew-symmetric on W and U (this follows from properties of the curvature tensor and the skew symmetry of the operator ∇V (AX )). If a three-linear form is symmetric in one pair of variables and skew-symmetric in another one, then obviously it is zero. Therefore, ψ(V,U) ≡ 0 and ∇V (AX ) = [∇V , AX ] = R(X,V ) for any vector field V , which proves the second assertion of the proposition. The third assertion follows from the previous one: g(R(X,U)V,W ) = g(∇U AX V,W ) − g(AX ∇U V,W ) = −g(∇U ∇V X,W ) + g(∇U V, AX W ) = −g(∇U ∇V X,W ) − g(∇U V, ∇W X), since the operator AX is skew-symmetric for the Killing vector field X.

t u

Theorem 3.1.7. Let X be a Killing vector field on a Riemannian manifold (M, g). Then the following assertions hold: 1) The restriction of X to any geodesic c(t), t ∈ R, in (M, g) is a Jacobi vector field along this geodesic, i.e. ∇2τ(t) X + R(X, τ(t))τ(t) = 0, where τ(t) = c0 (t).   2) If h(t) = 12 g X(c(t)), X(c(t)) , where c(t) (t ∈ R) is geodesic on (M, g), then h00 (t) = g(∇τ(t) X, ∇τ(t) X) − g(R(X, τ(t))τ(t), X). 3) For any point x ∈ M with the property gx (X, X) 6= 0, the integral trajectory of the field X, passing through the point x, is a geodesic on (M, g) if and only if x is a critical point of the squared length g(X, X) of the field X. Proof. Let us prove the first assertion. Let c = c(t), t ∈ [a, b], be any geodesic and X be a Killing vector field on (M, g). Then the formula C(t, s) := Ψs (c(t)), where Ψs , s ∈ R, is the 1-parameter isometry group of the manifold (M, g) defined by the field X, determines some geodesic variation in (M, g), moreover, TC(t, s)(∂ /∂ s) = X(C(t, s)). It remains to apply Proposition 1.4.10. The second assertion of the theorem follows from the first one and the computations 1 ∇ g(X, X) = g(∇τ(t) X, X), 2 τ(t) h00 (t) = g(∇2τ(t) X, X) + g(∇τ(t) X, ∇τ(t) X) h0 (t) =

= g(∇τ(t) X, ∇τ(t) X) − g(R(X, τ(t))τ(t), X). Let us prove the third assertion. Put f = g(X, X). According to 1) of Proposition 3.1.6, for every vector field Y and every point x, we have the equality Y f = 2g(∇Y X, X) = −2g(Y, ∇X X). If x is a non-critical point, then ∇X X 6= 0 at this point

3.2 The Curvature Tensor and Killing Vector Fields

123

(if ∇X X = 0, then for every Y we have Y f = −2g(Y, ∇X X) = 0 and x is critical), i.e. the integral curve of the field X through the point x is not geodesic. On the other hand, if x is a critical point, then ∇X X = 0 at x. Denoting by Ψt , t ∈ R, the isometric flow generated by X, we get TΨt (Xx ) = XΨt (x) . That is, ∇X X = 0 along the whole orbit Ψt (x), i.e. this orbit (= the integral curve of the field X) is a geodesic. t u

3.2 The Curvature Tensor and Killing Vector Fields Lemma 3.2.1. Let X,Y, Z be Killing vector fields on a Riemannian manifold (M, g). Then 2g(∇X Y, Z) = g([X,Y ], Z) + g([X, Z],Y ) + g(X, [Y, Z]). Proof. Since [U,V ] = ∇U V − ∇V U for all vector fields U,V and the operator AX (AX Y = −∇Y X) is skew-symmetric for every Killing vector field X, we have g([X, Z], X) = g(∇X Z, X) − g(∇Z X, X) = −g(AZ X, X) + g(AX Z, X) = −g(Z, AX X) = g(∇X X, Z). The polarization of this equality gives us g(∇X Y + ∇Y X, Z) = g([X, Z],Y ) + g([Y, Z], X). Now, adding the obvious equality g(∇X Y − ∇Y X, Z) = g([X,Y ], Z), we get the lemma. t u Lemma 3.2.2. Let X,Y, Z,W be Killing vector fields on a Riemannian manifold (M, g). Then g(R(X,Y )Z,W ) = g(∇Z X, ∇Y W ) − g(∇Y Z, ∇W X) 1h − g([Y, [Z, X]],W ) + g([Y, [Z,W ]], X) + g([Y, [X,W ]], Z) 2 i

+g([Z, X], [Y,W ]) + g([Z,W ], [Y, X]) + g([Y, Z], [X,W ]) .

Proof. Let us consider f := 2g(∇Z X,W ) = g([Z, X],W ) + g([Z,W ], X) + g(Z, [X,W ]) (see Lemma 3.2.1). Then Y · f = 2g(∇Y ∇Z X,W ) + 2g(∇Z X, ∇Y W ) and Y · f = g([Y, [Z, X]],W ) + g([Z, X], [Y,W ]) + g([Y, [Z,W ]], X) +g([Z,W ], [Y, X]) + g([Y, [X,W ]], Z) + g([Y, Z], [X,W ]).

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3 Isometric Flows and Killing Vector Fields on Riemannian Manifolds

Therefore, g(∇Y ∇Z X,W ) = −g(∇Z X, ∇Y W ) 1h + g([Y, [Z, X]],W ) + g([Y, [Z,W ]], X) + g([Y, [X,W ]], Z) 2 i

+g([Z, X], [Y,W ]) + g([Z,W ], [Y, X]) + g([Y, Z], [X,W ]) .

By 3) of Proposition 3.1.6, we get g(R(X,Y )Z,W ) = −g(∇Y ∇Z X,W ) − g(∇Y Z, ∇W X). t u

The lemma follows from the last two equalities. For smooth vector fields X,Y on Riemannian manifold (M, g), we set 1 U (X,Y ) := ∇X Y − [X,Y ]. 2

(3.5)

Since [X,Y ] = ∇X Y − ∇Y X, we have U (X,Y ) = U (Y, X), and the equality ∇X Y = 1 2 [X,Y ] + U (X,Y ) gives a decomposition into the sum of skew-symmetric and symmetric forms. By Lemma 3.2.1, 2(U (X,Y ), Z) = g([X, Z],Y ) + g(X, [Y, Z])

(3.6)

for all Killing vector fields X,Y, Z. Theorem 3.2.3. Let X,Y, Z,W be Killing vector fields on a Riemannian manifold (M, g). Then g(R(X,Y )Z,W ) = g(U (X, Z), U (Y,W )) − g(U (X,W ), U (Y, Z)) 1 1 1 + g([X,Y ], [Z,W ]) + g([X, Z], [Y,W ]) − g([X,W ], [Y, Z]) 2 4 4 i 1h + g([X, [Z,W ]],Y ) − g([Y, [Z,W ]], X) + g([Z, [X,Y ]],W ) − g([W, [X,Y ]], Z) . 4 Proof. Taking into account Lemma 3.2.2, the equations (3.5) and (3.6), and the Jacobi identity, we get g(R(X,Y )Z,W ) = g(∇Z X, ∇Y W ) − g(∇Y Z, ∇W X) 1h − g([Y, [Z, X]],W ) + g([Y, [Z,W ]], X) + g([Y, [X,W ]], Z) 2 i + g([Z, X], [Y,W ]) + g([Z,W ], [Y, X]) + g([Y, Z], [X,W ])

1 1 1 = g([Z, X], [Y,W ]) + g([Y, [Z, X]],W ) + g(Y, [W, [Z, X]]) 4 4 4 1 1 + g([Z, [Y,W ]], X) + g(Z, [X, [Y,W ]]) + g(U (X, Z), U (Y,W )) 4 4

3.3 Killing Vector Fields of Constant Length

125

1 1 1 − g([Y, Z], [W, X]) − g([W, [Y, Z]], X) − g(W, [X, [Y, Z]]) 4 4 4 1 1 − g([Y, [W, X]], Z) − g(Y, [Z, [W, X]]) − g(U (Y, Z), U (X,W )) 4 4 1 1 1 − g([Y, [Z, X]],W ) − g([Y, [Z,W ]], X) − g([Y, [X,W ]], Z) 2 2 2 1 1 1 − g([Z, X], [Y,W ]) − g([Z,W ], [Y, X]) − g([Y, Z], [X,W ]) 2 2 2 = g(U (X, Z), U (Y,W )) − g(U (X,W ), U (Y, Z)) 1 1 1 + g([X,Y ], [Z,W ]) + g([X, Z], [Y,W ]) − g([X,W ], [Y, Z]) 2 4 4 1h + g([X, [Z,W ]],Y ) − g([Y, [Z,W ]], X) 4 i + g([Z, [X,Y ]],W ) − g([W, [X,Y ]], Z) .

t u

Setting in this theorem Z = Y and W = X, we get the following important Corollary 3.2.4. Let X,Y be Killing vector fields on a Riemannian manifold (M, g). Then g(R(X,Y )Y, X) = g(U (X,Y ), U (X,Y )) − g(U (Y,Y ), U (X, X)) 3 1 1 − g([X,Y ], [X,Y ]) − g([X, [X,Y ]],Y ) − g(X, [Y, [Y, X]]). 4 2 2

3.3 Killing Vector Fields of Constant Length In this section we will establish some properties of Killing vector fields of constant length on Riemannian manifolds M n and point out their connections with Clifford– Wolf translations. Note that in the book [200], Killing vector fields of constant length are called infinitesimal translations. It is possible to characterize Killing vector fields of constant length on Riemannian manifolds in terms of some special coordinate systems. It is not difficult to prove the validity of the following statements. (1) The existence of the Killing vector field X in a neighborhood of a point x of a Riemannian manifold (M n , g) with condition X(x) 6= 0 is equivalent to the existence of a local chart (U, ϕ,V ), where x ∈ U, ϕ(x) = 0 ∈ Rn , with canonical coordinate functions xi ◦ ϕ, i = 1, . . . , n, such that all components gi j ◦ ϕ −1 of metric tensor g do not depend on xn and X has components ξ i = δni . Thus, the local 1-parameter group of local isometries γ(t), generated by the vector field X, has the form ϕ(γ(t)(ϕ −1 (x1 , . . . , xn ))) = (x1 , . . . , xn−1 , xn + t).

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3 Isometric Flows and Killing Vector Fields on Riemannian Manifolds

(2) Under the above conditions, a curve c(t), −ε < t < ε, with the coordinate functions xi (t) := xi (ϕ(c(t)) = δni t, is a geodesic if and only if the function gnn (ϕ −1 (x1 , . . . , xn−1 , 0))

(3.7)

has a critical value at the point 0 = (0, . . . , 0). (3) Points near x are shifted to locally constant distances under the action of γ(t) if and only if the function (3.7) is constant in some neighborhood of the point 0. Statements (1)–(3) are proved in Section 72 of [200]. In [200], the following statement is also proved: Proposition 3.3.1. A unit vector field X on (M, g) is a (unit) Killing vector field if and only if the angle between the field X and the tangent vectors along any (oriented) geodesic in (M, g) is constant along this geodesic. Corollary 3.3.2. The integral (geodesic) curves of two infinitesimal translations form a constant angle along any of them. Corollary 3.3.3. Any two different trajectories of a unit Killing vector field X on (M, g), together with a geodesic, joining them, and the orbit of this geodesic relative to the 1-parameter isometry group of (M, g) generated by X, constitute a (possibly virtual) 2-dimensional submanifold S with the induced Riemannian metric, that is locally isometric to Euclidean plane. Moreover, S together with any of its points z contains two geodesics of (M, g) passing through z. We have the following statement (see p. 499 in [107]). Proposition 3.3.4. Let X be a Killing vector field on a Riemannian manifold (M, g). Then the following properties are equivalent: 1) X has constant length; 2) ∇X X = 0; 3) Every integral curve of the field X is geodesic in (M, g). Proof. It is sufficient to note that the length of the field X is constant along any of its integral curves and that for any smooth vector field Y on (M, g) the following equality holds: 0 = (LX g)(X,Y ) = X · g(X,Y ) − g([X, X],Y ) − g(X, [X,Y ]) = g(∇X X,Y ) + g(X, ∇X Y ) − g(X, [X,Y ]) 1 = g(∇X X,Y ) + g(X, ∇Y X) = g(∇X X,Y ) + Y · g(X, X). 2 The proposition is proved.

t u

Now, we consider some other well-known properties of Killing vector fields of constant length.

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127

Proposition 3.3.5. For any Killing vector field Z of constant length and all vector fields X,Y on a Riemannian manifold (M, g), the following equalities hold: g(∇X Z, ∇Y Z) = g(R(X, Z)Z,Y ) = g(R(Z,Y )X, Z), g(R(X, Z)Z, ∇Y Z) + g(R(Y, Z)Z, ∇X Z) = 0, g(∇Z ∇Y Z, ∇X Z) = g(R(X, Z)Z, ∇Z Y ). Proof. Let us prove the first equality. Since g(Z, Z) = const, we have X · g(Z, Z) = 2g(∇X Z, Z) = 0. Therefore, 0 = Y · g(∇X Z, Z) = g(∇Y ∇X Z, Z) + g(∇X Z, ∇Y Z). According to 3) of Proposition 3.1.6, we get g(∇Y ∇X Z, Z) = −g(R(Z,Y )X, Z) − g(∇Y X, ∇Z Z) = −g(R(Z,Y )X, Z), since ∇Z Z = 0. Then g(∇X Z, ∇Y Z) = g(R(Z,Y )X, Z). The formula g(∇X Z, ∇Y Z) = g(R(X, Z)Z,Y ) follows from symmetries of the curvature tensor. Let us prove the second equality. By the first equality, g(R(X, Z)Z, ∇Y Z) + g(R(Y, Z)Z, ∇X Z) = g(∇X Z, ∇∇Y Z Z) + g(∇Y Z, ∇∇X Z Z) = g(U, ∇V Z) + g(V, ∇U Z), where U = ∇X Z and V = ∇Y Z. Now, using 1) of Proposition 3.1.6, we get that g(U, ∇V Z) + g(V, ∇U Z) = 0. Further, since ∇Z Z = 0 and for any smooth vector field W , the equalities ∇Z ∇W = ∇W ∇Z + ∇[Z,W ] + R(Z,W ),

∇Z W = ∇W Z + [Z,W ]

hold, we get (using the first equality) g(∇Z ∇Y Z, ∇X Z) = g(∇[Z,Y ] Z, ∇X Z) + g(R(Z,Y )Z, ∇X Z) = g(R(X, Z)Z, [Z,Y ]) + g(R(Z,Y )Z, ∇X Z) = g(R(X, Z)Z, ∇Z Y ) − g(R(X, Z)Z, ∇Y Z) − g(R(Y, Z)Z, ∇X Z). On the other hand, we have proved that g(R(X, Z)Z, ∇Y Z) + g(R(Y, Z)Z, ∇X Z) = 0, Consequently, we obtain the third equality.

t u

Notice that Killing vector fields of constant length naturally arose in the study of various geometric constructions, for instance, they are an important ingredient of the definitions of K-contact manifolds and Sasaki manifolds (see. e.g. [48, 110, 124]). There are many obstructions to the existence of nontrivial Killing vector fields of constant length on a given Riemannian manifold. So, according to the well-known

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H. Hopf theorem, the necessary condition for the existence of such field on a compact manifold (M, g) is the equality χ(M) = 0 for the Euler characteristic. Moreover, in this case Theorem 2 from the work [119] by R. Bott permits us to state that all the Pontryagin numbers of orientable coverings of M are equal to zero (cf. Theorem 4.12.1).

3.4 Killing Vector Fields of Constant Length and Clifford–Wolf Translations For any Riemannian manifold (M, g), there exists a remarkable connection between the Killing vector fields of constant length and the Clifford–Wolf translations. Definition 3.4.1. A Clifford–Wolf translation on a Riemannian manifold (M, g) with an intrinsic metric d is an isometry s, moving all points in M one and the same distance, i.e. d(x, s(x)) ≡ const for all x ∈ M. Note that Clifford–Wolf translations are often called Clifford translations (see e.g. [483] or [291]), but we follow here the terminology of the paper [216]. The Clifford–Wolf translations naturally appear in the study of homogeneous Riemannian coverings of homogeneous Riemannian manifolds [483, 291]. Clifford– Wolf translations are studied in various papers (see e.g. [144, 187, 216, 381, 475, 476] for the Riemannian case and [174, 175, 176] for the Finsler case). For a detailed discussion we refer to [80] and [490]. Let us consider some natural constructions of Clifford–Wolf translations. Example 3.4.2. The following simple observation gives many examples of Clifford– Wolf translations on homogeneous Riemannian manifolds. Suppose that the Lie group G acts transitively on a Riemannian manifold (M, g) by isometries, and an isometry f on (M, g) commutes with G (in particular, f could be in the center of G), then f is a Clifford–Wolf translation on (M, g). Indeed, for x, y ∈ M, one can choose a ∈ G such that a(x) = y, then d(x, f (x)) = d(a(x), a( f (x))) = d(a(x), f (a(x))) = d(y, f (y)). In particular, if the center Z of the group G is not discrete then any 1-parameter subgroup in Z is a 1-parameter group of Clifford–Wolf translations on (M, g). The following two examples are partial cases of Example 3.4.2. Example 3.4.3. Let Cn be the n-dimensional complex vector space, supplied with the following inner product: if u = (u1 , . . . , un ), v = (v1 , . . . , vn ) are vectors in Cn , then (u, v) = ℜ ∑nl=1 ul vl , where vl denotes the complex conjugation of vl . Then the unit sphere S2n−1 = {u ∈ Cn : (u, u) = 1} can be treated as the unit sphere in Euclidean space E2n .

3.4 Killing Vector Fields of Constant Length and Clifford–Wolf Translations

129 T

The unitary group U(n) of all complex (n × n)-matrices A such that AA = En , T where En is the unit (n × n)-matrix, A is obtained from A by complex conjugation of its elements with successive transposition, acting by left multiplication on column vectors in Cn , consists of isometries of the space (Cn , (·, ·)). The multiplications of vectors in Cn by complex numbers z such that |z| = 1 constitutes the isometry group S1 of (Cn , (·, ·)). By definition of the complex vector structure in Cn , S1 commutes with the mentioned action of U(n). Really S1 can be treated as the subgroup of diagonal matrices in U(n), which is the center of U(n). Then U(n) acts transitively on S2n−1 by isometries, so S1 is a group of Clifford–Wolf translations on S2n−1 . The next example is a variation of Example 3.4.3. Example 3.4.4. Let Hn be the left (respectively, right) n-dimensional quaternion vector space, supplied with the following inner product: if u = (u1 , . . . , un ), v = (v1 , . . . , vn ) are vectors in Hn , then (u, v) = ℜ ∑nl=1 ul vl , where vl denotes the quaternion conjugation of vl . Then the unit sphere S4n−1 = {u ∈ Hn : (u, u) = 1} can be treated as the unit sphere in the Euclidean space E4n . The compact symplectic group Sp(n) of all quaternion (n × n)-matrices A such T T that AA = En , where En is the unit (n × n)-matrix, A is obtained from A by quaternion conjugation of its elements with successive transposition, acting by right (respectively, left) multiplication on row vectors (respectively, column vectors) in Hn , consists of isometries of the space (Hn , (·, ·)). The left (respectively, right) multiplications of vectors in Hn by quaternions q such that |q| = 1 constitutes the isometry group F1 (respectively, F2 ) of (Hn , (·, ·)). Note that the formula g 7→ g−1 defines an isomorphism of the group F1 onto F2 and an isomorphism of the group F2 onto the group F1 . By definition of the left (respectively, right) quaternion vector structure in Hn , F1 (respectively, F2 ) commutes with the mentioned right (respectively, left) action of Sp(n). Then the mentioned right (respectively, left) action of Sp(n) is transitive and isometric on S2n−1 so F1 (respectively, F2 ) is a group of Clifford–Wolf translations on S2n−1 . Evidently, the groups F1 and F2 mutually commute with each other. Note that a few classical Riemannian manifolds admit 1-parameter groups of Clifford–Wolf translations. For instance, it is known that among compact simply connected irreducible symmetric spaces only the odd-dimensional spheres, the spaces SU(2m)/Sp(m), m > 1, and simple compact Lie groups, supplied with some bi-invariant Riemannian metrics, admit 1-parameter groups of Clifford–Wolf translations [476]. The following proposition is obvious. Proposition 3.4.5. Assume that a 1-parameter isometry group γ(t) on (M, g), generated by a Killing vector field X, consists of Clifford–Wolf translations. Then X has constant length.

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Example 3.4.6. Let K be a compact subgroup of a Lie group G, then H = diag(K) = {(a, a) | a ∈ K} is a compact subgroup of the group G × K. It is easy to see that even G in G × K acts transitively on the homogeneous space M = (G × K)/H. But K in G × K commutes with G, hence, K consists of Clifford–Wolf translations on M for every (G × K)-invariant Riemannian metric due to the arguments in Example 3.4.2. Note also that all Killing fields corresponding to K have constant length. The above example can be generalized as follows. Example 3.4.7. Consider a compact simple Lie group F and compact Lie groups G1 , G2 , . . . , Gl such that every Gi has a subgroup isomorphic to F × Ki . Then the group F × G1 × · · · × Gl acts naturally on M = G1 /K1 × · · · × Gl /Kl : (a, b1 , b2 , . . . , bl ) · (c1 K1 , c2 K2 , . . . , cl Kl ) → (b1 c1 K1 a−1 , b2 c2 K2 a−1 , . . . , bl cl Kl a−1 ). Hence, M = (F × G1 × · · · × Gl )/(K1 × · · · × Kl × diag(F)). It is clear that for any (F × G1 × · · · × Gl )-invariant Riemannian metric g, F consists of Clifford–Wolf translations on (M, g) and every Z ∈ f has constant length on (M, g), because the group G1 × · · · × Gl is transitive on (M, g). Proposition 3.4.5 admits a partial converse. More exactly, the following result holds. Proposition 3.4.8. Suppose that a Riemannian manifold (M, g) has the injectivity radius bounded from below by a positive constant (in particular, this condition is satisfied for any compact or any homogeneous manifold) and X is a nontrivial Killing vector field of constant length on (M, g). Then the isometries γ(t) from the 1-parameter isometry group, generated by the vector field X, are Clifford–Wolf translations if t is sufficiently close to 0. Proof. Without loss of generality we can assume that X is a unit Killing vector field. Assume that the injectivity radius of Riemannian manifold (M, g) is bounded from below by a number δ > 0. By Proposition 3.3.4, the integral trajectories of the vector field X or, equivalently, the orbits of the 1-parameter isometry group γ(t), t ∈ R, generated by this vector field, are geodesics in (M, g). Choose s such that |s| < δ . Then for any point x ∈ M, the geodesic segment γ(t)(x), 0 ≤ t ≤ s, joining the points x and γ(s)(x) is a shortest curve. Therefore, d(x, γ(s)(x)) = s for every point x ∈ M, i.e. the isometry γ(s) moves all points of the manifold one and the same distance. t u Remark 3.4.9. Notice that Proposition 3.4.8 is not true in general if the injectivity radius is not bounded from below by a positive constant. Indeed, one can construct a (non-compact) Riemannian manifold (M, g) with Killing field X of constant length such that: For the 1-parameter isometry group γ(t), t ∈ R, generated by X, the isometry γ(s) is not a Clifford–Wolf translation for every s > 0. The first step is to construct a manifold M that admits an effective action of S1 with no fixed point such that for any n ∈ N, there is a point x ∈ M with the isotropy

3.4 Killing Vector Fields of Constant Length and Clifford–Wolf Translations

131

subgroup Zn . It should be noted that an example of a manifold M with the above action of S1 (due to D. Montgomery) can be found in the paper [495]. Then we can consider a Riemannian metric g1 on M such that the above action of S1 is isometric. If X is a vector field, generated by the action of S1 , then it is a Killing vector field on (M, g1 ), hence, is a unit Killing vector field on (M, g),
−1 where g = g1 (X, X) g1 (compare with Theorem 3.7.3). Let us show that we get a suitable example. Let γ(t), t ∈ R, be a 1-parameter isometry group of (M, g), generated by X. It is easy to see that the isometry γ(s) is not a Clifford–Wolf translation for every s > 0. Indeed, it follows from the fact that M contains some points with the isotropy subgroup Zn , where 2π/n < s. This argument also implies that (M, g) has an injectivity radius that is not bounded from below by a positive number. Generally speaking, under the conditions of Proposition 3.4.8, not all isometries from the 1-parameter isometry group γ(t), generated by a Killing vector field X of constant length, are Clifford–Wolf translations. This particularly demonstrates results of the following sections of this chapter. Definition 3.4.10. Assume that a Killing vector field X on a Riemannian manifold (M, g) has constant length. Then X is called regular (quasiregular) if all integral curves of the vector field X are closed and have one and the same length (respectively, if there are integral curves of different length). It is clear that a Killing vector field of constant length on a Riemannian manifold is regular if and only if it is generated by a free isometric action of the group S1 . A Killing vector field of constant length on a Riemannian manifold is quasiregular if and only if it is generated by an almost free (i.e. when all points have finite stabilizers and there exist points with nontrivial stabilizers) isometric action of the group S1 . In this case any point with trivial (respectively, nontrivial) stabilizer is called regular (respectively, singular). Proposition 3.4.11. Let X be a nontrivial Killing vector field on a Riemannian manifold M. If the 1-parameter isometry group γ(t), t ∈ R, generated by the field X consists of Clifford–Wolf translations, then the field X has constant length and it is either regular or all of its integral trajectories are not closed. Proof. According to Proposition 3.4.5, the field X has constant length. It is clear that all integral trajectories of the field X coincide with orbits of the group γ(t), t ∈ R. Assume that γ(t)(x) = x for some point x ∈ M and t > 0. There exists the minimal number t∗ among all numbers τ > 0 such that γ(τ)(x) = x. Therefore, γ(t)(x) 6= x for t ∈ (0,t∗ ). Since γ(t∗ ) is a Clifford–Wolf translation, we have γ(t∗ )(y) = y for every point y ∈ M. Now, if γ(t)(z) = z for a point z ∈ M and t ∈ (0,t∗ ), then γ(t)(x) = x, since γ(t) is also a Clifford–Wolf translation. Thus, all points in the manifold (M, g) have one and the same positive period t∗ relative to the action of the group γ(t). Since the length of the field X is constant, this means that all integral trajectories of the field X have one and the same length, i.e. the field X is regular if at least one orbit is closed. t u

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One can easily construct examples that show that both situations in the statement of Proposition 3.4.11 are possible for one and the same space. For instance, consider a Riemannian manifold (M, g) with a group of Clifford–Wolf translations, isomorphic to S1 × S1 : the flat torus S1 × S1 or the direct metric product of two odd-dimensional round spheres are such manifolds. If we choose in S1 × S1 some subgroup, isomorphic to S1 (rational torus winding), then it is clear that the Killing vector field (M, g), tangent to this subgroup, is regular. If, however, we choose some non-closed subgroup R (an irrational torus winding) in S1 × S1 , then the Killing vector field on (M, g), generated by this subgroup, has no closed integral curve.

3.5 Killing Vector Fields and Curvatures In this section, we will give (based on the presentation in the paper [77]) some results (many of them are well known) about the dependence of curvature characteristics of a Riemannian manifold (M, g) on the presence of a Killing field of a certain form on (M, g). Theorem 3.5.1 ([77]). Let X be a Killing vector field on a Riemannian manifold (M, g), generating a 1-parameter isometry group γ(s), s ∈ R, of (M, g), x ∈ M a point of nonzero local minimum (maximum) of the length g(X, X)1/2 of the field X, and Y (s) (s ∈ R) be a vector field along geodesic c(s) = γ(s)(x), defined by the formula Y (s) = T (γ(s))(w), where w ∈ Mx . Then g(Y,Y ) ≡ g(w, w),

g(Y, X) ≡ g(w, X(x)),

(3.8)

the expressions g(∇X Y, ∇X Y ) and g(R(X,Y )Y, X) do not depend on s ∈ R, moreover, g(∇Y X, ∇Y X) = g(∇X Y, ∇X Y ) ≥ (≤) g(R(X,Y )Y, X)

(3.9)

on the geodesic c(s) = γ(s)(x), s ∈ R. Proof. The equalities (3.8) and the independence of indicated expressions on s are evident. Define a smooth mapping C : R × R → M,

C(s,t) = Exp(tY (s)).

For the 1-parameter isometry group γ(s), s ∈ R, on (M, g), generated by X, we have the obvious equality γ(s1 )(C(s,t)) = γ(s1 )(Exp(tY (s))) = γ(s1 )(Exp(tdγ(s)(w))) = Exp(dγ(s1 )(dγ(s)(tw))) = Exp(tdγ(s1 + s)(w))) = C(s + s1 ,t). Consequently, γ(s1 )(C(s,t)) = C(s + s1 ,t). Thus, we have

(3.10)

3.5 Killing Vector Fields and Curvatures

 TC

∂ ∂s

133



 = X(C(s,t)),

TC

∂ ∂t

 = Y (s,t),

where Y (s,t) is an extension of the vector field Y (s) = Y (s, 0). This means that vector fields X and Y are tangent vector fields along the mapping C; and also   ∂ ∂ ∇X Y − ∇Y X = T C , = 0. (3.11) ∂ s ∂t It is clear that for any fixed s, the curve cs (t) = C(s,t), t ∈ R, is a geodesic in (M, g) with tangent vector field Ys (t) = Y (s,t). According to 2) of Theorem 3.1.7, we have 1 2 ∇ g(X, X) = g(∇Y X, ∇Y X) − g(R(X,Y )Y, X) 2 Y for all points of the curve c(s) = γ(s)(x), s ∈ R. On the other hand, the point x ∈ M is a point of nonzero local minimum (maximum) of the squared length g(X, X) of the field X; each point c(s) = γ(s)(x), s ∈ R, has the same property. Therefore, 1 2 ∇ g(X, X) = g(∇Y X, ∇Y X) − g(R(X,Y )Y, X) ≥ (≤) 0 2 Y for all points c(s) = γ(s)(x), s ∈ R, as required.

t u

In the case of a unit Killing vector field, Theorem 3.5.1 immediately implies the following statement. Theorem 3.5.2. Let X be a unit Killing field on a Riemannian manifold (M, g), generating a 1-parameter isometry group γ(s), s ∈ R, of (M, g) and Y (s) (s ∈ R) be a vector field along the curve c(s) = γ(s)(x), x ∈ M, defined by the formula Y (s) = d(γ(s))(w), where w ∈ Mx . Then g(Y,Y ) ≡ g(w, w),

g(Y, X) ≡ g(w, X(x)),

g(∇Y X, ∇Y X) = g(∇X Y, ∇X Y ) = g(R(X,Y )Y, X) = const .

(3.12) (3.13)

In particular, if g(w, w) = 1 and w ⊥ X(x), then g(∇Y X, ∇Y X) = g(∇X Y, ∇X Y ) = K(X,Y ) = const .

(3.14)

Using Theorem 3.5.2, one can easily obtain Corollary 3.5.3. Under conditions of Theorem 3.5.2, for any point x ∈ M, we have K(X(x), w) ≥ 0,

where

w ∈ Mx ,

|w| = 1,

w ⊥ X(x),

(i.e. at each point x ∈ M, the sectional curvature of any two-plane containing the vector X(x) is nonnegative). Moreover, this inequality turns into an equality if and only if T γ(t)(w) is a parallel vector field along the geodesic γ(t)(x).

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Corollary 3.5.4. On a Riemannian manifold M of negative Ricci curvature, there is no nontrivial Killing vector field of constant length. Theorem 3.5.5. Every nontrivial parallel vector field X on a Riemannian manifold M n is a Killing vector field of constant length [200], and Ric(X, X) = 0. Moreover, M n is locally a direct metric product of a one-dimensional manifold, tangent to the field X, and its orthogonal complement. If the Riemannian manifold M n is complete, e n = Pn−1 × R, where the e n is a direct metric product M then its universal covering M e projecting to the field X by the natural projection M e n → M n , is tangent to field X, the R-direction. Proof. It is clear that the parallelism condition ξi, j = 0 for the field X (ξi are covariant components of the field X in some local coordinate system) implies the equality ξi, j + ξ j,i = 0. This means that X is a Killing vector (see Proposition 3.1.5). The fact that the length of the parallel vector field X has constant length and the equality for the Ricci curvature Ric(X, X) = 0 are evident. Since X is a parallel vector field, the distribution orthogonal to X is also parallel and involutive on M n . Now the last assertion of the theorem follows from the de Rham decomposition theorem (§6 in [290, Chapter IV]). t u Theorem 3.5.6. Let X be a unit Killing vector field on an n-dimensional Riemannian manifold M n . Then the Ricci curvature Ric of the manifold M n satisfies the condition Ric(X, X) ≥ 0. Moreover, the equality Ric(X, X) ≡ 0 is equivalent to the parallelism of the vector field X. Consequently, in this case all assertions of Theorem 3.5.5 hold. Proof. The first assertion of the theorem is an obvious consequence of the equality (3.14). Suppose now that Ric(X, X) ≡ 0. By the equality (3.14), for any point x ∈ M n and vector w ∈ Mx such that w ⊥ X(x), we have K(X(x), w) = 0 and ∇w X(x) = 0. Besides, ∇X X ≡ 0. Thus, the field X is parallel on M n . t u Theorem 3.5.6 can also be obtained using the formula n

(∆ f )x = ∑ g(∇Vi X, ∇Vi X) − Ric(X, X), i=1

where X is an infinitesimal affine transformation of the Riemannian manifold (M, g), f = 12 g(X, X), and V1 , . . . ,Vn is an orthonormal basis of Mx (see Lemma 2, §4, Chapter II in [290]). Theorem 3.5.6 immediately implies Corollary 3.5.7. For a Killing vector field X on a Riemannian manifold M with nonpositive Ricci curvature, the following two conditions are equivalent: 1) X has constant length; 2) X is a parallel vector field on M. For manifolds with non-positive sectional curvature the previous corollary can be strengthened.

3.5 Killing Vector Fields and Curvatures

135

Proposition 3.5.8. For a Killing vector field X on a complete Riemannian manifold (M, g) with non-positive sectional curvature, the following three conditions are equivalent: 1) the length of X is bounded on (M, g); 2) X has constant length; 3) X is a parallel vector field on (M, g). Proof. By Corollary 3.5.7, it suffices to show that any Killing vector field of bounded length has constant length. Suppose, that the length of a Killing vector field X is bounded on (M, g). According to 2) of Theorem 3.1.7, for each geodesic c(t), t ∈ R, in (M, g) the equality h00 (t) = g(∇τ(t) X, ∇τ(t) X) − g(R(X, τ(t))τ(t), X)   holds, where h(t) = 12 g X(c(t)), X(c(t)) . Since the sectional curvature of the manifold (M, g) is non-positive, h00 (t) ≥ 0 for t ∈ R, thus, h is a convex function. Moreover, since X has bounded length, h is also bounded on R. Consequently, h(t) = const for t ∈ R. There is a geodesic passing through given two points on a (complete) Riemannian manifold (M, g). Therefore, the Killing field X has constant length. t u Remark 3.5.9. Note that Proposition 3.5.8 can be easily deduced from Theorem 1 in [479], which, in particular, states that each bounded isometry on a simply connected Riemannian manifold M of nonpositive sectional curvature is a Clifford–Wolf translation. Using Theorem 3.5.1, we get the following well-known result. Theorem 3.5.10 (Berger’s theorem, [96]). Every Killing vector field on a compact even-dimensional Riemannian manifold (M, g) of positive sectional curvature vanishes at some point of M. Proof. Suppose that the field X has no zero on (M, g), and let γ(s), s ∈ R, be a 1parameter isometry group, generated by the field X. Since M is compact, the length of the Killing field X attains its absolute minimum at some point x ∈ M; moreover, g(X(x), X(x)) 6= 0. According to Theorem 3.5.1, for every vector w ∈ Mx , on the geodesic c(s) = γ(s)(x), s ∈ R, we have the inequality g(∇X Y, ∇X Y ) ≥ g(R(X,Y )Y, X), where Y = Y (s) = T γ(s)(w) (s ∈ R) is a vector field along the geodesic c = c(s) = γ(s)(x). Define the linear operator AX : (Mx , gx ) → (Mx , gx ) by the formula τ−s (dγ(s)(w)) − w , s→0 s

AX (w) = (∇X Y )(x) = lim

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where τ−s : (Mc(s) , gc(s) ) → (Mx , gx ) is the parallel translation along the segment c(σ ), 0 ≤ σ ≤ s, from the endpoint of the geodesic c to its origin. Since T γ(s) and τ−s are linear isometries of Euclidean vector spaces, preserving the restriction of the vector field X to c, it follows that AX is a skew-symmetric endomorphism; furthermore, AX (X(x)) = 0. Consequently, the restriction of the operator AX to the unit Euclidean sphere Sn−2 ⊂ (Mx , gx ) orthogonal to X(x) determines a Killing field on Sn−2 (n = dim M). However, since n is an even number, there exists a point w ∈ Sn−2 such that AX (w) = 0. This follows, for example, from the fact that χ(Sn−2 ) = 2 6= 0 or from the fact that each operator on an odd-dimensional real vector space has an eigenvector. For such a point w we get the inequality 0 = g(AX (w), AX (w)) ≥ g(R(X(x), w)w, X(x)), which is impossible, because the sectional curvature of (M, g) is positive.

t u

Corollary 3.5.11. Under the conditions of Theorem 3.5.2 and the notation of Theorem 3.5.10, the formula Z(w) = AX (w) = ∇X Yw (x), where Yw (s) = T γ(s)(w) (w ∈ Mx , |w| = 1, w ⊥ X(x)), determines a Killing vector field of rotations on the unit Euclidean sphere Sn−2 ⊂ (Mx , gx ) orthogonal to X(x), where n = dim M. Also we have the following constraint to the length of the Killing vector field Z on Sn−2 : |Z(w)| = K(X(x), w). In turn, this corollary implies Corollary 3.5.12. Let M be an n-dimensional Riemannian manifold with a unit Killing vector field X. If n is even then at each point x ∈ M, there exists a unit vector w ⊥ X(x) such that K(X(x), w) = 0. Consequently, if there is a point x ∈ M such that the sectional curvature in the direction of any two-plane, containing the vector X(x), is positive, then n is odd. Corollary 3.5.13. Each complete two-dimensional Riemannian manifold M with a unit Killing vector field X is locally Euclidean. Consequently, M is isometric to the Euclidean plane or to one of the flat complete surfaces: the cylinder, the torus, the M¨obius strip, or the Klein bottle. Moreover, the field X is parallel. In the last two cases X is quasiregular and has a unique singular circle trajectory, and X is determined up to multiplication by −1. In all other cases the field X can have any direction. Notice that in [290], Berger’s theorem is deduced from the following Theorem 3.5.14 (Chapter II, theorem 5.6 in [290]). Let M be a compact orientable Riemannian manifold of positive sectional curvature, and let f be an isometry of M. 1) If n = dim M is even and f preserves orientation, then f has a fixed point. 2) If n = dim M is odd and f changes orientation, then f has a fixed point. Remark 3.5.15. In [464], Theorem 3.5.14 is generalized to the case of an arbitrary conformal transformation f .

3.6 Isometric Flows and Points with Finite Period

137

Another proof of Berger’s theorem, close to the original proof in [96], is presented in the book [290]. One more proof is given in (Berger’s) Lemma 2.2 and Corollary 2.1 from [455]. As a corollary of Theorem 3.5.14, we easily obtain Synge’s theorem 1.5.80. Theorem 3.5.1 implies some other interesting consequences. Corollary 3.5.16 (Bochner’s theorem, [111]). If the Ricci curvature of a compact Riemannian manifold M is negative, then M admits no nontrivial Killing vector field. For the proof, it suffices (in Theorem 3.5.1) to consider as the point x ∈ M a point of the absolute maximum of the length of the Killing field X. Observe that Bochner’s theorem follows also from the more general Theorem 4.15.14. Corollary 3.5.17 (Corollary 4.2 of Chapter II in [290]). If X is a Killing vector field on a compact Riemannian manifold M with non-positive Ricci curvature, then X is parallel and, consequently, has constant length. From this corollary and Theorem 3.5.6 we get Corollary 3.5.18. Every compact Riemannian manifold M with non-positive Ricci curvature and a nontrivial Killing field has infinite fundamental group.

3.6 Isometric Flows and Points with Finite Period The content of this and the following sections in this chapter is based on [75] and [77]. In this section we obtain some results on the structure of the set of all points of finite period relative to isometric flows on Riemannian manifolds. We also present a detailed description of properties of isometric flows on Riemannian manifolds, generated by Killing vector fields all of whose trajectories are closed (Theorem 3.6.9). Consider a smooth action µ : R × M → M of the additive group of real numbers on a smooth manifold M. We define the periodicity function Pµ : M → R ∪ {∞} of the flow µ as follows:

Pµ (x) =

 

0, if µ(s, x) = x for all s > 0, t > 0, if µ(t, x) = x and µ(s, x) 6= x for all 0 < s < t,  ∞, if µ(s, x) 6= x for all s > 0.

(3.15)

Analogously, given a flow µ, we define the set Mµ (t) Mµ (0) = Pµ−1 (0),

Mµ (t) = {x ∈ M | µ(t, x) = x} for t > 0.

(3.16)

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Lemma 3.6.1. Let µ : R × M → M be a non-trivial isometric flow on a Riemannian manifold (M, g). Then for every t ∈ [0, ∞), the set Mµ (t) is either empty or a closed (possibly, disconnected) totally geodesic submanifold of (M, g). Moreover, for nonempty Mµ (t) the following assertions hold: 1) Each connected component of Mµ (0) has an even codimension in M. 2) If Mµ (t) 6= M for some t ∈ [0, ∞), then each connected component of Mµ (t) has codimension ≥ 1 in M. 3) If the manifold M is orientable then for every t ∈ [0, ∞) each connected component of submanifold Mµ (t) has an even codimension in M. Proof. The fact that Mµ (0) is a closed totally geodesic submanifold of (M, g) (with each connected component having even codimension) is proved in [288]. The idea of the proof is that Mµ (0) is the zero set of the Killing vector field V , generating the flow µ. The codimension of each connected component of Mµ (0) in M is even because the image of a skew-symmetric operator on Euclidean space must have even dimension. Consider now some t > 0. Since µ(t) is an isometry of the Riemannian manifold (M, g), the set of fixed points Mµ (t) of this isometry is a closed totally geodesic submanifold in M. This statement is also well known (see e.g. [291, Chapter VII, §8]), but we shall give a sketch of its proof to obtain Assertion 3) of the lemma. It is clear that the set Mµ (t) is closed in M. Consider some point x ∈ Mµ (t) and the differential Q : Mx → Mx of the isometry I = µ(t) at the point x. Consider now the subspace L in Mx , where Q is identical. Let L0 be the set of vectors in L of length less than r0 , where r0 > 0 is chosen so that the exponential mapping expx : Mx → M is injective on the ball of radius r0 with the center at the origin. Then V0 , the image of L0 relative to this exponential mapping, coincides with the set of points in Mµ (t) at distance < r0 from the point x. Since expx is a bijection between sets L0 and V0 , we see that Mµ (t) is indeed a submanifold in M. Take an arbitrary point y ∈ Mµ (t) such that d(x, y) < r0 . Then there exists a unique shortest arc joining x and y, moreover, this shortest arc is invariant under the isometry I = µ(t), i.e. its points belong to Mµ (t). Since the point x ∈ Mµ (t) can be chosen arbitrarily, Mµ (t) is a totally geodesic submanifold in M. If Mµ (t) 6= M then any connected component of Mµ (t) has codimension ≥ 1, because an isometry, identical on some open subset in M, is identical everywhere. It remains to prove the statement about orientable manifolds M. In this case the mapping I = µ(t) does not change orientation; therefore its differential Q also preserves orientation in Mx . Finally, note that the complementary subspace to L (on which Q is identical) must be even-dimensional. t u The following result clarifies the behavior of the periodicity function of an isometric flow on a Riemannian manifold. Theorem 3.6.2 (V. Ozols, [380]). Let µ : R × M → M be a non-trivial isometric flow on a Riemannian manifold (M, g) and let Pµ be the periodicity function of this flow. Then every point x ∈ M has a neighborhood on which the periodicity function Pµ takes only a finite number of values.

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Later we will need the following Lemma 3.6.3. Let Q be a non-trivial orthogonal transformation of Euclidean space Rn . Define the following subsets in Rn : Q f = {x ∈ Rn | Qm (x) = x for some m ∈ N},

Q∞ = Rn \ Q f .

Then the following assertions hold: 1) Q f is a Q-invariant linear subspace in Rn . 2) There is a unique p ∈ N such that Q p |Q f = Id and Qm |Q f 6= Id for 1 ≤ m < p. 3) If p > 1 then there exists a y ∈ Q f such that Q p (y) = y and Qm (y) 6= y for 1 ≤ m < p. 4) For each y ∈ Q∞ , the closure of the set {Qm (y) | m ∈ Z} in Rn contains a torus T l of dimension l ≥ 1. 5) If Q∞ 6= 0/ then Q f has codimension ≥ 2 in Rn . 6) For any natural number l < p, where p is the number from 2), the set Ql = {x ∈ Rn | Ql (x) = x} is a Q-invariant linear subspace in Q f and Ql 6= Q f . The subspace Ql has codimension ≥ 2 in Rn or Q(v) = −v for every vector v ⊥ Ql , where v ∈ Rn . Proof. The first assertion is obvious. Let e1 , e2 , . . . , es be a basis in Q f and s = dim(Q f ). For each 1 ≤ i ≤ s, let ai be the minimal natural number such that Qai (ei ) = ei . Let p ∈ N be the least common multiple of all numbers ai for 1 ≤ i ≤ s. It is clear that p is exactly the number whose existence and uniqueness constitutes the assertion 2) of the lemma. The assertion 3) follows from 2) and the fact that the set {x ∈ Q f | Qm (x) = x} for 1 ≤ m < p is an eigenspace in Q f . Now we shall prove the fourth assertion. Let G be the closure of the set of orthogonal transformations {Qm | m ∈ Z} in the group O(n). Since y ∈ Q∞ , G is a closed non-discrete commutative subgroup in O(n). Now, let Oy be the orbit of the point y ∈ Rn relative to the action of the group G. It is obvious that Oy is the closure of e be the unit component of the group G and the set {Qm (y) | m ∈ Z} in Rn . Now, let G n e e It let Oy be the orbit of the point y ∈ R with respect to the action of the group G. e e is clear that Oy ⊂ Oy . Since Oy is a homogenous space of the connected compact e Q ey is homeomorphic to the torus T l ; moreover, l ≥ 1, commutative Lie group G, ey reduces to a point and y ∈ Q f . since otherwise O Assume now that Q∞ 6= 0, / and let P be the (nontrivial) complement to Q f in Rn . It is clear that P is Q-invariant. If P is one-dimensional then Q2 (x) = x for every x ∈ P, i.e. P ⊂ Q f , which is impossible. Thus, P has dimension ≥ 2, which proves the fifth assertion. The assertion 6) can be easily proved using a similar argument. t u

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Theorem 3.6.4. Let µ : R × M → M be a non-trivial smooth isometric action of the additive group of real numbers on a connected Riemannian manifold (M, g) and P = Pµ be the periodicity function of the flow µ. Assume that M∞ = P−1 (∞) 6= 0. / Then the set M f = P−1 ([0, ∞)) is closed in M and has zero Lebesgue measure, while M∞ is open, connected and dense in M. Proof. Let us show that for every point x ∈ M, there exists an rx > 0 such that the intersection of M f with the closed ball B(x, rx ) = {y ∈ M | d(y, x) ≤ rx } is a closed set in M of zero measure. According to Theorem 3.6.2, there is an rx > 0 such that the periodicity function P takes in the ball B(x, rx ) only a finite set of values. Let 0 ≤ t1 < t2 < · · · < ts be all finite values of P in this ball. Now, consider the sets Mµ (ti ) (see (3.16)) for 1 ≤ i ≤ s. By Lemma 3.6.1, each set Mµ (ti ) is a closed totally geodesic submanifold in M; moreover, each connected component of Mµ (ti ) has codimension ≥ 1 in M. Put U = ∪si=1 Mµ (ti ). Then B(x, rx ) ∩ M f = B(x, rx ) ∩U =

s  [

 B(x, rx ) ∩ Mµ (ti ) .

i=1

Since B(x, rx ) ∩ Mµ (ti ) is closed and has zero measure in M for all 1 ≤ i ≤ s, we obtain the required result. In order to prove that the set M∞ is connected if M∞ 6= 0, / it suffices to show (by Theorem 3.6.2 and Lemma 3.6.1) that an arbitrary connected component of each totally geodesic submanifold Mµ (t), t > 0, has codimension ≥ 2 in M. Indeed, in this case the topological dimension of the set M f does not exceed n − 2, where n is the dimension of the manifold M (see e.g. [256, Chapter 9, §7]). If Mµ (t) has codimension ≤ 1 in M in a neighborhood of a point x ∈ Mµ (t), then it is not difficult to see that the set Mµ (2t) contains some neighborhood of the point x in M (see the proof of 5) in Lemma 3.6.3). But the latter means that M f = Mµ (2t) = M which contradicts the condition M∞ 6= 0. / t u Remark 3.6.5. In order to prove that the topological dimension of the set M f does not exceed n − 2, it is possible also to apply the third assertion of Lemma 3.6.1 to e of M and the flow µ e→M e generated by e : R×M the oriented two-fold covering M the flow µ. Observe that the conditions of the previous theorem do not necessarily imply that the set M f is a totally geodesic submanifold in (M, g). This can be demonstrated by the following example. Let µ(t), t ∈ R, be an isometric flow on the Euclidean space M = R4 given as follows: µ(t) is the orthogonal transformation defined in the standard basis {ei }, 1 ≤ i ≤ 4, by the matrix A(t) = diag(A1 (t), A2 (t)), where     cos(t) sin(t) cos(πt) sin(πt) A1 (t) = , A2 (t) = . − sin(t) cos(t) − sin(πt) cos(πt) It is easy to see that in this case M f is a union of two two-dimensional linear subspaces, one of which is spanned by the vectors e1 and e2 and the second by vectors e3 and e4 . This example shows that the existence of fixed points of the isometric

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flow under consideration can be an obstacle for M f to be totally geodesic. We will show in Theorem 3.6.8 that M f is indeed a totally geodesic submanifold in (M, g) when there is no such point. Theorem 3.6.6. Let µ : R × M → M be a non-trivial smooth action of the additive group of real numbers on a smooth manifold M. Suppose that g is a Riemannian metric on M such that for some t > 0, the diffeomorphism µ(t) is an isometry of the Riemannian manifold (M, g) and there is a point x ∈ M such that P(x) = t for the periodicity function P = Pµ of the flow µ. Consider M f = P−1 ([0, ∞)). Then there exist T > 0 and a neighborhood U of the point x in M such that the following hold: 1) T is divisible by t and so µ(T ) is an isometry of (M, g). 2) M f ∩ U = Mµ (T ) ∩ U, where Mµ (T ) = {y ∈ M | µ(T, y) = y} is a closed totally geodesic submanifold in (M, g). 3) Every neighborhood of x contains a point y ∈ M f such that P(y) = T . Proof. First we introduce some necessary notation. For any point y ∈ M, denote by Oy the orbit of this point under the action of the 1-parameter diffeomorphism group µ(s), s ∈ R. For the point x from the statement of the theorem, choose a number rx > 0 such that the following conditions are fulfilled: 1) the exponential mapping Expx : Mx → M is injective on the ball of radius rx with the center at the origin; 2) no orbit Oy , y ∈ M, lies entirely in the ball U = {y ∈ M | d(x, y) < rx }. The existence of such a number rx can be easily deduced from the conditions of the theorem. Now we will show that U is the neighborhood of the point x whose existence is proclaimed in the theorem. Since t = P(x) > 0, x is a fixed point of the isometry I := µ(t). Now, let Q : Mx → Mx be the differential of the isometry I at the point x. Identifying Mx with the Euclidean space Rn , we can use Lemma 3.6.3 for the orthogonal mapping Q. Set Q f = {z ∈ Mx | Qm (x) = x for some m ∈ N},

Q∞ = Mx \ Q f .

Consider the number p ∈ N whose existence is stated in Lemma 3.6.3 and put T = t · p. We will show that the just defined number is exactly the number whose existence is proclaimed in the theorem. Consider a nonzero vector z ∈ Mx of length < rx . This vector is mapped by the exponential mapping Expx : Mx → M to some point y ∈ M. It is clear that Expx (Qm (z)) = I m (y) for every integer m, because the exponential mapping of Riemannian manifolds commutes with isometries. If z ∈ Q f then by the choice of the number p we have Q p (z) = z, and therefore I p (y) = y; i.e. y ∈ Mµ (T ). Suppose now that z ∈ Q∞ . We will show that y = Expx (z) is not contained in the set M f . Assume the contrary. Then the orbit Oy of the point y with respect to the flow µ is compact. Let S be the closure in Mx of the orbit {Qm (z) | m ∈ Z}. It is clear that all vectors in this orbit have length < rx . According to Lemma 3.6.3, S contains some torus T l of dimension l ≥ 1. Since the orbit Oy is compact, Exp(T l ) ⊂ Expx (S) ⊂ Oy .

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By injectivity of Expx : Mx → M on the ball of radius rx with the center at the origin, Exp(T l ) is itself a torus of dimension l ≥ 1. Since the orbit Oy is homeomorphic to the circle, we obtain that l = 1 and Oy is a homeomorphic image of the circle T l = T 1 under Expx . Consequently, the orbit Oy entirely belongs to the neighborhood U of the point x, which is impossible by the choice of this neighborhood. Thus, the set U ∩ M f is the image of the set of vectors z in Q f of length < rx under the exponential mapping Expx . Consequently, M f ∩ U = Mµ (T ) ∩ U, where Mµ (T ) = {y ∈ M | µ(T, y) = y} is a closed totally geodesic submanifold in (M, g) by Lemma 3.6.1. It remains to prove that in any neighborhood of the point x, there is a point y ∈ M f such that P(y) = T . If p = 1 then T = t and all is obvious. Assume that p > 1. Then by point 3) of Lemma 3.6.3, there exists a vector z ∈ Q f such that Q p (z) = z and Qm (z) 6= z for 1 ≤ m < p. Using a homothety, we can make the length of the vector z arbitrary small and so less than rx > 0. It is clear that P(y) = T for the point y = Expx (z). The proof is complete. t u One can easily deduce from the last theorem the following one. Theorem 3.6.7. Suppose that in the conditions of Theorem 3.6.6, in some neighborhood of the point x in M, the periodicity function P takes only finite values. Then there exists a number T > 0 with the following properties. 1) The transformation µ(T ) is identical on M. 2) The periodicity function P takes only finite values; moreover, there are no more than countably many such values; and for every point y ∈ M, either P(y) = 0 or T is divisible by P(y). 3) In any neighborhood of x in M there are points y such that P(y) = T . 4) There is a correctly defined smooth effective action η : S1 × M → M of the circle S1 = R/(T · Z) on M whose orbits coincide with those of µ. Proof. By Theorem 3.6.6, we can choose a number T > 0 and a neighborhood U of the point x such that all the assertions of Theorem 3.6.6 are valid. In particular, the equality M f ∩U = Mµ (T ) ∩U holds, where Mµ (T ) = {y ∈ M | µ(T, y) = y} and µ(T ) is an isometry of the Riemannian manifold (M, g). By the condition of the theorem, all the points y from a neighborhood of the point x in M have finite periods. This means that the isometry µ(T ) is identical in some neighborhood of the point x. Since M is connected and complete, it is identical on the whole manifold. The first assertion of the theorem is proved. Since the isometry µ(T ) is identical on M, the number T is a period for every point in M. Therefore T is divisible by the number P(y) for the points y ∈ M such that P(y) 6= 0. Thus, the set of the least positive periods is at most countable. We have proved the second assertion. Assertion 3) follows from 3) of Theorem 3.6.6. The fourth assertion is an immediate corollary of the previous assertions. t u

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Theorem 3.6.8. Let X be a nonvanishing Killing vector field on a Riemannian manifold (M, g) and let µ(t), t ∈ R, be the 1-parameter isometry group generated by X, and P = Pµ the periodicity function of µ. Then M f = {x ∈ M | Pµ (x) < ∞} is a closed totally geodesic (possibly, disconnected) submanifold in (M, g). Proof. The theorem is obvious if M f = M. Therefore, consider the case when M f 6= M. Since the field X does not vanish on the manifold M, P−1 (0) = 0. / Consider now an arbitrary point x ∈ M f ; then P(x) = t > 0. According to Theorem 3.6.6, there are a neighborhood U of x in M and a number T > 0 such that M f ∩ U = Mµ (T ) ∩ U, where Mµ (T ) = {y ∈ M | µ(T, y) = y} is a closed totally geodesic submanifold of (M, g). Thus, in some neighborhood of any point x ∈ M f , the set M f coincides with a closed totally geodesic submanifold of (M, g). This proves the theorem. t u If the conditions of Theorem 3.6.8 are fulfilled and M 0 is a connected component of the set M f , then M 0 is a closed totally geodesic submanifold of (M, g), invariant with respect to the isometric flow µ. Therefore, it is possible to study in more detail the periodicity function Pµ of this flow restricted to M 0 . The following result gives us some information about this object. Theorem 3.6.9. Let µ : R × M → M be a nontrivial smooth isometric action of the additive group of real numbers on a Riemannian manifold (M, g) and P = Pµ be the periodicity function of the flow µ. If the function P takes only finite values then the following assertions hold: 1) The function P : M → R is lower semicontinuous and takes an at most countable set of positive values τi , τi+1 < τi for 1 ≤ i < κ, where κ ∈ N ∪ {∞}. The number T = τ1 is divisible by each of the numbers τi . 2) There is a smooth, isometric, and effective action η : S1 × M → M of the circle S1 = R/T · Z, whose orbits coincide with those of µ. 3) Let Mi be the set of points x ∈ M with the least positive period τi . Then M1 is an open, connected, and everywhere dense subset in M. 4) Each set Mi is contained in a (possibly disconnected) closed totally geodesic submanifold Mµ (τi ) = {x ∈ M | µ(τi , x) = x}. Proof. Consider an arbitrary point x ∈ M such that t := P(x) > 0. Let Ox be the orbit of the point x under the action of the flow µ. Choose a number rx > 0 such that the set T Ox = {y ∈ M | d(y, Ox ) < rx } is a tubular neighborhood of the orbit Ox , i.e. for each point y ∈ T Ox there is only one point on the orbit Ox nearest to y. Since the orbit Ox is compact, the existence of the needed number rx > 0 is easily proved. Now, let us show that the function P is lower semicontinuous at the point x ∈ M. It is clear that for every point z ∈ Ox , we have the equality P(z) = P(x) = t. Consider an arbitrary point y in the open ball {y ∈ M | d(x, y) < rx }. Let us show that l = P(y) (the least positive period for the point y) is divisible by t. Indeed, let z

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be the point of the orbit Ox nearest to y. Since µ(l)(y) = y and µ(l) is an isometry of (M, g), the point µ(l)(z) is also the nearest point to y of the orbit Ox . But since the neighborhood T Ox is tubular, µ(l)(z) = z, and l is divisible by t. Thus, the periodicity function P does not take values less that t in the ball {y ∈ M | d(x, y) < rx }. Hence the function P is lower semicontinuous at the point x. The fact that the function P is lower semicontinuous at points z ∈ M where P(z) = 0 is obvious. Thus, we have proved that the function P is lower semicontinuous at each point of the manifold. Now assertions 1) and 2) follow from Theorem 3.6.7. Let us prove assertion 3). Since the periodicity function P is lower semicontinuous on M, the inverse image P−1 (T ) = M1 is an open subset in M. As follows from Theorem 3.6.7, the set M1 = P−1 (T ) intersects any neighborhood of each point y ∈ M with positive value P(y). Moreover, Lemma 3.6.1 implies that any connected component of the set P−1 (0) is a totally geodesic submanifold of (M, g) of codimension ≥ 2. Thus, the points y ∈ M with the property P(y) = T constitute a dense open subset in M. Assume now that the set M1 is disconnected. It is easy to see that this is possible only when the set M \ M 0 , where M 0 is some connected component of the set Mi , i > 1, is disconnected. Notice that there exists at most one component M 0 with this property. It is clear that in this case M 0 has codimension 1 in M. Consider a point x ∈ M 0 . Then, according to assertion 6) of Lemma 3.6.3, the differential of the isometry µ(τi ) at the point x changes the orientation in Mx . This means that the manifold M is not orientable, and, consequently, the set M \ M 0 is connected, contrary to our assumption. The obtained contradiction proves that the set M1 is connected. Assertion 4) follows from Lemma 3.6.1. The proof is complete. t u

3.7 Regular and Quasiregular Killing Vector Fields 3.7.1 Orbits of Isometric S1 -actions on Riemannian Manifolds First we will give some standard information about smooth isometric actions of the circle group S1 = SO(2) on Riemannian manifolds. All facts below and their various generalizations can be found in [335] and [57]. Note that the existence of a smooth S1 -action with some prescribed properties on a given smooth manifold M gives partial information about the topology of the manifold M. So, for instance, it is shown in [383] that if a compact orientable smooth manifold M admits a smooth S1 -actions with isolated fixed points that are isolated as singularities as well, then all the Pontryagin numbers and the signature of M are zero, the Euler characteristic of the manifold M is even and is equal to the number of fixed points. Let τ : S1 × M → M be an effective almost free smooth isometric action on a Riemannian manifold (M, g). Consider the projection π : M → M/S1 = M,

(3.17)

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where M = M/S1 is the quotient space whose points are the orbits of the group S1 . The space M/S1 is equipped with the natural intrinsic metric d in the following way. Let Ox and Oy be the orbits of some points x and y relative to the S1 -action on (M, d). One can show that there exists d(Ox , Oy ) = min{d(u, v) | u ∈ Ox , v ∈ Oy }. One can check that the so-defined function d is a complete intrinsic metric on M = M/S1 . Moreover, if d(Ox , Oy ) = d(u, v), then any shortest arc in (M, g), joining the points u and v, is orthogonal to both orbits Ox and Oy . Therefore, if one considers the orbit space M jointly with the metric d, then the mapping (3.17) is a submetry. Also, it is shown in [397] that M/S1 is a Riemannian orbifold (a manifold with singularities). Assume that some point x ∈ M has the trivial stabilizer with respect to the S1 action on (M, g). Then every point in Ox , the orbit of x, has the same property. In this case the orbit Ox is called regular, and singular in the opposite case. Moreover, singular points of the orbifold M/S1 are characterized as the projections of singular orbits (relative to π). It is well known that the union of all regular orbits constitutes an open everywhere dense subset in M. Each point of a singular orbit has a nontrivial stabilizer group, isomorphic to Zn for some n ≥ 2 (see, e.g. [495] or [121], §9). If singular orbits are absent, i.e. the group S1 acts freely, then the projection (3.17) is a fibre bundle and M = M/S1 is a smooth manifold, which can be equipped with the natural Riemannian metric g such that the projection (3.17) is the Riemannian submersion [374]. Note that in this case d is generated by the Riemannian metric g. By Theorem 3.6.9, for a Riemannian manifold (M n , g) with a (quasi)regular Killing vector field X, there exists a smooth, isometric, and effective action η : S1 × M → M of the circle S1 = R/T · Z, whose orbits coincide with those of the flow µ generated by the vector field X. Let us give a coordinate description of a Riemannian manifold (M n , g) with a (quasi)regular Killing vector field X in the tubular neighborhood U = T Ox (U 0 = T Ox ) (from the proof of Theorem 3.6.9) of a regular (respectively, singular) orbit O = Ox . 1) The orbit O is regular if and only if there exists a “chart” (U, ϕ,Vrn−1 × S1 ), where Vrn−1 is an open ball of radius r > 0 with the center 0 in Rn−1 such that the components of the metric tensor g in this chart are independent of θ ∈ S1 , gnn ◦ ϕ −1 is constant, and for each unit vector v = (v1 , . . . , vn−1 ) ∈ Rn−1 and real number t ∈ [0, r) the following equalities hold gi j (tv, 0)vi v j = 1,

gn j (tv, 0)v j = 0

and gi j (tv, 0)vi w j = 0

for

w ∈ Rn ,

w ⊥ v.

Here we use the Einstein summation rule; gkl denote the functions gkl ◦ ϕ −1 . 2) The orbit O is singular if and only if the space U 0 (with intrinsic metric) is the quotient space of the space U from item (1) relative to the action of a cyclic isometry

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group I on U of finite order k ≥ 2 generated by an isometry h, where f = ϕ ◦ h ◦ ϕ −1 has the form   2π f (v, θ ) = f1 (v), θ + , k where f1 is the rotation of order k around the center of the ball Vrn−1 . Moreover, f1 is also an isometry of the Riemannian manifold (Vrn−1 , g0 ), where the metric tensor g0 has components g0 i j (x1 , . . . , xn−1 ) = gi j (ϕ −1 (x1 , . . . , xn−1 , 0)),

1 ≤ i, j ≤ n − 1.

Regular and quasiregular Killing vector fields of constant length on Riemannian manifolds possess several common properties. First of all, we will refine the results of Theorem 3.6.9 for the case of 1-parameter isometry groups generated by (quasi)regular unit Killing vector fields on a given Riemannian manifold. Theorem 3.7.1. Let µ(t), t ∈ R, be a 1-parameter motion group on a Riemannian manifold (M, g), generated by a unit (quasi)regular Killing vector field X. Then the periodicity function P of the flow µ takes only finite values and Theorem 3.6.9 is valid for the flow µ. Also the following assertions hold. 1) All integral curves of the field X are simple closed geodesics; moreover, for each point x ∈ M the number P(x) is the length of the integral curve of the field X passing through the point x. 2) The isometric action η : S1 × M → M of the circle S1 = R/T · Z is effective and almost free. Moreover, the orbits of this action with length T are regular, while all other orbits are singular. The action η is free if and only if the Killing vector field X is regular. 3) If the lengths of the integral curves of the field X are bounded from below by some number δ > 0 (which is always true for any Riemannian manifold (M, g) with the injectivity radius bounded from below by a positive constant), then κ from Theorem 3.6.9 is finite, i.e. the periodicity function P takes only finitely many values. 4) If M 0 is the set of singular points of M with respect to the action η then M 0 is the union of at most countably many (and even finitely many in the case of compact M) totally geodesic submanifolds of M whose dimension and codimension ≥ 1. If M is orientable then the codimension of each of these totally geodesic submanifolds is even. 5) If the manifold M is two-dimensional and orientable then all orbits of η are regular, i.e. have the same length T . 6) If the manifold M is three-dimensional and orientable then the singular orbits of η on M are isolated and, consequently, M is a Seifert fibre space (cf. [407, §3]). 7) If the manifold M is compact, three-dimensional and orientable then the set of the singular orbits of η is finite. Proof. All the integral curves of the field X (or, what is the same, the orbits of the flow µ) are circles, because X is (quasi)regular. Thus, the periodicity function of

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the flow µ takes only finite values. Consequently, in this case all the conditions of Theorem 3.6.9 hold. Further, the assertion 1) follows from Proposition 3.3.4. The second assertion follows from the fact that X has no zero on M. Let us prove the third assertion. Since each number τi is the length of some closed geodesic, we get the inequality τi ≥ δ > 0 for all 1 ≤ i < κ. Moreover, the number τ1 = T (the maximal period) is divisible by τi . Therefore κ is finite in this case. The fourth assertion follows from Lemma 3.6.1, 4) of Theorem 3.6.9, and from the fact that the field X has no zero. The remaining assertions are evident consequences of the fourth assertion. t u Thus, for an isometric flow µ(t), t ∈ R, on a Riemannian manifold (M, g) generated by some (quasi)regular Killing vector field X of constant length, the image of the one-parameter group µ(t) under the natural homomorphism into Isom(M, g) (the full isometry group of (M, g)) is a circle S1 = SO(2) with an effective (almost) free action. The last condition means that the stabilizer of an arbitrary point x ∈ M with respect to the action of S1 is a finite subgroup in S1 . The converse is also true. Let η : S1 × M → M be a smooth, effective, and almost free action of the circle S1 on a smooth manifold M. This action generates a nonvanishing vector field X on M. Then the manifold M can be endowed with a Riemannian metric g such that X is a Killing vector field of constant length on (M, g). Both these statements follow from results of A. Wadsley in [453]. In particular, in this paper he proved the following Theorem 3.7.2 (A. Wadsley, [453]). Let µ : R × M → M be a Cr -smooth action (3 ≤ r ≤ ∞) of the additive group of real numbers, each of whose orbits is a circle, and let M be a Cr -smooth manifold. Then the existence of a Cr -smooth action η : S1 × M → M whose orbits coincide with those of µ is equivalent to the existence of a Riemannian metric g on M such that the orbits of µ are embedded totally geodesic submanifolds in (M, g). Another proof of Theorem 3.7.2 is given by D. B. E. Epstein in Appendix A to the monograph [105]. In connection with Theorem 3.7.2, note that the fulfilment of only one condition that all orbits of the flow µ are circles, even for compact M and real analyticity of the flow µ, does not guarantee the existence of a smooth action η : S1 × M → M whose orbits coincide with those of µ. The corresponding counterexamples were presented in the paper [425] for dimension 5 (see also an essentially simpler example due to W. Thurston in Appendix A to [105]) and in [207] for dimension 4. The presence of such counterexamples in any dimension ≥ 4 is demonstrated by multiplying manifolds by tori. At the same time, it is proved in paper [206] that in dimension 3, such counterexamples are impossible even in the case of a compact manifold and a flow of the class C1 but possible if M is noncompact. Observe also that papers [453, 426] show the equivalence of the following two conditions for a one-dimensional foliation F on a smooth manifold M: 1) the manifold M admits a Riemannian metric g such that all the leaves of the foliation F are geodesics in (M, g);

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2) M admits a 1-form ω such that ω(X) = 0 and dω(T ) = 0, where X is an arbitrary vector and T is any two-plane, both tangent to F . Moreover, it is shown in [426] that all the orbits of a flow µ, generated by some unit vector field on a Riemannian manifold (M, g), are geodesics if and only if there exists a field of planes of codimension 1 on (M, g) which is transversal to the orbits of µ and invariant under the flow µ. The following theorem was essentially proved in [453]. We will need it later together with the construction of a Riemannian metric which is given in the proof. Theorem 3.7.3. Let η : S1 × M → M be a smooth effective almost free action of the circle S1 on a smooth manifold M, and let X be the vector field on M generated by this action. Then M can be endowed with a Riemannian metric g such that X is a Killing vector field of unit length on (M, g). Moreover, X is regular (quasiregular) if the action η on M is free (not free). Proof. It is clear that X has no zero on M. Fix some Riemannian metric g1 on M. Now, consider a new Riemannian metric g2 , obtained by averaging the metric g1 via integration with respect to the Haar measure on S1 . The action η is isometric on (M, g2 ). Therefore, X is the Killing vector field on (M, g2 ). It is clear that X need not have constant length on (M, g2 ). Hence, consider a new Riemannian metric g on M conformally equivalent to g2 . Namely, put g = f g2 , where f : M → R is defined by the formula f = (g2 (X, X))−1 . Since X is Killing on (M, g2 ), we have LX g2 = 0. Therefore, X f = −(g2 (X, X))−2 · X(g2 (X, X)) = −(g2 (X, X))−2 · (LX g2 )(X, X) = 0, and LX g = LX ( f g2 ) = (X f )g2 + f (LX g2 ) = 0. Thus, the field X is Killing on (M, g); moreover, g(X, X) ≡ 1. If the action η on M is free then all orbits of η are regular, i.e. the stabilizer of η is trivial at each point x ∈ M. This means that the integral curves of the field X have constant length on (M, g). If the action η is not free then there is a point x ∈ M whose stabilizer is isomorphic to Zn for some n ≥ 2. So, the length of the integral curve of X through x will be n times shorter than the length of the integral curve of X through each regular point y ∈ M, i. e., a point whose stabilizer with respect to η is trivial. Hence, in this case the field X is quasiregular. t u Remark 3.7.4. Note that [495] contains an example of an analytic almost free action of the group S1 on an analytic noncompact three-dimensional manifold M with infinitely many pairwise nonisomorphic stabilizers of points x ∈ M. Using Theorem 3.7.3, we can endow M with a Riemannian metric g such that the action of S1 on (M, g) is isometric and the Killing vector field X corresponding to this action has unit length, see also Remark 3.4.9. It is clear that in this case there are infinitely many least positive periods of the points in M and these periods are not bounded jointly from below by any positive number.

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Theorem 3.7.3 immediately implies the following version of Wadsley’s Theorem (Theorem 3.7.2). Theorem 3.7.5. Let M be a smooth manifold and X a nowhere vanishing vector field on M, whose integral trajectories are closed. The field X is generated by some smooth action η : S1 × M → M if and only if M admits a Riemannian metric g such that X is a unit Killing vector field on the Riemannian manifold (M, g). It should be noted that in the papers [327] by W. C. Lynge and [118] by C. C. Bosch, the authors established some necessary and sufficient conditions for a complete vector field on a manifold M to be a Killing vector field for a suitable Riemannian metric on M.

3.7.2 Flows on Simply Connected Manifolds Using Theorem 3.7.3, we can construct many examples of Riemannian manifolds with quasiregular Killing vector fields of constant length. In this subsection we consider simply connected examples of such manifolds. Consider a smooth action µ : S1 × M → M of the circle on a smooth manifold M. An almost free action µ is called pseudofree if the set of singular orbits is finite but not empty. By Theorem 3.7.3, each manifold M with a pseudofree action of the circle S1 carries a Riemannian metric g such that the vector field X generated by the action of S1 is a quasiregular Killing field of unit length. Since there are many examples of pseudofree actions of the circle on manifolds, we obtain many examples of Riemannian manifolds with quasiregular Killing fields of constant length. For instance, it was proven in [342] that all 28 smooth homotopic 7-spheres admit smooth pseudofree actions of S1 . Applying Theorem 3.7.3, we obtain Corollary 3.7.6. All 28 smooth homotopic 7-spheres carry Riemannian metrics with quasiregular Killing fields of unit length. There are well-known pseudofree actions of the circle S1 on the odd-dimensional spheres S2n−1 which are described below. In the space C n , n ≥ 2, with the standard Hermitian norm, consider the sphere S2n−1 = {z = (z1 , z2 , . . . , zn ) ∈ C n | kzk = 1}. Let q1 , q2 , . . . , qn be natural numbers having no common divisor > 1. Consider the following action of the circle S1 ⊂ C on S2n−1 : s(z) = (sq1 z1 , sq2 z2 , . . . , sqn zn ),

s ∈ S1 .

(3.18)

It is clear that this action is isometric with respect to the canonical metric on S2n−1 induced by the standard Euclidean metric on C n . If not all numbers qi are equal then there exists a singular orbit of this action; i. e., the action of S1 is pseudofree.

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Corollary 3.7.7. Each sphere S2n−1 for n ≥ 2 carries a Riemannian metric having a quasiregular unit Killing field such that only one integral trajectory of this field is singular. Proof. Consider the special case of (3.18) when q1 = q > 1 and qi = 1 for 2 ≤ i ≤ n. For this action, there exists only one singular orbit; namely, that passing through (1, 0, 0, . . . , 0) ∈ C n . By Theorem 3.7.3, the sphere S2n−1 (n ≥ 2) carries a Riemannian metric such that the vector field X generated by the considered action of S1 on S2n−1 is a Killing field of unit length with respect to this metric. t u Note that the metrics in Corollary 3.7.7 constructed along the lines of the proof of Theorem 3.7.3 have cohomogeneity 1. Recall that a smooth (compact Riemannian) manifold M has cohomogeneity 1 if the orbit space M/G of the full isometry group G of M is one-dimensional. Various information and a bibliography on the Riemannian manifolds of cohomogeneity 1 can be found in [15] and [246]. The following result is more interesting. Theorem 3.7.8. For every ε > 0, each sphere S2n−1 , n ≥ 2, carries a (real analytic) Riemannian metric g of cohomogeneity 1 such that there is a (real analytic) Killing vector field X of unit length on (S2n−1 , g) with closed integral trajectories such that 1) all sectional curvatures of (S2n−1 , g) differ from 1 by at most ε; 2) L/l > 1/ε, where L and l are the maximal and minimal lengths of integral trajectories of X. Proof. Let us consider the special case of (3.18) when q1 = q > 1 and qi = q + 1 for 2 ≤ i ≤ n. Observe that the least positive period with respect to the flow, corresponding to the action of S1 for the points x ∈ S2n−1 , is equal to either 2π/q, 2π/(q+1), or 2π. The squared length of the Killing vector field V , generated by the action (3.18) on the sphere S2n−1 with the canonical metric, is calculated by the formula 2 2 canx (V,V ) = q2 (x12 + x22 ) + (q + 1)2 (x32 + x42 + · · · + x2n−1 + x2n ), √ where x = (x1 , x2 , . . . , x2n−1 , x2n ), x2k−1 + −1x2k = zk ∈ C .q Now, consider the vec1 2n−1 tor field Ve = q V . It is Killing on (S , can), moreover, 1 ≤ can(Ve , Ve ) ≤ 1 + 1/q. These fields converge when q → ∞ to a unit Killing field on the sphere (S2n−1 , can). Let fq : S2n−1 → R be defined by formula fq (x) = canx (Ve , Ve ) = 1+ 2q+1 ϕ(x), where q2 2n

ϕ(x) = ∑ xi2 for x ∈ S2n−1 . i=3

Now, consider a new Riemannian metric gq on S2n−1 defined as follows:  −1 1 gq = can(Ve , Ve ) can = can . fq Thus, gq is obtained by some conformal deformation of the canonical metric on S2n−1 , and the field Ve is unit and Killing with respect to this metric.

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Now we show that the metrics gq on the spheres S2n−1 have cohomogeneity 1. Observe that the function ϕ together with the functions fq are invariant under the action of the group SO(2) × SO(2n − 2) ⊂ SO(2n) which acts isometrically on (S2n−1 , can) with cohomogeneity 1. The orbit space of this action is a segment of the real axis. The orbits of the action of this group are the sets 2 = 1 − t 2} Mt = {x ∈ R2n | x12 + x22 = t 2 , x32 + x42 + · · · + x2n

for t ∈ [0, 1]. It is obvious that the orbit M0 is diffeomorphic to the sphere S2n−3 , the orbit M1 is diffeomorphic to S1 , and any orbit Mt is diffeomorphic to S1 × S2n−3 for t ∈ (0, 1). By the invariance of the functions fq under the action of the group SO(2) × SO(2n − 2), the metrics gq are also invariant under the action of this group and, therefore, have cohomogeneity 1. Note that each orbit Mt , t ∈ (0, 1), endowed with the metric induced by the metric gq is isometric to the direct metric product of one-dimensional and (2n − 3)dimensional Euclidean spheres of appropriate radii. The singular orbits M1 and M0 are totally geodesic in (S2n−1 , gq ) for  every  natural q. Moreover, M1 is isometric to q S1 (1) and M0 is isometric to S2n−3 q+1 , where we denote by Sm (r) the sphere of radius r in the (m + 1)-dimensional Euclidean space with the induced Riemannian metric. Now we prove that the sectional curvatures of the metrics gq converge as q → ∞ to the sectional curvatures of the canonical metric can uniformly over all points of the sphere and all 2-planes. Recall how the sectional curvature of a Riemannian metric changes under conformal deformation. Let g and g = θ · g be two conformally equivalent Riemannian metrics on a manifold M, where θ : M → R is a positive smooth function. Suppose that σ is a tangent 2-plane at some point x ∈ M and the vectors V and W form an orthonormal basis for σ with respect to g. The sectional curvatures Kσ and K σ of the 2-plane σ with respect to the metrics g and g are connected by the formula (see §3.6 in [241]):   1 k∇ψk2 − (V ψ)2 − (W ψ)2 θ K σ = Kσ − hψ (V,V ) + hψ (W,W ) + , (3.19) 2 2

where ψ = ln θ , hψ is the Hessian of the function ψ defined as hψ (X,Y ) = g(∇X ∇ψ,Y ). In the case under consideration we have M = S2n−1 , g = can, and, moreover,   2q + 1 θ (x) = 1/ fq (x), ψ(x) = − ln fq (x) = − ln 1 + ϕ(x) . (3.20) q2 For every x ∈ S2n−1 , we have 0 ≤ ϕ(x) ≤ 1; therefore, fq converge uniformly on to the identically one function. By compactness of S2n−1 the norms of the covariant differentials ∇ϕ and ∇2 ϕ (with respect to the metric can) are bounded above by some constants on S2n−1 . S2n−1

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From here, (3.19), and (3.20) we obtain easily that K σ → Kσ as q → ∞ uniformly over all points of the sphere and all tangent 2-planes. Thus, we can consider as the metric g in the statement of the theorem the metric gq for sufficiently large q; moreover, we can take Ve as X. Indeed, for q sufficiently large all sectional curvatures of (S2n−1 , gq ) differ from 1 by at most ε. It is easy to see now that the length of each regular integral trajectory of the field Ve on (S2n−1 , gq ) is equal to 2πq. The lengths of singular integral trajectories q q are equal to 2π or 2π q+1 . Thus, L = 2πq and l = 2π q+1 , where L and l are the maximal and minimal lengths of the integral trajectories of the field Ve on (S2n−1 , gq ), respectively. Therefore, L/l = q + 1 > 1/ε for q sufficiently large. The theorem is proved. t u It is interesting to compare the results of Theorem 3.7.8 with those of [44]. It is proven in [44] that for every ε > 0 there exists a δ = δ (ε, m) > 0 such that every simple closed geodesic on the sphere Sm endowed with a Riemannian metric g with the condition 1 − δ < K < 1 + δ on the sectional curvature K has length l such that either l ∈ (2π −ε, 2π +ε) or l > 1/ε; i. e., closed geodesics are of two types, “short” and “long”. Among the closed geodesics representing the integral curves of X in the examples of metrics on the sphere S2n−1 from Theorem 3.7.8, there are geodesics of the first and second types. The following result can be easily obtained using the ideas of the proof of Theorem 3.7.8. Theorem 3.7.9. For every ε > 0, each sphere S2n−1 , n ≥ 2, carries a (real analytic) Riemannian metric g of cohomogeneity 1 and has a (real analytic) Killing field X of unit length on (S2n−1 , g) such that 1) all sectional curvatures of (S2n−1 , g) differ from 1 by at most ε; 2) the field X has both closed and nonclosed integral trajectories. Proof. We can represent the sphere S2n−1 with the canonical metric can of constant sectional curvature 1 as a submanifold S2n−1 = {z ∈ C n | kzk = 1} with the Riemannian metric induced by the standard Euclidean metric on C n . Consider the flow µ : R × S2n−1 → S2n−1 defined as follows: µ(s, z) = (es

√ −1

z1 , esq

√ −1

z2 , . . . , esq

√ −1

zn ),

where z = (z1 , z2 , . . . , zn ) ∈ S2n−1 , s ∈ R, and q is some irrational number. It is clear that the flow µ is isometric with respect to the canonical metric can on S2n−1 . The square of the length of the Killing field X on (S2n−1 , can) corresponding to the flow µ is calculated by the formula 2 2 canx (X, X) = (x12 + x22 ) + q2 (x32 + x42 + · · · + x2n−1 + x2n ), √ where x = (x1 , x2 , . . . , x2n−1 , x2n ), x2k−1 + −1x2k = zk ∈ C . These fields X converge as q → 1 to a unit Killing field on the sphere (S2n−1 , can). Consider the new Riemannian metric gq on S2n−1 defined as

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gq = (can(X, X))−1 can and presenting a conformal deformation of the canonical metric on S2n−1 . Now, X is a unit Killing field with respect to gq (see the proof of Theorem 3.7.3). It is clear that the metric gq is real analytic and has cohomogeneity 1 on S2n−1 with respect to the isometric action of the group SO(2) × SO(2n − 2) ⊂ SO(2n) (see the proof of Theorem 3.7.8). Using arguments similar to those in the proof of Theorem 3.7.8, we demonstrate that for q sufficiently close to 1 the metric gq satisfies the first condition of Theorem 3.7.9. The second condition of the theorem holds for each metric gq since q is irrational. Thus, we can choose as g of Theorem 3.7.9 the metric gq for some irrational q sufficiently close to 1. Note also that the set of points of finite period on (S2n−1 , gq ) with respect to the flow generated by the Killing field X consists of two connected components (of dimension 1 and 2n − 3) isometric to some Euclidean spheres. All points of each of these components have common period, but the periods of different components are incommensurable. t u The following theorem is interesting in connection with the above results: Theorem 3.7.10 (W. Tuschmann [443]). Let S(n, D) be the class of simply connected n-dimensional Riemannian manifolds (M, g) of sectional curvature |Kg | ≤ 1 and diameter diam(g) ≤ D (n ≥ 2, D > 0). Then there exists a positive number v = v(n, D) with the following property: if (M, g) ∈ S(n, D) satisfies vol(g) < v(n, D) then 1) there is a smooth locally free action of the circle S1 on M; 2) for each ε > 0 there exists an S1 -invariant metric gε on M such that e−ε g < gε < eε g,

|∇g − ∇gε | < ε,

|∇igε Rgε | < C(n, i, ε);

3) the quotient space of the induced Seifert bundle on M is a simply connected Riemannian orbifold. W. Tuschmann notes that, by the Bonnet theorem, the class PS(n, δ ) of all ndimensional Riemannian manifolds (M, √ g) with sectional curvature 0 < δ ≤ Kg ≤ 1 is a subclass of the class S(n, D = π/ δ ). Therefore, Theorem 3.7.10 is valid for the class PS(n, δ ) as well. If the above-indicated S1 -action for the manifold (M, g) from Theorem 3.7.10 is free (locally free) then by Theorem 3.7.3 M carries a Riemannian metric g1 such that the relevant S1 -action is induced by a regular (quasiregular) unit Killing field on M. If we have (M, g) ∈ PS(n, δ ) then by the Berger theorem (Theorem 3.5.10) n must be odd. Moreover, by Remark 0.6 in [443], M admits a δ /2-pinched metric ge invariant under the S1 -action. On the other hand, no condition is presently known guaranteeing the existence of free or, on the contrary, almost free smooth S1 -actions on manifolds in the class PS(n, δ ).

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3.7.3 Geodesic Flows and the Sasaki Metric on Tangent Bundles of Riemannian Manifolds An additional source of Killing fields of constant length are vector fields of geodesic flows on Riemannian manifolds all of whose geodesics are closed. The manifolds of this sort are the objects of the study in [105]. In [406] Sasaki defined and studied in local coordinates the natural and remarkable Riemannian metric gS on the tangent bundle T M over a Riemannian manifold (M, g) which is usually called the Sasaki metric. This metric was later described in terms of connectors as the metric g1 in Section 1K in [105] without any reference to [406]. Henceforth for each u > 0 we denote by Su M the spherical bundle of vectors of length u on a Riemannian manifold (M, g). We can give an implicit but complete characterization of gS by the following three properties: 1) The metric on each tangent space Mx ⊂ T M, x ∈ M, induced by the metric gS , coincides with the natural Riemannian metric on the Euclidean space (Mx , gx ). 2) The natural projection p : (T M, gS ) → (M, g) is a Riemannian submersion. 3) The horizontal geodesics of the Riemannian submersion p are precisely the parallel vector fields along geodesics in (M, g). As an easy consequence of these properties, we get the following property. 4) The vector field F of geodesic flow of a Riemannian manifold (M, g) defined on (T M, gS ) is a horizontal vector field of the Riemannian submersion p and is tangent to the bundle Su M of the tangent vectors of length u 6= 0; the integral curves of F are special horizontal geodesics in (T M, gS ) presenting the tangent vector fields to geodesics in (M, g). The proof of the following property is given in Proposition 1.102 of [105]. 5) The vertical fibers p −1 (x), x ∈ M, of the submersion p are totally geodesic with respect to gS . We know no reference to the following important result. Theorem 3.7.11. If f is an isometry of (M, g) then T f is an isometry of (T M, gS ). We are especially interested in Theorem E in [435]. Theorem 3.7.12 (S. Tanno, [435]). A Riemannian manifold (M, g) has constant sectional curvature k > 0 if and only if the restriction of the vector field of geodesic flow F to Su M, u = √1k , is a Killing vector field of constant length with respect to the induced metric (T M, gS ). Following the book [105], we mean by a Riemannian manifold all of whose geodesics are closed, or a P-manifold, a Riemannian manifold (M, g) such that each geodesic on (M, g) is periodic (in other words, closed). A special case of a P-manifold is an SC-manifold; i. e., a Riemannian manifold (M, g) such that all geodesics on (M, g) have common least period l, 0 < l < ∞, and each geodesic of

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length l in (M, g) is a simple closed curve. If we remove the last requirement of this definition, then we will obtain an intermediate notion of C-manifold. The classical examples of SC-manifolds are compact rank 1 symmetric spaces. Smooth Riemannian SC-surfaces of rotation (S2 , g) with noncanonical metric are known; we can point out among them the real analytic Zoll example (1903). Much later, using this example, Weinstein constructed noncanonical smooth Riemannian SC-metrics on each sphere Sn , n ≥ 3 (see [105]). The following observation is important for us: if (M, g) is a P-manifold and Γ is a finite free isometry group of (M, g), then the quotient space M/Γ supplied with the natural quotient Riemannian metric is also a P-manifold [105]. This construction is an additional source of P-manifolds Proposition 3.7.13. The following assertions hold: 1) The restriction X of the vector field F of geodesic flow of an arbitrary smooth Riemannian P-manifold M to (S1 M, gS ) is a unit tangent vector field and its integral trajectories are simple periodic geodesics on (S1 M, gS ) parameterized by arc length. 2) The field X is generated by an effective smooth action of the group S1 on S1 M and is a unit Killing field with respect to some smooth Riemannian metric g0 on S1 M. 3) If (M, g) has constant sectional curvature k = 1 then we can take g0 = gS . 4) Every Riemannian P-manifold is compact and all its geodesics have common (not necessarily least) period. 5) The above action of S1 is free if and only if (M, g) is a C-manifold. Proof. The first assertion follows from the second and fourth properties of the Sasaki metric gS on T M listed above. To prove the second assertion we need Theorem A.26 in [105], stating that each e such that all field Y of unit tangent vectors on an arbitrary Riemannian manifold M, integral trajectories of Y are geodesic circles, is generated by some smooth action of S1 (the circle S1 is identified here with R/cZ, where c > 0 is a constant not necessarily equal to 1). This result together with the first assertion of the proposition and Theorem 3.7.3 imply the second assertion of the proposition. The third assertion is a consequence of Theorem 3.7.12. We now prove the last two assertions. The manifold M is compact since M is the image under the corresponding (smooth) mapping p ◦ ϕ : S1 × S1 M → M restricted to the compact subset S1 × (S1 M ∩ Mx ), where x is an arbitrary point in M and ϕ : S1 × S1 M → S1 M is the flow of the vector field X. By the first two assertions of the proposition all geodesics on an arbitrary Pmanifold (M, g) have common (not necessarily least) period c. It is clear that the period c is least for all geodesics if and only if (M, g) is a C-manifold. t u

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Corollary 3.7.14. If a Riemannian P-manifold (M, g) is not a C-manifold, then in the notation of Proposition 3.7.13, X is a quasiregular unit Killing field on (S1 M, g0 ). If (M, g) is an arbitrary homogeneous Riemannian manifold of constant sectional curvature 1 other than the Euclidean sphere or real projective space then X is a quasiregular unit Killing field on (S1 M, gS ). Proof. The first assertion follows from 4) of Proposition 3.7.13. Let (M, g) be a homogeneous Riemannian manifold of constant sectional curvature 1 other than the Euclidean sphere or real projective space. By Theorem 4.8.17 there is a quasiregular Killing vector field X of constant length on (M, g). It is obvious that closed geodesics representing the integral curves of the field X do not have common least period. Consequently, (M, g) is not a C-manifold. The second assertion of Corollary 3.7.14 follows now from 1), 2), 5) of Proposition 3.7.13. t u In connection with the above considerations it would be appropriate to mention the following Theorem 3.1 in [114]. Theorem 3.7.15 (A. Bolsinov, B. Jovanovi´c, [114]). The geodesic flow on every Riemannian P-manifold is completely integrable. We see from this theorem and Corollary 3.7.14 that in general the existence of a quasiregular Killing field of constant length on a Riemannian manifold (M, g) is not an obstruction to the complete integrability of the geodesic flow on (M, g). Additional information on completely integrable geodesic flows as well as necessary definitions and constructions can be found in [114] and [113]. The reader can also find some arguments about the drawbacks of the results similar to Theorem 3.7.15 in [113, p. 4196].

3.8 Discrete Subgroups of Lie Groups Any consideration of complete non-simply connected smooth Riemannian manie In this case folds M is connected with its universal locally isometric covering M. the fundamental group of M emerges as a discrete isometry group Γ of the space e and M = Γ /M e is the orbit space of Γ with the quotient metric induced by the M e e is: 1) homogeneous, 2) symmetric, or 3) has conmetric of M. The cases when M stant sectional curvature, are progressively more interesting. In the very last case we consider the so-called Clifford–Klein problem of space forms. Case 2) is connected with Section 4.8. In the most general case 1) Γ is a discrete subgroup of a transitive e The manifold M is homogeneous if and only if there Lie isometry group Isom(M). e commuting with covering mappings. As exists a transitive isometry Lie group of M e a corollary the covering mappings are Clifford–Wolf isometries of M. e On the other hand, it is possible to begin with M and consider the orbit spaces, e where Γ is any discrete isometry group of including orbifolds, of the form Γ /M

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157

e For this approach, the situations when Γ /M e is compact or has a finite volume M. present special interest. The last properties are equivalent to the same properties of e such that various congruent copies fundamental domains (which are regions D in M e γ(D) with γ ∈ Γ cover M and have only boundary points in common). In this section we present without proofs some results in this direction using various sources. Let us begin with Hilbert’s problem 18 [338]: “Is there in n-dimensional Euclidean space En . . . only a finite number of essentially different kinds of groups of motions with a [compact] fundamental region?” Any such group is called an n-dimensional crystallographic group, or a Bieberbach group after Hilbert’s question was answered affirmatively by L. Bieberbach in 1910 [108]. Theorem 3.8.1. (1) Every discrete subgroup Γ ⊂ Isom(En ) with compact fundamental domains contains n linearly independent translations. (2) The subgroup T consisting of all translations in Γ forms a normal subgroup of finite index. That is, Γ admits a short exact sequence of the form i

1 → Zn → Γ → Φ → 1

(3.21)

with Φ finite. Furthermore, the image T = i(Zn ) is a maximal abelian subgroup of Γ . In other words, if we let Φ act on Zn by inner automorphisms, we obtain a faithful representation of Φ in GL(n, Z). (3) For each fixed n, there are only finitely many isomorphism classes of groups Γ which can be obtained in this way as extensions of Zn by finite subgroups of GL(n, Z). L. Bieberbach first proved the important statement (1). He then showed easily that (2) holds for any such Γ . It follows that this group T of translations in Γ can be characterized as the union of all abelian normal subgroups of Γ . Later proofs of (1) and (2) were given by L. Auslander and J. Wolf. Finally L. Bieberbach proved statement (3). For the proof of (3), Bieberbach first recalled Minkowski’s statement that there is an upper bound, depending only on n, for the orders of finite subgroups of GL(n, Z). Given a finite group Φ, one also needs to know that there are only finitely many embeddings of Φ in GL(n, Z), up to inner automorphism. This statement is known in the literature as “the Jordan–Zassenhaus theorem”. Finally, given the action of Φ on Zn , one needs to know that there are only finitely many extensions (3.21). Nowadays it is known that these extensions are classified by elements of the finite cohomology group H 2 (Φ; Zn ). For general information on Bieberbach groups, refer to the book [147] by L.S. Charlap. The embedding of Γ into the group Isom(En ) is not unique in general. However in 1912, inspired by a paper of G. Frobenius, L. Bieberbach showed that this embedding is uniquely determined up to conjugation by an element of the larger group of affine transformations of Rn . Many years later, H. Zassenhaus showed that a converse to Bieberbach’s theorem is also true:

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Theorem 3.8.2. Any group Γ which can be described as an extension of Zn by a finite subgroup of GL(n, Z) can actually be embedded as a discrete subgroup of the group Isom(En ) to act on En with compact fundamental domain. J. Milnor gives a table, listing the numbers of different Bieberbach groups Γ in dimensions up to 4. These numbers are 2, 17, 219, and 4783 respectively. The classification of Bieberbach groups in dimension 3 is of great importance in the study of crystals. This classification was carried out independently by E. S. Fedorov [211] and by A. Schoenflies [415] around 1890. A more modern presentation is given by H. Burkhardt. In dimension 4 a suitable computation was obtained by H. Brown, R. B¨ulow, J. Neub¨user, H. Wondratschek, and H. Zassenhaus [130]. It is natural to ask whether Bieberbach’s theorems would still hold if one forgets the metric on Euclidean space and simply looks for groups of one-to-one affine transformations of Rn which are discrete and have compact fundamental domain. A simple counter-example has been given by L. Auslander. Example 3.8.3. Let H be the 3-dimensional Heisenberg group of all real upper triangular (3 × 3)-matrices with units on the main diagonal and H(Z) its subgroup having integer elements. Then H(Z) acts by left multiplication on H. Identifying H with the coordinate space R3 in the obvious way, each element of H(Z) gives rise to a one-to-one affine transformation of R3 . This action is discrete, has no fixed points, and has the unit cube as a fundamental domain. Yet the group H(Z) does not contain three linearly independent translations. From the point of view of differential geometry, the most interesting Bieberbach groups are those which give compact flat manifolds as their quotient spaces. These are Γ with free action on En or, which is equivalent, torsion-free Γ . Since Γ contains a free abelian subgroup of finite index, it follows that every such compact flat manifold is finitely covered by a flat torus. The simplest example of a compact flat manifold which is not torus is the Klein bottle. In dimension 3 there are ten distinct examples [483]. Six of them are orientable, the quotient Φ = Γ /T being either cyclic of order 1, 2, 3, 4, 6, or dihedral of order 4. The orbit Λ of a zero point for any cocompact, i.e. having a compact fundamental domain, translation subgroup T of Isom(En ) is called a lattice. To find possible extensions Γ ⊂ Isom(En ) of the group T , it suffices to get all possible subgroups Φ ⊂ O(n) ⊂ Isom(En ) fixing the zero point and preserving the lattice Λ . For any such Φ, the corresponding group Γ is generated by T and Φ. So, it is important to find the maximal group Φ with the mentioned property. We shall call this group Aut(Λ ) the automorphism (or symmetry) group of the lattice Λ . It is clear that Aut(Λ ) is a finite group. Now it is appropriate to use some notions and results from the very interesting book [165] by J. H. Conway and N. J. A. Sloane on Euclidean lattices and related matters. A generating matrix M of a lattice Λ ⊂ En is an (n × n)-matrix whose columns are components (with respect to an orthonormal basis in En ) of some n generating

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elements of Λ considered as an additive group. Then all vectors of Λ have the form Mξ , where ξ is a vector-column in Rn with integer coordinates. The matrix A = M T · M, where T means the transposition, is called the Gram matrix of this lattice. The determinant of the lattice is by definition det(A). The lattice packing for Λ is a family of open congruent pairwise non-intersecting balls of maximal possible radius with centers in Λ . The density of such a lattice packing is p ∆ := the volume of one ball / det(Λ ). (3.22) An orthogonal matrix B is in Aut(Λ ) if and only if there exists a matrix U ∈ GL(n, Z) such that MU = BM. This implies the equality U = A−1 M T BM, where A is the Gram matrix. On the other hand, the set of all matrices U ∈ GL(n, Z) for which there exists an orthogonal matrix B with MU = BM forms the integer representation of the group Aut(Λ ). Table 1.1. in [165] contains all n-dimensional lattices in En for n ≤ 8 with maximal density of corresponding lattice packings. They are respectively Z, A2 , A3 , D4 , D5 , E6 , E7 , E8 with contact numbers for corresponding closed balls: 2, 6, 12, 24, 40, 72, 126, 240. The automorphism groups of these lattices have special interest: they have subgroups of small index, generated by reflections (the Coxeter groups). It is known that the densest lattice packing in dimensions from 1 to 8 is unique. Now we will follow Thurston’s book [436]. There are 32 subgroups of O(3) that occur as groups Φ of 3-dimensional crystallographic groups. They are known in crystallography as the point groups or the rotation subgroups of Γ in mathematics, and were first enumerated by J. F. C. Hessel. There are 14 crystallographic groups that are the full groups of symmetries of lattices in E3 up to affine equivalence. They were enumerated by A. Bravais [125]. There are 65 orientation preserving crystallographic groups, and they were classified by L. Sohncke [418]. The following general theorem is due to D. Gromoll and J. A. Wolf [240]: Theorem 3.8.4. If M is a compact manifold with K ≤ 0 and solvable fundamental group, then M = En /Γ is a flat manifold and Γ is a (torsion-free) Bieberbach group. For corresponding references to the literature or additional information concerning the crystallographic groups, see [338], [483], or [436]. Proofs of Bieberbach’s theorems and some other results on crystallographic groups can be found in the books [483] and [499]. Note that (1) in Theorem 3.8.1 is the most difficult part of this theorem. Bieberbach’s original proof of this statement is based on Minkowski’s theorem on simultaneous rational approximation and is difficult to read. Shortly after it came out, Frobenius gave a more accessible proof which is based on an argument using the commutativity of some unitary matrices. The original articles of Bieberbach and Frobenius are [108], [218]. A simplified version of Frobenius’ paper using minor amounts of Lie group theory is given in [33]. A proof of (3) in Theorem 3.8.1 may be found in [399, p. 375].

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Frobenius’s method has in one form or another became standard in the contemporary literature. But even for n = 3 the existing proofs are difficult. It is would be very desirable to get a simple proof of this statement since it takes a central place in all geometric crystallography. Such a proof is presented in the paper [133] where P. Buser applied a completely different approach to (1) in Theorem 3.8.1 which has its origin in Gromov’s work on almost flat manifolds [242], [135], [136]. The new idea is to start with those isometries which have a very small rotation part contained in the group Φ and then proceed to show that, in fact, these isometries are pure translations. The simplification which results from this approach is striking. P. Buser also gives a new proof of (3) in Theorem 3.8.1 which does not run via the usual algebraic characterization of crystallographic groups. Instead he uses a method which is more in the spirit of Minkowski’s geometry of numbers, from where Bieberbach’s original arguments departed. Remark 3.8.5. Notice that any group Φ ⊂ O(n) which appeared earlier is a discrete isometry group of the unit sphere Sn−1 and its orbit space is some so-called elliptic orbifold. Theorem 3.8.6 (4.4.14 in [436]). Let M be an elliptic three-manifold. (a) If π1 (M) is abelian, it is cyclic, and M is a lens space. Otherwise, M is the quotient of RP3 by a group H of one of the following type: (b) H = H1 × H2 , where H1 is a dihedral group, the tetrahedral group, the octahedral group, or the icosahedral group, and H2 is a cyclic group with order relatively prime to the order of H1 . (c) H is a subgroup of index 3 in T ×C3m , where m is odd, T is the tetrahedral group, and Ck is a cyclic group of order k. (d) H is a subgroup of index 2 in C2n × D2m , where n is even, m and n are relatively prime, and Ck (Dk ) is a cyclic (respectively, dihedral) group of order k. Theorem 3.8.7 (4.1.7 in [436]). Let G be a Lie group. (a) There exists a neighborhood U of e in G such that any discrete subgroup Γ of G generated by Γ ∩ U is a cocompact subgroup of a connected, closed, nilpotent subgroup N of G whose intersection with U is connected. (b) The closure of the set of discrete subgroups of G consists of closed subgroups (but not necessarily all closed subgroups) whose identity component is nilpotent. Corollary 3.8.8 (4.1.11 in [436]). For any dimension n, there is an integer m such that any discrete subgroup of O(n) contains a normal abelian subgroup with index at most m. Corollary 3.8.9 (4.1.13 in [436]). For each dimension n, there is an integer m such that any discrete subgroup of isometries of En has an abelian subgroup of index at most m.

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Proposition 3.8.10 (4.1.16 in [436]). Let G be a Lie group acting on a manifold M so that stabilizers of points are compact. Let x ∈ M be any point. There exists an integer m and an ε > 0 such that any discrete subgroup Γ of G generated by elements which move x a distance less than ε has a normal nilpotent subgroup of finite index no more than m. Furthermore, Γ is contained in a closed subgroup of G with no more than m components and with a nilpotent identity component. Corollary 3.8.11 (4.1.17 in [436]). For every dimension n there is an integer m and an ε > 0 such that any discrete subgroup Γ of Isom(H n ) generated by elements which move some point x a distance less than ε contains a normal abelian subgroup with index at most m (H n is the n-dimensional hyperbolic (Lobachevsky) space). Let M be a complete hyperbolic manifold, possibly with infinite volume. For any ε > 0, we can decompose M into the thick part M = M ≥ε , where   1 M ≥ε = x ∈ M : Radinj(x) ≥ ε 2 and its complement   1 M 0.

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Theorem 3.8.15 ([94, 416]). Let M be a complete simply connected irreducible Riemannian manifold of nonpositive curvature and x ∈ M. If the holonomy group at x does not act transitively on the unit sphere (S1 )x M in Mx , then M is a symmetric space of rank at least two. All the remaining results of this section are mainly proved or at least discussed in the book [43] by W. Ballmann, M. Gromov, and V. Schroeder, see also [332] and [394]. Let M be a complete Riemannian manifold of sectional curvature −a2 ≤ K ≤ 0, a > 0, of finite volume. The universal covering Riemannian space of M is denoted e The unit tangent bundles of M and M e are denoted by SM and SM e respectively. by M. p e we define J (v) to be the space of parallel Jacobi fields For v ∈ SM or v ∈ SM along the geodesic γv determined by v (including the field γ 0v (t)), and set rk(v) = dim J p (v),

rk(M) = min{rk(v) : v ∈ SM}

It is easy to see that this definition of the rank agrees with the usual one in the case when M is a locally symmetric space. If M has rank one, then it resembles in many respects a manifold of negative curvature. For example, there is a geodesic γ in M such that the set of γ 0 (t), t ∈ R, is dense in SM. e is irreducible, then M is either a space of Theorem 3.8.16 ([39, 40, 41, 42]). If M rank one or a locally symmetric space of noncompact type of rank at least two. Theorem 3.8.17 (the Margulis lemma). Given n ∈ N there are constants µ = µ(n) > 0 and I(n) ∈ N with the following property: Let M be an n-dimensional simply connected Riemannian manifold with intrinsic metric d and the curvature condition −1 ≤ K ≤ 0 and let Γ be a discrete group of isometries acting on M. For x ∈ M let Γµ (x) := h{γ ∈ Γ : dγ (x) = d(x, γ(x)) ≤ µ}i be the subgroup generated by elements γ ∈ Γ with dγ (x) ≤ µ. Then Γµ (x) is almost nilpotent, i.e. it contains a nilpotent subgroup of finite index. The index is bounded above by I(n). Let us state some immediate consequences of the Margulis lemma. Let M be a complete manifold of negative curvature. One says that the injectivity radius goes to zero (Radinj → 0) if for all ε > 0 the set {x ∈ M : Radinj(x) ≤ ε} is compact. We have the implications M is compact =⇒ Vol(M) < ∞ =⇒ Radinj → 0. Theorem 3.8.18 (G. Margulis–E. Heintze). Let M be a complete Riemannian manifold with curvature −1 ≤ K < 0 and Radinj → 0, then there is a point x ∈ M with Radinj(x) ≥ µ/2, where µ = µ(n) is the constant of the Margulis lemma.

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Corollary 3.8.19. There exists a constant ν(n) such that the following holds: let M be a complete Riemannian manifold of dimension n which satisfies the curvature condition −1 ≤ K < 0, then Vol(M) ≥ ν(n). Recall that an end E of a (noncompact) manifold M is an equivalence class of nested sequences of connected open sets U1 ⊃ U2 ⊃ · · · with the property that ∩Ui = 0/ and each Ui is a component of M \ Ci where Ci is a compact submanifold (with boundary). Two such sequences {Ui }, {Vi } are equivalent if for each i there exist j, k such that V j ⊂ Ui and Uk ⊂ Vi . Any open set Ui as above is called a neighborhood of E . Theorem 3.8.20 ([43]). Let M be an n-dimensional complete Riemannian manifold satisfying the curvature condition −1 ≤ K ≤ −a2 < 0 and Radinj → 0. Then (1) M is diffeomorphic to the interior of a compact manifold with boundary. In particular M has only finitely many ends and the fundamental group has a finite presentation. (2) There exists a constant ε(n) > 0 such that rk H∗ (M) ≤ ε(n)- ess - Vol(M). Here rk H∗ (M) denotes the sum of the Betti numbers bi (M, k) of the homology groups Hi (M, k), where k is any field of coefficients; ε- ess - Vol(M) (the ε-essential-volume of M) is the supremum of the number of points (x1 , x2 , . . . ) in M with Radinj(xi ) ≥ ε and d(xi , x j ) ≥ 2ε for i 6= j. Theorem 3.8.21 ([43]). Let M be analytic with curvature −1 ≤ K ≤ 0. Let us ase has no Euclidean de Rham factor and Radinj → 0. Then there exists a sume that M point x ∈ M with Radinj(x) ≥ µ/2. Remark 3.8.22. If M is locally symmetric with K ≤ 0 without Euclidean factor then, by the Kazhdan–Margulis theorem, there exists a point x ∈ M where Radinj(x) ≥ µ/2, here no additional assumption on Radinj at ∞ is needed. This result can be extended to more general manifolds. Theorem 3.8.23 ([348]). Let M ∗ be a compact irreducible locally symmetric Riemannian space of nonpositive curvature and rk(M ∗ ) ≥ 2. Let M be a compact Riemannian manifold with nonpositive curvature whose fundamental group is isomorphic to the fundamental group of M ∗ . Then M and M ∗ are isometric provided that one multiplies the metric of M or M ∗ by a suitable constant. The paper [283] by D. Kazhdan and G. Margulis settles affirmatively two outstanding conjectures. Let G be a connected linear semi-simple Lie group, without compact factor, Γ a discrete subgroup of G, µ a Haar measure on G, and µ(G/Γ ) the volume for the measure associated to µ. Theorem 3.8.24. Let ρ be a right invariant metric on G. There exists a neighborhood V of e, constants C1 > 0 and c > 1 such that, given Γ , one can find g ∈ G satisfying the two following conditions: (1) ρ(e, g) ≤ C1 , (2) for any γ ∈ Γ ∩ V , γ 6= e, we have ρ(e, gγg−1 ) ≥ cρ(e, γ).

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Corollary 3.8.25. There exists a constant µ0 > 0 such that µ(G/Γ ) > µ0 for any Γ . More precisely, there exists a neighborhood V1 of the identity such that any Γ has a conjugate intersecting V1 only at e. Theorem 3.8.26. If µ(G/Γ ) < ∞, but G/Γ is not compact, then Γ contains an element γ 6= e which is unipotent (i.e. 1 − γ is a nilpotent linear transformation). Remark 3.8.27. The unipotency of γ in Theorem 3.8.26 is equivalent to the fact that e is an accumulation point of the conjugacy class C(γ) of γ in G, and it is in fact this property which occurs in the proofs. The statement of Theorem 3.8.26 had been conjectured by A. Selberg. The statement in Corollary 3.8.25 was also a well-known conjecture. These results are essentially derived by elementary, but very ingenious, manipulations from Theorem 3.8.24.

3.9 Open Questions In light of the above results, we mention some questions, the answers to which are not known to us. Question 3.9.1. Let X be a unit Killing vector field on a compact homogeneous simply connected Riemannian manifold M (with nonnegative sectional curvature) such that all integral curves of this field are closed. Is it true that the field X is regular? We remark that the answer to the above question is positive for two-dimensional manifolds (even without the assumptions on the homogeneity and the positiveness of the sectional curvature). According to Corollary 4.8.6, the answer to this question is also positive when M is a compact symmetric space M. In contrast, we have constructed examples of simply connected cohomogeneity 1 manifolds (Theorem 3.7.8) and homogeneous manifolds of constant sectional curvature with nontrivial fundamental group (Theorem 4.8.17), which admit a quasiregular Killing vector field of constant length. Question 3.9.2. Let X be a unit Killing vector field on a compact homogeneous simply connected Riemannian manifold M such that there is at least one closed integral curve of this field. Is it true that all integral curves of this field are closed? As it follows from Theorem 3.7.9, the answer to this question is negative under weakening of the assumption of homogeneity (e.g., if we replace it by the assumption of cohomogeneity 1). It seems interesting to search for conditions on a Riemannian manifold which permit us to find a converse to Proposition 3.4.11. The following argument shows some obstructions to such a converse. Let us consider any Riemannian manifold M which admits some quasiregular Killing vector field of constant length. Then it is easy to see that there is a Killing vector field of constant length on the direct

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165

metric product R × M such that all its integral curves are not closed, and there is an isometry in the 1-parameter group generated by this field, which is not a Clifford– Wolf translation. The following is more interesting. Question 3.9.3. Let X be a regular Killing vector field of constant length on a homogeneous Riemannian manifold M. Is it true that the 1-parameter group, generated by the field X, consists of Clifford–Wolf translations? Remark 3.9.4. Questions 3.9.2 and 3.9.3 are also interesting under the additional assumption that M is simply connected. Note that without the homogeneity assumption Question 3.9.3 has a negative answer. The following example is due to P. D. Andreev. Example 3.9.5. Let N be a compact Riemannian manifold, admitting free action by isometries of the finite cyclic group Zn , n ≥ 2, such that its generator β is not a Clifford–Wolf translation of N. The simplest example of such N is a round torus (the torus isometric to the torus of revolution in three-dimensional Euclidean space). Consider now a direct metric product P = N × R . This space admits the following action of the group Z: the generator of Z acts on the N-factor as the generator β of Zn and on the R-factor by a translation with sufficiently large shift l. It is enough to take the shift l larger than the intrinsic diameter of the Riemannian manifold N. Finally, let M = P/Z be the corresponding quotient manifold. Evidently, the unit vector field Xe on P, tangent to the R-factor, is a Killing vector field and it projects to the unit regular Killing vector field X on M by the differential of the quotient map. Let γ(t), t ∈ R, be a 1-parameter isometry group of M generated by X. Then the isometry g = γ(l) is not a Clifford–Wolf translation on M. Indeed, if g is a Clifford–Wolf translation, then by the above assumptions we get that β (the generator of Zn ) is a Clifford–Wolf translation on N, since the distance between z and g(z) in M is equal to the distance between y and β (y) in N, where y is the first component of any preimage of z under the quotient map P = N × R → M.

Chapter 4

Homogeneous Riemannian Manifolds

Abstract This chapter is devoted to homogeneous spaces and homogeneous Riemannian manifolds. We discuss transitive isometry groups for a given homogeneous Riemannian manifold and topological properties of homogeneous spaces. We consider the infinitesimal structure of homogeneous Riemannian manifolds and the structure of the set of G-invariant Riemannian metrics on a homogeneous space G/H. Moreover, we derive useful formulas for the sectional curvature, the Ricci curvature, and the scalar curvature of a given homogeneous Riemannian space. Special attention is paid to Killing vector fields of constant length and the corresponding isometric flows on symmetric spaces. It is proved that such a flow on any symmetric space is free or induced by a free isometric action of the circle and consists of Clifford–Wolf translations. Various examples of the Killing vector fields of constant length, generated by the isometric effective almost free but not free actions of the circle on the Riemannian manifolds close in some sense to symmetric spaces, are constructed. Finally, we discuss various topological and algebraical restrictions for homogeneous Riemannian spaces with positive or negative sectional curvature, as well as positive or negative Ricci curvature, and the structure of compact homogeneous Riemannian spaces with Killing vectors fields of constant length.

4.1 The Isometry Group of a Riemannian Manifold In what follows, Isom(M, g) and Isom0 (M, g) mean respectively the full isometry group and the full connected isometry group (i.e. the identity component of the full isometry group) of a Riemannian manifold (M, g). The following result is very important. Theorem 4.1.1 (S. Myers–N. Steenrod, [350]). Let (M, g) be a connected smooth Riemannian manifold. Then the following assertions hold. (a) The full isometry group Isom(M, g) is a Lie group, acting smoothly on M. (b) For any x ∈ M, the isotropy subgroup (or the stabilizer) © Springer Nature Switzerland AG 2020 V. Berestovskii and Y. Nikonorov, Riemannian Manifolds and Homogeneous Geodesics, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-56658-6_4

167

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Ix (M, g) = { f ∈ Isom(M, g) : f (x) = x} is closed in Isom(M, g). Moreover, the isotropy representation ρ : Ix (M, g) → GL(Mx ), f 7→ ρ( f ) = (T f )x defines an isomorphism of the group Ix (M, g) onto a closed subgroup of the orthogonal group O(Mx , gx ) ⊂ GL(Mx ). In Corollary 4.2 in [421, Chapter VII], it is proved that the automorphism group of any G-structure of finite type (the Riemannian structure is a partial case of such G-structures) is a Lie group. The Myers–Steenrod theorem directly implies Corollary 4.1.2. Let (M n , g) be a Riemannian manifold, then the following assertions hold. (a) Ix (M, g) is a compact subgroup of the group Isom(M, g). Moreover, if M is compact then Isom(M, g) is compact too. (b) dim Isom(M, g) ≤ n(n+1) = dim O(n) + dim M, moreover, for the simply con2 nected case, the equality in this inequality holds if and only if (M, g) has constant sectional curvature. Remark 4.1.3. An isometry f of a connected complete Riemannian manifold (M, g) is uniquely determined by the value f (x) at a given point x and by the corresponding tangent mapping Tx f , since f ◦ Expx = Exp f (x) ◦Tx f . Moreover, Tx f is a linear isometry of the Euclidean space (Mx , gx ) onto the Euclidean space (M f (x) , g f (x) ). This directly implies assertion (b) of Corollary 4.1.2. Since an isometry preserves the Riemannian metric g, it also preserves the LeviCivita connection, geodesics, the volume element, and the curvatures. Theorems 3.1.4 and 4.1.1 imply Theorem 4.1.4. For any complete Riemannian manifold (M, g), a Killing vector field X is characterized by the condition LX g = 0; any vector field with this condition is complete, i.e. it generates a 1-parameter isometry group; and the Lie algebra of Killing vector fields is the Lie algebra of the Lie group Isom(M, g). It should be noted that the action of the full isometry group on a given Riemannian manifold is proper in the sense of R. S. Palais [382]. Definition 4.1.5. Let G be a topological group acting continuously on a topological space X by µ : G × X → X. The action µ is proper if the mapping G × X 3 (g, x) 7→ (µ(g, x), x) ∈ X × X

(4.1)

is proper, i.e. if the preimage of any compact subset of X × X under (4.1) is a compact subset of G × X. Note that µ(g, x) could be denoted simply by g(x) when µ is specified. For our goals, smooth actions of Lie group on smooth manifolds are of the most importance. It is easy to prove the following criterion, which implies (in particular) that all actions by compact groups are proper.

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Proposition 4.1.6. Let µ : G × M → M be an action of a Lie group G on a smooth manifold M. The action µ is proper if and only if for any sequence {gn } in G and any convergent sequence {xn } in M, such that {µ(gn , xn )} converges, the sequence {gn } admits a convergent subsequence. Proof. Suppose that µ : G × M → M is proper and consider a sequence {gn } in G and a sequence {xn } in M, such that xn → x and µ(gn , xn ) → y as n → ∞ for some x, y ∈ M. Let B be a compact set in M such y ∈ B and µ(gn , xn ) ∈ B for all n. Since µ is proper, then the preimage of B × {x} under (4.1) is compact. In particular, the sequence {gn } is situated in a suitable compact subset in G. Let us prove the converse. Suppose that K is a compact set in M × M such that its preimage P ⊂ G × M under (4.1) is not compact. Then we can find a sequence (gn , xn ) ∈ P, n ∈ N, that has no convergent subsequence. Passing, if necessary, to a subsequence, we can assume that the image {(µ(gn , xn ), xn )} of this sequence under (4.1) converges to a point (x, y) ∈ M × M as n → ∞. In particular, xn → y as n → ∞, hence, the sequence {gn } has no convergent subsequence. Taking into account that µ(gn , xn ) → x as n → ∞, we get the contradiction. t u The following proposition gives another description of proper actions on Riemannian manifolds. Proposition 4.1.7. Let (M, g) be a connected complete smooth Riemannian manifold and G a subgroup of Isom(M, g). Then the action µ : G × M → M of G on M is proper if and only if for any point x ∈ M and compact subset C ⊂ M, the inverse image (µ × pr2 )−1 (C × {x}), where pr2 is the projection of G × M onto the second factor, is compact. Proof. The necessity is evident. Let us prove the sufficiency. Let K be any compact subset K ⊂ M × M. There is nothing to prove if (µ × pr2 )−1 (K) is empty. In the opposite case, there is a point (x1 , x2 ) ∈ K such that x1 = µ(g, x2 ) = g(x2 ) for some g ∈ G. Let p1 , p2 be the projections from M ×M to the first and the second factors of M × M. Set Ki = pi (K) for i = 1, 2. Then Ki are compact and K ⊂ K1 × K2 . Let d be the intrinsic distance of the Riemannian manifold (M, g). There exist numbers ri > 0, i = 1, 2 such that K1 ×K2 ⊂ B1 ×B2 , where Bi = B(xi , ri ) for i = 1, 2. Then the triangle inequality in (M, d) and the inclusion G ⊂ Isom(M, g) implies that (µ × pr2 )−1 (B1 × B2 ) ⊂ (µ × pr2 )−1 (B(x1 , r1 + r2 ) × {x2 }). Since µ × pr2 is a smooth, hence continuous, mapping, (µ × pr2 )−1 (K) is a closed set. The set C = B(x1 , r1 + r2 ) is compact by the Hopf–Rinow theorem 1.5.1. Therefore, the last preimage is compact if (µ × pr2 )−1 (C × {x}) is compact for x = x2 , because (µ × pr2 )−1 (K) ⊂ (µ × pr2 )−1 (C × {x}). t u The following result is very important (we refer to [19, 123, 382, 497] for a more detailed discussion).

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Theorem 4.1.8. Let (M, g) be a complete Riemannian manifold and G a closed subgroup of Isom(M, g). Then the action µ : G × M → M, µ(g, x) = g(x), is proper. Proof. We shall use Proposition 4.1.7. Let C be a compact subset of (M, g), x0 ∈ M, and gk a sequence in G such that (gk , x0 ) ∈ (µ × pr2 )−1 (C × {x0 }), i.e. xk := gk (x0 ) ∈ C. Then there is a subsequence hl = gkl such that yl = hl (x0 ) converges to a point y0 ∈ C. Choose some orthonormal bases V (x0 ) in (Mx0 , gx0 ) and V (yl ) in (Myl , gyl ), l = 0, 1, . . . , such that V (yl ) converges in (T M)n , where n = dim M, to V (y0 ) when l → ∞. Then the matrix Al of (T hl )x0 with respect to V (x0 ) and V (yl ) is an element of the compact group O(n). Hence there is a subsequence lm such that Alm converges to a matrix A0 ∈ O(n). Set zm = ylm , fm = hlm . This implies that (T fm )x0 : Mx0 → Mzm converges (pointwise on Mx0 in T M ) to an isometric linear mapping L : (Mx0 , gx0 ) → (My0 , gy0 ) with the matrix A0 relative to the bases V (x0 ) and V (y0 ). Define a mapping F0 = Expy0 ◦L : Mx0 → M. Since fm are isometries, 


 d Expx0 (v1 ), Expx0 (v2 ) = d Expzm (T fm )x0 (v1 ) , Expzm (T fm )x0 (v2 ) for any v1 , v2 ∈ Mx0 , where d is the intrinsic metric of (M, g). Then by continuity,

d Expx0 (v1 ), Expx0 (v2 ) = d F0 (v1 ), F0 (v2 ) . This implies that formula f0 (x) = F0 (v) if x = Expx0 (v) for v ∈ Mx0 , defines an isometry of (M, g) and fm → f0 ∈ G when m → ∞. This means that (µ × pr2 )−1 (C × {x0 }) is compact and the action µ is proper according to Proposition 4.1.7. t u Theorem 4.1.8 implies the following result. Corollary 4.1.9. Let (M, g) be a Riemannian manifold and let Isom(M, g) be its full isometry group. Then the following assertions hold. 1) For any x ∈ M and any r ≥ 0, ∆x,r := {a ∈ Isom(M, g) | d(x, a(x)) ≤ r}, where d is the intrinsic metric on (M, g), is a compact subset in Isom(M, g). 2) If G is any subgroup of Isom(M, g) and O is the orbit of G through a point x ∈ e of O in M is the orbit of the group G, e the closure of G M, then the closure O in Isom(M, g), through the point x. In particular, if G is a closed subgroup of Isom(M, g), then all orbits of G on M are closed. Proof. By Theorem 4.1.8, the action of Isom(M, g) on M is proper. Hence, the first assertion of the corollary follows
immediately from Proposition 4.1.7, because ∆x,r = q1 (µ × pr2 )−1 (B(x, r) × {x}) , where q1 : Isom(M, g) × M → Isom(M, g) is the natural projection. e Let us prove the second assertion. Assume that y is an arbitrary
point in O and e on M. Then q1 (µ × pr2 )−1 (B(y, r) × {x}) ∩ G 6= 0, µ is the action of G / hence
Dr := q1 (µ × pr2 )−1 (B(y, r) × {x}) are compact nonempty sets for all r > 0 by Theorem 4.1.8 and Proposition 4.1.7. Therefore, the set D0 = ∩ r>0 Dr is compact e is and nonempty, and g(x) = y for every g ∈ D0 . This means that any point y ∈ O

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171

e through the point x. Now, assume that y is any contained in the orbit of the group G e e point in the orbit of the group G through the point x, i.e. g(x) = y for some g ∈ G. Then there exists a sequence gk ∈ G converging to g and y = g(x) = lim gk (x), where k→∞

e gk (x) ∈ O. This means that y ∈ O.

t u

The following theorem is a partial converse to Theorem 4.1.8. Theorem 4.1.10 (Theorem 3.4.4 in [454]). If µ : G × M → M is a smooth proper action of a Lie group G on a smooth manifold M, then M admits a smooth G-invariant Riemannian metric. After proving Theorem 4.1.10, C. T. C. Wall adds the following: “I expect that the existence of a complete invariant metric can be established, but have not found a proof”. Given a smooth action of the Lie group G on M, and a closed subgroup H of G, a smooth H-slice to the action is a smooth embedded submanifold V of M such that (we denote by O(G, z) the orbit of G through a point z ∈ M) (S1) For all y ∈ V , My = O(G, y)y +Vy ; (S2) V is H-invariant; (S3) If s ∈ V , g ∈ G, and g(s) ∈ V , then g ∈ H. Theorem 4.1.11 (Theorem 3.3.4 in [454], see also [382]). For any proper smooth action of a Lie group G on a smooth manifold M and any x ∈ M there exists a smooth Gx -slice V to the action with x ∈ V . Given a subgroup H of G and an action of H on V , we define G ×H V to be the quotient of G ×V by the relation (gh, y) ' (g, hy) for all g ∈ G, h ∈ H and y ∈ V . We denote the equivalence class containing (g, y) by [g, y]. Setting g0 ([g, y]) := [g0 g, y] defines an action of G on G ×H V . Theorem 4.1.12 (Theorem 3.3.5 in [454]). Let V be an H-slice at x to a smooth proper action of the Lie group G on a smooth manifold M with H = Gx . Then the action induces a smooth map j : G ×H V → M giving an equivariant diffeomorphism onto a neighborhood Y of O(G, x) in M. If V is an open disc, this gives a tubular neighborhood of O(G, x) in M.

4.2 Homogeneous Spaces Let us start with the definition of homogeneous spaces. Definition 4.2.1. A smooth manifold M is called a homogeneous (G-)space if there is a smooth transitive action µ : G × M → M (cf. Definition 2.2.3).

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The isotropy subgroups Gx and Gy at the points x, y of a homogeneous G-space M are conjugate in the group G. Indeed, there is a ∈ G such that a(x) = y. Then Gy = aGx a−1 . Note that several Lie groups can act smoothly and transitively on a given manifold M. Examples of such actions will be considered below. More detailed expositions of this subject can be found in the books [126], [377], and [261] (see also the references therein). The condition of being simply transitive for a given action (cf. Definition 2.2.3) is equivalent to the condition that the isotropy groups are trivial. Note that, for a simply transitive action, the manifold M is diffeomorphic to the group G (in order to construct a required diffeomorphism, we fix a point x ∈ M and then we associate an arbitrary point y ∈ M with the (unique) element of the group G that moves x to y). Now, let us consider the universal model of homogeneous spaces. Recall that according to Theorem 2.1.32, every closed subgroup H of the Lie group G is a Lie subgroup. Definition 4.2.2. Let G be a Lie group and H be its closed subgroup. Then the quotient space G/H = {aH | a ∈ G} of left cosets with the natural topology and the canonical projection π : G → G/H, defined by the equality π(a) = aH, is called a homogeneous space (of the group G by the subgroup H). Remark 4.2.3. Note, that in the same manner, one can define a homogeneous space of right cosets H \ G = {Ha | a ∈ G}. The homogeneous space G/H has a unique analytic structure such that the group G is a Lie group of transformations of the manifold G/H (for details, see Theorem 4.2 from [256]). The corresponding action of the Lie group G on G/H is generated by the left translations: (a, bH) → (ab)H = a(bH) =: la (bH),

a, b ∈ G.

(4.2)

The stabilizer (the isotropy subgroup, in other terms) of this action at the point H ∈ G/H is the group H, T (H) = {T la , a ∈ H} is the linear isotropy group, and the normal subgroup GG/H = ∩g∈G gHg−1 in the group G is the non-effectiveness kernel. The group G acts on G/H (almost) effectively if and only if H does not contain a (non-discrete) non-trivial normal subgroup of the group G. Note that in general, right translations in the group G do not generate a natural action on a homogeneous space G/H. Nevertheless, for the homogeneous space G/H, there is a natural (right) action of the group NG (H) = {n ∈ G | nHn−1 = H}, which is the normalizer of H in the group G. The normalizer NG (H) acts on the homogeneous space G/H by the following G-equivariant diffeomorphisms: (bH, n) → bnH = bHn =: rn (bH),

b ∈ G, n ∈ NG (H).

(4.3)

Moreover, every G-equivariant diffeomorphism of G/H has this form [126]. Also we have a natural action (aH, nH) → aHn = anH, a ∈ G, n ∈ NG (H), of the group

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173

NG (H)/H on G/H. The group NG (H)/H is often called the gauge group of the homogeneous space G/H. Remark 4.2.4. Note that for the trivial isotropy group H = {e}, the normalizer NG (H) coincides with G. The above construction just means that right translations by all elements of the group G are well-defined on the Lie group G = G/H. Example 4.2.5. Let us consider CPn = SU(n + 1)/S(U(1)U(n)). Then the group SU(n + 1)/Zn+1 , where Zn+1 is the center of the group SU(n + 1), acts effectively on CPn , whereas the group SU(n + 1) acts on CPn only almost effectively. It is clear that, in general, it suffices to deal only with effective spaces, but in some cases it is technically more convenient (cf. the above example) to deal with spaces that are not effective (e.g. almost effective). Example 4.2.6. In the paper [340] by D. Montgomery and H. Samelson and in the papers [115, 116] by A. Borel, all transitive and effective actions of compact (connected) Lie groups on spheres are classified (see also [106, Chapter 7], [377], [389, Chapter 21], or [502]). Namely, spheres can be realized as homogeneous spaces G/H in one of the following ways: 1) Sn−1 = SO(n)/SO(n − 1); 2) S2n−1 = SU(n)/SU(n − 1); 3) S2n−1 = U(n)/U(n − 1); 4) S4n−1 = Sp(n)/Sp(n − 1); 5) S4n−1 = Sp(n)Sp(1)/Sp(n − 1)Sp(1); 6) S4n−1 = Sp(n)U(1)/Sp(n − 1)U(1); 7) S15 = Spin(9)/Spin(7); 8) S7 = Spin(7)/G2 ; 9) S6 = G2 /SU(3). Example 4.2.7. Other important examples of homogeneous spaces are projective spaces: RP n = SO(n + 1)/O(n),

CP n = SU(n + 1)/S(U(1)U(n)),

HP n = Sp(n + 1)/Sp(n)Sp(1),

CaP 2 = F4 /Spin(9).

All Lie groups, acting transitively on the sphere Sn , also act transitively on the real projective space RPn , moreover, Sp(n) acts transitively on the complex projective space CP2n−1 (using the identification of Hn with C2n ) with the isotropy subgroup Sp(n − 1)U(1). This exhausts the list of compact Lie groups (up to local isomorphisms) that act transitively and almost effectively on projective spaces (for more details, see [377]). Example 4.2.8. The only compact connected Lie group acting transitively and effectively on the torus T n is T n itself [341].

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Example 4.2.9. Examples of the simplest noncompact homogeneous spaces are the following: 1) the hyperbolic space (the Lobachevsky space) H n = SO0 (n, 1)/SO(n), where SO0 (n, 1) is the identity component of the group O(n, 1), preserving a quadratic form with the signature (n, 1) in Rn+1 ; 2) complex, quaternion, and octonion analogues of the hyperbolic space CH n = SU(n, 1)/S(U(n)U(1)),

HH n = Sp(n, 1)/Sp(n)Sp(1),

CaH 2 = F4−20 /Spin(9). In what follows, g and h mean the Lie algebras of the group G and of its subgroup H (more precisely, LG and LH) respectively. Put o = eH = π(e), where e is the unit in the Lie group G. Next, we select a special important class of homogeneous spaces. Definition 4.2.10. A homogeneous space G/H is called reductive if there is a subspace p in the Lie algebra g such that g = h ⊕ p,

Ad(H)(p) = p.

(4.4)

In this case, the decomposition (4.4) is also called reductive, whereas p is called a reductive complement to h in g. The space p in Definition 4.2.10 is naturally identified with the tangent space to G/H at the point o = eH ∈ G/H by the mapping Te π, where π : G → G/H is the canonical projection. In this case, the restriction of the group Ad(H) = AdG (H) to p coincides with the action of the linear isotropy group. Theorem 4.2.11. A homogeneous space G/H is reductive if one of the following conditions holds: 1) There is an AdG (H)-invariant bilinear form Q on the Lie algebra g of the Lie group G such that its restriction to the Lie algebra h is non-degenerate; 2) The group H is compact; 3) The closure Ad(H) of the group Ad(H) = AdG (H) in GL(g) is compact. Proof. If Condition 1) holds, we can take as p the Q-orthogonal complement to h in g. Indeed, the restriction of Q to h is non-degenerate, hence, we get a linear space decomposition g = h ⊕ p. Moreover, AdG (H) preserves both the form Q and the Lie algebra h. Therefore, it preserves also the subspace p, and the space G/H is reductive. It is clear that Condition 2) implies Condition 3). Hence, it suffices to consider the latter condition. Let us consider an inner product Q on g that is obtained from an arbitrary inner product on g by using the averaging procedure with respect to the compact group Ad(H). All operators Ad(a), a ∈ H, are isometries with respect to Q. Therefore, Condition 1) holds, and G/H is reductive. t u

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175

Example 4.2.12. Let G be a semisimple (not necessarily compact) Lie group and H be its compact subgroup. Then for the homogeneous space G/H, there is a reductive decomposition g = h ⊕ p orthogonal with respect to the Killing form Bg of the Lie algebra g. This construction is well-defined due to Corollary 2.4.17 and Condition 1) in Theorem 4.2.11 (in this case, we take Q = Bg ). Remark 4.2.13. According to [404, Theorem 2], the statement of Theorem 4.2.11 is true if G is a connected simple Lie group and H is its closed semisimple subgroup. In [212], all non-reductive homogeneous pseudo-Riemannian spaces of dimension 4 are classified; such spaces do exist. Lemma 4.2 in the same paper implies the existence of non-reductive homogeneous spaces G/H with semisimple Lie group G. It should be noted that, in general, there are several reductive decompositions for a given reductive homogeneous space. Example 4.2.14. Suppose that G = F × F for a simple compact Lie group F, and H = diag(F) = {(x, x) ∈ F × F | x ∈ F}. Then we may choose the following subspaces as reductive complements to h = {(U,U) ∈ f ⊕ f} in g = f ⊕ f: {(U, 0) ∈ f ⊕ f |U ∈ f},

{(U, −U) ∈ f ⊕ f |U ∈ f},

{(0,U) ∈ f ⊕ f |U ∈ f}.

More generally, for every α, β ∈ R with α 2 + β 2 6= 0 and α 6= β , the subspace {(αU, βU) ∈ f ⊕ f |U ∈ f} is a reductive complement to h in g. Proposition 4.2.15. Let G/H be a homogeneous space such that the action of Ad(H) on g is completely reducible (cf. Definition 2.1.36). (For instance, this assumption is fulfilled if G/H admits an invariant Riemannian metric, see Proposition 4.9.1.) Then G/H is reductive and, moreover, the following conditions are equivalent: 1) A reductive decomposition g = h ⊕ p for G/H is not unique; 2) There is a nontrivial Ad(H)-equivariant module homomorphism ϕ : p 7→ h; 3) There are Ad(H)-isomorphic and Ad(H)-irreducible submodules p0 ⊂ p and h0 ⊂ h. Proof. Since h is Ad(H)-invariant, there is an Ad(H)-invariant complement to h in g, i.e. G/H is reductive. Let us prove the second assertion. 1) ⇒ 2) Let us consider a decomposition g = h⊕ e p such that e p is Ad(H)-invariant e e and p 6= p. Every X ∈ p has a unique representation as X = Xh + Xp , where Xh ∈ h and Xp ∈ p. Hence, we get two Ad(H)-equivariant homomorphisms ψ1 : e p 7→ p and ψ2 : e p 7→ h such that ψ1 (X) = Xp and ψ2 (X) = Xh for any X ∈ e p. Obviously, ψ1 is surjective, hence, we may take ϕ = ψ2 ◦ ψ1 −1 as an Ad(H)-equivariant homomorphism ϕ : p 7→ h. 2) ⇒ 3) Since the homomorphism ϕ : p 7→ h is Ad(H)-equivariant and nontrivial, there is an Ad(H)-irreducible submodule p0 ⊂ p such that the restriction of ϕ to p0 is nontrivial.

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3) ⇒ 1) Consider some Ad(H)-invariant decompositions p = p0 ⊕ p00 and h = and fix a non-trivial Ad(H)-isomorphism ϕ : p0 7→ h0 . Consider a subspace 0 e p = {X + ϕ(X) | X ∈ p}. It is clear that it is Ad(H)-invariant, hence, we get that e p0 ⊕ p00 is an Ad(H)-invariant complement to h in g and it is distinct from p. t u h0 ⊕ h00 ,

Finally we give some explicit examples of non-reductive homogeneous spaces. Example 4.2.16. Let us consider M = Rn \ {0}, n ≥ 2. This is a homogeneous space with respect to the transformation group GL(n, R). Let G0 be the identity component of GL(n, R) with the Lie algebra g = gl(n, R), that is also transitive on M. Let H be the isotropy subgroup at the point o = (1, 0, 0, . . . , 0) with respect to the action of G0 . Direct calculations show that    1α H= | det A > 0 . 0A The corresponding Lie subalgebra h is determined by the condition h = {X ∈ gl(n, R) | exp(tX) ∈ H for all t ∈ R}. It is easy to see that  exp(tX) =

1 y 0 e tY



 ∈H

for X =

1y 0Y

 , t ∈ R.

Obviously, the subspace  e h=

1y 0Y



 |Y ∈ gl(n − 1, R)

is a Lie subalgebra in gl(n, R) and e h ⊂ h. Since dim H = dim e h, we get e h = h. Any complementary subspace p to h in g = gl(n, R) has the form   x (x, z 0 )T p= , z R where T is a constant matrix with one row and (n − 1) columns, all entries of R are linear functions on x, z1 , z2 , . . . , zn−1 , and z 0 = (z1 , . . . , zn−1 ). Indeed, entries from the first column should be arbitrary, but all other entries should depend linearly on the entries of this column. Now, if we take     1α x (x, z 0 )T a= ∈ H, X = ∈ p, 0A z R then a direct calculation gives  x + αz Ad(a)X = aXa−1 = Az

−(x + αz)αA−1 + (x, z 0 )TA−1 + αRA−1 −Az αA−1 + ARA−1

 ,

4.3 On the Topology of Compact Homogeneous Spaces

177

hence, the subspace p is not Ad(H)-invariant. Indeed, let us take X ∈ p and a ∈ H such that A = diag(1, 1, . . . , 1), αz = 0, and z α is not a zero matrix. If p is Ad(H)invariant, then we get Ad(a)X = X, but this is not the case due to z α 6= 0. Therefore, M = G0 /H is not reductive.

4.3 On the Topology of Compact Homogeneous Spaces It is very important that the Euler characteristic χ(M) of a (compact) homogeneous space M = G/H is non-negative. Theorem 4.3.1 (H. Hopf–H. Samelson, [260], [377]). Let M = G/H be a homogeneous space, where G and H are connected compact Lie groups. Then χ(M) ≥ 0. Moreover, the following conditions are equivalent: (i) χ(M) > 0; (ii) rk(G) = rk(H). |WG | If χ(M) > 0, then χ(M) = |W , where |WG | (respectively, |WH |) is the order of the H| Weyl group WG (respectively, WH ) of the Lie group G (respectively, H).

Let us prove this theorem following [260]. Lemma 4.3.2. Let G × M → M be a smooth action of a compact connected Lie group G on a smooth compact manifold M n , g ∈ G, and a set F of fixed points of the mapping g on M consists of isolated points. Then the set F is finite and χ(M) = |F|, where |F| is the cardinality of the set F. If |F| > 0, then n is even. Proof. It is clear that the set F is finite. The last assertion follows from the fact that, for every point x ∈ F, (T g)x preserves the orientation of the space Mx and has no fixed non-zero vectors, which is possible only for even n due to compactness of the group G [222]. The following assertion is proved on p. 534 and p. 542 in [20]: if f is a continuous mapping, homotopic to the identical one, of an n-dimensional connected compact polyhedra P to itself, having a finite set F of fixed points in P, then the sum of the indices of all fixed points is equal to (−1)n χ(P). It is clear that all conditions of this assertion are fulfilled for P = M n and f = g. Moreover, in this case, the index of every point x ∈ F is equal to (−1)n . This observation implies the equality χ(M) = |F|. t u Proof (of Theorem 4.3.1). Let us suppose that rk(G) > rk(H). Then there is a U ∈ g such that the dimension of the closure of the 1-parameter subgroup exp(tU), t ∈ R, in G coincides with rk(G). We state that Ad(s)(U) 6∈ h for all s ∈ G. Indeed, suppose that V := Ad(s)(U) ∈ h. Since Ad(s) is an automorphism of the Lie algebra g, the dimension of the closure of the 1-parameter subgroup exp(tV ), t ∈ R, in G is also equal to rk(G). On the other hand, this closure is a torus in H, since H is a compact

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subgroup of the Lie group G. This contradicts the inequality rk(H) < rk(G). Since the stabilizers of points in G/H are pairwise conjugate, then, for any x ∈ G/H, U is not contained in the Lie algebra of the isotropy group of the point x. Now, according to Theorem 4.11.1, the space G/H admits a G-invariant Riemannian metric. Then there is a number t ∈ R such that g = exp(tU) has no fixed point in G/H due to Theorem 3.6.2 and the compactness of G/H. Now, Lemma 4.3.2 implies χ(M) = 0. Suppose that rk(G) = rk(H). According to Definition 2.5.24 and Theorem 2.5.22, there is a maximal torus T ⊂ G that is contained in H. Let t ⊂ h be the Cartan subalgebra tangent to the torus T and let U ∈ t be such that the 1-parameter subgroup exp(tU), t ∈ R, is dense in T . Let us determine when a point gH is fixed under the action of this subgroup. Let hgH = Ad(g)(h) be the Lie algebra of the isotropy group at the point gH. Obviously, U ∈ hgH ⇔ Ad(g−1 )(U) ∈ h. Then according to the inclusion t ⊂ h and Theorem 2.5.22, there is an h ∈ H such that Ad(hg−1 )(U) ∈ t or hg−1 ∈ NG (T ) ⇔ gh−1 ∈ NG (T ) ⇔ g ∈ NG (T )H ⇔ g ∈ NG (T )NH (T )H. It is clear that the converse is also true: the last inclusion implies U ∈ hgH . Therefore, the number of such fixed points gH is finite and is equal to |NG (T )/NH (T )| = |(NG (T )/T )/(NH (T )/T )| = |WG /WH | =

|WG | . |WH |

According to Theorem 3.6.2 and the compactness of G/H, there is a number t ∈ R such that g = exp(tU) has no other fixed points in G/H. Therefore, we get χ(M) = |WG | t u |WH | by Lemma 4.3.2. The theorem is proved.

4.4 General Results on Reductive Decompositions Let G/H be a homogeneous manifold, where G is a connected Lie group and H is a compact subgroup in G. We will suppose that G acts effectively on G/H (otherwise it is possible to factorize by U, the maximal normal subgroup of G in H). Since H is compact, there is an Ad(H)-invariant decomposition g = h ⊕ m,

(4.5)

where g = Lie(G) and h = Lie(H). We know that such a decomposition could be non-unique. In this section we discuss some natural choices of such decompositions. By [·, ·] we denote the Lie bracket in g, and by [·, ·]m its m-component according to (4.5). We denote the radical (the largest connected solvable normal subgroup) and the largest connected nilpotent normal subgroup of the Lie group G by R(G) and N(G), respectively. For the Lie algebra g = Lie(G), we denote by n(g) and r(g) the nilradical and the radical of g, respectively. A maximal semisimple subalgebra of g is

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179

called a Levi factor or a Levi subalgebra. The corresponding connected virtual Lie subgroup S in G is called a Levi subgroup. Recall that there is a semidirect decomposition g = r(g) o s, where s is an arbitrary Levi factor, moreover, any two Levi factors of g are conjugate by an automorphism exp(ad(Z)) of g, where Z is in the nilradical n(g) of g (see the Levi–Mal’tsev theorem 2.3.13). For any finite-dimensional representation ρ of a semisimpleTLie algebra f, the representation space V is the direct sum of the f-modules Ker ρ(x) and x∈f

∑ ρ(x)(V ), see Proposition 5.5.17 in [257]. Hence, if we consider r(g) as a repx∈f

resentation space for s, then we get r(g) = [s, r(g)] ⊕ Cr(g) (s) (the direct sum of linear subspaces), where Cr(g) (s) is the centralizer of s in r(g). By Theorem 2.13 in [271], we have [g, r(g)] ⊂ n(g), therefore, [s, r(g)] ⊂ [g, r(g)] ⊂ n(g). Moreover, by Theorem 3.7 in [271], D(r(g)) ⊂ n(g) for every derivation D of g. For any Levi factor s, we have [g, g] = [r(g) + s, r(g) + s] = [g, r(g)] o s ⊂ n(g) o s. Hence, [g, g] ∩ r(g) = [g, r(g)] ⊂ n(g). Note also that any semisimple subalgebra f of g is in n(g) o s, since f = [f, f] ⊂ [g, g] ⊂ n(g) o s. If B is the Killing form of g, then n(g) ⊂ {X ∈ g | B(X,Y ) = 0 ∀Y ∈ g}

(4.6)

⊂ {X ∈ g | B(X,Y ) = 0 ∀Y ∈ [g, g]} = r(g), see e.g. Proposition 1.4.6 in [122] and Theorem 3.5 in [271]. In particular, B([g, r(g)], g) = 0. Recall that a subalgebra k of a Lie algebra g is said to be compactly embedded in g if g admits an inner product relative to which the operators ad(X) : g 7→ g, X ∈ k, are skew-symmetric. This condition is equivalent to the following condition: the closure of AdG (exp(k)) in Aut(g) is compact, see e.g. [257, 487]. Note that for a compactly embedded subalgebra k, every operator ad(X) : g → g, X ∈ k, is semisimple and the spectrum of ad(X) lies in i R. A subgroup K of the Lie group G is compactly embedded if the closure of AdG (K) in Aut(g) is compact. It is known that a connected subgroup K is compactly embedded in G if and only if k = Lie(K) is compactly embedded in g = Lie(G), and the maximal compactly embedded subgroup of a connected Lie group is closed and connected [487]. Recall also that a subalgebra k of a Lie algebra g is said to be compact if it is compactly embedded in itself. This is equivalent to the fact that there is a compact Lie group with a given Lie algebra k. It is clear that any compactly embedded subalgebra k of g is compact. Lemma 4.4.1. The Killing form B of g is negative semi-definite on every compactly embedded subalgebra (in particular, on Lie algebras of compact subgroups in G) k of g. If the corresponding Lie subgroup K is the isotropy group in the isometry group G of an (almost) effective Riemannian space (G/K, ρ), then B is negative definite on k.

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Proof. There is an ad(k)-invariant inner product Q on g (by the definition of compactly embedded subalgebra), hence, every operator ad(X) : g → g is skewsymmetric with respect to Q and B(X, X) is a trace of a square skew-symmetric matrix. Hence B(X, X) ≤ 0 and B(X, X) = 0 if and only if X is in the center of g. The latter is impossible for the isotropy algebra k because G/K is assumed to be (almost) effective. t u Let G/H be an effective homogeneous space with compact H. Note that AdG (H) (which will be also denoted as Ad(H)) is a compact, hence completely reducible, group of automorphisms of the Lie algebra g = Lie(G). A completely reducible group of automorphisms of a Lie algebra keeps at least one maximal semisimple subalgebra invariant [347]. Therefore, there is a Levi factor s of g invariant with respect to AdG (H). Then we have the following Levi decomposition: g = r(g) + s,

(4.7)

where both r(g) and s are invariant with respect to AdG (H). It should be pointed out that, in the following, this AdG (H)-invariant Levi decomposition will be used. For any X ∈ h = Lie(H), there exists a unique decomposition X = Xr(g) + Xs ,

where Xr(g) ∈ r(g), Xs ∈ s.

(4.8)

Since [X, r(g)] ⊂ r(g), [X, s] ⊂ s, and [g, r(g)] ⊂ r(g), we get [Xr(g) , s] = 0 for all X ∈ h. Hence for any X,Y ∈ h we get [X,Y ] = [Xr(g) + Xs ,Yr(g) +Ys ] = [Xr(g) ,Yr(g) ] + [Xs ,Ys ]. Therefore, we get the Lie algebra homomorphisms ϕ : h → r(g) and ψ : h → s such that ϕ(X) = Xr(g) and ψ(X) = Xs . Note that ϕ(h) is a Lie subalgebra of r(g) such that [ϕ(h), s] = 0. Note that h is reductive, i.e. h = c(h) ⊕ [h, h], where c(h) is the center and [h, h] is a semi-simple ideal in h. Since r(g) is a solvable Lie algebra, it contains no semisimple Lie subalgebras, hence [h, h] ⊂ Ker(ϕ) ⊂ s. In particular, ϕ(h) is an abelian Lie algebra. It is clear also that h + s = s ⊕ ϕ(h) is a reductive Lie algebra. Indeed, ϕ(h) is the center of this algebra ([ϕ(h), s ⊕ ϕ(h)] = 0), and s is semisimple. Lemma 4.4.2. If a linear space q ⊂ r(g) is ad(h)-invariant, then we get the inclusions [h, q] ⊂ q ∩ [g, r(g)] ⊂ q ∩ n(g). In particular, q ∩ n(g) = 0 or q ∩ [g, r(g)] = 0 implies [h, q] = 0. Proof. It suffices to use [g, r(g)] ⊂ n(g).

t u

Lemma 4.4.3. We can choose an Ad(H)-invariant complement h1 to Ker(ψ) in h such that h1 ∩ s = h ∩ s, in particular, [h, h] = [h1 , h1 ]. Let h2 be an Ad(H)-invariant complement to h1 ∩ s in such h1 . Then h2 ⊂ c(h), i.e. h2 is central in h, the Lie algebras h2 , ϕ(h2 ), and ψ(h2 ) are pairwise isomorphic, and the following decompositions hold: h = h2 ⊕(h∩r(g))⊕(h∩s),

ϕ(h) = ϕ(h2 )⊕(h∩r(g)),

ψ(h) = ψ(h2 )⊕(h∩s).

4.4 General Results on Reductive Decompositions

181

Proof. The space h ∩ s is Ad(H)-invariant and (h ∩ s) ∩ Ker(ψ) = 0. Let l be an Ad(H)-invariant complement to (h ∩ s) ⊕ Ker(ψ) in h. Then we can take h1 := (h ∩ s) ⊕ l. Since [h, h] ⊂ s we see that l is in the center of h. From this we get [h, h] = [h1 , h1 ]. It is clear that Ker(ψ) and h1 are ideals in h, hence h = Ker(ψ) ⊕ h1 as Lie algebras. Obviously, the restriction of ψ to h1 is one-to-one. Moreover, Ker(ψ) ⊂ ϕ(h) and [Ker(ψ), h] ⊂ [ϕ(h), ϕ(h) ⊕ s] = 0, hence Ker(ψ) is central in h. Note also that the Lie algebras h1 and ψ(h) are isomorphic (under ψ). It is clear that the subalgebra h1 ∩ s = h ∩ s is Ad(H)-invariant. Since [h1 , h1 ] ⊂ h1 , [h, h] = [h1 , h1 ], and [h, h] ⊂ s, we have [h, h] = [h1 , h1 ] ⊂ h1 ∩ s. In particular, [h2 , h] ⊂ h1 ∩ s. Since h2 is an Ad(H)-invariant complement to h1 ∩ s in h1 , we have [h2 , h] ⊂ h2 , hence, h2 ⊂ c(h), i.e. h2 is central in h. It is clear also that h2 , ϕ(h2 ), and ψ(h2 ) are pairwise isomorphic. Indeed, ψ is one-to-one even on h1 , but if X ∈ h2 , X 6= 0, and ϕ(X) = 0, then X = ψ(X) ∈ h ∩ s, which is also impossible. Hence, we get h = h2 ⊕ (h ∩ r(g)) ⊕ (h ∩ s), ϕ(h) = ϕ(h2 ) ⊕ (h ∩ r(g)), and ψ(h) = ψ(h2 ) ⊕ (h ∩ s). t u

Now we can describe some useful types of reductive decomposition (5.9), using the above constructions. The subspaces Im(ψ) = ψ(h) and Ker(ψ) = h ∩ r(g) in g are Ad(H)-invariant. Now, consider any Ad(H)-invariant complement m1 to Ker(ψ) in r(g) and any Ad(H)-invariant complement m2 to Im(ψ) in s. Then m := m1 ⊕ m2 is an Ad(H)-invariant complement to h in g. Indeed, such m is Ad(H)-invariant. By Lemma 4.4.3, we have dim(m) + dim(h) = dim(m1 ) + dim(m2 ) + dim(h2 ) + dim(h ∩ r(g)) + dim(h ∩ s) = dim(m1 ) + dim(h ∩ r(g)) + dim(m2 ) + dim(ψ(h2 )) + dim(h ∩ s) = dim(r(g)) + dim(s) = dim(g). If X ∈ m ∩ h, then X = Xr(g) + Xs , where Xr(g) = ϕ(X) ∈ r(g) and Xs = ψ(X) ∈ s, since X ∈ h. On the other hand, X ∈ m implies Xr(g) = ϕ(X) ∈ m1 and Xs = ψ(X) ∈ m2 . Hence, Xs ∈ Im(ψ) ∩ m2 = 0 and X = Xr(g) ∈ Ker(ψ) ∩ m1 = 0. Therefore, m ∩ h = 0 and m is an Ad(H)-invariant complement to h in g. We will call such type of Ad(H)-invariant complements consistent with the invariant Levi decomposition (4.7). Sometimes, it would be better (for some technical reasons) to deal with the case when n(g) ⊂ m1 . It is possible to choose a complement with this property and consistent with the invariant Levi decomposition (4.7). Indeed, the Killing form B is negative on h and zero on n(g), therefore, h ∩ n(g) = 0. On the other hand, n(g) is a characteristic ideal in g, hence it is invariant with respect to Ad(H). Now, we can define an Ad(H)-invariant complement u to n(g) ⊕ Ker(ψ) in r(g) and put m1 := n(g) ⊕ u. Since [g, r(g)] ⊂ n(g) we get [h, u] = 0.

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Remark 4.4.4. There is another natural way to choose m1 such that n(g) ⊂ m1 : let us consider m01 := {X ∈ g | B(X,Y ) = 0 ∀Y ∈ s + h}. Since [g, g] ⊂ n(g) ⊕ s and B(n(g), g) = 0, we have r(g) = {X ∈ g | B(X,Y ) = 0 ∀Y ∈ [g, g]} = {X ∈ g | B(X,Y ) = 0 ∀Y ∈ s}. Taking into account (4.6) and s ⊂ s + h ⊂ g, we get n(g) ⊂ m01 ⊂ r(g). Further, s + h is Ad(H)-invariant, hence, m01 has the same property. The Killing form B is negative definite on Ker(ψ) = h ∩ r(g), therefore, m01 ∩ Ker(ψ) = 0. Now let us consider m001 , an Ad(H)-invariant complement to m01 ⊕ Ker(ψ) in r(g). Then we can put m1 := m01 ⊕ m001 . It is clear that m01 = {X ∈ r(g) | B(X,Y ) = 0 ∀Y ∈ ϕ(h)}. Let l be the B-orthogonal complement to Ker(ψ) = h∩r(g) in h. Note that for any Y ∈ Ker(ψ) and any X ∈ l, we get 0 = B(Y, X) = B(Y, ψ(X) + ϕ(X)) = B(Y, ϕ(X)), because B(r(g), s) = 0. Example 4.4.5. It should be noted that there are reductive decompositions (5.9) such that n(g) 6⊂ m (in particular, the corresponding complements are not consistent with the invariant Levi decomposition (4.7)). Let us consider the homogeneous space G/H = U(2)/S1 = (SU(2) × S1 )/S1 , where the isotropy group is embedded as follows: H = S1 = diag(S1 ) ⊂ S1 × S1 ⊂ SU(2) × S1 , where the embedding of the second multiple S1 in the product S1 × S1 is identical, and the embedding of the first multiple is defined by some fixed embedding i : S1 → SU(2). Since all circles (maximal tori) in the group SU(2) are pairwise conjugate under the adjoint action of SU(2), we get a unique homogeneous space (up to homogeneous space morphism) diffeomorphic to S3 . In the Lie algebra g = su(2) ⊕ R, we choose vectors e1 , . . . , e4 such that ei ∈ su(2) for 1 ≤ i ≤ 3, e4 ∈ R, h = Lie(H) = Lin{e3 + e4 }, [e1 , e2 ] = e3 , [e2 , e3 ] = e1 , [e3 , e1 ] = e2 . Let us fix an Ad(G)-invariant inner product h·, ·i on g such that the vectors ei (i = 1, . . . , 4) form an orthonormal basis with respect to this inner product. For any α > 0 consider a non-degenerate Ad(G)-invariant quadratic form (·, ·)α = h·, ·i|su(2) + αh·, ·i|R on g. Now, consider the orthogonal complement m to h in g with respect to (·, ·)α , which defines a reductive decomposition of g. The restriction of (·, ·)α to m defines a normal homogeneous, hence, a geodesic orbit metric, on the space G/H = S3 for all positive α, see a more detailed discussion e.g. in [362]. Note that n(g) = Lin{e4 } 6⊂ m for any α > 0, since (h, e4 )α 6= 0. It is clear that there are similar reductive decompositions for all compact homogeneous spaces G/H with non-semisimple groups G. Sometimes it is reasonable to use the following Ad(H)-invariant complement to h in g: m = {X ∈ g | B(X,Y ) = 0 for all Y ∈ h}. (4.9)

4.4 General Results on Reductive Decompositions

183

Since B is negative definite on h, m is a complement to h in g. Note also that n(g) ⊂ m due to (4.6). We will call such type of Ad(H)-invariant complements consistent with the Killing form of g. Advantages of this reductive decomposition follow from the next lemma. Lemma 4.4.6. If H is a compact subgroup in G, then m := {X ∈ g | B(X, h) = 0} is an Ad(H)-invariant complement to h in g. Let us consider any Ad(H)-invariant inner product (·, ·) on m and the operator A : m → m such that B(X,Y ) = (AX,Y ), X,Y ∈ m. Then the eigenspaces of A are Ad(H)-invariant and pairwise orthogonal both with respect to B and (·, ·). Moreover, there is a (possibly, non-unique) B-orthogonal and (·, ·)-orthogonal decomposition m = ⊕si=1 pi , where pi are Ad(H)invariant and Ad(H)-irreducible (ad(h)-invariant and ad(h)-irreducible) submodules. Moreover, [pi , p j ] ⊂ m, i 6= j, for any such decomposition. Proof. Since the Killing form B is negative definite on h, the first assertion is obvious. Let us prove the second assertion. It is clear that the operator A is Ad(H)equivariant and symmetric (with respect to (·, ·)), in particular, the eigenspaces of A are Ad(H)-invariant. Let Aα and Aβ be two eigenspaces of A with the eigenvalues α 6= β . Consider any X ∈ Aα and Y ∈ Aβ and suppose that β 6= 0. Then we have β (X,Y ) = (X, AY ) = B(X,Y ) = (AX,Y ) = α(X,Y ), which implies (X,Y ) = B(X,Y ) = 0. Hence, the eigenspaces of A are pairwise orthogonal with respect to B and (·, ·). Further, one can decompose every such eigenspace into a direct (·, ·)-orthogonal sum of Ad(H)-invariant and Ad(H)irreducible (or ad(h)-invariant and ad(h)-irreducible) submodules. Since B is a multiple of (·, ·) on every eigenspace of A, we get the required decomposition. Finally if [pi , p j ] 6⊂ m, then B([X,Y ], Z) 6= 0 for some X ∈ pi , Y ∈ p j , Z ∈ h. This implies B(Y, [Z, X]) 6= 0 and pi is not ad(h)-invariant, which is impossible. t u Remark 4.4.7. Note that A0 is an ideal of g. Indeed, A0 = {X ∈ g | B(X, g) = 0}, the equality B(A0 , g) = 0 implies B(A0 , [g, g]) = 0 and B([g, A0 ], g) = 0. Moreover, n(g) ⊂ A0 ⊂ r(g) according to (4.6). It is natural to consider the reductive complement (4.9) in order to study various problems related to homogeneous spaces. However, sometimes this complement is not the most helpful. Finally, we describe a possible choice of a reductive complement m by using maximal compactly embedded subalgebras in g. For a compact subgroup H of the Lie group G we can choose a maximal compactly embedded subgroup K in G such that H ⊂ K, since all maximal compactly embedded subgroups of G are conjugate. Similarly, all maximal compactly embedded subalgebras of g are conjugate under Aut(g). Recall also that the radical r(g) is a characteristic ideal in g. Now, we can choose a Levi subalgebra s in g, which is invariant under AdG (K). Then we get

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Lemma 4.4.8 (Lemma 14.3.3 in [257]). For every maximal compactly embedded subalgebra k of g, there exists an AdG (K)-invariant Levi decomposition g = r(g) o s with the following properties: 1) [k, s] ⊂ s; 2) [k ∩ r(g), s] = 0; 3) k = (k ∩ r(g)) ⊕ (k ∩ s); 4) [k, k] ⊂ s; 5) k ∩ s is a maximal compact subalgebra in s. A similar result for Lie groups is obtained in [487, Theorem 2]. Now, in order to choose an Ad(H)-invariant complement m to h in g one needs to choose some Ad(H)-invariant complement m1 to h in k, some Ad(K)-invariant complement m2 to k∩s in s, and some Ad(K)-invariant complement m3 to k∩r(g) in r(g), and put m := m1 ⊕m2 ⊕m3 . It is reasonable to call such complement consistent with a maximal compactly embedded subalgebra in g. It should be noted that a given Ad(H)-invariant complement to h in g could be simultaneously consistent with the invariant Levi decomposition (4.7) and consistent with the Killing form of g, and in some cases it could also be consistent with a maximal compactly embedded subalgebra in g, see e.g. Proposition 4.2.15.

4.5 Invariant Affine Connections on Reductive Homogeneous Spaces In this section we consider the description of invariant affine connections on reductive homogeneous spaces obtained by K. Nomizu in [369], see also [291]. K. Nomizu defined an affine connection as a rule which assigns to each X ∈ F ∞ (M) an C∞ (M)-endomorphism t(X) of F ∞ (M) satisfying the conditions t(X1 + X2 ) = t(X1 ) + t(X2 )

t( f X)(Y ) = f t(X)(Y ) + (Y f )X,

where f ∈ C∞ (M). Hence, that t(X) should be a C∞ (M)-endomorphism means that t(X)( fY ) = f t(X)(Y ). In particular, if we define t(X) by Y → ∇Y X for any covariant derivative ∇ on M, then we get an affine connection in the sense of Nomizu. Below we consider some important results from [369]. Let M = G/H be a reductive homogeneous space with a fixed reductive decomposition g = h ⊕ p of the Lie algebra g = LG, i.e. Ad(H)(p) ⊂ p. Let π : G → G/H be the natural projection, and o := eH ∈ M = G/H. Theorem 4.5.1. Under the above conditions and notation, there exists a one-to-one correspondence between the set of all invariant affine connections on G/H, and the set of all bilinear functions Λ on p × p, with values in p which are invariant with respect to Ad(H), i.e. Ad(a)(Λ (X,Y )) = Λ (Ad(a)(X), Ad(a)(Y )) for every X,Y ∈ p and every a ∈ H. The correspondence is given by

4.5 Invariant Affine Connections on Reductive Homogeneous Spaces

185


Λ (X,Y ) = t(Y ∗ )(X ∗ ) (o), where X,Y ∈ p, X ∗ and Y ∗ are suitable invariant vector fields (for details, see [369]) defined on a neighborhood of the point o and such that X ∗ (o) = X, Y ∗ (o) = Y , under the natural identification of p with the tangent space Mo . Notice that a Levi-Civita connection of a given G-invariant Riemannian metric on G/H (see Section 4.6) is also an invariant affine connection. Theorem 4.5.2. Under the above conditions and notation, let x(s), s ∈ R, be the 1-parameter subgroup of G generated by X ∈ p ⊂ g, and let x∗ (s) = π(x(s)) ⊂ M, the image of x(s) under the natural projection π : G → G/H = M. There exists a e on M with respect to the reductive decomposition unique G-invariant connection ∇ e is trivial and the curve x∗ (s) is a geodesic g = h⊕p such that the torsion tensor of ∇ e for every X ∈ p. Moreover, it corresponds to the connection function in (G/H, ∇) Λ (X,Y ) = 12 [X,Y ]p , Y ∈ p. Theorem 4.5.3. In the above notation and conditions, let x(s), s ∈ R, be the 1parameter subgroup of G generated by X ∈ p ⊂ g, and let x∗ (s) = π(x(s)) ⊂ M, the image of x(s) under the natural projection π : G → G/H = M. There exists a e on M with respect to the reductive decomposition unique G-invariant connection ∇ g = h ⊕ p such that the parallel translation of Y along x∗ (s), s ∈ R, coincides with the translation of Y induced by the subgroup x(s) for every X ∈ p. Moreover, it corresponds to the connection function Λ : p × p → p, that vanishes identically. These results lead to the following definition. e from Theorem 4.5.2 (respectively, TheDefinition 4.5.4. The unique connection ∇ orem 4.5.3) is called the canonical connection (in the sense of Nomizu) of the first kind (respectively, of the second kind) on M = G/H with respect to the reductive complement p. In other terminology (see e.g. [291]), the canonical connection of the second kind is called the canonical connection, whereas the canonical connection of the first kind is called the natural connection without torsion. It should be noted that in [369], the general formulas for the torsion tensor and the curvature tensor of any invariant affine connection on a given homogeneous space G/H are established (in particular, for the Levi-Civita connections of a given G-invariant Riemannian metric). Here we state only the following result. Theorem 4.5.5. Let M = G/H be a reductive homogeneous space with a reductive decomposition g = h ⊕ p. Then the torsion tensor Te and the curvature tensor Re of e of the second kind with respect to p have the following the canonical connection ∇ properties: e Te(X,Y ) = −[X,Y ]p , R(X,Y )Z = [[X,Y ]h , Z], as well as e Te = ∇ e Re = 0, ∇ for every X,Y, Z ∈ p.

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Note that for some reductive homogeneous spaces, the canonical connection of the first kind is the Levi-Civita connection for a suitable invariant Riemannian metric. These are exactly naturally reductive metrics, see Section 4.10.

4.6 Main Properties of Homogeneous Riemannian Manifolds In this section, we discuss the definitions and main properties of homogeneous Riemannian manifolds and homogeneous Riemannian spaces. Definition 4.6.1. A Riemannian manifold (M, g) is called a homogeneous Riemannian manifold (briefly, HRM), if its isometry group Isom(M, g) is transitive on M (i.e. for all points x, y ∈ M, there is an isometry that moves x to y). It is easy to see that some proper subgroups of the group Isom(M, g) can also act transitively on a given homogeneous Riemannian manifold (M, g). For instance, it is easy to prove that Isom0 (M, g), the identity component of Isom(M, g), acts transitively on every connected homogeneous Riemannian manifold (M, g). In this case, Isom0 (M, g) is called the full connected isometry group of (M, g). Euclidean spaces En give other examples of such kind, since the group of parallel translations, being a proper subgroup of the full connected isometry group of En , acts transitively on En . For a more detailed analysis, the following definition is useful. Definition 4.6.2. A Riemannian manifold (M, g) is called G-homogeneous (or homogeneous with respect to the action of the Lie group G) if G is a (possibly, virtual) Lie subgroup in Isom(M, g) that acts transitively on M. Sometimes in this definition, only Lie subgroups (excluding all other virtual Lie subgroups) of the group Isom(M, g) are considered as G (see e.g. [106]). This requirement does not significantly restrict the generality. Indeed, if G is a transitive and effective group of transformations of the manifold M, preserving the Riemannian metric g, then its closure G in the group Isom(M, g) acts transitively on M, being in this case a Lie subgroup of the Lie group Isom(M, g). For a detailed discussion see [185]. Note that a Riemannian manifold (M, g) together with a transitive effective isometry group G on it is called a (G-)homogeneous Riemannian space. In this case, M is diffeomorphic to the homogeneous space G/H, where H = Hx is the isotropy group (with respect to the action of G) at a point x ∈ M. If G is a closed subgroup of the group Isom(M, g), then according to Corollary 4.1.2, the isotropy group H of a given point x ∈ M is compact. If, moreover, G is transitive on M, then M is compact if and only if G compact. The linear isotropy representation χ( f ) = (T f )x of the isotropy group H in GL(Mx ) (cf. Theorem 4.1.1) is faithful (i.e. injective). The local compactness of manifolds easily implies

4.6 Main Properties of Homogeneous Riemannian Manifolds

187

Theorem 4.6.3. A homogeneous Riemannian manifold is complete. The following theorem gives a characterization of simply connected homogeneous Riemannian manifolds in terms of a special linear connection. Theorem 4.6.4 (W. Ambrose–I. M. Singer, [22]). A complete, connected and simply connected Riemannian manifold (M, g) is homogeneous if and only if there exists a linear connection ∇ with torsion T and curvature R satisfying ∇g = 0,

∇T = 0,

∇R = 0.

Remark 4.6.5. It is known (in the notation of Theorem 4.6.4) that (M, g) is a symmetric space for T = 0. If R = 0, then (M, g) is either a Lie group with a bi-invariant metric or the round 7-sphere [481, 482]. Note that Theorem 4.6.4 was generalized for manifolds with linear connections by B. Kostant in [295]. The proof of Theorem 4.6.4 can also be found in [442]. An interesting discussion of related results can be found in the paper [441] by F. Tricerri. In particular, in [441], examples of locally homogeneous spaces which are not locally isometric to a globally homogeneous space are provided. Now we are going to discuss an important property of geodesics in homogeneous Riemannian manifolds. We start with a more general situation. Proposition 4.6.6. Let γ = γ(t), t ∈ R, be a unit-speed geodesic in a Riemannian manifold (M, g). Suppose that γ(0) = γ(τ) with some τ > 0 and there is a Killing field X on (M, g) with X(γ(0)) = γ 0 (0). Then γ is a closed geodesic with period τ. Proof. We give two short proofs of this proposition. The first proof. By the Killing equation we have gx (∇Y X,Y ) = 0 for every vector Y ∈ Mx at every point x ∈ M, see e.g. 1) in Proposition 3.1.6. If we take Y = γ 0 (t) for t ∈ R, we get d g(X(γ(t)), γ 0 (t)) = g(∇γ 0 (t) X, γ 0 (t)) ≡ 0, dt

g(X(γ(t)), γ 0 (t)) = const .

The value of this inner product at t = 0 is equal to 1. Therefore, its value between X(γ(τ)) = X(γ(0)) = γ 0 (0) and γ 0 (τ) is also equal to 1, and it follows from the Cauchy–Schwarz inequality that γ 0 (0) = γ 0 (τ), which shows that γ is a closed geodesic with period τ. The second proof. Let φ (s), s ∈ R, be the 1-parameter isometry group of (M, g) generated by X. Then γs (t) = φ (s)γ(t), 0 ≤ t ≤ τ, is a geodesic variation with geodesic segments of length l(s) ≡ τ. Therefore, by (1.58) 0 = l 0 (0) = g(X(γ(τ)), γ 0 (t1 )) − g(X(γ(0)), γ 0 (0)) = g(γ 0 (0), γ 0 (τ)) − 1. Then g(γ 0 (τ), γ 0 (0)) = 1 and γ 0 (τ) = γ 0 (0) by the Cauchy–Schwarz inequality. This shows that γ is a closed geodesic with period τ. t u

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From Proposition 4.6.6 we easily get the following important result (see e.g. Theorem 5.1 in [315], Corollary 2 in [334], Lemma 1 in [170] or Remark 2.2 in [101]). Proposition 4.6.7. In a homogeneous Riemannian manifold (M, g), any geodesic loop must be a closed geodesic. Proof. Since (M, g) is homogeneous, for any unit-speed geodesic γ : R → M, there is a Killing field X on (M, g) with X(γ(0)) = γ 0 (0). Now it suffices to apply Proposition 4.6.6. t u Theorem 1.5.44 and Proposition 4.6.7 imply the following corollary for the homogeneous case. Corollary 4.6.8. Let (M, g) be a homogeneous Riemannian manifold, x ∈ M, and s0 (x) = d(x, Conjx ). Denote by L0 (x) the infimum of the lengths of nontrivial simple closed geodesics with origin and end at x. Then Radinj(M) = Radinj(x) = min(s0 (x), L0 (x)/2), and the functions M 3 x 7→ s0 (x) and M 3 x 7→ L0 (x) are constant.

4.7 Symmetric and Locally Symmetric Spaces Important examples of homogeneous Riemannian manifolds are Riemannian symmetric spaces. Definition 4.7.1. A Riemannian manifold (M, g) is called symmetric, or a Riemannian symmetric space, if for every point x ∈ M, there is an isometry sx of the manifold (M, g) such that x is an isolated fixed point of sx . The isometry sx is called the (central) symmetry with the center at the point x. Proposition 4.7.2. For a given point x in a Riemannian symmetric space (M, g), the isometry sx reverses every geodesic through the point x and, moreover, we have (T sx )x = − Id Mx . In particular, the isometry sx is uniquely defined for connected M. Proof. Let A = (T sx )x . Then A Mx ⊂ Mx and A2 = Id. It is easy to see that for X ∈ Mx 2X = (X −A X)+(X +A X) and Mx = V − ⊕V + , where V ± := {X ∈ Mx | A X = ±X}. Take any nontrivial X ∈ V + and consider a geodesic γ tangent to X. Then sx will leave γ pointwise fixed. This contradicts the assumption that x is an isolated fixed point. Therefore, Mx = V − and A = − Id. It is clear also that for any geodesic γ tangent to some nontrivial X ∈ V − , the isometry sx reverses γ with x fixed. t u Many important properties of Riemannian symmetric spaces are discussed in Subsection 4.8.1. Let us prove the following result. Theorem 4.7.3. A Riemannian symmetric space (M, g) is homogeneous.

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189

Proof. First, (M, g) is complete, since any segment of the geodesic can be extended by the symmetry with the center at its ends. Further, for all points x, y ∈ M, the symmetry with the center in the middle of any segment of a geodesic, connecting x and y (that does exist due to the completeness), rearranges x and y. Thus, the isometry group acts transitively on M. t u Let G be a full connected isometry group of a Riemannian symmetric space (M, g) and let H be the isotropy subgroup at a point x ∈ M. The central symmetry sx at the point x ∈ M generates an involutive automorphism σ of the group G: σ σ (a) = sx ◦ a ◦ s−1 x . Let us consider G = {a ∈ G | σ (a) = a} and its identity comσ ponent G0 . Proposition 4.7.4. In the above notation, Gσ0 ⊂ H ⊂ Gσ . Proof. Indeed, if a ∈ H, then a(x) = x and σ (a)(x) = sx ◦ a ◦ s−1 x (x) = x. Moreover, −1 ) = (Ta) , hence, σ (a) = a and (T σ (a))x = (T sx ◦a◦s−1 ) = (T s ) ◦(Ta) ◦(T s x x x x x x x x H ⊂ Gσ . Further, let ψ(t), t ∈ R, be a 1-parameter group in Gσ0 . Then σ (ψ(t)) = ψ(t) and sx ◦ ψ(t) = ψ(t) ◦ sx for every t ∈ R. In particular, sx (ψ(t)(x)) = ψ(t)(x), and ψ(t)(x) is a fixed point for sx . On the other hand, ψ(0)(x) = x and x is an isolated fixed point for sx . Therefore, ψ(t)(x) = x for all t ∈ R and Gσ0 ⊂ H. t u Let g be a Lie algebra of G and let h be a common Lie algebra of the subgroups Gσ0 , H, Gσ . Consider the automorphism T σ = AdG (sx ) of g, induced by the automorphism σ of G. It is clear that h = {X ∈ g | T σ (X) = X}. Let us consider p = {X ∈ g | T σ (X) = −X}. Then p is a reductive complement to h in g, in particular, [h, p] ⊂ p. Moreover, the following result holds. Proposition 4.7.5. In the above notation, the following inclusions hold: [h, h] ⊂ h,

[h, p] ⊂ p,

[p, p] ⊂ h.

The submodule p is AdG (H)-invariant, and p is a reductive complement for the homogeneous space G/H. The corresponding reductive decomposition g = h ⊕ p is called the Cartan decomposition for the symmetric Riemannian manifold (M, g). Remark 4.7.6. If G in Proposition 4.7.5 is a semisimple Lie group, then [p, p] = h. Proof. The inclusion [h, h] ⊂ h is obvious. If X,Y ∈ p and U ∈ h, then T σ (X) = −X, T σ (Y ) = −Y , T σ (U) = U, hence T σ ([U,Y ]) = [T σ (U), T σ (Y )] = [U, −Y ] = −[U,Y ], T σ ([X,Y ]) = [T σ (X), T σ (Y )] = [−X, −Y ] = [X,Y ], i.e. [h, p] ⊂ p and [p, p] ⊂ h. The second assertion is a simple consequence of the first one. t u This proposition shows that symmetric spaces, considered as homogeneous Riemannian manifolds, have very remarkable properties. It is known that any Ad(H)invariant inner product on p, as in Proposition 4.7.5, generates a Riemannian (homogeneous) metric on G/H that is symmetric, for details, see e.g. [256].

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´ CarThe classification of Riemannian symmetric spaces was obtained by E. tan [145], it can be found in many books (see, e.g. [106, 256, 326, 483]). Theorem 4.7.7 (Theorem 8.1.1 in [483]). Let R be the curvature tensor of a Riemannian manifold M. Then the following conditions are equivalent: 1) sectional curvature is invariant under parallel translation of tangent 2-planes of M; 2) R is parallel, i.e. ∇R = 0, where ∇ is the Levi-Civita connection on M; 3) if a linear isometry Mx → Mz of tangent spaces sends Rx to Rz then it extends to an isometry of normal neighborhoods; 4) if x ∈ M then the geodesic symmetry Expx (X) → Expx (−X) is an isometry on a normal neighborhood of x. The Riemannian manifold M is called locally symmetric (or a locally symmetric space) if it satisfies the conditions in Theorem 4.7.7. An important characterization of locally symmetric spaces is given in Proposition 7.3.1. It should be noted that a locally symmetric Riemannian manifold is not necessarily homogeneous. On the other hand the following result is known, see e.g. [483, Theorem 8.3.1]. Theorem 4.7.8. A complete simply connected locally symmetric Riemannian manifold M is symmetric. This theorem extends to any locally symmetric Busemann G-space, [53, 55]. Remark 4.7.9. A Riemannian homogeneous space with the property [p, p] ⊂ h, as in Proposition 4.7.5, is not necessarily symmetric, but it is locally symmetric and its universal covering space is symmetric, see e.g. [106, 7.70].

4.8 Killing Vector Fields of Constant Length on Locally Symmetric Manifolds This section is devoted to the study of Killing vector fields of constant length on locally symmetric Riemannian manifolds. Proposition 3.4.5 establishes connections between such fields and 1-parameter groups γ(t), t ∈ R, (flows) of Clifford–Wolf translations: every such flow is generated by a Killing vector field X of constant length on (M, g). The following question is natural: when is the converse statement true? If X is a unit Killing vector field on a compact (or on a homogeneous) Riemannian manifold M, then all isometries from the 1-parameter group γ(t) generated by X, t ∈ R, with sufficiently small |t| are Clifford–Wolf translations on M (Proposition 3.4.8). But this property is not valid in general for any t ∈ R . This is clear in the case when the unit Killing field X is quasiregular or has both closed and nonclosed integral curves.

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It is remarkable that the flow on a symmetric space, generated by a unit Killing vector field, consists of Clifford–Wolf translations (see Theorem 4.8.1), whereas there exist quasiregular Killing vector fields of unit length on homogeneous spherical space forms (hence, locally symmetric Riemannian manifolds) (Theorem 4.8.17). The presentation in this section (consisting of four subsections) is based on the paper [74].

4.8.1 Symmetric Spaces In this subsection we show that there is no quasiregular Killing field of constant length on Riemannian symmetric spaces (see Definition 4.7.1). Moreover, one has the following Theorem 4.8.1. Let M be a symmetric Riemannian space, and X be a Killing vector field of constant length on M. Then the 1-parameter isometry group µ(t), t ∈ R, of the space M, generated by the field X, consists of Clifford–Wolf translations. Moreover, if the space M has positive sectional curvature, then the flow µ(t), t ∈ R, induces a free isometric S1 -action on M. Beforehand, we will prove a series of auxiliary results. Lemma 4.8.2. Let M be a compact symmetric Riemannian space with the full isometry group Isom(M), Isom0 (M) the identity component of Isom(M), and X a Killing field of constant length on M, which generates the 1-parameter isometry group µ(t), t ∈ R. Then Zµ , the centralizer of the flow µ in Isom0 (M), acts transitively on M. Proof. The basis of the proof is the following observation. For an arbitrary Clifford– Wolf translation γ on a compact symmetric space M, let us consider Zγ , the identity component of the centralizer of γ in the group Isom(M). According to the results of V. Ozols [381, Corollary 2.7], the group Zγ acts transitively on M. As a result of the homogeneity of M, the injectivity radius of M is a positive constant s. According to the proof of Proposition 3.4.8, for all t such that |t| < s, the isometry µ(t) is a Clifford–Wolf translation on M. Put si = s/i! for any i ∈ N, and let Zi be the identity component of the centralizer of µ(si ) in the group Isom(M). It is clear that the inclusion Zi+1 ⊂ Zi holds for all i. Besides, all the groups Zi act transitively on M. Therefore, the sequence {Zi }, being decreasing by inclusion, is stabilized beginning with some number. Let Z be the intersection of all Zi for every i ∈ N. Then, as stated above, Z acts transitively on M. On the other hand, all the elements of the group Z commute with every µ(si ). Hence, as is easy to see, the group Z centralizes every isometry from the flow µ. It is obvious that Z ⊂ Zµ . Therefore, Zµ acts transitively on M. t u Further, we investigate the properties of a Killing field of constant length on a simply connected Riemannian symmetric space M. For such a space one has the decomposition into the direct metric product

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M = M0 × M+ × M− , where M0 is a Euclidean space, and M+ (M− ) is a simply connected symmetric space of compact (respectively, noncompact) type. According to a theorem of J. I. Hano [291, Theorem 3.5 of Chapter VI], one has the decomposition Isom0 (M) = Isom0 (M0 ) × Isom0 (M+ ) × Isom0 (M− ) for the identity components of the full isometry groups of these spaces, and, moreover, one has the corresponding decomposition g = g0 ⊕ g+ ⊕ g− for their Lie algebras that, in a natural way, are identified with the Lie algebras of Killing fields. Recall that a transvection on a symmetric space is an isometry, which induces a parallel transport along some geodesic [483]. Equivalently, one can define a transvection on a symmetric space M as a composition of two geodesic symmetries sx and sy for x, y ∈ M [291]. Let G be a closed isometry subgroup, generated by all transvections on a simply connected symmetric space M. Then G = V × Isom0 (M+ ) × Isom0 (M− ), where V is the vector translation group of the Euclidean space M0 [483, Theorem 8.3.12]. It is well known that the group G acts transitively on M. Moreover, we have the following lemma. Lemma 4.8.3. Let X be a Killing field of constant length on a simply connected Riemannian symmetric space M, which generates the 1-parameter group of isometries µ(t), t ∈ R. Then the image of the flow µ is situated in the group G of transvections of the space M, and ZµG , the centralizer of the flow µ in the group G, acts transitively on M. Proof. For the Killing field X, one has a decomposition X = X0 ⊕ X+ ⊕ X− , where Xi is a Killing field on the symmetric space Mi , i ∈ {0, +, −}. Since X is of constant length on M, every field Xi generates a 1-parameter subgroup of bounded isometries, i. e., isometries with bounded displacement function, on Mi . Then the field X0 is parallel on the Euclidean space M0 , while the field X− is trivial on M− , because M− has negative Ricci curvature (see Proposition 3.5.8, Corollary 3.5.4, and also [479]). Let µi (t), t ∈ R, be a 1-parameter isometry group on Mi , generated by the field Xi . It is clear that µ0 (t) is a 1-parameter group of parallel translations, and µ− (t) is the trivial group. Therefore, every isometry µ(t), t ∈ R, is situated in the group G of transvections, and µ(t) = µ0 (t) × µ+ (t) × e, where e is the unit of the group Isom0 (M− ). Note now that the centralizer of the flow µ0 in the group V coincides with the group V , but the centralizer of the flow µ+ in the group Isom0 (M+ ) is transitive on M+ according to Lemma 4.8.2. Therefore, the centralizer of the flow µ in the group G of transvections is transitive on the symmetric space M. t u

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Lemma 4.8.4. Let N be a symmetric Riemannian space, and Y a Killing vector field of constant length on N, which generates the 1-parameter isometry group ν(t), t ∈ R. Then there is a subgroup S of the full isometry group of the space N such that it is transitive on the space N, and it commutes with every isometry from the flow ν. Proof. Let M be the universal covering manifold for the symmetric space N. Then M is a simply connected symmetric space. Denote by G the group of transvections of M, and consider ∆ , the centralizer of the group G in the full isometry group Isom(M) of the symmetric space M. It is known that there is a discrete commutative subgroup Γ of ∆ , such that N = M/Γ , see [483, Theorem 8.3.11]. Now, consider a Killing field X on M, obtained by lifting the field Y on N, and the 1-parameter isometry group µ(t), t ∈ R, on M, generated by X. It is obvious that the field X is of constant length. According to Lemma 4.8.3, there is a subgroup K in the group G of transvections which acts transitively on M and which centralizes every isometry from the flow µ. Let us now consider the centralizer H of the group Γ in the group Isom(M). Since every element of the group Γ centralizes the group G of transvections in the group Isom(M), we have K ⊂ H. Since Γ is normal in H, we can define e It is clear that the group H e is a subgroup a natural epimorphism π : H → H/Γ = H. of the full isometry group of the symmetric space N. Let S = π(K). Since the group K is transitive on M, then the group S = π(K) is transitive on N. Since K commutes with every isometry from the flow µ, the Lie algebra of the group K commutes with the Killing field X. The Lie algebra of the group S is isomorphic to the Lie algebra of K and, obviously, it commutes with the Killing field Y on N. This means that the group S commutes with every isometry from the flow ν, generated by the Killing field Y . t u Proof (of Theorem 4.8.1). According to Lemma 4.8.4, there is a subgroup S in the full isometry group of the symmetric space M which acts transitively on M and commutes with every isometry from the flow µ. Let us show that for every t ∈ R, the isometry µ(t) is a Clifford–Wolf translation on M. Indeed, if we take x, y, ∈ M, then there is an s ∈ S such that s(x) = y. Hence d(x, µ(t)(x)) = d(s(x), s(µ(t)(x))) = d(s(x), µ(t)(s(x))) = d(y, µ(t)(y)). Let us prove the second assertion of Theorem 4.8.1. If the sectional curvature of the space M is positive, then M is compact and has rank 1 [256]. Let K be the image of the 1-parameter group µ(t), t ∈ R, under the natural homomorphism into the full (compact) isometry group Isom(M) of the space M. Consider the closure e of the group K in Isom(M). Each isometry, which is the limit of a sequence of K Clifford–Wolf translations, is a Clifford–Wolf translation itself. Therefore, the group e consists of Clifford–Wolf translations on M and, therefore, it acts freely on M. In K e is connected and commutative. Consequently, if dim(K) e ≥ 2, addition, the group K then the rank of the symmetric space M is at least 2 [256]. But in this case M does e = 1 and K e = K is isomorphic not have positive sectional curvature. Therefore, dim K 1 to the group S by the compactness of the group Isom(M). The theorem is proved. t u

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Remark 4.8.5. Let us note that in [208], the authors classified all free isometric actions of the group S1 on compact simply connected irreducible symmetric Riemannian spaces. Corollary 4.8.6. Let M be a symmetric Riemannian space, and X a Killing field of constant length on M. Then either the field X is regular or all its integral curves are not closed. To prove this corollary, it suffices to apply Theorem 4.8.1 and Proposition 3.4.11. Remark 4.8.7. The results of Subsection 4.8.3 demonstrate that the above statements are not valid for locally symmetric spaces. In Subsection 4.8.4 we show that there exist quasiregular Killing fields of constant length on some compact homogeneous locally symmetric Riemannian manifolds. It seems to be of interest to find some more extensive (natural) classes of Riemannian manifolds with the property that they do not admit a quasiregular Killing field of constant length.

4.8.2 Non-simply Connected Manifolds In this subsection we will show how one can reduce the study of the behavior of integral trajectories of Killing vector fields of constant length on non-simply connected Riemannian manifolds to the study of the same question on their universal Riemannian coverings. In particular, we will get some sufficient conditions for the existence of quasiregular Killing vector fields of constant length on non-simply connected Riemannian manifolds. It is not difficult to prove the following well-known result. e Theorem 4.8.8. Let M = M/Γ be a Riemannian quotient manifold of a simply e by its discrete free isometry group Γ , and let connected Riemannian manifold M e p : M → M be the canonical projection. A vector field X on M is Killing (of constant length) if and only if it is p-connected with a Γ -invariant Killing vector field Y on e (of constant length). Moreover, each element f ∈ Γ commutes with all elements M in γe(t), t ∈ R, the 1-parameter isometry group generated by Y . The following theorem permits us to understand the behavior of integral curves of Killing vector fields of constant length on non-simply connected Riemannian manifolds. Theorem 4.8.9. In the notation of Theorem 4.8.8, the field X on M is quasiregular or it has both closed and nonclosed trajectories if and only if one of the following conditions is fulfilled: 1) The corresponding vector field Y has this property.

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195

2) There exist nontrivial elements f ∈ Γ and γe(t) (for some t ∈ R), and a point e such that f (y) = γe(t)(y), but f 6= γe(t). y∈M In the latter case, the orbit of the field X, passing through the point x = p(y), is closed. Moreover, the field X is quasiregular and this orbit is singular (respectively, the field X has both closed and nonclosed orbits) if and only if the isometry f −1 γe(t) has finite (respectively, infinite) order. Proof. We have to consider the following three possible cases: e f (y) does not lie on 1) For every nontrivial element f ∈ Γ and every point y ∈ M, the integral trajectory of the field Y , passing through the point y. 2) There exist nontrivial elements f ∈ Γ and γe(t) (for some t ∈ R), and a point e such that f (y) = γe(t)(y); but in this case necessarily f = γe(t). y∈M 3) There exist nontrivial elements f ∈ Γ and γe(t0 ) (for some t0 ∈ R), and a point e such that f (y) = γe(t0 )(y), but f 6= γe(t0 ). y∈M Now, we will examine each of these cases separately. e condition 1) is equivalent to the condition 1) Since the group Γ acts freely on M, e γe, where γe = {γe(t) |t ∈ R}, that the natural induced action of Γ on the orbit space M/ is also free. It is not difficult to see that in this case the field X on M is quasiregular (has both closed and nonclosed trajectories) if and only if the field Y has this property. 2) It follows from the last assertion of Theorem 4.8.8 that under condition 2), a group ∆ = Γ ∩ γe is a discrete nontrivial central subgroup both in the group Γ and in the group γe; moreover, the natural induced isometric actions of quotient groups e are defined with Γ /∆ and γe/∆ ∼ = S1 on the Riemannian quotient manifold P = M/∆ e quotient spaces P/(Γ /∆ ) = M and P/(γe/∆ ) = M/γe. It is clear that the vector field Y induces a Killing vector field Z of constant length on P, and also Z is induced by an isometric smooth action of the group S1 on P. Thus the vector field Z is regular or quasiregular. Then the vector field X is also regular or quasiregular. Moreover, e γe is free. So the vector field X on the action of the group Γ /∆ both on P and on M/ M is regular (quasiregular) if and only if the field Z is regular (quasiregular). In this case it remains to prove that the vector field Z is regular if and only if the vector field Y is induced by a free action of the groups R or S1 . There exists the least number t0 > 0 such that γe(t0 ) ∈ ∆ (i.e. γe(t0 ) generates ∆ ). It is not difficult to see that t0 is the maximal value of the periodicity function for the vector field Z. Suppose that Z is regular and Y is generated by a free action of neither the group e and a number T > 0 such that R nor the group S1 . Then there exist a point y ∈ M e e γ (T )(y) = y and γ (T ) 6= I, where I is an identical mapping. Hence γ(T )(z) = z, where γ is the corresponding induced action of the group γe on P, and z ∈ P is the projection of the point y. Since the vector field Z is regular, T = kt0 for some natural number k. Thus I 6= γe(T ) = (γe(t0 ))k ∈ ∆ ⊂ Γ ,

γe(T )(y) = y.

e But this contradicts the free action of the group Γ on M.

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Suppose that Z is quasiregular. Then there exists a number t1 > 0 and a point z ∈ P e which projects to such that t1 = tk0 , k ≥ 2, and γ(t1 )(z) = z. Then for a point y ∈ M, t0 the point z, there is the least positive number of the form k + nt0 (with integer n) such that γe( tk0 + nt0 )(y) = y. If one now supposes that Y is generated by the action of one of the groups R or S1 , then clearly it must be the group S1 , and the periods of all orbits of the field Y would be equal to tk0 + nt0 . It follows from the definition of the number t0 that it must be a divisor of the number tk0 + nt0 , a contradiction. 3) Suppose that condition 3) is fulfilled. Evidently one can assume that t0 > 0. The group γe(t), t ∈ R, covers the 1-parameter isometry group γ(t), t ∈ R, of the space M, which generates the vector field X, i.e. p ◦ γe(t) = γ(t) ◦ p,

t ∈ R.

(4.10)

Moreover, p ◦ f = p.

(4.11)

Then p(y) = p( f (y)) = p(γe(t0 )(y)). This means that the orbit of the vector field X, passing through the point x = p(y) ∈ M, is closed and the curve c = c(t) = γ(t)(x),

0 ≤ t ≤ t0 ,

(4.12)

is (closed), generally speaking, a nonsimple, geodesic trajectory of X. Let v = {v1 = T p(y)(e1 ), . . . , vn = T p(y)(en )} be an orthonormal basis in the tangent Euclidean vector space Mx , obtained by the ey . It follows from relations projection of an orthonormal basis e = {e1 , . . . , en } in M (4.10) and (4.11) that T γ(t0 )(x) ◦ T p(y) = (T γ(t0 ) ◦ T p)(y) = T (γ(t0 ) ◦ p)(y) = T (p ◦ γe(t0 ))(y) = T ((p ◦ f ) ◦ ( f −1 ◦ γe(t0 ))(y) = T (p ◦ ( f −1 ◦ γe(t0 ))(y) = T p(y) ◦ T ( f −1 ◦ γe(t0 ))(y). This means that the matrix of the linear isometry T γ(t0 )(x) of the space Mx in its basis v coincides with the matrix B of the linear isometry T ( f −1 ◦ γe(t0 ))(y) in the ey . Since f 6= γe(t0 ), we have A 6= E, where E is the identity basis e of the space M matrix. It is clear that X is quasiregular (respectively, has both closed and nonclosed orbits) if and only if the matrix A has finite (respectively, infinite) order. Clearly, this is equivalent to the property that the order of the element f −1 γe(t0 ) is finite (respectively, infinite). The theorem is proved. t u Corollary 4.8.10. If, under the conditions of the previous theorem, all orbits of the field Y are closed and there are nontrivial elements f ∈ Γ and γe(t) (for some t ∈ R), e such that f (y) = γe(t)(y), then the element f −1 γe(t) has finite order. and a point y ∈ M Now we shall consider one more construction of non-simply connected Riemannian manifolds with quasiregular Killing vector fields of constant length. Let us suppose that a group S1 of Clifford–Wolf translations acts (freely) on a compact Riemannian manifold M. Then one can define a Riemannian manifold

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197

N = M/S1 ; moreover, the natural projection p : M → N = M/S1

(4.13)

is a Riemannian submersion. Now let Γ be a finite free isometry group on the manifold M. Then the quotient space M = M/Γ is a Riemannian manifold. We suppose further that the groups S1 and Γ commute. Then the action of the group Γ on M induces an isometric action on the manifold N. More exactly, we consider the group Γ1 = S1 ∩ Γ . It is normal in Γ . Thus one can define the quotient group Γ = Γ /Γ1 . The natural projection (4.13) defines an isometric action of the group Γ on the manifold N. Moreover, the quotient space N = N/Γ is a Riemannian orbifold (see Subsection 3.7.1). Proposition 4.8.11. Suppose that under the conditions mentioned above, the orbifold N = N/Γ is not a Riemannian manifold. Then the Riemannian manifold M admits a quasiregular Killing vector field of constant length. Proof. The factorization of the fibre bundle projection (4.13) with respect to the action of the group Γ gives a projection p1 : M → N,

(4.14)

which is also a submetry. According to Proposition 3.4.5 and Theorem 4.8.8, the unit vector field X, tangent to the fibres of the projection (4.14), is a Killing vector field. If, moreover, the orbifold N is not a Riemannian manifold (i. e., has singularities), then among the circle fibres of the projection (4.14) there are circles of different length (see Subsection 3.7.12). This is equivalent to the quasiregularity of the vector field X. t u

4.8.3 Locally Euclidean Spaces The structure of complete locally Euclidean spaces, i. e., of Riemannian manifolds with zero sectional curvature, is well known. Every such manifold is a quotient space Rn /Γ of the Euclidean space Rn by the action of some discrete free isometry group Γ [483]. In this subsection we will give some (mainly well known) information on Killing vector fields on locally Euclidean spaces, in particular, on Killing vector fields of constant length. On these grounds we will deduce a criterion for the existence of a quasiregular (or possessing both closed and nonclosed trajectories) Killing vector field of constant length on a locally Euclidean space (Theorem 4.8.16). Now, we recall the structure of Killing vector fields on the Euclidean space Rn (we suppose that it is supplied with the standard metric). The full isometry group Isom(Rn ) of Euclidean space is isomorphic to a semidirect product O(n) n V n , where O(n) is the group of orthogonal transformations, and V n is the vector group of parallel translations on Rn . So, each isometry g ∈ Isom(Rn ) acts by the following

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rule: g(x) = A(x) + a, where A ∈ O(n), a ∈ Rn . Let us agree to use the following notation for this: g = (A, a). If we have two isometries g = (A, a) and f = (B, b) of Euclidean space, then their composition is g· f = (AB, A(b)+ a). Since the group V n of parallel translations is a normal subgroup in Isom(Rn ), one can define the natural epimorphism d : Isom(Rn ) → Isom(Rn )/V n = O(n). (4.15) It is not difficult to see that an arbitrary Killing vector field X on Rn is defined by a pair (W, w), where w ∈ Rn , and W is a skew-symmetric mapping in Rn . Moreover, X(x) = W (x) + w for all x ∈ Rn . Notice that the field X is bounded on Rn if and only if the map W is zero. Now, we deduce a criterion for a Killing vector field X to be invariant under an isometry g = (A, a) ∈ Isom(Rn ). Proposition 4.8.12. An isometry g = (A, a) ∈ Isom(Rn ) preserves a Killing vector field X = (W, w) if and only if the equalities [A,W ] = AW −WA = 0

and

W (a) + w = A(w)

hold. In particular, for W = 0, these conditions are equivalent to the fact that the vector w is fixed by the transformation A. Proof. It is clear that the invariance condition of the field X with respect to the isometry g is equivalent to the equality dg(X(x)) = X(g(x)), i. e., [A,W ](x) = W (a) − A(w) + w for all x ∈ Rn . Hence, g preserves the Killing field X if and only if [A,W ] = 0 and W (a) + w = A(w). The second assertion of the proposition is an evident consequence of the first one. t u Now, let M be a locally Euclidean space, i.e. M = Rn /Γ , where Γ is a discrete free isometry group of Euclidean space Rn . Denote by dΓ the image of Γ under the epimorphism d (see (4.15)). Proposition 4.8.13. There is a nontrivial Killing vector field of constant length on the locally Euclidean space M if and only if there is a nonzero vector a ∈ Rn which is invariant relative to all transformations of the group dΓ . Moreover, every such field on M is the projection of a parallel Killing field X = (0, a) on Rn , where the vector a is invariant under all transformations in dΓ . Proof. Let Y be a Killing vector field of constant length on M. It is lifted to a unique Killing field X on Rn , which also has constant length, therefore has the form X = (0, a). Furthermore, according to Proposition 4.8.12, the vector a must be invariant under transformations in the group dΓ . And if some nonzero vector a ∈ Rn is invariant under the group dΓ , then one can define the projection to M of the Killing field X on Rn , which evidently will be a Killing vector field of constant length on M. t u Proposition 4.8.14. There exists a three-dimensional compact orientable locally Euclidean space M 3 without nontrivial Killing vector fields.

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199

Proof. One can consider a type G6 space from Theorem 3.5.5 in [483] as a space M 3 . The corresponding group dΓ for this space is generated by transformations di ∈ O(3), 1 ≤ i ≤ 3; moreover, in the appropriate orthonormal basis of the space R3 , these transformations have the form d1 = diag(1, −1, −1),

d2 = diag(−1, 1, −1),

d3 = diag(−1, −1, 1).

Hence, there is no nonzero vector a ∈ R3 , invariant under all transformations in the group dΓ . According to Proposition 4.8.13 and Corollary 4.2 of Chapter II in [290], the space M 3 has no nontrivial Killing vector field. t u Proposition 4.8.15. There exists a three-dimensional noncompact orientable locally Euclidean space M 3 with a unit Killing vector field such that exactly one integral trajectory of this field is closed. Proof. Consider the product metric on R2 × [0, 1], where R2 is the Euclidean plane. Let f be the rotation of R2 about the origin by an angle α which is incommensurable with π. Now identify each point of the form (x, 0) with the point ( f (x), 1). The obtained manifold is a locally Euclidean space and belongs to the class J1α by the Wolf classification [483, Theorem 3.5.1]. Now, consider the unit Killing vector field which is orthogonal to the fibres R2 . Evidently, exactly one integral trajectory of this field is closed. t u Note that among locally Euclidean spaces only spaces of the form Rm ×T l , where is an l-dimensional flat torus, are homogeneous [483, Theorem 2.7.1]. Since all these manifolds are symmetric Riemannian spaces, they admit no quasiregular Killing vector fields of constant length (Corollary 4.8.6). On the other hand, many nonhomogeneous locally Euclidean spaces admit quasiregular Killing vector fields of constant length. For example, the M¨obius band and the Klein bottle admit such fields, as well as many three-dimensional locally Euclidean spaces. Since the classification of locally Euclidean spaces in dimensions ≥ 5 is not known, the following result is of some interest. Tl

Theorem 4.8.16. A locally Euclidean space M = Rn /Γ admits a quasiregular (respectively, having both closed and nonclosed trajectories) Killing vector field of constant length if and only if Γ contains an element of the form f = (A, (I −A)b+a), where I is the identity map, the vector a 6= 0 is invariant under all transformations of the group dΓ , the vector b is orthogonal to the vector a, and A 6= I has finite (respectively, infinite) order. Moreover, the corresponding Killing vector field is the field d p(0, a); its singular trajectory passes through the point p(b), where p : Rn → M is the canonical projection. Proof. We will prove both assertions simultaneously. Necessity. Suppose that M admits a quasiregular (respectively, having both closed and nonclosed trajectories) Killing field X of constant length. Then, by Proposition 4.8.13, the field X is the projection of some parallel Killing field Y = (0, a) on Rn , and the vector a 6= 0 is invariant under all transformations in the group dΓ .

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e = Rn is induced by a free isometric action of the group Since the field Y on M γe(t) = (I,ta), t ∈ R, it follows from the proof of Theorem 4.8.9 that case 2) of Theorem 4.8.9 must hold, i. e., there exist nontrivial elements f = (A, c) ∈ Γ and γe(t0 ), t0 ∈ R, and a vector b ∈ Rn such that f (b) = Ab + c = γe(t0 )(b) = b + t0 a, but f = (A, c) 6= γe(t0 ) = (I,t0 a). Then c = (I − A)b + t0 a. If A = I, then c = t0 a and (A, c) = (I,t0 a), which is impossible. Thus A 6= I. Since A(a) = a, one can take an arbitrary vector of the form b + τa instead of b. Thus one can suppose that the vector b is orthogonal to a. Also, all the previous relations are preserved. Finally, by scaling the vector field X, one can replace t0 a by a. If one takes the canonical basis of the space Rn at the point y = b as the basis e, then the matrix B from the proof of Theorem 4.8.9 will coincide with the matrix A−1 . Then by Theorem 4.8.9 the vector field X on M is quasiregular, and its orbit through the point x = p(b) is singular (respectively, the field X has both closed and nonclosed orbits) if and only if the matrix A has finite (respectively, infinite) order. Sufficiency. Suppose that Γ contains an element of the form f = (A, (I −A)b+a), where the vector a 6= 0 is invariant under all transformations in the group dΓ , the vector b is orthogonal to the vector a, and A 6= I has finite (respectively, infinite) order. By Proposition 4.8.13, the parallel nontrivial vector field Y = (0, a) on Rn is pconnected with a Killing field X of constant length |a| > 0 on M, i. e., d p◦Y = X ◦ p, while the corresponding 1-parameter isometry groups γ(t), t ∈ R, and γe(t) = (I,ta), t ∈ R of spaces M and Rn are connected by relations (4.10). Evidently, f (b) = A(b) + (I − A)b + a = b + a = γe(1)(b) = (I, a)(b), but f 6= γe(1). Then by Theorem 4.8.9 one of the following two conditions holds: 1) the field X is quasiregular; 2) the field X has both closed and nonclosed orbits. Since the matrix A has the same sense as in the proof of the necessity, case 1) (respectively, 2)) holds if and only if A has finite (infinite) order; the corresponding singular orbit passes though the point x = p(b). The theorem is proved. t u

4.8.4 Homogeneous Spherical Space Forms According to the results of Subsection 4.8.2, we construct here examples of compact homogeneous Riemannian manifolds with quasiregular Killing fields of constant length. The main result of this section is the following Theorem 4.8.17. A homogeneous Riemannian manifold M of constant positive sectional curvature does not admit a quasiregular Killing field of constant length if and only if M is either a Euclidean sphere or a real projective space. Without loss of generality, we can deal with homogeneous Riemannian manifolds of constant sectional curvature 1 (the general case is reduced to this one by a scaling of the metric).

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In the simply connected case, it is natural to consider the sphere Sm with the Riemannian metric of constant sectional curvature 1 as a submanifold Sm = {x ∈ Rm+1 | kxk = 1} in the Euclidean space Rm+1 with the induced Riemannian metric. The isometry group of Sm coincides with O(m+1), the group of orthogonal transformations of the Euclidean space Rm+1 . According to Theorem 7.6.6 in [483], a homogeneous Riemannian manifold M of dimension m ≥ 2 has constant positive sectional curvature 1 if and only if it is a quotient manifold Γ \Sm of the sphere Sm of constant sectional curvature 1 by some discrete group Γ of Clifford– Wolf translations. J. A. Wolf obtained the following classification of homogeneous Riemannian manifolds with constant positive curvature (see [483, Theorem 7.6.6]): e be the round sphere of dimension m. Then M = Theorem 4.8.18 (J. A. Wolf). Let M e Γ \M is homogeneous only in the following cases (up to a conjugacy in O(m + 1)): 1) Γ is trivial; 2) Γ = {±I(m+1) } ⊂ O(m + 1); e ⊂ Cn , where m + 1 = 2n, Γ = {diag(e2πi(p/q) , ..., e2πi(p/q) )} ⊂ U(n), 3) M 0 ≤ p < q, p an integer and q ≥ 3 a fixed natural number; e ⊂ Hl , where Γ = {diag(z, ..., z)} ⊂ Sp(l), and z are elements 4) m + 1 = 4l, M of the binary dihedral or the binary polyhedral subgroup of the multiplicative group Sp(1) of unit quaternions in H, and Γ acts on the row vectors by right multiplication. A description of binary dihedral and binary polyhedral groups can be found e.g. in [483, §2.6]. The following result gives a characterization of Clifford–Wolf translations on the Euclidean sphere Sm . Lemma 4.8.19 (Lemma 7.6.1 in [483]). A linear transformation A ∈ O(m + 1) is a Clifford–Wolf translation of Sm if and only if either A = ±I or there is a unimodular complex number λ such that half the eigenvalues of A are λ and the other half are λ . Proof. Let r be the radius of Sm and let eib be an eigenvalue of A where |b| ≤ π. Then there is a point x ∈ Sm such that the distance from x to A(x) is r|b|. If A is a Clifford–Wolf translation of Sm , it follows that either A = ± Id or all eigenvalues of A have one and the same real part cos(b), moreover, the eigenvalues are divided into the pairs of pairwise conjugate eigenvalues. If A = ± Id, then A is of constant displacement 0 or πr. If there is a unimodular complex number λ such that half of the eigenvalues of A are λ and the other half are λ , then we may assume that λ = eib with 0 ≤ b ≤ π, and A is obviously of constant displacement br on Sm . t u By using Theorem 4.8.9, we obtain the following result. Proposition 4.8.20. Let Γ be a cyclic group of Clifford–Wolf translations on an odd-dimensional sphere S2n−1 , n ≥ 2, of constant sectional curvature, which is distinct from the groups {I} and {±I}. Then the homogeneous Riemannian space M = Γ \S2n−1 admits a quasiregular Killing field of constant length.

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Proof. Let p : S2n−1 → M = Γ \S2n−1 be the canonical projection. According to Theorem 7.6.6 in [483], the group Γ is generated by an element f ∈ O(2n), which has the following form in some orthonormal basis of the Euclidean space R2n :   cos(2π/q) sin(2π/q) f = diag(B, B, . . . , B), where B = , − sin(2π/q) cos(2π/q) for some q ∈ N , q ≥ 3. Now, let us consider a 1-parameter group of Clifford–Wolf translations (see Lemma 4.8.19), which is defined in the same basis as follows: µ(t) = diag(A(t), . . . , A(t), A−1 (t)), where   cos(2πt) sin(2πt) A(t) = , t ∈ R. − sin(2πt) cos(2πt) Let Y be a Killing field of constant length on S2n−1 , generating the flow µ. Note that each element of the group Γ commutes with every element of the flow µ. It is obvious that µ(1/q) 6= f , but µ(1/q)(y) = f (y), where y = (1, 0, . . . , 0) ∈ S2n−1 . Moreover, it is clear that the element g = f −1 µ(1/q) ∈ O(2n) has finite order. Therefore, by Theorem 4.8.9, we obtain that there is a quasiregular Killing field of constant length on the homogeneous space M = Γ \S2n−1 . Namely, as the required field we can consider a Killing field X on M = Γ \S2n−1 , which is p-connected with the field Y on S2n−1 . t u Remark 4.8.21. Note that for the quasiregular Killing field of constant length constructed in the proof of Proposition 4.8.20, one component of the set of singular points with respect to the corresponding action of S1 has codimension 2 in the manifold Γ \S2n−1 . Indeed, such a component is the image of the set S under the canonical projection p : S2n−1 → Γ \S2n−1 , where S = {y = (y1 , y2 , . . . , y2n−1 , y2n ) ∈ S2n−1 | y2n−1 = 0, y2n = 0}. We also need one more auxiliary result. Lemma 4.8.22. In the full isometry group of the Euclidean sphere S4n−1 , n ≥ 1, there are two pairwise commuting subgroups F1 and F2 of Clifford–Wolf translations, both isomorphic to the multiplicative unit quaternion group F. Moreover, the intersection of these two subgroups is the group {±I}, generated by the antipodal map of the sphere. Proof. Note that Example 3.4.4 contains in fact the proof of this lemma. But we give here the proof based on Lemma 4.8.19 in order to show various tools in the study of Clifford–Wolf translations Let us consider an n-dimensional (left) vector space H n over the quaternion field H . There is a standard H -Hermitian inner product in this space, whose real part coincides with the standard inner product in R4n (we fix an embedding H → R4 , defined by the formulae x1 + ix2 + jx3 + kx4 → (x1 , x2 , x3 , x4 ), and the induced em-

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bedding H n → R4n ). In terms of the norm k · k generated by this inner product, the sphere S4n−1 is defined by the equation kzk = 1, where z = (z1 , z2 , . . . , zn ) ∈ H n . Note that the multiplication of the quaternion y = y1 + iy2 + jy3 + ky4 by the quaternion x = x1 + ix2 + jx3 + kx4 from the left corresponds to a linear transformation in R4 , defined by the formula      y1 x1 −x2 −x3 −x4 y1  y2   x2 x1 −x4 x3   y2   →   (4.16)  y3   x3 x4 x1 −x2   y3  , y4 x4 −x3 x2 x1 y4 whereas the multiplication by the quaternion x = x1 + ix2 + jx3 + kx4 from the right corresponds to the transformation      y1 x1 −x2 −x3 −x4 y1  y2   x2 x1 x4 −x3   y2   →   (4.17)  y3   x3 −x4 x1 x2   y3  . y4 x4 x3 −x2 x1 y4 Note q that the matrices in both these transformations have exactly two eigenvalues x1 ± i x22 + x32 + x42 with multiplicity 2.

Further, consider the following action of the group F on S4n−1 :   f (z1 , z2 , . . . , zn ) = ( f z1 , f z2 , . . . , f zn ).

(4.18)

It is easy to see that this action is isometric and we obtain a subgroup F1 of the full isometry group of the sphere S4n−1 , which is isomorphic to the group of unit quaternions F. Let us now consider the following action of the group F on S4n−1 :   f (z1 , z2 , . . . , zn ) = (z1 f , z2 f , . . . , zn f ). (4.19) This action is also isometric and we obtain another subgroup F2 of the full isometry group of the sphere S4n−1 , which is isomorphic to the group F of unit quaternions. Since multiplications by quaternions from the left always commute with multiplications by quaternions from the right, the group F1 commutes with the group F2 . Now, let us verify that the groups F1 and F2 consist of Clifford–Wolf translations. Every element of the group F1 (F2 ) generates an orthogonal transformation e = diag(D, D, . . . , D), where D is the matrix of of the space R4n with the matrix D e has the the transformationq(4.16) (respectively, (4.17)). It is clear that the matrix D

eigenvalues f1 ± i f22 + f32 + f42 , each of multiplicity 2n (we perform a multiplication by the quaternion f = f1 + i f2 + j f3 + k f4 ). According to Lemma 4.8.19, this transformation is a Clifford–Wolf translation on the sphere S4n−1 . Therefore, the groups F1 and F2 consist of Clifford–Wolf translations.

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Note that the multiplication by the quaternion f = ±1 from the left coincides with the multiplication by this quaternion from the right. It is not difficult to verify that the intersection of the subgroups F1 and F2 is exactly the group {±I} = Z2 . Indeed, this intersection is a central subgroup in each of the groups Fi . But the center of the group of unit quaternions is isomorphic to Z2 . The lemma is proved. t u Using the above lemma, it is easy to get the following Proposition 4.8.23. There is a quasiregular Killing vector field of constant length on every homogeneous Riemannian manifold Γ \S4n−1 , n ≥ 1, where Γ is a discrete group of Clifford–Wolf translations on S4n−1 , which is isomorphic either to a binary dihedral group or to a binary polyhedral group. Proof. Let us consider the subgroups of Clifford–Wolf translations F1 and F2 from Lemma 4.8.22 in the full isometry group of the sphere S4n−1 . Let us choose any subgroup S1 of F1 and a finite subgroup Γ of F2 , which is isomorphic to one of binary dihedral or binary polyhedral groups. Consider now the construction of the fibre bundle (4.13), where M = S4n−1 , and S1 ⊂ F1 is chosen as above. In this case we obtain a Hopf fibre bundle, where N = S1 \S4n−1 = CP2n−1 is a complex projective space of real dimension 4n−2. It is clear that the group Γ1 = S1 ∩ Γ is isomorphic to {±I}, and the group Γe = Γ /Γ1 is either a dihedral or polyhedral group. In any case, the order of the group Γe is at least 3. Factorizing by the action of S1 , we obtain that the group Γe acts isometrically on the complex projective space N = CP2n−1 . According to Theorem 9.3.1 in [483], there are only two finite groups, the group Z2 and the trivial one, which act freely by isometries on a complex projective space. Since the group Γe is not such a group, then its action on N is not free. Therefore, the quotient space Γ˜ \N is not a manifold. According to Proposition 4.8.11, there is a quasiregular Killing field of constant length on the space Γ \S4n−1 . t u Proof (of Theorem 4.8.17). By Corollary 4.8.6, there is no quasiregular Killing field of constant length on Euclidean spheres or on real projective spaces. Let M = Γ \Sm be a homogeneous Riemannian manifold, obtained by the factorizing of Sm by a finite group of Clifford–Wolf translations Γ , which differs from the groups {I} and {±I}. Then, according to Theorem 4.8.18, either m is odd, and Γ is isomorphic to Zk for some k ≥ 3, or m + 1 ≡ 0 (mod 4), and Γ is isomorphic to a binary dihedral group or to a binary polyhedral group. In the first case, the existence of a quasiregular Killing field of constant length follows from Proposition 4.8.20, in the second case one gets the same from Proposition 4.8.23. t u

4.9 Infinitesimal Structure of a Homogeneous Riemannian Space Let G be a (possibly, virtual) Lie subgroup of the group Isom(M, g), that is transitive on a given homogeneous Riemannian manifold (M, g), and let H = Gx be its

4.9 Infinitesimal Structure of a Homogeneous Riemannian Space

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isotropy subgroup at the point x ∈ M. As already noted, M is diffeomorphic to the homogeneous space G/H, where G and H are equipped with the topology induced from Isom(M, g). In this section, we describe the infinitesimal structure of the homogeneous Riemannian space (G/H, g). We denote by g and h the Lie algebra LG of the group G and the Lie algebra LH of the group H respectively (h ⊂ g by the inclusion H ⊂ G). For a given X ∈ g, we consider exp(tX), t ∈ R, the 1-parameter subgroup of the group G, generated by the vector X. The action of exp(tX) on M allows us to consider it as a 1-parameter group ϕ(t) = exp(tX) of diffeomorphisms of the manifold M, where ϕ(t)(y) = exp(tX)y. Let us identify X ∈ g with the vector field on M, generated by ϕ(t). Then g is identified with the set of Killing vector fields on (M, g) that are generated by 1-parameter subgroups in G. The Lie subalgebra h ⊂ g of the subgroup H is identified with the subalgebra of Killing vector fields in g that vanish at the point x ∈ M. First, we prove the following important property of homogeneous Riemannian spaces. Proposition 4.9.1. Let (M = G/H, g) be a homogeneous Riemannian space. Then the action of the isotropy subgroup Ad(H) = AdG (H) on the Lie algebra g is completely reducible (cf. Definition 2.1.36). Proof. Put G∗ = Isom(M, g) and let H ∗ be the isotropy subgroup of the point o = eH ∈ M in the group G∗ . Then H ∗ is compact by Theorem 4.1.1. Therefore, the group AdG∗ (H ∗ ) is compact in GL(g∗ ) (g∗ = LG∗ ), whereas its subgroup AdG∗ (H) is precompact in GL(g∗ ). Since g ⊂ g∗ and G ⊂ G∗ , Ad(H) = AdG (H) is also precompact in GL(g), i.e. its closure Ad(H) in GL(g) is compact. By averaging of an arbitrary inner product Q0 on g with respect to the action of the compact group Ad(H), we get an Ad(H)-invariant inner product Q on g. Now, if V is an Ad(H)-invariant submodule in g, then the Q-orthogonal complement V ⊥ to V in g is a complementary Ad(H)-invariant submodule for V in g. Indeed, it suffices to check that V ⊥ is Ad(H)-invariant. Take any a ∈ H, then for any X ∈ V and any Y ∈ V ⊥ we get 0 = Q(Ad(a−1 )X,Y ) = Q(X, Ad(a)Y ), as required. Therefore, we get the proposition. t u Proposition 4.9.2. Every G-homogeneous Riemannian space (M = G/H, g) is reductive. Moreover, the Lie group G admits a left-invariant and H-right-invariant Riemannian metric g0 such that the natural projection π : (G, g0 ) → (G/H, g) is a Riemannian submersion with totally geodesic fibers. Proof. Obviously, the Lie algebra h is an Ad(H)-invariant subspace in g. Hence, according to Proposition 4.9.1, there is an Ad(H)-invariant complement p to h in g, i.e. the homogeneous space G/H is reductive. Further, we identify p with the tangent space Mx , associating the element X ∈ p with the vector T π(X), where π : G → G/H = M is the canonical projection. Then the isotropy representation χ of the group H in Mx is identified with the restriction of the adjoint representation AdG of the subgroup H on p. Now, let (·, ·) = gx be

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an Ad(H)-invariant inner product on the space p. It can be extended to an Ad(H)invariant inner product on the Lie algebra g, assuming (p, h) = 0 and (·, ·)|h = Q|h (see the proof of Proposition 4.9.1). This inner product generates a left-invariant and H-right-invariant Riemannian metric g0 on the Lie group G. It is not difficult to see that the natural projection π : (G, g0 ) → (G/H, g) is really a Riemannian submersion with totally geodesic fibers in this case, since any 1-parameter subgroup of H is a geodesic in (G, g0 ). t u Using the symbol p for the Ad(H)-invariant complement to h in g (which is identified with the tangent space Mx as in the proof of the previous proposition), we will denote by [·, ·]p and [·, ·]h respectively the p-component and the h-component of the Lie bracket in the Lie algebra g. It should be noted that if an inner product (·, ·) : p × p → R is Ad(H)-invariant, then it is also ad(h)-invariant, which means that the equality ([Z, X],Y ) + (X, [Z,Y ]) = 0 is satisfied for all Z ∈ h and all X,Y ∈ p. The latter condition can be interpreted as the skew-symmetry of the operators ad(Z) : p → p with respect to (·, ·). We also note that for a connected group H, the property of being Ad(H)-invariant is equivalent to the property of being ad(h)-invariant, and the property of being Ad(H)-irreducible is equivalent to the property of being ad(h)-irreducible for all modules in g.

4.10 Special Invariant Metrics Now we describe some special classes of invariant metrics that have several remarkable properties. We deal with a homogeneous Riemannian space (G/H, g) with a fixed reductive decomposition g = h ⊕ p, where g is determined by the inner product (·, ·) on p. It will be convenient for us to use the mapping U : p × p → p, defined by the equality 2(U(X,Y ), Z) = ([Z, X]p ,Y ) + (X, [Z,Y ]p )

(4.20)

for all Z ∈ p. Definition 4.10.1. Let (M = G/H, g) be a homogeneous Riemannian space and let p be an Ad(H)-invariant complement to h in g. The metric g is called naturally reductive with respect to p, if U ≡ 0. In this case, (M = G/H, g) is called a (G-)naturally reductive space with respect to a given p. For a naturally reductive space, we can compute explicitly the Levi-Civita connection ∇: 1 e e 1 (∇XeYe )x = [X, Y ](x) = − [X,Y ]p , 2 2 e is the Killing field on (M = G/H, g) induced by W ∈ p. In fact, this identity where W follows directly from (4.20) and the formula (4.32) below. We see that the Levi-

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Civita connection ∇ coincides with the canonical connection of the first kind for G/H in the sense of K. Nomizu, see Section 4.5. We note that naturally reductive metrics appear very often in various geometrical topics, for instance, the majority of examples of invariant Einstein metrics on compact homogeneous spaces are naturally reductive. A convenient characterization of naturally reductive metrics was obtained by B. Kostant in [293] (see also [169]). Theorem 4.10.2 (B. Kostant, [293]). An invariant Riemannian metric g on a homogeneous space M = G/H is naturally reductive with respect to p if and only if there is an Ad(G1 )-invariant non-degenerate quadratic form Q on the ideal g1 := p + [p, p] (see Proposition 2.5.15) of the Lie algebra g, where G1 is a (transitive on M) subgroup of the group G corresponding to the subalgebra g1 , such that the Riemannian metric g is generated by the restriction of the form Q to p. Note that it is possible to assume in advance that G1 = G to study the majority of geometric problems. The classification of naturally reductive homogeneous spaces of dim ≤ 5 was obtained by O. Kowalski and L. Vanhecke in 1985 [303]. A new approach to study naturally reductive homogeneous spaces was suggested by I. Agricola, A. C. Ferreira, and T. Friedrich in the paper [10]. This approach made it possible to get the classification of naturally reductive homogeneous spaces of dim ≤ 6 [10]. The classification in dimensions seven and eight was recently completed in [424]. Recent interesting results on naturally reductive homogeneous spaces can be found in [9, 423] and in the references therein. Note that the description of all left-invariant Riemannian metrics on simple compact Lie groups, that are naturally reductive (possibly, with respect to more extensive isometry groups) is obtained by J. E. D’Atri and W. Ziller in [169], see also Theorem 5.12.15 in Subsection 5.12.5. We give two additional definitions of more narrow classes of invariant metrics. Definition 4.10.3. A G-homogeneous Riemannian manifold (M, g) is called normal if there is an Ad(G)-invariant inner product Q on the Lie algebra g such that the metric g is generated by the restriction of Q to p, where p is the Q-orthogonal complement to h in g. It is easy to see that normal homogeneous spaces are naturally reductive. At the same time, it is important to note that a naturally reductive metric is not necessarily normal homogeneous. The only exceptions are compact spaces of positive Euler characteristic (see Theorem 4.12.13). A particular case of normal homogeneous Riemannian manifolds are Killing or, in other terminology, standard homogeneous Riemannian manifolds. The corresponding homogeneous Riemannian metric is also called standard. Definition 4.10.4. A normal G-homogeneous Riemannian manifold (M, g) is called standard or Killing if in Definition 4.10.3 the group G is semisimple, and the Ad(G)invariant inner product Q is the minus Killing form on the Lie algebra LG (this implies the compactness of the group G by Theorem 2.5.11).

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The description of the full isometry groups of naturally reductive (in particular, normal homogeneous) Riemannian spaces is obtained in [372, 373, 376, 396]. It is naturally connected with the description of affine transformations with respect to the canonical connection for naturally reductive Riemannian spaces. In particular, in [396], the identity component of the group of affine transformations with respect to the canonical connection for a normal homogeneous space is obtained. In [372], C. Olmos and S. Reggiani proved that the canonical connection of a compact naturally reductive space is unique, provided the space does not split off, locally, a sphere or a compact Lie group with a bi-invariant metric. A key point in proving this result was a decomposition theorem which is specific to compact homogeneous spaces and false in the non-compact case. In [373], C. Olmos and S. Reggiani extend the previous result to the non-compact case, proving that the canonical connection of a simply connected irreducible naturally reductive non-compact space is unique, except when the space is the symmetric dual of a compact Lie group with a bi-invariant metric. One of the most interesting consequences is the following Theorem 4.10.5 (C. Olmos–S. Reggiani, [373]). Let M be a simply connected and irreducible naturally reductive space. Assume that M is not (globally) isometric to a sphere, nor to a Lie group with a bi-invariant metric or its symmetric dual. Then, M has a unique naturally reductive presentation. Note also that results of [372, 373] generalized well-known classification results by A. L. Onishchick [376], for normal homogeneous spaces with simple groups of isometries. Definition 4.10.6. A reductive homogeneous space G/H (with an Ad(H)-invariant decomposition g = h ⊕ p) is called isotropy irreducible (strongly isotropy irreducible) if the linear isotropy group AdG (H) (respectively, the subgroup AdG (H0 ) of the linear isotropy group, where H0 is the identity component of the group H) acts irreducibly on p. A G-homogeneous Riemannian manifold (G/H, g) is called isotropy irreducible (strongly isotropy irreducible) if the homogeneous space G/H (which is necessarily reductive by Proposition 4.9.2) has this property. We note that on each isotropy irreducible space, the invariant Riemannian metrics form exactly one 1-parameter family of pairwise proportional metrics. For a compact space G/H, the Lie group G is semisimple and every such metric is proportional to the standard one, in particular, it is normal. Irreducible symmetric spaces are well-known examples of strongly isotropy irreducible Riemannian manifolds. It is not difficult to show (see Proposition 4.76 in [106]) that symmetric examples are exhausted by all noncompact isotropy irreducible spaces. Compact strongly isotropy irreducible spaces are classified independently by O. V. Manturov [330] and J. A. Wolf [480] (with some omissions in their original lists, but the union of these lists gives the complete classification). Later, by another method, the complete classification was obtained by M. Kr¨amer in [309] (see also Chapter 7 in [106]). The classification of all simply connected isotropy irreducible spaces was obtained by M. Wang and W. Ziller [460].

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Note also that there is an extension of the normality notion for Finsler homogeneous spaces, see [488], which mainly considers compact normal homogeneous Finsler manifolds. The latter are very special cases of compact normal homogeneous Finsler manifolds from Theorems 6.12.4 and 6.12.7 which are characterized as δ homogeneous compact manifolds with intrinsic metric. The corresponding continuous Finsler norms on tangent bundles of these manifolds and appropriate Lie groups generally are not smooth and do not satisfy condition (3) in Subsection 2.1 from [488].

4.11 The Structure of the Set of Invariant Metrics In this section, we describe the structure of G-invariant Riemannian metrics on homogeneous spaces G/H with compact groups H (a survey on more general spaces of Riemannian metrics can be found in [417]). Recall that a homogeneous space M = G/H with this property is reductive, i.e. the Lie algebra g of the Lie group G admits an Ad(H)-invariant decomposition (see Section 4.4) g = h ⊕ p,

Ad(H)(p) = p.

Moreover, many results in this section demand only the above decomposition, hence, the compactness of the group H can be weakened in some cases. If o = eH ∈ G/H, then the space p could be naturally identified with the tangent e space Mo to M at the point o. Take any U ∈ p, then we get the vector U(o) ∈ Mo , where e = d exp(tU)(x) |t=0 , x ∈ M, U(x) dt e is the corresponding Killing field on M = G/H. Obviously, the map U → U(o) d is linear. Moreover, dt exp(tU)(x)|t=0 = 0 for all U ∈ h, since exp(tU) ⊂ H, and exp(tU)(x) = x for all t ∈ R. Therefore, we get a linear bijection between p and Mo . If we have two reductive complements p1 and p2 , then vectors U1 ∈ p1 and U2 ∈ p2 determine one and the same tangent vector in Mo if and only if U2 is a projection of U1 along h, i.e. U2 − U1 ∈ h. This observation allows us to identify some geometric values (e.g. inner products) defined for different complements. It is clear that any Ad(H)-invariant inner product (·, ·) on p is carried by the differentials of left translations over the space G/H, generating a G-invariant Riemannian metric g on G/H. Indeed, we assume go = (·, ·). Now, take any x ∈ G/H, then there is an a ∈ G such that aH = x. Consider the left translation la : G/H → G/H, la (yH) = ayH, and its differential at the point o: dla (o) : (G/H)o → (G/H)x , and put gx (dla ·, dla ·) = go . Since the inner product (·, ·) is Ad(H)-invariant, this definition is well-defined and the obtained Riemannian metric is G-invariant. Conversely, every G-invariant Riemannian metric g on the space G/H generates an Ad(H)-invariant

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4 Homogeneous Riemannian Manifolds

inner product (·, ·) = go (·, ·) if we identify (G/H)o with p. Therefore, we get the following Theorem 4.11.1. There is a natural one-to-one correspondence (as it is described above) between the set of G-invariant Riemannian metrics on G/H and the set of Ad(H)-invariant inner products on p. Due to this theorem, we can identify (and we will often do so) the set of Ginvariant Riemannian metrics on a homogeneous space G/H with the set of Ad(H)invariant inner products on some Ad(H)-invariant complement p to h in g. Now, let us study in more detail the structure of an arbitrary Ad(H)-invariant inner product (·, ·) on p. For this goal we use the fact that the representation Ad(H) on p is completely reducible by Proposition 4.9.1 and Proposition 2.1.37. It could also be shown using the fact that the inner product (·, ·) is Ad(H)-invariant. Indeed, if q ⊂ p is an arbitrary Ad(H)-invariant subspace, then its (·, ·)-orthogonal complement in p is also Ad(H)-invariant. Now, consider an arbitrary (·, ·)-orthogonal Ad(H)-invariant decomposition p = p0 ⊕ p1 ⊕ · · · ⊕ pr

(4.21)

such that Ad(H)|p0 = Idp0 and Ad(H)|pi is irreducible for 1 ≤ i ≤ r. This decomposition is not unique in the case when the representations of Ad(H) on some Ad(H)invariant modules pi and p j (1 ≤ i < j ≤ r) are equivalent. On the other hand, the subspace p0 and the set of numbers di = dim(pi ) do not depend on the choice of the decomposition. Definition 4.11.2. Two irreducible Ad(H)-modules (ad(h)-modules) pi and p j are called Ad(H)-isomorphic (respectively, ad(h)-isomorphic), if the corresponding representations on them are equivalent, i.e. there is an Ad(H)-equivariant (respectively, ad(h)-equivariant) isomorphism between pi and p j . There is a way to parameterize all invariant Riemannian metrics by Ad(H)equivariant symmetric operators (endomorphisms) on p. Definition 4.11.3. Let K be a Lie group and k be its Lie algebra. Let us consider any representation Π : K → GL(p) of the Lie group K and the induced representation π : k → gl(p) of the Lie algebra k. Then, a linear operator L : p → p is called Π (K)-equivariant (π(k)-equivariant), if it commutes with any element from Π (K) (respectively, π(k)). Denote by MG the set of all Ad(H)-invariant inner products on p. This set could be characterized in terms of a fixed decomposition. Let us fix h·, ·i, some Ad(H)invariant inner product on p. Now, consider an arbitrary Ad(H)-invariant symmetric bilinear form (·, ·) on p. Then, as it is easy to see, there is an Ad(H)-equivariant h·, ·i-symmetric operator A : p → p such that (·, ·) = hA·, ·i. If the form (·, ·) is positive definite, then the operator A is also positive definite. In this case, the operator A is called a metric endomorphism or metric operator (for

4.11 The Structure of the Set of Invariant Metrics

211

Table 4.1 Transitive actions on spheres and invariant metrics G

SO(n)

U(n)

SU(n)

Sp(n)Sp(1)

Sp(n)U(1)

H

SO(n-1)

U(n-1)

SU(n-1)

Sp(n-1)Sp(1)

Sp(n-1)U(1)

Sphere

Sn−1

S2n−1

S2n−1

S4n−1

S4n−1

dim MG

1

2

2

2

3

G

Sp(n)

G2

Spin(7)

Spin(9)

H

Sp(n-1)

SU(3)

G2

Spin(7)

Sphere

S4n−1

S6

S7

S15

dim MG

7

1

1

2

the inner product (·, ·) with respect to the inner product h·, ·i). Therefore, one can establish a one-to-one correspondence between the set of invariant Riemannian metrics on a homogeneous space G/H and the set of h·, ·i-symmetric positive definite Ad(H)-equivariant operators on p. The latter set is convex in the space of all linear operators, so it (and, hence, the set MG ) admits a C∞ -smooth structure. Moreover, it is not difficult to see that this set is diffeomorphic to the space of positive definite elements in ! Rr ×

∏

HomAd(H) (pi , p j ) × S2 (p0 ),

1≤i< j≤r

where HomAd(H) (pi , p j ) means the set of Ad(H)-equivariant homomorphisms from pi to p j with the natural differential structure and S2 (p0 ) is the space of bilinear forms on the subspace p0 [459]. Therefore, the set MG could be supplied with the structure of a smooth (C∞ -smooth) manifold. Example 4.11.4. In Example 4.2.6, the Montgomery–Samelson–Borel classification of compact connected Lie groups, acting transitively and effectively on spheres, is given. In Table 4.1, we complete this classification with the information on dim MG , the dimension of the space of invariant Riemannian metrics, for each realization. There are some special families of invariant metrics, so-called “diagonal” families, that are the most convenient for various simultaneous calculations. Let us consider the decomposition (4.21) for the inner product h·, ·i. We represent the subspace p0 as a direct sum of pairwise h·, ·i-orthogonal onedimensional subspaces p0 = pr+1 ⊕ · · · ⊕ ps . Now, consider the following family of Ad(H)-invariant Riemannian metrics: (·, ·) = x1 h·, ·i|p1 ⊕ · · · ⊕ xr h·, ·i|pr ⊕ xr+1 h·, ·i|pr+1 ⊕ · · · ⊕ xs h·, ·i|ps , where xi > 0 for i = 1, . . . , s. It is easy to calculate simultaneously curvatures of the metrics from such a “diagonal” family, see e.g. [459].

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4 Homogeneous Riemannian Manifolds

Remark 4.11.5. Note that any automorphism τ of the Lie group G, preserving the isotropy group H of a homogeneous space G/H, induces a mapping τe : MG → MG , moving every G-invariant metric ρ from MG to an isometric invariant metric. In the remainder of this section we will consider only homogeneous spaces G/H with a compact Lie algebra g (for a connected Lie group G this means that g admits an Ad(G)-invariant inner product h·, ·i, which is equivalent to the fact that the group G admits a Riemannian bi-invariant metric). Remark 4.11.6. Note that for a connected compact Lie group G, the group Ad(G) is isomorphic to the group of inner automorphisms Int(g) of the Lie algebra g (see Proposition 2.1.25 and Section 2.7). Remark 4.11.7. If a connected Lie group G is compact and semisimple, then one can take the minus Killing form of the Lie algebra g as an Ad(G)-invariant inner product on g (for a simple g, every such inner product is a multiple of the Killing form). For a commutative Lie algebra g, any inner product is Ad(G)-invariant. Let us fix an Ad(G)-invariant inner product h·, ·i on the Lie algebra g, that is generated by some bi-invariant Riemannian metric on the Lie group G. Further, let us consider the h·, ·i-orthogonal decomposition g = h ⊕ p. As already noted, the set of invariant metrics MG on a homogeneous space G/H can be identified with the set of positive h·, ·i-symmetric Ad(H)-equivariant endomorphisms (i.e. metric endomorphisms) of the space p: an inner product (·, ·) on p corresponds to the operator A : p → p according to the formula (·, ·) = hA·, ·i. For a homogeneous space G/H, it is natural to consider a special action of the group NG (H)/H on MG , where NG (H) is the normalizer of H in the group G. The normalizer NG (H) acts on the homogeneous space G/H by G-equivariant diffeomorphisms Rn , where n ∈ NG (H) (see (4.3)): Rn (bH) = bnH = bHn,

b ∈ G.

Since NG (H) normalizes H, we can consider the restriction of the adjoint action NG (H) on p. Therefore, we get the action of NG (H) on MG , determined for every n ∈ NG (H) as follows: e = Ad(n)|p ◦ A ◦ Ad(n)−1 |p , (A, n) → A

(4.22)

where A is in S2 (p)Ad(H) , the set of positive h·, ·i-symmetric Ad(H)-equivariant endomorphisms of p. Since the inner product h·, ·i is Ad(G)-invariant, the equality Ad(n)−1 |p = e ∈ S2 (p)Ad(H) . Obviously, invariant metrics that correspond Ad(n)t |p holds, and A e are isometric to each other. Therefore, in the study to the endomorphisms A and A of invariant metrics on the homogeneous space G/H, it suffices to study only one of these two metrics. As a rule, it is convenient to choose the metric generated by the e among all operators of simplest (in a certain informal but natural sense) operator A −1 the form Ad(n)|p ◦ A ◦ Ad(n) |p , where n ∈ NG (H).

4.12 Compact Homogeneous Riemannian Manifolds of Positive Euler Characteristic

213

Since we are only interested in Ad(H)-equivariant metric endomorphisms, we can reduce the action of the group NG (H) on MG (4.22) to the action of the gauge group NG (H)/H.

4.12 Compact Homogeneous Riemannian Manifolds of Positive Euler Characteristic Compact homogeneous Riemannian manifolds of positive Euler characteristic play an important role in this book. First, we give some structural results on homogeneous spaces of positive Euler characteristic (cf. Section 19.5 in [377]). Unless otherwise stated, everywhere in this section we assume that G/H is an almost effective compact homogeneous space of a connected Lie group G. Theorem 4.12.1. Let M = (G/H, µ) be a compact simply connected homogeneous Riemannian manifold with compact Lie groups G and H, and suppose G is connected. Then the following conditions are equivalent: 1) χ(M) = 0; 2) rk G > rk H; 3) There is a right-invariant vector field on G that projects under the canonical mapping p : G → M on a Killing vector field without zeros; 4) All characteristic numbers of the Riemannian manifold M, determined for the principal bundle π : SO(M) → M of orthonormal oriented bases on M, are zero. Proof. Since G is connected and G/H is simply connected, the homotopy sequence of the fibration p : G → G/H implies that the group H is connected. Hence, all conditions of Theorem 4.3.1 are fulfilled. Hence, Conditions 1) and 2) above are equivalent by Theorem 4.3.1. Theorem 3.6.2 and Lemma 4.3.2 imply that 3) ⇒ 1). Let us show that 2) ⇒ 3). Let U be the element from the first part of the proof of Theorem 4.3.1. Then, it is clear that a right-invariant vector field W on G satisfying the condition W (e) = U projects under the mapping p to a Killing vector field on M without zeros. The characteristic numbers from Condition 4) are defined only for even-dimensional Riemannian manifolds. In this case, the Euler characteristic is also the characteristic number (corresponding to the characteristic Euler class) by the Gauss– Bonnet theorem. Then in this case Condition 1) follows from Condition 4). Condition 4) follows from Condition 3) (even from the more weak condition of the existence of a Killing vector field without zeros on an arbitrary compact smooth oriented Riemannian manifold of even dimension) by the Bott theorem [119] (the proof is also given in [290, Theorem 6.1 of Chapter 2]). In the odd-dimensional case, χ(M) = 0 and Condition 1) is fulfilled, which implies Conditions 2) and 3), as was said earlier. If we assume that the characteristic numbers of the odd-dimensional (compact Riemannian) manifold are equal to zero

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4 Homogeneous Riemannian Manifolds

by definition, then Condition 4) is automatically satisfied. Therefore in this case all 4 conditions are equivalent and always hold. t u Proposition 4.12.2 ([457]). If a compact connected Lie group G acts effectively on the space M = G/H of positive Euler characteristic, then the center of G is trivial; the space M = G/H is simply connected if and only if H is connected. Proof. The Lie algebra g of the compact Lie group G is compact. According to Proposition 2.5.6 and Theorem 2.5.5, g is represented as a direct sum of ideals g = z ⊕ l, where z is the center of the Lie algebra g and l is a semisimple Lie algebra. Let us consider an arbitrary G-invariant Riemannian metric µ on the manifold M = G/H. If the Lie algebra g is not semisimple, then every non-trivial vector from its center z generates a Killing vector field without zeros on (M, µ) (see the proof of Theorem 4.12.1). Then, M has zero Euler characteristic in this case due to Theorem 4.12.1. Therefore, the Lie group G (as well as its Lie algebra g) is semisimple (this assertion remains in force even for an almost effective action of the group G). According to Theorem 4.3.1, there is a maximal torus T ⊂ G such that T ⊂ H. The center Z(G) of the group G is trivial, since, according to the first two assertions of Theorem 2.5.23, Z(G) is a subset of every maximal torus of the group G (in particular, Z(G) ⊂ T ⊂ H), but the action of G is effective. The second assertion follows from the first one and from the homotopy sequence of the fibration G → G/H. Indeed, the necessity of the second assertion is obtained immediately even without the requirement on the positivity of the Euler characteristic of the space G/H. To prove the sufficiency, we must replace the group G in the e and H by its preimage under the homotopy sequence by its universal covering G, e universal covering G → G; the latter group is connected because of the positivity of the Euler characteristics of the space G/H and analogues of the arguments from the previous paragraph. t u Theorem 4.12.3 ([294]). Let (G/H, µ) be a connected and simply connected compact almost effective homogeneous Riemannian space of positive Euler characteristic. Then (G/H, µ) is indecomposable if and only if the group G is simple. In particular, two Lie groups, one simple and one non-simple, could not act simultaneously transitively and effectively on a compact Riemannian manifold M of positive Euler characteristic. Now we consider the case of transitive actions of simple Lie groups. Theorem 4.12.4 ([413, 414, 377]). Let G and G0 be connected simple compact Lie groups, H ⊂ G and H 0 ⊂ G0 be its connected subgroups of maximal rank, and G and G0 act (naturally) on M = G/H and on M 0 = G0 /H 0 almost effectively. If the spaces M = G/H and M 0 = G0 /H 0 are diffeomorphic to each other, then either the pairs (G, H) and (G0 , H 0 ) are locally isomorphic, or they are isomorphic (up to permutation) to some pairs from the following list:

4.12 Compact Homogeneous Riemannian Manifolds of Positive Euler Characteristic

215

G = SU(2n) (n ≥ 2), H = S(U(1) ×U(2n − 1)); G0 = Sp(n), H 0 = U(1) · Sp(n − 1); M = M 0 = CP2n−1 . G = SO(7), H = SO(6); G0 = G2 , H 0 = SU(3); M = M 0 = S6 . G = SO(7), H = SO(5) × SO(2); G0 = G2 , H 0 = SU(2) · SO(2); + M = M 0 = Gr7,2 . G = SO(2n) (n ≥ 4), H = U(n); G0 = SO(2n − 1), H 0 = U(n − 1); C . M = M 0 = I 0 Gr 2n,n Theorem 4.12.4 easily implies the classification of transitive actions of compact connected Lie groups on simply connected homogeneous spaces of positive Euler characteristic. Theorem 4.12.5 ([414]). Let M and M 0 be homogeneous spaces of connected compact Lie groups, χ(M) > 0, χ(M 0 ) > 0, and suppose M is homotopically equivalent to M 0 , then M is diffeomorphic to M 0 . The results of [376] and [413] imply Theorem 4.12.6. Let (G/H, µ) be a simply connected homogeneous Riemannian manifold of positive Euler characteristic, and G be a simple connected Lie group. Then the full connected isometry group of (G/H, µ) is G/C (C is the center of G), excepting the cases when (G/H, µ) is one of the following manifolds: 1) G/H = Sp(n)/U(1) · Sp(n − 1) (n ≥ 2), µ is the symmetric (Fubini) metric on CP2n−1 = SU(2n)/S(U(1) ×U(2n − 1)); 2) G/H = SO(2n−1)/U(n−1) (n ≥ 4), µ is the symmetric metric on SO(2n)/U(n) (the space of orthogonal complex structures on R2n ); + 3) G/H = G2 /SU(2) · SO(2), µ is the symmetric metric on Gr7,2 = SO(7)/SO(5) × SO(2); 4) G/H = G2 /SU(3) (strongly isotropy irreducible), µ is an arbitrary G-invariant metric. In the first three cases the metric µ is not G-normal, in the last case µ is a metric of constant curvature on S6 = SO(7)/SO(6). Proof. Using Proposition 4.12.2 and Theorem 4.12.4, we easily get the main assertions. We need only to show that in Cases 1), 2), and 3) the metric µ is not G-normal. This follows from results of [376] (see also [284]). Indeed, in that paper the author proved that the full connected isometry group of a simply connected G-normal homogeneous space M = G/H of a connected simple compact Lie group G is G · AutG (M)0 (a locally direct product), where AutG (M) = { f ∈ Diff(M) | f (gx) = g f (x), g ∈ G, x ∈ M}, excepting the following cases: G2 /SU(3) = S6 , Spin(7)/G2 = S7 , Spin(8)/G2 = S7 × S7 . Only one of these spaces (namely, G2 /SU(3) = S6 ) has positive Euler characteristic. Moreover, it is strongly isotropy irreducible. We need to note also that

216

4 Homogeneous Riemannian Manifolds

AutG (M)0 is trivial for spaces M = G/H of positive Euler characteristic (this easily follows from Theorem 4.12.3). t u Proposition 4.12.7 ([455]). Every even-dimensional homogeneous Riemannian manifold M of positive sectional curvature has positive Euler characteristic. Proof. According to the Myers theorem 1.5.79, M is compact. According to the Berger theorem (Theorem 3.5.10), any Killing vector field on an even-dimensional homogeneous Riemannian manifold of positive sectional curvature vanishes at some point. If M = G/H has zero Euler characteristic, then according to Theorem 4.12.1, M admits a Killing vector field without zeros. Therefore, χ(M) > 0 due to the Hopf– Samelson theorem (Theorem 4.3.1). t u Remark 4.12.8. An example of a flat even-dimensional torus, having zero Euler characteristic, shows that the statement of Proposition 4.12.7 is not true under the condition that sectional curvature is nonnegative. Note also that any compact odddimensional triangulated (in particular, smooth) manifold has zero Euler characteristic due to the Poincar´e duality. Corollary 4.12.9. All compact symmetric spaces of rank one, excepting odd-dimensional (i.e. S2k+1 and RP2k+1 ), have positive Euler characteristic. A. Borel and J. de Siebenthal obtained in [117] the classification of subgroups with maximal rank of compact Lie groups (see also Section 8.10 in [483] and [297]). This classification gives us a description of compact homogeneous spaces with positive Euler characteristic. The complete description of homogeneous spaces of classical Lie groups with positive Euler characteristic has been obtained by H. C. Wang in [457]. There is one very important subclass of compact homogeneous manifolds of positive Euler characteristic, namely, the (generalized) flag manifolds. They can be described as the orbits M of compact connected Lie groups G under the adjoint representation. In other words, M = G/H, where H = ZG (S) is the centralizer of a non-trivial torus S ⊂ G; the Lie group H is always connected. In this case, orbits of regular elements in the Lie algebra g of the Lie group G are called (full) flag manifolds, for details, see e.g. [25]. Chapter 8 in [106] contains the following results: simply connected compact homogeneous K¨ahler manifolds are exactly (generalized) flag manifolds. Any such manifold (admitting a canonical K¨ahler–Einstein structure, unique in a sense) is a rational complex algebraic (hence, complex projective) manifold. In the special case G = Sp(l), the stabilizer subgroups, whose centers are 1-dimensional, are Sp(l) subgroups U(l − m) × Sp(m). Among the corresponding orbits Ml−m , the only Sp(l)

ones for which the normal metric is K¨ahler (hence, K¨ahler-symmetric) are M1 , Sp(l) that is CP2l−1 = Sp(l)/U(1) × Sp(l − 1), and Ml , isomorphic to Sp(l)/U(l), which is the manifold of totally isotropic complex l-subspaces of C2l . The space SO(2l+1) Ml = SO(2l + 1)/U(l) is the manifold of complex flags of type l.

4.12 Compact Homogeneous Riemannian Manifolds of Positive Euler Characteristic

217

Using Chapter 15 in [377], we can say more. Any (compact generalized) flag manifold M, supplied with the above-mentioned canonical K¨ahler–Einstein structure, is isomorphic to G/H, where G is a complex connected Lie group and H is a closed complex parabolic Lie subgroup in G. We recall that a connected complex Lie subgroup of G is called parabolic if it contains a Borel subgroup of G. A Borel subgroup in G is any its maximal connected solvable complex Lie subgroup. Thus M is a so-called flag homogeneous space. In this case, the corresponding complex structure on M is induced by complex structure on G. Any parabolic subgroup of G contains Rad(G), a normal subgroup in G. Hence M is a flag homogeneous space of the semisimple complex Lie group G0 := G/ Rad(G). In this case M = G0 /H0 , where G0 is any compact real form of G0 and H0 = G0 ∩ H0 for H0 = H/ Rad(G). It is proved in [339, Corollary 7.12 on p. 301] that a maximal connected Lie subgroup H of maximal rank in a compact connected Lie group G is a connected component of the normalizer (= of the centralizer) of some element g ∈ G. From this corollary and related results, the list of all maximal connected compact subgroups H of maximal rank (more precisely, with corresponding Lie subalgebras of maximal rank) in a compact connected simple Lie groups G is given [339, Table 5.1]. In particular, G/H is an orbit of the above-mentioned element g ∈ G with respect to the action of the group I(G) of all inner automorphisms of the Lie group G. (Generalized) flag manifolds can also be considered as such orbits, when g ∈ G is taken in a diffeomorphic image expG (U), where U is an open ball with the center 0 ∈ g with respect to an Ad(G)-invariant Euclidean metric on g. Proposition 4.12.10. Let h be a Lie subalgebra of the Lie algebra g of a connected Lie group G such that Ng (h) = h, where Ng (h) is the normalizer of the Lie algebra h in g. Then h is the Lie algebra of a unique connected Lie subgroup H in G. Proof. Consider H1 = {g ∈ G : Ad(g)(h) ⊂ h}. It is clear that H1 is a closed subgroup of the group G. Hence, its identity component H is also closed. By the Cartan theorem 2.1.32, H is a Lie subgroup of the Lie group G. Obviously, the Lie algebra of H is Ng (h), which coincides with h by the assumptions. Hence, H is the required Lie subgroup. t u It is easy to prove the following Proposition 4.12.11. If h is a reductive Lie subalgebra (i.e. its radical coincides with its center) of the Lie algebra g, that contains a maximal commutative Lie subalgebra t of g, then Ng (h) = h. Propositions 4.12.2, 4.12.10, and 4.12.11 easily imply Theorem 4.12.12. Let G be a simple compact connected Lie group and let t be the Lie algebra of a maximal torus T ⊂ G. Then every proper Lie subalgebra h ⊂ g with the condition t ⊂ h is the Lie algebra of a unique connected Lie subgroup H ⊂ G. Moreover, G/H is a connected and simply connected compact homogeneous space of positive Euler characteristic.

218

4 Homogeneous Riemannian Manifolds

Now, we consider a simple description of naturally reductive homogeneous manifolds of positive Euler characteristic, which follows from Theorem 4.12.3. Theorem 4.12.13 ([79]). Let M be a compact naturally reductive homogeneous Riemannian manifold of positive Euler characteristic. Then M is G1 -normal homogeneous for some (transitive on M) semisimple Lie subgroup G1 ⊂ G, where G is the full connected isometry group of M. Proof. The group G is semisimple, since χ(M) > 0 (see Proposition 4.12.2). Without loss of generality, we can assume that M is simply connected. Indeed, the e of M has a semisimple transitive group of motions universal Riemannian covering M e e have one and the same Lie algebra, G e G, which is a covering of G. Since G and G e e is compact, therefore, M is compact too. If M is normal homogeneous with respect e then M is G1 -normal homogeneous, where e1 ⊂ G, to some semisimple subgroup G e e → G. G1 ⊂ G is the image of G1 under the natural covering epimorphism π : G Moreover, we can assume in addition that M is indecomposable. Indeed, if M = M1 × · · · × Ms is the de Rham decomposition of M then every Mi is a naturally reductive homogeneous manifold ([293, Corollary 7]; see also [291, Chapter X, Theorem 5.2]). If we prove that every Mi is normal homogeneous (with respect to some transitive subgroup of its full connected isometry group), then M is normal homogeneous too. Let M be a compact simply connected indecomposable naturally reductive homogeneous manifold with χ(M) > 0, and G is its (semisimple) full connected isometry group. From the Kostant theorem (Theorem 4.10.2) we get that there is a subgroup G1 ⊂ G, transitive on M, with the following property: there is an Ad(G1 )-invariant non-degenerate quadratic form Q on the Lie algebra g1 of the group G1 such that the Riemannian metric of M is generated by the restriction of Q to the Q-orthogonal complement p of h1 in g1 (H1 is the stabilizer at some point of M with respect to the action of G1 , and h1 is the corresponding Lie subalgebra of the Lie algebra g1 ). Note that the group G1 is simple according to Theorem 4.12.3. This implies that Q is a multiple of the Killing form of the Lie algebra g1 . Therefore, Q is positive definite on g1 , and M is G1 -normal. The theorem is proved. t u We conclude this section with an example of an important homogeneous space of positive Euler characteristic with a two-parameter family of invariant metrics. Example 4.12.14. Let us consider the homogeneous space SO(5)/U(2), where U(2) ⊂ SO(4) ⊂ SO(5), and the pairs (SO(5), SO(4)), (SO(4),U(2)) are irreducible symmetric. Let us provide more details. Consider the following Ad(SO(5))invariant inner product 1 hA, Bi = − trace(A · B) 2 √ on A=  the Lie  algebraso(5).For matrices of the type A + −1B ∈ u(2),where  √ 0c ad AB and B = , we consider the embedding A + −1B 7→ in −c 0 d b −B A so(4) that corresponds to the symmetric pair (so(4), u(2)) (see e.g. [106]). Let us use also the standard embedding of so(4) in so(5): A 7→ diag(A, 0).

4.13 Some Special Compact Homogeneous Spaces

219

Consider the following h·, ·i-orthogonal decomposition: g = so(5) = so(4) ⊕ p1 = u(2) ⊕ p2 ⊕ p1 , where

p = p1 ⊕ p2 ,

   0 c ad0          −c 0 d b 0       u(2) =  −a −d 0 c 0  ; a, b, c, d ∈ R ,      −d −b −c 0 0        0 0 000     0 0 0 0 k         0 0 0 0 l        p1 = X =  0 0 0 0 m  ; k, l, m, n ∈ R ,      0 0 0 0 n       −k −l −m −n 0     0 e 0 f 0         −e 0 − f 0 0        ; e, f ∈ R . 0 f 0 −e 0 p2 = Y =        −f 0 e 0 0       00 0 00

It is clear that the modules p1 and p2 are Ad(U(2))-invariant and Ad(U(2))irreducible. Note that we have the equality hX, Xi = k2 + l 2 + m2 + n2 for all vectors X in p1 and the equality hY,Y i = 2e2 + 2 f 2 for vectors Y ∈ p2 . An arbitrary SO(5)-invariant Riemannian metric on SO(5)/U(2) is a metric g = gx1 ,x2 , generated by the inner product (·, ·) = x1 h·, ·i|p1 + x2 h·, ·i|p2 on p for some positive numbers x1 and x2 . It is easy to see that the homogeneous space SO(5)/U(2) coincides with the homogeneous space Sp(2)/U(1) · Sp(1) and initiates two series of important homogeneous spaces SO(2n + 1)/U(n) and Sp(n)/U(1) · Sp(n − 1), n ≥ 2.

4.13 Some Special Compact Homogeneous Spaces In this section we consider some important classes of compact homogeneous spaces. Along with symmetric, naturally reductive and normal homogeneous spaces, they are convenient examples of homogeneous spaces, on which it is possible to work out various methods of research and check the current working hypotheses. In addition, these spaces give many examples of remarkable properties of homogeneous Riemannian spaces.

220

4 Homogeneous Riemannian Manifolds

4.13.1 Aloff–Wallach Spaces Let us consider the Aloff–Wallach spaces Wk,l = SU(3)/SO(2), which are defined by an embedding of the circle SO(2) = S1 into SU(3) of the type   ik,l : e2πiθ 7→ diag e2πikθ , e2πilθ , e2πimθ , √ where i = −1, k, l, m are integers with greatest common divisor 1 and k + l + m = 0. By using the Weyl group of SU(3) one can assume that k ≥ l ≥ 0 and k > 0. These spaces were investigated by S. Aloff and N. R. Wallach in [21]. Now we examine the sets of invariant metrics on Aloff–Wallach spaces. Let us fix the Ad(SU(3))-invariant inner product hX,Y i = − 12 ℜ trace(XY ) on the Lie algebra su(3). Note that Bsu(3) (·, ·) = −12h·, ·i for the Killing form Bsu(3) of su(3). Let t be the Lie algebra of the standard maximal torus n  o T = e2πia , e2πib , e2πic | a, b, c ∈ R, a + b + c = 0 in SU(3) and let h = hk,l be the Lie algebra of the Lie group ik,l (S1 ) = Hk,l = H. It is useful to define the value L := k2 + l 2 + m2 and to check that k2 + l 2 + m2 − kl − km − ml = 3L/2. Now, let us consider the vectors    √ l −m 0 k0 0 0 2i Z = i  0 l 0  , X0 = √  0 m − k 0  , 3L 00m 0 0 k−l 

     0 10 0 i 0 0 01 X1 =  −1 0 0  , X2 =  i 0 0  , X3 =  0 0 0  , 0 00 000 −1 0 0       00 i 0 0 0 000 X4 =  0 0 0  , X5 =  0 0 1  , X6 =  0 0 i  i 00 0 −1 0 0 i 0 in the Lie algebra su(3). Note, that the subalgebra h is defined by the vector Z. Moreover, all the vectors Xi have a unit length with respect to the chosen inner product, are mutually h·, ·i-orthogonal and are orthogonal to the subalgebra h. Moreover, we have the following h·, ·i-orthogonal decomposition g = t ⊕ p1 ⊕ p2 ⊕ p3 = h ⊕ p4 ⊕ p1 ⊕ p2 ⊕ p3 , where the modules p1 = Lin(X1 , X2 ),

p2 = Lin(X3 , X4 ),

p3 = Lin(X5 , X6 ),

p4 = Lin(X0 )

are Ad(H)-invariant. Direct calculations show that [Z, X0 ] = 0, [Z, X1 ] = (k − l)X2 , [Z, X2 ] = (l − k)X1 , [Z, X3 ] = (k − m)X4 , [Z, X4 ] = (m − k)X3 , [Z, X5 ] = (l − m)X6 ,

4.13 Some Special Compact Homogeneous Spaces

221

[Z, X6 ] = (m − l)X5 . This implies that the modules pi are ad(h)-irreducible, hence, Ad(H)-irreducible (recall that the group H = S1 = exp(tZ), t ∈ R, is connected) for pairwise distinct k, l and m. Let us examine the cases when there are some pairwise isomorphic Ad(H)modules pi . For example, let ϕ : p1 → p2 be any Ad(H)-isomorphism, i.e. a linear map with the condition ϕ([Z, X]) = [Z, ϕ(X)] (recall that H = exp(tZ), t ∈ R) for every X ∈ p1 . There are real numbers a, b, c and d that satisfy the equations ϕ(X1 ) = aX3 + bX4 , ϕ(X2 ) = cX3 + dX4 . Since ϕ([Z, X1 ]) = [Z, ϕ(X1 )], ϕ([Z, X2 ]) = [Z, ϕ(X2 )], the following equalities are satisfied: (k − l)c = (m − k)b, (k − l)d = (k − m)a, (k − l)a = (k − m)d, (l − k)b = (k − m)c. Therefore, a necessary condition for the existence of the required isomorphism is the equation (k − l)2 = (m − k)2 , or equivalently, (l − m)(2k − l − m) = 0. The last relationship implies either l = m, or k = 0 and l = −m. Consequently, |l| = |m|. We have assumed that k ≥ l ≥ 0 and k > 0. Therefore, the equality |l| = |m| is impossible, and p1 is never Ad(H)-isomorphic to p2 . By similar computations one can prove that p1 ' p3 if and only if (k, l, m) = (1, 0, −1) and p2 ' p3 if and only if (k, l, m) = (1, 1, −2). If (k, l) 6= (1, 1) and (k, l) 6= (1, 0), then the irreducible Ad(H)-modules pi are not pairwise isomorphic for the space Wk,l . Hence, an arbitrary inner product on p = p1 ⊕ p2 ⊕ p3 ⊕ p4 for these spaces has the form (·, ·) = x1 h·, ·i|p1 + x2 h·, ·i|p2 + x3 h·, ·i|p3 + x4 h·, ·i|p4 , where x1 , x2 , x3 , and x4 are some positive real numbers. It should also be noted that the set of invariant metrics on W1,0 and W1,1 depends on 6 and 10 parameters respectively. For details, see e.g. [359].

4.13.2 Generalized Wallach Spaces Let us discuss a remarkable class of compact homogeneous spaces known as generalized Wallach spaces (cf. [368, pp. 6346–6347] and [365]). These spaces have been studied by various authors from the point of view of sectional curvature, Einstein metrics, Ricci flow and special geodesics. Consider a homogeneous almost effective compact space G/H with a (compact) semisimple connected Lie group G and its closed subgroup H. Let h·, ·i be the minus Killing form of the Lie algebra g and consider the orthogonal complement p of h in g with respect to h· , ·i. Suppose that G/H is such that the Ad(H)-invariant module p is decomposed as a direct sum of three Ad(H)-invariant irreducible modules pairwise orthogonal with respect to h· , ·i, i.e. p = p1 ⊕ p2 ⊕ p3 , such that [pi , pi ] ⊂ h,

i ∈ {1, 2, 3}.

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4 Homogeneous Riemannian Manifolds

Since this condition on each module resembles the condition of local symmetry for homogeneous spaces a locally symmetric homogeneous space G/H is characterized by the relation [p, p] ⊂ h, where g = h ⊕ p and p is Ad(H)-invariant), spaces with this property were called three-locally-symmetric in (cf. [160]). But we will use the term generalized Wallach spaces, as in [368]. There are many examples of these spaces, e.g. the Wallach spaces, that are the manifolds of complete flags in the complex, quaternionic, and Cayley projective planes (a complete flag in any of these planes is a pair (p, l) where p is a point in the plane and l a line (complex, quaternionic or octonionic) containing the point p): W6 = SU(3)/Tmax ,

W12 = Sp(3)/Sp(1) × Sp(1) × Sp(1),

W24 = F4 /Spin(8).

These spaces are interesting because they admit invariant Riemannian metrics of positive sectional curvature as was proved by N. Wallach in [455]. The Lie
group SU(2) H = {e} is another example of a generalized Wallach space. Other examples of generalized Wallach spaces are some K¨ahler C-spaces (compact simply connected homogeneous K¨ahlerian manifolds) such as
 SU(n1 + n2 + n3 ) S U(n1 ) ×U(n2 ) ×U(n3 ) , SO(2n)/U(1) ×U(n − 1),

E6 /U(1) ×U(1) × Spin(8).

There are two other 3-parameter families of generalized Wallach spaces:  SO(n1 + n2 + n3 ) SO(n1 ) × SO(n2 ) × SO(n3 ),  Sp(n1 + n2 + n3 ) Sp(n1 ) × Sp(n2 ) × Sp(n3 ). Let us consider any generalized Wallach space. The inclusion [pi , pi ] ⊂ h implies that ki := h ⊕ pi is a subalgebra of g for any i, and the pair (ki , h) is irreducible symmetric (it could be non-effective, of course). We easily get from the definition that [p j , pk ] ⊂ pi for pairwise distinct i, j, k. Therefore, [p j ⊕ pk , p j ⊕ pk ] ⊂ h ⊕ pi = ki ,

{i, j, k} = {1, 2, 3},

and all the pairs (g, ki ) are also irreducible symmetric.  Denote by di the dimension of pi . Let eij be an orthonormal basis in pi with respect to h· , ·i, where i ∈ {1, 2, 3}, 1 ≤ j ≤ di = dim(pi ). Consider the expression [i jk] defined by the equality [i jk] =

∑

D

E β γ 2 eαi , e j , ek ,

(4.23)

α,β ,γ

where α, β , and γ range from 1 to di , d j , and dk respectively. The symbols [i jk] are symmetric in all three indices by the bi-invariance of the metric h· , ·i. Moreover, for

4.13 Some Special Compact Homogeneous Spaces

223

the spaces under consideration, we have [i jk] = 0 if two indices coincide. Therefore, the quantity A := [123] (4.24) plays an important role. It is easy to see that di ≥ 2A for all i = 1, 2, 3 (see e.g. Lemma 7 in [365]). Hence the following constants ai =

A , di

i ∈ {1, 2, 3},

(4.25)

are such that (a1 , a2 , a3 ) ∈ [0, 1/2]3 . Note that these constants completely determine some important properties of a generalized Wallach space G/H, e.g. the equation of the Ricci flow on G/H, see [2, 3]. Of course, not every triple (a1 , a2 , a3 ) ∈ [0, 1/2]3 corresponds to some generalized Wallach space. We will get a complete description of suitable triples together with the classification of generalized Wallach spaces, that was obtained independently in the papers [160] by Zh. Chen, Y. Kang, K. Liang (for simple G) and [365] by Yu. G. Nikonorov. Theorem 4.13.1 ([160, 365]). Let G/H be a connected and simply connected compact homogeneous space. Then G/H is a generalized Wallach space if and only if it is of one of the following types: 1) G/H is a direct product of three irreducible symmetric spaces of compact type (A = a1 = a2 = a3 = 0 in this case); 2) The group G is simple and the pair (g, h) is one of the pairs in Table 4.2 (the embedding of h to g is determined by the following requirement: the corresponding pairs (g, ki ) and (ki , h), i = 1, 2, 3 are symmetric); 3) G = F × F × F × F and H = diag(F) ⊂ G for some connected simply connected compact simple Lie group F, with the following description on the Lie algebra level:
(g, h) = f ⊕ f ⊕ f ⊕ f, diag(f) = {(X, X, X, X) | X ∈ f} , where f is the Lie algebra of F, and (up to permutation) p1 = {(X, X, −X, −X) | X ∈ f}, p2 = {(X, −X, X, −X) | X ∈ f}, p3 = {(X, −X, −X, X) | X ∈ f} (a1 = a2 = a3 = 1/4 in this case). Generalized Wallach spaces are interesting from the point of view of invariant Einstein metrics. We note only that every of these spaces admits at least one and at most four such metric, for details and historical notes we refer to [157]. In [2, 3], N. A. Abiev, A. Arvanitoyeorgos, Yu. G. Nikonorov, and P. Siasos studied the normalized Ricci flow equation (see e.g. [249] and [112])

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4 Homogeneous Riemannian Manifolds

Table 4.2 The pairs (g, h) defined generalized Wallach spaces G/H with simple G, S = k + l + m h

d1

d2

d3

a1

a2

a3

1 so(k + l + m) so(k) ⊕ so(l) ⊕ so(m)

kl

km

lm

m 2(S−2)

l 2(S−2)

k 2(S−2)

2 su(k + l + m) s(u(k) ⊕ u(l) ⊕ u(m))

2kl

2km

2lm

m 2S

l 2S

k 2S

3 sp(k + l + m) sp(k) ⊕ sp(l) ⊕ sp(m) 4kl

4km

4lm

m 2(S+1)

l 2(S+1)

k 2(S+1)

N

g

4

su(2l), l ≥ 2 u(l)

l(l − 1) l(l + 1) l 2 − 1

l+1 4l

l−1 4l

1 4

5

so(2l), l ≥ 4 u(1) ⊕ u(l − 1)

2(l − 1) 2(l − 1) (l−1)(l−2)

l−2 4(l−1)

l−2 4(l−1)

1 2(l−1)

6

e6

su(4) ⊕ 2sp(1) ⊕ R

16

16

24

1 4

1 4

1 6

7

e6

so(8) ⊕ R2

16

16

16

1 6

1 6

1 6

8

e6

sp(3) ⊕ sp(1)

14

28

12

1 4

1 8

7 24

9

e7

so(8) ⊕ 3sp(1)

32

32

32

2 9

2 9

2 9

10 e7

su(6) ⊕ sp(1) ⊕ R

30

40

24

2 9

1 6

5 18

11 e7

so(8)

35

35

35

5 18

5 18

5 18

12 e8

so(12) ⊕ 2sp(1)

64

64

48

1 5

1 5

4 15

13 e8

so(8) ⊕ so(8)

64

64

64

4 15

4 15

4 15

14 f4

so(5) ⊕ 2sp(1)

8

8

20

5 18

5 18

1 9

15 f4

so(8)

8

8

8

1 9

1 9

1 9

Sg ∂ g(t) = −2Ricg + 2g(t) ∂t n on generalized Wallach spaces. Some general properties of this equation were established. In [1], N. A. Abiev and Yu. G. Nikonorov studied the evolution of positively curved metrics on the Wallach spaces W6 , W12 , and W24 . It is proved that for all Wallach spaces, the normalized Ricci flow evolves all generic invariant Riemannian metrics with positive sectional curvature into metrics with mixed sectional curvature. Moreover, it is proved that for the spaces W12 and W24 , the normalized Ricci flow evolves all generic invariant Riemannian metrics with positive Ricci curvature into metrics with mixed Ricci curvature. These results generalized some previously known results from [112] and [162], see the detailed discussion in [1].

The description of geodesic orbit Riemannian metrics for generalized Wallach spaces is provided in Subsection 5.12.6.

4.13 Some Special Compact Homogeneous Spaces

225

4.13.3 Ledger–Obata Spaces The spaces F m / diag(F) are called Ledger–Obata spaces, where F is a connected compact simple Lie group, F m = F × F × · · · × F (m factors and m ≥ 2), and diag(F) = {(X, X, . . . , X)|X ∈ F}. Ledger–Obata spaces were first introduced in [318] as a natural generalization of symmetric spaces, since F 2 / diag(F) is an irreducible symmetric space, and have been actively studied since then (see the book [299] by O. Kowalski). In particular, invariant Einstein metrics on Ledger– Obata spaces are studied in several papers, the classification of such metrics on F m / diag(F), m ≤ 4, is known (see [161] and references therein). Let f be the Lie algebra of F and B the Killing form of f. Then h·, ·i = −B(·, ·) is a bi-invariant inner product on f. Let p = dim(f) = dim(F). We will also use the same notations B and h·, ·i for the Killing form and the minus Killing form on the Lie algebra g := nf = f ⊕ f ⊕ · · · ⊕ f | {z } n summands

G := F n ,

of the Lie group and h := diag(f) = {(X, X, . . . , X) | X ∈ f} ⊂ g for the Lie algebra of H := diag(F) ⊂ G = F n . e = F n+1 acts transitively on the Lie group G = F n by Note that the group G −1 −1 (a1 , a2 , . . . , an , an+1 ) ◦ (x1 , x2 , . . . , xn ) = (a1 x1 a−1 n+1 , a2 x2 an+1 , . . . , an xn an+1 )

e = F n+1 at the unit of G = F n , which e = diag(F) ⊂ G with the isotropy group H n identifies F with the Ledger–Obata space F n+1 / diag(F). Remark 4.13.2. It is easy to see that every copy of F in F n+1 consists of Clifford– Wolf translations, and every copy of the Lie algebra f in f ⊕ f ⊕ · · · ⊕ f = (n + 1) · f consists of Killing fields of constant length, on (F n+1 / diag(F), g) for any invariant Riemannian metric g. Indeed, as we have seen, the subgroup of F n+1 consisting of all copies of F, but not a given one, is isomorphic to F n and acts transitively on F n+1 / diag(F). According to Example 3.4.2, a given copy of F consists of Clifford– Wolf translations and the corresponding Lie subalgebra f ⊂ (n + 1) · f consists of Killing vector fields of constant length. Since the adjoint representation of every real compact simple Lie algebra is real and irreducible, we easily get the following lemma. Lemma 4.13.3. Let f be a real compact simple Lie algebra. Then every linear mapping Q : f → f commuting with all operators of the adjoint action ad(X) : f → f, where X ∈ f, is proportional to the identity map. Proof. This lemma could be proved by using splitting operators of the (irreducible) adjoint representation of the Lie algebra f, but we give a more direct proof. Suppose that Q has a nontrivial real eigenvalue λ . Then the corresponding eigenspace Qλ := {X ∈ f | Q(X) = λ X} is a nontrivial ideal in f. Indeed, for any X ∈ Qλ and any Y ∈ f,

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4 Homogeneous Riemannian Manifolds

we get Q([Y, X]) = [Y, Q(X)] = λ [X,Y ], hence, [Y, X] ∈ Qλ . Therefore, since f is simple, we get Qλ = f and Q = λ Id. Therefore, it suffices to prove that Q has a real eigenvalue. It is clear that the centralizer of any element in f is preserved by Q. Hence, any Cartan subalgebra t in f is invariant under Q (f is a centralizer of a regular element in f). For a given Cartan subalgebra t, there are a two-dimensional root space V with a basis {E, F} and a nontrivial lineal form τ : t → R such that [X, E] = τ(X)F and [X, F] = −τ(X)E for all X ∈ t (see e.g. Proposition 2.5.29 or (2.34)). Take X such that τ(X) 6= 0, then τ(Q(X))E = −[Q(X), F] = [F, Q(X)] = Q([F, X]) = τ(X)Q(E), hence τ(Q(X)) · τ(X)−1 is a real eigenvalue of Q with the eigenvector E. The lemma is proved. t u e It should be noted that there are several Ad(H)-invariant complements to e h in e g. To see this, the following lemma is useful. e Lemma 4.13.4. Every irreducible Ad(H)-invariant submodule in e g has the form {(α1 X, α2 X, . . . , αn+1 X) ⊂ e g | X ∈ f}

(4.26)

for some fixed αi . e Proof. Let m be an irreducible Ad(H)-invariant submodule in e g. Consider the linear maps θi : m → f, i = 1, 2, . . . , n + 1, defined by θi (X) = Xi , where X = (X1 , X2 , . . . , Xn , Xn+1 ) ∈ m ⊂ (n + 1)f. Since m is ad(e h)-invariant, for any Ze = e e (Z, Z, · · · , Z) ∈ h where Z ∈ f, we see that θi ([Z, X]) = [Z, θi (X)]. In particular, the image of θi is an ideal in f, hence it is either 0 or f itself. At least one of such image should be non-zero. Without loss of generality, we may assume that this is the −1 case for i = n + 1. Hence, we may define the maps κi : f → f by κi = θi ◦ θn+1 for any i = 1, 2, . . . , n. By the above discussion we see that κi is a linear map commuting with all operators of the adjoint action ad(Z) : f 7→ f for any Z ∈ f. By Lemma 4.13.3, κi is proportional to the identity map, which implies Lemma 4.13.4. t u Clearly, the isotropy subalgebra e h is a submodule as in Lemma 4.13.4 with α1 = α2 = · · · = αn+1 6= 0. Denote by p the union of all submodules (4.26) with α1 + α2 + · · · + αn+1 = 0 (it could be represented also as a direct sum of n suitable submodules of this type). Note that p is the h·, ·i-orthogonal complement to e h in e g, where h·, ·i is the minus Killing form of e g, i.e. ( ) n+1
p = Lin (α1 X, α2 X, . . . , αn+1 X) ⊂ e g | X ∈ f, ∑ αi = 0 . i=1

On the other hand, all submodules from Lemma 4.13.4 are pairwise isomorphic e with respect to Ad(H). This allows us to construct various Ad(H)-invariant come e plements to h in g.

4.13 Some Special Compact Homogeneous Spaces

227

Define the linear map ϕn+1 : p → g = nf by
(α1 X, α2 X, . . . , αn+1 X) 7→ (α1 − αn+1 )X, . . . , (αn − αn+1 )X .

(4.27)

It is clear that

−1 ϕn+1 (α1 X, α2 X, . . . , αn X, 0) = (α1 − s)X, (α2 − s)X, . . . , (αn − s)X, −sX , where s = (α1 + α2 + · · · + αn )/(n + 1). The map ϕn+1 induces an identification e H e e= between the set of Ad(H)-invariant metrics on the Ledger–Obata space G/ n+1 n F / diag(F) and the set of Ad(H)-invariant metrics on G = F (it is assumed that Z ∈ p and ϕn+1 (Z) ∈ g have the same length in the induced norms). Now we will deal with Ad(H)-invariant or, equivalently, with ad(h)-invariant inner products on the Lie algebra g, taking into account the above identification e of such inner products with Ad(H)-invariant metrics on the Ledger–Obata space e H e = F n+1 / diag(F). G/ This allows us to give a useful parametrization of all invariant Riemannian metrics on a given Ledger–Obata space. Let (·, ·) be an ad(h)-invariant inner product on g. First of all, we choose a (·, ·)-orthonormal basis of g. By Lemma 4.13.4, there exists (a1 , . . . , an ) ∈ Rn such that a given ad(h)-invariant irreducible submodule g˜ is of the form e g = {(a1 X, a2 X, . . . , an X)|X ∈ f}. Furthermore, there are n orthogonal ad(h)-invariant irreducible submodules g1 , . . . , gn of g such that g = g1 ⊕ · · · ⊕ gn and we can diagonalize (·, ·) and h·, ·i simultaneously, where, for i = 1, . . . , n, gi = {(bi1 X, bi2 X, . . . , bin X)|X ∈ f}. For any X ∈ f with hX, Xi = 1, define a vector space LX by LX := {(a1 X, a2 X, a3 X, . . . , an−1 X, an X) | ai ∈ R, i = 1, . . . , n} ⊂ g,

(4.28)

which is naturally identified with Rn . Moreover, h·, ·i is identified with the standard inner product on Rn . Clearly, {(bi1 X, bi2 X, . . . , bin X)}i=1,2,...,n is an orthogonal basis of LX , which gives a (·, ·)-orthonormal basis { f1 , . . . , fn } of LX . For some real numbers ci 6= 0 and αi j , i, j = 1, . . . , n, fi = fi (X) = ci (bi1 X, bi2 X, . . . , bin X) = (αi1 X, αi2 X, αi3 X, . . . , αin X).

(4.29)

Let {X1 , . . . , X p } be a h·, ·i-orthogonal basis of f. Then we have a (·, ·)-orthonormal basis of g fi j = fi (X j ) = (αi1 X j , αi2 X j , . . . , αin X j ),

1 ≤ j ≤ p, 1 ≤ i ≤ n.

(4.30)

Obviously, this basis is determined by a non-degenerate (n×n)-matrix A = (αi j ). e := QA also determines For any matrix Q ∈ O(n), it is easy to see that the matrix A X a (·, ·)-orthonormal basis in L . Without loss of generality, we may suppose that A is lower triangular with positive elements in the main diagonal, i.e. αi j = 0 for j > i

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4 Homogeneous Riemannian Manifolds

and αii > 0. On the other hand, we can construct an ad(h)-invariant inner product on g from a non-degenerate matrix (αi j ) satisfying αi j = 0 for j > i and αii > 0. That is, there is a one-to-one correspondence between ad(h)-invariant inner products on g and non-degenerate (n × n)-matrices (αi j ) satisfying αi j = 0 for j > i and αii > 0. Hence, the latter set of matrices can be used for the parameterization of ad(h)-invariant inner products on g. General properties of invariant metrics on Ledger–Obata spaces F m / diag(F) were studied in [358] (see also Subsection 5.12.2), where the following results are obtained: all naturally reductive metrics are classified (notice that in the case m = 3, any invariant metric is naturally reductive by Proposition 5.12.3), it is proved that a Ledger–Obata space supplied with an invariant metric is a geodesic orbit space if and only if this metric is naturally reductive, and it is proved that a Ledger–Obata space is reducible if and only if it is isometric to the product of Ledger–Obata spaces of smaller dimensions.

4.14 Curvatures of Homogeneous Riemannian Spaces For the first time special formulas for calculating the sectional curvature of homogeneous Riemannian manifolds were obtained by K. Nomizu in [369]. Later, there appeared simpler variants to derive such formulas (see e.g. [51], [106], or [150]). Here we obtain the corresponding formula directly from Corollary 3.2.4, which relates to Killing vector fields on an arbitrary Riemannian manifold (see also [361]). It is easy to see that in view of the invariance of the curvature under isometries, it suffices to calculate it only at some point x of the Riemannian manifold (M = G/H, g), therefore, the curvature tensor R at the point x ∈ M is identified with an AdG (H)-invariant tensor on the space p (see Definition 4.2.10 and Theorem 4.2.11). We denote by (·, ·) the Ad(H)-invariant inner product on p corresponding p to the Riemannian metric g (we will also use the notation |X| instead of (X, X), X ∈ p). This inner product can be extended to an Ad(H)-invariant inner product on the Lie algebra g (it suffices to choose some Ad(H)-invariant inner product on the Lie algebra h and put (p, h) = 0), which we will denote by the same symbol. The curvatures of the homogeneous Riemannian manifold (M, g) can be expressed in terms of (·, ·) and the Lie bracket in the Lie algebra g. In this section, we denote by [·, ·]g the Lie bracket in the Lie algebra g, and also use the notations [·, ·]p and [·, ·]h for the p-component and for the h-component of the bracket [·, ·]g ; preserving the notation [·, ·] for the Lie bracket of vector fields on the manifold M. We note that the differential T π of the canonical projection π : G → G/H = M is a monomorphism of the Lie algebra RG of right-invariant vector fields on G into the Lie algebra of Killing vector fields on (M, g) (and even an isomorphism, if G contains the full connected group of isometries of the space (M, g)). Identifying X ∈ RG with its value X(e) ∈ Ge = g, we obtain that [X,Y ] = −[X,Y ]g for any X,Y ∈ RG according to Theorem 2.1.29, which must be taken into account in all calculations.

4.14 Curvatures of Homogeneous Riemannian Spaces

229

First, we define the operator U : p × p → p (we have used it in (4.20)) by the formula 2(U(X,Y ), Z) = ([Z, X]p ,Y ) + (X, [Z,Y ]p ) ∀ Z ∈ p. (4.31) It is easy to see that U(X,Y ) = U (X,Y )|x (cf. (3.5) and (3.6)), using the identification of Mx with p. Recall that the formula 1 ∇X Y = − [X,Y ]p +U(X,Y ) 2

(4.32)

holds at the point x ∈ M (Lemma 3.2.1). Now, taking into account everything mentioned above, we immediately obtain the following important result (see e.g. [106, 291]) from Corollary 3.2.4. Theorem 4.14.1. Let (M = G/H, g) be a Riemannian homogeneous space (Ghomogeneous Riemannian manifold), X,Y ∈ p = Mx . Then 3 1 1 (R(X,Y )Y, X) = − |[X,Y ]p |2 − ([X, [X,Y ]g ]p ,Y ) − ([Y, [Y, X]g ]p , X) 4 2 2 2 +|U(X,Y )| − (U(X, X),U(Y,Y )). Let {Xi } be an orthonormal basis of the space (p, (·, ·)). Consider the vector Z = ∑ U(Xi , Xi ).

(4.33)

i

The equality (4.31) implies that (Z, X) = trace(ad(X)) for every X ∈ p (the latter equality uniquely determines the vector Z, which, therefore, does not depend on the choice of basis {Xi }). Consequently, Z = 0 if and only if the Lie algebra g is unimodular. Note also that trace(ad(Z)) = (Z, Z). With the notation introduced, the following assertions hold. Theorem 4.14.2. The Ricci curvature Ric at the point x of the Riemannian homogeneous space (M = G/H, g) is determined by the formula 1 1 1 Ric(X, X) = − B(X, X) − ∑ |[X, Xi ]p |2 + ∑([Xi , X j ]p , X)2 − ([Z, X]p , X), 2 2 i 4 i, j where B is the Killing form of the Lie algebra g. In order to prove this theorem, we need two lemmas. Further, for X ∈ g, we denote by ad∗X the operator adjoint to the operator adX = ad(X) : g → g with respect to the inner product (·, ·), i.e. (ad∗X (Y ), Z) = (Y, adX (Z)) for all Y, Z ∈ g. Lemma 4.14.3. For any X ∈ p, the following equality holds: 4 ∑ |U(X, Xi )|2 = ∑ |[X, Xi ]p |2 + ∑([Xi , X j ]p , X)2 i

i

i, j

+2 ∑([Xi , [Xi , X]g ]p , X) − 2 ∑([X, [X, Xi ]h ]p , Xi ). i

i

230

4 Homogeneous Riemannian Manifolds

Proof. According to formula (4.31), we get  2 4 ∑ |U(X, Xi )|2 = 4 ∑(U(X, Xi ), X j )2 = ∑ [X j , X]p , Xi ) + (X, [X j , Xi ]p ) i, j

i

i

2

2

= ∑ |[X j , X]p | + ∑([Xi , X j ]p , X) + 2 ∑([X j , X]p , Xi )(X, [X j , Xi ]p ). j

i, j

i, j

Now, it suffices to note that

∑([X j , X]p , Xi )(X, [X j , Xi ]p ) = ∑([X j , X]g , Xi )(ad∗X j (X), Xi ) i, j

i, j

([X j , X]g , ad∗X j (X)) −

=∑ j

∑([X j , X]h , ad∗X j (X)) j

= ∑([X j , [X j , X]g ]p , X) + ∑([[X j , X]h , X j ]p , X) j

j

= ∑([X j , [X j , X]g ]p , X) − ∑([X, [X, X j ]h ]p , X j ). j

j

t u

The lemma is proved.

Let {V j } be an orthonormal basis of the Lie subalgebra (h, (·, ·)). Note that the Killing form satisfies the equality B(X, X) = ∑([X, [X, Xi ]g ]g , Xi ) + ∑([X, [X,V j ]g ]g ,V j ). i

j

In particular, since the inner product (·, ·) on g is Ad(H)-invariant, we get B(X, X) = − ∑([X, Xi ]g , [X, Xi ]g ) − ∑([X,V j ]g , [X,V j ]g ) i

j

for every X ∈ h. Therefore, the Killing form B is negative definite on h (indeed, B(X, X) = 0 for X ∈ h means that X is in the center of the Lie algebra g, i.e. X = 0 according to the (almost) effectiveness of the space G/H). Lemma 4.14.4. For every X ∈ p, the following equality holds:

∑([X, [X, Xi ]h ]p , Xi ) = ∑([X, [X,V j ]g ]h ,V j ). i

j

Proof. Since the inner product (·, ·) on p is Ad(H)-invariant and [X,V j ] ∈ [p, h] ⊂ p, we get

∑([X, [X, Xi ]h ]p , Xi ) = ∑([X, Xi ]h , ad∗X (Xi )) i

i

= ∑([X, Xi ]g ,V j )(ad∗X (Xi ),V j ) = ∑(Xi , ad∗X (V j ))(Xi , [X,V j ]g ) i, j

(ad∗X (V j ), [X,V j ]g ) =

=∑ j

i, j

∑(V j , [X, [X,V j ]g ]g ) = ∑([X, [X,V j ]g ]h ,V j ). j

j

4.14 Curvatures of Homogeneous Riemannian Spaces

231

t u

The lemma is proved.

Proof (of Theorem 4.14.2). Using Theorem 4.14.1, Lemma 4.14.3, Lemma 4.14.4, and the formula (4.33), we get Ric(X, X) = ∑(R(X, Xi )Xi , X) = − i

3 |[X, Xi ]p |2 4∑ i

1 1 − ∑([X, [X, Xi ]g ]p , Xi ) − ∑([Xi , [Xi , X]g ]p , X) 2 i 2 i   + ∑ |U(X, Xi )|2 − U(X, X), ∑ U(Xi , Xi ) i

i

3 1 = − ∑ |[X, Xi ]p |2 − ∑([X, [X, Xi ]g ]p , Xi ) 4 i 2 i − +

1 1 1 ([Xi , [Xi , X]g ]p , X) + ∑ |[X, Xi ]p |2 + ∑([Xi , X j ]p , X)2 2∑ 4 4 i, j i i

1 1 ([Xi , [Xi , X]g ]p , X) − ∑([X, [X, Xi ]h ]p , Xi ) − (U(X, X), Z) ∑ 2 i 2 i =−

1 1 |[X, Xi ]p |2 + ∑([Xi , X j ]p , X)2 2∑ 4 i, j i

−

1 1 ([X, [X, Xi ]g ]p , Xi ) − ∑([X, [X,Vk ]g ]h ,Vk ) − (U(X, X), Z) 2∑ 2 k i

=−

1 1 1 |[X, Xi ]p |2 + ∑([Xi , X j ]p , X)2 − B(X, X) − ([Z, X]p , X), ∑ 2 i 4 i, j 2 t u

as required.

Theorem 4.14.5. The scalar curvature sc at the point x of the Riemannian homogeneous space (M = G/H, g) is determined by the formula sc = −

1 1 |[Xi , X j ]p |2 − ∑ B(Xi , Xi ) − |Z|2 . 4∑ 2 i i, j

Proof. By Theorem 4.14.2, we get sc = ∑ Ric(Xi , Xi ) = − i

+ =−

1 1 B(Xi , Xi ) − ∑ ([Xi , X j ]p , Xk )|2 2∑ 2 i, j,k i

1 ([Xi , X j ]p , Xk )2 − ∑([Z, Xi ]p , Xi ) 4 i,∑ i j,k

1 1 |[Xi , X j ]p |2 − ∑ B(Xi , Xi ) − |Z|2 , ∑ 4 i, j 2 i

since ∑i ([Z, Xi ]p , Xi ) = trace(ad(Z)) = (Z, Z) according to the definition of the vector Z. t u

232

4 Homogeneous Riemannian Manifolds

The expressions for the sectional curvature and the Ricci curvature are substantially simplified in the case of naturally reductive (in particular, normal) homogeneous Riemannian manifolds. Assume that the inner product (·, ·) is the restriction to p of some Ad(G)-invariant non-degenerate quadratic form Q on g. Then U ≡ 0 (see the formula (4.31), Proposition 2.1.25, and Theorem 2.1.27), in particular, Z = 0 (the formula (4.33)). Theorems 4.14.1 and 4.14.2 implies the following statement. Theorem 4.14.6. Let (M = G/H, g) be a naturally reductive homogeneous Riemannian manifold, i.e. (·, ·) is the restriction to p of some Ad(G)-invariant nondegenerate quadratic form Q on g such that Q(p, h) = 0. Then the following formulas hold: 1 (R(X,Y )Y, X) = |[X,Y ]p |2 + Q([X,Y ]h , [X,Y ]h ) , 4 1 1 Ric(X, X) = − B(X, X) + ∑ Q([X, Xi ]h , [X, Xi ]h ) , 4 2 i where B is the Killing form of the Lie algebra g. In particular, for any normal homogeneous metric (equivalently, when the form Q is positive definite on g) the sectional curvature is non-negative. Proof. Since ([X, [X,Y ]g ]p ,Y ) = ([X, [X,Y ]p ]p ,Y ) + ([X, [X,Y ]h ]g ,Y ) −([X,Y ]p , [X,Y ]p ) − Q([X,Y ]h , [X,Y ]g ) = −|[X,Y ]p |2 − Q([X,Y ]h , [X,Y ]h ) and, analogously, ([Y, [Y, X]g ]p , X) = −|[X,Y ]p |2 − Q([X,Y ]h , [X,Y ]h ), Theorem 4.14.1 and the equality U ≡ 0 imply 3 1 1 (R(X,Y )Y, X) = − |[X,Y ]p |2 − ([X, [X,Y ]g ]p ,Y ) − ([Y, [Y, X]g ]p , X) 4 2 2 1 = |[X,Y ]p |2 + Q([X,Y ]h , [X,Y ]h ). 4 Since (here we use Lemma 4.14.4 in addition to the definitions)

∑([Xi , X j ]p , X)2 = ∑(X j , [Xi , X]p )2 = ∑ |[X, Xi ]p |2 , i, j

i, j

i

−B(X, X) = − ∑ Q([X, [X, Xi ]g ]g , Xi ) − ∑([X, [X,V j ]g ]g ,V j ) i

j

= ∑ Q([X, Xi ]g , [X, Xi ]g ) − ∑([X, [X, Xi ]h ]p , Xi ) i

i

= ∑ Q([X, Xi ]g , [X, Xi ]g ) + ∑ Q([X, Xi ]h , [X, Xi ]h ) i

i

4.15 Homogeneous Riemannian Manifolds with Restrictions on Curvatures

233

= ∑ |[X, Xi ]p |2 + 2 ∑ Q([X, Xi ]h , [X, Xi ]h ) , i

i

then according to Theorem 4.14.2, we get 1 1 1 Ric(X, X) = − B(X, X) − ∑ |[X, Xi ]p |2 + ∑([Xi , Xk ]p , X)2 2 2 i 4 i,k 1 1 1 = − B(X, X) − ∑ Q([X, Xi ]g , [X, Xi ]g ) + ∑ Q([X, Xi ]h , [X, Xi ]h ) 2 4 i 4 i 1 1 = − B(X, X) + ∑ Q([X, Xi ]h , [X, Xi ]h ), 4 2 i as required.

t u

4.15 Homogeneous Riemannian Manifolds with Restrictions on Curvatures There are many different topological, geometric, and analytical constraints on a homogeneous Riemannian manifold with some given restrictions on the curvature. Some of them will be considered in this section. Note also that there are some wellknown restrictions on the structure of a homogeneous space, all of whose invariant metrics have a common condition on the curvature. To illustrate the latter, we consider the characterization of homogeneous spaces, all invariant Riemannian metrics on which have a positive scalar curvature. The corresponding result in the final form was obtained in [61], although in a few other versions were known before (see e.g. [51] and [459]). Theorem 4.15.1. Let G/H be a compact simply connected effective homogeneous space of a compact connected Lie group G by its closed subgroup H. Then the following conditions are equivalent: 1) Every G-invariant Riemannian metric on G/H is G-normal; 2) All G-invariant Riemannian metrics on G/H have positive Ricci curvature; 3 ) All G-invariant Riemannian metrics on G/H have positive scalar curvature; 4) The space G/H is a direct product of several compact strongly isotropy irreducible spaces; 5) Every invariant distribution on G/H is involutive; 6) The space G/H does not admit invariant sub-Riemannian metrics [49]. If any of these conditions are violated, then G/H admits an invariant Riemannian metric of arbitrary scalar curvature. Later in this section, we provide an exposition on homogeneous Riemannian manifolds of positive (non-positive) sectional curvature, and of positive (non-posi-

234

4 Homogeneous Riemannian Manifolds

tive) Ricci curvature. All these results are of undoubted independent interest, but some of them will also be needed for further discussions. It should be noted also that there are interesting results on homogeneous Finsler spaces with positive curvature, see e.g. the book [172], the survey [177], and references therein.

4.15.1 Homogeneous Riemannian Spaces of Positive Sectional Curvature The most famous examples of homogeneous manifolds of positive sectional curvature are CROSSes, compact rank one symmetric spaces. Among these spaces (all of them are isotropy irreducible and admit a unique invariant metric, up to proportionality), there are the spheres Sn = SO(n + 1)/SO(n), as well as the projective spaces RPn = SO(n + 1)/O(n), CPn = SU(n + 1)/S(U(n) · U(1)), HPn = Sp(n + 1)/Sp(n) · Sp(1), and CaP2 = F4 /Spin(9) (see e.g. [256, 483, 291]). We recall that spheres, real and complex projective spaces can be realized as homogeneous spaces in various ways (for details, see [502]). Obviously, the δ -pinching of symmetric metrics on spheres and real projective spaces is equal to 1. For symmetric metrics on the remaining projective spaces, the δ -pinching is equal to 1/4. Despite the fact that in the recent paper [446] by L. Verdiani and W. Ziller, all invariant metrics of positive sectional curvature on spheres are classified, the question of the δ -pinching of these metrics remains open in the general case (the same remark applies to real projective spaces). For complex projective spaces there is only one additional possibility to represent them as a homogeneous space: CP2n−1 = Sp(n)/S(n − 1) · U(1)). The homogeneous space Sp(n)/S(n − 1) · U(1)) admits a two-parameter family of invariant metrics (see Section 6.9.3). Metrics of positive sectional curvature among this family (together with the corresponding values of the δ -pinching) were found in the paper [452] by D. E. Volper. We also note that the δ -pinching of homogeneous Riemannian manifolds, diffeomorphic to CROSSes, has been calculated by many mathematicians (see e.g. [444, 86, 450, 451, 452, 403, 446]). In the paper [95] by M. Berger, the classification of simply connected compact normal homogeneous spaces of positive sectional curvature was obtained. It should be noted that in this classification one normal homogeneous space was missed, as was later noted by B. Wilking (see [466] for more details). Later, by the efforts of N. R. Wallach and L. Berard-Bergery, a classification of general homogeneous Riemannian manifolds of positive sectional curvature (up to diffeomorphism) was obtained [455, 50]. Earlier N. R. Wallach constructed in [455] invariant Riemannian metrics of positive sectional curvature on the spaces (Wallach spaces) SU(3)/Tmax ,

Sp(3)/Sp(1) × Sp(1) × Sp(1),

F4 /Spin(8),

4.15 Homogeneous Riemannian Manifolds with Restrictions on Curvatures

235

whereas S. Aloff and N. R. Wallach constructed homogeneous Riemannian metrics of positive sectional curvature on the spaces (Aloff–Wallach spaces) Wk,l = 1 [21]. SU(3)/Sk,l 1 are defined by an emRecall that the Aloff–Wallach spaces Wk,l = SU(3)/Sk,l bedding of the circle SO(2) = S1 into SU(3) of the type e2πiθ 7→ diag(e2πikθ , e2πilθ , e2πimθ ), where k, l, m are integers with greatest common divisor 1 and k + l + m = 0, see Subsection 4.13.1. These spaces were investigated by S. Aloff and N. R. Wallach in [21], where they showed that Wk,l admits an invariant metric of positive sectional curvature if and only if kl(k + l) 6= 0. Moreover, H 4 (Wk,l ; Z) = Z/|k2 + l 2 + kl|Z, and hence there are infinitely many different homotopy types among these spaces. Later M. Kreck and S. Stolz found in [311] that there are homeomorphic but nondiffeomorphic spaces among the Wk,l . Taking into account the action of the Weyl group of SU(3), one can assume also that k ≥ l ≥ 0. We note that the space of invariant metrics on Wk,l is four-dimensional, with the exception of the cases (k, l) = (1, 1) and (k, l) = (1, 0), when the dimension of the corresponding spaces is 10 and 6 respectively. A detailed description of invariant metrics for all Aloff–Wallach spaces Wk,l is given in [359], see also Subsection 4.13.1. We note that there exists the following connection between any Aloff–Wallach 1 and the Wallach space SU(3)/T space Wk,l = SU(3)/Sk,l max : the first of them is the total space and the second one is the base of the natural fiber bundle 1 → SU(3)/T 1 π : SU(3)/Sk,l max with the fiber S . Theorem 4.15.2 (L. B´erard-Bergery–N. R. Wallach). Let M = (G/H, g) be a simply connected homogeneous Riemannian space of positive sectional curvature. Then either M is diffeomorphic to a CROSS, or M is one of the spaces from the following list: Sp(2)/SU(2), SU(5)/Sp(2) × S1 , 1 (SU(3) ×U(2)/S1,1 )/U(2), 1 SU(3)/S1,1 ,

SU(3)/Tmax ,

1 (SU(3) × T 2 /S1,1 )/T 2 ,

1 (SU(3) × T 2 /Sk,l )/T 2 ,

1 SU(3)/Sk,l ,

Sp(3)/Sp(1) × Sp(1) × Sp(1),

F4 /Spin(8).

It should be noted that the spaces 1 (SU(3) ×U(2)/S1,1 )/U(2),

1 (SU(3) × T 2 /S1,1 )/T 2 ,

1 (SU(3) × T 2 /Sk,l )/T 2

1 for suitable pairs (k, l). We explain this with the are diffeomorphic to SU(3)/Sk,l 1 )/T 2 . In this case S1 is embedded into the example of the space (SU(3) × T 2 /Sk,l k,l 2 torus T , where

T 2 = diag(eiα , eiβ , e−i(α+β ) ),

α, β ∈ R,

236

4 Homogeneous Riemannian Manifolds

1 ) acts transitively on SU(3)/S1 , where SU(3) acts on the group SU(3) × (T 2 /Sk,l k,l 1 1 acts from the right. Then the isotropy group at SU(3)/Sk,l from the left, and T 2 /Sk,l 1 is a diagonally embedded torus T 2 . the point Sk,l

As was recently observed by J. A. Wolf and M. Xu [489], there was a gap in the original proof of Theorem 4.15.2 in the case of the Stiefel manifold Sp(2)/U(1) = SO(5)/SO(2). A refined proof of the suitable result was obtained by B. Wilking, see Theorem 5.1 in [489]. The recent paper [469] by B. Wilking and W. Ziller gives a new and short proof of the classification of homogeneous manifolds of positive curvature. The isometry groups of all homogeneous Riemannian spaces of positive sectional curvature were found in the paper [410] by K. Shankar. In the case of homogeneous Riemannian manifolds, it is also convenient to introduce the following value [392].

Definition 4.15.3. The δ -pinching of a pair (G, H) is the number δ (G, H) = sup{δ (G/H, g) | g ∈ M (G, H)}, where M (G, H) is the space of all G-invariant metrics on G/H.

In the even-dimensional homogeneous Riemannian case (with the exception of spaces diffeomorphic to CROSSes) the δ -pinchings of all pairs were found by F. M. Valiev in [444]. The results of this paper for Wallach spaces are given in Table 4.3. Note also that in [444], the author found the values of the δ -pinchings for all invariant metrics on each of three even-dimensional Wallach spaces (the space of invariant metrics for each of these spaces is three-dimensional).

Table 4.3 δ -pinchings of the pair for the Wallach spaces G

H

dim G/H

dim M (G/H)

δ (G, H)

SU(3)

T2

6

3

1 64

Sp(3)

Sp(1)3

12

3

1 64

F4

Spin(8)

24

3

1 64

In what follows, we need a special function (cf. [392]).

4.15 Homogeneous Riemannian Manifolds with Restrictions on Curvatures

237

Definition 4.15.4. The continuous function ∆ : [0, 1] → [0, 1] is defined by the equal1 ) ∈ (0, 1) (see Definition 4.15.3) for rational numbers ity ∆ (k/l) = δ (SU(3), Sk,l l/k ∈ (0, 1), whereas at other points of the interval [0, 1], it is defined by using the passage to the limit. Note that ∆ (0) = 0 and the function ∆ : [0, 1] → [0, 1] is strictly increasing according to [392]. Its value at the point 1, namely, ∆ (1) = 1/37, gives a lower 1 ) (recall that the dimension of the bound for the δ -pinching of the pair (SU(3), S1,1 1 set of invariant metrics on W1,1 = SU(3)/S1,1 is greater than for all other spaces 1 ). Wk,l = SU(3)/Sk,l In the odd-dimensional case, the δ -pinchings of pairs was investigated by H. Eliasson [205], E. Heintze [253], H.-M. Huang [265], and T. P¨uttmann [392]. H. Eliasson 1 showed that δ (Sp(2), SU(2)) = 37 for the Berger pair (Sp(2), SU(2)). Further, E. Heintze and H.-M. Huang found the δ -pinching of the Berger pair (SU(5), Sp(2) × S1 )) under the restriction to some special classes of invariant Riemannian metrics. T. P¨uttmann completed these studies, showing, that 1 δ (SU(5), Sp(2) × S1 )) = 37 . Moreover, in the paper [392] by T. P¨uttmann, the question of the δ -pinchings of pairs for Aloff–Wallach spaces, as well as other nonsymmetric spaces, admitting homogeneous Riemannian metrics of positive sectional curvature, was studied. The results of this paper are given in Table 4.4.

Table 4.4 δ -pinchings of pairs for odd-dimensional manifolds G

H

dim G/H

dim M (G/H)

δ (G, H)

Sp(2)

SU(2)

7

1

1 37

SU(5)

Sp(2) × S1

13

2

1 37

1 SU(3) ×U(2)/S1,1

U(2)

7

2

∆ (1) =

1 37

1 SU(3) × T 2 /S1,1

T2

7

4

∆ (1) =

1 37

SU(3)

1 S1,1

7

7

≥ ∆ (1) =

1 ,k>l SU(3) × T 2 /Sk,l

T2

7

4

1 37

∆ (k/l) < ∆ (1), lim ∆ (k/l) = ∆ (1) =

k/l→1

SU(3)

1 ,k>l Sk,l

1 37

∆ (k/l) < ∆ (1), 7

4

lim ∆ (k/l) = ∆ (1) =

k/l→1

1 37

The last column of Table 4.4 contains information on the δ -pinching of the corresponding pair (for the last two pairs, the asymptotic of the δ -pinching function is provided). A detailed exposition of various results on the set of invariant metrics with positive sectional curvature, the best pinching constant, and full connected isometry groups can be found in [392, 410, 444, 446, 450, 451, 452].

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4.15.2 Homogeneous Riemannian Spaces of Positive Ricci Curvature Homogeneous Riemannian metrics of positive or non-negative Ricci curvatures were studied in [51, 337, 60]. In particular, in [337], the following result was proved. Theorem 4.15.5 (J. Milnor). A connected Lie group G admits a left-invariant Riemannian metric of positive Ricci curvature if and only if G is compact and the fundamental group π1 (G) is finite. In this case, any bi-invariant Riemannian metric is suitable. The following theorem is a generalization of the above theorem to the case of an arbitrary homogeneous space. Theorem 4.15.6 (V. N. Berestovskii, [60]). An effective homogeneous space M = G/H, where G is connected and H is compact, admits a G-invariant Riemannian metric of positive Ricci curvature if and only if any of the following conditions holds: 1) M is compact and its fundamental group π1 (M) is finite; 2) G is compact and a Levi subgroup L of G (i.e. a maximal connected semisimple subgroup of G) acts transitively on M. In this case, G is compact and any normal G-invariant metric on M is suitable. Proof. Necessity of Condition 1). Let h·, ·i be any G-invariant Riemannian metric on M = G/H with positive Ricci curvature ric. Let us fix a point x ∈ M. The function ric(v) is continuous and, hence, attains its positive minimum ricmin > 0 on the compact set Tx1 M of unit tangent vectors v to M at the point x ∈ M. Since the space (M, h·, ·i) is homogeneous, for all unit tangent vectors v ∈ T 1 (M) to M, we have ric(v) ≥ ricmin > 0. By the Myers theorem 1.5.79, the space M is compact and its fundamental group π1 (M) is finite. This observation, Proposition 4.9.2, and Theorem 1.6.14 easily imply that the group G is compact. Sufficiency of Condition 1). Suppose that the homogeneous space M = G/H is compact and its fundamental group π1 (M) is finite. Then the group G is compact and, according to Proposition 2.5.6, admits a bi-invariant Riemannian metric g0 , generated by some Ad(G)-invariant inner product Q on g = LG. It suffices to prove that the Ricci curvature ric of the induced Riemannian metric g = (·, ·) (i.e. a metric such that the canonical projection q : (G, g0 ) → (G/H, g) is a Riemannian submersion) on M is positive. Note that the metric g = (·, ·) is normal homogeneous, and by Theorem 4.14.6, we get 1 1 Ric(X, X) = − B(X, X) + ∑ Q([X, Xi ]h , [X, Xi ]h ) , 4 2 i where B is the Killing form of the Lie algebra g, X ∈ p = Mx , x = eH, and p is the Q-orthogonal complement to h = LH in g. Since the Lie algebra g is compact, g = c ⊕ l, where c is the center and l is a maximal semisimple ideal (which coincides

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with a Levi subalgebra, that is unique in this case). It is clear that the Killing form vanishes on c and is negative definite on l (Theorem 2.5.11). Put C(p) = {X ∈ p | Ric(X, X) = 0}. Obviously, C(p) = c ∩ p. It suffices to prove that this space is trivial. Since the modules p and c are Ad(H)-invariant (the second one is even Ad(G)invariant, since it is the center of g), then the module C(p) = c ∩ p and its Qorthogonal complement C(p)⊥ in p are also Ad(H)-invariant. Moreover, C(p) is an ideal in g, and its Q-orthogonal complement h ⊕C(p)⊥ in g is also an ideal. Note that the subspace C(p)⊥ is non-trivial due to effectiveness of the space G/H. Therefore, g is decomposed into the following Ad(H)-invariant and Q-orthogonal sum of two vector subspaces: g = C(p) ⊕ (h ⊕C(p)⊥ ), that are ideals in g. These ideals determine G-left-invariant and H-right-invariant mutually orthogonal transversal integrable distributions, hence, some foliations A and B on G. The Riemannian submersion q : (G, g0 ) → (M = G/H, g) projects these foliations on G-invariant mutually orthogonal and transversal foliations A, B on (M, g). Moreover, every leaf of the foliation A is locally isometrically mapped on some leaf of the foliation A. Since the metric g0 is bi-invariant, every 1-parameter subgroup in G as well as its left and right translations are geodesics in (G, g0 ). This implies that the foliations A and B in (G, g0 ) are totally geodesic. Moreover, according to Corollary 1.6.9, every 1-parameter subgroup in G, tangent to p at the point e, is entirely a horizontal geodesic in (G, g0 ), i.e. is orthogonal to leaves of the Riemannian submersion q at each of its points, and its image under q is a geodesic in (M, g). Hence, the foliations A and B are also totally geodesic. Moreover, by the definition of the spaces C(p) and C(p)⊥ , all leaves of the foliation A are flat and all leaves of the foliation B have positive Ricci curvature, hence, (due to the homogeneity and Theorem 1.5.79) are compact and have finite fundamental groups. Moreover, the Riemannian space (M, g) is locally isometric to the direct metric product of (arbitrarily chosen) leaves of the foliations A and B, supplied with the induced metrics. This cannot be true for a compact and simply connected manifold M, a contradiction. Therefore, the space C(p) = {X ∈ p | Ric(X, X) = 0} = c ∩ p is trivial, and the normal homogeneous Riemannian space (M = G/H, g) has positive Ricci curvature. Necessity of Condition 2). If an effective homogeneous space M = G/H admits a G-invariant Riemannian metric of positive Ricci curvature, then, according to the necessity of Condition 1), M and G are compact and π1 (M) is finite. Since the Lie group G is compact, then, according to Proposition 2.5.6 and Theorem 2.5.5, its Lie algebra g has the form g = l ⊕ c, where c is the center of the Lie algebra g and l is a semisimple Lie algebra. Moreover, the Lie algebra of L is l and G = L · C, where C is a central subgroup in G with the Lie algebra c. Since the group G is connected and compact, G = exp(g). Therefore, if a Levi subgroup L in G is not transitive on M, then G 6= L · H and g 6= l + h. Hence, in the notation from the proof

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of the sufficiency of Condition 1), {0} = 6 (l + h)⊥ ⊂ c ∩ p for every Ad(G)-invariant (=ad(g)-invariant) inner product Q. As in the proof of the sufficiency of Condition 1), we get a contradiction. Sufficiency of Condition 2) follows from the sufficiency of Condition 1) and the following lemma. t u Lemma 4.15.7. A homogeneous space M = G/H of a connected compact semisimple Lie group G by its closed subgroup H has finite fundamental group. Proof. It is well known that the canonical projection q : G → G/H = M is a locally trivial fibre bundle (see e.g. Proposition 4.9.2 and Theorem 1.6.14). Let us consider the following segment of the exact homotopy sequence of this fibre bundle ([420], [263], [268]): π1 (H) → π1 (G) → π1 (G/H) → π0 (H) → π0 (G). Here, π0 (G) = G/Ge = {1}, and the group π0 (H) = H/He , since the group H is compact (He and Ge are the identity components, in particular, open normal subgroups, in the groups H and G respectively). By Theorem 2.5.11, the group π1 (G) is finite. Therefore, the π1 (G/H) is also finite. t u Theorem 4.15.8. 1) Under the conditions of Theorem 4.15.6 with non-semisimple Lie group G, for any G-invariant normal homogeneous Riemannian metric h·, ·i on M = G/H, all orbits of a maximal central subgroup C ⊂ G in M are flat torus, totally geodesic fibers of the G-invariant non-trivial principal fiber bundle p : M → M/C. Here, M/C is the space of orbits of the group C in M, supplied with the natural Riemannian metric h·, ·i1 , such that p is a Riemannian submersion. 2) Moreover, (M/C, h·, ·i1 ) is a homogeneous Riemannian space G/HC = (G/C)/(HC/C) with the compact semisimple Lie group (G/C) and with a (G/C)-invariant normal homogeneous Riemannian metric of positive Ricci curvature. Proof. 1) Proposition 2.5.6 and Theorem 2.5.5 imply that the Lie algebra g of the non-semisimple Lie group G has the decomposition g = l ⊕ z with the non-trivial center z and a semisimple ideal l. Moreover, z ∩ h = {0} = z ∩ p. The first equality is the result of the effectiveness of the space G/H, the second one is the result of the finiteness of the group π1 (G/H). If dim z = m, then C is isomorphic to an mdimensional torus Tm = Rm /Zm , where Zm is an integer lattice in Rm . Since C is a central subgroup in G, then all its elements are Clifford–Wolf translations in the space (M, h·, ·i), and all its orbits are flat totally geodesic tori Tm in (M, h·, ·i), on which C acts simply transitively. These orbits determine a G-invariant foliation on M, since C is a central subgroup in G. Since G acts transitively on M, the natural projection p : M = G/H → M/C = G/HC = (G/C)/(HC/C)

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determines a principal G-invariant fiber bundle with the structure group C. This fiber bundle is non-trivial, otherwise the space M has infinite fundamental group. 2) The Lie group G/C is compact and semisimple. Hence, according to Lemma 4.15.7, the fundamental group π1 (M/C) = π1 ((G/C)/(HC/C)) is finite. Consider the following natural projections: p1 : G → G/C;

q1 : G/C → M/C .

We get the commutative diagram q

(G, (·, ·)) −→ (M, h·, ·i) ↓ p1

p↓ q1

(G/C, (·, ·)1 ) M −→ (M/C, h·, ·i1 ). Here, the metric (·, ·)1 is such that p1 is a Riemannian submersion. Since q, p, p1 are Riemannian submersions, q1 is a Riemannian submersion too. Since the (·, ·) is bi-invariant on G, it follows that (·, ·)1 and h·, ·i1 are respectively a bi-invariant and a normal G/C-invariant Riemannian metric on G/C and M/C. Then, according to Theorem 4.15.6, (M/C, h·, ·i1 ) has positive Ricci curvature, as required. t u Example 4.15.9. In the case of the sphere S2n+1 = U(n + 1)/U(n), we have u(n + 1) = su(n + 1) ⊕ z with the one-dimensional center z. This example is interesting because the canonical Riemannian metric on S2n+1 is not normal for U(n + 1)/U(n). At the same time, the group U(n + 1) contains a simple subgroup SU(n + 1) of special unitary matrices, which is transitive on S2n+1 . The presence of a one-dimensional connected central subgroup C in U(n + 1) gives the principal U(n + 1)-invariant nontrivial fiber bundle (the Hopf fibration) of the sphere S2n+1 into totally geodesic tori (in this case, circles) that are orbits of the group C. The base CPn = U(n + 1)/U(n)C of the bundle is a non-effective homogeneous space with transitive effective simple Lie group SU(n + 1) = U(n + 1)/C, admitting an invariant normal Riemannian metric of positive sectional curvature 14 ≤ K ≤ 1 (the Fubini–Study metric). The classification problem for homogeneous Riemannian manifolds of positive Ricci curvature is very hard. It is not solved even in the case of homogeneous Riemannian manifolds of constant positive Ricci curvature (homogeneous Einstein manifolds). Note also that [337] studied the behavior of the Ricci curvature, as well as the sectional curvature of Lie groups with left-invariant Riemannian metrics. In [403], the authors found some two-sided estimates for the Ricci curvature, and other types of curvatures on three-dimensional Lie groups with left-invariant Riemannian metrics. The signature of the Ricci curvature of left-invariant metrics on the four-dimensional

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Lie groups is studied in the papers [313, 314], which also provide an extensive bibliography on relevant topics.

4.15.3 Homogeneous Riemannian Spaces of Non-positive Sectional Curvature Important examples of Riemannian manifolds of non-positive sectional curvature are symmetric spaces of Euclidean and noncompact types. Moreover, symmetric spaces of rank 1 of noncompact type have negative sectional curvatures. Nevertheless, there are other homogeneous Riemannian manifolds with this property. They have been studied in sufficient detail, but here we give only some important results. We have the following result. Theorem 4.15.10 (J. A. Wolf, [479]). A homogeneous Riemannian manifold M of non-positive sectional curvature is isometric to the direct metric product of a simply connected Riemannian manifold M 0 with the same property and a flat torus T m . This theorem generalizes the following result by S. Kobayashi: a homogeneous Riemannian manifold of non-positive sectional curvature and of negative Ricci curvature is simply connected [289]. As a corollary of Theorem 4.15.10 in [479], it is shown that on an arbitrary Riemannian manifold of non-positive sectional curvature, some solvable group acts transitively. This result was (independently) refined in papers [254] by E. Heintze and [14] by D. V. Alekseevsky: a simply connected homogeneous Riemannian manifold of non-positive sectional curvature admits a simply transitive solvable group of motions, i.e. can be represented as a solvable Lie group with a left-invariant Riemannian metric. In the proof of the latter result, a ´ Cartan, stating key role was played by the classical theorem of J. Hadamard and E. that any compact isometry group of a simply connected Riemannian manifold of non-positive sectional curvature has a fixed point [256]. We note that a solvable Lie group G with a left-invariant Riemannian metric g of non-positive sectional curvature must have a very special algebraic structure. Let (·, ·) be an inner product on g = LG, which generates such a metric g. Theorem 4.15.11 (D. V. Alekseevsky, [14]). In the notation as above, there are the following (·, ·)-orthogonal decompositions: g = Z(g) ⊕ s = Z(g) ⊕ N(s) ⊕ a, where Z(g) is the center of g, s is a solvable ideal in g, N(s) = [s, s] is the nilradical of s, and a is an abelian subalgebra in s. The dimension of the space a is called the rank of the metric Lie algebra (g, (·, ·)) of non-positive sectional curvature. In the case of negative sectional curvature, the center Z(g) is trivial, dim a = 1 and there exists an A0 ∈ a such that the symmetric

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part of the operator ad(A0 )|g0 : g0 → g0 (where g0 = [g, g] = N(s)) is positive definite [14, 254]. Moreover, the following theorem holds. Theorem 4.15.12 (E. Heintze, [254]). Let g be a solvable Lie algebra, and g0 = [g, g] be its derived Lie algebra. Then the following two conditions are equivalent: 1) g admits an inner product of negative sectional curvature; 2) dim g = dim g 0 + 1 and there exists an A0 ∈ g such that all eigenvalues of the operator ad(A0 )|g 0 have a positive real part. It should be noted that a homogeneous Riemannian manifold of non-positive sectional curvature can be represented in several forms as a solvable Lie group with left-invariant metric (many examples of this kind are discussed in [14]). It is possible to study all such representations, knowing the structure of the full (connected) isometry group of the manifold under consideration. Partial results in this direction are obtained in [14]. Necessary and sufficient conditions for solvable Lie algebras that admit inner products of non-positive sectional curvature are found in [34] (formulated in Definition 6.2 of this paper). The paper [35] contains a detailed study of the isometry groups of homogeneous Riemannian manifolds of non-positive sectional curvature. Any isometry group Isom(En ), Isom(H n ), Isom(Sn ) is a closed subgroup of the M¨obius group M¨ob(n) consisting of homeomorphisms of Rn = Rn ∪ {∞} which are compositions of finitely many reflections with respect to spheres or hyperplanes in Rn . The group Isom(En ) is generated by reflections with respect to hyperplanes. The group Isom(H n ) is isomorphic to the subgroup of elements g ∈ M¨ob(n) such that g(U(O, 1)) ⊂ U(O, 1) and g−1 (U(O, 1)) ⊂ U(O, 1). The group M¨ob(n) itself is realised as Isom(H n+1 ) if any of its elements continuously extends to the unique conformal transformation of the open half-space Rn+1 e metric ds2 = + = {x = (x1 , . . . , xn , xn+1 ) | xn+1 > 0} supplied by the H. Poincar´ n+1 2 2 ∑k=1 (dxk ) /xn+1 . For this interpretation, the (n + 1)-dimensional subgroup of M¨ob(n) acting by parallel translations and homotheties in Rn = {x ∈ Rn+1 | xn+1 = 0} is a simply transitive solvable isometry Lie group G of H n+1 . So we get the Lie group G with left invariant Riemannian metric isometric to H n+1 . Its Lie algebra g is solvable, noncommutative, has n-dimensional commutative nilradical, and any vector subspace of g is its Lie subalgebra. The Lie algebra g and the Lie group G is entirely characterized by these properties. Any left invariant Riemannian metric on G has constant negative sectional curvature [337]. Also for the above isometric action of M¨ob(n) on H n+1 , the subgroup of M¨ob(n) n fixing the point (0, . . . , 0, 1) ∈ Rn+1 + is isomorphic to O(n + 1) = Isom(S ). The considerations of the special Example 1.7 and Theorem 2.5 in [337] imply the following

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Theorem 4.15.13. Let Gn+1 be an (n + 1)-dimensional Lie group generated by parallel translations and the homotheties (with zero center) in Rn , n ≥ 1. Then any leftinvariant Riemannian metric on Gn+1 has constant negative sectional curvature. These Lie groups Gn+1 (for different n ≥ 1) are characterized up to isomorphism as connected Lie groups such that every left-invariant Riemannian metric on any of them has negative sectional curvature. As a corollary, any such Lie group Gn+1 has trivial center. Moreover, these groups are the only connected noncommutative Lie groups that admit no left-invariant sub-Riemannian metric.

4.15.4 Homogeneous Riemannian Spaces of Non-positive Ricci Curvature The condition of non-positive Ricci curvature allows us to draw some conclusions about the algebraic and topological structure of a homogeneous Riemannian manifold. First, we will find out how restrictive the compactness condition is. We have the following result, improving Corollary 3.5.17. Theorem 4.15.14 (Theorem 2.10 in [496]). If a Killing vector field X on a compact Riemannian manifold (M, g) satisfies the condition Ric(X, X) ≤ 0, then X is parallel on (M, g), and, consequently, Ric(X, X) ≡ 0. This theorem easily implies Corollary 4.15.15. If a compact homogeneous Riemannian manifold (M, g) has non-positive Ricci curvature, then it is a Euclidean torus. In particular, its full connected isometry group is commutative. From Corollary 4.15.15 we get that any homogeneous Riemannian manifold (M, g) of negative Ricci curvature must be noncompact (as well as its transitive isometry groups). Homogeneous Riemannian manifolds with zero Ricci curvature have a simple description. Theorem 4.15.16 (D. V. Alekseevsky–B. N. Kimel’fel’d, [17]). A homogeneous Riemannian manifold of zero Ricci curvature is flat, i.e. isometric to a direct metric product of a Euclidean space and a flat torus. Natural examples of homogeneous Riemannian manifolds of negative curvature are symmetric spaces of noncompact type. In addition to these spaces, there are many examples of homogeneous Riemannian manifolds of Ricci negative curvature with semisimple transitive group of motions. Invariant metrics of negative Ricci curvature are constructed on several semisimple Lie groups (see, for example, the construction of left-invariant Riemannian metrics of negative Ricci curvature on the groups SL(n, R), n ≥ 3, in [320]) and on more general homogeneous spaces

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of semisimple Lie groups (for example, in [51], invariant Riemannian metrics of negative Ricci curvature are constructed on the spaces SO0 (n, 2)/SO(n), n ≥ 2). A much more abundant source of examples of homogeneous Riemannian metrics of negative Ricci curvature are non-unimodular solvable Lie groups with special left-invariant metrics. For example, a large number of Einstein left-invariant metrics on solvable non-unimodular Lie groups (the Ricci curvature is necessarily negative in this case). More detailed information on these manifolds (so-called Einstein solvmanifolds) can be found in the paper [251] and in Chapter 4 of the book [37]. At this time, there is no convenient description of homogeneous Riemannian manifolds of negative Ricci curvature. We do not even know the answer to the following question. Question 4.15.17. Which homogeneous spaces (in particular, Lie groups) admit invariant Riemannian metrics of negative Ricci curvature? Some results on Lie groups, admitting left-invariant metrics of negative Ricci curvature, are provided in Subsection 4.16.3.

4.16 Signature of the Ricci Curvature on Homogeneous Spaces Let us consider a homogeneous space G/H with compact H. Informally, we are interested in the following question: What kind of invariant Riemannian metric does this space admit? For instance, we may focus on some natural curvature characteristics of invariant Riemannian metrics, that can attain several values in general, and try to classify all realizable values on a given homogeneous space. There are a lot of well-known results in this direction, as shown in the previous sections. Here we consider more closely some results related to the Ricci curvature. For a given invariant metric on G/H we consider the signature of the Ricci curvature (n, p, z), that is a triple of natural numbers, where n (respectively, p and z) means the quantity of negative (respectively, positive and zero) eigenvalues of the Ricci form (or the Ricci operator) of the metric under consideration. Then we may emphasize our informal question as follows: What triples (n, p, z) could be realized as the signatures of the Ricci form of some invariant Riemannian metrics on a given homogeneous space G/H? Recall that according to Theorem 4.15.6 by V. N. Berestovskii, a homogeneous space admits an invariant metric of positive Ricci curvature if and only if it is compact and has finite fundamental group. Unfortunately, we have no analogous result for negative Ricci curvature (or for other possible signatures). But there are some interesting partial results that we shall discuss below. All these results concern the Ricci curvature of left-invariant Riemannian metrics on Lie groups. The Ricci curvature of a left-invariant metric on a Lie group G can be entirely computed from the algebraic data: the structure of the Lie algebra g of G and the inner product Q on g. With a slight abuse of terminology, we can speak of the Ricci curvature of the metric Lie algebra (g, Q).

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4.16.1 Signature of the Ricci Operator on Lie Groups As noted before, the signatures of the Ricci curvature of left-invariant metrics on 3-dimensional Lie groups were classified by J. Milnor in [337]. Some refinements of this result were obtained by M. S. Chebarykov in [149] and K. Y. Ha and J. B. Lee in [248]. The signatures of the Ricci curvature of left-invariant metrics on 4-dimensional Lie groups were classified in two papers by A. G. Kremlyov and Yu. G. Nikonorov [313, 314]. It should be noted that some partial results in this direction were also obtained in the paper [153] by D. Chen. For dimensions n ≥ 5 we have various partial results. It is necessary to mention the paper [184] by I. Dotti-Miatello, where Ricci signatures of left-invariant Riemannian metrics on two-step solvable unimodular Lie groups are determined, and the paper [312] by A. G. Kremlyov, where he determined all possible signatures of the Ricci operators for left invariant metrics on nilpotent five-dimensional Lie groups. Interesting results on the possible signatures of left-invariant Riemannian metrics on nilpotent Lie groups were obtained in [182] by M. B. Djiadeu Ngaha, M. Boucetta, and J. Wouafo Kamgaa. Recall also the following result: Theorem 4.16.1 (V. Kaiser [278]). Each left invariant Riemannian metric on an N-dimensional Lie group G (N > 2) with a one-dimensional commutator group possesses only the signatures (n, p, z) of the Ricci form with p = 1, n = 2m
m = 1, 2, . . . , [ N−1 2 ] , if the group G is unimodular, and with p = 1, n = 2 or with p = 0, n = 2, if G is non-unimodular. All these signatures are accessible, that is, for a prescribed signature of the Ricci form, there exists one of the above-mentioned groups which admits a left invariant metric with that signature. Note that one of important tools in the above-mentioned research was the possibility of choosing “a good basis” for computation of the Ricci curvature. Details on this technical problem can be found in [149], [250], [313], [314], [337], and references therein.

4.16.2 On Two Negative Eigenvalues of the Ricci Operator The above results lead to various natural conjectures. We consider one of them. It is shown in [273, 337] that the scalar curvature of every non-flat left-invariant Riemannian metric on a given solvable Lie group is negative, therefore, the Ricci operator of this metric has at least one negative eigenvalue. It is convenient to study the curvature properties of left-invariant Riemannian metrics on Lie groups in terms of metric Lie algebras (i.e. Lie algebras supplied with inner products), see e.g. Subsection 2.4.1. Indeed, let G be a Lie group with the Lie algebra g. Then every inner product (·, ·) on g uniquely determines a left-invariant Riemannian metric ρ on G, and vice versa.

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There are various examples of solvable metric Lie algebras such that the Ricci operator has at least two negative eigenvalues. The authors of [314] proved (in particular) that the Ricci operator of every non-unimodular solvable metric Lie algebra of dimension ≤ 4 has at least two negative eigenvalues. Moreover, the authors of [314] posed the following conjecture: The Ricci operator of every solvable nonunimodular metric Lie algebra has at least two negative eigenvalues. This conjecture was completely confirmed in [363] as a corollary of a general result, which will be discussed below (see the discussion in [363] for earlier results). As usual, we denote by [s, s] the derived algebra of a Lie algebra s. For every solvable Lie algebra s, [s, s] is a nilpotent ideal of s and [s, s] 6= s. Recall that an operator A (acting on a given Euclidean space) is called normal if it commutes with its adjoint A0 . Theorem 4.16.2 (Yu. G. Nikonorov, [363]). Let (s, Q) be a solvable metric Lie algebra and let n = [s, s], and a be a Q-orthogonal complement to n in s. Then one of the following mutually exclusive assertions holds: 1. The ideal n is commutative, a is a commutative subalgebra of s, and for every X ∈ a the operator ad(X)|n is skew-symmetric with respect to Q (in this case the Ricci operator of (s, Q) is zero); 2. The ideal n is commutative, a is a commutative subalgebra of s, and for every X ∈ a the operator ad(X)|n is trace-free and normal with respect to Q, but the subspace b = {X ∈ a | ad(X)|n is skew-symmetric with respect to Q} has codimension 1 in a (in this case the Ricci operator of (s, Q) has only one negative eigenvalue, while all other eigenvalues are zero); 3. The Ricci operator of the metric Lie algebra (s, Q) has at least two negative eigenvalues. Obviously, the first and second cases of Theorem 4.16.2 are impossible for nonunimodular Lie algebras. Hence, Theorem 4.16.2 implies immediately a confirmation of the above-mentioned conjecture: Theorem 4.16.3. Let s be a non-unimodular solvable Lie algebra. Then for every inner product Q on s, the Ricci operator of the metric Lie algebra (s, Q) has at least two negative eigenvalues. Theorem 4.16.2 also implies the following result. Theorem 4.16.4. Let s be a non-commutative nilpotent Lie algebra. Then for every inner product Q on s the Ricci operator of the metric Lie algebra (s, Q) has at least two negative eigenvalues. From Theorem 4.16.2 we easily get the following two corollaries. Corollary 4.16.5. If the Ricci operator of a solvable metric Lie algebra (s, Q) has at least one positive eigenvalue, then it has at least two negative eigenvalues.

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Corollary 4.16.6. Let (s, Q) be a solvable metric Lie algebra such that a Q-orthogonal complement a to n = [s, s] in s is not a commutative subalgebra of s, then the Ricci operator of (s, Q) has at least two negative eigenvalues. The following (open) question is natural: Question 4.16.7. Which solvable metric Lie algebras have exactly two negative eigenvalues of the Ricci operator?

4.16.3 Lie Groups with Metrics of Negative Ricci Curvature Now we will discuss the following question (see Question 4.15.17): Which Lie groups admit a left-invariant metric of negative Ricci curvature? By the following result, no unimodular solvable Lie group (in particular, no nilpotent group) admits a left-invariant metric with Ric < 0. Theorem 4.16.8 (I. Dotti Miatello, [184]). Any left-invariant metric with nonpositive Ricci curvature on a solvable unimodular Lie group is Ricci-flat. Further in [186] it was proved that a unimodular Lie group which admits a leftinvariant metric with negative Ricci curvature is noncompact and semisimple. Examples of such metrics were constructed on SL(n, R), n ≥ 3, in [320] and on some complex simple Lie groups in [186]. In [470, 471], C. Will constructed many examples of Lie groups G with compact Levi factor, admitting a left-invariant metric with negative Ricci curvature. We consider two results by C. Will. Theorem 4.16.9 ([471]). Let V be the standard real representation of so(m) on the space of complex-valued homogeneous polynomials of degree n on Rm . If g = (RZ ⊕ so(m)) n V is the Lie algebra such that [Z, so(m)] = 0 and Z acts as the identity on V , then g admits an inner product with negative Ricci curvatures for any n, m ≥ 3. Theorem 4.16.10 ([471]). Let g = (RZ ⊕ su(2)) n n be a Lie algebra where n is any nilpotent Lie algebra and [Z, su(2)] = 0. If [su(2), n] 6= 0 and ad(Z) is a positive multiple of the identity on each su(2)-irreducible subspace of n, then g admits an inner product with negative Ricci curvature. Note that the partial case of this result for commutative n was obtained earlier in [470]. Some sufficient conditions for a Lie algebra g to admit left-invariant Riemannian metrics of negative Ricci curvature are obtained in [471] also for some other types of g, in particular, for g = (a ⊕ h) n n, where h is a semisimple Lie algebra with no compact factor, n is nilpotent and [a, a ⊕ h] = 0. In the rest of this subsection we deal with solvable Lie groups supplied with leftinvariant Riemannian metrics, or equivalently, with metric solvable Lie algebras, i.e. solvable Lie algebras supplied with inner products.

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Let us consider the following theorem, which gives some necessary conditions and some sufficient conditions for the existence of an inner product of negative Ricci curvature on a solvable Lie algebra. Theorem 4.16.11 (Y. Nikolayevsky–Yu. G. Nikonorov, [357]). Let g be a solvable Lie algebra, n the nilradical of g, and z the center of n. Then 1) If g admits an inner product of negative Ricci curvature, then there exists Y ∈ g such that trace ad(Y ) > 0 and all the eigenvalues of the restriction of the operator ad(Y ) to z have positive real part; 2) If there exists a Y ∈ g such that all the eigenvalues of the restriction of ad(Y ) to n have positive real part, then g admits an inner product of negative Ricci curvature. The following is an immediate consequence of Theorem 4.16.11. Corollary 4.16.12. A solvable Lie algebra g with an abelian nilradical n admits an inner product of negative Ricci curvature if and only if there exists a Y ∈ g such that all the eigenvalues of the restriction of ad(Y ) to n have positive real part. Note that the latter property of g is equivalent to the following one: there exists a Y 0 ∈ g such that the restriction of ad(Y 0 ) to n is a stable linear operator, that is, all its eigenvalues have negative real part. In [357], the authors also obtained necessary and sufficient conditions for the existence of left-invariant metrics of negative Ricci curvature for a Lie group whose Lie algebra has either a Heisenberg or standard filiform nilradical. In a more recent paper [356] by Y. Nikolayevsky, analogous necessary and sufficient conditions were found for a Lie group whose Lie algebra has a filiform nilradical. In the paper [178] by J. Der´e and J. Lauret, some structural conditions on a solvable Lie group, related to the existence of negative Ricci curvature metrics, were obtained. J. Der´e and J. Lauret studied the question: Which nilpotent Lie algebras can be the nilradical of some solvable Lie algebra admitting inner products with negative Ricci curvature? They called such a nilpotent Lie algebra a Ricci negative nilradical (RN-nilradical for short). Since the existence of a positive derivation (i.e. a derivation whose eigenvalues have positive real part) is sufficient, even for inner products with negative sectional curvature (see Theorem 4.15.12), any nilpotent Lie algebra which is 2-step or has dimension ≤ 6 is a RN-nilradical. It is proved in [178] also that any nilpotent Lie algebra of dimension 7 having a non-nilpotent derivation is an RN-nilradical. Further, using Condition 1) from Theorem 4.16.11, the authors of [178] obtained some explicit examples of nilpotent Lie algebras that are not RN-nilradicals. All these examples are nilpotent Lie algebras n with derivations of positive trace, moreover, examples with the following properties are obtained (for the first three examples any diagonalizable derivation has a zero eigenvalue on the center): • dim n = 8. • n is 3-step nilpotent.

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• A continuous family of pairwise non-isomorphic algebras of dimension 13. • n has a non-singular derivation but any diagonalizable derivation has a negative eigenvalue on the center. All the above results show that the structure of the set of RN-nilradicals is quite complicated.

4.17 Homogeneous Riemannian Spaces with Killing Vector Fields of Constant Length Throughout this section all manifolds are assumed to be connected. Let us consider any Lie group G acting (non-necessarily transitively) on the Riemannian manifold (M, g) by isometries. The action of a ∈ G on x ∈ M we denote by a(x). As usual, we will identify the Lie algebra g of G with the corresponding Lie algebra of Killing vector fields on (M, g) as follows. For any U ∈ g we consider a 1-parameter isometry e by the usual group exp(tU) ⊂ G, t ∈ R, of (M, g) and define a Killing vector field U formula e = d exp(tU)(x) |t=0 . U(x) (4.34) dt ^]. e is linear and injective, but [U, e Ve ] = −[U,V It is clear that the map U → U The above identification is standard and allows us to use Lie algebra structure results to study particular properties of Killing vector fields. The most important case for us is that when G is a transitive isometry group of (M, g). M. Xu and J. A. Wolf obtained the classification of normal Riemannian homogeneous spaces G/H with nontrivial Killing vector field of constant length [490]. More precisely, the following result holds. Theorem 4.17.1 (M. Xu–J. A. Wolf, [490]). Let G/H be a compact homogeneous space G/H supplied with a G-normal Riemannian metric ρ. Suppose that there is a nontrivial Killing vector field of constant length (from the Lie algebra g of the Lie group G) on (G/H, ρ) and dim(G) > dim(H) > 0. Then (G/H, ρ) is locally symmetric and its universal Riemannian cover is either an odd-dimensional sphere of constant curvature, or a Riemannian symmetric space SU(2n)/Sp(n). This result is very important in the context of the study of general homogeneous Riemannian manifolds with nonzero Killing fields of constant length. In a more recent paper [485], this result was extended to the class of pseudo-Riemannian normal homogeneous spaces. Now we discuss some results from [364] on the Lie algebras of transitive isometry groups of a general compact homogenous Riemannian manifold with nontrivial Killing vector fields of constant length. Let (M, g) be a compact connected Riemannian manifold and G a transitive isometry group of (M, g). We identify elements of the Lie algebra g of G with Killing vector fields on (M, g) as above. Since G is compact, we have a decomposition

4.17 Homogeneous Riemannian Spaces with Killing Vector Fields of Constant Length

g = c ⊕ g1 ⊕ g2 ⊕ · · · ⊕ gk ,

251

(4.35)

where c is the center and gi , i = 1, . . . , k, are simple ideals in g. We are going to state the main results of [364], that are formulated under the above assumptions and notations. Theorem 4.17.2. Let Z = Z0 + Z1 + Z2 + · · · + Zl ∈ g be a Killing vector field of constant length on (M, g), where 1 ≤ l ≤ k, Z0 ∈ c, Zi ∈ gi and Zi 6= 0 for 1 ≤ i ≤ l. Then the following statements hold: 1) For every i 6= j, 1 ≤ i, j ≤ l, we have g(gi , g j ) = 0 at every point of M. In particular, g(Zi , g j ) = 0 and g(Zi , Z j ) = 0. 2) Every Killing field of the type Z0 + Zi , 1 ≤ i ≤ l, has constant length. Conversely, if for every i, 1 ≤ i ≤ l, the Killing field Z0 + Zi has constant length, and for every i 6= j, 1 ≤ i, j ≤ l, the equality g(gi , g j ) = 0 holds, then the Killing field Z = Z0 + Z1 + Z2 + · · · + Zl has constant length on (M, g). Corollary 4.17.3. If (under the assumptions of Theorem 4.17.2) we have c = 0 and k = l, then every gi , 1 ≤ i ≤ k, is a parallel distribution on (M, g). Moreover, if (M, g) is simply connected, then it is a direct metric product of k Riemannian manifolds. Theorem 4.17.2 allows us to restrict our attention to Killing vector fields of constant length of the following special type: Z = Z0 + Zi , where Z0 is in the center c of g and Zi is in the simple ideal gi in g. Without loss of generality we will assume that i = 1. Theorem 4.17.4. Let Z = Z0 + Z1 ∈ g be a Killing vector field of constant length on (M, g), where Z0 ∈ c, Z1 ∈ g1 and Z1 6= 0, and let k be the centralizer of Z (and Z1 ) in g1 . Then either any X ∈ g1 is a Killing field of constant length on (M, g), or the pair (g1 , k) is one of the following irreducible Hermitian symmetric pair: 1) 2) 3) 4)

(su(p + q), su(p) ⊕ su(q) ⊕ R), p ≥ q ≥ 1; (so(2n), su(n) ⊕ R), n ≥ 5; (so(p + 2), so(p) ⊕ R), p ≥ 5; (sp(n), su(n) ⊕ R), n ≥ 2.

In the latter four cases the center of k is a one-dimensional Lie algebra spanned by the vector Z1 . If a Killing vector field of constant length Z = Z0 + Z1 is as in cases 1)–4) in Theorem 4.17.4, we will say that it has Hermitian type. Recall that a vector U ∈ g is regular in g if its centralizer has minimal dimension among all the elements of g. For Z of Hermitian type, Z1 is not a regular element in g1 , since k is not commutative in cases 1)–4) of Theorem 4.17.4. Hence, we get Corollary 4.17.5. If Z1 is a regular element in the Lie algebra g1 (under the assumptions of Theorem 4.17.4), then every X ∈ g1 is a Killing vector field of constant length on (M, g).

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Moreover, the following result holds. Theorem 4.17.6. If Z = Z0 + Z1 + · · · + Zk is a regular element of g and has constant length on (M, g), then the following assertions hold: 1) g(gi , g j ) = 0 for every i 6= j, i, j = 1, . . . , k, at every point of M; 2) g(Z0 , gi ) = 0 for every i = 1, . . . , k, at every point of M; 3) every X ∈ gs , where gs = g1 ⊕ · · · ⊕ gk is a semisimple part of g, has constant length on (M, g). Moreover, if M is simply connected, then Z0 = 0 and (M, g) is a direct metric product of (Gi , µi ), 1 ≤ i ≤ k, where Gi is a connected and simply connected compact simple Lie group with the Lie algebra gi and µi is a bi-invariant Riemannian metric on Gi .

4.17.1 Examples of Killing Vector Fields of Constant Length In what follows, we will consider examples of cases 1)–4) in Theorem 4.17.4. We say that a Lie algebra g of Killing vector fields is transitive on a Riemannian manifold (M, g) if g generates the tangent space to M at every point x ∈ M, or, equivalently, the connected isometry group G with the Lie algebra g acts transitively on (M, g). Example 4.17.7. The following simple observation gives many examples of Killing vector fields of constant length on homogeneous Riemannian manifolds. Suppose that a Lie algebra g of Killing vector fields is transitive on a Riemannian manifold (M, g) and a Killing field Z on (M, g) commutes with g (in particular, Z could be in the center of g), then Z has a constant length on (M, g). Indeed, for any X ∈ g we have X · g(Z, Z) = 2g([X, Z], Z) = 0. Since g is transitive on M, we get g(Z, Z) = const. Example 4.17.8. Consider the irreducible symmetric space M = SU(2n)/Sp(n), n ≥ 2. It is known that the subgroup SU(2n − 1) · S1 ⊂ SU(2n) acts transitively on M, see e.g. [377] or [483]. Therefore, the Killing vector Z generated by S1 is a Killing vector field of constant length on M = SU(2n)/Sp(n). The centralizer k of Z in g1 is obviously su(2n − 1) ⊕ R, i.e. (g1 , k) = (su(2n), su(2n − 1) ⊕ R), see case 1) in Theorem 4.17.4. Example 4.17.9. Consider the sphere S2n−1 , n ≥ 2, as the symmetric space S2n−1 = SO(2n)/SO(2n − 1). It is known that the subgroup U(n) = SU(n) · S1 ⊂ SO(2n) acts transitively on S2n−1 . Therefore, the Killing vector Z generating S1 is a Killing vector field of constant length on S2n−1 . The centralizer k of Z in g1 is u(n) = su(n)⊕ R, i.e. (g1 , k) = (so(2n), su(n) ⊕ R), see case 2) in Theorem 4.17.4. Example 4.17.10. Consider the sphere S4n−1 , n ≥ 2, as the symmetric space S4n−1 = SO(4n)/SO(4n − 1). It is known that the subgroup Sp(n) · S1 ⊂ SO(4n) acts transitively on S4n−1 . Therefore, the Killing vector Z generated by S1 is a Killing vector

4.17 Homogeneous Riemannian Spaces with Killing Vector Fields of Constant Length

253

field of constant length on S4n−1 . On the other hand, the centralizer k of Z in g1 is su(2n) ⊕ R, since sp(n) ⊕ R ⊂ su(2n) ⊕ R (for details, see e.g. Section 6.11). Therefore, (g1 , k) = (so(4n), su(2n) ⊕ R), see case 2) in Theorem 4.17.4. Note that the full connected isometry group of the sphere Sn−1 with the canonical Riemannian metric gcan of constant curvature 1 is SO(n), but there are some subgroups G of SO(n) with transitive action on Sn−1 . It is interesting that Sn−1 is G-Clifford–Wolf homogeneous for some of them: S2n−1 = SO(2n)/SO(2n − 1) = U(n)/U(n − 1) is SO(2n)-Clifford–Wolf homogeneous and U(n)-Clifford–Wolf homogeneous; S4n−1 = Sp(n)/Sp(n − 1) = Sp(n) · S1 /Sp(n − 1) · S1 = SU(2n)/SU(2n − 1) is Sp(n)-Clifford–Wolf homogeneous, Sp(n) · S1 -Clifford–Wolf homogeneous, and SU(2n)-Clifford–Wolf homogeneous; S7 = Spin(7)/G2 is Spin(7)-Clifford–Wolf homogeneous; S15 = Spin(9)/Spin(7) is Spin(9)-Clifford–Wolf homogeneous (for details, see Section 6.11). Each of these results gives an example of a Killing vector field of constant length in the Lie algebra g corresponding to the group G. Example 4.17.11. There exists a Killing vector field of constant length Z on (S4p−1 , gcan ) such that Z ∈ su(2p) and the centralizer k of Z in su(2p) is su(p) ⊕ su(p) ⊕ R, i.e. (g1 , k) = (su(2p), su(p) ⊕ su(p) ⊕ R), see 1) in Theorem 4.17.4 and Proposition 7.11.7. Note, that we may take Z = diag(i, . . . , i , −i, . . . , −i ) ∈ su(2p). | {z } | {z } p

p

Example 4.17.12. There exists a Killing vector field of constant length Z on (S2(p+q)−1 , gcan ) such that Z ∈ u(p + q) and the centralizer k of Z in u(p + q) is u(p) ⊕ u(q), i.e. (g1 , k) = (su(p + q), su(p) ⊕ su(q) ⊕ R), see case 1) in Theorem 4.17.4 and Proposition 7.11.7. We may take Z = diag(i, . . . , i , −i, . . . , −i ) ∈ | {z } | {z } p

q

u(p + q). Note also that the Lie algebra u(p + q) has a 1-dimensional center, and Z0 = 0 if and only if p = q, when we get the previous example. If p 6= q, then the Killing vector field Z1 does not have a constant length on (S2(p+q)−1 , gcan ), see [364] for details. Example 4.17.13. There exists a Killing vector field of constant length Z on (S7 , gcan ) such that Z ∈ spin(7) ' so(7) and the centralizer k of Z in spin(7) ' so(7) is so(5) ⊕ R, i.e. (g1 , k) = (so(7), so(5) ⊕ R), see case 3) in Theorem 4.17.4 and Remark 6 in [83]. Example 4.17.14. There exists a Killing vector field of constant length Z on (S15 , gcan ) such that Z ∈ spin(9) and the centralizer k of Z in spin(9) ' so(9) is so(7) ⊕ R, i.e. (g1 , k) = (so(9), so(7) ⊕ R), see case 3) in Theorem 4.17.4 and Proposition 7.11.13. Example 4.17.15. There exists a Killing vector field of constant length Z on (S4n−1 , gcan ) such that Z ∈ sp(n) and the centralizer k of Z in sp(n) is u(n), i.e. (g1 , k) = (sp(n), su(n) ⊕ R), see case 4) in Theorem 4.17.4 and Proposition 7.11.5. Note, that we may take Z = diag(i, i, . . . , i) ∈ sp(n).

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4.17.2 Additional Properties of Killing Vector Fields of Constant Length We recall some properties of Killing vector fields of constant length in addition to those already mentioned in Chapter 3. We give some proofs in order to show the standard tools for studying such objects. In particular, they provide tools for proving Theorems 4.17.2, 4.17.4, and 4.17.6. Proposition 4.17.16. If a Killing vector field X ∈ g has constant length on (M, g), then for any Y, Z ∈ g the equalities g([Y, X], X) = 0 ,

(4.36)

g([Z, [Y, X]], X) + g([Y, X], [Z, X]) = 0

(4.37)

hold at every point of M. If G acts transitively on (M, g), then condition (4.36) implies that X has constant length. Moreover, condition (4.37) also implies that X has constant length for compact M and transitive G. Proof. If g(X, X) = const, then 2g([Y, X], X) = Y · g(X, X) = 0 at every point of M for every Y ∈ g, which proves (4.36). From this we have 0 = Z · g([Y, X], X) = g([Z, [Y, X]], X) + g([Y, X], [Z, X]) for any Z ∈ g, which proves (4.37). In the case of transitive action, we obviously get g(X, X) = const from the equality Y · g(X, X) = 2g([X,Y ], X) = 0, Y ∈ g. Note also that in the case of transitive action the condition (4.37) means that for any Y ∈ g we have g([Y, X], X) = C = const on M. If M is compact then there is an x ∈ M where g(X, X) has its maximal value. Obviously, gx ([Y, X], X) = 0. Therefore, C = 0 and X has constant length by the previous assertion. t u

Proposition 4.17.17. Let X be a Killing field and let Y be a Killing field of constant length on a Riemannian manifold (M, g). Then the formula R(X,Y )Y = −∇Y ∇Y X holds on M, where R(X, Z) := ∇X ∇Z − ∇Z ∇X − ∇[X,Z] . Proof. All integral curves of the field Y are geodesics by Proposition 3.3.4. On the other hand, the restriction of the Killing field X to any geodesic is a Jacobi field by Theorem 3.1.7. Hence, ∇Y ∇Y X + R(X,Y )Y = 0 on M. t u We will use the identification (4.34) of elements of Lie algebras g of G with corresponding Killing vector fields on (M, g) in the proof of the following Proposition 4.17.18. Let X,Y ∈ g such that g(X,Y ) = C = const on M. Then for every inner automorphism A of g we get g(A(X), A(Y )) = C at every point of M. In particular, if X ∈ g has constant length on (M, g), then A(X) has the same property. Proof. If La : M → M is the action of a ∈ G on M, then for any U ∈ g we have

4.17 Homogeneous Riemannian Spaces with Killing Vector Fields of Constant Length

255



d d −1 e dLa (U(x)) = a exp(tU) (x) = a exp(tU)a (a(x)) dt dt t=0 t=0
 d ^ = exp Ad(a)(U)t + o(t) (a(x)) = Ad(a)(U)(a(x)). dt t=0 Since La is an isometry of (M, g), we have ^ Ad(a)(Y ^ )) e Ye ) = gL (x) (dLa (X), e dLa (Ye )) = ga(x) (Ad(a)(X), gx (X, a at every point x ∈ M. Recall that the inner automorphism group of g coincides with the adjoint group of G, therefore we get the proposition. t u

4.17.3 Unsolved Questions Results of this section and [364] can be used for the classification of homogeneous Riemannian manifolds with Killing vector fields of some special kinds. We state some unsolved questions and problems in this direction. Problem 4.17.19. Classify homogeneous Riemannian spaces (G/H, g) with nontrivial Killing vector fields of constant length, where G is simple. Recall that normal homogeneous Riemannian spaces (G/H, g) with Killing fields of constant length, where G is simple, are classified in [490]. On the other hand, the set of G-invariant metrics on a space G/H could have any dimension even for simple G. Question 4.17.20. Let (M, g) be a homogeneous Riemannian manifold, G be its full connected isometry group, and Z ∈ g be a Killing vector field of constant length on (M, g). Does k, the centralizer of Z in g, act transitively on M? Note that for symmetric spaces (M = G/H, g) we have an affirmative answer to this question by Theorem 4.8.1. Of course, Question 4.17.20 is interesting even under the additional assumption that the group G is simple. There is a natural generalization of normal homogeneous spaces. Recall, that a Riemannian manifold (M = G/H, g), where H is a compact subgroup of a Lie group G and g is a G-invariant Riemannian metric, is called a geodesic orbit space if any geodesic γ of M is an orbit of 1-parameter subgroup of the group G. It is known that all normal homogeneous spaces are geodesic orbit (see Chapter 5). Hence, the following problem is natural. Problem 4.17.21. Classify geodesic orbit Riemannian spaces with nontrivial Killing vector fields of constant length.

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Recall the assertion of Theorem 5.5.1: If (M = G/H, g) is a geodesic orbit Riemannian space, g is its Lie algebra of Killing fields, and a is an abelian ideal in g, then every X ∈ a has constant length on (M, g). Hence it suffices to study only homogeneous spaces G/H with semisimple G in Problem 4.17.21. Note that the examples in Subsection 4.17.1 cover all possibilities in Theorem 4.17.4, excepting the case 3), where we have examples only for specific values of p. Question 4.17.22. Is there a homogeneous Riemannian space (M = G/H, g) with a Killing vector field of constant length Z ∈ g, corresponding to the case 3) in Theorem 4.17.4, i.e. (g1 , k) = (so(p + 2), so(p) ⊕ R), for p = 6 and p ≥ 8? For p = 5 and p = 7 see Examples 4.17.13 and 4.17.14.

Chapter 5

Manifolds With Homogeneous Geodesics

Abstract This chapter is devoted to geodesic orbit Riemannian spaces and manifolds. Geodesic orbit Riemannian manifolds are characterized by the condition that every geodesic is an orbit of some 1-parameter isometry subgroup (geodesics with this property are called homogeneous). It should be noted that normal homogeneous, naturally reductive, isotropy irreducible, symmetric, and weakly symmetric spaces constitute subclasses of geodesic orbit spaces. We consider some general properties of geodesic orbit spaces and some deep structural results. In particular, we prove that the isometry group of a geodesic orbit space has some important features. This, in particular, is related to the structure of the radical and the nilradical of the full connected isometry group. We get also some partial classification results. For instance, we discuss in detail the classification of simply connected compact geodesic orbit Riemannian spaces of positive Euler characteristic. A useful homogeneity criterion for a geodesic in a Riemannian manifold is obtained, which implies a new proof of the fact that all weakly symmetric spaces are geodesic orbit. In the final part of the chapter we consider various recent results related to homogeneous geodesics and geodesic orbit spaces.

5.1 Homogeneous Geodesics on Riemannian Manifolds Definition 5.1.1. A geodesic γ of a Riemannian manifold (M, g) is called homogeneous if it is an orbit of some 1-parameter subgroup γ(t) ⊂ Isom(M, g) (and, consequently, an integral curve of a Killing vector field, generating by this subgroup). Let (M, g) be a complete Riemannian manifold and let X be a Killing vector field on (M, g). Note that integral curves of the vector field X are exactly the orbits of a 1-parameter isometry group of (M, g), generated by X. Let us consider an arbitrary point x ∈ M, where X(x) 6= 0. Recall that, according to 3) in Theorem 3.1.7, an integral curve of the vector field X, passing through the point x, is a geodesic if and only if the point x is a critical point of the function y ∈ M 7→ gy (X, X). © Springer Nature Switzerland AG 2020 V. Berestovskii and Y. Nikonorov, Riemannian Manifolds and Homogeneous Geodesics, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-56658-6_5

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` B. VinThe following important result was established by B. Kostant [292], E. berg [447], O. Kowalski and L. Vanhecke [304]. Theorem 5.1.2. Let (M = G/H, g) be a homogeneous Riemannian space with a reductive decomposition g = h ⊕ p, Y ∈ h, X ∈ p. Then γ(t) = exp(t(X +Y ))(o), t ∈ R, is a geodesic if and only if one of the following conditions are fulfilled: 1) [X,Y ] = U(X, X); 2) ([Y, X], Z) = (X, [X, Z]p ) for all Z ∈ p; 3) ([X +Y, Z]p , X) = 0 for all Z ∈ p. Proof. It is clear that 1) is equivalent to 2) due to the formula (4.31). The equality 3) is equivalent to the equality 2), since ([Y, Z], X) = −(Z, [Y, X]) due to the skew symmetry of the operator adY on the space p. On the other hand, the equality 3) is fulfilled if and only if the curve γ(t) = exp(t(X + Y ))(o) is a geodesic by Theorem 3.1.7. t u Note that homogeneous geodesics correspond to relative equilibria of the geodetic flow, considered as a Hamiltonian system on the cotangent bundle (see e.g. papers [315, 440] and the references therein). Theorem 5.1.2 implies the following useful definition. Definition 5.1.3. Let (M = G/H, g) be a homogeneous Riemannian space. A vector of the type X +Y , where X 6= 0 ∈ p and Y ∈ h, is called a geodesic vector, if one of the following three equivalent conditions holds: 1) the curve γ(t) = exp(t(X +Y ))(o) is a geodesic; 2) ([X +Y, Z]p , X) = 0 for all Z ∈ p; 3) [X,Y ] = U(X, X), where U is defined by the equality (4.31). The following result is well known. Theorem 5.1.4 ([301]). Every homogeneous Riemannian manifold has at least one homogeneous geodesic passing through a given point. Proof. Let G be the full connected isometry group of a given Riemannian homogeneous manifold (M = G/H, g), where H is the isotropy subgroup at a given point x ∈ M. Consider the Lie algebras h ⊂ g of the Lie groups H ⊂ G and the Killing form B of g. The B-orthogonal complement m to h in g is Ad(H)-invariant (cf. Lemma 4.4.6). Suppose that the Riemannian metric g is generated by an inner product (·, ·) on m. Consider the operator A : m → m such that B(X,Y ) = (AX,Y ), X,Y ∈ m. By Lemma 4.4.6, the eigenspaces of A are Ad(H)-invariant and pairwise orthogonal both with respect to B and (·, ·). Take some nontrivial X from any eigenspace of A with non-zero eigenvalue λ . Then for any Y ∈ m we have λ (X, [X,Y ]m ) = B(X, [X,Y ]m ) = B(X, [X,Y ]) = −B([X, X],Y ) = 0 by the invariance of the Killing form B. Hence, exp(tX)(x), t ∈ R, is a homogeneous geodesic in (M = G/H, g) through the point x by Theorem 5.1.2.

5.1 Homogeneous Geodesics on Riemannian Manifolds

259

Now, let us suppose that the operator A has no non-zero eigenvalue. This means that B(m, m) = B(m, g) = 0. Since B is negative definite on h (cf. Lemma 4.4.1), we get that m coincides with the subalgebra {X ∈ g | B(X,Y ) = 0 ∀Y ∈ g} in g. This subalgebra is solvable since it is a subset of the radical r(g) of g due to (4.6). Therefore, [m, m] ⊂ m and [m, m] 6= m, and we may take a non-trivial X ∈ m such that (X, [m, m]) = 0, in particular, (X, [X,Y ]m ) = (X, [X,Y ]) = 0 for any Y ∈ m. Applying Theorem 5.1.2 again, we see that exp(tX)(x), t ∈ R, is a homogeneous geodesic. t u In the partial case of Lie groups, the result of Theorem 5.1.4 was obtained earlier in [279]. For compact homogeneous spaces this result was obtained in [315]. Special examples of 3-dimensional non-unimodular Lie groups admit exactly one homogeneous geodesic [333]. On the other hand, in a three-dimensional unimodular Lie group G endowed with a left-invariant metric g, there always exist three mutually orthogonal homogeneous geodesics through each point. Moreover, for generic metrics, there are no other homogeneous geodesics [333]. In the recent paper [494] by Z. Yan and L. Huang, it was proved that any homogeneous Finsler space admits a homogeneous geodesic. Interesting results on homogeneous geodesics in Riemannian manifolds of negative sectional curvature were obtained in [109, pp. 19–22], where such geodesics were called Killing geodesics. Interesting results on the behavior of geodesics in homogeneous Riemannian spaces are obtained in [400, 401, 402]. The rest of this section is based on [84].

5.1.1 Homogeneity Criterion for a Geodesic In the paper [84], the authors obtained the following homogeneity criterion for a geodesic in a Riemannian manifold. Theorem 5.1.5 ([84]). Let (M, g) be a complete Riemannian manifold, and let γ : R → M be a geodesic parameterized by arc length. Then the geodesic γ is homogeneous, i.e. an orbit of a 1-parameter isometry group, if and only if for any s ∈ R there is an isometry η(s) ∈ Isom(M, g), such that η(s)(γ(t)) = γ(t + s) for all t ∈ R. Proof. The necessity is obvious. Let us prove the sufficiency. Let us consider Gγ (s) := {a ∈ Isom(M, g) | a(γ(t)) = γ(t + s) ∀t ∈ R}, Gγ := ∪s∈R Gγ (s).

s ∈ R;

(5.1) (5.2)

We know that Gγ (s) 6= 0/ for all s ∈ R and Gγ = ∪s∈R Gγ (s). It is clear that Gγ (0) is compact, since it is the intersection of the isotropy subgroups in Isom(M, g) at all points of γ(R). It is clear that Gγ is a subgroup in Isom(M, g). The most crucial step in this proof is to prove that Gγ is a Lie group with respect to a natural topology. We prove this fact separately in Proposition 5.1.6 below.

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Let G0γ be the identity component of Gγ . Since Gγ is a Lie group, G0γ is also a Lie group, hence, a Lie (possibly, virtual) subgroup in Isom(M, g). Therefore, for any U ∈ g, where g is the Lie algebra of the group G0γ , we get that exp(tU) ⊂ G0γ . Clearly, there is a U ∈ g such that exp(tU) 6⊂ Gγ (0). Hence, exp(RU)(γ(0)) = γ(R) as required. t u Proposition 5.1.6. The group Gγ defined by (5.2) is a Lie group with respect to a natural topology. Proof. If γ is non-injective then the properties of Gγ (see (5.1) and (5.2)) imply that γ is periodic i.e. there is a smallest s > 0 such that γ(t + s) = γ(t) for all t ∈ R (recall that the motion of a point γ(t) along γ under the action of Gγ depends essentially only on its position in M, but not on the value of t). Then Gγ is compact, hence, a Lie subgroup in Isom(M, g). If γ(R) is a closed subset in (M, g), then Gγ is a closed subgroup of Isom(M, g). Therefore, Gγ is a Lie group by the Myers–Steenrod theorem 4.1.1 and Cartan’s theorem 2.1.32. From this point we suppose that γ(R) is not a closed subset in (M, g). In this case we give two proofs of the fact that Gγ is a Lie group. The first proof. Let us supply the group Gγ with the natural topology. Define the natural projection η : Gγ 7→ R, as follows: η(a) = s if and only if a(γ(t)) = γ(t + s) for all t ∈ R. Clearly, such s is unique. Let Πs be the parallel transport of Mγ(0) to Mγ(s) along γ with respect to the Levi-Civita connection of the Riemannian manifold (M, g). Consider any a ∈ Gγ and put s = η(a). Let us define the map a ∈ Gγ 7→ ϕ(a) ∈ O(Mγ(0) ) as follows: ϕ(a) = Πs−1 ◦ (T a)γ(0) . Let us supply Gγ with the topology induced by the product topology on the space R × O(Mγ(0) ) under the mapping a ∈ Gγ 7→ (η(a), ϕ(a)) ∈ R × O(Mγ(0) ), which is obviously injective. This topology makes Gγ a locally compact topological group. It is clear that η(a1 · a2 ) = η(a1 ) + η(a2 ) for a1 , a2 ∈ Gγ . Since η is an open surjective homomorphism of topological groups, then Ker(η) is a normal subgroup in Gγ . If fact, Ker(η) = Gγ (0) is compact. The following important result is known in the theory of topological groups: If G is a topological group and H is a closed invariant subgroup such that H and G/H are Lie groups, then G is a Lie group (see Theorem 1 in [224], Theorem 7 in [270], or pp. 153–154 in [343]). Since R = Gγ /Gγ (0), R and Gγ (0) are Lie groups, Gγ is a Lie group by the above result. The second proof. We will use the so-called development of the geodesic γ in p−1 (γ), where p : T M → M is the canonical projection. Let Πt 0 ,t be the parallel transport of Mγ(t) to Mγ(t 0 ) along γ with respect to the Levi-Civita connection of the Riemannian manifold (M, g). Obviously, Πt 0 ,t γ 0 (t) = γ 0 (t 0 ). A similar statement is valid for all T ψ, ψ ∈ Gγ .

5.1 Homogeneous Geodesics on Riemannian Manifolds

261

Let W = ∪t∈RWγ(t) , Wγ(t) = {w ∈ Mγ(t) : w ⊥ γ 0 (t)}. Choose any orthonormal basis e2 , . . . , en in Wγ(0) and define a basis e2 (t), . . . , en (t) of Wγ(t) by the equalities e j (t) = Πt,0 (e j ), j = 2, . . . , n. Then W is a smooth vector bundle over γ (the topology on γ is defined by the parameter t) with respect to the restriction of p to W , and Πt 0 ,t , T ψ, where ψ ∈ Gγ , are smooth linear isomorphisms on W . In addition, for any t,t 0 , s ∈ R and ψ ∈ Gγ (s), (T ψ)γ(t 0 ) ◦ Πt 0 ,t = Πt 0 +s,t+s ◦ (T ψ)γ(t) . Since the matrix of any parallel transport Πr0 , r with respect to the bases e2 (r), . . . , en (r) and e2 (r 0 ), . . . , en (r 0 ) is always the unit matrix, this implies that the matrix of (T ψ)γ(t) with respect to the bases e2 (t), . . . , en (t) and e2 (t + s), . . . , en (t + s) coincides with the matrix of (T ψ)γ(t 0 ) with respect to the bases e2 (t 0 ), . . . , en (t 0 ) and e2 (t 0 + s), . . . , en (t 0 + s). Denote this matrix, which is independent of t, by (ψ). The above argument shows that any element ψ ∈ Gγ (s) and the action of T ψ on W is  uniquely defined by the pair (s, (ψ)). In this notation, the product in the group T ψ, ψ ∈ Gγ is written as (s, (ψ))(s 0 , (ψ 0 )) = (s + s 0 , (ψ)(ψ 0 )).

(5.3)

Thus, if A(s) = {(ψ) : ψ ∈ Gγ (s)} then A(s + s 0 ) = A(s)A(s 0 ) = A(s 0 )A(s),

A(−s) = A(s)−1 ,

A(s) = As A(0)

(5.4)

for s, s 0 ∈ R, As ∈ A(s). Then A(s) is compact for any s ∈ R, since A(0) is compact, and A(−s) = A(0)−1 A−1 = A(0)A−1 . The last equalities implies that A(s) = AA(0) = A(0)A for any A ∈ A(s), s ∈ R, and A(s + t) = As A(t) for any As ∈ A(s) and s,t ∈ R. In particular, A(0) is a compact normal subgroup in the group A = {A(s), s ∈ R} ⊂ O(n − 1). Let d be the intrinsic metric on (M, g), δ any bi-invariant metric on the group O(n − 1), whose restriction to SO(n − 1) coincides with the intrinsic metric defined by a bi-invariant Riemannian metric on SO(n − 1) and δH the corresponding Hausdorff metric on the family of compact subsets in O(n − 1). Note that δH (A(s), A(0)) = δ (As , A(0)) = δ (A0 , A(s)) for any As ∈ A(s) and A0 ∈ A(0) because of the last equality in (5.4) and the right invariance of the metric δ . We state that δH (A(s), A(0)) → 0 when s 6= 0 and s → 0. Otherwise, there is a sequence ψsn ∈ Gγ (sn ) such that sn 6= 0, sn → 0, δ ((ψsn ), A(0)) > ε for some ε > 0 and all n ∈ N, and d(ψsn (x), ψ(x)) → 0 for some ψ ∈ Isom(M, g) uniformly on x ∈ B(γ(0), r), where 0 < r < ∞. Then ψ ∈ Gγ (0) but (ψ) ∈ / A(0), a contradiction. Therefore, for any s0 ∈ R and As0 ∈ A(s0 ),


δH A(s0 + s), A(s0 ) = δH As0 A(s), As0 A(0) = δH A(s), A(0) → 0 (5.5) if s 6= 0 and s → 0. The orthonormal basis e2 (t), . . . , en (t) in Wγ(t) permits us to consider W = ∪sWγ(s) as a direct product R × Wγ(0) and supply the last manifold with the direct product of the standard Riemannian metrics on its factors. Then W is isometric to ndimensional Euclidean space. If ψ ∈ Gγ (s) we define the Euclidean motion in W by the formula

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ψ(t, w) = s + t, (ψ)w ,

(5.6)

where the vector w ∈ Wγ(0) is considered as a vector-column with components in the base e2 (0), . . . , en (0). In consequence of (5.5) and compactness of the sets A(s) ⊂ O(n − 1), the correspondence ψ ∈ Gγ (s) → (s, (ψ)) and formulae (5.3), (5.6) gives the exact representation of Gγ as a closed, hence a Lie, subgroup of Isom(W ) for n-dimensional Euclidean space W . t u Remark 5.1.7. In fact, “the natural topology” on Gγ in the proof of Proposition 5.1.6 in all cases is nothing else than the so-called “leaf topology” on Gγ ⊂ Isom(M, g). It has a base consisting of connected components of intersections of open sets in Isom(M, g) with Gγ . This topology does not coincide with the induced topology if and only if Gγ is not a closed (that is, virtual) subgroup in Isom(M, g), in other words, when γ(R) is not a closed subset in (M, g). Therefore the equality Gγ = Gγ (0) × R is possible even in the case when Isom(M, g) is compact. This happens, for example, when γ is an irrational winding of a flat torus M of dimension ≥ 2. Remark 5.1.8. We gave two different proofs of Proposition 5.1.6 because they are entirely different: the first one is short but uses results on positive solution of Hilbert’s fifth problem, while the second proof uses nothing besides initial notions from Riemannian and Euclidean geometries, but it is longer than the first proof. Now, we discuss some results related to Theorem 5.1.5. Let (M, g) be a Riemannian manifold, and let γ : R → M be a geodesic parameterized by arc length. The geodesic γ is called invariant under the isometry a ∈ Isom(M, g) if there is a τ ∈ R such that a(γ(t)) = γ(t + τ) for all t ∈ R. If γ is a homogeneous geodesic, then, according to Proposition 5.1.6, the isometry group of all a ∈ Isom(M, g) such that γ is invariant under a is a Lie group Gγ defined by (5.2). Moreover, by the proof of Proposition 5.1.6, Gγ = Gγ (0) × R if γ(R) is not a closed subset in (M, g), where Gγ (0) = {a ∈ Isom(M, g) | a(γ(t)) = γ(t) ∀t ∈ R}. Geodesics invariant under a distinguished isometry are studied in various papers, see e.g. [244, 45] and references therein. In particular, K. Grove proved the following result. Theorem 5.1.9 ([244]). If M is compact and the isometry A ∈ Isom(M, g) has a nonclosed invariant geodesic then there are uncountably many A-invariant geodesics on M. The following recent result by V. Bangert should also be noted. Theorem 5.1.10 ([45]). Let γ be a non-closed and bounded geodesic in a complete Riemannian manifold (M, g) and assume that γ is invariant under an isometry A of (M, g), but is not contained in the set of fixed points of A. Then for some k ≥ 2, the geodesic line flow γ 0 corresponding to γ is dense in a k-dimensional torus T k embedded in T M and, in particular, every geodesic with initial vector in T k is Ainvariant.

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263

5.1.2 On the Closure of a Homogeneous Geodesic Now we are going to discuss important properties of an arbitrary homogeneous geodesic γ on a given Riemannian manifold (M, g). The main object of our interest is the closure of a given homogeneous geodesic. The following result is well known (see e.g. [234]). Proposition 5.1.11. Let ψ(s) = exp(sU), s ∈ R, be a 1-parameter group in a given Lie group G, where U is from g, the Lie algebra of G. Then ψ(R) is a connected abelian subgroup of G and there are three possibilities: 1) ψ(R) is a closed subgroup of G, diffeomorphic to R (ψ is a diffeomorphism); 2) ψ(R) is a closed subgroup of G, diffeomorphic to the circle S1 (ψ is a covering map); 3) ψ(R) is not a closed subgroup of G, and its closure is a torus T k of dimension k ≥ 2. Proof. It is clear that K, the closure of ψ(R) in G, is a connected abelian Lie group. If K = ψ(R), then we get either the first or the second possibility. Suppose that K 6= ψ(R). If K is not a torus, then K = R × K1 for some abelian connected group K1 . If π : R × K1 → R is the projection to the first factor, then π ◦ ψ : R → R is a Lie group isomorphism. Now, consider any point (a, b) ∈ R × K1 . There is a sequence (an , bn ) ∈ ψ(R) ⊂ R × K1 such that lim (an , bn ) = (a, b). It is clear that (an , bn ) ∈ n→∞

ψ([−M, M]) for some positive M ∈ R. Indeed, an → a as n → ∞ and π ◦
ψ is a Lie group automorphism of R, hence, the set (π ◦ ψ)−1 (an ) = ψ −1 (an , bn ) , n ∈ N, is bounded. Since ψ([−M, M]) is compact, then (a, b) ∈ ψ(R) and K = ψ(R), which is not possible. Hence, K is a torus T k of dimension k ≥ 1. Obviously, K 6= ψ(R) implies k ≥ 2. t u Now we consider the structure of the closure of homogeneous geodesics in Riemannian manifolds. Let (M, g) be a Riemannian manifold, and let γ : R → M be a geodesic parameterized by arc length, γ(0) = x ∈ M. Suppose that γ is homogeneous, i.e. there is a U in the Lie algebra g of the Lie group G = Isom(M, g) such that γ(t) = ψ(t)(x), where ψ(t) = exp(tU), t ∈ R. All orbits of any closed subgroup of Isom(M, g) on M are closed (Corollary 4.1.9). Therefore, for the closed set ψ(R) in G, the geodesic γ(R) is closed as a set in M. In case 2) of Proposition 5.1.11, the geodesic γ = γ(t) is periodic. The same is possible in case 3). For instance, one can find several examples of Killing vector fields of constant length that have both compact and non-compact integral curves (such curves are homogeneous geodesics), see e.g. Theorems 3.7.9, 4.8.9, 4.8.16, and Proposition 4.8.15. Now, we assume that ψ(R) is not closed in G. The following result had been proved in [244, Theorem 3.2] for any complete Riemannian manifold (M, g) with compact isometry group Isom(M, g).

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In [84], the following more general version is proved. Theorem 5.1.12. Let (M, g) be a complete Riemannian manifold and let γ : R → M be a homogeneous geodesic, i.e. it is an orbit of some 1-parameter isometry group ψ(t) = exp(tU), t ∈ R, for some U ∈ g, the Lie algebra of the Lie group G = Isom(M, g). Assume that γ(R) is a non-closed subset in M. Then γ(R) lies in a submanifold of (M, g) diffeomorphic to a l-dimensional torus T l with l ≥ 2 and any orbit of the group ψ(t), t ∈ R, through a point of T l is a geodesic which is dense in T l . Furthermore, the sectional curvature of any 2-plane, tangent to T l at γ(0) ∈ T l and containing γ 0 (0), is nonnegative. Proof. Put x = γ(0) ∈ M. Consider the closure K of ψ(R) in G and the closure N of γ(R) in M. By Proposition 5.1.11, K is a torus T k of dimension k ≥ 2. Note that N is invariant under the action of every ψ(t), t ∈ R. Indeed, if x0 ∈ N, then there are tn ∈ R such that γ(tn ) = ψ(tn )(x) → x0 as n → ∞. Hence, ψ(t)(γ(tn )) = ψ(t + tn )(x) → ψ(t)(x0 ) as n → ∞, i.e. ψ(t)(x0 ) ∈ N. Now, it is easy to see that N is invariant even under the action of K. Moreover, this action is transitive. Indeed, for every two points a, b ∈ N, there are a sequence tn such that an := ψ(tn )(x) → a as n → ∞ and a sequence sn such that bn := ψ(sn )(x) → b as n → ∞. It is clear that a sequence ϕn = ψ(sn − tn ) ∈ ψ(R) ⊂ K is such that ϕn (an ) = bn for all n. Passing, if necessary, to a subsequence, we can assume that ϕn → ϕ as n → ∞ for some ϕ ∈ K ⊂ Isom(M, g). Hence, b = lim bn = lim ϕn (an ) = ϕ(a), K acts transitively n→∞ n→∞ on N, and N is the orbit of K through the point x. Hence, N is a homogeneous space of a torus K = T k , therefore, N is a torus itself, i.e. N = T l with l ≥ 2 (since γ(R) is not a closed subset in M). e U), e where U e is a Note that x is a critical point for the function y ∈ M 7→ gy (U, Killing field, corresponding to U ∈ g. It is easy to see that the value of this function is constant on N. Hence, any point y of N is a critical point for the same function e through y is a homogeneous geodesic. Since the distance and the integral curve of U between the points ψ(t1 )(x) and ψ(t2 )(y) is equal to the distance between the points ψ(t1 + t)(x) and ψ(t2 + t)(y) for every t,t1 ,t2 ∈ R, the geodesic ψ(t)(y), t ∈ R, is dense in N for all y ∈ N. Finally, let us prove the assertion on the sectional curvature. Due to the previous assertion, we may (without loss of generality) consider only points of the geodesic γ. Suppose that p = γ(s) for some s ∈ R. Let Yp be a unit tangent vector to N = ep . We define the vector field Y along γ by setting Y (t) = T l at p orthogonal to U T (ψ(t)) p (Yp ). By the construction of Y and the previous discussion, Y is obtained by a 1-parameter geodesic variation of γ, i.e. Y is a Jacobi field, therefore, ∇2Y + R(Y, γ 0 )γ 0 = 0.

(5.7)

Taking the inner product on both sides of (5.7) with Y we obtain that the sectional e on γ) satisfies curvature of the 2-plane spanned on Y and γ 0 (the latter is parallel to U 2 0 0 the equality Ksec = −g(∇ Y,Y )/g(γ , γ ). Since g(Y,Y ) is constant along γ, we have

5.1 Homogeneous Geodesics on Riemannian Manifolds

2g(∇Y,Y ) = 0=

265

d g(Y,Y ) = 0, dt

d g(∇Y,Y ) = g(∇2Y,Y ) + g(∇Y, ∇Y ), dt

hence, Ksec = g(∇Y, ∇Y )/g(γ 0 , γ 0 ) ≥ 0.

t u

Remark 5.1.13. Let Kx be the isotropy subgroup of K at the point x ∈ N ⊂ M. Then N = K/Kx . It is interesting to study explicit examples with non-discrete Kx . Note that, for any a ∈ Kx (and, moreover, for any a from the isotropy subgroup of
Isom(M, g) at the point x), the orbit of the group ξ (t) = exp t(Ad(a)(U)) , t ∈ R, through x is a geodesic. We have the following obvious corollary. Corollary 5.1.14. If a homogeneous geodesic γ is not bounded in (M, g) then γ(R) is a closed subset in M. Theorem 5.1.12 leads to the following natural questions (we use the above notation). Question 5.1.15. Is it true that N = T l is totally geodesic in (M, g)? Question 5.1.16. Is it true that for any W ∈ Lie(K = T k ), the orbit of the group exp(tW ), t ∈ R, through every point of N = T l is a geodesic? The above two questions are especially interesting for the case of homogeneous Riemannian manifolds. It is well known that these questions have positive answer for symmetric spaces M. In this case, the closure of a given geodesic is either the geodesic itself or a totally geodesic torus N = T l , where 1 ≤ l ≤ rk M and rk M is the rank of the symmetric space M. In the next subsection we will show that generally both these questions have negative answers even for the case of Lie groups with left-invariant Riemannian metrics. There is another proof of the last assertion in Theorem 5.1.12, which gives additional information connected with Questions 5.1.15 and 5.1.16. It is based on the famous Gauss formula and equation. We shall use corresponding results from [291], with a slightly different notation. Let N be any smooth submanifold of a smooth Riemannian manifold (M, g) with the induced metric tensor g0 and Levi-Civita connection ∇0 . If X,Y are vector fields on N then the Gauss formula is ∇X Y = ∇0X Y + α(X,Y ), where α is the second fundamental form of N (in M) and α(X,Y ) is orthogonal to N. By [291, Proposition 4.5] we have

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Proposition 5.1.17. Let X and Y be a pair of orthonormal vectors in Nx , where x ∈ N. For the 2-plane X ∧Y , spanned by X and Y , we have KM (X ∧Y ) = KN (X ∧Y ) + g(α(X,Y ), α(X,Y )) − g(α(X, X), α(Y,Y )), where KM (respectively, KN ) denotes the sectional curvature in M (respectively, N). Under the conditions of Theorem 5.1.12, we can take X = γ 0 (0) and any unit vector Y at x, tangent to N = T l and orthogonal to X (or even corresponding parallel vector fields X, Y on N = T l ). Then α(X, X) = 0 and we immediately get the last statement of Theorem 5.1.12 and the following corollary. Corollary 5.1.18. If (M, g) has negative sectional curvature then γ(R) is a closed subset in M. In the general case, N is a totally geodesic submanifold in (M, g) if and only if α ≡ 0 on N. Therefore, using in our case the above parallel vector fields X, Y on N = T l for l ≥ 2, we get the following corollary. Corollary 5.1.19. If (M, g) has positive sectional curvature and l ≥ 2 then N = T l is not a totally geodesic submanifold in (M, g). There are Riemannian manifolds of positive sectional curvature that have homogeneous geodesics with the closure N = T l for l ≥ 2. For instance, we can consider the sphere S3 = U(2)/U(1) supplied with U(2)-invariant Riemannian metrics (the Berger spheres), that are sufficiently close (but not equal) to the metric of constant curvature in order to have positive sectional curvature, see [501, pp. 587–589]. Such metrics are naturally reductive, hence geodesic orbit, i.e. all their geodesics are homogeneous (see Section 5.3). It is easy to choose a geodesic that has a 2dimensional closure. Indeed, there is a countable number of periodic geodesics through a given point. By Proposition 4.6.7, any self-intersecting geodesic in a homogeneous Riemannian manifold is periodic. Then for all other geodesics, the closure N = T l is such that l = 2 (since U(2) does not contain a 3-dimensional torus), see [500, Example 1] for details. Similar examples could be constructed for the sphere S2n−1 = U(n)/U(n − 1) with any n ≥ 2. By Corollary 5.1.19, they provide counterexamples to Question 5.1.15, hence, Question 5.1.16. Note that the Berger spheres are weakly symmetric (see Section 5.3). This shows that the behavior of geodesics in weakly symmetric spaces is more complicated than in symmetric spaces.

5.1.3 Examples of Homogeneous Geodesics Here we consider some examples of homogeneous geodesics on Riemannian manifolds. We restrict ourself to Lie groups with left-invariant Riemannian metrics. Let us show that a small deformation of a given inner product could seriously change the set of homogeneous vectors.

5.1 Homogeneous Geodesics on Riemannian Manifolds

267

Example 5.1.20. Suppose that a compact semisimple Lie algebra g is supplied with the minus Killing form h·, ·i. For any nontrivial U ∈ g, we have a h·, ·i-orthogonal decomposition g = Cg (U) ⊕ [U, g], where Cg (U) is the centralizer of U in g. Indeed, the operator ad(U) is skew symmetric with respect to h·, ·i, hence X ∈ Cg (U) implies hX, [U, g]i = 0 and the converse is also true. Let us take any vector V ∈ g that is not orthogonal both to U and to [U, g] with respect to h·, ·i. Note that a generic vector in g has this property. Let us define the following inner product: (X,Y ) = hX,Y i + α · hX,V i · hY,V i, where α > 0. If hX,V i = 0, then (X,Y ) = hX,Y i for every Y ∈ g, hence such X is a geodesic vector for the inner product (·, ·). On the other hand, there is a W ∈ g such that h[U,W ],V i 6= 0, hence, ([W,U],W ) = h[W,U],W i + α · h[W,U],V i · hU,V i = α · h[W,U],V i · hU,V i 6= 0, U is not a geodesic vector for (·, ·). If rk g ≥ 2, then we can choose the vector X ∈ Cg (U) with the property hX,V i = 0. Moreover, let us fix a Cartan (i.e. maximal abelian) subalgebra t ⊂ g. We may consider the vector X ∈ t such that it is a regular vector in g (Cg (X) = t) and the closure of the 1-parameter group exp(sX), s ∈ R, coincides with a maximal torus T := exp(t) in G. If U ∈ t with hU, Xi = 0 and V ∈ g such that hV, Xi = 0, hV,Ui 6= 0 and hV, [U, g]i 6= 0, then X is (U is not) a geodesic vector for (·, ·) as above. Hence, the closure of a homogeneous geodesic exp(sX), s ∈ R, is the torus T , but the orbit exp(sU), s ∈ R, is not geodesic, although U ∈ t = Lie(T ). This gives the negative answer to Question 5.1.16. For the Levi-Civita connection (elements of g are considered as left-invariant vector fields on G, see e.g. [106]) we have 2(∇X U,Y ) = 2(∇U X,Y ) = ([Y,U], X) + (U, [Y, X]) = h[Y,U], Xi + hU, [Y, X]i + α · h[Y,U],V i · hX,V i + α · hU,V i · h[Y, X],V i = α · hU,V i · h[Y, X],V i 6= 0 for some Y ∈ [X, g] = [t, g]. Indeed, [U, g] ⊂ [X, g] (due to Cg (X) = t ⊂ Cg (U)), hence there is a Y ∈ g such that h[X,Y ],V i 6= 0. This implies the required result. We may assume also (without loss of generality) that this Y is in [X, g] = [t, g] (since we have a h·, ·i-orthogonal decomposition g = Cg (X) ⊕ [X, g] = t ⊕ [X, g] and the t-component of Y commutes with X). Further, since the subspace [X, g] = [t, g] is h·, ·i-orthogonal to Cg (X) = t, the torus T is not totally geodesic with respect to the left-invariant metric generated by the inner product (·, ·). This gives a negative answer to Question 5.1.15. Now we are going to study geodesic vectors for some left-invariant Riemannian metrics on the Lie group G = SU(2) × SU(2). Example 5.1.21. Let us consider the basis Ei , i = 1, . . . , 6, in g = su(2) ⊕ su(2) such that the first three vectors are in the first copy of su(2) while other vectors are in the second copy of su(2) and

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[E1 , E2 ] = E3 ,

[E2 , E3 ] = E1 ,

[E1 , E3 ] = −E2 ,

[E4 , E5 ] = E6 ,

[E5 , E6 ] = E4 ,

[E4 , E6 ] = −E5 .

It is clear that this basis is orthonormal with respect to the bi-invariant Riemannian metric h·, ·i = −1/2 · B(·, ·), where B is the Killing form of g. Let us consider a non-degenerate (6 × 6)-matrix A = (ai j ) and the inner product (·, ·) such that the vectors Fi = ∑6j=1 ai j E j , i = 1, . . . , 6, constitute a (·, ·)-orthonormal basis. It is easy to see that there is a one-to-one correspondence between the set of all inner products on g and the set of lower triangle matrices A = (ai j ) with positive elements on the principal diagonal. Let us consider the inner product (·, ·)d that is generated by the matrix   100000 0 1 0 0 0 0   0 0 1 0 0 0  A=  1 0 0 1 0 0 ,   1 0 0 0 1 0 d1100d where d > 0. Direct calculations (that could be performed using any computer algebra system) show that the vector V = ∑6i=1 vi Ei is a geodesic vector for the inner product (·, ·)d if and only if one of the following conditions holds: • v1 ∈ R, v2 = v3 = 0, v4 ∈ R, v5 ∈ R, v6 = d · v1 ; • v1 ∈ R, v2 ∈ R, v3 ∈ R, v4 ∈ R, v5 = 2v1 − v4 , v6 = d · v1 + v2 + v3 ; • v1 = d · v3 , v2 = v3 , v3 ∈ R, v4 = d · v3 , v5 = d · v3 , v6 ∈ R. Hence, the set of geodesic vectors for the metric (·, ·)d is the union of one 2dimensional, one 3-dimensional, and one 4-dimensional linear subspace in g = su(2) ⊕ su(2). The set of vectors V = ∑6i=1 vi Ei with v2 = v3 = v4 = v5 = 0 determines a Cartan subalgebra t in g. We see that V ∈ t is a geodesic vector if either v1 = 0, or v6 = d ·v1 . It is clear that in the latter case for any irrational d we get the vector V such that the closure of the 1-parameter group exp(sV ), s ∈ R, coincides with a maximal torus T := exp(t) in G. Therefore, we again get a negative answer to Question 5.1.16. If D is the discriminant of the characteristic polynomial of the Ricci operator Ricd of the metric (·, ·)d , then D=


318 · 172 1 + o(d) as 8 44 2 ·d

d → 0.

Therefore, for sufficiently small positive d, Ricd has 6 distinct eigenvalues. This implies that the full connected isometry group of the metric (·, ·)d is G = SU(2) × SU(2). Indeed, if the dimension of full isometry group is > 6, then the isotropy subgroup is non-discrete and has (due to the effectiveness) not only one-dimensional (hence, trivial) irreducible subrepresentations in the isotropy representation, hence

5.1 Homogeneous Geodesics on Riemannian Manifolds

269

Ricd should have some coinciding eigenvalues. Therefore, every operator ad(X), X ∈ g = su(2) ⊕ su(2), is not skew symmetric with respect to (·, ·)d for sufficiently small d > 0.

5.1.4 A Special Quadratic Mapping We have discussed that the set of geodesic vectors on a given Lie algebra depends on the chosen inner product (·, ·). Let us consider this problem in a more general context. Let g be a Lie algebra, then every inner product (·, ·) on g determines a special quadratic mapping ξ = ξ(·,·) : g 7→ g (5.8) as follows: For any X ∈ g we put ξ (X) := V , where V is a unique vector in g with the equality ([X,Y ], X) = (V,Y ) for all Y ∈ g. This mapping is well known, see e.g. (4.31) on p. 229 for the more general case of homogeneous Riemannian spaces (where one should put ξ (X) := −U(X, X)). The set of zeros of the mapping ξ(·,·) is exactly the set of geodesic vectors in g with respect to the inner product (·, ·). In particular, this set always contains the center of the Lie algebra g. For any bi-invariant inner product (·, ·), the map ξ(·,·) is obviously trivial (ξ(·,·) ≡ 0). On the other hand, this map could have unexpected properties for some special inner products (·, ·). Example 5.1.22. Let us consider a basis Ei , i = 1, 2, 3, in g = su(2) such that [E1 , E2 ] = E3 ,

[E2 , E3 ] = E1 ,

[E1 , E3 ] = −E2 .

Fix some positive numbers a, b, c and consider the inner product (·, ·) on g = su(2) that has an orthonormal basis F1 = aE1 , F2 = bE2 , F3 = cE3 . Direct calculations give us an explicit form of the mapping ξ(·,·) (we use coordinates of all vectors with respect to the original basis E1 , E2 , E3 ):  
a(b − c) b(c − a) c(a − b) ξ(·,·) x1 , x2 , x3 = x2 x3 , x1 x3 , x1 x2 . bc ac ab It is easy to see that for a 6= b 6= c 6= a, any geodesic vector should be a multiple of one of the vectors E1 , E2 , E3 . On the other hand, for a = b 6= c, geodesic vectors are exactly the vectors either with x3 = 0 or with x1 = x2 = 0. For a = b = c we have a bi-invariant inner product. Obviously, x1 x2 > 0 and x1 x3 > 0 imply x2 x3 > 0, hence, ξ(·,·) is not surjective. This correlates with Theorem 8 in [24], stating (in particular) that any surjective quadratic mapping q : R3 → R3 has no non-trivial zero. It could be an interesting problem to study general properties of the quadratic mapping (5.8) for general Lie algebras and general inner products. Below we consider some results in this direction. It should be recalled that the mapping ξ(·,·) al-

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ways has at least one non-trivial zero according to [279]. Nevertheless, this property does not imply directly the non-surjectivity of ξ(·,·) for dim g ≥ 6, see Example 3 and the corresponding discussion in [24]. Recall that a Lie algebra g is unimodular if trace ad(Y ) = 0 for all Y ∈ g (here, as usual, the operator ad(Y ) : g → g is defined with ad(Y )(Z) = [Y, Z]). All compact and semisimple Lie algebras are unimodular. Any Lie algebra g contains the unimodular kernel, the maximal unimodular ideal, that could be described as follows: u := {Y ∈ g | trace ad(Y ) = 0}. For a non-unimodular Lie algebra g, the ideal u has codimension 1 in g. The following result is also well known (compare with the discussion of (4.33) on p. 229). Proposition 5.1.23. Let g be a Lie algebra supplied with an inner product (·, ·), dim g = n. Then the quadratic map ξ(·,·) (see (5.8)) has the following properties: 1) For a given Y ∈ g, the operator ad(Y ) is (·, ·)-skew symmetric if and only if (ξ (X),Y ) = 0 for all X ∈ g. 2) Let us define ∆ = ∆(·,·) ∈ g by the equation (∆ ,Y ) = trace ad(Y ), Y ∈ u. Then ∆ is (·, ·)-orthogonal to the unimodular kernel u of g. In particular, ∆ = 0 if g is unimodular. 3) ∆ = − ∑ni=1 ξ (Ei ) for any (·, ·)-orthonormal basis Ei , i = 1, . . . , n, in g. Proof. Let us prove the first assertion. Recall that the operator ad(Y ) is (·, ·)-skew symmetric if and only if ([X,Y ], X) = 0 for all X ∈ g. By the definition of ξ(·,·) we get (ξ (X),Y ) = ([X,Y ], X) for every X ∈ g, which proves the required assertion. The second assertion follows directly from the definition of the unimodular kernel u. Let us prove the third assertion. Fix any Y ∈ g. By the definition of ξ(·,·) we have (ξ (Ei ),Y ) = ([Ei ,Y ], Ei ). Therefore, ! n

∑ ξ (Ei ),Y

i=1

n

n

= ∑ (ξ (Ei ),Y ) = − ∑ ([Y, Ei ], Ei ) = − trace ad(Y ) = −(∆ ,Y ), i=1

which proves the required result.

i=1

t u

We hope that the further study of the mapping (5.8) will lead to a deeper understanding of the set of geodesic vectors for general metric Lie algebras.

5.2 Geodesic Orbit Spaces Definition 5.2.1. A Riemannian manifold (M, g) is called a manifold with homogeneous geodesics or a geodesic orbit manifold (shortly, GO-manifold) if any geodesic γ of M is an orbit of some 1-parameter subgroup of the full isometry group of (M, g) (i.e. γ is a homogeneous geodesic).

5.2 Geodesic Orbit Spaces

271

It is obvious that any connected geodesic orbit Riemannian manifold is homogeneous. Since a given Riemannian manifold can admit transitive actions of different isometry groups, it is useful to consider the following Definition 5.2.2. A Riemannian homogeneous space (G/H, g), where H is a compact subgroup of the Lie group G and g is a G-invariant Riemannian metric, is called a geodesic orbit space (shortly, GO-space) if any of its geodesics γ is an orbit of some 1-parameter subgroup of the group G. In this case, the invariant metric g is called a geodesic orbit metric (shortly, GO-metric). Hence, a Riemannian manifold (M, g) is a geodesic orbit Riemannian manifold if it is a geodesic orbit space with respect to its full connected isometry group. The notions of geodesic orbit manifolds and geodesic orbit spaces were introduced by O. Kowalski and L. Vanhecke in [304], who initiated a systematic study of such spaces. Among the nice geometric properties of geodesic orbit manifolds, we recall the following one. Theorem 5.2.3 ([302]). For every geodesic orbit manifold, the local geodesic symmetries are volume preserving up to sign. Riemannian manifolds with the property that the local geodesic symmetries are volume preserving up to sign are known as “D’Atri spaces”. A detailed survey of this interesting class of Riemannian manifolds can be found in [306]. Note also the following result (for details, see [276]): Theorem 5.2.4 ([276]). The geodesic flow on an arbitrary geodesic orbit space is completely integrable in the non-commutative sense. As usual, the symbol Mx denotes the tangent space to the manifold M at the point x ∈ M. Lemma 5.2.5. Let (M, g) be a Riemannian manifold and g be the Lie algebra of its Killing vector fields. Then (M, g) is a geodesic orbit manifold if and only if for every point x ∈ M and every vector v ∈ Mx , there exists a Killing vector field X ∈ g such that X(x) = v and x is a critical point of the function y ∈ M 7→ gy (X, X). If (M, g) is homogeneous, then the latter condition is equivalent to the following one: for any Killing field Y ∈ g the equality gx ([Y, X], X) = 0 holds. Proof. According to 3) in Theorem 3.1.7, an integral curve of X, passing through the point x ∈ M, is geodesic if and only if x is a critical point of the function y ∈ M 7→ gy (X, X). If the manifold (M, g) is homogeneous (i.e. the full isometry group of (M, g) is transitive on M), then, according to Theorem 3.1.4, this condition is equivalent to the equality Y · g(X, X)|x = 2gx ([Y, X], X) = 0 for all Y ∈ g.

t u

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Let (M = G/H, g) be an arbitrary homogeneous Riemannian space. Let us fix a reductive decomposition g = h⊕p

(5.9)

of the Lie algebra g = LG, where p is naturally identified with the tangent space Mo , o = eH, and an inner product (·, ·) on p corresponds to go (·, ·) on Mo . For every subspace m ⊂ p, we will denote by m⊥ its orthogonal complement in p with respect to (·, ·). Since we identify the elements X,Y ∈ p with the corresponding Killing fields on (M, g), the formulas (4.31) and (4.32) are satisfied. As a direct consequence of Theorem 5.1.2 and Definition 5.1.3, we obtain Theorem 5.2.6. A homogeneous Riemannian space (M = G/H, g) is a geodesic orbit space if and only if for every X ∈ p, there exists a Z ∈ h such that X + Z is a geodesic vector, i.e. ([X + Z,Y ]p , X) = 0 for all Y ∈ p. Remark 5.2.7. If (M = G/H, g) is a homogeneous Riemannian manifold with a naturally reductive metric g with respect to p, then for every X ∈ p, the vector X itself is a geodesic vector (i.e. we can take Z = 0 in this case). Hence, naturally reductive spaces are the simplest geodesic orbit spaces in this sense. Theorem 5.2.6 shows that the property of being a GO-space is completely determined by the reductive decomposition (5.9) and the Euclidean metric go on p. In particular, if (M = G/H, g) is a geodesic orbit space, then each of its Riemannian coverings (M 0 = G0 /H 0 , g0 ) is also geodesic orbit. Moreover, the direct metric product of Riemannian spaces is a GO-space if and only if each of the factors in this product has the same property. Example 5.2.8. A Lie group G supplied with a left-invariant Riemannian metric g is a geodesic orbit space if and only if the metric g is bi-invariant, i.e. the corresponding inner product (·, ·) on the Lie algebra g is AdG -invariant. In particular, the Lie algebra g must be compact in this case. Indeed, the condition for (G, g) to be a GO-space could be rewritten in the form 0 = (X, [X,Y ]) = −(adY X, X) = 0 (since the isotropy subalgebra is trivial). This proves that the inner product (·, ·) is AdG -invariant. It should be noted that a compact Lie group G can also admit non-bi-invariant left-invariant metrics g with homogeneous geodesics. But in this case the corresponding GO-spaces have the form L/H, where the group L contains G as a proper subgroup (in other words, G is not the full isometry group of such metrics), see, for example, [169] for details. Theorem 4.15.13 shows that similar statements are true for some noncompact Lie groups. Further, we consider several useful properties of general geodesic orbit spaces. For every subspace l ⊂ p and every vector V ∈ g we will use the symbol adVl to denote the restriction of the operator adV to l, i.e. adVl : l → l, adVl (X) = [V, X]l . Here, Zl means the (·, ·)-orthogonal projection to l of the p-component (with respect to (5.9)) of a vector Z ∈ g.

5.2 Geodesic Orbit Spaces

273

Lemma 5.2.9. Let (M = G/H, g) be a geodesic orbit space, m be an Ad(H)invariant subspace in p, and let m⊥ be its (·, ·)-orthogonal complement in p. Then ⊥ for every V ∈ m, the operator adVm is skew symmetric, which is equivalent to the ⊥ ⊥ ⊥ inclusion U(m , m ) ⊂ m . If, moreover, [h, m] = 0, then the operator adVp is also skew symmetric. Proof. For any X ∈ m⊥ , there is a Z ∈ h such that ([Z + X,Y ]p , X) = 0 for all Y ∈ p (see Theorem 5.2.6). Consequently, 0 = ([Z + X,V ]p , X) = ([Z,V ]p , X) + ([X,V ]p , X) = ([X,V ]p , X), since [Z,V ] ∈ m and X ∈ m⊥ . If, moreover, [h, m] = 0, then the same argument is valid for any X ∈ p (in this case [Z,V ] ⊂ [h, m] = 0). t u We note a remarkable property of the isometry group of a GO-space (cf. [230]). Lemma 5.2.10. For every geodesic orbit space (M = G/H, g), the Lie group G is unimodular. Proof. Here we give a proof from [360], which is more direct than the original one in [230]. Suppose that the Lie algebra g = LG of the Lie group G is not unimodular and consider its proper subspace u = {X ∈ g | trace(adX ) = 0}. Note that h ⊂ u (for every Y ∈ h, the adjoint operator adY : X 7→ [Y, X] on h is skew-symmetric with respect to every ad(h)-invariant inner product on h, moreover, adY |p are skew-symmetric with respect to a given ad(h)-invariant inner product on p). Since ad[X,Y ] = [adX , adY ]

and

trace(ad[X,Y ] ) = trace([adX , adY ]) = 0,

[u, g] ⊂ [g, g] ⊂ u, hence, u is an ideal of g. Consider p1 = p ∩ u and p2 = p⊥ 1, the (non-trivial) g-orthogonal complement to p1 in p. Since u is an ideal of g and the metric g is ad(h)-invariant, p1 and p2 are ad(h)-invariant. On the other hand, [h, p2 ] ⊂ [g, g] ⊂ u. Therefore, [h, p2 ] = 0. Now, consider any non-trivial Y ∈ p2 . By our construction, trace(adY ) 6= 0. On the other hand, by Theorem 5.2.6, for every X ∈ p, there is a Z ∈ h such that ([Z + X,Y ]p , X) = 0. Since [h, p2 ] = 0, we get ([X,Y ]p , X) = 0 (see also Lemma 5.2.9), which implies trace(adY ) = 0, a contradiction. t u Next, we will study geodesic vectors in more detail. It should be noted that in the general case, for a given X ∈ p, a geodesic vector X + Z (where Z ∈ h) is not uniquely defined (see Definition 5.1.3). Let us fix a homogeneous Riemannian space (M = G/H, g) and the decomposition (5.9). For a given X ∈ p, we define (according to [429], [304], and [300]) the sets

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5 Manifolds With Homogeneous Geodesics

qX = {A ∈ h | [A, X] = 0}, NX = {B ∈ h | [B, qX ] ⊂ qX },

c0X

= {B ∈ h | [B, qX ] = 0}.

(5.10) (5.11)

J. Szenthe obtained in his paper [429] some important results for manifolds with linear connections, which constitute the content of the following theorem. For the Riemannian case the exposition is adapted in [304]. Theorem 5.2.11. In the above notation, the following assertions hold. 1) If X +Y is a geodesic vector, where X ∈ p and Y ∈ h, then the set of all geodesic vectors with the same p-component has the form X +Y + qX . 2) If X +Y is a geodesic vector with the p-component X, then Y ∈ NX . 3) Consider the Q-orthogonal decomposition NX = qX ⊕ cX with respect to any Ad(H)-invariant inner product Q on h. Then cX ⊂ c0X . 4) If there exists a geodesic vector X + Y with the p-component X, then there is a geodesic vector of the form X +Y 0 , where Y 0 ∈ cX ⊂ c0X . Proof. 1) The condition that X +Y is a geodesic vector can be rewritten in the form [X,Y ] = U(X, X), which implies the required assertion. 2) We should prove the inclusion [Y, qX ] ⊂ qX , which is equivalent to the equality [[X,Y ], qX ] = [[X,Y ], qX ] + [[qX , X],Y ] = −[[Y, qX ], X] = 0 (we have used the Jacobi identity and the definition of qX ). Since [X,Y ] = U(X, X) ∈ p, it suffices to show that for any Z ∈ p and any A ∈ qX the equality 0 = ([A, [X,Y ]], Z) = ([A,U(X, X)], Z) = −(U(X, X), [A, Z]) = −([[A, Z], X]p , X) holds (here we have used the definition of the form U and the fact that the inner product (·, ·) is ad(h)-invariant). Now, the Jacobi identity implies [[A, Z], X]p = −[[Z, X], A]p − [[X, A], Z]p = −[[Z, X]p , A], since [X, A] = 0. Therefore, it suffices to check the identity 0 = ([[Z, X]p , A], X). Since A ∈ qX ⊂ h, we get ([[Z, X]p , A], X) = ([Z, X]p , [A, X]) = 0. Therefore, the assertion is proved. 3) Obviously, qX is a Lie algebra and an ideal in the Lie algebra NX (the latter easily follows from the Jacobi identity). Therefore, since the inner product Q is Ad(H)-invariant, cX is also an ideal in NX (see Proposition 2.5.3), consequently, [cX , qX ] = 0, i.e. cX ⊂ c0X . 4) Let Y 0 be the Q-orthogonal projection of the vector Y ∈ NX on cX (⊂ c0X ). Then Y −Y 0 ∈ qX , and, according to 1), the vector X +Y 0 is a geodesic vector. t u Now, let us assume that the homogeneous Riemannian space (M = G/H, g) is geodesic orbit. Then for every non-trivial vector X ∈ p, there is Y ∈ h such that the vector X +Y is a geodesic vector, and Y ∈ NX according 2) of Theorem 5.2.11. Let ξ (X) be the Q-orthogonal projection of the vector Y on cX , as in the proof of the item 4) in Theorem 5.2.11. Let us put also ξ (0) = 0. In other words, ξ (X) is an unique element of the affine subspace Y + cX in h which has the smallest norm with respect to Q. Hence, we obtain the mapping

5.2 Geodesic Orbit Spaces

275

ξ :p→h

(5.12)

Since Q is Ad(H)-invariant, the geodesic graph defined in this way is obviously Ad(H)-equivariant, i.e. ξ (Ad(a)(X)) = Ad(a)(ξ (X)) for all a ∈ H and X ∈ p. Moreover, by construction, the vector X + ξ (X) is a geodesic vector for all non-trivial X ∈ p. The mapping (5.12) is called a geodesic graph for the geodesic orbit space (M = G/H, g) [429, 304]. Obviously, the geodesic graph is homogeneous of degree one. Hence, it is linear if and only if it is differentiable at the origin. There are some immediate observations (independently of the choice of the inner product (·, ·) on p): 1) If there is exactly one element Y ∈ h for which X + Y is a geodesic vector, then qX = 0. 2) If X ∈ p itself is a geodesic vector, then ξ (X) = 0. We note that the dependence of the construction of a geodesic graph on the choice of a reductive decomposition is minimal (a geodesic graph is easily recalculated for another decomposition). It should also be noted that this circumstance allows us to avoid considering various reductive decompositions for the pair of Lie algebras (g, h) when checking the property of a given invariant metric to be naturally reductive. More precisely, the following result holds (cf. [429, 304]). Theorem 5.2.12 (J. Szenthe). A homogeneous Riemannian space (M = G/H, g) is naturally reductive with respect to some decomposition g = h ⊕ p0 if and only if there is a linear Ad(H)-equivariant map τ : p → h such that X + τ(X) is a geodesic vector for every X ∈ p. Proof. Suppose that (M = G/H, g) is naturally reductive with respect to a decomposition g = h ⊕ p0 . Hence, any vector X 0 ∈ p0 is a geodesic vector, i.e. exp(tX 0 )(0 = eH), t ∈ R, is a geodesic. For any X 0 ∈ p0 we have a unique decomposition X 0 = X +Y , where X ∈ p and Y ∈ h, hence, we have some linear maps ϕ1 : p0 → p and ϕ2 : p0 → h, where X = ϕ1 (X 0 ) and Y = ϕ2 (X 0 ). Clearly, the maps ϕ1 and ϕ2 are Ad(H)-equivariant and ϕ1 is one-to-one. Now, consider the map τ : p → h, where τ = ϕ2 ◦ ϕ1−1 . Obviously, it is a linear and Ad(H)-equivariant. Moreover, for any X ∈ p, X + τ(X) is a geodesic vector. On the other hand, if (M = G/H, g) admits a linear and Ad(H)-equivariant map τ : p → h such that X + τ(X) is a geodesic vector for any X ∈ p, then p0 = {X + τ(X) | X ∈ p} is an Ad(H)-invariant complement to h in g and every vector X 0 ∈ p0 is geodesic. Therefore, U(p0 , p0 ) = 0 by Theorem 5.1.2 (U is defined by the equality (4.31)) and (M = G/H, g) is naturally reductive with respect to the decomposition g = h ⊕ p0 . t u We note that explicit constructions of geodesic graphs for some geodesic orbit spaces are given in [190, 188, 189, 300, 304, 305]. In the same papers, one can find more detailed information on the geodesic graphs and become acquainted with a number of open questions in this area.

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It should be noted also that there is a natural extension of the notion of geodesic orbit manifolds to the pseudo-Riemannian case, see e.g. [47, 141, 171, 195] and the references therein.

5.3 Important Subclasses of Geodesic Orbit Spaces There are some important subclasses of geodesic orbit manifolds. For instance, GOspaces may be considered as a natural generalization of Riemannian symmetric ` Cartan in [145]. spaces, introduced and classified by E. Indeed, a simply connected symmetric space can be defined as a Riemannian manifold (M, g), every geodesic γ ⊂ M of which is an orbit of a 1-parameter group g(t) of transvections, i.e. a 1-parameter isometry group that preserves γ and induces the parallel transport along γ. If we reject the latter assumption of parallel transport, we obtain the definition of a GO-space. On the other hand, the class of geodesic orbit spaces are much wider than the class of symmetric spaces. For instance, every homogeneous space M = G/H of a compact Lie group G admits a G-invariant GO-metric, i.e. a metric g such that (M, g) is a geodesic orbit space. To prove this, it suffices to take any G-normal Riemannian metric g on M = G/H. Moreover, every naturally reductive homogeneous Riemannian space is geodesic orbit. The first example of a non-naturally reductive GO-manifold had been constructed by A. Kaplan [280]. It was a 6-dimensional two-step nilpotent Lie group, equipped with a suitable left-invariant Riemannian metric, see also Theorems 5.3.10 and 5.3.11. In [304], O. Kowalski and L. Vanhecke classified all geodesic orbit spaces of dimension ≤ 6. The main result of this paper is the following theorem. Theorem 5.3.1 (O. Kowalski–L. Vanhecke, [304]). The following assertions hold. 1) All Riemannian GO-spaces of dimension up to 4 are naturally reductive. 2) Every 5-dimensional Riemannian GO-space is either naturally reductive, or has an isotropy subalgebra of the type su(2). 3) Every 6-dimensional Riemannian GO-space is either naturally reductive or one of the following: (a) A two-step nilpotent Lie group with two-dimensional center, equipped with a left-invariant Riemannian metric such that the maximal connected isotropy group (that corresponds to the maximal isometry group) is isomorphic to either SU(2) or U(2). Corresponding GO-metrics depend on three real parameters. (b) The universal covering space of a homogeneous Riemannian manifold of the form (M = SO(5)/U(2), g) or (M = SO(4, 1)/U(2), g), where SO(5) or SO(4, 1) is the identity component of the full isometry group, respectively. In each case, all corresponding invariant metrics are GO-metrics g depending on two real parameters.

5.3 Important Subclasses of Geodesic Orbit Spaces

277

As pointed out by the authors in [304, p. 190], the GO-spaces (a) and (b) are in no way naturally reductive in the following sense: whatever the representation of (M, g) as a quotient of the form G0 /K 0 , where G0 is a connected transitive group of isometries of (M, g), and whatever is the Ad(K 0 )-invariant decomposition g0 = k0 ⊕ p0 , the curve γ(t) = exp(tX) · o is not a geodesic for some X ∈ p0 \ {0}. The description of a 5-dimensional simply connected Riemannian GO-space of isotropy type SU(2) is obtained in [304, Theorem 4.4]. In particular, all such spaces are either naturally reductive or become naturally reductive after extending the isotropy subgroup SU(2) to U(2). Therefore, every geodesic orbit Riemannian manifold of dimension ≤ 5 is naturally reductive. At the same time, in dimension 6, there are 3- and 2-parameter families of geodesic orbit metrics that are not naturally reductive, as follows from Theorem 5.3.1. The compact homogeneous space SO(5)/U(2) with a 2-parameter family of invariant GO-metrics is an example of the latter. It should be noted that the class of weakly symmetric spaces, introduced by A. Selberg [408], is closely related to the class of geodesic orbit spaces. Definition 5.3.2. A Riemannian manifold (M, g) is called weakly symmetric if for every two points p, q ∈ M, there is an isometry of M interchanging the points p and q. Riemannian manifolds with this property are also called weakly symmetric Riemannian spaces or simply weakly symmetric spaces. Theorem 5.3.3. Let (M, g) be a complete Riemannian manifold. Then the following conditions are equivalent. 1) (M, g) is weakly symmetric. 2) There is a subgroup G of the full connected isometry group Isom(M, g), that acts transitively on (M, g), and f ∈ Isom(M, g) such that f G f −1 = G, f 2 ∈ G, and for all p, q ∈ M there exists an a ∈ G with a(p) = f (q) and a(q) = f (p). 3) For every geodesic γ : R → M and every point m on γ(R), there exists an isometry s := s(m, γ) ∈ Isom(M, g) such that s is a nontrivial involution on γ(R) and m is a fixed point of s. 4) For all x ∈ M and all ξ ∈ Mx , there exists a t ∈ Isom(M, g) such that t(x) = x and (T t)x (ξ ) = −ξ . Remark 5.3.4. It is clear that if the second condition holds, then we may consider the whole Isom(M, g) as G. Note also that in [408], the second condition in the above theorem was taken as a definition of weakly symmetric spaces. Proof. 2) ⇒ 1) Take two points p, q ∈ M. There exists an a ∈ G with a(p) = f (q) and a(q) = f (p). Therefore, b(p) = q and b(q) = p, where b = f −1 ·a ∈ Isom(M, g). 3) ⇒ 2) Let G be the full isometry group Isom(M, g) and f be the identity transformation on M. Clearly, f G f −1 = G and f 2 ∈ G in this case. Let p, q ∈ M and γ : R → M be a geodesic in (M, g) parameterized by arc-length with γ(0) = p and

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γ(2t) = q for some t > 0. Then s := s(m, γ) with m = γ(t) is an isometry of (M,g) mapping p to q and q to p. 4) ⇒ 3) Let us consider any geodesic γ : R → M and fix any point m = γ(u), u ∈ R. Then there exists a t ∈ Isom(M, g) such that t(m) = m and Tx t(ξ ) = −ξ , where ξ = γ 0 (u). It is clear that t is a nontrivial involution on γ(R). 1) ⇒ 4) It is clear that any manifold satisfying 1) is homogeneous, hence, has a positive injectivity radius. Let us consider a geodesic γ : R → M parameterized proportionally to arc length such that γ(0) = x and γ 0 (0) = ξ . Fix t > 0 and put p = γ(t) and q = γ(−t). There is an isometry f of (M, g), interchanging the points p and q. If t is small enough, then f preserves γ(R) (otherwise, the points p and q are connected with different shortest arcs). It is clear also that t preserves the point x and Tx t(ξ ) = −ξ . t u This theorem and the structural results on Riemannian homogeneous spaces easily imply Corollary 5.3.5. A Riemannian manifold (M, g) is weakly symmetric if and only if it is homogeneous and for some (hence, every) point x ∈ M and any reductive decomposition g = h ⊕ p of the Lie algebra of the Lie group G = Isom(M, g), where h is the Lie subalgebra of the isotropy subgroup H at the point x, the following property holds: for every U ∈ p there is an s ∈ H such that Ad(s)(U) = −U. In particular, every G-invariant Riemannian metric on the homogeneous space G/H makes it a weakly symmetric space. It is clear that any symmetric space is weakly symmetric and naturally reductive. On the other hand, there exist weakly symmetric spaces that are not even naturally reductive. For instance, geodesic spheres in the Cayley projective plane CaP2 have this property. Note also that all geodesic spheres in all symmetric spaces of rank 1 are weakly symmetric spaces [503]. Theorem 5.1.5 implies the following important result, which was obtained in [100] by J. Berndt, O. Kowalski, and L. Vanhecke (using other methods). Theorem 5.3.6 ([100]). Every weakly symmetric Riemannian space (M, g) is geodesic orbit. Proof. Let us fix a geodesic γ : R → M in a weakly symmetric Riemannian manifold (M, g). For any p ∈ γ(R), there is a non-trivial isometry η(p) ∈ Isom(M, g) that is a nontrivial involution on γ(R) fixing the point p ∈ γ(R) by Theorem 5.3.3. For a given s ∈ R, the isometry ψ(s) := η(γ(s/2)) ◦ η(γ(0)) preserves γ(R) and its orientation, and moves the point γ(0) to γ(s). Therefore, the geodesic γ is homogeneous by Theorem 5.1.5. t u

5.3 Important Subclasses of Geodesic Orbit Spaces

279

A. Selberg proved that every weakly symmetric Riemannian manifold M is commutative, i.e. admits a transitive Lie isometry group G such that the algebra of all G-invariant differential operators on M is commutative [408]. If G is connected and M = G/H, then this is equivalent to the property that the functional space L1 (H\G/H) is commutative, i.e. (G, H) is a Gelfand pair, or to the property that for every unitary irreducible representation of the group G, the dimension of the H-fixed set is not more than 1, i.e. (G, H) is a spherical pair. J. Lauret constructed a noncompact commutative, but not weakly symmetric Riemannian manifold [317]. At the same time, under certain restrictions on the transitive isometry group, the classes of weakly symmetric spaces and commutative spaces coincide [11]. In particular, if (G, H) is a spherical pair with a connected compact simple Lie group G and a closed subgroup H, then G/H is weakly symmetric [354]. All such pairs are known from [310] and [354]. Note also the following result. Theorem 5.3.7 (Y. Mao, [331]). If the algebra of isometrically invariant differential operators on a complete connected Riemannian manifold M is commutative, M is a homogeneous Riemannian manifold. Many examples of weakly symmetric spaces were built by W. Ziller [503]. The classification of weakly symmetric reductive homogeneous Riemannian spaces was obtained by O. S. Yakimova [492] on the basis of the paper [11]. More details on properties of weakly symmetric spaces can be found in the book [484] by J. A. Wolf. Interesting results on geometry of weakly symmetric spaces can be found in [102], [99], and [449]. It should also be noted that there are many interesting results for pseudoRiemannian weakly symmetric manifolds, see e.g. [158, 159] and references therein. In the papers [66, 87, 88, 89, 90, 91, 92], the authors found invariant geodesic orbit sub-Riemannian metrics, their geodesics, distances, conjugate and cut loci on the Selberg weakly symmetric spaces (SO(3) × SO(2))/SO(2), (SO0 (2, 1) × SO(2))/SO(2), (SL(2) × SO(2))/SO(2) and all their coverings. In the paper [67], the author suggested methods to calculate sectional and Ricci curvatures and find geodesics for some homogeneous sub-Riemannian manifolds and applied them in some cases. Using these methods, he found in [68] curvatures and geodesics of natural left-invariant sub-Riemannian metrics on the full connected semisimple isometry groups of Riemannian symmetric spaces; notice that these metrics are geodesic orbit. An original approach to the study of geodesic orbit spaces, based on the concept of a geodesic graph, is proposed by O. Kowalski, S. Nik˘cevi´c, and Z. Vl´as˘ek [300], [305]. We consider it in more detail in Section 5.2. The structure of geodesic orbit spaces was also studied by C. Gordon, who obtained several important structural and classification results [230]. In particular, she obtained a useful description of nilpotent groups with left-invariant Riemannian metrics, which are GO-manifolds. It turned out that such metrics exist only on

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commutative or two-step nilpotent Lie groups (see Proposition 5.4.5). Also in [230], there are some constructions of geodesic orbit spaces with compact and noncompact semisimple isometry groups. Another result (which we will consider separately) due to C. Gordon is the classification of geodesic orbit manifolds of non-positive Ricci curvature (see Theorem 5.6.1). Among other publications it is necessary to point out papers [434, 433] by H. Tamaru, the results by Z. Du˘sek in [188, 189], and the paper [16] by D. V. Alekseevsky and A. Arvanitoyeorgos, devoted to the description of invariant geodesic orbit metrics on (generalized) flag manifolds. In particular, the following classification result was obtained in [16]. Theorem 5.3.8 (D. V. Alekseevsky–A. Arvanitoyeorgos, [16]). Let (G/H, g) be a Riemannian flag manifold with a simple Lie group G and an invariant metric g, that is geodesic orbit but not naturally reductive. Then either G/H = SO(2l + 1)/U(l), or G/H = Sp(l)/U(1) · Sp(l − 1), where l ≥ 2. In both these cases, the set of invariant metrics with such property is a 1-parameter family (up to similarity). We note that the classification of general flag manifolds reduces to the classification of simply connected irreducible flag manifolds that have simple full isometry groups. We also note that the geodesic orbit metrics, mentioned in the above theorem, are weakly symmetric, which was shown earlier by W. Ziller in [503]. In Section 5.10, we provide the classification of compact geodesic orbit Riemannian manifolds of positive Euler characteristic from [18], which generalizes the result of Theorem 5.3.8. Many groups of Heisenberg type, introduced by A. Kaplan in [280], are geodesic orbit Riemannian manifolds. Definition 5.3.9. A Lie algebra n endowed with an inner product h·, ·i is said to be of Heisenberg type, or simply an H-type algebra, if it admits an orthogonal decomposition n = v ⊕ z, dim z ≥ 1, such that k ad(X)∗ Y k = kX1 k kY2 k, where kV k = hV,V i for V ∈ n, ∗ denotes the adjoint relative to h·, ·i, and the subscripts 1, 2 denote the orthogonal projections onto v and z respectively. An Htype Lie group (generalized Heisenberg group, in other terms) is a connected, simply connected Lie group N whose Lie algebra n is an H-type algebra. In this case N is assumed to be supplied with the left-invariant Riemannian metric induced by h·, ·i. p

Theorem 5.3.10 (A. Kaplan, [280]). Let n be an H-type Lie algebra and N be the corresponding H-type Lie group, then the following assertions hold. 1) n is a 2-step nilpotent Lie algebra with center z. 2) Putting JV (X) = ad(V )∗ (X), V ∈ z, X ∈ v we get the linear maps JV : z → End(v) and the structure of a Clifford Clm -module on v, where m = dim z; and vice versa.

5.4 On the Radical and the Nilradical of the Lie Algebra of the Isometry Group . . .

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3) The isometry group of (N, h·, ·i) is G = A(N) n N, where A(N) is the group of automorphisms of N, whose differential at e is an isometry of (n, h·, ·i). 4) N is naturally reductive with respect to G if and only if m = 1 (the Heisenberg group) or m = 3 (its quaternion analog). 5) If m = 2 then N is a geodesic orbit Riemannian manifold, and the minimal dimension of such N is 6. 6) There are infinitely many Lie algebras of type H with arbitrary natural number m and n = dim v = kl, where l = 24α+β , if 8α + 2β −1 − 1 < m ≤ 8α + 2β − 1 with non-negative integer α and β = 0, 1, 2, 3, and k is an arbitrary natural number. In other words, a pair (m, n) appears if and only if Sn−1 admits m continuous tangent vector fields, linearly independent at every point. Theorem 5.3.11 (C. Riehm, [398]). An H-type Lie group (N, h·, ·i) is a geodesic orbit Riemannian manifold if and only if m = 1, 2, 3, 5 for arbitrary n from 3) of Theorem 5.3.10; m = 6, n = 8; and m = 7, n = 8, 16, 24. We say that an H-type group N is weakly symmetric in the broad sense (resp. in the narrow sense) if given any point x of N and any geodesic γ through x, there is an element g of G = A(N) n N (resp. of the identity component G0 of G) that fixes x and inverts the orientation of the geodesic. Theorem 5.3.12 (J. Berndt–F. Ricci–L. Vanhecke, [103]). An H-type Lie group (N, h·, ·i) is weakly symmetric in the broad sense (resp. in the narrow sense) if and only if m = 1, 2, 3 (resp. m = 2, 3) for arbitrary n from 3) of Theorem 5.3.10; m = 5, 6, 7, n = 8 (resp., m = 6, 7) and m = 7, n = 16, v is isotypic. An important proper subclass of the class of geodesic orbit manifolds are the generalized normal homogeneous Riemannian manifolds, that are studied in Chapter 6.

5.4 On the Radical and the Nilradical of the Lie Algebra of the Isometry Group of a Geodesic Orbit Space Now we consider some structural results on the radical r(g) and the nilradical n(g) of the Lie algebra g of the isometry group G for a given geodesic orbit space (G/H, g). The exposition is mainly based on the paper [366]. Let us consider an arbitrary geodesic orbit Riemannian space (G/H, g) and an Ad(H)-invariant decomposition (5.9) with B(h, m) = 0, where B is the Killing form of g, and g is generated by an Ad(H)-invariant inner product (·, ·) on m. It should be noted that n(g) ⊂ m, since n(g) ⊂ {X ∈ g | B(X,Y ) = 0 ∀ Y ∈ g}, see e.g. Proposition 1.4.6 in [122]. Let Cg (h) be the centralizer of h in g. It is clear that Cg (h) = Cg (h) ∩ h +Cg (h) ∩ m,

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where Cg (h) ∩ h = Ch (h) is the center of h. Obviously, Cg (h) and Cg (h) ⊕ [h, h] are subalgebras in the Lie algebra g with [Cg (h), [h, h]]=0. The centers of G and g we denote by C(G) and C(g) respectively. The following results explain the correlations among the above objects. Theorem 5.4.1. Under the above notations and assumptions, the following assertions hold: 1) For any Y ∈ Cg (h) the operator adYm = ad(Y )|m is skew-symmetric. If X ∈ Cg (h)∩ m, then ([X, Z]m , X) = 0 for any Z ∈ g. 2) The Lie algebra k := Cg (h) ⊕ [h, h] is compactly embedded in g. The Killing form B of g is non-positive on k. Moreover, B(Y,Y ) = 0 for Y ∈ k if and only if Y is in the center of g. 3) We have the decomposition m = (Cg (h) ∩ m) ⊕ [h, m], which is both B-orthogonal and (·, ·)-orthogonal. Moreover, Cg (h) is B-orthogonal to [h, g] = [h, h] ⊕ [h, m]. 4) Let us consider the normalizer NG (h) = {a ∈ G | Ad(a)(h) ⊂ h} of h in the Lie group G. Then it is a closed subgroup in G with the Lie algebra Cg (h) ⊕ [h, h]. Its unit component NG (h)0 has the same property. 5) The inner product (·, ·) is not only Ad(H)-invariant but also Ad(NG (H0 ))invariant, where NG (H0 ) is the normalizer of the unit component H0 of the group H in G. This theorem is a compilation of Lemma 8, Corollary 2, Proposition 9, Proposition 11, and Corollary 4 in [366]. Proof. 1) Note that [Y, m] ⊂ m for any Y ∈ Cg (h), which follows from B([Y, m], h) = −B(m, [Y, h]) = −B(m, 0) = 0. Obviously, ad(Y )|m is skew-symmetric for any Y ∈ h. Now, take any Y ∈ Cg (h) ∩ m. By Theorem 5.2.6, for any X ∈ m there is a Z ∈ h such that X + Z is a geodesic vector, hence, 0 = ([X + Z,Y ]m , X) = ([X,Y ]m , X) + ([Z,Y ], X) = ([X,Y ]m , X), because [h,Y ] = 0. Therefore, ad(Y )|m is also skew-symmetric in this case. Further, according to Theorem 5.2.6, for any X ∈ Cg (h) ∩ m, there is a Z ∈ h such that 0 = ([X + Z,Y ]m , X) = ([X,Y ]m , X) + ([Z,Y ], X) = ([X,Y ]m , X)−(Y, [Z, X]) = ([X,Y ]m , X) for any Y ∈ m, since [h, X] = 0. For Y ∈ h, the equality ([X,Y ]m , X) = 0 is obvious. 2) We know that the operator ad(Y )|m is skew-symmetric for all Y ∈ Cg (h). By the definition, we get also [Y, h] = 0. If we extend (·, ·) to the Ad(H)-invariant product on g with (h, m) = 0, then ad(Y ) is skew-symmetric. We can say the same about any Y ∈ h, hence for all Y ∈ k. This means that k is compactly embedded in g. It is also obvious that B(Y,Y ) = trace(ad(Y ) ad(Y )) = ∑([Y, [Y, Ei ]], Ei ) = − ∑([Y, Ei ], [Y, Ei ]) ≤ 0, i

i

where {Ei } is any (·, ·)-orthonormal basis in g, with B(Y,Y ) = 0 if and only if ad(Y ) = 0. 3) Since m is an ad(h)-module, we get the decomposition into the sum of linear spaces m = (Cg (h) ∩ m) ⊕ [h, m], which is well known (see e.g. Lemma 14.3.2 in [257]). Since the inner product (·, ·) is ad(h)-invariant,

5.4 On the Radical and the Nilradical of the Lie Algebra of the Isometry Group . . .

283

0 = −([h,Cg (h) ∩ m], m) = (Cg (h) ∩ m, [h, m]). The same is valid for the Killing form B: 0 = −B([h,Cg (h) ∩ m], m) = B(Cg (h) ∩ m, [h, m]). Moreover, we have 0 = −B([h,Cg (h)], g) = B(Cg (h), [h, g]) = B(Cg (h), [h, h] ⊕ [h, m]). 4) The closeness of NG (h) follows directly from its definition. Hence, NG (h) is a Lie subgroup of G. Let us show that NG (h) corresponds to the Lie subalgebra Cg (h) ⊕ [h, h] in g. Indeed, the Lie algebra of NG (h) could be characterized as Lie(NG (h)) = {X ∈ g | [X, h] ⊂ h}. It is clear that h ⊂ Lie(NG (h)). Further, consider any X ∈ Lie(NG (h)). There are U ∈ h and V ∈ m such that X = U + V . Since U ∈ Lie(NG (h)), we have V ∈ Lie(NG (h)). Recall that [h, m] ⊂ m. Hence, V ∈ Lie(NG (h)) is equivalent to [V, h] = 0, i.e. to V ∈ Cg (h). It is also obvious that V ∈ Cg (h) implies X = U +V ∈ Lie(NG (h)), therefore, Lie(NG (h)) = Cg (h)⊕[h, h]. 5) This easily follows from 1) and 4), since NG (H0 ) = NG (h). t u Remark 5.4.2. Note that property 5) in Theorem 5.4.1 is well known for weakly symmetric spaces (see e.g. Lemma 2 in [492]) and for generalized normal homogeneous spaces (Corollary 6 in [73]). Theorem 5.4.3. Under the above notations and assumptions we get the following assertions: 1) Any ad(h)-invariant complement p to [g, r(g)] in r(g) is in Cg (h); 2) Cg (h) ∩ n(g) is the center of g; 3) n(g) = {X ∈ g | [B(X, g) = 0} and we have the following (·, ·)-orthogonal sum: n(g) = [g, r(g)] ⊕ l, where l is a central subalgebra of g.
4) Let q be the (·, ·)-orthogonal complement to [r(g), g] ⊂ n(g) in r(g) ∩ m, then ad(Y )|m is skew-symmetric for any Y ∈ q, q commutes both with h and with the orthogonal complement to [r(g), g] in m; in particular, [q, q] = 0. This theorem is a compilation of Proposition 5, Corollary 3, and Theorem 1 in [366]. Proof. 1) It is clear that [h, p] ⊂ [g, r(g)]. On the other hand, [h, p] ⊂ p, which proves p ⊂ Cg (h). 2) Take any Y ∈ Cg (h) ∩ n(g). By 1) of Theorem 5.4.1, we get that the operator ad(Y )|m is skew-symmetric, whereas [Y, h] = 0. On the other hand, Y ∈ n(g) and ad(Y ) should be nilpotent. Hence, Y is in the center of g. It is clear also that any central elements of g are situated both in n(g) and in Cg (h), hence we get the required assertion. 3) By 1), any ad(h)-invariant complement p to [g, r(g)] in r(g) is in Cg (h). By 2) of Theorem 5.4.1, B(X, X) ≤ 0 for X ∈ p, whereas B(X, X) = 0 if and only if X is in the center of g. Since the center is a subset of the nilradical, B is negative definite

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on any ad(h)-invariant complement q to n(g) (recall that [g, r(g)] ⊂ n(g)) in r(g). From this we get also n(g) = {X ∈ g | [B(X, g) = 0},

since

n(g) ⊂ {X ∈ g | [B(X, g) = 0} ⊂ r(g).

If l is an arbitrary ad(h)-invariant complement to [g, r(g)] in n(g), then l ⊂ Cg (h) by 1), and moreover, l is a central subalgebra in g by 2). 4) Indeed, q ⊂ Cg (h) by 1) and ad(Y )|m is skew-symmetric for any Y ∈ q by 1) of Theorem 5.4.1. Since [q, g] ⊂ [r(g), g] and ad(Y )|m is skew-symmetric for any Y ∈ q, we see that such Y commutes with any X ∈ m orthogonal to [r(g), g]. t u Remark 5.4.4. Note that Ad(X), X ∈ q (see 4) of Theorem 5.4.3) could act nontrivially only on [r(g), g] ⊂ n(g), but in general q is a non-trivial subspace, see Example 5.4.8 below. This is a good place to recall the structure of the nilradical in g for a geodesic orbit space (G/H, ρ). The following result was obtained by C. Gordon [230, Theorem 2.2], see also [484, Proposition 13.1.9] in general case. Proposition 5.4.5 (C. Gordon–J. A. Wolf). Let (G/H, ρ) be a geodesic orbit space. Then the nilradical n(g) of the Lie algebra g = Lie(G) is commutative or two-step nilpotent. Proof. Suppose that n(g) is not commutative, i.e. [n(g), n(g)] 6= 0. Let a be the orthogonal complement to [n(g), n(g)] in n(g) ⊂ m. It is easy to see that for any X ∈ a the operator ad(X)|[n(g),n(g)] is skew-symmetric. Indeed, by Theorem 5.2.6, there is a Z ∈ h such that 0 = ([X + Z,Y ], X) = ([X,Y ], X) + ([Z,Y ], X) = ([X,Y ], X) for any Y ∈ [n(g), n(g)], since [Z,Y ] ∈ [n(g), n(g)] ⊥ a. On the other hand, ad(X)|[n(g),n(g)] is nilpotent by the Engel theorem. Therefore, [X,Y ] = 0 for any X ∈ a and any Y ∈ [n(g), n(g)]. It is easy to see that [a, a], [a, [a, a]], . . . , [a, [. . . , [a, [a, a]] . . . ]], . . . also commute with [n(g), n(g)]. But the subspace a generates n(g), since n(g) is nilpotent (see e.g. [122], exercise 4 to Section 4 of Chapter I). Hence, [n(g), [n(g), n(g)]] = 0 and n(g) is two-step nilpotent. t u Remark 5.4.6. From Theorem 5.4.3 and Proposition 5.4.5 we see that [g, r(g)] is either commutative or a two-step nilpotent Lie algebra for any geodesic orbit space. There is one more important result on the radical r(g) for a geodesic orbit space (G/H, ρ). Recall that by the Levi–Mal’tsev theorem 2.3.13, there is a semisimple Levi subalgebra s ⊂ g such that g = r(g) o s. Such subalgebra is unique up to an automorphism of g. We can write s as a direct sum of its subalgebras s = s c ⊕ s nc , where s c is the maximal compact (semisimple) ideal in s and s nc is the maximal semisimple ideal of noncompact type in s. The following proposition is asserted in [230], although the proof is not included there. We reproduce the proof from [232]. Proposition 5.4.7 (C. Gordon, [230]). Let (G/H, ρ) be a geodesic orbit space. Then the radical r(g) of the Lie algebra g = Lie(G) commutes with s nc for any Levi subalgebra s in g, i.e. s nc is an ideal in g and g = (r(g) o s c ) ⊕ s nc as a Lie algebra.

5.4 On the Radical and the Nilradical of the Lie Algebra of the Isometry Group . . .

285

Proof. Let m be the orthogonal complement to h in g with respect to the Killing form of g. Then the metric g is determined by some inner product (·, ·) on m. It should be noted that the nilradical n(g) satisfies n(g) ⊂ m. Let o be the subalgebra of g defined by o = {Y ∈ g | ad(Y )|n(g) is (·, ·)-skew-symmetric}.

(5.13)

We first show that g = o + n(g).

(5.14)

Let p be the (·, ·)-orthogonal complement to n(g) in m and let Y ∈ p. For a given X ∈ n(g), there is a Z ∈ h such that ([X + Z,Y ]m , X) = ([X + Z,Y ], X) = 0 for all Y ∈ p according to Theorem 5.2.6. We have ([Y, X], X) = ([Y, X + Z], X) = 0, where the first equation follows from the fact that [h, p] ⊂ p. Thus p ⊂ o. Since trivially h ⊂ o, Equation (5.14) follows. We next show that o contains a semisimple Levi factor of g. Let us consider the homomorphic projection π : g → s := g/r(g). By Equation (5.14) and the fact that n(g) ⊂ r(g), we have π(o) = s. Since the Lie algebra s is semisimple, Levi’s Theorem (see e.g. Theorem 5.6.6 in [257]), yields a homomorphism ϕ : s → o such that π ◦ ϕ = Ids . The image of ϕ is the desired Levi factor es. Let es nc be the noncompact part of es. The map Y 7→ ad(Y )|r(g) is a representation of es nc which acts by semisimple endomorphisms on the invariant subspace n(g). Since the only representation of a semisimple Lie algebra of noncompact type by skew-symmetric endomorphisms is the trivial representation, we conclude that [es nc , n(g)] = 0. Finally, since [es nc , r(g)] ⊂ n(g) and since every representation of a semisimple Lie algebra is completely reducible, we have [es nc , r(g)] = 0. In particular, es nc is an ideal in g. Finally, any semisimple Levi factor s is conjugate to es via an element of n(g) by the Levi–Mal’tsev theorem 2.3.13. Hence s nc = es nc due to [es nc , r(g)] = 0. t u Theorem 1.15 in [230] claims that N(G) · S acts transitively on a geodesic orbit Riemannian space (G/H, ρ), where N(G) is the largest connected nilpotent normal subgroup and S is any semisimple Levi factor in G, which is equivalent to n(g) + s + h = g. Examples 5.4.8 and 5.4.12 below refute this assertion (nevertheless, it is valid for the case of naturally reductive metrics, see Theorem 3.1 in [229]). In particular, we get n(g) 6= r(g) ∩ m in these examples. Proposition 5.4.11 shows how to construct examples of this kind. Example 5.4.8. Let us consider the standard action of the Lie algebra u(n) on R2n (by skew-symmetric matrices), n ≥ 2. We will use the notation A(X) for the action of A ∈ u(n) on X ∈ R2n . Note that for any X ∈ R2n there is a non-zero A ∈ u(n) such that A(X) = 0 and A is not contained in su(n) (because the standard action of U(n) on S2n−1 has the isotropy group U(n − 1) 6⊂ SU(n)).

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Now consider r = R2n+1 = R2n o z, a semidirect sum of Lie algebras, where z acts on R2n as the center of u(n) on R2n . This Lie algebra is solvable (but it is not nilpotent) with the abelian nilradical R2n . Moreover, we also have a natural action of su(n) on r = R2n+1 : [Z, X + Y ] = Z(X) for every Z ∈ su(n), X ∈ R2n , Y ∈ z. Hence, we get semidirect sum g = r o su(n) of Lie algebras. Note also that r could be defined as the radical of the Lie algebra R2n o u(n) = r o su(n). Supply r = R2n+1 with the standard Euclidean inner product. Let us prove that G/H with the corresponding invariant Riemannian metrics, where H = SU(n) and G has the Lie algebra R2n o u(n) = r o su(n), is a geodesic orbit Riemannian space. It suffices to prove that for any X ∈ R2n and for any Y ∈ z = R there is a Z ∈ su(n) such that ([X + Y + Z, X1 + Y1 ], X + Y ) = ([X,Y1 ] + [Y + Z, X1 ], X) = 0 for every X1 ∈ R2n and Y1 ∈ z = R. Since su(n) and z = R act on R2n by skew-symmetric operators, this is equivalent to [Y + Z, X] = 0. If Y = 0, then we can take Z = 0. Now, suppose that Y 6= 0. We know that there exists a non-zero A ∈ u(n) such that A(X) = 0 and A 6∈ su(n) for the standard action of u(n) on R2n . Let A = Y2 + Z, where Y2 ∈ z = R and Z ∈ su(n). Since Y2 6= 0 we may assume (without loss of generality) that Y2 = Y by multiplying A by a suitable constant. Hence, we have found a desirable Z ∈ su(n). Hence, we get an example of a geodesic orbit Riemannian space (G/H, µ), where n(g) 6= r(g) ∩ m. Remark 5.4.9. Note that the constructed GO-space is simply the Euclidean space R2n+1 . See also case 4) of Theorem 4.4 in [304]. Now we consider another construction of geodesic orbit spaces. Let n be a twostep nilpotent Lie algebra with an inner product (·, ·). Denote by z and a the Lie subalgebra [n, n] and the (·, ·)-orthogonal complement to it in n. It is clear that z is central in n. For any Z ∈ z we consider the operator JZ : a → a defined by the formula (JZ (X),Y ) = ([X,Y ], Z). It is clear that J : z → so(a) is an injective map. The Lie algebra of the isometry group of the corresponding Lie group N with the corresponding left-invariant Riemannian metric µ is a semidirect sum of n with D(n), the algebra of skew-symmetric derivations of n, see e.g. Theorem 4.2 in [476] or [472]. We state the following important result. Theorem 5.4.10 (C. Gordon, [230]). (N, µ) is a geodesic orbit Riemannian manifold if and only if for any X ∈ z and Y ∈ a there is a D ∈ D(n) such that [D, X] = D(X) = 0, [D,Y ] = D(Y ) = JX (Y ). In [230], this result had been used to construct low-dimensional examples of nilpotent Lie groups N supplied with left-invariant Riemannian metric µ such that (N, µ) are geodesic orbit Riemannian manifolds. Now suppose that D(n) = c ⊕ d, where c is a one-dimensional central ideal in D(n). Hence we can define a solvable Lie algebra sol = n ⊕ c, which is a semidirect sum ([U, X] = U(X) for X ∈ n and U ∈ c). We can extend (·, ·) to sol assuming (n, c) = 0 and choosing any inner product on c. Since [c, d] = 0, it follows that d is an algebra of skew-symmetric derivations of sol.

5.4 On the Radical and the Nilradical of the Lie Algebra of the Isometry Group . . .

287

Proposition 5.4.11. Under the above assumptions and notation, suppose that 1) for every X ∈ z and every Y ∈ a, there is a D1 ∈ d ⊂ D(n) such that [D1 , X] = D1 (X) = 0, [D1 ,Y ] = D1 (Y ) = JX (Y ); 2) the set {D ∈ D(n) | D(X) = D(Y ) = 0} does not lie in d for every X ∈ z and every Y ∈ a. Then the Lie group Sol with a left-invariant Riemannian metric, corresponding to the metric Lie algebra (sol, (·, ·)), is a geodesic orbit Riemannian manifold. Proof. Since d is an algebra of skew-symmetric derivations of sol, it suffices to prove the following: for any X ∈ z, Y ∈ a, and Z ∈ c there is a D ∈ d such that ([X +Y + Z + D,U], X +Y + Z) = 0

for any

U ∈ sol.

Since X is central in n, [c, c] = 0, [c, d] = 0, and [sol, sol] ⊂ n, this is equivalent to the following: 0 = ([Y + Z + D,U], X +Y ) = ([Y + Z + D,U1 ], X +Y ) + ([X +Y,U2 ], X +Y ), where U1 ∈ n, U2 ∈ c, U = U1 + U2 . Note that ([X + Y,U2 ], X + Y ) = 0 because U2 ∈ c ⊂ D(n) is skew-symmetric on n. On the other hand, ([Y + Z + D,U1 ], X + Y ) = ([Y,U1 ], X) + ([Z + D,U1 ], X +Y ) = (JX (Y ),U1 ) − ([Z + D, X +Y ],U1 ) by the definition of JX , and we get the equivalent equation 0 = (JX (Y ),U1 ) − ([Z + D, X + Y ],U1 ). Since U is arbitrary, it suffices to prove that for any X ∈ z, Y ∈ a, and Z ∈ c there is a D ∈ d such that [Z + D, X +Y ] = JX (Y ). From assumption 1) we know that there is a D1 ∈ d such that [D1 , X] = D1 (X) = 0 and [D1 ,Y ] = D1 (Y ) = JX (Y ). Let consider the set C(X +Y ) := {D ∈ D(n) | D(X) = D(Y ) = 0}. This set is not trivial and is not in d by assumption 2). Now it is clear that for any W ∈ c there is a V ∈ d such that V +W ∈ C(X +Y ). Indeed, such a V exists at least for one nontrivial W by assumption 2). Now, it suffices to use multiplication by a constant, because c is 1-dimensional. For a given Z ∈ c we can choose V ∈ d such that Z +V ∈ C(X +Y ). Now consider D = V + D1 . Since D1 ∈ d, we have D ∈ d and [Z + D, X +Y ] = JX (Y ). t u Example 5.4.12. Consider the 13-dimensional metric Lie algebra of Heisenberg type n (for details, see [194]). In this case dim(z) = 5, dim(a) = 8, D(n) = so(5)⊕R. It it known that for any X ∈ z and Y ∈ a there is (a unique) D1 ∈ so(5) such that [D1 , X] = D1 (X) = 0 and [D1 ,Y ] = D1 (Y ) = JX (Y ), see p. 92 in [194]. In fact this was proved earlier in [398]. In [194], a suitable D1 is determined by a (unique) solution d of a linear system Bd = b, where B is a (10 × 10)-matrix with elements that depend linearly on X and Y . It is easy to see also that the set {D ∈ D(n) | D(X) = D(Y ) = 0} is not situated in so(5) for any X ∈ z and any Y ∈ a, because any nontrivial D ∈ so(5), such that D(X) = D(Y ) = 0, is determined by a non-trivial solution d of the homogeneous linear system Bd = 0, where B is nondegenerate, which is impossible (for details, see [194]). Therefore, according to Proposition 5.4.11, we get a 14-dimensional solvmanifold that is a geodesic orbit Riemannian space.

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5.5 Killing Vector Fields of Constant Length on GO-spaces In this section, we describe the relationship between Killing vector fields of constant length on a given geodesic orbit Riemannian space (M = G/H, g) and the structure of its isometry group G. As usual, the Lie algebra RG of the isometry group G of the space (M = G/H, g) is identified with the Lie algebra g of Killing vector fields on (M = G/H, g). The following remarkable result holds (cf. [360]). Theorem 5.5.1. Let (M = G/H, g) be a geodesic orbit Riemannian space and g be its Lie algebra of Killing vector fields. Suppose that a is a commutative ideal in g. Then every Killing field X ∈ a has constant length on (M = G/H, g). As a corollary, g(X,Y ) ≡ const on M for every X,Y ∈ a. Proof. Let x be any point in M. We will prove that x is a critical point of the function y ∈ M 7→ gy (X, X). Since (M = G/H, g) is homogeneous, it suffices to prove that gx ([Y, X], X) = 0 for every Y ∈ g (see Lemma 5.2.5). Consider any Y ∈ a, then Y · g(X, X) = 2g([Y, X], X) = 0 on M, since the ideal a is abelian. Now, consider Y ∈ g such that gx (Y,U) = 0 for every U ∈ a. We will prove that gx ([Y, X], X) = 0. By Lemma 5.2.5, for the vector X(x) ∈ Mx , there is a Killing field Z ∈ g such that Z(x) = X(x) and gx ([V, Z], Z) = 0 for any V ∈ g. In particular, gx ([Y, Z], Z) = 0. Now, W = X − Z vanishes at x and we get gx ([Y, X], X) = gx ([Y, Z +W ], Z +W ) = gx ([Y, Z +W ], Z) = gx ([Y, Z], Z) + gx ([Y,W ], Z) = gx ([Y,W ], Z). Note that gx ([Y,W ], Z) = −gx ([W,Y ], Z) = gx (Y, [W, Z]) = 0 because W (x) = 0 (0 = W · g(Y, Z)|x = gx ([W,Y ], Z) + gx (Y, [W, Z])) and [W, Z] = [X, Z] ∈ a. Therefore, gx ([Y, X], X) = 0. Hence, x is a critical point of the function y ∈ M 7→ gy (X, X). Since every x ∈ M is a critical point of the function y ∈ M 7→ gy (X, X), the Killing field X has constant length on (M = G/H, g). The last assertion follows from the equality 2g(X,Y ) = g(X + Y, X + Y ) − g(X, X) − g(Y,Y ). t u Note that Theorem 5.5.1 could be applied to a given geodesic orbit Riemannian manifold (M, g). To do so, one should use a representation M = G/H, where G is the full connected isometry group Isom0 (M, g) of the manifold (M, g). In this case, g is the Lie algebra of all Killing vector fields on (M, g).

5.6 Geodesic Orbit Manifolds of Non-positive Ricci Curvature In this section we prove Theorem 5.6.1, characterizing geodesic orbit Riemannian manifolds of non-positive Ricci curvature. Examples of such manifolds are Euclidean spaces, as well as symmetric spaces of noncompact type. It turns out that

5.6 Geodesic Orbit Manifolds of Non-positive . . .

289

the list of GO-manifolds of non-positive Ricci curvature is exhausted by symmetric spaces. Theorem 5.6.1 (C. Gordon, [230]). Every geodesic orbit Riemannian manifold with non-positive Ricci curvature is a symmetric space. Remark 5.6.2. It should be noted that the original proof of this theorem (Theorem 5.1 in [230]) has an error in the claim “Since U ∗ /L∗ is a compact homogeneous space, its Ricci curvature Ric∗ is nonnegative”. Nevertheless, this error can be corrected, and the proof in [230] requires only a little modification. But here we give a simpler proof [360], in which some constructions from [230] are essentially used. It suffices to prove Theorem 5.6.1 for simply connected manifolds. Indeed, if a Riemannian homogeneous manifold M has a Riemannian symmetric space of nonpositive Ricci curvature (equivalently, non-positive sectional curvature) as a universal covering space, then it is symmetric too [483, 256]. Let (M, g) be a simply connected geodesic orbit Riemannian manifold with nonpositive Ricci curvature, and let G be its full connected isometry group. We know that G is unimodular (Lemma 5.2.10), and the isotropy subgroup H must be connected. First, we reduce the problem to the case when G is semisimple. Proposition 5.6.3. Let (M, g) be a simply connected GO-manifold with non-positive Ricci curvature. Then it is a direct metric product of a Euclidean space Em , m ≥ 0, and a simply connected GO-manifold (M1 , g1 ) (of non-positive Ricci curvature) with semisimple full isometry groups. Proof. Recall that a Lie algebra g is semisimple if and only if it has no nontrivial abelian ideal. Let g be the Lie algebra of Killing fields on (M, g). Now, suppose that a is a nontrivial abelian ideal of g. By Theorem 5.5.1, any nontrivial Killing field X ∈ a has constant length on (M, g). Since the Ricci curvature of (M, g) is non-positive, by Theorem 3.5.6 we get that Ric(X, X) = 0, moreover, the Killing field X is parallel on (M, g), and the Riemannian manifold (M, g) is a direct metric product of two Riemannian manifolds, one of which is a one-dimensional manifold e ge), is a simply connected E1 tangent to the Killing field X, and another one, say (M, GO-manifold with non-positive Ricci curvature. e ge) etc., unless the last obtained RieThis procedure could be repeated with (M, mannian manifold has a semisimple full isometry group (i.e. the Lie algebra of all Killing fields on the latter manifold has no nontrivial abelian ideal). This proves the proposition. t u In what follows we suppose that the group G is semisimple. Now, we consider a reductive decomposition (see (5.9)) g = h ⊕ p, where g = LG, h = LH, and p is an orthogonal complement to h in g with respect to the Killing form Bg of the (semisimple) Lie algebra g (see Example 4.2.12). The

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5 Manifolds With Homogeneous Geodesics

Riemannian metric g is G-invariant and is determined by an Ad(H)-invariant inner product g = (·, ·) on the space p. Now we consider a maximal compactly embedded subalgebra k ⊂ g such that h ⊂ k. If h = k, then the manifold under consideration is a symmetric space [483, 256]. Suppose now that h 6= k. Then there are Ad(H)-invariant subspaces p1 , p2 ⊂ p such that (p1 , p2 ) = 0, p = p1 ⊕ p2 , and k = h ⊕ p1 . Let K ∗ be a compact connected Lie group with the Lie algebra k and H ∗ be its subgroup, corresponding to the subalgebra h ⊂ k. We have k = h ⊕ p1 , therefore, p1 could be identified with the tangent space at the point eH ∗ of a compact homogeneous manifold M ∗ = K ∗ /H ∗ . We consider a K ∗ -invariant Riemannian metric g∗ on M ∗ that is generated by the inner product (·, ·)|p1 . Note that K ∗ may not act effectively on M ∗ = K ∗ /H ∗ , but this is not important for the calculation of the Ricci curvature of (M ∗ , g∗ ). We choose a (·, ·)-orthonormal basis X1 , X2 , . . . , Xr , r = dim(p1 ), in p1 , and (·, ·)orthonormal basis Y1 ,Y2 , . . . ,Ys , s = dim(p2 ), in p2 . Denote the Ricci curvature of (M ∗ , g∗ ) by Ric∗ and the Killing forms of k and g by Bk and Bg respectively. Using the standard formula for the Ricci curvature (see Theorem 4.14.2) and the fact that g is unimodular (Lemma 5.2.10 is not necessary in this case, since every semisimple Lie algebra is unimodular), we get 1 1 1 Ric(X, X) = − Bg (X, X) − ∑ |[X, Xi ]p |2 − ∑ |[X,Y j ]p |2 2 2 i 2 j +

1 1 1 ([Xi , X j ]p , X)2 + ∑([Yi ,Y j ]p , X)2 + ∑([Xi ,Y j ]p , X)2 4∑ 4 2 i, j i, j i, j

and 1 1 1 Ric∗ (X, X) = − Bk (X, X) − ∑ |[X, Xi ]p1 |2 + ∑([Xi , X j ]p1 , X)2 2 2 i 4 i, j for every X ∈ p1 (Ric∗ (X, X) = 0 for dim(p1 ) = 1). According to Lemma 5.2.9, Bg (X, X) = Bk (X, X) + ∑([X, [X,Yi ]]p ,Yi ) = Bk (X, X) − ∑ |[X,Yi ]p2 |2 , i

i

then, using Lemma 5.2.9 once more (([Xi ,Y j ]p , X)2 = ([Y j , Xi ]p1 , X)2 = ([Y j , X]p1 , Xi )2 , ∑([Xi ,Y j ]p , X)2 = ∑ |[X,Y j ]p1 |2 ), we get i, j

j

Proposition 5.6.4 ([230]). For every X ∈ p1 the equality Ric∗ (X, X) = Ric(X, X) − holds.

1 ∑ ([Yi ,Y j ]p1 , X)2 2 1≤i< j≤s

5.7 Compact Geodesic Orbit Spaces

291

Since (M = G/H, g) has non-positive Ricci curvature, by Proposition 5.6.4, we get Ric∗ (X, X) ≤ 0 for every X ∈ p1 , and the equality Ric∗ (X, X) = 0 implies Ric(X, X) = 0 and ([p2 , p2 ]p1 , X) = 0. Since (M ∗ = K ∗ /H ∗ , g∗ ) is a compact homogeneous Riemannian manifold with non-positive Ricci curvature, it is a Euclidean torus by Corollary 4.15.15. Therefore, Ric∗ (X, X) = Ric(X, X) = 0 for all X ∈ p1 , p1 lies in the center of k, and [p2 , p2 ] ⊂ h ⊕ p2 by Proposition 5.6.4. Let m be the Bg -orthogonal complement to k in g. It is well known (see e.g., § 2 of Chapter VI in [256]) that [m, m] ⊂ k and [k, m] = m (the latter equality follows, for instance, from the fact that a symmetric space of noncompact type is a direct metric product of (strongly isotropy) irreducible symmetric spaces); if h1 := [m, m], then g1 := h1 ⊕ m is a maximal semisimple ideal of noncompact type in the Lie algebra g. Now we will prove that m = p2 . By the last assertion of Lemma 5.2.9 we get that for any U ∈ p1 , the operator adUp is skew symmetric ([h, p1 ] = 0 since p1 lies in the center of k). The same is true for any U ∈ h, and, therefore, for any U ∈ k = h ⊕ p1 . From the relations m ⊂ p, h1 ⊂ k, [k, p1 ] = 0, and [k, m] = m we get (p1 , m) = (p1 , [k, m]p ) = ([k, p1 ]p , m) + (p1 , [k, m]p ) = 0, since the operators adUp are skew symmetric for all U ∈ k. This proves the equality m = p2 . Therefore, h1 := [m, m] = [p2 , p2 ] ⊂ h ⊕ p2 , and we get h1 ⊂ h. Let g2 be the Bg -orthogonal complement to g1 in g. Then g2 is a maximal compact semisimple ideal in the Lie algebra g. If h2 is the Bg -orthogonal complement to h1 in h, then g2 = h2 ⊕ p1 . Recall that [h2 , p1 ] ⊂ [h, p1 ] = 0. Therefore, h2 is an ideal in the Lie algebra g. Since the space G/H is effective, h2 is trivial. Since p1 = g2 is commutative (p1 lies in the center of k), it is also trivial (otherwise, g2 is not semisimple). Hence, h = k, and (M = G/H, g) is a symmetric space. Theorem 5.6.1 is completely proved.

5.7 Compact Geodesic Orbit Spaces Let (M = G/H, g) be a compact homogeneous Riemannian space. This means that the isometry group Isom(M, g) of (M = G/H, g) is compact and the closure of G in Isom(M, g) is compact. In particular, there exists a (possibly, non-unique) Ad(G)invariant inner product b = h·, ·i on the Lie algebra g of the group G. Let us fix such inner product (for instance, if G is semisimple, then one can take the minus Killing form of the Lie algebra g), and further we will assume that the reductive decomposition g = h ⊕ p is b-orthogonal. For any subspace m in g, Xm means the b-orthogonal projection prm (Xm ) of a vector X ∈ g to m, bm means the restriction of the form b on m, and Lm = prm ◦L ◦ prm is the projection to m of an endomorphism L, defined on an enclosing space.

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5 Manifolds With Homogeneous Geodesics

An inner product (·, ·) on p, corresponding to the Riemannian metric g (we will also denote it here by g for brevity) is determined by a unique b-symmetric positive definite Ad(H)-equivariant metric endomorphism A on p, given by the relations g = (·, ·) = bp (A·, ·) = hA·, ·i|p . As in the previous sections, for a subspace m ⊂ p, we denote by m⊥ its (·, ·)orthogonal complement in p, and 1m will denote the identical operator on m. Now, we write down the homogeneity criterion for geodesics and the criterion for compact homogeneous spaces to be geodesic orbit in terms of the operator A (cf. [16] and [419]). Theorem 5.7.1. Let (M = G/H, g) be a compact homogeneous Riemannian manifold, Y ∈ h, X ∈ p. Then the curve γ(t) = exp(t(X +Y ))(o) is a geodesic if and only if [Y + X, AX] = 0. Consequently, (M = G/H, g) is a geodesic orbit Riemannian space if and only if for every X ∈ p, there exists a Z ∈ h such that [X + Z, AX] = 0. Proof. According to Theorem 5.1.2, the curve γ(t) is a geodesic if and only if ([Y +X,W ]p , X) = 0 for all W ∈ p. Using the Ad(G)-invariance of the form b = h·, ·i, we get 0 = ([Y + X,W ]p , X) = h[Y + X,W ]p , AXi = −hW, [Y + X, AX]p i. Since W ∈ p can be chosen arbitrarily, [Y +X, AX] ∈ h. If we prove that [Y +X, AX] ∈ p, then we get the first assertion of the theorem. Since p is Ad(H)-invariant and AX ∈ p, we get [Y, AX] ∈ p. Therefore, it remains to prove that [X, AX] ∈ p. Chose any vector V ∈ h and prove that h[X, AX],V i = 0. By using the ad(h)-equivariance and the b-symmetry of A, we get h[X, AX],V i = −hX, [V, AX]i = −hX, A[V, X]i = −hAX, [V, X]i = −h[X, AX],V i, which implies that h[X, AX],V i = 0. In the above arguments we have used the fact that every operator ad(W ), W ∈ g, is skew symmetric on g with respect to b due to the Ad(G)-invariance of b = h·, ·i. Therefore, [X, AX] ∈ p, and the first assertion is proved. The second assertion of the theorem is an obvious consequence of the first one. t u Lemma 5.7.2. Let (M = G/H, g) be a compact homogeneous Riemannian space with the metric endomorphism A and let p = p1 ⊕ p2 ⊕ · · · ⊕ pk

(5.15)

be a decomposition of p as the sum of the eigenspaces of the operator A, where A|pi = λi · 1pi , λi 6= λ j for i 6= j. Then (pi , p j ) = hpi , p j i = 0

(5.16)

5.7 Compact Geodesic Orbit Spaces

293

and the Ad(H)-modules pi satisfy the inclusion [pi , p j ] ⊂ p for i 6= j. Proof. Since A commutes with Ad(H), the eigenspaces pi are Ad(H)-invariant, and for X ∈ pi , Y ∈ p j , i 6= j, we get λi hX,Y i = hAX,Y i = (X,Y ) = hX, AY i = λ j hX,Y i. This implies the equality (5.16). The inclusion [pi , p j ] ⊂ p follows from the fact that p j is Ad(H)-invariant and h[pi , p j ], hi = hpi , [p j , h]i = 0. t u Proposition 5.7.3. Let (M = G/H, g) be a compact geodesic orbit space with the metric endomorphism A. Then the following assertions hold: 1) Let X,Y ∈ p be eigenvectors of the metric endomorphism A with different eigenvalues λ , µ. Then [X,Y ] =

λ µ [Z, X] + [Z,Y ] λ −µ λ −µ

for some Z ∈ h. 2) Assume that the vectors X,Y belong to pλ , the λ -eigenspace of A, and X is gorthogonal to the subspace [h,Y ]. Then [X,Y ] ∈ h + pλ . Proof. 1) Let us chose the vector Z ∈ h as in Theorem 5.7.1. Then [Z + X +Y, A(X +Y )] = [Z + X +Y, λ X + µY ] = λ [Z, X] + µ[Z,Y ] + (µ − λ )[X,Y ] ∈ h. By Lemma 5.7.2, we get [Z, X], [Z,Y ], [X,Y ] ∈ p, and the right-hand side is zero. 2) Let us chose an eigenvector W of the endomorphism A with an eigenvalue µ 6= λ . Then µ ([X,Y ]p ,W ) = µh[X,Y ],W i = µhX, [Y,W ]i = (X, [Y,W ]p ) λ   µ λ µ = X, [Z,Y ] + [Z,W ] = 0, λ λ −µ λ −µ where Z is chosen as in 1) for the vectors Y,W . This shows that [X,Y ] ∈ h + pλ . t u From this proposition we immediately obtain Corollary 5.7.4. Let (M = G/H, g) be a compact geodesic orbit space with the reductive decomposition (5.15) and the metric endomorphism A. Then for any AdH submodules mi ⊂ pi , m j ⊂ p j , i 6= j, we have [mi , m j ] ⊂ mi + m j .

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5 Manifolds With Homogeneous Geodesics

Moreover, if m and m0 are g-orthogonal AdH -submodules of pi , then [m, m0 ] ⊂ h + pi . Ad(H)-modules m and l are called disjunctive if they do not contain nontrivial equivalent submodules. If an Ad(H)-module p can be decomposed into a direct sum p = p1 ⊕ · · · ⊕ pk of disjunctive submodules, then it is not difficult to show that any Ad(H)-invariant inner product g and the corresponding metric endomorphism A are as follows: g = gp1 ⊕ · · · ⊕ gpk ,

A = Ap1 ⊕ · · · ⊕ Apk .

5.8 Applications of the Totally Geodesic Property In this section, we consider relationships between the geodesic orbit and totally geodesic properties of submanifolds of a geodesic orbit manifold, and discuss also some related questions. Note that a totally geodesic submanifold M 0 of a geodesic orbit Riemannian space M = G/H could have an isometry group that is not a subgroup of G. Example 5.8.1. The round sphere S2n−1 can be realized as the homogeneous Riemannian space U(2n)/U(2n − 1), but its totally geodesic submanifold S2n−2 admits only one connected transitive isometry group SO(2n − 1), which is not a subgroup of U(2n). It is possible to produce various generalizations of this example. This explains some troubles with the study of totally geodesic submanifolds of a given geodesic orbit Riemannian space. According to Theorem 6.6.2 (Theorem 11 in [73]), every closed totally geodesic submanifold of a geodesic orbit manifold is geodesic orbit itself. The disadvantage of this theorem is that it does not give any information about the isometry group of a submanifold. Here we will obtain some results, partially eliminating this gap. Further we consider a homogeneous Riemannian space (M = G/H, g) with the reductive decomposition g = h ⊕ p (5.9). Recall that m⊥ means the orthogonal complement to a given subspace m ⊂ p in p with respect to g = (·, ·). Definition 5.8.2. We say that a Lie subalgebra k ⊂ g is compatible with a given reductive decomposition if k = h0 ⊕ m, where h0 = k ∩ h and m = k ∩ p. A connected (possibly virtual) Lie subgroup K ⊂ G is said to be compatible with a given reductive decomposition if its Lie algebra k ⊂ g has this property.

5.8 Applications of the Totally Geodesic Property

295

Definition 5.8.3. A subspace m ⊂ p is called totally geodesic if there is a connected (possibly virtual) Lie subgroup K ⊂ G such that its Lie algebra has the form k = h0 ⊕ m, where h0 ⊂ h (in particular, it is compatible with a given reductive decomposition), and the orbit Ko ⊂ G/H = M of a subgroup K, passing through the point o = H, is a (possibly virtual) totally geodesic submanifold in (M = G/H, g). Proposition 5.8.4. A subspace m ⊂ p is totally geodesic if and only if the following two conditions hold: 1) m generates a Lie subalgebra of the type h0 ⊕ m, where h0 is a Lie subalgebra in h; m ⊥ 2) the endomorphism adm Z ∈ End(m), adZ (X) = [Z, X]m , for any Z ∈ m is g-skewsymmetric or, equivalently, U(m, m) ⊂ m. Proof. It is clear that Condition 1) is equivalent to the following: m ⊂ p is the tangent space at the point o to the orbit N := Ko of a minimal connected (possibly virtual) Lie subgroup K ⊂ G of the form mentioned in the previous definition. Let us identify the Lie algebra g of the Lie group G with the (corresponding) subalgebra of the Lie algebra of Killing vector fields on (M = G/H, g). Then it is easy to see that every element X of the Lie algebra k of the Lie group K is tangent to the submanifold N, i.e. X(x) ∈ Nx for every point x ∈ N. Hence, according to properties of the Levy-Civita connection ∇ on (M, g) and Proposition 1.6.8, the orbit N is totally geodesic if and only if ∇X Y is tangent to N for all X,Y ∈ k. According to (3.5), the latter property is equivalent to the following: U (X,Y ) is tangent to N. Since K is an isometry group both for M and N, it suffices to check the latter condition only for X,Y ∈ m at the point o. Since U(X,Y ) = U (X,Y )(o) (see the discussion right after the formula (4.31)), this is equivalent to the inclusion U(m, m) ⊂ m. t u Corollary 5.8.5. Let (M = G/H, g) be a Riemannian homogeneous space with the reductive decomposition (5.9) and K be a connected Lie subgroup of G, compatible with this decomposition. Then the orbit P = Ko = K/H 0 is a totally geodesic submanifold in (M = G/H, g) if and only if U(m, m) ⊂ m, or, equivalently, the endom ⊥ morphisms adm Z ∈ End(m), adZ (X) = [Z, X]m , for all Z ∈ m are g-skew-symmetric. We note that any connected Lie group K ⊂ G containing H as a subgroup is compatible with an arbitrary reductive decomposition. In the case of GO-spaces we can say more. Proposition 5.8.6. Let (M = G/H, g) be a geodesic orbit Riemannian space with the reductive decomposition (5.9). Then any connected Lie subgroup K ⊂ G, containing H, has a totally geodesic orbit P = Ko = K/H, which is a geodesic orbit space (with the induced metric). Proof. Let us consider a Lie algebra decomposition k = h ⊕ m, where m = k ∩ p. It is clear that m is an Ad(H)-invariant module. Since (M = G/H, g) is a geodesic orbit space (see Theorem 5.2.6 and Definition 5.1.3), for any X ∈ p, there exists a Y ∈ h such that [X,Y ] = U(X, X), where U is defined by the equality (4.31). If, moreover, X ∈ m, then U(X, X) = [X,Y ] ∈ m due to the Ad(H)-invariance of the module m.

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Hence, this module is totally geodesic by Proposition 5.8.4. Moreover, by virtue of the same equality [X,Y ] = U(X, X), the space P = Ko = K/H with the induced Riemannian metric is geodesic orbit (see Theorem 5.2.6). t u Now, we apply Corollary 5.8.5 to get a well-known result (see e.g. Theorem 1.14 in [230]). Corollary 5.8.7. Let (M = G/H, g) be a geodesic orbit Riemannian space and let K be a connected closed subgroup of G, normalized by H. Then the orbit K/K ∩ H of K through the point o = eH is a totally geodesic submanifold of M, hence, a geodesic orbit Riemannian manifold. In particular, the orbits of the radical R(G), of the nilradical N(G), and of a Levi group S, that are stable under H, through the point o = eH are geodesic orbit Riemannian manifolds. Proof. Since K is normalized by H, its Lie algebra k is Ad(H)-invariant. The Lie algebra h ∩ k (being Ad(H)-invariant) admits an Ad(H)-invariant complement m in k. On the other hand, the Ad(H)-invariant Lie subalgebra h + k admits an Ad(H)invariant complement q in g. Hence, we can take p := m ⊕ q as an Ad(H)-invariant complement to h in g. Without loss of generality, we may assume that q = m⊥ in p with respect to g = (·, ·). It is easy to see that k = (k ∩ h) ⊕ (k ∩ p) = (k ∩ h) ⊕ m, and [m, m] ⊂ m ⊕ (k ∩ h) ⊂ m ⊕ h. Hence, it suffices to apply Corollary 5.8.5 and ⊥ Theorem 6.6.2, since adm u Z is g-skew-symmetric for all Z ∈ m by Lemma 5.2.9. t In general, we do not know the details of the reductive complement m in the proof of this corollary (since m depends on the subgroup K). Hence, we could not obtain general results on the isometry group G0 with respect to which a given orbit of K is geodesic orbit space. But it is possible to get such results for some subgroups K of special types. It should be noted also that the space K/K ∩ H in Corollary 5.8.7 is not necessarily a geodesic orbit Riemannian space (in this case the full isometry group is more extensive than K). On the other hand, it is easy to show that the geodesic orbit Riemannian manifold K/K ∩ H could be realized as geodesic orbit space K · H/H (it need not be effective, but one can always obtain an effective realization by dividing by the effective kernel). Example 5.8.8. Let us consider the following weakly symmetric, hence, geodesic orbit, space G/H = Sp(n + 1)U(1)/Sp(n)U(1) (diffeomorphic to the sphere S4n+3 ). This space admits a 3-parameter family of invariant Riemannian metrics that are geodesic orbit. The subalgebra sp(n + 1) is stable under AdG (H), hence Sp(n + 1)/Sp(n) with induced metrics is a totally geodesic (in fact it coincides with the original manifold S4n+3 ) and hence geodesic orbit Riemannian manifold. On the other hand, the space of Sp(n + 1)-invariant geodesic orbit metrics on Sp(n + 1)/Sp(n) is only 2-dimensional. Therefore, for a suitable choice of Sp(n + 1)U(1)-invariant metrics (in fact, for almost all such metrics) on S4n+3 , the orbit of Sp(n + 1) through the point o = eSp(n)U(1) is not a geodesic orbit space. One can find details in [362], see also Remark 2 in [196].

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297

We emphasize in particular the fact that the study of general geodesic orbit Riemannian manifolds does not reduce completely to the study of nilmanifolds and homogeneous spaces G/H with semisimple G. In what follows, we consider the case of compact homogeneous Riemannian spaces (M = G/H, g), which often allows us to get more detailed information. We assume that the reductive decomposition (5.9) is orthogonal with respect to some fixed Ad(G)-invariant inner product b = h·, ·i on the Lie algebra g, and the inner product g = (·, ·), is generated by a metric endomorphism A : p → p, i.e. g = (·, ·) = bp (A·, ·) = hA·, ·i|p . Definition 5.8.9. Let (M = G/H, g) be a compact homogeneous Riemannian manifold with the reductive decomposition (5.9) and K a connected Lie subgroup of the Lie group G, compatible with this decomposition (i.e. k = h0 ⊕ m, where h0 ⊂ h, m ⊂ p). We say that the Lie subgroup K ⊂ G is compatible with the invariant Riemannian metric g if the ad(h0 )-module m is A-invariant, i.e. Am = m. Proposition 5.8.10. If a Lie subgroup K is compatible with the invariant Riemannian metric g, then (in the notation of Definition 5.8.9) the following assertions hold. 1) The subspace m⊥ (the g-orthogonal complement to m in p) is A-invariant and b-orthogonal to the subspace m. 2) The subspace m is totally geodesic. 3) If (M = G/H, g) is a geodesic orbit space, then the (totally geodesic) orbit P = Ko = K/H 0 (supplied with the induced metric) is also a geodesic orbit space. e e is a connected 4) If m is ad(h)-invariant, then m⊥ is Ad(K)-invariant, where K (possibly virtual) Lie subgroup in G, corresponding to the Lie subalgebra ek = h ⊕ m ⊂ g. Proof. The first assertion follows from the equalities 0 = (m, m⊥ ) = hAm, m⊥ i = hm, m⊥ i and hm, Am⊥ i = hAm, m⊥ i = hm, m⊥ i = 0. Further, since Am = m, Am⊥ = m⊥ and hm, m⊥ i = 0, we have hZ, [X, AX]i = 0 for all X ∈ m and Z ∈ m⊥ . This observation and Proposition 5.8.4 imply the second assertion: 0 = h[Z, X], AXi = h[Z, X]p , AXi = ([Z, X]p , X) = (U(X, X), Z). Now let us prove the third assertion. According to Theorem 5.7.1, for every X ∈ p, there exists a Y ∈ h such that [X + Y, A(X)] = 0. Put X ∈ m and let Y 0 be the borthogonal projection of Y to h0 = h ∩ k. Since hY −Y 0 , ki = 0, we have [Y −Y 0 , m] ⊂ m⊥ (since h[Y −Y 0 , m], mi ⊂ hY −Y 0 , [m, m]i ⊂ hY −Y 0 , ki = 0). On the other hand, [Y 0 , A(X)] ⊂ [k, m] ⊂ k and [X, A(X)] ⊂ [m, m] ⊂ k. Hence, [Y − Y 0 , A(X)] = 0 and [X + Y 0 , A(X)] = 0. Applying Theorem 5.7.1 again, we get that the homogeneous space K/H 0 with the induced Riemannian metric is geodesic orbit. Note now that the endomorphisms of the type ad(Z), where Z ∈ ek = h ⊕ m, are b-skew symmetric and preserve the Lie subalgebra ek. Therefore, they also preserve e is connected, its b-orthogonal complement in g, which coincides with m⊥ . Since K we get the fourth assertion of the proposition. t u We strengthen Proposition 5.8.6.

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Proposition 5.8.11 ([18]). Let (M = G/H, g) be a compact geodesic orbit Riemannian space with the reductive decomposition (5.9). Then every connected Lie subgroup K ⊂ G, containing H, has a totally geodesic orbit P = Ko = K/H, which is a geodesic orbit space (with the induced metric). Moreover, if the Lie subgroup K is compatible with the invariant Riemannian metric g, then [k, m⊥ ] ⊂ m⊥ , and the inner product g|m⊥ = (·, ·)|m⊥ is Ad(K)invariant and defines an invariant geodesic orbit metric g¯ on the homogeneous space N = G/K. In this case, the projection π : G/H → G/K is a Riemannian submersion with totally geodesic fibers, where both the fibers and the base are geodesic orbit spaces. Proof. The first assertion is proved in Proposition 5.8.6. The correctness of the definition of the invariant Riemannian metric g¯ on N = G/K follows from the fourth e = K) and Lemma 5.2.9. Moreover, assertion in Proposition 5.8.10 (in this case K from the first assertion in Proposition 5.8.10 we obtain that Am⊥ = m⊥ , and the restriction of the operator A to m⊥ is a metric endomorphism for (G/K, g). ¯ Further, according to Theorem 5.7.1, for each X ∈ p, there exists a Y ∈ h such that [X +Y, A(X)] = 0. It is clear that for X ∈ m⊥ , Y chosen in this manner satisfies the condition [X +Y, A(X)] = 0, i.e. (G/K, g) ¯ is a GO-space. t u In what follows, we shall need the following result (see e.g. [13]). Proposition 5.8.12. Let (M = G/H, g) be a homogeneous Riemannian space, T a connected commutative Lie subgroup in the Lie group H, and C(T ) the unit component of its centralizer in G. Then the orbit P = C(T )o = C(T )/H 0 (H 0 = C(T ) ∩ H) is totally geodesic. If, moreover, (M = G/H, g) is a compact geodesic orbit space of positive Euler characteristic, then the space P = C(T )o = C(T )/H 0 with the induced Riemannian metric is also geodesic orbit. Proof. Let us fix a reductive decomposition (5.9). Let u be the Lie algebra of the group C(T ) and t ⊂ u the Lie algebra of the Lie group T ⊂ C(T ). Note that [t, u] = 0. Put X = Xh + Xp ∈ u; since [t, X] ⊂ u, [t, Xh ] ⊂ h and [t, Xp ] ⊂ p, the group C(T ) is compatible with the decomposition (5.9). Put u = h0 ⊕ m, where h0 = h ∩ u and m = p∩u. Let us show that U(X, X) ∈ m for every X ∈ m. For this, it suffices to show that ([Y,U(X, X)], Z) = 0 for all Y ∈ t and all Z ∈ m⊥ . Using the ad(h)-invariance of the inner product g = (·, ·) and the definition of the operator U (cf. the equality (4.31)), we get ([Y,U(X, X)], Z) = −(U(X, X), [Y, Z]) = −([[Y, Z], X]p , X) = −([Y, [Z, X]]p , X) = −([Y, [Z, X]p ], X) = ([Z, X]p , [Y, X]) = 0. In the above calculations, we used the equality [Y, X] = 0 and the Jacobi identity ([[Y, Z], X] = −[[Z, X],Y ] − [[X,Y ], Z] = −[[Z, X],Y ] = [Y, [Z, X]]). Thus, U(X, X) ∈ m and we obtain the first assertion by Proposition 5.8.4. Let us prove the second assertion. We may assume that the reductive decomposition is orthogonal with respect to an Ad(G)-invariant inner product b = h·, ·i on

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the Lie algebra g and the inner product g = (·, ·) is generated by some metric endomorphism A : p → p. By Assertion 3) from Proposition 5.8.10, it suffices to show that the subgroup C(T ) is compatible with the invariant Riemannian metric g, i.e. the subspace m is invariant under the metric endomorphism A. Let Tmax be a maximal torus in H containing the group T . Since the Euler characteristic of the space G/H is positive, Tmax is also a maximal torus in the group G (see Theorem 4.3.1). The subspace p ⊂ g is an Ad(Tmax )-invariant direct sum of some two-dimensional root subspaces in g with respect to a Cartan subalgebra LTmax , and all these subspaces are not pairwise Ad(Tmax )-isomorphic (see Proposition 2.5.29). The metric endomorphism A, being Ad(H)-equivariant, is also Ad(Tmax )-equivariant. Thus, all two-dimensional root subspaces in p are invariant with respect to the endomorphism A. Since Tmax ⊂ C(T ), it follows that m is also a direct sum of some two-dimensional root subspaces in g with respect to the Cartan subalgebra LTmax . Therefore, m is invariant under the metric endomorphism A. This proves the second assertion. t u We note that a connected Lie subgroup K ⊂ G, containing H, is compatible with any invariant metric on G/H if the Ad(H)-modules m and p/m (m = k ∩ p) are disjunctive. This observation implies Proposition 5.8.13. Let M = G/H be a homogeneous space with compact H. Then the unit component N0 (Z) of the normalizer of any central subgroup Z of the group H as well as the unit component N0 (H) of the normalizer of H are subgroups that are compatible with any invariant Riemannian metric on M = G/H. Using arguments analogous to those used in the proof of the second assertion of Proposition 5.8.12, we easily get Proposition 5.8.14. Let M = G/H be a compact homogeneous space of positive Euler characteristic and Tmax be some common maximal torus of the groups H and G. If some connected Lie subgroup K ⊂ G contains a torus Tmax , then K is compatible with any invariant Riemannian metric on the homogeneous space M = G/H.

5.9 Geodesic Orbit Riemannian Metrics on Spheres In this section, the complete classification of geodesic orbit Riemannian metrics on spheres Sn is provided, see [362]. We discuss in detail some properties of these metrics. Finally, we consider the classification of geodesic orbit Riemannian metrics on projective spaces, see Example 5.9.4. Recall that the classification of all transitive and effective actions of connected compact Lie groups on spheres is obtained in [340], see Examples 4.2.6 and 4.11.4. We collect in Table 5.1 all variants to represent Sn as a homogeneous space G/H. In this table, by dim(G/H) we denote the dimension of the corresponding space (and the corresponding sphere), and dim(M ) (respectively, dim(MGO )) means the

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Table 5.1 Invariant and geodesic orbit Riemannian metrics on spheres G

H

dim(G/H)

dim(M )

dim(MGO ) Cond.

1

SO(n + 1)

SO(n)

n

1

1

2

G2

SU(3)

6

1

1

3

Spin(7)

G2

7

1

1

4

SU(2)

{e}

3

6

1

5

SU(n + 1)

SU(n)

2n + 1

2

2

n≥2

6

U(n + 1)

U(n)

2n + 1

2

2

n≥1

7

Spin(9)

Spin(7)

15

2

2

8

Sp(n + 1)Sp(1)

Sp(n) diag(Sp(1))

4n + 3

2

2

n≥1

9

Sp(n + 1)U(1)

Sp(n) diag(U(1))

4n + 3

3

3

n≥1

10

Sp(n + 1)

Sp(n)

4n + 3

7

2

n≥1

n≥1

dimension of the space of G-invariant (the space of G-invariant geodesic orbit) Riemannian metrics on the homogeneous space G/H. Note that Riemannian metrics of constant sectional curvature constitute a 1parameter family of metrics on each sphere Sn , n ≥ 2 (there is no such notion for the trivial case n = 1). These are exactly the metrics invariant under the action of the orthogonal group O(n + 1) and under its connected unit component SO(n + 1). These groups are respectively the full isometry group and the full connected isometry group of each metric of constant curvature on Sn . We list all inclusions between the isometry groups in Table 5.1: G2 ⊂ SO(7), SU(k) ⊂ U(k) ⊂ SO(2k), Sp(k) ⊂ Sp(k)U(1) ⊂ Sp(k)Sp(1) ⊂ SO(4k), Sp(k) ⊂ Sp(k)U(1) ⊂ U(2k), SU(4) ⊂ Spin(7) ⊂ SO(8), Spin(9) ⊂ SO(16) (see Example 4.2.6 and Chapter 4 of [377]). Of course, geodesic orbit metrics on some spaces in Table 5.1 are well known. Below we describe all the cases in this table. Case 1). The homogeneous space SO(n + 1)/SO(n) is irreducible symmetric, all SO(n + 1)-invariant Riemannian metrics are SO(n + 1)-normal homogeneous (hence, geodesic orbit) and constitute the set of Riemannian metrics of constant sectional curvature on Sn . This set is a part of any other family of invariant metrics from Table 5.1. Cases 2) and 3). The spaces G2 /SU(3) and Spin(7)/G2 are isotropy irreducible. All invariant metrics on these spaces are normal homogeneous (hence, GO-metrics) and have constant sectional curvature (see Section 7 in [106]). Case 4). All left-invariant metrics on a compact Lie group G, that are geodesic orbit with respect to G, should be bi-invariant (see Example 5.2.8 or Proposition 8 in [18]). Since the group SU(2) is simple, then all bi-invariant Riemannian metrics on SU(2) constitute a one-parameter family of metrics. Since SU(2)2 / diag(SU(2))

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301

= SO(4)/SO(3), these metrics are exactly metrics of constant curvature on S3 = SU(2). Cases 5) and 6). Note that the set of U(n + 1)-invariant metrics on S2n+1 coincides with the set of SU(n + 1)-invariant metrics and constitutes a 2-parametric family of metrics. Every such metric is naturally reductive and weakly symmetric [501, 502, 503]. Therefore, in both these cases we have a two-parameter family of geodesic orbit metrics. Case 7). The family of invariant metrics on Spin(9)/Spin(7) is 2-parametric. All these metrics are weakly symmetric but there are only two 1-parametric families of naturally reductive metrics among them, namely, all metrics proportional either to the SO(16)-normal metric or to the Spin(9)-normal metric on Spin(9)/Spin(7) [502, 503]. Therefore, we have a two-parameter family of geodesic orbit metrics. Case 8). The family of invariant metrics on Sp(n + 1)Sp(1)/Sp(n) diag(Sp(1)) is 2-parametric (we use the standard inclusion Sp(n) × Sp(1) ⊂ Sp(n + 1) to define diag(Sp(1))). Every such metric is naturally reductive and weakly symmetric [502, 503]. Therefore, in both these cases we have a 2-parameter family of geodesic orbit metrics. More details on this case can be found in Sections 4.8.18. Case 9). Note that the previous family is a part of this one. The family of Sp(n + 1)U(1)-invariant metrics on Sp(n + 1)U(1)/Sp(n) diag(U(1)) = S4n+3 is 3-parametric (we use the inclusions Sp(n) ×U(1) ⊂ Sp(n) × Sp(1) ⊂ Sp(n + 1) to define diag(U(1))). Every such metric is weakly symmetric (see e.g. 12.9.2 in [484] or Table 1 in [492]). Therefore, in this case we have a 3-parameter family of geodesic orbit metrics. Details on the normality and the natural reductivity of Sp(n + 1)U(1)invariant metrics can be found in Remark 5.9.2. Some explicit forms of geodesic vectors for Sp(n + 1) ×U(1)-invariant metrics can be found in [362]. Case 10). In the last case we get a 7-dimensional space of Sp(n + 1)-invariant metrics on Sp(n + 1)/Sp(n) = S4n+3 . We deal with this case in Subsections 5.9.1 and 5.9.2. As was proved in Proposition 5.9.1, a homogeneous Riemannian space (S4n+3 = Sp(n + 1)/Sp(n), g) is geodesic orbit if and only if the metric g is invariant under Sp(n + 1)Sp(1).

5.9.1 On Invariant Metrics on S4n+3 for Transitive Actions of the Groups Sp(n + 1), Sp(n + 1)U(1), and Sp(n + 1)Sp(1) Let H be the field of quaternions. Denote by i, j, k the quaternionic units in H (ij = −ji = k, jk = −kj = i, ki = −ik = j, ii = jj = kk = −1). For X = x1 + ix2 + jx3 + kx√ 4 , xi ∈ R, define ℜ(X) = x1 (the real part of X), X = x1 − ix2 − jx3 − kx4 and kXk = XX. If ℜ(X) = x1 = 0, then the quaternion X is called pure imaginary. Let us consider a (left) vector space Hl over H. For X = (X1 , X2 , . . . , Xn+1 ) ∈ n+1

Hn+1 and Y = (Y1 ,Y2 , . . . ,Yn+1 ) ∈ Hn+1 we define (X,Y )1 = ∑ XsY s . Then s=1

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Sp(n + 1) is the group of all H-linear operators A : Hn+1 → Hn+1 with the property (A(X), A(Y ))1 = (X,Y )1 for every X,Y ∈ Hn+1 . If we choose some (·, ·)1 orthonormal quaternionic basis in Hn+1 , then we can identify Sp(n + 1) with a group of matrices A = (ai j ), ai j ∈ H with the property A−1 = A∗ , where a∗i j = a ji
for 1 ≤ i, j ≤ l. In this case sp(n + 1) consists of quaternionic (n + 1) × (n + 1) matrices A with the property A∗ = −A. Later on we shall use these identifications. We have a natural embedding H 7→ R4 via x1 + ix2 + jx3 + kx4 7→ (x1 , x2 , x3 , x4 ) and the induced embedding Hn+1 7→ R4n+4 . It is well known that the group G := Sp(n + 1) acts transitively on the sphere S4n+3 = {(X1 , X2 , . . . , Xn+1 ) ∈ Hn+1 | kX1 k2 + kX2 k2 + · · · + kXn+1 k2 = 1}. Let us consider the natural embedding diag(Sp(1), Sp(n)) ⊂ Sp(n + 1), and let K and H be the images of Sp(1) and Sp(n) respectively under this embedding. Then H is the isotropy subgroup of a point (1, 0, . . . , 0) ∈ Hn+1 under the above action of Sp(n + 1). Since K = Sp(1) is a normal subgroup in the group diag(Sp(1), Sp(n)), we have the following (almost effective) transitive action of G × K on S4n+3 = G/H: (a, b)(cH) = acHb−1 = acb−1 H,

a, c ∈ G, b ∈ K.

(5.17)

The isotropy group of this action at the point (1, 0, . . . , 0) ∈ Hn+1 is Sp(n) × Sp(1) = Sp(n) × diag(Sp(1)) ⊂ Sp(1) × Sp(n + 1) × Sp(1) ⊂ Sp(n + 1) × Sp(1) = G × K. We also get an effective representation S4n+3 = Sp(n + 1)Sp(1)/Sp(n) diag(Sp(1)) (after dividing by the noneffectiveness kernel). Let L be any subgroup U(1) = S1 in K = Sp(1). Then we get a transitive (and almost effective) action of G × L on S4n+3 = G/H: (a, b)(cH) = acHb−1 = acb−1 H,

a, c ∈ G, b ∈ L,

(5.18)

that is a part of the action (5.17). In this case we get the following isotropy group at the point (1, 0, . . . , 0) ∈ Hn+1 : Sp(n) ×U(1) = Sp(n) × diag(U(1)) ⊂ U(1) × Sp(n) ×U(1) ⊂ Sp(n + 1) ×U(1) = G × L. We also get an effective representation S4n+3 = Sp(n + 1)U(1)/Sp(n) diag(U(1)). For A, B ∈ sp(n + 1) we define hA, Bi =

1 trace(ℜ(AB∗ )). 2

(5.19)

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303

It is easy to see that h·, ·i is an Ad(Sp(n + 1))-invariant inner product on the Lie algebra g = sp(n + 1). We write Ei j for the skew-symmetric matrix with 1 in the i j-th entry and −1 in the ji-th entry, and zeros elsewhere. We denote by Fi j the symmetric matrix with 1 in both √ the i j-th and ji-th entries, and zeros elsewhere. Denote also by Gi the matrix with 2 in ii-th entry, and zeros elsewhere. It is easy to check that the matrices iGi , jGi , kGi , Ei j , iFi j , jFi j , kFi j , where 1 ≤ i, j ≤ n + 1 and i < j, constitute a h·, ·iorthonormal (see (5.19)) basis in sp(n + 1). Let us consider the following h·, ·i-orthogonal decomposition: sp(n + 1) = k ⊕ sp(n) ⊕ p1 = sp(n) ⊕ p,

k = l ⊕ p2 ,

(5.20)

where k and l are the Lie algebras of the Lie subgroups K and L (see above). Therefore, the embedding of k ⊕ sp(n) = sp(1) ⊕ sp(n) in sp(n + 1) is defined by (A, B) 7→ diag(A, B), where A ∈ sp(1) and B ∈ sp(n). Without loss of generality we may suppose that the Lie subalgebra l = u(1) (u(1) ⊕ sp(n) ⊂ sp(1) ⊕ sp(n) ⊂ sp(n + 1)) is spanned by the vector iG1 . Then p2 = Lin{jG1 , kG1 }. Any Sp(n + 1)-invariant metric on S4n+3 is defined by an Ad(Sp(n))-invariant inner product (·, ·) on p. Note that Ad(Sp(n)) acts irreducibly on p1 and trivially on k. Therefore, any such inner product is generated by a metric endomorphism of the type e ⊕ x1 Id |p A=A (5.21) 1 e : k → k (see for some x1 > 0 and some symmetric and positive definite operator A Section 4.11). In particular, this implies that the space of Sp(n + 1)-invariant Riemannian metrics on S4n+3 is 7-dimensional. e = x2 Id |p ⊕ x3 Id |l , then the inner product (·, ·) generates Sp(n + 1) × LIf A 2 e = x2 Id |k , then the inner product (·, ·) generates a invariant metric on S4n+3 . If A Sp(n + 1) × K-invariant metric on S4n+3 .

5.9.2 Sp(n + 1)-invariant Geodesic Orbit Metrics on the Sphere S4n+3 Here we classify all geodesic orbit metrics on the sphere S4n+3 with respect to the group Sp(n + 1). Proposition 5.9.1. A homogeneous Riemannian space (S4n+3 = Sp(n+1)/Sp(n), g) is geodesic orbit with respect to Sp(n + 1) if and only if the metric g is invariant under Sp(n + 1) × Sp(1). Proof. Originally this result was obtained in [362]. Here we provide simpler arguments.

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Suppose that a metric endomorphism A (see (5.21)) generates a Riemannian geodesic orbit metric g with respect to Sp(n + 1). Then by 5) of Theorem 5.4.1, we have that the inner product g = (·, ·) on p is not only Ad(Sp(n))-invariant but also Ad(NG (H0 ))-invariant, where NG (H0 ) is the normalizer of the unit component H0 of the group Sp(n) in G. It is clear that H0 = Sp(n) and NG (H0 ) = NG (Sp(n)) ⊃ Sp(1)×Sp(n) = K ×Sp(n). Obviously, an Ad(K ×Sp(n))-invariant inner product is a Sp(n+1)×Sp(1)-invariant inner product on Sp(n+1), since Ad(Sp(1)) = Ad(K) acts irreducibly on sp(1) = k. t u It is clear that Proposition 5.9.1 completes the classification for Case 10) in the Introduction. Therefore, we have verified completely all data from Table 5.1.

Remark 5.9.2. Let us note that for x1 = x2 6= x3 we get Sp(n + 1)U(1)-naturally reductive metrics on the sphere S4n+3 (they are even Sp(n + 1)U(1)-normal homogeneous for x3 < x1 = x2 ). All these metrics are also U(2n + 2)-invariant and U(2n + 2)-naturally reductive. By analogy, for x3 = x2 6= x1 we get Sp(n + 1)Sp(1)naturally reductive metrics on the sphere S4n+3 (for x3 = x2 < x1 these metrics are even Sp(n + 1)Sp(1)-normal homogeneous). Obviously, for x1 = x2 = x3 we get Sp(n + 1)-normal homogeneous metrics. For all other values of parameters xi , i = 1, 2, 3, Sp(n + 1)U(1)-invariant metrics are not naturally reductive. For details, see [502] and [503]. Remark 5.9.3. Consider the homogeneous spaces Sp(n + 1)/Sp(n) · U(1), where n ≥ 1, U(1) ⊂ Sp(1), and Sp(1) is the first factor in the group Sp(1) × Sp(n) ⊂ Sp(n + 1). The Lie algebra of the group Sp(n) · U(1) is l ⊕ sp(n) ⊂ sp(n + 1) in the decomposition (5.20). It is known that the homogeneous space Sp(n + 1)/Sp(n) · U(1) is diffeomorphic to CP2n+1 , hence we get a representation of an odd-dimensional complex projective space. It is also known that the space Sp(n + 1)/Sp(n) · U(1) admits a two-parameter family of Sp(n + 1)-invariant Riemannian metrics [502]. All these metrics are weakly symmetric [503]. It is interesting that only Sp(n + 1)-normal and SU(2n + 2)-normal metrics in this family are naturally reductive. Note that explicit expressions of geodesic vectors for Sp(n + 1)invariant Riemannian metrics on CP2n+1 = Sp(n + 1)/Sp(n) · U(1) can be found in [93]. Remark 5.9.4. All geodesic orbit Riemannian metrics from Table 5.1 induce geodesic orbit Riemannian homogeneous metrics on corresponding real projective spaces RPn . The metrics obtained in such a way, metrics from Remark 5.9.3 together with the normal homogeneous metrics on the projective spaces CPn = SU(n + 1)/S(U(n) × U(1)), HPn = Sp(n + 1)/Sp(n) × Sp(1), and CaP2 = F4 /Spin(9) exhaust all geodesic orbit Riemannian metrics on projective spaces (for details, see [502] and [503]).

5.10 Compact GO-spaces of Positive Euler Characteristic

305

5.10 Compact GO-spaces of Positive Euler Characteristic In this section, we obtain the classification of connected and simply connected geodesic orbit Riemannian manifolds of positive Euler characteristic.

5.10.1 Basic Facts on Compact Homogeneous Manifolds of Positive Euler Characteristic Here we recall some properties of compact connected homogeneous spaces of positive Euler characteristic (see Section 4.12 or [377] for more details). The homogeneous space M = G/H of a compact connected Lie group G has positive Euler characteristic χ(M) > 0 if and only if the isotropy group H has maximal rank (rk(H) = rk(G)). If the group G acts almost effectively on M, then it is semisimple and the univere H e = G/ e for M is a direct product sal covering space M e = G1 /H1 × · · · × Gk /Hk , M e = G1 × G2 × · · · × Gk is the decomposition of the group G e (which is a where G e is the universal covering group of G) into a direct product of simple factors, H e → G, inverse image of the unit component of the group H under the epimorphism G e ∩ Gi . and Hi = H e and An arbitrary invariant metric g on M defines an invariant metric ge on M, e e e the homogeneous Riemannian manifold (M = G/H, ge) is a direct metric product of homogeneous manifolds (Mi = Gi /Hi , gi ), i = 1, . . . , k, with simple Lie groups Gi . More precisely, according to Theorem 4.12.3, a compact simply connected homogeneous Riemannian manifold (M = G/H, g) of positive Euler characteristic is indecomposable if and only if the group G is simple. Hence it follows that a simply connected compact GO-space (M = G/H, g) of positive Euler characteristic is a direct metric product of simply connected GOspaces (Mi = Gi /Hi , gi ) of positive Euler characteristic with simple isometry groups. Thus, it suffices to classify simply connected GO-spaces of positive Euler characteristic with simple isometry groups. The description of simply connected homogeneous manifolds G/H of positive Euler characteristic reduces to the description of connected subgroups H of maximal rank of simple groups G or, equivalently, to the description of Lie subalgebras of maximal rank in simple compact Lie algebras g, see [117] and Section 8.10 in [483]. An important subclass of compact homogeneous manifolds of positive Euler characteristics are flag manifolds. They are orbits of the adjoint action M = Ad(G)x of compact connected semisimple groups G or, in other words, the quotient spaces M = G/H of semisimple groups G by the centralizers H = ZG (T ) of nontrivial tori T ⊂ G.

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5.10.2 The Classification Theorem Let G be a simple compact connected Lie group and H ⊂ K ⊂ G its closed connected subgroups. Denote by b = h·, ·i the minus Killing form of the Lie algebra g = LG and consider a b-orthogonal decomposition g = h ⊕ p = h ⊕ p1 ⊕ p2 , where k = h ⊕ p2 is the Lie algebra LK of the Lie group K. Obviously, [p2 , p1 ] ⊂ p1 . Let g = gx1 ,x2 be a G-invariant Riemannian metric on M = G/H, generated by an inner product g = (·, ·) on p of the following form: g = x1 · bp1 + x2 · bp2

(5.22)

where x1 and x2 are some positive numbers. Equivalently, we can say that the metric is generated by the metric endomorphism A = x1 · 1p1 + x2 · 1p2 .

(5.23)

Let us consider the following two families of such homogeneous Riemannian manifolds (M = G/H, gx1 ,x2 ): Example 5.10.1. (G, K, H) = (SO(2n + 1), SO(2n), U(n)), n ≥ 2. The group G = SO(2n + 1) acts transitively on the symmetric space Com(R2n+2 ) = SO(2n + 2)/U(n)) of complex structures on R2n+2 with stabilizer H = U(n), see [256]. So we can identify M = G/H with this symmetric space, but the metric gx1 ,x2 is not SO(2n + 2)-invariant if x2 6= 2x1 [284]. Example 5.10.2. (G, K, H) = (Sp(n), Sp(1) · Sp(n − 1),U(1) · Sp(n − 1)), n ≥ 2. The group G = Sp(n) acts transitively on the complex projective space CP2n−1 = SU(2n + 2)/U(2n + 1) with stabilizer H = U(1) · Sp(n − 1). So we can identify M = G/H with CP2n−1 , but the metric gx1 ,x2 is not SU(2n + 2)-invariant if x2 6= 2x1 , see [256, 284]. It should be noted that for n = 2 the spaces from these two families coincide: SO(5)/U(2) = Sp(2)/U(1) · Sp(1), see Example 4.12.14. This is a six-dimensional space with an invariant Riemannian metric gx1 ,x2 , the first compact example of a GO-space that is not naturally reductive [304]. Definition 5.10.3. A geodesic orbit space (M = G/H, g) of a simple compact Lie group G is called a proper geodesic orbit space if its metric g is not G-normal, i.e. the corresponding metric endomorphism A is not proportional to the identity operator. Now we can state the main theorem on compact geodesic orbit spaces of positive Euler characteristic.

5.10 Compact GO-spaces of Positive Euler Characteristic

307

Theorem 5.10.4 (D. V. Alekseevsky–Yu. G. Nikonorov, [18]). Let (M = G/H, g) be a simply connected proper geodesic orbit space of positive Euler characteristic with a simple compact Lie group G. Then either M = G/H = SO(2n + 1)/U(n), n ≥ 2, or M = G/H = Sp(n)/U(1) × Sp(n − 1), n ≥ 2, and g = gx1 ,x2 is any G-invariant metric which is not G-normal homogeneous. The metric g is G-normal homogeneous (respectively, symmetric) when x2 = x1 (respectively, x2 = 2x1 ). Moreover, all these manifolds are weakly symmetric flag manifolds. The full connected isometry group of any of the considered geodesic orbit spaces (M = G/H, gx1 ,x2 ) with non-symmetric metric gx1 ,x2 is the quotient of the group G by its center (see the discussions in [284], [377], [16]). The assertion that all these Riemannian manifolds are weakly symmetric spaces is proved in [503].

5.10.3 Proof of the Classification Theorem We reduce the proof to a description of some special decompositions of the root system of the Lie algebra g of the isometry group G. Let M = G/H be a homogeneous space of positive Euler characteristic of a compact simple Lie group G and let g = h+m be a suitable reductive decomposition. The subgroup H contains a maximal torus T of G. We consider the root space decomposition g C = tC ⊕

M

gα

α∈R

of the complexification gC of the Lie algebra g, where tC is a Cartan subalgebra, corresponding to T , and R is the root system. For any subset P ⊂ R we denote by g(P) =

∑ gα α∈P

the subspace spanned by corresponding root spaces gα . Then the Ad(H)-module mC is decomposed into a direct sum pC = g(R1 ) ⊕ · · · ⊕ g(Rk ) of mutually non-equivalent submodules (see Proposition 2.5.29), where R = R1 ∪ · · · ∪ Rk is a disjoint decomposition of R and the subsets Ri are symmetric, i.e. −Ri = Ri . Moreover, all real Ad(H)-modules g∩g(Ri ) are irreducible. Any invariant metric on M is defined by some metric endomorphism A on m, whose extension to mC has the form

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A = diag(x1 · 1m1 , · · · , x` · 1m` ), where xi are arbitrary positive numbers, xi 6= x j , and every mi is a direct sum of some modules g(Rm ), m = 1, . . . , k. We will assume that A is not a scalar operator (i.e. ` > 1) and it defines an invariant metric with homogeneous geodesics. We say that a root α corresponds to the eigenvalue xi of A if gα ⊂ mi . Lemma 5.10.5. There are two roots α, β , where α corresponds to the eigenvalue x1 and β corresponds to some eigenvalue xi for 2 ≤ i ≤ `, such that α + β is also a root. Proof. Let us suppose the contrary, then [m1 , mi ] = 0 for all i 6= 1, then, as it is easy to see, g1 := m1 + [m1 , m1 ] is a proper ideal of the simple Lie algebra g, which is impossible. t u Now, consider α and β as in the previous lemma. Since R(α, β ) := R∩span{α, β } is a rank 2 root system, we can always choose roots α, β ∈ R, that form a basis of the root system R(α, β ). Then the subalgebra gα,β := tC +

gγ

∑ γ∈R(α,β )

of the Lie algebra gC is the centralizer of the Lie algebra t0 = ker α ∩ ker β ⊂ tC . Then the orbit Gα,β o ⊂ M of the corresponding Lie subgroup Gα,β = T 0 · G0α,β ⊂ G is a totally geodesic submanifold of the manifold under consideration, hence, is a proper GO-space (supplied with the induced Riemannian matric) with effective action of the rank two simple group G0α,β , corresponding to the root system R(α, β ) (see Proposition 5.8.12). Note that the obtained space has positive Euler characteristic, since the stabilizer of the point o contains the two-dimensional torus generated by the vectors Hα , Hβ ∈ 2 tC associated with the roots α, β . Recall that Hα = b−1 ·α (the symbol b−1 ·α denotes a vector in tC conjugate to the root α with respect to b). The following proposition enables us to understand the structure of the obtained homogeneous manifold. Proposition 5.10.6. Every proper GO-space (M = G/H, g) of positive Euler characteristic, where G is a simple compact group of rank 2, is locally isometric to the manifold M = SO(5)/U(2) supplied with an invariant Riemannian metric, defined by the metric endomorphism A = x1 · 1g(Rs ) + x2 · 1g(R` ) ,

x1 6= x2 > 0

where Rs = {±ε1 , ±ε2 },

R` = {±ε1 ± ε2 },

are the sets of short and, respectively, long roots of the Lie algebra so(5). We may also assume that

5.10 Compact GO-spaces of Positive Euler Characteristic

mC = g(Rs ∪ {ε1 + ε2 }) and

309

hC = tC + gε1 −ε2 .

Proof. The proof of this proposition follows from results of [16]. Indeed, the Lie algebra g of the group G is one of the following: su(3) = A2 , so(5) = sp(2) = B2 = C2 , g2 . Since the universal Riemannian covering of a GO-space is a GO-space itself, we may assume without loss of generality that G/H is simply connected. If g = su(3), then G/H = SU(3)/S(U(2) ×U(1)) (a symmetric space) or G/H = SU(3)/T 2 , where T 2 is a maximal torus in SU(3). Both these spaces are flag manifolds, and from results of [16] we get that any GO-metric on these spaces is SU(3)normal homogeneous. If g = so(5) = sp(2), then (g, h) = (so(5), R2 ), (g, h) = (so(5), R ⊕ su(2)l ), (g, h) = (so(5), R ⊕ su(2)s ), or (g, h) = (so(5), su(2)l ⊕ su(2)l ), where su(2)l (respectively, su(2)s ) means a three-dimensional Lie subalgebra of g generated by all long (respectively, short) roots. The last pair corresponds to the irreducible symmetric space SO(5)/SO(4), which admits only normal invariant metrics. All other spaces are flag manifolds. Results of [16] imply the only possible pair is (g, h) = (so(5), R ⊕ su(2)l ), which corresponds to the space SO(5)/U(2) = Sp(2)/U(1) · Sp(1). Now, let us consider the root system R of the Lie algebra g2 : R(G2 ) = {±α, ±β , ±(α + β ), ±(α + 2β ), ±(α + 3β ), ±(2α + 3β )}. It should be noted that ±α, ±(α + 3β ), and ±(2α + 3β ) are all long roots. Let us list all proper closed symmetric root subsystems (recall that the root subsystem R0 of a root system R is said to be closed (symmetric) if the inclusions ϕ, ψ ∈ R0 and ϕ + ψ ∈ R imply ϕ + ψ ∈ R0 (respectively, if the inclusion ϕ ∈ R0 implies −ϕ ∈ R0 )): 0, /

{±α},

{±β },

{±β , ±(2α + 3β )},

{±α, ±(α + 3β ), ±(2α + 3β )}.

The first three cases give us respectively the following (generalized) flag manifolds: G2 /T 2 , G2 /SU(2)SO(2), and G2 /A1,3 SO(2), where A1,3 is a Lie group with Lie subalgebra of the type A1 of index 3, see [377]. It is proved in [16] that all G2 invariant Riemannian GO-metrics on them are G2 -normal. The last two closed symmetric root subsystems are maximal, hence, they correspond to maximal Lie subalgebras in g2 , that are respectively isomorphic to su(2) ⊕ su(2) and su(3) with the corresponding homogeneous spaces G2 /SO(4) and G2 /SU(3) = S6 . We note that G2 /SO(4) is an irreducible symmetric space, and the second space is isotropically irreducible. Therefore, they admit only normal homogeneous metrics. t u Remark 5.10.7. In this proof we applied results from the paper [16] only in order to show that there are no proper GO-metrics on the flag manifolds SU(3)/T 2 , SO(5)/T 2 , SO(5)/U(1)SU(2)s , G2 /T 2 , G2 /SU(2)SO(2), and G2 /A1,3 SO(2). For the same goal one can use results from the paper [434] by H. Tamaru (see also Subsection 5.12.3), where he classified geodesic orbit Riemannian spaces fibered over

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irreducible symmetric spaces. It is also possible to apply simpler arguments for each of the above listed spaces. Thus, the obtained orbit Gα,β o ⊂ M is locally isometric to the geodesic orbit space (SO(5)/U(2), gx1 ,x2 ) with some x1 6= x2 . This observation works for any choice of the roots α and β satisfying the conditions of Lemma 5.10.5. Therefore, we get the following corollary. Corollary 5.10.8. Let M = G/H be a proper GO-space of positive Euler characteristic, where G is a simple compact Lie group. Then the root system R of gC admits a disjoint decomposition R = R0 ∪ R1 ∪ R2 , where R0 is the root system of the complexified isotropy subalgebra hC with the following properties: 1) If α ∈ R1 , β ∈ R2 , and α + β ∈ R, then α − β ∈ R and the rank 2 root system R(α, β ) has type B2 = C2 . 2) Moreover, if α, β is a basis of R(α, β ) (i.e. hα, β i < 0), then one of the roots α, β is short, the other is long, one of the long roots α ± β belongs to R0 , and the other belongs to R1 ∪ R2 . 3) If both roots α, β are short, then one of the long roots α ± β belongs to R0 and the other belongs to R1 ∪ R2 . 4) If α ∈ R1 and β ∈ R2 are long roots, then α ± β ∈ / R. We call a decomposition with such properties a special decomposition. In order to get such a decomposition, it suffices (see Lemma 5.10.5) to consider the set R1 , consisting of all the roots α corresponding to the eigenvalue x1 of the metric endomorphism A, and the set R2 , consisting of all the roots β corresponding to the eigenvalues xi of the operator A for i > 1. Corollary 5.10.8 implies the following result. Corollary 5.10.9. There is no proper GO-space of positive Euler characteristic with simple isometry group G = SU(n), SO(2n), E6 , E7 , E8 (these are all simple Lie groups with Lie algebras having simply laced root systems). Proposition 5.10.6 implies (in particular) that any GO-space (G/H, g) of positive Euler characteristic with G = G2 is normal homogeneous. Now we are going to describe all special decompositions of the root systems of types Bn , Cn , and F4 . We will use notation from [228] (see also Table 2.1 in Chapter 2) for root systems and simple roots. Lemma 5.10.10. The root system R(F4 ) = {±εi ± ε j , ±εi , 1/2(±ε1 ± ε2 ± ε3 ± ε4 ), i, j = 1, 2, 3, 4, i 6= j} does not admit a special decomposition.

5.10 Compact GO-spaces of Positive Euler Characteristic

311

Proof. Assume that such a decomposition does exist. Then we can choose roots α ∈ R1 , β ∈ R2 such that α ± β is a root. Then α and β have different lengths and we may assume that |α| < |β | and hα, β i < 0. Then we can include α, β into a system of simple roots δ , α, β , γ, see [228]. Since all such systems are conjugate, we may assume that α = ε4 and β = ε3 − ε4 , see Table 2.1 or [228]. Then we get a contradiction, since α − β is not a root. t u Now we describe two special decompositions for the root systems R(Bn ) = {±εi , ±εi ± ε j , i, j = 1, . . . , n} and R(Cn ) = {±2εi , ±εi ± ε j , i, j = 1, . . . , n} of types Bn and Cn (that actually correspond to proper GO-spaces). There are the following special decompositions R = R0 ∪ R1 ∪ R2 of the systems R(Bn ) and R(Cn ), that we call the standard decompositions: R(Bn ) = R0 (Bn ) ∪ R1 (Bn ) ∪ R2 (Bn ),

R(Cn ) = R0 (Cn ) ∪ R1 (Cn ) ∪ R2 (Cn ),

where R0 (Bn ) = {±(εi − ε j )}, R1 (Bn ) = {±εi }, R2 (Bn ) = {±(εi + ε j )}, 1 ≤ i, j ≤ n, and R0 (Cn ) = {±2εi , ±εi ± ε j }, R1 (Cn ) = {±ε1 ± εi }, R2 (Cn ) = {±ε1 }, 1 < i, j ≤ n. It is clear that R0 (Bn ) generates the Lie algebra u(n) = An ⊕ R, whereas R0 (Bn ) ∪ R2 (Bn ) generates the Lie algebra so(n) = Dn . Hence, the special decomposition R(Bn ) leads to the following reductive decomposition of the homogeneous spaces SO(2n + 1)/U(n): so(2n + 1) = h ⊕ (p1 ⊕ p2 ) = g(R0 (Bn )) ⊕ (g(R1 (Bn )) ⊕ g(R2 (Bn )), where p1 and p2 are ad(h)-irreducible submodules in p = p1 ⊕ p2 , h = u(n), and h ⊕ p2 = so(2n). Note also that R0 (Dn ) generates the Lie algebra sp(n − 1) ⊕ R = Cn−1 ⊕ R, whereas R0 (Cn ) ∪ R2 (Cn ) generates the Lie algebra sp(n − 1) ⊕ sp(1) = Cn−1 ⊕C1 . Hence, the special decomposition R(Cn ) leads to the following reductive decomposition of the homogeneous space Sp(n)/U(1) · Sp(n − 1): sp(n) = h ⊕ (p1 ⊕ p2 ) = g(R0 (Cn )) ⊕ (g(R1 (Cn )) ⊕ g(R2 (Cn )), where p1 and p2 are ad(h)-irreducible submodules in p = p1 ⊕p2 , h = sp(n−1)⊕R, and h ⊕ p2 = sp(n − 1) ⊕ sp(1). It is well known that a metric endomorphism of the type A = diag(x1 ·1p1 , x2 ·1p2 ) defines an invariant Riemannian metric with homogeneous geodesics on the corre-

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sponding manifold M = G/H (see the discussion before the statement of Theorem 5.10.4 and [16]). Now, the proof of Theorem 5.10.4 follows from the following proposition. Proposition 5.10.11. Any special decomposition of the root system R(Bn ) or of the root system R(Cn ) is conjugate (with respect to the action of the Weyl group of the root system) to the standard one. Proof. We give a proof for the root system R(Bn ). The proof for R(Cn ) is similar. Let us consider any special decomposition R(Bn ) = R0 ∪ R1 ∪ R2 of the root system R(Bn ). We may assume that there are roots α ∈ R1 and β ∈ R2 such that hα, β i < 0 and |α| < |β |. Then we can include α and β into a system of simple roots, which, without loss of generality, can be written as ε1 − ε2 ,

··· ,

εn−2 − εn−1 ,

εn−1 − εn = β ,

εn = α.

Then (εn−1 − εn ) ∈ R0 . We need the following Lemma 5.10.12. Let R(Bn ) = R0 ∪ R1 ∪ R2 be the special decomposition as above, V 0 = εn⊥ the orthogonal complement to the vector εn , and R(Bn−1 ) = R 0 := R ∩ V 0 the induced root system in the hyperspace V 0 . Then the induced decomposition R 0 = R 00 ∪ R 01 ∪ R 02 , where R 0i := Ri ∩V 0 , is a special decomposition. Proof. It is sufficient to check that the subsets R 01 and R 02 are not empty. We say that two roots γ and δ are R0 -equivalent (γ ∼ δ ) if their difference belongs to R0 . It is clear that equivalent roots belong to one and the same component Ri . The root εn−1 = εn − (εn−1 − εn ) is R0 -equivalent to α = εn . Hence it belong to R1 . We say that a pair of roots γ, δ with hγ, δ i < 0 is special if one of the roots belongs to R1 and the other to R2 . Then they have different length (say, |γ| > |δ |) and the root γ + δ is short and belongs to the same part Ri , i = 1, 2, as the short root δ , and the root 2γ + δ is a long root from R0 . Consider the roots σ± = ±εn−2 − εn−1 . They have negative inner product with εn−1 ∈ R1 and β = εn−1 − εn ∈ R2 . None of the roots σ± can belong to R1 since then we get a special pair σ± , β which consists of long roots. Note also that the roots σ± cannot belong to R0 simultaneously, since otherwise ±εn−2 ∼ εn−1 ∈ R1 and ±εn−2 − εn ∼ εn−1 − εn ∈ R2 , and we get a special pair γ = εn−2 ∈ R1 ,

δ = −εn−2 − εn ∈ R2 ,

such that 2γ + δ = εn−2 − εn ∈ R2 , which is impossible. We conclude that one of the roots σ± = ±εn−2 − εn−1 ∈ R 0 must belong to R2 . Since the root εn−1 ∈ R 0 belongs to R1 , the lemma is proved. t u

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313

Now, we finish the proof of Proposition 5.10.11 by induction on n. The proposition is true for n = 2 according to Proposition 5.10.6. Suppose that it is true for R(Bn−1 ) and let R(Bn ) = R0 ∪ R1 ∪ R2 be a special expansion as above. By Lemma 5.10.12, the decomposition R 0 = R 00 ∪ R 01 ∪ R 02 , induced in the hyperplane V 0 , orthogonal to the vector εn , is special. By the inductive hypothesis we may assume that the special decomposition of R 0 = R(B(n − 1)) has the standard form: R0 = {±(εi − ε j )},

R1 = {±εi },

R2 = {±(εi + ε j ), i, j = 1, · · · , n − 1}.

Together with the previous information, this observation implies that the special decomposition of R is also standard. t u

5.11 GO-spaces and Representations with Non-trivial Principal Isotropy Algebras In this section, we deal with an arbitrary geodesic orbit Riemannian space (G/H, g) (the action of G on G/H is assumed to be almost effective) with the Ad(H)-invariant decomposition g = h ⊕ m, where m := {X ∈ g | B(X, h) = 0}, B is the Killing form of g, and the Riemannian metric g is generated by an inner product (·, ·) on m. Let us consider the operator A : m → m, related with the Killing form by B(X,Y ) = (AX,Y ), see Lemma 4.4.6 and Remark 4.4.7. All results in this section are obtained under these assumptions. The presentation is based on [366]. Note, that A is a symmetric operator and A[U, X] = [U, AX] for every X ∈ m and U ∈ h (i.e. A is ad(h)-equivariant). Indeed, we have (AX,Y ) = B(X,Y ) = B(Y, X) = (AY, X) = (X, AY ), (A[U, X],Y ) = B([U, X],Y ) = −B(X, [U,Y ]) = −(AX, [U,Y ]) = ([U, AX],Y ) for every X,Y ∈ m, U ∈ h. It is clear that the operator A is invertible on the subspace m± := ⊕α6=0 Aα , where Aα is the eigenspace of the operator A with the eigenvalue α. In what follows we assume that A−1 X ∈ m± for all X ∈ m± . Proposition 5.11.1. The following assertions hold: 1) For X ∈ m± , the condition to be a homogeneous geodesic from Theorem 5.1.2 (there is a Y ∈ h such that ([X +Y, Z]m , X) = 0 for all Z ∈ m) is equivalent to the following one: there is a Z ∈ h such that [X + Z, A−1 X] ⊂ A0 . 2) For any X ∈ Aα and Y ∈ Aβ , where 0 6= α 6= β 6= 0, there is a Z ∈ h such that (β − α)[X,Y ] = β [Z, X] + α[Z,Y ]. In particular, [Aα , Aβ ] ⊂ Aα ⊕ Aβ .

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3) Suppose that X,Y ∈ Aα , α 6= 0, are such that ([h, X],Y ) = 0, then [X,Y ] ∈ A0 ⊕ Aα . Proof. Let us prove 1). It is easy to see that 0 = ([X + Z,Y ]m , X) = B([X + Z,Y ], A−1 X) = −B(Y, [X + Z, A−1 X]). Since Y ∈ m is arbitrary and B is non-degenerate on m± , we get [X + Z, A−1 X] ⊂ A0 ⊕h. Taking into account that B is non-degenerate on h by Lemma 4.4.1, it suffices to prove that B([X + Z, A−1 X],U) = 0 for any U ∈ h. Since [Z, A−1 X] ∈ m, we need only to prove B([X, A−1 X],U) = 0. By using the properties of the Killing form and the operator A, we get (we put Y := A−1 X, hence, X = AY ) −B([Y, AY ],U) = B(Y, [U, AY ]) = B(Y, A[U,Y ]) = B(AY, [U,Y ]) = B([Y, AY ],U), which implies that B([X, A−1 X],U) = B([AY,Y ],U) = 0. This completes the proof of 1). Let us prove 2). By 1), for the vector X + Y there is a Z ∈ h such that [X + Y + Z, A−1 (X +Y )] ⊂ A0 . It is clear that [X +Y + Z, A−1 (X +Y )] = [X +Y + Z, α −1 X + β −1Y ] 1 = ((α − β )[X,Y ] + β [Z, X] + α[Z,Y ]), αβ hence, (α − β )[X,Y ] + β [Z, X] + α[Z,Y ] ∈ A0 . On the other hand, A0 is an ideal in g and the restriction of ad(X) to ⊕γ6=α Aγ is skew-symmetric by Lemma 5.2.9, hence, ad(X)(Aβ ) ⊂ m± = ⊕γ6=0 Aγ . Moreover, [Z, X] ⊂ Aα and [Z,Y ] ∈ Aβ by Lemma 4.4.6 (or by the fact that A is ad(h)-equivariant). Consequently, (α − β )[X,Y ] + β [Z, X] + α[Z,Y ] = 0, which proves the assertion. Let us prove 3). Take any U ∈ Aβ , β 6∈ {0, α}. Then, using 2), we get ([X,Y ],U) = β −1 B([X,Y ],U) = −β −1 B(Y, [X,U]) = β −1 (α − β )−1 B(Y, β [Z, X] + α[Z,U]) = α(α − β )−1 (Y, [Z, X]) = 0 for some Z ∈ h, since [Z,U] ∈ Aβ and ([h, X],Y ) = 0. Note also that ([h, X],Y ) = 0 implies 0 = B([h, X],Y ) = B(h, [X,Y ]). Therefore, we see that [X,Y ] ∈ A0 ⊕ Aα , which proves the third assertion. t u Remark 5.11.2. Note that A0 = n(g) by Theorem 5.4.3. If we have m = Aα for some α < 0, then the space (G/H, ρ) is normal homogeneous, because (·, ·) = α −1 B on m and ρ is generated by the invariant positive definite form α −1 B on the Lie algebra g. Of course, such space is geodesic orbit. On the other hand, if m = Aα for some α > 0, then the space (G/H, ρ) is symmetric of non-compact type (see e.g. [256]), hence, naturally reductive. Indeed, r(g) is trivial by Theorem 5.4.3 (n(g) = A0 is trivial, hence r(g) is also trivial) and h should coincide with a maximal compactly embedded subalgebra in the semisimple Lie algebra g.

5.11 GO-spaces and Representations with Non-trivial Principal Isotropy Algebras

315

Now, we consider the case when there are several eigenspaces of the operator A with non-zero eigenvalues. For any V ∈ g we denote the centralizer of V in h by Ch (V ). Theorem 5.11.3 ([366]). For every X ∈ Aα and Y ∈ Aβ , where 0 6= α 6= β 6= 0, there are Z1 ∈ Ch (X) and Z2 ∈ Ch (Y ) such that [X,Y ] = [Z2 , X] + [Z1 ,Y ]. In particular, if Ch (Y ) = {0} (respectively, Ch (X) = {0}), then [X,Y ] ∈ Aβ (respectively, [X,Y ] ∈ Aα ). Consequently, the equality Ch (Y ) = Ch (X) = {0} implies [X,Y ] = 0. Proof. By 2) of Proposition 5.11.1, for given X ∈ Aα and Y ∈ Aβ , there is a Z ∈ h such that (β − α)[X,Y ] = β [Z, X] + α[Z,Y ]. Similarly, for X ∈ Aα and −Y ∈ Aβ , there is a Z 0 ∈ h such that (β − α)[X, −Y ] = β [Z 0 , X] + α[Z 0 , −Y ]. From the above two equalities we get β [Z + Z 0 , X] + α[Z − Z 0 ,Y ] = 0. It is clear that Z + Z 0 ∈ Ch (X) and Z − Z 0 ∈ Ch (Y ). Now, if we consider Z1 := 2(βα−α) (Z + Z 0 ) β −α and Z2 := 2(ββ−α) (Z − Z 0 ), then Z = β −α α Z1 + β Z2 and (β − α)[X,Y ] = β [Z, X] + α[Z,Y ] implies [X,Y ] = [Z2 , X] + [Z1 ,Y ]. t u

This theorem allows us to apply new tools to the study of geodesic orbit Riemannian manifolds. These tools are related to the problem of classifying principal orbit types for linear actions of compact Lie groups. If a compact linear Lie group K acts on some finite-dimensional vector space V (in other terms, we have a representation of a Lie group on the space V ), then almost all points of V are situated on the orbits of K, which are pairwise isomorphic as K-manifolds. Such orbits are called orbits in general position. The isotropy groups of all points on such orbits are conjugate in K, the class of these isotropy groups is called a principal isotropy group for the linear group K and the corresponding Lie algebra is called a principal isotropy algebra or a stationary subalgebra of points in general position. Roughly speaking, the principal isotropy algebra is trivial for general linear Lie groups K, but this is not the case for some special linear groups. The classification of linear actions of simple compact connected Lie groups with non-trivial connected principal isotropy subgroups has already been carried out in [308], [262], and [203]. For details, see [261, §3 of Chapter 5]. In §4 and §5 of Chapter 5 in [261], one can find a description of more general compact connected Lie groups with non-trivial connected principal isotropy subgroups (see e.g. Theorem (V.70 ) in [261]). ` In 1972, A. G. Elashvili described all stationary subalgebras of points in general position for the case in which G is simple and V is arbitrary (see [203]) and also if G is semisimple and V is irreducible (see [204]). According to [269], a linear action of a semisimple group K on a finite-dimensional vector space V is said to be locally strongly effective if every simple normal subgroup of K acts nontrivially on every irreducible submodule W ⊂ V . In [269], D. G. Il’inskii obtained a complete

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description of the locally strongly effective actions of a semisimple complex algebraic group K on a complex vector space V with nontrivial stationary subalgebra of points in general position. The following result from [366] gives serious restrictions on the algebraic structure of geodesic orbit spaces. Theorem 5.11.4. Let χ : H → O(m) be the isotropy representation for (G/H, ρ) and let χp be its irreducible subrepresentation on a submodule p ⊂ Aα , α 6= 0. Then, the following assertions hold: 1) For any Aβ , where α 6= β 6= 0, we get [Aβ , p] ⊂ Aβ ⊕ p; 2) If p⊥ is the B-orthogonal (or, equivalently, (·, ·)-orthogonal) complement to p in Aα , then [p, p⊥ ] ⊂ A0 ⊕ Aα ; 3) If the principal isotropy algebra of the representation χp is trivial, then [Aβ , p] ⊂ p. Proof. By Theorem 5.11.3, for any X ∈ p ⊂ Aα and Y ∈ Aβ there are Z1 ∈ Ch (X) and Z2 ∈ Ch (Y ) such that [X,Y ] = [Z2 , X] + [Z1 ,Y ]. Therefore, [X,Y ] ∈ Aβ ⊕ p. The inclusion [p, p⊥ ] ⊂ A0 ⊕ Aα follows from 3) of Proposition 5.11.1, since p is ad(h)-invariant. If the principal isotropy algebra of the representation χp is trivial, then for almost all X ∈ p, the linear space Ch (X) is trivial, hence Z1 = 0 and [X,Y ] ∈ p. By continuity, [X,Y ] ∈ p holds for all X ∈ p. t u Corollary 5.11.5. If, in the notation of Theorem 5.11.4, [Aβ , p] 6⊂ p, then the principal isotropy algebra of the representation χp : H → O(p) is non-trivial, hence, has dimension ≥ 1. Corollary 5.11.5 shows that the algebraic structure of geodesic orbit Riemannian spaces is very special. This observation could help us to understand geodesic orbit Riemannian spaces on a deeper level.

5.12 Miscellaneous In this section we collect various recent results related to homogeneous geodesics and geodesic orbit spaces.

5.12.1 Compact GO-spaces with Two Isotropy Summands It is well known that every isotropy irreducible Riemannian space is naturally reductive, and hence geodesic orbit, see e.g. [106] and Remark 5.11.2. A natural problem is to classify geodesic orbit Riemannian spaces (G/H, ρ) such that the isotropy

5.12 Miscellaneous

317

representation χ : H → O(m), χ(a) = Ad(a)|m has exactly two irreducible components. This problem was solved for the case of compact and simply connected G/H in the paper [156]. Recall that the geodesic orbit property is related to classes of locally isomorphic homogeneous spaces due to Theorem 5.1.2. The first step to the mentioned classification is the following proposition. Proposition 5.12.1. Suppose that a compact homogeneous space G/H with connected compact H has two irreducible components in the isotropy representation. Then one of the following possibilities holds: 1. g = f ⊕ f ⊕ f and h = diag(f) ⊂ g for a compact simple Lie algebra f; 2. g = g1 ⊕ g2 and h = h1 ⊕ h2 , where hi ⊂ gi and the pair (gi , hi ) is isotropy irreducible with simple compact Lie algebras gi for i = 1, 2; 3. g = f ⊕ f ⊕ g1 for simple compact Lie algebras g1 and f, h = diag(f) ⊕ h1 , where h1 ⊂ g1 and the pair (g1 , h1 ) is isotropy irreducible; 4. g = l⊕k, where l is a simple compact Lie algebra, k is either a simple compact Lie algebra or R, and there exists a Lie algebra k1 such that k ⊕ k1 is a subalgebra in l such that the pair (l, k ⊕ k1 ) is isotropy irreducible, whereas h = diag(k) ⊕ k1 ⊂ l ⊕ k; 5. g = R2 , h = 0, G/H = G = S1 × S1 ; 6. g = R ⊕ g1 , where g1 is a semisimple compact Lie algebra, h ⊂ g1 and the pair (g1 , h) is isotropy irreducible; 7. g is a simple compact Lie algebra. Furthermore, we have the following theorem. Theorem 5.12.2 ([156]). Assume that G/H is a compact and simply connected homogeneous space with non-simple G and the isotropy representation is the direct sum of two irreducible representations (this corresponds to cases (1)–(6) in Proposition 5.12.1). Then G/H, supplied with any G-invariant Riemannian metric, is naturally reductive, hence, geodesic orbit. In fact, those spaces G/H in cases 2, 3, 5, and 6 of Proposition 5.12.1 are normal homogeneous. Note that for a simply connected compact homogeneous space G/H cases 5 and 6 are impossible. It is known also that G/H in case 4 are naturally reductive, see [156, Proposition 2] for details. Case 1 of Proposition 5.12.1 is more complicated. Here we deal with the Ledger–Obata spaces G/H = F 3 / diag(F), see Subsection 4.13.3. Note also that these spaces give examples of generalised Wallach spaces, see Subsection 4.13.2 and [365]. It is not difficult to show that the spaces of all invariant metrics on F 3 / diag(F) and naturally reductive metrics both have dimension 3. This hints to the assertion of the following proposition, which covers case 1 in Proposition 5.12.1. Proposition 5.12.3. For any compact connected simple Lie group F, every invariant Riemannian metric on the Ledger–Obata space G/H = F 3 / diag(F) is naturally reductive, hence geodesic orbit.

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Finally, we have the following theorem for simple G. Theorem 5.12.4 ([156]). Assume that G/H is a compact and simply connected homogeneous space with simple G and the isotropy representation is the direct sum of two irreducible representations. If G/H, supplied with an G-invariant Riemannian metric which isn’t normal homogeneous with respect to G, is a geodesic orbit space, then there exists a K ⊂ G such that H ⊂ K ⊂ G, G/K is symmetric, and (H, K, G) is one of the following cases: 1. G2 ⊂ Spin(7) ⊂ Spin(8); 2. SO(2) × G2 ⊂ SO(2) × SO(7) ⊂ SO(9); 3. U(k) ⊂ SO(2k) ⊂ SO(2k + 1) for k ≥ 2; 4. SU(2r + 1) ⊂ U(2r + 1) ⊂ SO(4r + 2) for r ≥ 2; 5. Spin(7) ⊂ SO(8) ⊂ SO(9); 6. SU(m) × SU(n) ⊂ S(U(m)U(n)) ⊂ SU(m + n) for m > n ≥ 1; 7. Sp(n)U(1) ⊂ S(U(2n)U(1)) ⊂ SU(2n + 1) for n ≥ 2; 8. Sp(n)U(1) ⊂ Sp(n) × Sp(1) ⊂ Sp(n + 1) for n ≥ 1; 9. Spin(10) ⊂ Spin(10)SO(2) ⊂ E6 . Moreover, every such space G/H, supplied with any G-invariant Riemannian metric, is geodesic orbit. The space of G-invariant Riemannian metrics on G/H has dimension 2 for all spaces except case 1, where it has dimension 3. Note that the spaces of cases 1, 3–9 in Theorem 5.12.4 are weakly symmetric, whereas the space of case 2 is not, see [354], [484] or [492] for details. The spaces of cases 3, 5, 6 for n = 1, and 7 in the above theorem admit generalized normal homogeneous metrics, see Chapter 6 or [73] and [83] for details. Finally, the spaces of cases 6 and 8 in the above theorem are naturally reductive according to Theorem 3 in [501]. It should be noted also that simply connected compact homogeneous spaces G/H with simple G and two components in the isotropy representation (corresponding to case 7 of Proposition 5.12.1) were classified by W. Dickinson and M. Kerr in the paper [180]. In order to prove Theorem 5.12.4, the authors of [156] made substantial use of this classification together with special new methods, based on the result presented in Section 5.11. Another important ingredient was the classification of geodesic orbit spaces fibered over irreducible symmetric spaces by H. Tamaru [434].

5.12.2 Geodesic Orbit Metrics on Ledger–Obata Spaces In [358], the authors classified all geodesic orbit metrics on Ledger–Obata spaces F m / diag(F), see Subsection 4.13.3 for general properties of Ledger–Obata spaces. Let f be the Lie algebra of the Lie group F, g = mf = f ⊕ f ⊕ · · · ⊕ f the Lie algebra of G = F m , and h = diag(f) = {(X, X, . . . , X) | X ∈ f} ⊂ g the Lie algebra of

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319

H = diag(F) ⊂ G = F m . We denote by h·, ·i the minus Killing form on g, and by h·, ·ii its restriction to the i-th summand f. We have the following characterization of naturally reductive metrics on Ledger– Obata spaces F m / diag(F). Theorem 5.12.5 ([358]). Let F be a connected, compact, simple Lie group. An invariant metric on a Ledger–Obata space F m / diag(F) is naturally reductive if and only if it is induced by a pair (p, (·, ·)) such that either 1. p = (m − 1)f is an ideal in g and (·, ·) is an ad(p)-invariant inner product on it, so that (·, ·) = ∑m−1 i=1 βi h·, ·ii , where βi > 0, or 2. p is the orthogonal complement to h = diag(f) relative to an ad(g)-invariant quadratic form Q = ∑m i=1 αi h·, ·ii on g and (·, ·) = Q|p , where either a. αi > 0 for all i = 1, . . . , m, or b. there exists j = 1, . . . , m such that α j < 0 and αi > 0 for all i 6= j, and S := ∑m i=1 αi < 0. It is known that every invariant Riemannian metric on a Ledger–Obata space F 3 / diag(F), where F is a connected, compact, simple Lie group, is naturally reductive and hence is geodesic orbit, see Proposition 5.12.3. The claim is no longer true for F m / diag(F) with m > 3, by the parameter count. Theorem 5.12.6 ([358]). A Ledger–Obata space with an invariant Riemannian metric is a geodesic orbit space if and only if it is naturally reductive. The question of characterizing Ledger–Obata spaces which are geodesic orbit manifolds is a natural one. The following decomposition theorem gives the answer to this question as well as the description of the full connected isometry group of a Ledger–Obata space. Theorem 5.12.7 ([358]). Let F be a connected, compact, simple Lie group. A Ledger–Obata space F m / diag(F) with an invariant metric is isometric to the product of Ledger–Obata spaces F mi / diag(F), i = 1, . . . , s, with invariant metrics which are irreducible (as Riemannian manifolds), where s ≥ 1, mi > 1, and ∑si=1 (mi − 1) = m − 1; such a decomposition is unique, up to relabelling. Furthermore, the full connected isometry group of F m / diag(F) is the direct product ∏si=1 F mi , where each factor F mi acts from the left on the corresponding factor F mi / diag(F) and trivially on the other factors. It follows that the full connected isometry group of an invariant metric on a Ledger–Obata space F m / diag(F) is F k , where m ≤ k ≤ 2(m − 1). The cases k = m and k = 2(m − 1) correspond to irreducible and bi-invariant metrics respectively. For a special class of metrics on Ledger–Obata spaces, the full connected isometry group was computed in [298]. Theorem 5.12.7 and Theorem 5.12.6 imply the following result. Corollary 5.12.8. Let F be a connected, compact, simple Lie group. A Ledger– Obata space F m / diag(F) is a geodesic orbit manifold if and only if each of the factors F mi / diag(F) in its irreducible decomposition is naturally reductive (so that the metric on each of them has the form given in Theorem 5.12.5).

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5.12.3 GO-spaces Fibered over Irreducible Symmetric Spaces In the paper [434], H. Tamaru classified homogeneous spaces M = G/K fibered over irreducible symmetric spaces G/H and admitting invariant GO-metrics. More precisely, for a connected compact and semisimple Lie group G and its compact subgroups H, K, such that G ⊃ H ⊃ K, let us consider the fibration F = H/K → M = G/K → B = G/H and the G-invariant Riemannian metrics ga,b on M are determined by the inner products h , i = −a B|f − bB|b , a, b > 0. Here B is the Killing form of the Lie algebra g (it is negative definite on g) and f and b are the tangent spaces of F and B respectively, so that the tangent space of M at the origin is identified with f ⊕ b. Theorem 5.12.9 (H. Tamaru, [434]). In the above notation, the following two conditions are equivalent: 1) G-invariant metrics ga,b on M are geodesic orbit for all a, b > 0; 2) The triple of Lie algebras (g, h, k), such that (g, h) is a symmetric pair and (g, k) corresponds to the homogeneous space G/K, is in Table 5.2. Note that the main technical tools in the proof of this theorem were special properties of polar representations, see [434] for details. A similar result was obtained in [434] for the case of connected noncompact semisimple G∗ with compact subgroups K ⊂ H, where H is a maximal compact subgroup in G∗ . In this case, we can consider the fibration F = H/K → M ∗ = G∗ /K → B = G∗ /H and the G∗ -invariant Riemannian metrics g∗a,b on M ∗ determined by the inner products h , i = −a B|f + bB|b , a, b > 0. Here B is the Killing form of the Lie algebra g∗ of the Lie group G∗ , and f and b are the tangent spaces of F and B respectively. For the triple (G∗ , H, K) we can consider a triple of compact Lie groups (G, H, K), where the symmetric pair (G, H) is dual to the symmetric pair (G∗ , H). Theorem 5.12.10 (H. Tamaru, [434]). In the above notation, the following two conditions are equivalent: 1) G∗ -invariant metrics g∗a,b on M ∗ are geodesic orbit for all a, b > 0; 2) The triple of the Lie algebras (g, h, k), where (g, h) is the symmetric pair dual to the symmetric pair (g∗ , h), is in Table 5.2.

5.12 Miscellaneous

321

Table 5.2 Riemannian GO-spaces G/K fibered over irreducible symmetric spaces G/H g

h

k

1

so(2n + 1), n ≥ 2

so(2n)

u(n)

2

so(4n + 1), n ≥ 1

so(4n)

su(2n)

3

so(8)

so(7)

g2

4

so(9)

so(8)

so(7)

5

su(n + 1), n ≥ 2

u(n)

su(n)

6

su(2n + 1), n ≥ 2

u(2n)

u(1) ⊕ sp(n)

7

su(2n + 1), n ≥ 2

u(2n)

sp(n)

8

sp(n + 1), n ≥ 1

sp(1) ⊕ sp(n)

u(1) ⊕ sp(n)

9

sp(n + 1), n ≥ 1

sp(1) ⊕ sp(n)

sp(n)

10

su(2r + n), r ≥ 2, n ≥ 1

su(r) ⊕ su(r + n) ⊕ R

su(r) ⊕ su(r + n)

11

so(4n + 2), n ≥ 2

u(2n + 1)

su(2n + 1)

12

e6

R ⊕ so(10)

so(10)

13

so(9)

so(7) ⊕ so(2)

g2 ⊕ so(2)

14

so(10)

so(8) ⊕ so(2)

spin(7) ⊕ so(2)

15

so(11)

so(8) ⊕ so(3)

spin(7) ⊕ so(3)

5.12.4 Geodesic Orbit Riemannian Structures on Rn In [232], the authors studied geodesic orbit spaces diffeomorphic to Rn . We shall discuss some results of this paper. Recall that if U is a connected semisimple Lie group of noncompact type and K ⊂ U is a subgroup such that k = Lie(K) is a maximal compactly embedded subalgebra of U, then U admits an Iwasawa decomposition U = KS, where S is a simply connected solvable Lie group and K ∩ S is trivial. We will say that a solvable Lie group is an Iwasawa group if it is a solvable Iwasawa factor of some semisimple Lie group U. In this case, U/K with a left-invariant Riemannian metric is a Riemannian symmetric space. Theorem 5.12.11. Let (M, g) be an n-dimensional simply connected geodesic orbit manifold. Then the following are equivalent: 1) (M, g) is a Riemannian solvmanifold, i.e. it admits a simply transitive solvable group of isometries. 2) M is diffeomorphic to Rn . 3) In the above notation, there exists an Iwasawa group S, a simply connected nilpotent Lie group N, and a left-invariant metric h on S × N such that (S × N, h) is isometric to (M, g). N is at most 2-step nilpotent.

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Every n-dimensional simply connected solvable Lie group is diffeomorphic to Rn , so the implication “1) implies 2)” is trivial. However the converse is not true for arbitrary homogeneous Riemannian manifolds, as the following example illustrates. e of SL(2, R) is diffeomorphic to R3 . For a Example 5.12.12. The universal cover G e generic left-invariant metric g on G, the identity component of the full isometry is e itself. Thus (G, e g) satisfies 2) but not 1). SL(2, R) has Iwasawa decomposition just G KS with K = SO(2) and S the upper triangular matrices of determinant one. The e is KS, e where K e ' R is the universal cover of SO(2). Iwasawa decomposition of G e×K e e acts almost If g is a left-invariant metric that is also Ad(K)-invariant, then G e effectively and isometrically on (G, g) and the simply connected solvable Lie group e acts simply transitively. Thus condition 1) of Theorem 5.12.11 does hold in S×K this case. e Remark 5.12.13. If g in Example 5.12.12 is left invariant and Ad(K)-invariant, then e e e e (G, g) is isometric to the weakly symmetric space (G × K/ diag(K), g). Theorem 5.12.14. Assume that (M, g) satisfies the equivalent conditions of Theorem 5.12.11. Then (M, g) is isometric to one of the following: 1) A Riemannian symmetric space of noncompact type; 2) A simply connected Riemannian geodesic orbit nilmanifold (N, h) (necessarily of step size at most 2); 3) The total space of a Riemannian submersion π : M → P with totally geodesic fibers, where the base is a Riemannian symmetric space of noncompact type and the fibers are isometric to a simply-connected Riemannian geodesic orbit nilmanifold. In the statement of Theorem 5.12.11, the metric on S × N need not be a product metric, and the induced metric on S need not be symmetric. We only have that S is a global section of the submersion in Theorem 5.12.14 and N is isometric to the fiber. As such, S is diffeomorphic to the base but the induced metric on S is in general not isometric to that of the base.

5.12.5 On Left-invariant Einstein Riemannian Metrics that are not Geodesic Orbit Now, we discuss left-invariant Einstein Riemannian metrics that are not geodesic orbit. It should be noted that there are many homogeneous examples of Einstein metrics, in particular, normal homogeneous and naturally reductive, see e.g. [106]. Let us consider the following problem: Find an example of a compact Lie group G supplied with a left-invariant Riemannian metric g such that (G, g) is Einstein but is not a geodesic orbit Riemannian manifold. If we replace the geodesic orbit property with the stronger condition of natural reductivity, then we get a well-known problem. Let us recall some details.

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323

In [169], J. E. D’Atri and W. Ziller have investigated naturally reductive metrics among the left invariant metrics on compact Lie groups, and have given a complete classification in the case of simple Lie groups. Let G be a compact connected semisimple Lie group, H a closed subgroup of G, and let g be the Lie algebra of G and h the subalgebra corresponding to H. Denote by h·, ·i the minus Killing form B of g. Let m be an orthogonal complement of h with respect to B. Then we have g = h ⊕ m,

Ad(H) m ⊂ m.

Let h = h0 ⊕ h1 ⊕ · · · ⊕ h p be the decomposition into ideals of h, where h0 is the center of h and hi , i = 1, · · · , p, are simple ideals of h. Let A0 be an arbitrary inner product on h0 . Theorem 5.12.15 (J. E. D’Atri–W. Ziller, [169]). Under the notations above, a left invariant Riemannian metric on G of the form (·, ·) = xh·, ·i|m + A0 + u1 h·, ·i|h1 + · · · + u p h·, ·i|h p

( x, u1 , . . . , u p ∈ R+ ) (5.24)

is naturally reductive with respect to G × H, where G × H acts on G by (a, b)(c) = acb−1 . Conversely, if a left-invariant metric (·, ·) on a compact simple Lie group G is naturally reductive, then there exists a closed subgroup H of G such that the metric (·, ·) is given by the form (5.24). In [169], J. E. D’Atri and W. Ziller found a large number of left-invariant Einstein metrics, which are naturally reductive, on the compact Lie groups SU(n), SO(n) and Sp(n). In the same paper, the authors posed the problem of existence of left-invariant Einstein metrics, which are not naturally reductive, on compact Lie groups. The first examples were obtained by K. Mori in [344], where he constructed non-naturally reductive Einstein metrics on the Lie group SU(n) for n ≥ 6. Further, in [29] A. Arvanitoyeorgos, Y. Sakane, and K. Mori proved the existence of nonnaturally reductive Einstein metrics on the compact Lie groups SO(n) (n ≥ 11), Sp(n) (n ≥ 3), E6 , E7 and E8 . In [155] Z. Chen and K. Liang found three naturally reductive and one non-naturally reductive Einstein metric on the compact Lie group F4 . Also, in [30], the authors obtained new left-invariant Einstein metrics on the symplectic group Sp(n) (n ≥ 3), and in [163] I. Chrysikos and Y. Sakane obtained new non-naturally reductive Einstein metrics on exceptional Lie groups. In the recent paper [31], the authors obtained new left-invariant Einstein metrics on the compact Lie groups SO(n) (n ≥ 7) which are not naturally reductive. Note that there are examples of homogeneous Einstein manifolds (distinct from compact Lie groups with left-invariant Riemannian metrics), that are neither naturally reductive, nor geodesic orbit. It is well known that all homogeneous Einstein Riemannian manifolds of dimension ≤ 4 are locally symmetric, hence, naturally reductive and geodesic orbit, see e.g. [106]. On the other hand, there are many examples of Einstein homogeneous manifolds of dimension ≥ 5 that are not naturally reductive,for details, see e.g. [106] or [367].

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One of the non-naturally reductive Einstein metrics, obtained in [163] by I. Chrysikos and Y. Sakane, is a left-invariant metric on the Lie group G2 , see [163, Section 4.4]. It is possible to prove that this metric is not geodesic orbit, therefore, we get the following Theorem 5.12.16 ([367]). There is a left-invariant Riemannian metric g on the compact simple Lie group G2 such that (G2 , ρ) is Einstein but is not a geodesic orbit Riemannian manifold. It should be noted also that in [367], some special tools were developed in order to prove that the corresponding metric is not geodesic orbit. Note also that the parameters of this Einstein metric satisfy some polynomial system, where the resulting polynomial equation with respect to one parameter is of degree 39. In the recent article [154], H. Chen, Z. Chen, and S. Deng proved that the compact simple Lie groups SU(n) for n ≥ 6, SO(n) for n ≥ 7, Sp(n) for n ≥ 3, E6 , E7 , E8 , and F4 admit left-invariant Einstein metrics that are not geodesic orbit. The construction of suitable metrics is based on the study of special properties of generalized Wallach spaces, see Subsection 4.13.2.

5.12.6 Various Results on Homogeneous Geodesics and GO-spaces In this subsection we consider some results on homogeneous geodesics and geodesic orbit spaces that were not discussed earlier. We should also mention the recent survey [27] by A. Arvanitoyeorgos as well as the paper [193] by Z. Duˇsek on the topic. In the paper [196], the author constructed a 2-parameter family of compact irreducible 7-dimensional Riemannian GO-manifolds (together with their “noncompact duals”) which are in no way naturally reductive. The compact examples are defined as suitable invariant Riemannian metrics on the spaces SO(5) × SO(2)/U(2). It should be noted that these metrics constitute a subset of a 3-parameter family of geodesic orbit (and even weakly symmetric) metrics, that are Sp(2)U(1)-invariant metrics on the sphere S7 , see Section 5.9. Moreover, noncompact dual examples correspond to a 3-dimensional (2-dimensional up to homothety) family of GOmetric on the space Sp(1, 1)U(1)/Sp(1)U(1). Note that all invariant metrics (a 3-dimensional family) on the space Sp(n, 1)U(1)/Sp(n)U(1), n ≥ 1, are weakly symmetric and, therefore, geodesic orbit, see the book [484] for details. Using Theorem 5.4.10, some examples of 7-dimensional geodesic orbit not naturally reductive spaces are constructed in [230]. There are many papers devoted to the study of homogeneous geodesics in various homogeneous spaces. O. Kowalski, S. Nikˇcevi´c and Z. Vl´asˇek studied properties of homogeneous geodesics in homogeneous Riemannian manifolds [305], G. Calvaruso and R. Marinosci studied homogeneous geodesics in three-dimensional Lie groups in [333] and [141]. Homogeneous geodesics were also studied by J. Szenthe in [429], [431], [430], [432]. Also, D. Latifi studied homogeneous geodesics

5.12 Miscellaneous

325

in homogeneous Finsler spaces [316], and A. Arvanitoyeorgos investigated homogeneous geodesics in the flag manifold SO(2l + 1)/U(l − m) × SO(2m + 1) [26]. Homogeneous geodesics in the affine setting were studied in [189] and [197] (and in particular for any non-reductive pseudo-Riemannian manifold). Let G/K be a generalized flag manifold with K = C(S) = S × K1 , where S is a torus in a compact simple Lie group G and K1 is the semisimple part of K. Then the associated M-space is the homogeneous space G/K1 . These spaces were introduced and studied by H. C. Wang in [458]. In the paper [32] A. Arvanitoyeorgos, Y. Wang, and G. Zhao investigated homogeneous geodesics in a class of homogeneous spaces called M-spaces. It is proved that for various classes of M-spaces, the only GO-metric is the standard metric. For other classes of M-spaces the authors give either necessary or necessary and sufficient conditions so that a G-invariant metric on G/K1 is a GO-metric. The analysis is based on properties of the isotropy representation of the flag manifold G/K, in particular on the dimension of the submodules mi , where m = m1 ⊕ · · · ⊕ ms . Let us consider some results in more detail. Let g and k be the Lie algebras of the Lie groups G and K ⊂ G respectively, B the Killing form of g. Let g = k ⊕ m be the B-orthogonal reductive decomposition of the Lie algebra g, where m ∼ = (G/K)o , o = eK. Assume that m = m1 ⊕ · · · ⊕ ms

(5.25)

is a B-orthogonal decomposition of m into pairwise inequivalent irreducible ad(k)modules. Let G/K1 be the corresponding M-space and s and k1 be the Lie algebras of S and K1 respectively. Denote by n the tangent space (G/K1 )o , where o = eK1 . A G-invariant metric g on G/K1 induces an inner product (·, ·) on n which is Ad(K1 )-invariant. Such an Ad(K1 )-invariant inner product (·, ·) on n can be expressed in terms of the decomposition (5.25) and the inner product h·, ·i = −B(·, ·). The main results are the following: Theorem 5.12.17 ([32]). Let G/K be a generalized flag manifold with s ≥ 3 in the decomposition (5.25). Let G/K1 be the corresponding M-space. 1) If dim mi 6= 2 (i = 1, . . . , s) and (G/K1 , g) is a GO-space, then g = (·, ·) = µh·, ·i |s +λ h·, ·i |m1 ⊕m2 ⊕···⊕ms , (µ, λ > 0). 2) If there exists some j ∈ {1, . . . , s} such that dim m j = 2, then (G/K1 , g) is a GOspace if and only if g is the standard metric. Theorem 5.12.18 ([32]). Let G/K be a generalized flag manifold with two isotropy summands m = m1 ⊕ m2 and let (G/K1 , g) be the corresponding M-space. 1) If dim m2 = 2 then the standard metric is the only GO-metric on M, unless G/K1 = SO(5)/SU(2) or Sp(n)/Sp(n − 1).

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2) If dim mi 6= 2 (i = 1, 2) and the corresponding M-space (G/K1 , g) is a GO-space, then g = (·, ·) = µh·, ·i |s +λ h·, ·i |m1 ⊕m2 , (µ, λ > 0), unless G/K1 = SO(2n + 1)/SU(n), n > 2. Note that the spaces SO(5)/SU(2) and Sp(n)/Sp(n − 1) do admit GO-metrics (see e.g. Table 5.2). It is well known that any homogeneous Riemannian manifold is reductive, but this is not the case for pseudo-Riemannian manifolds in general. In fact, there exist homogeneous pseudo-Riemannian manifolds which admit no reductive decomposition. Therefore, there is a dichotomy in the study of geometrical problems between reductive and non-reductive pseudo-Riemannian manifolds. Due to the existence of null vectors in a pseudo-Riemannian manifold the definition of a homogeneous geodesic γ(t) = exp(tX) · o needs to be modified by requiring that ∇γ˙ γ˙ = k(γ)γ˙ (cf. the relevant discussion in [336, pp. 90–91]). It turns out that k(γ) is a constant function (cf. [195]). Even though an algebraic characterization of geodesic vectors (see Theorem 5.1.2 for the Riemannian case) was known to physicists ([213], [388]), a formal proof was given by Z. Duˇsek and O. Kowalski in [195]. Proposition 5.12.19 ([195]). Let M = G/H be a reductive homogeneous pseudoRiemannian space with reductive decomposition g = h ⊕ m, and X ∈ g. Then the curve γ(t) = exp(tX) · o is a geodesic with respect to some parameter s if and only if ([V, Z]m ,Vm ) = k (Vm , Zm ), for all Z ∈ m, where k is some real constant. Moreover, if k = 0, then t is an affine parameter for this geodesic. If k 6= 0, then s = ekt is an affine parameter for the geodesic. This occurs only if the curve γ(t) is a null curve in a (proper) pseudo-Riemannian space. The existence of homogeneous geodesics in homogeneous pseudo-Riemannian spaces (for both reductive and non-reductive) was answered positively only recently by Z. Duˇsek in [191]. Other applications of Proposition 5.12.19 are obtained in [192]. All 2-dimensional and 3-dimensional homogeneous pseudo-Riemannian manifolds are reductive ([139], [212]). All 4-dimensional non-reductive homogeneous pseudo-Riemannian manifolds were classified by M. E. Fels and A. G. Renner in [212] in terms of their non-reductive Lie algebras. Their invariant pseudo-Riemannian metrics, together with their connection and curvature, were explicitly described by G. Calvaruso and A. Fino in [140]. The 3-dimensional pseudo-Riemannian GO-spaces were classified by G. Calvaruso and R. A. Marinosci in [141]. In the recent work [142], G. Calvaruso, A. Fino and A. Zaeim found explicit examples of 4-dimensional non-reductive pseudoRiemannian geodesic orbit spaces. They obtained an explicit description in coordinates for all invariant metrics of non-reductive homogeneous pseudo-Riemannian four-manifolds. For 4-dimensional non-reductive pseudo-Riemannian spaces that are not geodesic orbit, they determined the homogeneous geodesics through a point.

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327

In [28] A. Arvanitoyeorgos and Y. Wang investigated geodesic orbit Riemannian metrics on generalized Wallach spaces, see Subsection 4.13.2. Let us consider the main result, using the notation from Theorem 4.13.1. Theorem 5.12.20 ([28]). Let (G/K, g) be a generalized Wallach space as listed in Theorem 4.13.1. Then 1) If (G/K, g) is a space of type 1) then this is a GO-space for any Ad(K)-invariant Riemannian metric. 2) If (G/K, g) is a space of type 2) or 3) then this is a GO-space if and only if g is the standard metric. Moreover, the authors of [28] determined all homogeneous geodesics (for various invariant metrics) for the generalized Wallach space SU(2)/{e} (recovering a result of R. A. Marinosci [333, P. 266]) and SO(n)/SO(n − 2), n ≥ 4. In [419], N. P. Souris studied geodesic orbit metrics on compact homogeneous spaces with equivalent isotropy submodules. For this goal, a special approach was suggested. In particular, the following result was obtained. Theorem 5.12.21. Let Vk Cn be a complex Stiefel manifold. Then Vk Cn admits exactly one (up to scalar) family of U(n)-GO metrics gt , t > 0. The metrics gt are smooth deformations of the normal metric g1 , along the center of the group NG (H)/H, where G = U(n), H = U(n − k) and NG (H) is the normalizer of H in G. The following question is interesting: Does a given metric Lie algebra (g, Q) admit a basis that consists of geodesic vectors (a geodesic basis)? This question was studied in [137, 138, 143, 301]. In [301] it is shown that semisimple Lie algebras possess an orthonormal basis comprised of geodesics vectors, for every inner product. Results for certain solvable algebras are given in [143]. It is easy to see that if g possesses an orthonormal geodesic basis (with respect to some inner product), then g is unimodular [138]. The authors of [138] proved that every unimodular Lie algebra of dimension at most 4, equipped with an inner product, possesses an orthonormal basis comprised of geodesic vectors, whereas there is an example of a solvable unimodular Lie algebra of dimension 5 that has no orthonormal geodesic basis, for any inner product. The authors of [137] were interested in giving conditions on the Lie algebra g to admit a basis (not necessarily orthonormal) which is a geodesic basis with respect to some inner product. The main results of [137] show that the following Lie algebras admit an inner product having a geodesic basis: • • • •

unimodular solvable Lie algebras with abelian nilradical; some Lie algebras with abelian derived algebra; Lie algebras having a codimension one ideal of a particular kind; unimodular Lie algebras of dimension 5.

The authors of [137] also obtained some negative results. For instance, they found the list of nonunimodular Lie algebras of dimension 4 admitting no geodesic basis.

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5.12.7 Open Questions The classification of geodesic orbit Riemannian spaces in any dimension greater than seven is far from being accomplished. In dimension seven there are several examples but a complete classification is still unknown. However, as follows from the previous sections, we have comprehensive descriptions of some large classes of homogeneous Riemannian spaces with homogeneous geodesics. Finally, we propose some open questions that (at least in our opinion) are quite interesting and important for the theory of geodesic orbit Riemannian spaces. Question 5.12.22. Produce new examples of geodesic orbit Riemannian solvmanifolds by using Proposition 5.4.11. Compare with Example 5.4.12. Question 5.12.23. Classify geodesic orbit Riemannian spaces (G/H, ρ) with a small number of eigenvalues of the operator A (see Section 5.11). This question is interesting even in the case of two different eigenvalues (one of which could be zero). Question 5.12.24. Classify geodesic orbit Riemannian spaces (G/H, ρ) with a small number (2, 3, . . . ) of irreducible components in the isotropy representation. It should be recalled that all isotropy irreducible spaces are naturally reductive and, hence, geodesic orbit. The case of compact GO-spaces with two isotropy summands is discussed in Subsection 5.12.1. Question 5.12.25. Classify all homogeneous spaces G/H such that all G-invariant Riemannian metrics on G/H are geodesic orbit. Note that isotropy irreducible spaces give obvious examples of a required type. Abelian Lie groups, groups Gn+1 , and homogeneous spaces from Theorem 4.15.1 constitute other types of examples. The spaces mentioned in the previous two sentences, and symmetric spaces, are partial cases of homogeneous spaces such that any of their invariant distributions is involutive (see Theorem 4.15.1, [58], [59], [226], [70]). All homogeneous spaces of the last kind are characterized by the property that they admit no invariant sub-Riemannian metric (some other homogeneous spaces of this sort were discovered by V. V. Gorbatsevich, see [70]). Perhaps all homogeneous spaces with this property also have the property from Question 5.12.25. More interesting examples are weakly symmetric spaces, see e.g. [11] for more details.

Chapter 6

Generalized Normal Homogeneous Manifolds With Intrinsic Metrics

Abstract In this chapter, we prove that δ -homogeneous (or, generalized normal homogeneous, in other words) Riemannian manifolds 1) constitute a proper subclass of geodesic orbit spaces with non-negative sectional curvature, which properly includes the class of all normal homogeneous Riemannian spaces, 2) include naturally reductive compact homogeneous Riemannian manifolds of positive Euler characteristic, 3) are exactly homogeneous spaces Sp(l + 1)/U(1) · Sp(l) = CP2l+1 , l ≥ 1, with invariant Riemannian metrics of positive sectional curvature with the pinching constants in the interval (1/16, 1/4) if they are simply connected, compact of positive Euler characteristic, indecomposable, and not normal. Generalized normal homogeneous metrics on all spheres and projective spaces are classified. For the connected isometry Lie group G of a compact homogeneous Finsler manifold, a natural bi-invariant intrinsic Finsler metric, defined by the Chebyshev norm on the Lie algebra of G, is found. The Chebyshev norms for the Euclidean spheres, the Berger spheres, and homogeneous Riemannian metrics on CP3 are found.

6.1 δ -homogeneous and Clifford–Wolf Homogeneous Spaces We begin with the definitions of important classes of isometries of general metric spaces. Definition 6.1.1 ([85], [483]). Let (X, d) be a metric space and x ∈ X. An isometry f : X → X is called a δ (x)-translation (a Clifford–Wolf translation) if x is a point of maximal displacement of f , i.e. for every y ∈ X the inequality d(y, f (y)) ≤ d(x, f (x)) holds (respectively, f displaces all points of (X, d) the same distance, i.e. d(y, f (y)) = d(x, f (x)) for all x, y ∈ X. Definition 6.1.2. A metric space (X, d) is called (G)-δ -homogeneous (respectively, (G)-Clifford–Wolf homogeneous) if for every x, y ∈ X there exists a δ (x)-translation (respectively, Clifford–Wolf translation) of (X, d) (from an isometry group G), moving x to y. © Springer Nature Switzerland AG 2020 V. Berestovskii and Y. Nikonorov, Riemannian Manifolds and Homogeneous Geodesics, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-56658-6_6

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The following result is evident. Lemma 6.1.3. For a given metric space (X, d), any Clifford–Wolf translation is a δ (x)-translation for all x ∈ X, any (G)-Clifford–Wolf homogeneous space is (G)-δ homogeneous, and the latter space is (G)-homogeneous. Example 6.1.4. Every group with a bi-invariant metric (G, r) and every odd-dimensional Euclidean sphere (of unit radius) S2n+1 ⊂ E 2(n+1) with the induced inner (Riemannian) metric is a Clifford–Wolf homogeneous space. In the first case it is enough to use left translations by elements of the group. The second statement is proved essentially by Clifford himself. Obviously, a direct metric product of δ - (respectively, Clifford–Wolf) homogeneous spaces is again δ - (respectively, Clifford– Wolf) homogeneous. It is not difficult to show that a δ -homogeneous Riemannian manifold (M, µ) is G-δ -homogeneous for some connected transitive (possibly, not unique) isometry Lie group of the space (M, µ); as a G one can take the unit component of the full isometry group Isom(M, µ) of the space (M, µ). Therefore, in the Riemannian case, we can consider as G a connected transitive isometry Lie group. Definition 6.1.5. A locally compact intrinsic metric or a homogeneous Riemannian space (M = G/H, d) with a transitive locally compact in the compact-open topology topological or Lie group G and a stabilizer subgroup H at a point x ∈ M is called G-normal homogeneous in the generalized (respectively, usual) sense if G admits a bi-invariant (respectively, Riemannian bi-invariant) intrinsic metric r such that the natural projection (G, r) → (G/H, d) is a submetry. Notice that according to the paper [58], any bi-invariant intrinsic metric on a Lie group is Finslerian. It should be noted that the notions of δ -homogeneity and generalized normal homogeneity are applied for the case of Finsler manifolds, see [498, 491] and the references therein.

6.2 δ -homogeneous and Restrictively Clifford–Wolf Homogeneous Riemannian Manifolds Definition 6.2.1. An intrinsic metric space (M, d) is called restrictively (G)-δ homogeneous (respectively, restrictively (G)-Clifford–Wolf homogeneous) if for every x ∈ M there exists a number r(x) > 0 such that for every two points y, z in the open ball U(x, r(x))) there exists a δ (y)-translation (respectively, a Clifford–Wolf translation) of the space (M, d) (from the isometry group G), moving y to z. The supremum R(x) of all such numbers r(x) is called the (G)-δ -homogeneity radius (respectively, the (G)-Clifford–Wolf homogeneity radius) of the space (M, d) at the point x. In particular, all these characteristics could be used for homogeneous Riemannian spaces as in Definition 6.1.5.

6.2 δ -homogeneous and Restrictively Clifford–Wolf Homogeneous . . .

331

Proposition 6.2.2. Every connected restrictively (G)-δ -homogeneous locally compact complete intrinsic metric space is (G)-δ -homogeneous. Proof. It is clear that (in the notation of Definition 6.2.1) the function R(x), x ∈ M, is equal identically to +∞, i.e. the space (M, d) is (G)-δ -homogeneous, or it satisfies the inequality |R(x1 ) − R(x2 )| ≤ d(x1 , x2 ), where the function R(x), x ∈ M, is positive. Let us consider arbitrary points x, y of a metric space (M, d), and suppose that this space satisfies the above-stated condition. Then one can join the points x and y by some shortest arc [x, y]. According to the above discussion, one can divide sequentially this shortest arc by points x0 = x, x1 , . . . , xm = y such that for every l, where 0 ≤ l ≤ m − 1, there exists a δ (xl )-translation fl of the space (M, ρ) (from the group G), moving the point xl to the point xl+1 . Now the triangle inequality implies that the composition f := fm−1 ◦ · · · ◦ f0 is a δ (x)-translation of the space (M, d) (from the group G), such that f (x) = y. t u Corollary 6.2.3. Any connected restrictively (G)-Glifford–Wolf homogeneous locally compact complete intrinsic metric space is (G)-δ -homogeneous. Definition 6.2.4. An isometry f of a metric space (M, d) onto itself is called bounded if displ( f ) := sup d(x, f (x)) < +∞. x∈M

The number displ( f ) is called the displacement of the isometry f . Using the triangle inequality in (M, d) it is easy to prove Proposition 6.2.5. The set BI(M) of all bounded isometries of a metric space (M, d) is a group with bi-invariant metric η( f , g) := displ( f −1 g). Lemma 6.2.6. Let (M, d) be a Riemannian manifold (with the intrinsic metric d) with a transitive isometry Lie group G closed in the full isometry group of (M, d); and fk ∈ G be a sequence of isometries with displacements uniformly bounded from above by a number r, where 0 < r < +∞. Then there exists a subsequence of the sequence fk converging in G to some isometry f . Moreover, displ( f ) ≤ r. As a consequence, the group (BI(M) ∩ G, η) is metrically complete. Proof. The space (M, d) can be represented as a homogeneous space G/H of the full connected isometry Lie group G by its compact subgroup H (the stabilizer of some point x ∈ M). By Proposition 4.9.2 and Theorem 1.6.14, the corresponding projection p : G → G/H = M is a locally trivial fiber bundle. It is easy to deduce from here that this projection is a proper mapping, i.e. the inverse image of every compact subset in (M, d) relative to p is compact. In consequence of the condition of the lemma, for all k, d(x, fk (x)) ≤ displ( fk ) ≤ r, i.e. fk (x) ∈ B(x, r). By the Hopf– Rinow theorem, the ball B(x, r) is compact. Then all isometries fk lie in the compact set p −1 (B(x, r)) ⊂ G. Therefore, there exists a subsequence of the sequence fk converging in G to some isometry f . It is clear that displ( f ) ≤ r by continuity of the action of the group G in M. Now the last assertion is obvious. t u

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Lemma 6.2.7. Let (M, d) and G be the same as in Lemma 6.2.6 and let U be an arbitrary neighborhood of the unit in G. Then there exists a number r > 0 such that f ∈ U if f ∈ G and displ( f ) ≤ r. Proof. Assume that the lemma is not true. Then for every natural number k there exists an isometry fk ∈ G such that displ( fk ) ≤ 1/k and fk ∈ / U. Hence η( fk , e) = displ( fk ) ≤ 1/k and fk → e in G by Lemma 6.2.6, a contradiction. t u Lemma 6.2.8. Let (M, d) be a complete Riemannian manifold, Radinj(x) = l the injectivity radius of M at a point x ∈ M; f ∈ I(M), 0 < 2 displ( f ) = 2d(x, f (x)) = 2r < l. Then displ( f k ) ≤ |k|r for all k ∈ Z; d(x, f 2 (x)) = 2r; all points of the orbit of the point x relative to the group f k , k ∈ Z, lie on a unique geodesic γ, joining points x and f (x). Proof. The first statement is a consequence of the triangle inequality in (M, d). By the condition on the injectivity radius, there exists a unique point y such that d(x, y) = 2r and d( f (x), y) = r, i.e. f (x) is the middle point of a unique shortest arc [x, y]. Assume that the second assertion is not valid. Then d(x, f 2 (x)) < 2r, f 2 (x) 6= y, and the point f 2 (x) is the middle point of a unique shortest arc [ f (x), f (y)]. Hence d( f (x), y) = r and f 2 (x) 6= y imply that d(y, f (y)) > r. This contradicts the equality displ( f ) = d(x, f (x)) = r. Thus, we must have d(x, f 2 (x)) = 2r and f 2 (x) = y. It follows from here that for all k ∈ Z, fk+1 (x) is the middle point of a unique shortest arc [ fk (x), fk+2 (x)], which implies the last assertion of the lemma. t u Theorem 6.2.9. For an arbitrary connected complete Riemannian manifold (M, d) and an isometry Lie group G of (M, d), the following conditions are equivalent: 1) (M, d) is G-δ -homogeneous (respectively, restrictively G-Clifford–Wolf homogeneous); 2) every geodesic γ in (M, d) with an initial point x ∈ M is the integral curve of some Killing vector field on the manifold (M, d) from the Lie algebra of the Lie group G attaining the maximal length value at the point x (respectively, having a constant length); 3) for any point x ∈ M and every vector v ∈ Mx , there exists a Killing vector field X on the manifold (M, d) from the Lie algebra of the Lie group G with condition X(x) = v attaining the maximal length value at the point x (respectively, having a constant length). Proof. We will perform the proof for both cases simultaneously. Obviously, condition 2) implies condition 3). The converse statement follows from 3) of Theorem 3.1.7. Now, it suffices to prove the equivalence of conditions 1) and 2). Since (M, d) is connected and complete, any of the four conditions mentioned in 1) and 2) implies that the Lie group G is transitive on (M, d). Then the injectivity radius of the manifold M is not less than some constant L > 0.

6.2 δ -homogeneous and Restrictively Clifford–Wolf Homogeneous . . .

333

1) ⇒ 2). Let U = exp(V ) be a diffeomorphic image of some bounded convex symmetric open neighborhood of zero in Ge and γ a (nontrivial) geodesic in (M, d) starting at x ∈ M. Then there exists a number r > 0, for which the assertion of Lemma 6.2.7 and the condition of Lemma 6.2.8 (respectively, the condition from Definition 6.2.1 of restrictively Clifford–Wolf homogeneous manifold) are fulfilled simultaneously. For every natural number k, there exists an isometry (respectively, a Clifford–Wolf translation) fk ∈ G such that displ( fk ) = r/2k−1 = d(x, fk (x)) and fk (x) = γ(sk ), where sk = c/2k−1 for some fixed number c > 0 independent of k. It follows from the choice of the number r and Lemmas 6.2.8 and 6.2.7 that the orbit O(x, k) of the point x with respect to any isometry group generated by fk lies on γ, fkl ∈ U, and displ( fkl ) = d(x, fkl (x)) = |l|r/2k−1 (respectively, displ( fkl ) = d(y, fkl (y)) = |l|r/2k−1 for all points y ∈ M) for all l ∈ [−2k−1 , 2k−1 ]. This implies that O(x, k + 1) ⊃ O(x, k) and fkl = exp((l/2k−1 )Xk ) for some Xk ∈ V and all natural k and l ∈ [−2k−1 , 2k−1 ]. One can choose a subsequence from the sequence Xk converging to some vector X from the closure of the (bounded) region V . By construction, for any binary-rational numbers s,t ∈ [0, 1], d(exp(tX)x, exp(sX)x) = |s − t|r (respectively, d(exp(tX)y, exp(sX)y) = |s − t|r for all points y ∈ M) and exp(τX)x) = γ(cτ) for all binary-rational numbers τ. Therefore, this is true for all numbers s,t ∈ [0, 1], τ ∈ R and the vector X, considered as the Killing vector field on (M, d), has maximal length at the point x (respectively, constant length), which is equal to r. Now it is clear that γ is an integral curve of the desired Killing vector field Y = (1/c)X. 1) ⇐ 2). Assume now that every geodesic γ, starting at a point x ∈ M, is an integral curve of some Killing vector field X on the manifold (M, d) from the Lie algebra of the Lie group G attaining the maximal length value at the point x (respectively, having constant length). It is possible to suggest that any such geodesic γ is parameterized by arc length. Then for every number t ∈ (0, L) and every point y ∈ M, Z t

t = d(x, exp(tX)(x)) = 0

|c0x (s)|ds ≥

Z t 0

|c0y (s)|ds ≥ d(y, exp(tX)(y)),

where cx and cy denote respectively the integral curves of the field X passing through the points x and y (in consequence of 3) of Theorem 3.1.7, for the second condition instead of inequalities in the formulas above one can and must put equalities). It follows from here that exp(tX) is δ (x)-translation (respectively, is the Clifford–Wolf translation) and (M, d) is restrictively δ -homogeneous, therefore, δ -homogeneous (respectively, restrictively Clifford–Wolf homogeneous). The theorem is proved. t u

Corollary 6.2.10. Every (G)-δ -homogeneous Riemannian manifold is a geodesic orbit Riemannian manifold (with respect to the isometry group G).

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6.3 Some Structural Results From Theorems 6.2.9 and 3.5.1 we get Corollary 6.3.1. Every δ -homogeneous Riemannian manifold (M, µ) with the corresponding intrinsic metric d has nonnegative sectional curvature. Note that this corollary also follows from the next theorem. Theorem 6.3.2 (V. N. Berestovskii–C. Plaut, [85]). Every locally compact δ -homogeneous space of curvature bounded below in the sense of Alexandrov has nonnegative curvature. Lemma 6.3.3. Assume that a Riemannian manifold (M, µ) is isometric to a direct metric product (K, µ1 ) × (E m , µ2 ), where (K, µ1 ) is a compact homogeneous Riemannian manifold, and (E m , µ2 ) is a Euclidean space. Then every isometry f of the space (M, µ) has the form f = f1 × f2 , where f1 (respectively, f2 ) is an isometry of the space (K, µ1 ) (respectively, (E m , µ2 )). Proof. It is easy to see that a geodesic in (M, µ) is a metric line if and only if it is situated in some Euclidean subspace {k} × E m . Therefore, any isometry f of the space (M, µ) transposes such subspaces. Since f keeps the orthogonality, f must also transpose all fibers of the form K × {e}. This proves the lemma. t u Lemma 6.3.4. If M = M1 × M2 is a direct metric product of Riemannian manifolds, then each of its isometries of the form f = f1 × f2 is a δ (x)-translation for the point x = (x1 , x2 ) ∈ M if and only if both isometries f1 : M1 → M1 and f2 : M2 → M2 are δ -translations at the points x1 ∈ M1 and x2 ∈ M2 respectively. Analogous assertions are fulfilled for Clifford–Wolf translations. q Proof. Let us recall that d((x1 , x2 ), (y1 , y2 )) = d12 (x1 , y1 ) + d22 (x2 , y2 ), where d, d1 , d2 are the intrinsic metrics of the spaces M, M1 and M2 respectively. This easily implies the sufficiency. Suppose that f = f1 × f2 is a δ -translation of the space M at the point x = (x1 , x2 ), but, for instance, f1 is not a δ -translation at the point x1 . Then there is a point x10 such that d1 (x10 , f1 (x10 )) > d1 (x1 , f1 (x1 )). Therefore, q d((x1 , x2 ), f (x1 , x2 )) = d12 (x1 , f1 (x1 )) + d22 (x2 , f2 (x2 ))
0. If 0 ≤ skXk < r; t, s ∈ [0, 1], then it implies that for g(s) = exp(sX), g(t) = exp(tX), the point γ(t) is the point of maximal displacement on (M, ρ) for the motion g(s), since ρ(g(s)(γ(t)), γ(t)) = skXk according to the equalities g(s)(γ(t)) = g(s)(g(t)(y)) = g(s + t)(y) = γ(s + t). Hence, d(g(t), g(t + s)) = skXk, the length of the curve g(t), 0 ≤ t ≤ 1, in (G, d) is equal to kXk. Therefore, one can join any two points in (G, d) by a curve of finite length (with respect to the metric d). Let D be the intrinsic metric corresponding to d. There exists a positive number s0 such that exp : g → G is a diffeomorphism

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of some open subset V of g, containing the zero, onto the open ball U(e, s0 ) with radius s0 in (G, d). Then the above reasonings imply that the curve g(t), 0 ≤ t ≤ 1, is a geodesic in (G, D) and D(g, h) = d(g, h) if d(g, h) < min(r, s0 ). Also, d ≤ D. From the above calculations of the length of the geodesic g(t) = exp(tX), 0 ≤ t ≤ 1, in (G, D) it is clear that D is the bi-invariant Finsler (intrinsic) metric on G determined by the Ad(G)-invariant norm k · k on g, which is defined by the formula (6.2). It is easy to check that this formula defines a norm on g. t u Question 6.4.4. Do the metrics d and D coincide on G? Theorem 6.4.5. Let (M, µ) be a compact homogeneous Riemannian manifold. Then there exists a positive number s > 0 such that for an arbitrary motion f of the space (M, µ) with maximal displacement δ , which p is less than s, there is a unique Killing vector field X on (M, µ) such that maxx∈M µ(X(x), X(x)) = 1 and γX (δ ) = f , where γX (t), t ∈ R, is the 1-parameter isometry group of (M, µ) generated by the field X. If, moreover, f is a Clifford–Wolf translation, then the Killing field X has unit length on (M, µ). Proof. Let us supply the identity component G of the full isometry group of (M, µ) with the bi-invariant metric d as in Theorem 6.4.3. There is a sufficiently small number s > 0 (which we can suppose smaller than the injectivity radius r of the manifold (M, µ)) such that the exponential map exp : g → G is a diffeomorphism of some neighborhood V of the zero in g onto an open ball U(e, s) in (G, d). Then for every motion f of the space (M, µ) with the condition d( f , e) = δ < s there exists a unique vector Y ∈ V such that exp(Y ) = f . It was shown in the proof of Theorem 6.4.3 that for all such motions f we have D( f , e) = d( f , e). This common value is equal also to the length of the path exp(τY ), 0 ≤ τ ≤ 1, which joins elements e and f , with respect to the bi-invariant norm k · k on p g from Theorem 6.4.3, and to the length kY k. By the definition, kY k = maxx∈M µ(Y (x),Y (x)). Now it is clear that X = (1/δ )Y is the desired vector. The uniqueness of X follows from the above arguments. Let us suppose also that f is a Clifford–Wolf translation. By the above construction, p p kXk = 1 = max µ(X(x), X(x)) = µ(X(x1 ), X(x1 )) (6.3) x∈M

p for some point x1 ∈ M. We state that µ(X(x), p X(x)) ≡ 1. Indeed, in the opposite case there would be a point x0 ∈ M such that µ(X(x0 ), X(x0 )) = ε < 1. Then the path c(t) = exp(tX)(x0 ), 0 ≤ t ≤ δ , joins the point x0 with the point f (x0 ) and has length δ ε. Therefore, ρ(x0 , f (x0 )) ≤ δ ε < δ = ρ(x1 , f (x1 )), because, according to the condition (6.3), the orbit of the point x1 under the action of the 1-parameter group exp(tX), t ∈ R, is a geodesic (see 3) of Theorem 3.1.7), and δ < r. This contradicts the fact that f is a Clifford–Wolf translation. t u

6.4 Generalized Normal Homogeneous and δ -homogeneous Riemannian Manifolds

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Theorem 6.4.6. Let (M, µ) be a compact connected G-δ -homogeneous Riemannian manifold with the intrinsic metric ρ, and let G be a closed connected (Lie) subgroup of the full isometry group of (M, µ), supplied with the bi-invariant intrinsic metric D as in Theorem 6.4.3 (more exactly, by its restriction to G). Then D is an intrinsic bi-invariant metric on G. Let us fix a point x0 ∈ M and define a projection p : G → M by the formula p(g) = g(x0 ) such that under the usual identification of M with G/H, where H is the stabilizer of G at the point x0 , p coincides with the canonical projection p : G → G/H. Then the map p : (G, D) → (M, ρ) is a submetry. Proof. The first statement easily follows from the arguments in the last two paragraphs in the proof of Theorem 6.4.3, applied to G. Now it is enough to check properties 1) and 2) from the proof of Theorem 6.4.1. 1) Let g, h ∈ G. Then ρ(p(g), p(h)) = ρ(g(x0 ), h(x0 )) ≤ max ρ(g(x), h(x)) = d(g, h) ≤ D(g, h), x∈M

i.e. p does not increase distances. 2) Consider some points x, y in M and put ρ(x, y) = a. Let us choose an arbitrary shortest arc K in (M, ρ) joining points x and y; consider a geodesic γ(s), s ∈ R, in (M, µ) parameterized by arc length such that γ(0) = x, γ(a) = y and γ(s) ∈ K, 0 ≤ s ≤ a. Since (M, ρ) is G-δ -homogeneous, there is a δ (x)-translation gt ∈ G of (M, ρ), moving the point x to the point γ(t), 0 < t ≤ a. Now if t is small enough, then by Theorems 6.4.3 and 6.4.5, there is a one-parameter p group of motions g(s) = γ (s) ∈ G, s ∈ R, such that g(t) = g and max µ(X(y), X(y)) = X t y∈M p µ(X(x), X(x)). Then g(s)(x) = γ(s), s ∈ R. Therefore, D(e = g(0), g(s)) = d(e, g(s)) = s for 0 ≤ s ≤ a. Suppose that p(h) = h(x0 ) = x for some element h ∈ G. Then y = γ(a) = g(a)(x) = g(a)(h(x0 )) = p(g(a)h), D(h, g(a)h) = D(e, g(a)) = a = ρ(x, y), which proves the theorem.

t u

From Corollary 6.4.2 and Theorem 6.4.6 we obtain Corollary 6.4.7. A compact connected Riemannian manifold is (G)-δ -homogeneous if and only if it is (G)-normal in the generalized sense. Let (G/H, µ) be a G-δ -homogeneous (or compact homogeneous) Riemannian manifold with a connected Lie group G and h·, ·i be an Ad(G)-invariant inner product on the Lie algebra g of the Lie group G (which exists for the G-δ -homogeneous case by Theorem 6.3.5). Consider the h·, ·i-orthogonal decomposition g = h ⊕ p. As usual, we identify µ with the corresponding Ad(H)-invariant inner product (·, ·) on p. Now, the previous corollary can be restated as follows.

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Theorem 6.4.8. A compact Riemannian manifold (G/H, µ) is G-δ -homogeneous for a Lie group G if and only if there exists an Ad(G)-invariant convex body B in g which is centrally symmetric (with respect to zero) and such that P(B) = {v ∈ p | (v, v) ≤ 1}, where P : g → p is the h·, ·i-orthogonal projection. Corollary 6.4.9. The vector space p and the inner product (·, ·) are invariant under Ad(NG (H0 )), where NG (H0 ) is the normalizer of the connected unit component H0 of H in G. Proof. Obviously, h is Ad(NG (H0 ))-invariant. Then p is Ad(NG (H0 ))-invariant, since h·, ·i is Ad(G)-invariant. The Ad(NG (H0 ))-invariance of the inner product (·, ·) follows from Theorem 6.4.8. t u Theorem 6.4.10. Every simply connected compact homogeneous Riemannian manifold (M, g) admits a semisimple compact transitive isometry group. Furthermore, if the identity component of the full isometry group of (M, g) is not semisimple, then χ(M) = 0 and (M, g) is the total space of some Riemannian submersion, which is a nontrivial principal bundle with simply connected Riemannian manifold (M1 , g1 ) as the base and pairwise isometric totally geodesic tori as fibers. If (M, g) is δ homogeneous then (M1 , g1 ) is δ -homogeneous too. Proof. The proof is carried out in the spirit of Theorem 4.15.8. The first assertion of the theorem follows from Theorem 4.15.6. Moreover, G0 , the identity component of the group of all isometries of (M, g), is not semisimple if and only if G0 has a nontrivial connected unit component C of its center. Then C acts on (M, g) as a nontrivial connected group of Clifford–Wolf translations. Consequently, χ(M) = 0. Clearly, the orbits of the one-parameter subgroups of C in (M, g) are geodesics (also see the proof of Theorem 4.15.6). Therefore, the orbits of C are pairwise isometric flat totally geodesic tori in (M, g). The simple connectedness of M and the connectedness of the fibers of the Riemannian submersion q : (M, g) → (C\M, g1 ) := (M1 , g1 ) imply the nontriviality of q and the simple connectedness of M1 . Theorem 6.4.1 implies that the metric quotient space (C\M, g1 ) := (M1 , g1 ) is a δ -homogeneous Riemannian manifold if (M, g) is a δ -homogeneous Riemannian manifold. t u Remark 6.4.11. If (M, g) is a homogeneous compact Riemannian manifold with χ(M) > 0 then by Theorem 6.4.10, the identity component of the full isometry group of (M, g) is semisimple. The converse fails: the connected unit component of the full isometry group of the round sphere S2l−1 , l ≥ 3, is a simple Lie group SO(2l) and a semisimple Lie group SO(4) with Lie algebra so(4) = so(3) ⊕ so(3) for S3 . Remark 6.4.12. The well-known Berger spheres S2n+1 = U(n + 1)/U(n) show that, in the general case, the identity component of the group G of all isometries of (M, g)

6.5 Additional Symmetries of δ -homogeneous Metrics

341

is not semisimple (even if (M, g) is normal); in this case the universal covering Lie group of G is noncompact. It is also worth noting that, for the Berger spheres S2n+1 = U(n + 1)/U(n) (with the normal metrics), the Lie algebra of the stabilizer U(n) is not orthogonal to the center of the Lie algebra u(n + 1) with respect to the corresponding Ad(U(n + 1))-invariant inner product. See [502] for details. Every generalized normal homogeneous (in other words, δ -homogeneous) Riemannian manifold has nonnegative sectional curvature. In Theorem 4.15.6 (see also [351]) it is proved that every compact normal homogeneous Riemannian manifold with finite fundamental group has positive Ricci curvature. There naturally arises the following Question 6.4.13. Is it true that every compact generalized normal homogeneous Riemannian manifold with finite fundamental group has positive Ricci curvature? Obviously, Theorem 4.12.13 implies Corollary 6.4.14. Every compact naturally reductive homogeneous Riemannian manifold with positive Euler characteristic is δ -homogeneous. According to Corollary 6.4.14 and Theorem 4.3.1, any compact naturally reductive homogeneous Riemannian manifold M, which is not δ -homogeneous, has zero Euler characteristic.

6.5 Additional Symmetries of δ -homogeneous Metrics Recall that a group G acts on a homogeneous space G/H by the transformations Lb : G/H → G/H, b ∈ G, where Lb (cH) = bcH. Let NG (H) be the normalizer of H in G. Given a ∈ NG (H), we can correctly define G-equivariant diffeomorphism Ra : G/H → G/H, acting by the following rule: Ra (cH) = cHa−1 = ca−1 H. Theorem 6.5.1. Let (G/H, ρ) be a compact G-δ -homogeneous Riemannian manifold with a connected transitive Lie group of isometries G and let NG (H) be the normalizer of H in G. Then, for every a ∈ NG (H), the diffeomorphism Ra : G/H → G/H is a Clifford–Wolf translation on (G/H, ρ). Proof. Clearly, Ra is an isometry if and only if for all c ∈ G the differential dra−1 (c) preserves the length of every vector u ∈ horc ⊂ Gc , where horc is the horizontal subspace of the corresponding Riemannian submersion pr : (G, ν) → (G/H, µ) in Gc and dra−1 (horc ) = horca−1 (see Proposition 4.9.2). Here r, l stand for right and left translations in G. We have the obvious equality

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ra−1 = lc ◦ la−1 ◦ (la ◦ ra−1 ) ◦ lc−1 , and the appropriate equality for the composition of the differentials of mappings. Clearly, lc−1 (c) = e, dlc−1 (horc ) = hore = p, and d(la ◦ ra−1 )(e) = Ad(a). The last mapping preserves p and (·, ·) by Corollary 6.4.9 and the obvious inclusion NG H ⊂ NG (H0 ). All differentials of left translations preserve the horizontal distribution and the length of horizontal vectors. Thus, Ra is an isometry. Clearly, this mapping is a Clifford–Wolf translation, since it is generated by the right translation ra on G commuting with all left translations on G which generate a transitive isometry group on (G/H, ρ). t u Lemma 6.5.2. The transformation Ra of an (effective) homogeneous space G/H for a ∈ NG (H) coincides with the transformation Lb for some b ∈ G if and only if a is equal to the product of a central element in G and some element in H. Proof. Suppose that Ra = Lb for some b ∈ G. Since Ra commutes with each Ld , d ∈ G, we see that b lies in the center of G. Now, the condition Ra = Lb is equivalent to the following: ca−1 H = bcH = cbH for all c ∈ G. Therefore, a = e bd, where e b = b−1 is a central element in G and d is an element in H. The converse assertion is obvious. t u Theorem 6.5.3. Let (G/H, ρ) be a compact δ -homogeneous Riemannian manifold whose full connected isometry group is a transitive semisimple Lie group G. Then NG (H)/H is a finite group. Proof. By Theorem 6.5.1, for every a ∈ NG (H), the diffeomorphism Ra := G/H → G/H acting by the rule Ra (cH) = cHa−1 = ca−1 H is an isometry of (G/H, ρ). If dim(NG (H)) > dim(H) then we can choose a continuous family of isometries of the form Ra not lying in G. Indeed, consider a vector U, lying in the Lie algebra of NG (H) rather than in h. Consider a = exp(tU) ∈ NG (H) for some real t. The transformation Ra is isometric on (G/H, ρ). Since the center of G is discrete, using Lemma 6.5.2, we infer that, for some open set O ⊂ R, all transformations Ra for a = exp(tU),t ∈ O, are outside of G. However, this contradicts the fact that G is the full connected isometry group of (G/H, ρ). Therefore, we conclude that dim(NG (H)) = dim(H) while NG (H)/H is finite since it is compact. t u Example 6.5.4. Let G be a connected compact semisimple Lie group and let µ be a left-invariant Riemannian metric on G such that G is the identity component of the full isometry group of (G, µ). Then (G, µ) is not δ -homogeneous. Indeed, if (G, µ) is δ -homogeneous then, by Theorem 6.5.3, NG (H)/H is finite. However, in our case H = {e} is trivial and NG (H)/H = G is not discrete. According to this example it is necessary to discuss δ -homogeneous left-invariant Riemannian metrics on Lie groups. Clearly, each bi-invariant metric ρ on a Lie group G is a G-δ -homogeneous. However, there exist δ -homogeneous left-invariant non-bi-invariant Riemannian metrics on Lie groups. This can be demonstrated as follows.

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Example 6.5.5. Let G be a connected compact semisimple Lie group and let K be a connected subgroup in G. Among all left-invariant Riemannian metrics on G, consider the subclass MG,K of the metrics right-invariant with respect to K. It is easy to see that MG,K consists of the (G × K)-invariant Riemannian metrics on the homogeneous space M = (G × K)/ diag(K) (we use the natural inclusion K ⊂ G). Indeed, each metric in MG,K has the transitive motion group G × K with stabilizer diag(K) at the unit e ∈ G. On the other hand, it is clear that G is transitive on M = (G × K)/ diag(K). Now, consider a (G×K)-normal homogeneous Riemannian metric ρ on M. Then the Riemannian homogeneous space (M, ρ) is (G × K)-δ -homogeneous (see Corollary 6.4.7). The above argument shows that (M, ρ) is isometric to the Lie group G with some left-invariant metric (M, ρ). This metric may be bi-invariant but it is easy to see that the set of (G × K)-normal homogeneous Riemannian metrics ρ on M is greater than the set of bi-invariant metrics on G (see [169] for details). Therefore, we obtain δ -homogeneous left-invariant non-bi-invariant Riemannian metrics on G. Example 6.5.6. Suppose that F is a connected compact simple Lie group, G = F k , k ≥ 2, H = diag(F) ⊂ G. Consider the Ledger–Obata space G/H = F k / diag(F) endowed with the metric ρ generated by the (minus) Killing form on F k . Then the homogeneous Riemannian manifold (G/H, ρ) is δ -homogeneous. On the other hand, it is isometric to the Lie group F k−1 with some left-invariant Riemannian metric ρ1 (for example, see [460]). If k ≥ 3 then ρ1 is not biinvariant. Remark 6.5.7. Obviously, for a compact G-δ -homogeneous Riemannian manifold (G/H, ρ) with positive Euler characteristic, all conditions of Theorem 6.5.3 are fulfilled. Indeed, every connected one-dimensional central subgroup in G would induce a vector field without zeros on G/H, which is impossible in view of the inequality χ(G/H) > 0. On the other hand, in this case Theorem 6.5.3 is well known since H and G have the same rank.

6.6 Totally Geodesic Submanifolds In this section we investigate some totally geodesic submanifolds of δ -homogeneous and geodesic orbit Riemannian manifolds. Proposition 6.6.1 (Theorem 8.9 of Chapter VII in [291]). Let (M, µ) be a Riemannian manifold, N be its totally geodesic submanifold, and X be a Killing field on M. Consider a smooth vector field Xe on N, which is the tangent (to N) component of the field X. Then Xe is a Killing field on the Riemannian manifold N. In [291] this proposition is used to prove that every closed totally geodesic submanifold of a homogeneous Riemannian manifold is homogeneous itself [291, Corollary 8.10 of Chapter VII]. Here we give some specification of this classical result.

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Theorem 6.6.2 ([73]). Every closed totally geodesic submanifold of a δ -homogeneous (geodesic orbit) Riemannian manifold is δ -homogeneous (respectively, geodesic orbit) itself. Proof. Let N be a closed totally geodesic submanifold of a δ -homogeneous (geodesic orbit) Riemannian manifold M. Since M is homogeneous, it is complete. Since N is a closed submanifold of M, it is complete too. Let U 6= 0 be a tangent vector at some point x ∈ N. First suppose that M is δ -homogeneous. By Theorem 6.2.9 to prove the δ homogeneity of N it is enough to show that there is a Killing field Y on N, whose value at the point x is U, and the maximal length value of Y is attained at the point x. Since M is a δ -homogeneous Riemannian manifold, there is a Killing field X on M such that its value at the point x is U, and the maximal value of its length is attained e the tangent comat the point x. Now as the required Killing field Y we can take X, ponent of the field X to N. According to Proposition 6.6.1, this field is Killing on N e = X(x) obviously. Since at the point x the length of the field X is maximal and X(x) among all points y ∈ M, x is a point of the maximal length value of the field Xe (the length of the field Xe does not exceed the length of the field X at all points of the manifold N). Now consider the case when M is a geodesic orbit space. It suffices to prove that there is a Killing field Y on N with the following properties: 1) the value Y at the point x is U; 2) x is a critical point of the length of the field Y on N. Indeed, in this case a geodesic passing through x in the direction U is an orbit of a one-dimensional motion group generated by the Killing field Y (this one-parameter group is well-defined because of the completeness of N). Since M is a geodesic orbit space, there is a Killing field X on M, whose value at the point x is U, and such that x is a critical point of the length of the field X e the tangent (see Lemma 5.2.5). Now as the required Killing field Y , one can take X, component of the field X to N. According to Proposition 6.6.1, it is a Killing field e on N, and, moreover, X(x) = X(x). Now we need to prove only that x is a critical point of the length of the field Xe on N. Let Z = X − Xe be the normal component of the field X on the manifold N, and let µ be the metric tensor on M. It is clear that e X) e = µ(X, X) − µ(Z, Z). The point x is a zero point for µ(Z, Z), therefore, x µ(X, is a point of the minimal value of µ(Z, Z) on N. Consequently, x is a critical point both of the function µ(X, X) and of the function µ(Z, Z) on the manifold N. But in e X). e Therefore, x is a critical this case x is also a critical point for the function µ(X, e = U 6= 0)). The theorem is completely point of the length of the field Xe (since X(x) proved. t u Corollary 6.6.3. Every closed totally geodesic submanifold of a normal homogeneous Riemannian manifold is δ -homogeneous.

6.7 Properties of δ -vectors

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Remark 6.6.4. Let M be a Riemannian manifold and F be some set of its isometries. Then every connected component of the set of points of M, which are fixed under every isometry in F, is a closed totally geodesic submanifold of M. In the same manner, if K is some set of Killing fields on M, then every connected component of the set of points of M, which are zeros for every Killing field in K, is a closed totally geodesic submanifold of M [291]. According to Lemma 6.3.4, the metric product of δ -homogeneous spaces is δ homogeneous itself. In the Riemannian case we have a converse to this statement: Theorem 6.6.5. Let M = M0 × M1 × · · · × Mk be a direct metric decomposition of a δ -homogeneous (respectively, geodesic orbit) Riemannian manifold M with the maximal Euclidean factor M0 . Then all factors of this product are δ -homogeneous (respectively, geodesic orbit). If M is δ -homogeneous, then Mi are compact for i 6= 0. Moreover, an isometry f = f0 × · · · × fk of the manifold M, which is a product of δ -translations, is a δ -translation itself. Proof. Since every fiber of the product under consideration is a complete totally geodesic submanifold, then according to Theorem 6.6.2 applied to M, all factors are δ -homogeneous (respectively, geodesic orbit), which proves the first statement. The second assertion follows from the maximality of the Euclidean factor M0 , Proposition 6.3.1 and Theorem 1.7.10. The latter assertion of the theorem follows from Lemma 6.3.4. t u Proposition 6.6.6. Let (G/H, µ) be a G-δ -homogeneous Riemannian manifold and K be a connected Lie subgroup of the Lie group G such that H ⊂ K ⊂ G. Then the orbit of the point x = H in G/H with respect to the group K is a totally geodesic submanifold in (G/H, µ). In particular, K/H with metric induced by the tensor µ is a δ -homogeneous space. Proof. By Corollary 6.2.10, any G-δ -homogeneous Riemannian manifold is a GGO-space. Therefore the first assertion immediately follows from Proposition 5.8.6. The second assertion follows from Theorem 6.6.2. t u

6.7 Properties of δ -vectors Further, on the basis of Corollary 6.4.7, we will (as a rule) use the term generalized normal homogeneous Riemannian manifold instead of δ - homogeneous Riemannian manifold. Let us consider a homogeneous Riemannian manifold (M = G/H, µ) with connected transitive isometry group G and compact isotropy subgroup H, whose invariant Riemannian metric µ is generated by some Ad(H)-invariant inner product (·, ·) on p for a fixed reductive p decomposition g = h ⊕ p (see Definition 4.2.10 and Proposition 4.9.2), |V | := (V,V ) for V ∈ p. An expression for the covariant derivatives in terms of Killing vector fields X,Y ∈ p is given by formulas (4.31) and (4.32).

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Suppose that the Killing vector field X + Y , where X ∈ p and Y ∈ h, admits the maximum of its length at the point eH ∈ M. Proposition 6.7.1. Under the above conditions, the function ϕ : G → R, defined by the formula ϕ(g) = |(Ad(g)(X +Y ))p |, where g ∈ G, has its absolute maximum at the point g = e. Moreover, (X, [W, X +Y ]p ) = 0 for all W ∈ g, 2

(X, [W, [W, X +Y ]]p ) + |[W, X +Y ]p | ≤ 0 for all W ∈ g.

(6.4) (6.5)

Proof. The first assertion is clear. Let us consider an arbitrary W ∈ g. Then the function f (t) = |(Ad(etW )(X +Y ))p |2 has its absolute maximum at the point t = 0. Now the proposition follows from the relation
f (t) = |X|2 +2(X, [W, X +Y ]p )t + |[W, X +Y ]p |2 + (X, [W, [W, X +Y ]]p ) t 2 +o(t 2 ) when t → 0.

t u

Now from 3) in Theorem 6.2.9 and Proposition 6.7.1, we get the following result. Theorem 6.7.2. Let (G/H, µ) be a G-δ -homogeneous Riemannian manifold with connected Lie group G. Then for every X ∈ p, there is a Y ∈ h with the conditions (6.4) and (6.5). Let us suppose that M = (G/H, µ) is a compact homogeneous connected Riemannian manifold with connected compact Lie group G. Let g = h ⊕ p, h·, ·i, and (·, ·) be the same as in Section 6.4, i.e. h·, ·i is an Ad(G)-invariant inner product on the Lie algebra g of the Lie group G, the Riemannian metric µ is generated by an Ad(H)-invariant inner product (·, ·) on p, where p is determined by the h·, ·iorthogonal decomposition g = h ⊕ p. We use an Ad(G)-invariant norm k · k on g and corresponding bi-invariant intrinsic metric D on G from Theorem 6.4.3. From Section 6.4 we get the following Proposition 6.7.3. The map p : (G, D) → (G/H, µ) does not increase distances. It is a submetry if and only if M is G-δ -homogeneous. Definition 6.7.4. A vector w ∈ g is called a δ -vector for the Riemannian homop geneous manifold (M = G/H, µ) if |P(w)| := (P(w), P(w)) = kwk, where P is as in Theorem 6.4.8. (This is equivalent to the condition that for any a ∈ G, (wp , wp ) ≥ (Ad(a)(w)|p , Ad(a)(w)|p ).) Proposition 6.7.5. Let us suppose that for a vector v ∈pp, the set W (v) of all δ vectors of the form w = v + u, u ∈ h (such that kwk = (v, v)) is nonempty. Then W (v) is compact and convex. p Moreover, there is a unique vector w = w(v) ∈ W (v) with the smallest distance hw − v, w − vi. p Proof. We can suggest that (v, v) = 1. Since p in Proposition 6.7.3 does not increase distances, then P in Theorem 6.4.8 has the same property, and indeed

6.7 Properties of δ -vectors

347

kwk = 1. Let suppose that w1 , w2 ∈ W (v), 0 ≤ t ≤ 1, and w = tw1 + (1 − t)w2 . Then by the triangle inequality, kwk = ktw1 + (1 − t)w2 k ≤ tkw1 k + (1 − t)kw2 k = t + (1 − t) = 1. Since P is a linear map, P(w) = P(tw1 + (1 − t)w2 ) = tP(w1 ) + (1 − t)P(w2 ) = tv + (1 − t)v = v. Once more, because P does not increase distances, it follows from the last two relations that kwk = 1 and w ∈ W (v). So, the set W (v) is convex. Evidently, it is compact, and we have proved the first statement. The compactness of W (v) implies the existence of a vector w ∈ W (v) with the p smallest distance |w − v|1 = hw − v, w − vi. If we have another such a vector w0 6= w, then by the previous statement, w00 := 12 (w + w0 ) ∈ W (v), and we get a contradiction, because 2|w00 − v|1 = |(w − v) + (w0 − v)|1 < |w − v|1 + |w0 − v|1 = 2|w − v|1 . t u Let v ∈ p with W (v) 6= 0/ (see Proposition 6.7.5). According to Proposition 6.7.5, there is a unique vector w ∈ W (v) with the smallest distance |w − v|1 . In the next two propositions we will use the notation w(v) for this vector and the notation u(v) for the vector w(v) − v ∈ h. Proposition 6.7.6. Consider any vector v ∈ p satisfying W (v) 6= 0. / The following four conditions are equivalent (see the notation above): w(v) = v, u(v) = 0, kvk = |v|, and the corresponding vector field X(v) on M attains the maximum of the length at the point x0 = p(e). Proposition 6.7.7. If W (v) 6= 0, / then the inequalities u(v) 6= 0 and kvk > |v| are equivalent. In this case the following statements hold: for every element g ∈ G, such that Ad(g)(h) = h, the equality Ad(g)(v) = v (respectively, Ad(g)(v) = −v) implies that Ad(g)(u(v)) = u(v) (respectively, Ad(g)(u(v)) = −u(v)). Proof. This follows directly from 6.7.5, 6.7.3, and the fact that k · k, h·, ·ie are Ad(G)-invariant and invariant under central symmetry. t u From Theorem 6.4.8 we get the following Proposition 6.7.8. A homogeneous Riemannian manifold (G/H, µ) with connected Lie group G is G-generalized normal homogeneous if and only if for every vector v ∈ p, there exists a vector u ∈ h such that the vector v + u is a δ -vector. Now, for a fixed X ∈ p we consider the set U(X) = {Y ∈ h , X + Y is δ -vector}. If U(X) 6= 0, / then U(X) is a compact and convex subset of h, and there is a unique Ye ∈ U(X) such that hYe , Ye i ≤ hY,Y i for all Y ∈ U(X) by Proposition 6.7.5. We will denote such Ye ∈ U(X) by u(X).

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Lemma 6.7.9. Consider any X ∈ p with U(X) 6= 0/ and Zh (X) = {Z ∈ h , [Z, X] = 0}. Then for any Z ∈ Zh (X) the equality [Z, u(X)] = 0 holds. Proof. Let us consider a = exp(tZ) ∈ H for small t. It is clear that Ad(a)(X) = X and Ad(a)(u(X)) ∈ U(X) (the set of δ -vectors is invariant under the action of Ad(H) : g → g). But hAd(a)(u(X)), Ad(a)(u(X))i = hu(X), u(X)i and, therefore, Ad(a)(u(X)) = u(X) by the discussion right before the lemma. On the other hand, Ad(a)(u(X)) = u(X) + t[Z, u(X)] + o(t) when t → 0, hence [Z, u(X)] = 0. t u Corollary 6.7.10. If X ∈ p with U(X) 6= 0/ is such that the Lie algebra Zh (X) has trivial center and rk(Zh (X)) = rk(h), then u(X) = 0, i.e. X is a δ -vector. Moreover, (X, [Z, [Z, X]]p ) + ([Z, X]p , [Z, X]p ) ≤ 0 for all Z ∈ g. Proof. Assume that u(X) 6= 0. Note that u(X) 6∈ Zh (X) since [u(X), Zh (X)] = 0 by Lemma 6.7.9 and Zh (X) has trivial center. Therefore, the Lie algebra Ru(X) + Zh (X) ⊂ h has rank rk(Zh (X)) + 1 = rk(h) + 1, which is impossible. Hence, u(X) = 0. It is proved in Proposition 6.7.1 that for any δ -vector X +Y (X ∈ p and Y ∈ h) the inequality (X, [Z, [Z, X +Y ]]p ) + ([Z, X +Y ]p , [Z, X +Y ]p ) ≤ 0 holds for any Z ∈ g. If we put Y = 0 in this inequality, then we get the last assertion of the corollary. t u

6.7.1 The Case of Positive Euler Characteristic Now, suppose that (M = G/H, µ) is a compact almost effective G-δ -homogenous Riemannian manifold of positive Euler characteristic. Then the group G is semisimple by Proposition 4.12.2, and there exists a maximal torus T in the group G such that T ⊂ H ⊂ G. We take the minus Killing form of the Lie algebra g as an Ad(G)invariant inner product h·, ·i on g. Consider a decomposition of the root system g C = tC ⊕

M

gα

α∈R

for the complexification gC of the Lie algebra g, where tC is the Cartan subalgebra corresponding to T and R is the root system. For any subset P ⊂ R denote by g(P) = ∑ gα the subspace spanned by the α∈P

corresponding root spaces gα . It is not difficult to see that for every α ∈ R, the real two-dimensional Ad(T )module Vα = V−α := (gα ⊕ g−α ) ∩ g is irreducible. Let R = R1 ∪ R2 , where h = t ⊕ g(R2 ) and p = g(R1 ). By means of the inner product h·, ·i|t on t, any root α ∈ R is identified with some vector in t.

6.7 Properties of δ -vectors

349

Proposition 6.7.11. Let α1 , . . . , αk ∈ R be linearly independent roots. Then there is a unique (up to multiplication by a constant) vector tc ∈ Lin{α1 , . . . αk } such that Ad(exp(stc )) = − Id on ⊕ki=1Vαi for some positive number s. Proof. One can easily prove this by using the dual basis in the Euclidean space Lin{α1 , . . . αk }. t u Proposition 6.7.12. Let α1 , . . . , αk ∈ R1 be linearly independent roots and v = ∑ki=1 vi ∈ p, where vi ∈ Vαi , i = 1, . . . , k, are non-zero vectors. Let u ∈ h be a nontrivial vector such that the vector u + v is a δ -vector and Cu the set of roots γ ∈ R2 such that the h·, ·i-orthogonal projection of the vector u on them is nontrivial. Then Cu 6= 0, / Cu ⊂ Lin{α1 , . . . αk } − tc⊥ , where tc⊥ is the orthogonal complement in Lin{α1 , . . . αk } to the vector tc from Proposition 6.7.11. Proof. Indeed, if Cu = 0, / then u ∈ t⊥ , and by Proposition 6.7.11 we get Ad(exp(stc ))(v) = −v,

Ad(exp(stc ))(u) = u,

(6.6)

since [u,tc ] = 0. This contradicts Proposition 6.7.7. Therefore, Cu 6= 0. / Now, if some γ ∈ Cu is not in Lin{α1 , . . . αk }, then there is a vector w ∈ t which is orthogonal to all α1 , . . . αk and satisfies the condition hw, γi 6= 0. Then [w, v] = 0 and [w, u] 6= 0, which contradicts Proposition 6.7.7. Finally, if γ ∈ Cu ∩ tc⊥ , then Ad(exp(stc )(uγ )) = uγ for a nonzero component uγ of the vector u from Vγ , which contradicts Proposition 6.7.7. t u Since the roots α ∈ R1 and γ ∈ R2 are noncollinear to each other, Proposition 6.7.12 implies p Proposition 6.7.13. If v ∈ Vα , α ∈ R1 , then kvk = |v| = (v, v), i.e. v is a δ -vector. Proof. Assume that there exists a nontrivial vector u ∈ h such that v + u is a δ vector. Apply Proposition 6.7.12 for k = 1 (α1 = α). But there is no root in γ ∈ R2 , lying in Lin{α}. Therefore u = 0. t u Note that the action of the Weyl group W (T ) of the Lie algebra g on t is generated by elements Ad(n), where n ∈ NG (T ) (NG (T ) is the normalizer of the torus T in the group G). Since the Chebyshev norm k · k and the inner product h·, ·i are Ad(G)invariant, it follows from Proposition 6.7.13 that, for any roots α, β ∈ R1 from a joint orbit of the action by W (T ), the inner product (·, ·) is proportional to the inner product h·, ·i on modules Vα and Vβ with one and the same proportion coefficient. In particular, if the group G is simple, then the Weyl group W (T ) acts transitively on roots of one and the same length. Since all roots of any simple Lie algebra either have the same length or may have exactly two different lengths (so there are “long” and “short” roots), in the first case the G-δ -homogeneous metric µ must be normal, while in the second case it has a special form as in the next section 6.8. This observation plays an important role in the paper [73]. But in this book we have presented a somewhat simpler version of the relevant matter.

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6 Generalized Normal Homogeneous Manifolds With Intrinsic Metrics

6.8 Generalized Normal Homogeneous Manifolds of One Special Type Let G be a compact connected Lie group and H ⊂ K ⊂ G be its closed subgroups. Fix some Ad(G)-invariant inner product h·, ·i on the Lie algebra g of the group G. Consider the h·, ·i- orthogonal decomposition g = h ⊕ p = h ⊕ p1 ⊕ p2 ,

(6.7)

where k = h ⊕ p2 is the Lie algebra of the group K. Obviously, [p2 , p1 ] ⊂ p1 . Let µ = µx1 ,x2 be a G-invariant Riemannian metric on G/H, generated by the inner product (·, ·) = x1 h·, ·i|p1 + x2 h·, ·i|p2 (6.8) on p for some x1 > 0, x2 > 0 with x1 6= x2 . For any vector V ∈ g, we denote by Vp and Vh its h·, ·i-orthogonal projections to h and p respectively. Proposition 6.8.1 ([230]). The vector W = X + Y + Z, where X ∈ p1 , Y ∈ p2 , and Z ∈ h, is a geodesic vector on (G/H, µ) if and only if [Z,Y ] = 0,

[X,Y ] =

x1 [X, Z]. x2 − x1

(6.9)

Proof. By Theorem 5.1.2, the vector X + Y + Z is geodesic if and only if (X + Y, [U, X + Y + Z]p ) = 0 for any U ∈ p. Then the proposition follows from the equalities (X +Y, [U, X +Y + Z]p ) = x1 hX, [U, X +Y + Z]i + x2 hY, [U, X +Y + Z]i = x1 h[X +Y + Z, X],Ui + x2 h[X +Y + Z,Y ],Ui = h(x2 − x1 )[X,Y ] + x1 [Z, X] + x2 [Z,Y ],Ui , because [Z,Y ] ∈ p2 and [X,Y ], [Z, X] ∈ p1 .

t u

Proposition 6.8.2. Let W = X + Y + Z be a δ -vector on (G/H, µ), where X ∈ p1 , Y ∈ p2 , Z ∈ h. Then for any U ∈ p1 the following inequality holds: −x1 h[U, X]h , [U, X]h i + (x2 − x1 )h[U, X]p2 , [U, X]p2 i +(x1 − x2 )h[U,Y ], [U, X]i + (x1 − x2 )h[U,Y ], [U,Y ]i (6.10) +x1 h[U, X], [U, Z]i + (2x1 − x2 )h[U,Y ], [U, Z]i + x1 h[U, Z], [U, Z]i ≤ 0. Proof. As a consequence of Definition 6.7.4 and Proposition 6.7.1, we get (X +Y, [U, [U, X +Y + Z]]p ) + ([U, X +Y + Z]p , [U, X +Y + Z]p ) ≤ 0. It is clear that [Z, X], [Z,U], [Y, X], [Y,U] ∈ p1 , [Z,Y ] ∈ p2 . Therefore, using the Ad(G)-invariance of h·, ·i, we obtain

6.8 Generalized Normal Homogeneous Manifolds of One Special Type

351

0 ≥ (X +Y, [U, [U, X +Y + Z]]p ) + ([U, X +Y + Z]p , [U, X +Y + Z]p ) = −x1 h[U, X], [U, X +Y + Z]i − x2 h[U,Y ], [U, X +Y + Z]i +x1 h[U, X]p1 + [U,Y + Z], [U, X]p1 + [U,Y + Z]i + x2 h[U, X]p2 , [U, X]p2 i = −x1 h[U, X], [U, X]i − x1 h[U, X], [U,Y ]i − x1 h[U, X], [U, Z]i − x2 h[U,Y ], [U, X]i −x2 h[U,Y ], [U,Y ]i − x2 h[U,Y ], [U, Z]i + x1 h[U, X]p1 , [U, X]p1 i +x1 h[U,Y ], [U,Y ]i + x1 h[U, Z], [U, Z]i + 2x1 h[U,Y ], [U, X]i + 2x1 h[U, X], [U, Z]i +2x1 h[U,Y ], [U, Z]i + x2 h[U, X]p2 , [U, X]p2 i = −x1 h[U, X]h , [U, X]h i +(x2 − x1 )h[U, X]p2 , [U, X]p2 i + (x1 − x2 )h[U,Y ], [U, X]i + (x1 − x2 )h[U,Y ], [U,Y ]i +x1 h[U, X], [U, Z]i + (2x1 − x2 )h[U,Y ], [U, Z]i + x1 h[U, Z], [U, Z]i, which proves the proposition.

t u

Corollary 6.8.3. If under the conditions of Proposition 6.8.2, X = 0, then for any U ∈ p1 we have (x1 − x2 )h[U,Y ], [U,Y ]i + (2x1 − x2 )h[U,Y ], [U, Z]i + x1 h[U, Z], [U, Z]i ≤ 0. (6.11) Proposition 6.8.4. For any δ -vector X + Y + Z on (G/H, µ), the vector Y + Z is a δ -vector on K/H (with the induced metric). In particular, if (G/H, µ) is G-δ homogeneous, then K/H with the induced metric is K-δ -homogeneous. Proof. For any Ad(a), where a ∈ K, we have Ad(a)(p1 ) = p1 . Moreover, Ad(a)|p1 is an orthogonal transformation. Since (X, X) + (Y,Y ) = (X +Y, X +Y ) ≥ (Ad(a)(X +Y + Z)|p , Ad(a)(X +Y + Z)|p ) = (X, X) + (Ad(a)(Y + Z)|p , Ad(a)(Y + Z)|p ) for any a ∈ K, the vector Y + Z is a δ -vector for K/H. Note that the Riemannian subspace K/H of (G/H, µ) is K-normal in fact, because k = h ⊕ p2 . t u Proposition 6.8.5. If the vectors Xe + Y + Z and X + Y + Z are both δ -vectors on (G/H, µ), then e X]h , [X, e X]h i ≥ (x2 − x1 )h[X, e X]p , [X, e X]p i. x1 h[X, 2 2 e ] = x1 /(x2 − x1 )[X, e Z]. Proof. From Proposition 6.8.1 we have the equality [X,Y e Putting U = X in the inequality (6.10) and using the above equality, we prove the proposition. t u Proposition 6.8.6. Suppose that (G/H, µ) is G-δ -homogeneous. Consider some X ∈ p1 , Y ∈ p2 , a = exp(tY ) for some t ∈ R, Xe = Ad(a)(X). Then e X]h , [X, e X]h i ≥ (x2 − x1 )h[X, e X]p , [X, e X]p i. x1 h[X, 2 2 Proof. Let Z ∈ h be such that X + Y + Z is a δ -vector. From Proposition 6.8.1 we have [Z,Y ] = 0. This implies that Ad(a)(Z) = Z. Besides this, Ad(a)(Y ) = Y and

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6 Generalized Normal Homogeneous Manifolds With Intrinsic Metrics

e X), e since Ad(a)|p is (·, ·)-orthogonal. Therefore, the vector Xe + Y + (X, X) = (X, 1 Z = Ad(a)(X +Y + Z) is a δ -vector too. Now we can apply Proposition 6.8.5. t u Since for a = exp(tY ), we have Ad(a)(X) = X + [Y, X]t + o(t) when t → 0, we get the following infinitesimal version of Proposition 6.8.6. Proposition 6.8.7. Suppose that (G/H, µ) is G-δ -homogeneous. If X ∈ p1 and Y ∈ p2 , then x1 h[[Y, X], X]h , [[Y, X], X]h i ≥ (x2 − x1 )h[[Y, X], X]p2 , [[Y, X], X]p2 i.

6.9 Classification of Generalized Normal Homogeneous Riemannian Manifolds of Positive Euler Characteristic In this section, we obtain a classification of compact simply connected generalized normal homogeneous Riemannian manifolds (G/H, µ) with positive Euler characteristic. As a consequence of Theorem 4.12.3, any such manifold is indecomposable if and only if the isometry group G is simple. Therefore, we can consider only simple isometry groups. Since all normal metrics are generalized normal, we may assume by Theorem 6.6.5 that G-invariant metrics µ are not G-normal homogeneous. Moreover, according to Corollary 6.2.10, a generalized normal homogeneous space (G/H, µ) is geodesic orbit. According to the classification of geodesic orbit spaces of positive Euler characteristic (Theorem 5.10.4), we get either G/H = Sp(n)/U(1) · Sp(n − 1) (n ≥ 2), or G/H = SO(2n + 1)/U(n) (n ≥ 2). Recall that for n = 2 these spaces coincide. Both these families admit (orthogonal with respect to the Killing form) reductive decompositions of the form g = h ⊕ p = h ⊕ p1 ⊕ p2 , furthermore, k = h ⊕ p2 is a subalgebra in g (sp(1) ⊕ sp(n − 1) in the first case and so(2n) in the second case). Since the Ad(H)-modules p1 and p2 are irreducible and non-equivalent, every invariant metric µ on G/H is of the following special kind (see (6.8)): (·, ·) = x1 h·, ·i|p1 + x2 h·, ·i|p2 for some x1 > 0 and x2 > 0 (here h·, ·i is the minus Killing form of the Lie algebra g). These manifolds are discussed in more detail in Examples 5.10.1 and 5.10.2. Notice that the GO-space (SO(5)/U(2) = Sp(2)/U(1) · Sp(1), µ) can be isometrically embedded as a totally geodesic submanifold in (Sp(n)/U(1) · Sp(n − 1), µ) and (SO(2n + 1)/U(n), µ) under the coincidence of parameters x1 and x2 of invariant metrics (A ∈ SO(5) → diag(A, 0) ∈ SO(2n + 1) and A ∈ Sp(2) → diag(A, 0) ∈ Sp(n)), see Lemma 6.9.15 and Proposition 6.9.19.

6.9 Classification of Generalized Normal Homogeneous Riemannian Manifolds . . .

353

It turns out that the generalized normal homogeneous condition implies serious a priori restrictions on the values of parameters x1 and x2 of an invariant metric µ = µx1 ,x2 . Proposition 6.9.1. Suppose that a compact simply connected homogeneous manifold (G/H, µ) of positive Euler characteristic with a compact connected simple Lie group G is G-generalized normal but not G-normal homogeneous. Then the parameters x1 and x2 of the metric µ satisfy the inequality x1 < x2 ≤ 2x1 .

(6.12)

Proof. For x1 = x2 we have a G-normal metric. Therefore, we consider later the case x2 6= x1 . It is obvious that [p1 , p2 ] 6= 0. By Proposition 6.7.13, every vector of the form Y ∈ Vα ⊂ p2 is a δ -vector, so there exist a δ -vector Y ∈ p2 and a vector U ∈ p1 such that [U,Y ] 6= 0. Now, the inequality (6.11) implies that (x1 − x2 )h[U,Y ], [U,Y ]i ≤ 0 and, therefore, x1 < x2 . To prove the inequality x2 ≤ 2x1 it suffices to verify that there are two vectors X ∈ p1 and Y ∈ p2 such that the h·, ·i-orthogonal projections of the vector [[Y, X], X] to p2 and h are nontrivial and have equal lengths (relative to the inner product h·, ·i). Indeed, by Proposition 6.8.7, we get the following condition in this case: (2x1 − x2 )h[[Y, X], X]h , [[Y, X], X]h i = x1 h[[Y, X], X]h , [[Y, X], X]h i − (x2 − x1 )h[[Y, X], X]p2 , [[Y, X], X]p2 i ≥ 0. The existence of appropriate vectors X ∈ p1 and Y ∈ p2 can be easily justified with the help of root systems (which is how it was done in the paper [73]), on the other hand, it is possible to use the simple fact that all vectors X ∈ p1 and Y ∈ p2 for the space (G/H = SO(5)/U(2), µ) satisfy the equality 1 h[[Y, X], X]h , [[Y, X], X]h i1 = h[[Y, X], X]p2 , [[Y, X], X]p2 i1 = hX, Xi21 hY,Y i1 4 for the inner product hA, Bi1 = −1/2 trace(AB), since (G/H = SO(5)/U(2) = Sp(2)/U(1) · Sp(1), µ) is isometrically embedded (as a totally geodesic submanifold) into any manifold (G/H, µ) from the statement of the proposition (see the above discussion, Lemma 6.9.15 and Proposition 6.9.19). t u Therefore, it remains to clarify which of the homogeneous manifolds (SO(2n + 1)/U(n), µx1 ,x2 ) and (Sp(n)/U(1) · Sp(n − 1), µx1 ,x2 ) (n ≥ 2) for some x1 < x2 ≤ 2x1 are generalized normal homogeneous. It is necessary to note that for x2 = 2x1 they are symmetric, hence, normal homogeneous (with respect to more extensive isometry groups). Therefore, it suffices to consider only the metrics µx1 ,x2 for x1 < x2 < 2x1 .

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6 Generalized Normal Homogeneous Manifolds With Intrinsic Metrics

The main result of this section is the following classification theorem Theorem 6.9.2 (V. N. Berestovskii–E. V. Nikitenko–Yu. G. Nikonorov, [93]). Let (G/H, µ) be a compact simply connected indecomposable generalized normal homogeneous Riemannian manifold of positive Euler characteristic, which is not normal homogeneous. Then it is isometric to (Sp(l)/U(1) · Sp(l − 1), µ = µx1 ,x2 ), l ≥ 2, for x1 < x2 < 2x1 . The condition x1 < x2 < 2x1 for the metric µx1 ,x2 is equivalent to the fact that µx1 ,x2 has a positive sectional curvature with δ -pinching in the interval (1/16, 1/4). The full connected motion group of such a manifold is Sp(l)/{±I}. This theorem follows from Theorem 6.9.3 and Theorem 6.9.4. The proofs of these theorems are given respectively in Subsection 6.9.2 and Subsection 6.9.3. Theorem 6.9.3. The Riemannian manifold (SO(2l + 1)/U(l), µ = µx1 ,x2 ), where l ≥ 3, is not generalized normal homogeneous for x1 < x2 < 2x1 . Theorem 6.9.4. The Riemannian manifold (Sp(l)/U(1) · Sp(l − 1), µ = µx1 ,x2 ), l ≥ 2, is generalized normal homogeneous if and only if x1 ≤ x2 ≤ 2x1 . For x2 = x1 it is Sp(l)-normal homogeneous; for x2 = 2x1 it is symmetric and SU(2l)-normal homogeneous; for x2 ∈ (x1 , 2x1 ) it is not normal homogeneous for any isometry group. Recall that by Theorem 4.12.6, the full connected isometry group of the manifold (Sp(l)/U(1) · Sp(l − 1), µ) is Sp(l)/{±I} besides the case x2 = 2x1 , when the full connected motion group is the quotient group of the group SU(2l) by its center, moreover, the metric µ is SU(2l)-normal and the space (Sp(l)/U(1) · Sp(l − 1), µ) is isometric to the complex projective space CP2l−1 = SU(2l)/U(1) · S(U(2l − 1)) with the Fubini–Study metric. In the paper [73], the generalized normality of the Riemannian manifolds (M = SO(5)/U(2) = Sp(2)/U(1) · Sp(1), µx1 ,x2 ) for x1 < x2 < 2x1 was proved. The whole classification problem was solved in the paper [93], which we follow below. First of all, let us mention some important properties of spaces from two families which will be of interest to us. Both families have many common properties. They start with the same space SO(5)/U(2) = Sp(2)/U(1) · Sp(1) = CP3 , and admit a two-parametric family of invariant Riemannian metrics µx1 ,x2 ; all these metrics are geodesic orbit and weakly symmetric, and almost all are not normal [503, 434, 16]. Supplied with these metrics, the spaces from both these families are total spaces of Riemannian submersions, hence (nontrivial) fiber bundles, with irreducible symmetric Riemannian spaces with positive Euler characteristic as bases and (totally geodesic) fibers, SO(2l + 1)/SO(2l) = S2l and SO(2l)/U(l) respectively for the first family and Sp(l)/Sp(1) · Sp(l − 1) = HPl−1 and Sp(1)/U(1) = S2 for the second family. The bases in all cases are two-point homogeneous. Note that the space SO(2l)/U(l) is usually treated as the set of complex structures on R2l or the set of the metric-compatible fibrations S1 → RP2l−1 → CPl−1 [106], another interpretation is presented in Proposition 7.11.4. Notice that historically,

6.9 Classification of Generalized Normal Homogeneous Riemannian Manifolds . . .

355

SO(5)/U(2) = Sp(2)/U(1) · Sp(1) = CP3 was the first example of a compact nonnaturally reductive homogeneous space admitting invariant geodesic orbit Riemannian metrics [304]. Maybe it is appropriate to notice that the underlying manifold CP3 of the above homogeneous space is the Penrose twistor space [384], which can be interpreted, for instance, as the space of all compatible complex structures on the round 4-dimensional sphere [106]. Remark 6.9.5. Nevertheless, there is one known essential distinction between the families SO(2l + 1)/U(l), l ≥ 3 and Sp(l)/U(1) · Sp(l − 1) = CP2l−1 , l ≥ 2: the spaces of the first family admit no invariant Riemannian metrics of (strongly) positive sectional curvature, while all the spaces from the second family admit such metrics. Remark 6.9.6. The result of Theorem 6.9.4 implies the existence of the following unusual geometric situation: for every l ≥ 2, there are an irreducible orthogonal representation r : Sp(l) → SO(l(2l + 1)) (actually, the adjoint representation Ad of Sp(l)) in Euclidean space El(2l+1) and a convex body D bounded by an ellipsoid (not a ball!) in E2(2l−1) ⊂ El(2l+1) such that D is (the image under) the orthogonal projection (onto E2(2l−1) ) of an r(Sp(l))-invariant centrally symmetric convex body B in El(2l+1) . As a corollary of this, such B cannot be bounded by an ellipsoid in El(2l+1) . The authors do not know the answer to the following question. Question 6.9.7. Is it true that every compact simply connected indecomposable generalized normal homogeneous Riemannian manifold is either normal homogeneous, or weakly symmetric? Remark 6.9.8. In the decomposable case the answer to this question is negative.

6.9.1 General Ideas and Constructions Now we will describe a situation which is common to both Subsections 6.9.2 and 6.9.3. Let G be a compact connected Lie group and H ⊂ K ⊂ G be its closed subgroups. Let us fix any Ad(G)-invariant inner product h·, ·i on the Lie algebra g of the group G (recall that there is an unique, up to multiplication by a constant, such inner product for any simple Lie group G) and consider the following h·, ·i-orthogonal decomposition g = h ⊕ p = h ⊕ p1 ⊕ p2 , where k = h ⊕ p2 is the Lie algebra of the group K. Obviously, [p2 , p1 ] ⊂ p1 . Let µ = µx1 ,x2 be a G-invariant Riemannian metric on G/H, generated by the inner product of the form

356

6 Generalized Normal Homogeneous Manifolds With Intrinsic Metrics

(·, ·) = x1 h·, ·i|p1 + x2 h·, ·i|p2

(6.13)

on p for positive real numbers x1 and x2 . For any vector V ∈ g we denote by Vh and Vp its h·, ·i-orthogonal projection to h and p respectively. Recall (see Definition 6.7.4), that the vector W ∈ g is a δ -vector on (G/H, µ) if and only if (W |p ,W |p ) ≥ (Ad(a)(W )|p , Ad(a)(W )|p ), (6.14) for every a ∈ G. We will use repeatedly the result of Proposition 6.7.8: A homogeneous Riemannian manifold (G/H, µ) with connected Lie group G is G-generalized normal homogeneous if and only if for every vector V ∈ p there exists a vector U ∈ h such that the vector V +U is a δ -vector. Also we will need the assertion of Proposition 6.8.1, according to which, the vector W = X +Y + Z (X ∈ p1 ,Y ∈ p2 , Z ∈ h) is a geodesic vector on (G/H, µx1 ,x2 ) for x1 6= x2 if and only if the equalities [Z,Y ] = 0 and [X,Y ] = x1 /(x2 − x1 )[X, Z] hold. In Section 6.9.2, we deal with the spaces (G/H, µ = µx1 ,x2 ), where G = SO(2l + 1),

H = U(l),

K = SO(2l),

and

l ≥ 3,

with the embeddings U(l) ⊂ SO(2l) ⊂ SO(2l + 1), described below, and µ = µx1 ,x2 defined by the inner product (6.13). We define an Ad(SO(2l + 1))-invariant inner √ product on so(2l + 1), setting hA, Bi = −1/2 trace(A · B). We embed a matrix A + −1B ∈ u(l) into so(2l) via   √ AB A + −1B 7→ −B A in order to get the irreducible symmetric pair (so(2l), u(l)) (see e.g. [256]). Also we use the standard embedding of so(2l) into so(2l +1): A 7→ diag(A, 0). The inclusions u(l) ⊂ so(2l) ⊂ so(2l + 1), constructed above, induce the corresponding inclusions of connected matrix Lie groups τl : U(l) 7→ SO(2l) and τl0 : U(l) 7→ SO(2l + 1). The modules p1 and p2 , described above for a general situation, are Ad(τl0 (U(l)))invariant and Ad(τl0 (U(l)))-irreducible in this particular case. In Subsection 6.9.3 we find all generalized normal homogeneous metrics on the spaces G/H = Sp(l)/U(1) · Sp(l − 1), where H = U(1) · Sp(l − 1) ⊂ K = Sp(1) · Sp(l − 1) ⊂ Sp(l), with embedding (described below) such that the pairs (Sp(l), Sp(1) · Sp(l − 1)), (Sp(1),U(1)) are irreducible symmetric. Consider a (left-side) vector space Hl over the quaternion field H. For X = l

(X1 , X2 , . . . , Xl ) ∈ Hl and Y = (Y1 ,Y2 , . . . ,Yl ) ∈ Hl we define (X,Y )1 = ∑ XsY s . s=1

Then the group Sp(l) is a group of R-linear operators A : Hl → Hl with the property

6.9 Classification of Generalized Normal Homogeneous Riemannian Manifolds . . .

357

(A(X), A(Y ))1 = (X,Y )1 for any X,Y ∈ Hl . If we choose some (·, ·)1 -orthonormal quaternionic basis in Hl , then we can identify Sp(l) with a group of matrices A = (ai j ), ai j ∈ H with the property A−1 = A∗ , where a∗i j = a ji for 1 ≤ i, j ≤ l. In this case sp(l) consists of (l × l)-quaternion matrices A with the property A∗ = −A. Later on we shall use this identifications. For A, B ∈ sp(l) we define hA, Bi =

1 trace(ℜ(AB∗ )). 2

(6.15)

It is easy to see that h·, ·i is an Ad(Sp(l))-invariant inner product on the Lie algebra g = sp(l). In the sequel we will suppose (without loss of generality) that the embedding of sp(1) ⊕ sp(l − 1) in sp(l) is defined by (A, B) 7→ diag(A, B). We know that every invariant Riemannian metric µ = µx1 ,x2 on Sp(l)/U(1) · Sp(l − 1), corresponding to the inner product (6.13), is a geodesic orbit metric.

6.9.2 The Spaces SO(2l + 1)/U(l), l ≥ 3 First we will show that the Riemannian manifold (SO(7)/U(3), µ = µx1 ,x2 ), x1 < x2 < 2x1 , is not generalized normal homogeneous. Using the above notation, we have in this particular case 

 0 ab A =  −a 0 c  , −b −c 0  0 a b d e      −a 0 c e g        −b −c 0 f h u(3) =   −d −e − f 0 a      −e −g −h −a 0    − f −h −k −b −c   0 0 0 0 0

p1 =

                  

 d e f B =  e g h , f h k

 f 0 h 0  k 0  b 0 ; c 0  0 0 00



a, b, c, d, e, f , g, h, k ∈ R

        



X

 0 0 0 0 0 0 s1  0 0 0 0 0 0 s2     0 0 0 0 0 0 s3     =  0 0 0 0 0 0 s4  ;  0 0 0 0 0 0 s5     0 0 0 0 0 0 s6  −s1 −s2 −s3 −s4 −s5 −s6 0

         

si ∈ R

                  

,

,

358

6 Generalized Normal Homogeneous Manifolds With Intrinsic Metrics

  0 l      −l 0       −m −n   p2 = Y =   0 p    −p 0       −q −r    0 0

 m 0 p q0 n −p 0 r 0   0 −q −r 0 0   q 0 −l −m 0  ; r l 0 −n 0   0 m n 0 0 0 0 0 00

l, m, n, p, q, r ∈ R

         

.

        

Note that for vectors X from p1 as above we have hX, Xi = s21 + s22 + s23 + s24 + s25 + s26 , and for vectors Y ∈ p2 we have hY,Y i = 2l 2 + 2m2 + 2n2 + 2p2 + 2q2 + 2r2 . Let Ei, j be a (7 × 7)-matrix, whose (i, j)-th entry is equal to 1, and all other entries are zero. For any 1 ≤ i < j ≤ 7, we put Fi, j = Ei, j − E j,i . Proposition 6.9.9. The Riemannian manifold (SO(7)/U(3), µ = µx1 ,x2 ) is not generalized normal homogeneous for x1 < x2 < 2x1 . To prove this assertion, we need several lemmas. For W ∈ so(7), we denote by O(W ) the orbit of W under the action of Ad(SO(7)), i.e. O(W ) = {V ∈ so(7) | ∃ Q ∈ SO(7),V = QW Q−1 }. Lemma 6.9.10. Let us consider W = X + Y + Z, where X = s1 F1,7 ∈ p1 (s1 6= 0), Y = q(F1,6 − F3,4 ) + r(F2,6 − F3,5 ) ∈ p2 (q 6= 0, r 6= 0), Z ∈ h = u(3) (see above). If W is a geodesic vector on (SO(7)/U(3), µ) for x1 < x2 < 2x1 , then 

0 0 0 0 0

0 0 0 0 0

0 0 0

0 0

0 0

x2 x1 q s1 x2  x1 r 0 



    x2 −2x1 x2 −2x1   x1 q x1 r 0 0   2x1 −x2  W = q 0 0 0 0   x1   2x1 −x2 r 0 0 0 0   x1  x2  x2 −x q −x r  0 0 0 0 0 1 1 −s1 0 0 0 0 0 0 x2 x2 x2 − 2x1 x2 − 2x1 = s1 F1,7 + q F1,6 + r F2,6 + q F3,4 + r F3,5 . x1 x1 x1 x1 Proof. Since W is a geodesic vector, from Proposition 6.8.1 we get [Z,Y ] = 0, [X,Y ] = x1 /(x2 − x1 )[X, Z]. Direct calculations show that [Z,Y ] = (qh − f r)(F1,2 − F4,5 ) + (qd + qk + er)(F1,3 − F4,6 ) +(rk +rg+eq)(F2,3 −F5,6 )+(cq−br)(F1,5 −F2,4 )+ar(F1,6 −F3,4 )+aq(F3,5 −F2,6 ), [X,Y ] = s1 q F6,7 ,

[X, Z] = s1 (a F2,7 + b F3,7 + d F4,7 + e F5,7 + f F6,7 ).

The vectors F1,2 − F4,5 , F1,3 − F4,6 , F2,3 − F5,6 , F1,5 − F2,4 , F1,6 − F3,4 , F3,5 − F2,6 are linearly independent in p2 , and the vectors Fi,7 , 2 ≤ i ≤ 6 are linearly independent

6.9 Classification of Generalized Normal Homogeneous Riemannian Manifolds . . .

in p1 . Therefore, a = b = d = e = c = k = g = 0, f = lemma is proved.

x2 −x1 x1 q,

and h =

359 x2 −x1 x1 r.

The t u

Remark 6.9.11. The structure of all geodesic vectors on (SO(7)/U(3), µ) is studied in [190]. Lemma 6.9.12. If the Riemannian manifold (SO(7)/U(3), µ), x1 < x2 < 2x1 , is SO(7)-generalized normal homogeneous, then for every s1 6= 0, q 6= 0, and r 6= 0, W = s1 F1,7 +

x2 x2 x2 − 2x1 x2 − 2x1 q F1,6 + r F2,6 + q F3,4 + r F3,5 x1 x1 x1 x1

is a δ -vector for (SO(7)/U(3), µ). Proof. If (SO(7)/U(3), µ) is SO(7)-generalized normal homogeneous, then for every vector of the form V = X + Y , where X = s1 F1,7 ∈ p1 (s1 6= 0), Y = q(F1,6 − F3,4 ) + r(F2,6 − F3,5 ) ∈ p2 (q 6= 0, r 6= 0), there is a Z ∈ h such that e = X + Y + Z is a δ -vector (see Proposition 6.7.8). In particular, such the vector W e W should be a geodesic vector. According to Lemma 6.9.10, we get that e = W = s1 F1,7 + x2 q F1,6 + x2 r F2,6 + x2 − 2x1 q F3,4 + x2 − 2x1 r F3,5 . W x1 x1 x1 x1 Therefore, this W is a δ -vector.

t u

Lemma 6.9.13. If A, B ∈ so(7), then A and B are in the same orbit of Ad(SO(7)) if and only if their characteristic polynomials coincide. Proof. It is obvious that if A and B are in the same orbit of Ad(SO(7)), then their characteristic polynomials coincide. Suppose that the characteristic polynomials of A and B coincide. The standard Weyl chamber of the Lie algebra so(7) is the following (see e.g. [106]): 

 diag

       0 −z1 0 −z2 0 −z3 , , , 0 z1 ≥ z2 ≥ z3 ≥ 0 . z1 0 z2 0 z3 0

If A and B are conjugate to distinct elements of the Weyl chamber, then, as it is easy to see, their characteristic polynomials are distinct. Hence, A and B are conjugate to one and the same element of the Weyl chamber. This implies that A and B are in one and the same orbit of Ad(SO(7)). The lemma is proved. t u In what follows we need the value λ = x2 /x1 . Now we consider the following e (see Lemma 6.9.10): two geodesic vectors W and W

W = s1 F1,7 + where

x2 x2 − 2x1 x2 − 2x1 x2 q F1,6 + r F2,6 + q F3,4 + r F3,5 , x1 x1 x1 x1

(6.16)

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6 Generalized Normal Homogeneous Manifolds With Intrinsic Metrics

q

λ 2 ((2 − λ )2 + λ 3 − 1), s s (λ 3 − 1)(1 − (2 − λ )2 ) λ 3 (2 − λ )2 q= , r= , 2 3 (2 − λ ) + λ − 1 (2 − λ )2 + (λ 3 − 1) s1 =

x2 − 2x1 x2 − 2x1 e = se1 F1,7 + x2 qeF1,6 + x2 e e W r F2,6 + qeF3,4 + r F3,5 , x1 x1 x1 x1 and

(6.17)

q (2 − λ )2 + λ 4 (λ − 1), s s λ 2 (λ − 1)(λ 4 − (2 − λ )2 ) λ 3 (2 − λ )2 e qe = , r = . (2 − λ )2 + λ 4 (λ − 1) (2 − λ )2 + λ 4 (λ − 1) se1 =

Lemma 6.9.14. The vector W (see (6.16)) is not a δ -vector on (SO(7)/U(3), µ) for x1 < x2 < 2x1 .

e Proof. Direct calculations show that the characteristic polynomials P(z) and P(z) e (see (6.17)) are the following: of the matrices W and W P(z) = z7 + (a + b(λ 2 + (2 − λ )2 ))z5 +(ab(2 − λ )2 + acλ 2 + b2 λ 2 (2 − λ )2 )z3 + abcλ 2 (2 − λ )2 z, e = z7 + (e P(z) a+e b(λ 2 + (2 − λ )2 ))z5 +(e ae b(2 − λ )2 + aeceλ 2 + e b2 λ 2 (2 − λ )2 )z3 + aee be cλ 2 (2 − λ )2 z, where λ=

x2 , x1

a = s21 ,

b = q2 + r2 ,

c = r2 ,

ae = se 21 ,

e b = qe 2 + e r 2,

ce = e r 2.

e e |p , W e |p ). Since Now, we going to show that P(z) = P(z) and (W |p ,W |p ) < (W x1 < x2 < 2x1 , we have 1 < λ < 2. It is easy to check that b = 1,

a = λ 2 ((2 − λ )2 + λ 3 − 1),

c=

λ 3 (2 − λ )2 , (2 − λ )2 + (λ 3 − 1)

e b = λ 2,

ae = (2 − λ )2 + λ 4 (λ − 1),

ce =

λ 3 (2 − λ )2 . (2 − λ )2 + λ 4 (λ − 1)

e is equivalent to the following system of equations: The equality P(z) = P(z)

6.9 Classification of Generalized Normal Homogeneous Riemannian Manifolds . . .

 2 2 2 e 2   a + b(λ +2 (2 − λ2) ) =2 ae2+ b(λ +2 (2 − λ ) ),  ab(2 − λ ) + acλ + b λ (2 − λ ) =  aee b(2 − λ )2 + aeceλ 2 + e b2 λ 2 (2 − λ )2 ,   2 2 abcλ (2 − λ ) = aee be cλ 2 (2 − λ )2 .

361

(6.18)

It is easy to verify that system (6.18) is fulfilled for the considered a, b, c, ae, e b, ce. e Therefore, P(z) = P(z). e |p , W e |p ) = x1 (e Since (W |p ,W |p ) = x1 (a + 2λ b) and (W a + 2λ e b), the inequality e e (W |p ,W |p ) < (W |p , W |p ) is equivalent to the following one: a + 2λ b < ae + 2λ e b. It is easy to see that ae + 2λ e b − a − 2λ b = (2 − λ )2 + λ 4 (λ − 1) + 2λ 3 − λ 2 ((2 − λ )2 + λ 3 − 1) − 2λ = 2(2 − λ )(λ 2 − 1)(λ − 1) > 0. e |p , W e |p ). Therefore, (W |p ,W |p ) < (W e e ∈ O(W ). On the other hand, Since P(z) = P(z), by Lemma 6.9.13 we get W e e (W |p ,W |p ) < (W |p , W |p ). Consequently, the vector W is not a δ -vector, because otherwise the inequality e |p , W e |p ) (W |p ,W |p ) ≥ (W must hold (see the formula (6.14) for δ -vectors above). The lemma is proved.

t u

Now, it suffices to note that the proof of Proposition 6.9.9 follows from Lemmas 6.9.12 and 6.9.14. For 1 ≤ m < l, we consider a special embedding σm,l : SO(2m + 1) × SO(2k) 7→ SO(2l + 1), where k = l − m. This embedding is completely determined by the embedding dσm,l : so(2m + 1) ⊕ so(2k) 7→ so(2l + 1) for the corresponding Lie algebras. Note that so(2m + 1) consists of matrices of the following type   V U E Q1 =  −U t W F  , −E t −F t 0 where V and W are skew-symmetric (m × m)-matrices, U is an arbitrary (m × m)matrix, and E and F are arbitrary (m × m)-matrices. The Lie algebra so(k) consists of matrices of the following form   A B Q2 = , −Bt C where A and C are skew-symmetric (k × k)-matrices and B is an arbitrary (k × k)matrix. Now we define dσm,l as follows:

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6 Generalized Normal Homogeneous Manifolds With Intrinsic Metrics



V  O  t dσm,l (Q1 , Q2 ) =   −U  O −E t

O A O −Bt O

U O W O −F t

O B O C O

 E O  F , O 0

where the Os denote zero matrices. Note that for the considered embeddings we have   σm,l τm0 (U(m)) × τk (U(k)) ⊂ τl0 (U(l)),     σm,l τm0 (U(m)) × Id = σm,l SO(2m + 1) × Id ∩ τl0 (U(l)). Now we suppose that G = SO(2l  l ≥ 3 and 1 < m  < l. Let us consider   + 1), e = σm,l SO(2m + 1) × Id , and H e = σm,l τm0 (U(m)) × Id . H = τl0 (U(l)), G It is clear that e ⊂ G, G

e ∩ H; e=G H

e ∩ SO(2l) = σm,l (SO(2m)), G

(6.19)

dσm,l (so(2m)⊥ ) = dσm,l (so(2m + 1)) ∩ (so(2l))⊥ ,

(6.20)

dσm,l (dτm (u(m))⊥ ) = dσm,l (so(2m + 1)) ∩ (dτl (u(l)))⊥ .

(6.21)

e H e through the point e¯ = eH in e of the group G Lemma 6.9.15. The orbit G/ (G/H, µ = µx1 ,x2 ), supplied with the induced Riemannian metric η, is a totally geodesic submanifold of (G/H, µ = µx1 ,x2 ). Moreover, the map e H, e η) is an isometry. (SO(2m + 1)/U(m), µx1 ,x2 ) → (G/   Proof. Let us consider T , a maximal (k-dimensional) torus in σm,l Id ×τk (U(k)) .   Note that T is also a maximal torus in σm,l Id ×SO(2k) and T ⊂ H. Let C be the e By Proposition 5.8.12, centralizer of T in SO(2l + 1). It is easy to see that C = T · G. the orbit of C through the point eH ∈ G/H, with the induced Riemannian metric η, is a totally geodesic submanifold of (G/H, µ = µx1 ,x2 ). But T ⊂ H and, consee H. e The embeddings (6.19), (6.20), quently, this orbit coincides with the space G/ e H, e η) is an isomeand (6.21) imply that the map (SO(2m + 1)/U(m), µx1 ,x2 ) → (G/ try. t u Proof (of Theorem 6.9.3). The case l = 3 has been considered in Proposition 6.9.9. Let us suppose that l ≥ 4 and (SO(2l + 1)/U(l), µ = µx1 ,x2 ), where x1 < x2 < 2x1 , is generalized normal homogeneous. Then by Lemma 6.9.15, (SO(7)/U(3), µ = µx1 ,x2 ) is a totally geodesic submanifold of the generalized normal homogeneous manifold (SO(2l +1)/U(l), µ = µx1 ,x2 ), and by Theorem 6.6.2, it must be generalized normal homogeneous itself. We get a contradiction with Proposition 6.9.9. t u

6.9 Classification of Generalized Normal Homogeneous Riemannian Manifolds . . .

363

6.9.3 The Spaces Sp(l)/U(1) · Sp(l − 1), l ≥ 2 The following result is obtained (by using another parametrization of inner products) in Corollary 6.11.10. Proposition 6.9.16. The Riemannian manifold (G/H = Sp(l)/U(1) · Sp(l − 1), µ = µx1 ,x2 ) is Sp(l)-generalized normal homogeneous for every positive x1 and x2 such that x1 ≤ x2 ≤ 2x1 . Remark 6.9.17. If x1 = x2 , then the metric µ is Sp(l)-normal and, therefore, it is Sp(l)-generalized normal homogeneous. For the case x2 ∈ (x1 , 2x1 ), the original proof of this result was obtained by the methods applied in Subsection 6.9.2, and published in [93]. One can easily get the statement for x2 = 2x1 by using a limiting process. On the other hand, the proof of Corollary 6.11.10 is based on the use of Hopf fibrations and some results on generalized normal homogeneous Riemannian metrics on odd-dimensional spheres. Now we are ready to prove Theorem 6.9.4 and, hence, complete the proof of Theorem 6.9.2, the main result of the section. Proof (of Theorem 6.9.4). First, let us suppose that the Riemannian manifold (Sp(l)/U(1) · Sp(l − 1), µ = µx1 ,x2 ) is generalized normal homogeneous. Then we have x1 ≤ x2 ≤ 2x1 by Proposition 6.9.1. On the other hand, for x2 = x1 and x2 = 2x1 the metric µ is Sp(l)-normal homogeneous and SU(2l)-normal homogeneous respectively. From Proposition 6.9.16 we get that the Riemannian manifold (Sp(l)/U(1) · Sp(l − 1), µ = µx1 ,x2 ) is generalized normal homogeneous for 2x1 > x2 > x1 . The theorem is proved. t u Remark 6.9.18. The Riemannian manifolds (Sp(l)/U(1) · Sp(l − 1), µ = µx1 ,x2 ), l ≥ 2, have positive sectional curvatures and their (exact) pinching constant is x2 2 ε = ( 4x ) if 0 < x2 ≤ 2x1 . For all other values of x1 , x2 this statement is not true 1 and sectional curvature is not necessarily nonnegative [452]. In the Lie algebra g = sp(l), we consider the following subalgebra e g = {diag(A, 0) ∈ sp(l) | A ∈ sp(2), 0 ∈ sp(l − 2)}. e = Sp(2) be a connected (closed) subgroup in G = Sp(l), corresponding to e Let G g, e ∩ H. It is clear that H e=G e = U(1) × Sp(1), where U(1) × Sp(1) ⊂ Sp(1) × and H e Moreover, we get Sp(1) ⊂ Sp(2) = G. e H, e e that is an orbit of the group G, Proposition 6.9.19. The homogeneous space G/ passing through the point e¯ = eH in (G/H, µ = µx1 ,x2 ), supplied with the induced Riemannian metric, is a totally geodesic submanifold of (G/H, µ = µx1 ,x2 ). Proof. Let us consider the torus T = diag(1, 1, S1 , . . . , Sl−2 ) ⊂ Sp(l), where Si are e × T is the identity circles (one-dimensional tori). It is easy to see that T ⊂ H and G

364

6 Generalized Normal Homogeneous Manifolds With Intrinsic Metrics

component of the centralizer of T . Proposition 5.8.12 implies that the orbit of this group (through the point eH) is a totally geodesic submanifold in (G/H, µ). On the e H. e other hand, this orbit coincides with G/ t u Note that Proposition 6.9.19 leads to an alternative proof of Theorem 6.9.4, see [93] for details.

6.10 Generalized Normal Homogeneous Metrics and 2-means in the Sense of W. J. Firey In this section, we develop a new method to study generalized normal homogeneous Riemannian manifolds. It is based on the notion of (dual) 2-means due to W. J. Firey [214, P. 18]. The exposition is based on the paper [83]. The following two facts are due to H. Minkowski. For any convex body K, having the origin as an interior point, in the Euclidean vector space Rn supplied with the standard inner product (·, ·), its support function h(u) = max{(u, v) | v ∈ K} is nonzero, nonnegative, convex, and positively homogeneous. This means that h((1 − θ )u1 + θ u2 ) ≤ (1 − θ )h(u1 ) + θ h(u2 ), h(λ u) = λ h(u),

λ ≥ 0,

θ ∈ [0, 1],

u1 , u2 ∈ Rn ,

u ∈ Rn .

Conversely, any real-valued nonzero, nonnegative, convex, and positively homogeneous function on Rn is the support function of a unique convex body K ⊂ Rn with the origin as an interior point. Consider two convex bodies K1 and K2 in Rn with the origin as a common interior point and the support functions hi , i = 1, 2. Let us fix some real numbers p ≥ 1 and θ ∈ [0, 1]. For all nonnegative numbers a1 and a2 define the value
1/p M p (a1 , a2 ) = (1 − θ )a1p + θ a2p . Then (for all p ≥ 1 and θ ∈ [0, 1]) the function u 7→ M p (h1 (u), h2 (u)) is the support function of some convex body in Rn (with the origin as an interior point) since this function is nonzero, nonnegative, convex, (p) (p) and positively homogeneous. We shall denote this body by Kθ = Kθ (K1 , K2 ) (see (p)

details in [214]). We shall call this body Kθ the dual p-mean of the bodies K1 and K2 (with the parameter θ ∈ [0, 1]). For θ = 1/2 it is also called the dual p-sum of K1 and K2 . Let Lm be an m-dimensional subspace of Rn , m < n. Denote by K ∗ the orthogonal projection of a convex body K ⊂ Rn to Lm . It is clear that the support function of K ∗ is the restriction of the support function of K to Lm . Therefore, we obtain
∗ (p) (p) Kθ (K1 , K2 ) = Kθ (K1∗ , K2∗ )

(6.22)

6.10 Generalized Normal Homogeneous Metrics and 2-means in the Sense of W. J. Firey

365

for every p ≥ 1 and θ ∈ [0, 1] (see also [214, p. 22]). Let {·, ·} be any inner product on Rn . There is a unique positive definite symmetric (with respect to (·, ·)) linear operator A : Rn → Rn such that {·, ·} = (·, A·). The following statement has been proved in [215, pp. 53–54]. Proposition 6.10.1. Let E be the unit ball in {Rn , (·, ·)}. Then the support function of E is q h(u) =

(u, A−1 u).

(6.23)

Proof. It follows from the definition that h(u) = max{(u, v) | hv, Avi = 1}. Any such v can be represented in the form v = λ A−1 u + w, where λ ∈ R and (u, w) = 0. Then (v, Av) = λ 2 (u, A−1 u) + (w, Aw); q p (u, v) = λ (u, A−1 u) = (u, A−1 u) 1 − (w, Aw), hence, the maximum of (u, v) is attained at w = 0 and is equal to (6.23).

t u

Proposition 6.10.2. Let {·, ·}1 , {·, ·}2 be two inner products on (Rn , (·, ·)) with the (2) corresponding operators A1 , A2 and the unit balls E1 , E2 . Then Kθ (E1 , E2 ) is the unit ball of the inner product {·, ·} = (·, A·), where −1 A = (1 − θ )A−1 1 + θ A2


−1

.

(6.24) (2)

Proof. Let h1 , h2 , h be the support functions for E1 , E2 , Kθ (E1 , E2 ). Then by (2)

definition of Kθ (E1 , E2 ) and Proposition 6.10.1, q h(u) = (1 − θ )h21 (u) + θ h22 (u) = q q −1 −1 (1 − θ )(u, A−1 u) + θ (u, A u) = (u, [(1 − θ )A−1 1 2 1 + θ A2 ]u). −1 It is clear that (1−θ )A−1 1 +θ A2 is a positive definite and symmetric linear operator n on (R , (·, ·)) and the operator (6.24) has the same properties. Now the assertion follows from the above calculations for h and Proposition 6.10.1. t u

Let M = G/H be a homogeneous space of a compact connected Lie group G. Let us denote by h·, ·i a fixed Ad(G)-invariant Euclidean metric on the Lie algebra g of G (for example, the minus Killing form if G is semisimple) and by g = h⊕p

(6.25)

the associated h·, ·i-orthogonal reductive decomposition, where h = Lie(H). An invariant Riemannian metric g on M is determined by an Ad(H)-invariant inner product go = (·, ·) on the space p, which is identified with the tangent space Mo at the initial point o = eH.

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6 Generalized Normal Homogeneous Manifolds With Intrinsic Metrics

Note that the inner product h·, ·i|p generates a normal metric on G/H. Any invariant Riemannian metric g on G/H is generated by some Ad(H)-invariant inner product (·, ·) = h·, A·i|p on p that corresponds to a symmetric positive definite operator A : (p, h·, ·i|p ) → (p, h·, ·i|p ). Now, consider the inner product (·, ·)0 = h·, A−1 ·i. We will call a G-invariant metric gb on G/H that corresponds to this new inner product c = α −1 gb dual to the metric g (with respect to h·, ·i|p ). It is easy to see that b gb = g, αg for α > 0. Theorem 6.10.3 ([83]). Suppose that two G-invariant metrics g1 = h·, A1 ·i and g2 = h·, A2 ·i are G-generalized normal homogeneous on a compact homogeneous space G/H. Then for every θ ∈ [0, 1], the G-invariant metric gθ = h·, A·i, where −1 −1 A = [(1 − θ )A−1 1 + θ A2 ] , is also a G-generalized normal homogeneous metric on G/H. In particular, the space of metrics that are dual to G-generalized normal homogeneous metrics is convex. Proof. Fix the decomposition (6.25). Let (·, ·)1 and (·, ·)2 be Ad(H)-invariant inner products on p that generate the G-invariant Riemannian metrics g1 and g2 respectively. Consider Ad(H)-equivariant symmetric (with respect to h·, ·i) positive definite operators A1 , A2 : p → p such that (u, v)1 = hu, A1 vi and (u, v)2 = hv, A2 vi for all u, v ∈ p. By Theorem 6.4.8, there are centrally symmetric (relatively to the origin) convex bodies B1 and B2 in g such that P(B1 ) = {v ∈ p | (v, v)1 = hv, A1 vi ≤ 1} := E1 and P(B2 ) = {v ∈ p | (v, v)2 = hv, A2 vi ≤ 1} =: E2 . (2) Let us fix θ ∈ [0, 1]. Consider the dual 2-mean Kθ (B1 , B2 ) in g and the dual 2
(2) (2) (2) mean Kθ (E1 , E2 ) in p. It is clear that P Kθ (B1 , B2 ) = Kθ (E1 , E2 ) by (6.22). On (2)

the other hand, by Proposition 6.10.2, Kθ (E1 , E2 ) is the unit ball E of the (clearly, Ad(H)-invariant) inner product (u, v) = hu, Avi on p, where A is defined by formula (2) (6.24). It is clear also that Kθ (B1 , B2 ) is a centrally symmetric convex body in g, because both B1 and B2 have this property. Therefore, again by Theorem 6.4.8, the G-invariant metric gθ on G/H, generated by the inner product (·, ·) on p, is Ggeneralized normal homogeneous. t u If p = p1 ⊕ p2 ⊕ · · · ⊕ pl , where pi are irreducible Ad(H)-invariant submodules of p, then for every positive real xi , i = 1, . . . , l, we get Ad(H)-invariant inner products x1 h·, ·i|p1 + x2 h·, ·i|p2 + · · · + xl h·, ·i|pl

(6.26)

on the space p. Moreover, if the submodules pi are mutually inequivalent, then every Ad(H)-invariant inner product (·, ·) on p has the above form (6.26). We can identify G-invariant metrics on G/H with the corresponding inner products on p. It is easy to see that the metric x1−1 h·, ·i|p1 + x2−1 h·, ·i|p2 + · · · + xl−1 h·, ·i|pl is dual to the metric (6.26). From this observation and Theorem 6.10.3 we get

6.11 Generalized Normal Homogeneous Riemannian Metrics on Spheres and . . .

367

Corollary 6.10.4. Suppose that x1 h·, ·i|p1 + x2 h·, ·i|p2 + · · · + xl h·, ·i|pl and y1 h·, ·i|p1 + y2 h·, ·i|p2 + · · · + yl h·, ·i|pl are G-generalized normal homogeneous metrics on a compact space G/H. Then for any θ ∈ [0, 1], the metric (1 − θ )x1−1 + θ y−1 1


−1

h·, ·i|p1 + · · · + (1 − θ )xl−1 + θ y−1 l


−1

h·, ·i|pl

has the same property. As a simple partial case we have Corollary 6.10.5. If the metrics αh·, ·i|p1 + β h·, ·i|p2 , αh·, ·i|p1 + γh·, ·i|p2 , where α, β , γ > 0 (l = 2), are G-generalized normal homogeneous on a
compact space G/H, then for any r ∈ [0, 1], the metric αh·, ·i|p1 + (1 − r)β + rγ h·, ·i|p2 has the same property. Proof. It suffices to use Corollary 6.10.4 and the fact that the image of the function
−1 θ ∈ [0, 1] 7→ (1 − θ )β −1 + θ γ −1 is the closed interval of R with the endpoints β and γ. t u Remark 6.10.6. One can easily check that one needs to take θ = rγ/[(1 − r)β + rγ].

6.11 Generalized Normal Homogeneous Riemannian Metrics on Spheres and Projective Spaces In this section, we obtain a complete classification of generalized normal homogeneous Riemannian metrics on the spheres Sn . We prove that for any connected (almost effective) transitive on Sn compact Lie group G, the family of G-invariant Riemannian metrics on Sn contains generalized normal homogeneous but not normal homogeneous metrics if and only if this family depends on more than one parameters and n ≥ 5. Any such family (which exists only for n = 2k + 1) contains a metric gcan of constant sectional curvature 1 on Sn . We also obtain the classification of generalized normal homogeneous Riemannian metrics on projective spaces. The exposition is based on the paper [83]. It should be noted that many of the results presented here were obtained earlier in [63, 64] by V. N. Berestovskii, using more direct methods. The classification of generalized normal homogeneous metrics on spheres follows from Theorems 6.11.3, 6.11.9, 6.11.11, 6.11.24, and Proposition 6.11.6 below. We collect all these results in Table 6.2. To compare this classification with the classification of (usual) normal homogeneous metrics on spheres, we reproduce here Table 6.1 with the latter classification [246, Table 2.4], replacing the symbols a2 , b2 and c2 in [246] with (positive) parameters s, t and r respectively. We denote by gcan a Riemannian metric of constant curvature 1 on a given sphere. The space p3 appears only in the last line of Tables 6.1 and 6.2; it is u(1) ⊂ sp(n + 1) ⊕ u(1), which is naturally isomorphic to the space, tangent to the fiber of the Hopf fibration pr1 (see Section 6.11.1) at the point (1, 0, . . . , 0) ∈ S4n+3 .

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6 Generalized Normal Homogeneous Manifolds With Intrinsic Metrics

Table 6.1 Normal homogeneous metrics on spheres (·, ·)|p3

(·, ·)|p2

G

H

1

SO(n + 1)

SO(n)

rgcan

2

G2

SU(3)

rgcan

3

Spin(7)

G2

rgcan

4

SU(2)

{e}

rgcan

5

SU(n + 1)

SU(n)

n+1 2n

(·, ·)|p1

rgcan

rgcan n+1 2n 1 2

6

U(n + 1)

U(n)

t rgcan , t
0 on fibers, while all fibrations are still Riemannian submersions. As a result we obtain the following homogeneous Riemannian manifolds: (S2n+1 = U(n + 1)/U(n), ξt ), (S4n+3 = Sp(n + 1) × Sp(1)/(Sp(n) × Sp(1)), µt ), (S15 = Spin(9)/Spin(7), ψt ), (CP2n+1 = Sp(n + 1)/(U(1) × Sp(n)), νt ). Now, the metric µt,s , s > 0, on S4n+3 is defined such that pr1 : (S4n+3 , µt,s ) → (CP2n+1 , νt )

(6.27)

is a Riemannian submersion while the metric √ tensor is multiplied by s on the circlesfibers, so all these circles have length 2π s. Thus we get the homogeneous Riemannian manifolds (S4n+3 = Sp(n + 1) × U(1)/(Sp(n) × U(1)), µt,s ). Note that µ1,s = ξs . This is connected with the natural inclusions Sp(n + 1) ⊂ SU(2(n + 1)),

Sp(n + 1) ×U(1) ⊂ U(2(n + 1)).

(6.28)

Note that all invariant Riemannian metrics on the corresponding homogeneous spaces above are proportional to the mentioned metrics with one exception: leftinvariant metrics on the group SU(2) = S3 (see Proposition 6.11.6). Moreover, all Riemannian manifolds (S2n+1 , ξt ), (S4n+3 , µt ), (S15 , ψt ) are homothetic to distance spheres in (CPn+1 , gcan ), (HPn+1 , gcan ), and the Caley plane (CaP2 , gcan ) respectively; the converse statement is also true [502]. The normal homogeneous spaces (S2n+1 , ξt ), where 0 < t ≤ (n + 1)/2n, n ≥ 1, are also known as Berger’s spheres [95, 501, 502]. Finally, note that both Sp(4) and Spin(9) have dimension 36, which is minimal among the dimensions of all Lie groups, acting transitively on S15 .

6.11 Generalized Normal Homogeneous Riemannian Metrics on Spheres and . . .

371

All compact generalized normal homogeneous spaces studied here have positive sectional curvature. Omitting details, we refer to the papers [450], [451], and [452] by D. E. Volper, where he calculated the exact upper and lower bounds of sectional curvatures for all suitable one-parameter families. In all these cases, the bounds are some functions of the parameter t and do not depend on dimension. It is very interesting that these functions coincide for the spaces (S4n+3 , µt ) and (S15 , ψt ). There are some related results for spheres in the paper [446] by L. Verdiani and W. Ziller and in some other papers. As far as we know, the corresponding bounds for the family (S4n+3 , µt,s ) have not been calculated in the literature. Nevertheless, with the help of some criteria for positivity of sectional curvature from the paper [446], it is proved in [60] that all generalized normal homogeneous spaces from this family also have positive sectional curvatures. It should be noted that generalized normal homogeneous Riemannian metrics show a great diversity of properties. For example, for the families in Theorem 6.11.1, µt and ξt are simultaneously weakly symmetric and naturally reductive, and νt and ψt are weakly symmetric but not naturally reductive [502, 503]. The metrics µt,s are weakly symmetric (see e.g. 12.9.2 in [484]), but not naturally reductive [502].

6.11.2 G-generalized Normal Homogeneous Metrics on Spheres For compact matrix groups G we will always use the Ad(G)-invariant inner product 1 hU,V i = ℜ(trace(UV ∗ )), 2

U,V ∈ g,

(6.29)

T

where V ∗ = V and (·), (·)T are the operations of quaternionic or complex conjugation and transposition, respectively. In all these cases we will use representations by skew-Hermitian matrices and, therefore, we may suppose that hU,V i = − 12 ℜ(trace(UV )). Let G/H be one of the following homogeneous spaces: SO(n + 1)/SO(n) = Sn , G2 /SU(3) = S6 , and Spin(7)/G2 = S7 . In the first case G/H is irreducible symmetric and in the other two cases it is isotropy irreducible, therefore, in all these cases the homogeneous space G/H admits only one (up to scaling) G-invariant metric that is G-normal (see e.g. Chapter 7 in [106]). Now, we consider SU(n + 1)-invariant metrics on the homogeneous sphere S2n+1 = SU(n + 1)/SU(n). First, we prove the following result. Theorem 6.11.2. Any Euclidean sphere S2n+1 , n ≥ 1, is SU(n + 1)-generalized normal homogeneous. Proof. By Proposition 6.7.8, it is sufficient to prove that for any tangent vector v ∈ Sx2n+1 , x0 = (1, 0, . . . , 0)T ∈ S2n+1 , there exists a Killing vector field K(x) = Ux, 0 x ∈ S2n+1 , where U ∈ su(n + 1), such that Ux0 = v and which is a δ -vector. The group G = U(n + 1) is transitive on the sphere S2n+1 with the isotropy subgroup

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6 Generalized Normal Homogeneous Manifolds With Intrinsic Metrics

H = U(n) at the point x0 . Therefore, one can consider S2n+1 as a homogeneous manifold (G/H, µ) with the corresponding invariant Riemannian metric µ. One obtains the Ad(U(n))-invariant h·, ·i-orthogonal decomposition u(n + 1) = p ⊕ u(n) = p1 ⊕ p2 ⊕ u(n) = p1 ⊕ u(1) ⊕ u(n),

(6.30)

where  p1 =  p2 =

0 −uT u 0nn



u1 0Tn 0n 0nn



 , u ∈ Cn ,

(6.31)

 , u1 ∈ ℑ(C) .

(6.32)

The lower indices above indicate the size of the zero block-matrices. Our goal is to find for every vector-matrix X + Y ∈ p1 ⊕ p2 a vector-matrix Z ∈ u(n) of the form    0 0Tn Z= ,Unn ∈ u(n) (6.33) 0n Unn such that U := X + Y + Z ∈ su(n + 1) and the Killing vector field K(x) := Ux, x ∈ S2n+1 , is a δ -vector. We characterize all δ -vectors U ∈ u(n + 1) for the point x0 ∈ S2n+1 . If x ∈ S2n+1 , then (Ux,Ux) = (U ∗Ux, x) = −(U 2 x, x). A matrix U ∈ u(n + 1) is a δ -vector for the point x0 ∈ S2n+1 if and only if (Ux0 ,Ux0 ) = −(U 2 x0 , x0 ) ≥ −(U 2 x, x) = (Ux,Ux) for all x ∈ S2n+1 . Since the matrix −U 2 is symmetric and positive definite, we can reformulate this in the following form: −U 2 = diag(λ 2 , B), where λ ∈ R and the symmetric matrix B is such that the matrix λ 2 · Idn −B is positive semi-definite. We will use this characterization in the last part of the proof. Below we shall find such fields U in two cases. In both these cases we should choose a matrix Unn that defines the vector Z by formula (6.33). Denote by B(u1 ) a (n × n)-matrix such that its (1, 1)-entry is equal to −u1 and all other entries are zero. 1) Suppose that u = (u2 , 0, . . . , 0)T ∈ Cn , u2 ≥ 0. In this case we may take Unn = B(u1 ). Then −U 2 = diag(|u1 |2 + |u2 |2 , |u1 |2 + |u2 |2 , 0, 0, . . . , 0). Obviously, U ∈ su(n + 1). p 2) Now, suppose that u ∈ Cn is arbitrary and consider |u| = (u, u) ≥ 0. The group U(n) acts transitively on every sphere in Cn with the center at the origin. Therefore, there exists an element g ∈ U(n) such that g(|u|, 0, . . . , 0)T = u. In this e case it suffices to take the matrix Unn = Ad(g)(B(u1 )). Let us take U = Ad(e g)(U), e where g˜ = diag(1, g) ∈ U(n + 1), and U is the matrix U constructed in the previous case with u2 = |u|. Since the group Ad(U(n)) preserves the set of δ -vectors, U is indeed a δ -vector. It is also clear that U ∈ su(n+1), since trace(Unn ) = trace(B(u1 )) = −u1 . t u

6.11 Generalized Normal Homogeneous Riemannian Metrics on Spheres and . . .

373

Further, consider the Ad(SU(n))-invariant h·, ·i-orthogonal decomposition g = su(n + 1) = p ⊕ h = p1 ⊕ p2 ⊕ h,

(6.34)

where p1 is the same as in (6.30), h consists of elements of the form (6.33) with Unn ∈ su(n), and p2 = {ia · diag(n, −1, . . . , −1), a ∈ R} . (6.35) It should be noted that the module p1 is Ad(SU(n))-irreducible only for n ≥ 2, because SU(1) is a trivial group. Now we consider the family ξt of SU(n+1)-invariant metrics on the sphere S2n+1 that correspond to the inner products (·, ·)t = h·, ·i|p1 +

2nt h·, ·i|p2 n+1

(6.36)

on p for t > 0. We consider also a representation of the Riemannian manifold (S2n+1 , ξt ) as the homogeneous space U(n + 1)/U(n) with the (U(n + 1)-invariant) metric ξt generated by the inner product (·, ·)t = h·, ·i|p1 + 2th·, ·i|p2 ,

(6.37)

where we have used decomposition (6.30) and h·, ·i is the Ad(U(n + 1))-invariant inner product (6.29) on the Lie algebra u(n + 1). Theorem 6.11.3. The Riemannian manifold (S2n+1 , ξt ) is U(n+1)-generalized normal homogeneous if and only if t ∈ (0, 1]. Moreover, this Riemannian manifold is SU(n + 1)-generalized normal homogeneous if and only if t ∈ [(n + 1)/2n, 1]. Proof. By Table 6.1, ξt is SU(n + 1)-normal homogeneous if and only if t = n+1 2n , n+1 and U(n+1)-normal homogeneous if and only if t < 2n . The metric ξt has constant sectional curvature 1 on S2n+1 for t = 1. This observation and Theorem 7.11.2 from Chapter 7 imply that (S2n+1 = SU(n+1)/SU(n), ξ1 ) is U(n+1)-generalized normal homogeneous (since it is even U(n + 1)-Clifford–Wolf homogeneous). The space (S2n+1 = SU(n + 1)/SU(n), ξt ), t < n+1 2n , is U(n + 1)-normal homogeneous, hence U(n + 1)-generalized normal homogeneous by Theorem 6.4.8. It is also SU(n + 1)generalized normal homogeneous for t = 1 by Theorem 6.11.2 and for t = n+1 2n by Theorem 6.4.8. Now, Corollary 6.10.5 implies the sufficiency in both assertions. Let us prove the necessity in the first assertion of the theorem. Let ξt be a U(n + 1)-generalized normal homogeneous metric on S2n+1 = U(n + 1)/U(n). By Proposition 6.8.7, for every X ∈ p1 , Y ∈ p2 the inequality x1 h[[Y, X], X]h , [[Y, X], X]h i ≥ (x2 − x1 )h[[Y, X], X]p2 , [[Y, X], X]p2 i holds, where x1 = 1 and x2 = 2t (see (6.37)). If we take    0 i Y = diag(i, 0, . . . , 0) and X = diag , 0, . . . , 0) , i 0

374

6 Generalized Normal Homogeneous Manifolds With Intrinsic Metrics

then [[Y, X], X] = −2Y + 2Z, where Z = diag(0, i, 0, . . . , 0) ∈ h. Since h−2Y, −2Y i = h2Z, 2Zi = 4, we get 4 ≥ (2t − 1) · 4, i.e. t ≤ 1. Finally, we prove the necessity in the second assertion of the theorem. Let ξt be a SU(n + 1)-generalized normal homogeneous metric on S2n+1 . This metric is generated by the inner product (6.36). For any non-trivial X ∈ p2 we see that Zh (X) = h (see Lemma 6.7.9). By Corollary 6.7.10 we get that X is a δ -vector and (X, [U, [U, X]]p ) + ([U, X]p , [U, X]p ) ≤ 0 for all U ∈ su(n + 1). Let us take X = i · diag(n, −1, . . . , −1)  ∈ p2Tand  any nonzero U ∈ p1 as in (6.31) 0 iu with F = C. Then 0 6= [U, X] = (n + 1) ∈ p1 . Therefore, iu 0nn 0 ≥ (X, [U, [U, X]]p ) + ([U, X]p , [U, X]p ) 2nt = hX, [U, [U, X]]i + h[U, X]p , [U, X]p i n+1 2nt =− h[U, X], [U, X]i + h[U, X], [U, X]i n+1   2nt = 1− h[U, X], [U, X]i. n+1 2nt Since [U, X] 6= 0, we get 0 ≥ 1 − n+1 , which is equivalent to t ≥ (n + 1)/2n. The inequality t ≤ 1 follows from the previous assertion, since every SU(n + 1)-generalized normal homogeneous Riemannian manifold is also U(n + 1)-generalized normal homogeneous. t u

Remark 6.11.4. It follows from equality µ1,s = ξs and the second inclusion in (6.28) that Theorem 6.11.11 contains a stronger statement for odd n than the first assertion of Theorem 6.11.3. Remark 6.11.5. For n = 1 the second assertion of Theorem 6.11.3 also follows from the results of Subsection 6.13.3 (see also [78]), where we prove (in particular) that any U(2)-generalized normal homogeneous metric on S3 is either U(2)-normal homogeneous, or SU(2)-normal homogeneous. The case SU(2) = S3 is very special. There is a 6-dimensional space of leftinvariant Riemannian metrics on SU(2). But we have the following Proposition 6.11.6. If a left-invariant metric µ on S3 = SU(2) is SU(2)-generalized normal homogeneous, then it is a metric of constant sectional curvature. Proof. By Corollary 6.4.9 (or by Example 5.2.8), such a left-invariant metric µ should be bi-invariant. Therefore, (S3 , µ) has constant sectional curvature, since there are only metrics of constant curvature among the invariant Riemannian metrics on the homogeneous space SU(2) × SU(2)/ diag(SU(2)) = SO(4)/SO(3). t u We need the following general proposition.

6.11 Generalized Normal Homogeneous Riemannian Metrics on Spheres and . . .

375

Proposition 6.11.7. Let p : (M, µ) → (N, ν) be a Riemannian submersion which is a homogeneous fibration with respect to some isometry Lie group G of the space (M, µ), and suppose the space (M, µ) is G-generalized normal homogeneous. Then (N, ν) is G-generalized normal homogeneous too. Proof. Let ρM and ρN be the intrinsic metrics on M and N (induced by the metric tensors µ and ν). Consider any points x, y in N. In view of homogeneity, the space (M, ρM ) is finitely compact. Then there exist points x˜ ∈ p−1 (x), y˜ ∈ p−1 (y) such that ρM (x, ˜ y) ˜ = ρN (x, y). Since (M, ρM ) is G-generalized normal homogeneous, there is some δ (x)-translation ˜ g˜ ∈ G of the space (M, ρM ) such that g( ˜ x) ˜ = y. ˜ Since G preserves the fibers of the Riemannian submersion p, then g˜ induces some isometry g of the space (N, ρN ). Let z be an arbitrary point in M and z˜ be any point in the fiber p−1 (z). Then g(x) = y and ρN (x, g(x)) = ρN (x, y) = ρM (x, ˜ y) ˜ = ρM (x, ˜ g( ˜ x)) ˜ ≥ ρM (˜z, g(˜ ˜ z)) ≥ ρN (p(˜z), p(g(˜ ˜ z))) = ρN (z, g(z)). Therefore, (N, ν) is G-generalized normal homogeneous.

t u

The following proposition is a partial case of Proposition 6.11.7. Proposition 6.11.8. Let (M = G/H, µ) and (M1 = G/H1 , ν) be homogeneous connected compact Riemannian manifolds, H ⊂ H1 , and the canonical projection p : (M, µ) → (M1 , ν), induced by the inclusion H ⊂ H1 , be a Riemannian submersion. Then the space (G/H1 , ν) is G-generalized normal homogeneous if the space (G/H, µ) is G-generalized normal homogeneous. Now we consider the metrics µt on S4n+3 = Sp(n + 1)/Sp(n). Such metrics are generated by the inner products (·, ·)t := h·, ·i |p1 + 2t h·, ·i |p2 ,

(6.38)

where we have used the Ad(Sp(n))-invariant h·, ·i-orthogonal decomposition sp(n + 1) = p ⊕ sp(n) = p1 ⊕ p2 ⊕ sp(n), and the modules p1 , p2 = sp(1) are defined by formulas (6.31), (6.32) (see (6.30)). Theorem 6.11.9. The Riemannian manifold (S4n+3 , µt ) is Sp(n + 1) × Sp(1)-generalized normal homogeneous if and only if t ∈ (0, 1]. Moreover, this manifold is Sp(n + 1)-generalized normal homogeneous if and only if t ∈ [1/2, 1]. Proof. The metric µt is defined by the Ad(Sp(n))-invariant inner product (6.38). By Theorem 7.11.2 from Chapter 7, the sphere (S4n+3 , µ1 ) is Sp(n + 1)-Clifford– Wolf homogeneous, hence, Sp(n + 1)-generalized normal homogeneous, and hence Sp(n + 1) × Sp(1)-generalized normal homogeneous. By Table 6.1, (S4n+3 , µt ) is Sp(n + 1) × Sp(1)-normal homogeneous for all t ∈ (0, 1/2) and Sp(n + 1)-normal homogeneous for t = 1/2. Hence, the metric µt is Sp(n + 1) × Sp(1)-generalized

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6 Generalized Normal Homogeneous Manifolds With Intrinsic Metrics

normal homogeneous for t ∈ (0, 1] and Sp(n + 1)-generalized normal homogeneous for all t ∈ [1/2, 1] by Theorem 6.4.8 and Corollary 6.10.5. Let us suppose that (S4n+3 , µt ) is Sp(n + 1)-generalized normal homogeneous. Then we have the canonical Riemannian submersion pr1 : (S4n+3 = Sp(n + 1)/Sp(n), µt ) → (CP2n+1 = Sp(n + 1)/U(1) · Sp(n), νt ), and by Proposition 6.11.8, (CP2n+1 , νt ) is also Sp(n + 1)-generalized normal homogeneous. Now, Proposition 6.9.1 and Table 6.1 imply t ∈ [1/2, 1]. Now, let (S4n+3 = Sp(n+1)/Sp(n), µt ) be Sp(n+1)×Sp(1)-generalized normal homogeneous. We have the following Ad(Sp(n))-invariant h·, ·i-orthogonal decomposition sp(n + 1) = p ⊕ sp(n) = p1 ⊕ p2 ⊕ sp(n); p2 = sp(1), (see (6.30)). Now we consider the Ad(Sp(n) × Sp(1))-invariant decomposition sp(n + 1) ⊕ sp(1) = p ⊕ sp(n) ⊕ diag(sp(1)),

(6.39)

where (we identify elements (X, 0) ∈ sp(n + 1) ⊕ sp(1) with X ∈ sp(n + 1)) sp(n) ⊕ diag(sp(1)) ⊂ (sp(1) ⊕ sp(n)) ⊕ sp(1) ⊂ sp(n + 1) ⊕ sp(1), p = p1 ⊕ p2 ,

p2 = {(X, −X) , X ∈ sp(1)} ⊂ p2 ⊕ sp(1).

It should be noted that any vector (X, −X) ∈ p2 is projected to the vector 2X ∈ p2 when we project sp(n + 1) ⊕ sp(1) to sp(n + 1) along diag(sp(1)).
We consider elements of the Lie algebra sp(n + 1) ⊕ sp(1) as (n + 2) × (n + 2) -matrices. Then the metric µt is generated by the inner product (·, ·) = h·, ·i|p1 + 4th·, ·i|p2

(6.40)

on p (compare with (6.38)). By Proposition 6.8.6 we know that for every X ∈ p1 , Y ∈ p2 the inequality x1 h[[Y, X], X]h , [[Y, X], X]h i ≥ (x2 − x1 )h[[Y, X], X]p2 , [[Y, X], X]p2 i holds, where x1 = 1, x2 = 4t and h = sp(n) ⊕ diag(sp(1)). Now, if we take      0 1 X = diag , 0, . . . , 0 , diag(0, 0, . . . , 0) ∈ p1 −1 0 and Y = (diag(i, 0, . . . , 0), − diag(i, 0, . . . , 0)) ∈ p2 , then we get [[Y, X], X] = −U1 −U2 + 2U3 ,

where

6.11 Generalized Normal Homogeneous Riemannian Metrics on Spheres and . . .

377

U1 = (diag(i, 0, . . . , 0), − diag(i, 0, . . . , 0)) ∈ p2 , U2 = (diag(i, 0, . . . , 0), diag(i, 0, . . . , 0)) ∈ h, U3 = (diag(0, i, 0, . . . , 0), diag(0, 0, . . . , 0)) ∈ sp(n) ⊂ h. Since hU1 ,U1 i = hU2 ,U2 i = 1, hU3 ,U3 i = 1/2 and hU2 ,U3 i = 0 we get 1 · (1 + 4 · 1/2) ≥ (4t − 1) · 1, which proves the last assertion of the theorem. t u Corollary 6.11.10. The space (CP2n+1 = Sp(n + 1)/Sp(n) ·U(1), νt ) is Sp(n + 1)generalized normal homogeneous for t ∈ [1/2, 1]. Proof. As mentioned in Subsection 6.11.1, the Hopf fibration (6.27) (also for µt = µt,s=t ) is a Riemannian submersion. Now the statement follows from the sufficiency in the second assertion of Theorem 6.11.9 and Proposition 6.11.8. t u Theorem 6.11.11. The Riemannian manifold (S4n+3 , µt,s ) (in particular, µt,t = µt ) is Sp(n + 1) ×U(1)-generalized normal homogeneous if and only if t ∈ [1/2, 1] and s ∈ (0,t]. Proof. First, we prove the sufficiency. The metric µt,s is defined by the Ad(Sp(n))invariant inner product (·, ·)t,s := h·, ·i |p1 + 2t h·, ·i |p2,1 + 2s h·, ·i |p2,2

(6.41)

on p for an Ad(Sp(n))-invariant h·, ·i-orthogonal decomposition, which is defined by decomposition (6.30) and formulas (6.31), (6.32), with the additional subdivision p2 = p2,1 ⊕ p2,2 . Formula (6.32) shows that any element U ∈ p2 has the form of an ((n + 1) × (n + 1))-matrix with unique non-zero element U11 = u1 ∈ ℑ(H) if U 6= 0. Similarly, any element U in the subspace p2,1 ⊂ p2 (respectively, p2,2 ⊂ p2 ) is defined uniquely by U11 = u1 ∈ Rj ⊕ Rk (respectively, U11 = u1 ∈ Ri). By Theorem 7.11.2 from Chapter 7, the sphere (S4n+3 , µt,s ) for s = t = 1 is Sp(n + 1)-Clifford–Wolf homogeneous, hence, Sp(n + 1)-generalized normal homogeneous, and hence Sp(n + 1) × U(1)-generalized normal homogeneous. By Table 6.1, (S4n+3 , µt,s ) is Sp(n + 1) × U(1)-normal homogeneous if and only if 0 < s < t = 1/2. Hence, by Theorem 6.4.8, it is Sp(n + 1) ×U(1)-generalized normal homogeneous if t = 1/2 and 0 < s < t. So we can assume that 1/2 < t. If t = 1, the statement follows from Corollary 6.10.5. Now, consider a given pair (t, s), where 1/2 < t < 1 and 0 < s < t. It is clear that t = (1 − r) 21 + r · 1 for r = 2t − 1. Further, it follows from Remark 6.10.6 that
−1 t = (1−θ )( 12 )−1 +θ 1−1 for θ = (2t −1)/t. Now it is enough to prove that there
−1 −1 for θ = (2t − 1)/t, i.e. exists an s1 ∈ (0, 1/2) such that s = (1 − θ )s−1 1 +θ1  s=

(1 − t) 2t − 1 + ts1 t

−1 =

ts1 . (1 − t) + (2t − 1)s1

s(1−t) From this equality it is not difficult to show that s1 = t−(2t−1)s . It follows from the conditions for the pair (t, s) that both the numerator and the denominator of the

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6 Generalized Normal Homogeneous Manifolds With Intrinsic Metrics

above fraction are positive, so s1 > 0. Now, the inequality s1 < 1/2 is equivalent to the inequality 2(1 − t)s < t − (2t − 1)s or s < t, which is satisfied by conditions of the theorem. Therefore, (S4n+3 = Sp(n + 1)/Sp(n), µt,s ) is Sp(n + 1) ×U(1)-generalized normal homogeneous for all t ∈ [1/2, 1] and s ∈ (0,t). The homogeneous Riemannian space (S4n+3 = Sp(n+1)/Sp(n), µt = µt,s ) when s = t is even Sp(n+1)-generalized normal homogeneous for all t ∈ [1/2, 1] by Theorem 6.11.9 (the assertion for the case s = t could also be easily obtained by passing to the limit). Now, suppose that the Riemannian space (S4n+3 = Sp(n + 1)/Sp(n), µt,s ) is Sp(n + 1) ×U(1)-generalized normal homogeneous for some t > 0 and s > 0. Clearly, (6.27) is a homogeneous fibration with respect to Sp(n + 1) ×U(1), and also a Riemannian submersion, as was explained in Section 6.11.1. Moreover, the subgroup Id ×U(1) ⊂ Sp(n + 1) ×U(1) induces a trivial action on the base CP2n+1 of the fibration (6.27). Then (CP2n+1 , νt ) is Sp(n + 1)-generalized normal homogeneous by Proposition 6.11.7. Theorem 6.9.3 implies t ∈ [1/2, 1] (as in the proof of Theorem 6.11.9). Now, let us prove that s ≤ t. For this we consider S4n+3 = G/H, where G = Sp(n + 1) × U(1), H = Sp(n) × U(1), and the embeddings of Sp(1) and Sp(n) in G are defined by the symmetric pair (Sp(n + 1), Sp(1) × Sp(n)). The embedding of U(1) ⊂ H in G is diagonal: a 7→ (a, a) ⊂ Sp(1) ×U(1) ⊂ G. Recall the result of Proposition 5.8.12: For a homogeneous Riemannian manifold (M = G/H, µ), the orbit MT = C(T )(eH) of an arbitrary torus T ⊂ H is a totally geodesic submanifold of (M, µ). Let T be a maximal torus in H and consider its centralizer C(T ) in G. It is easy to see that U(1) × U(1) ⊂ C(T ) and Sp(1) ⊂ C(T ). Moreover, the orbit MT = C(T )(eH) with the induced Riemannian metrics is totally geodesic in (S4n+3 , µt,s ) by Proposition 5.8.12 and, therefore, is generalized normal homogeneous itself by Theorem 6.6.2. But it is easy to see that this orbit is isometric to the Riemannian space (U(2)/U(1), µ), where µ is generated by the inner product th·, ·i|p1 + 2sh·, ·i|p2 (we have used decomposition (6.30) and the Ad(U(2))invariant inner product (6.29) h·, ·i on the Lie algebra u(2)). Using a homothety and Theorem 6.11.3, we get s ≤ t. t u

6.11.3 Spin(9)-generalized Normal Homogeneous Metrics on the Sphere S15 In this section, we study the most difficult, but at the same time the most interesting case of transitive actions of a compact Lie groups on spheres, which involves essentially, besides other tools, the Clifford algebras Cl n and the Cayley algebra Ca of octonions. First we shall discuss briefly only notions and properties of a very general nature, which will be needed here. The exposition is based on the paper [83]. Let us recall some important definitions. Let (Rn , (·, ·)) be n-dimensional Euclidean space with the standard inner (scalar) product (·, ·) and orthonormal ba-

6.11 Generalized Normal Homogeneous Riemannian Metrics on Spheres and . . .

379

sis {e1 , . . . , e
n }. Then the Clifford algebra Cl n for (Rn , (·, ·)) more exactly, for (Rn , −(·, ·)) is an (unique) associative algebra over field R, containing the field R (as a subalgebra) with product operation · , the extension of a bilinear product · over Rn × Rn with relation x · y + y · x = −2(x, y)1,

1 ∈ R,

(6.42)

such that any possible relation in (Cl n , ·) is a corollary of the relation (6.42). Note that Cl 1 ∼ (6.43) = C, Cl 2 ∼ = H, Cl 8 ∼ = R(16), where R(16) is the algebra of real (16 × 16)-matrices [268]. It is clear from the definition that for any m, where 1 ≤ m ≤ n, Cl m is a subalgebra of Cl n . The algebra Cl n admits a Z2 -grading Cl n = Cl0n ⊕ Cl1n , where Cl0n is its subalgebra, generated by elements x · y, where x, y ∈ Rn . For n > 1 there exists a unique isomorphism In : Cl n−1 ∼ = Cl0n such that In (x) = x · en if x ∈ Rn−1 and In coincides on Cl0n−1 with composition of natural inclusions Cl0n−1 ⊂ Cl n−1 and Cl n−1 ⊂ Cl n . Proposition 6.11.12. Let v, w and { f1 , . . . , fn } be respectively two orthogonal vectors and an orthonormal basis in (Rn , (·, ·)). Then v · w = −w · v,

(6.44)

the product v · w is uniquely represented in the form v·w =

∑

γi j ( fi · f j ),

αi j ∈ R,

(6.45)

1≤i< j≤n

and the components γi j are calculated by the same rule as the components of the bivector v ∧ w. Proof. Formula (6.44) follows from the relation (6.42). Let us suppose that v = ∑ni=1 αi fi and w = ∑nj=1 β j f j . Then, using the rules (6.44) and (6.42), we get n

v·w =

∑ αi fi

!

!

n

·

i=1

∑ βj fj

=

j=1

n

∑

1≤i< j≤n

(αi β j ( fi · f j ) + βi α j ( f j · fi )) + ∑ αi βi ( fi · fi ) = i=1

n

∑

(αi β j − βi α j )( fi · f j ) − ∑ αi βi =

1≤i< j≤n

i=1

∑

(γi j := αi β j − βi α j )( fi · f j ).

1≤i< j≤n

t u The following well-known proposition easily follows from Proposition 6.11.16.

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6 Generalized Normal Homogeneous Manifolds With Intrinsic Metrics

Proposition 6.11.13. The linear span of elements x · y in Cl n , where x, y are orthogonal elements in (Rn , (·, ·)), is a Lie algebra (with the Lie bracket [W,V ] = W ·V −V ·W ), isomorphic to the Lie algebra spin(n) ∼ = so(n). In force of Propositions 6.11.13 and 6.11.12, we shall denote the linear span, mentioned in Proposition 6.11.13, as spin(n) and sometimes call its elements bivectors. Definition 6.11.14. An element W ∈ spin(n) is called simple if it can be represented in the form W = v · w, where v, w are orthogonal vectors in (Rn , (·, ·)). Lemma 6.11.15. An element W ∈ spin(n) is simple if and only if W 2 = −C2 · 1 for some real number C ≥ 0. In addition, if W = v · w, then C is equal to the area of the rectangle constructed on the vectors v, w. Proof. Let us first suppose that W = v · w for orthogonal vectors v, w ∈ (Rn , (·, ·)). Then, using the relation (6.42), we get W 2 = (v · w) · (v · w) = −(v · v) · (w · w) = −(−(v, v))(−(w, w)) · 1 = −(v, v)(w, w)1. This proves the necessity in the proposition and its last statement. Let W ∈ spin(n) be not simple. Then by a statement from the paper [307], formulated there for bivectors, there exists an orthonormal basis f1 , . . . , fn in (Rn , (·, ·)) such that m

W=

∑ al ( f2l−1 · f2l ),

where

al 6= 0

and

m > 1.

l=1

Then one can easily check that W 2 6= −C2 · 1 for any real C, since W 2 will contain a nonzero “four-vector” besides a real number. For example, if m = 2 then W 2 = −(a21 + a22 ) + 2a1 a2 ( f1 · f2 · f3 · f4 ). t u The algebra Cl n contains in itself not only the Lie algebra spin(n) ⊂ Cl0n , but also the spinor group (Spin(n), ·) as a Lie subgroup in the group of all invertible elements ((Cl n )× , ·), for details see, for example, Onishchik’s book [377]. Using the special action of the Caley algebra Ca = (R8 , (·, ·)) on R16 = Ca ⊕ Ca, one gets (see, for example, [377]) the special isomorphism φ : Cl 8 ∼ = R(16). The composition of the isomorphisms (I9 )−1 : Cl09 ∼ = Cl 8 and φ naturally induces an exact representation θ : spin(9) → gl(16) of the Lie algebra spin(9). It is very important that 1) θ (spin(9)) ⊂ so(16); 2) θ is a spinor representation, i.e. θ is induced by a (unique) exact representation Θ : Spin(9) → SO(16); 3) Spin(9) := Θ (Spin(9)) acts transitively on S15 ;

6.11 Generalized Normal Homogeneous Riemannian Metrics on Spheres and . . .

381

4) the isotropy subgroup H of Spin(9) at the point x0 = (1, 0, . . . , 0)T ∈ S15 is isomorphic to the Lie group Spin(7) [377]; 5) the Lie algebra h := spin(7) of the Lie subgroup H is not the standard inclusion of so(7) into spin(9) = so(9), but its image τ(so(7)) under an outer automorphism τ of the Lie algebra so(8), the so-called triality automorphism of order 3, where so(7) ⊂ so(8) ⊂ so(9) = spin(9) are the standard inclusions; 6) τ is induced by a rotation symmetry σ ∈ S3 of the Dynkin diagram D4 (which is a tripod) of the Lie algebra so(8); 7) τ can be defined, using Cartan’s triality principle (see e.g. a very clear presentation in the paper [225] by H. Gluck and W. Ziller), which is also based on Ca. Thus one gets the homogeneous space S15 = Spin(9)/Spin(7). We consider the following inner product hU,V i =

1 trace(UV T ), 2

U,V ∈ so(16),

(6.46)

where V T is the transposed matrix V . Since we are working with the representation of spin(9) in so(16), we can consider the (Ad(Spin(9))-invariant) restriction to spin(9) of the inner product (6.46) h·, ·i on so(16). Then we have an Ad(Spin(7))invariant h·, ·i-orthogonal decomposition g = spin(9) = spin(8) ⊕ p1 = spin(7) ⊕ p2 ⊕ p1 = h ⊕ p,

(6.47)

where the modules pi are Ad(Spin(7))-irreducible, [p2 , p1 ] ⊂ p1 and [p2 , p2 ] ⊂ spin(7). A convenient h·, ·i-orthogonal basis in spin(9), compatible with this decomposition, which we shall discuss shortly and use later, can be found in [451]. Proposition 6.11.16. Let { f1 , . . . , fn } be any orthonormal basis in (Rn , (·, ·)). Then the unique linear map L : spin(n) → so(n) such that L( fi · f j ) = 2Fji := 2(E ji − Ei j ) (where E ji is the (n × n)-matrix with a 1 in the ( j, i)th position and zeros elsewhere) is an isomorphism of Lie algebras. Proof. It follows from Proposition 6.11.12 that fi · f j , where 1 ≤ i < j ≤ n, constitute a basis in spin(n); Fji for the same indices constitute a basis in so(n). Therefore, L is a linear isomorphism of vector spaces. Since fi · f j = − f j · fi for i 6= j, we don’t need to care further about the order of indices i, j. It is known that if for all indices i, j, k, l, where i 6= j and k 6= l, there are no equal indices or there are two pairs of equal indices, then [Fji , Flk ] = 0; one can check directly that in this case also [ fi · f j , fk · fl ] = 0. So it is sufficient to consider the case of indices i, j, i, k. Then [ fi · f j , fi · fk ] = fi f j fi fk − fi fk fi f j = f j fk − fk f j = 2 f j · fk , [2Fji , 2Fki ] = 4[(E ji − Ei j )(Eki − Eik ) − (Eki − Eik )(E ji − Ei j )] = 4(−E jk + Ek j ) = 4Fk j = L(2( f j · fk )). These calculations imply the following proposition.

t u

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6 Generalized Normal Homogeneous Manifolds With Intrinsic Metrics

Proposition 6.11.17. Let { f1 , . . . , f9 } be an orthonormal basis in (R9 , (·, ·)) and let U=

∑

γiUj ( fi · f j ),

V=

1≤i< j≤9

∑

γiVj ( fi · f j )

1≤i< j≤9

be two elements in spin(9). Then hU,V i = 8

∑

γiUj γiVj .

(6.48)

1≤i< j≤9

Proof. Since SO(9) is a simple Lie group, any two Ad(SO(9))-invariant inner products on so(9) are proportional. Then it follows from known facts that Fji and Flk are h·, ·i-orthogonal if and only if { j, i} = 6 {l, k}. This, together with Proposition 6.11.16, implies that fi · f j and fk · fl , where 1 ≤ i < j ≤ 9 and 1 ≤ k < l ≤ 9, are not orthogonal if and only if they coincide. Then we need to check (6.48) only for the cases when U = V = fi · f j . Let us identify fi · f j with θ ( fi · f j ). Then by Lemma 7.11.9 and Proposition 7.11.1 ( fi · f j )2 = − Id. Therefore by definition (6.46) of h·, ·i, we have 1 1 h fi · f j , fi · f j i = − trace(( fi · f j )2 ) = trace(Id) = 8. 2 2 Here we have only one nonzero component γiUj = γiVj = 1, so equality (6.48) and the proposition are proved. t u To prove Proposition 6.11.18 and Lemma 6.11.22 below, we need some special bases of the subspaces h and p2 . For this goal we use two very symmetric orthogonal bases, constructed by D. E. Volper in [451, pp. 226–227]. We need only add 1 to all lower indices for the bivectors there and interchange p1 and p2 . These bases have an interesting property: if one multiplies the vector X4 ∈ p2 by −1, then, after adding to any vector of the basis with number i, 1 ≤ i ≤ 7, for p2 all three vectors in the i-th line for the basis in h, one gets a simple bivector, i.e. an element in spin(8) ⊂ so(9), which gives a Killing vector field of constant length on S15 . Proposition 6.11.18. For any vector X in p2 , (X, X)t = 2t hX, Xi, where (·, ·)t is the inner product on p, corresponding to the Riemannian metric ψt on S15 . Proof. Since Ad(Spin(7)) acts irreducibly on p2 , it is sufficient to check the statement for any nonzero vector X ∈ p2 . Let us take the first vector X1 = e7 e8 − e1 e2 − e3 e4 − e5 e6 of the basis in p2 , constructed in [451], see also [217]. Using the equality (6.48), we get hX1 , X1 i = 32. Adding to X1 the sum Y := (e1 e2 + e7 e8 ) + (e3 e4 + e7 e8 ) + (e5 e6 + e7 e8 ) of three vectors from the basis in h = spin(7), we get the vector

6.11 Generalized Normal Homogeneous Riemannian Metrics on Spheres and . . .

383

V = X1 +Y = 4e7 e8 ∈ spin(8) ⊂ spin(9) of constant length C on S15 . By Lemma 7.11.9, C2 = 16. It is clear that X1 x0 = V x0 for the initial point x0 ∈ S15 . Taking into account that X1 x0 is tangent to the (7dimensional) fiber at x0 of the Hopf fibration pro : S15 → S8 , we get that (X1 , X1 )t = 16t. This proves the proposition. t u Corollary 6.11.19. The inner product (·, ·)t on p, corresponding to the Riemannian metric ψt on Spin(9)/Spin(7), is defined by the formula 1 t (·, ·)t = h·, ·i|p1 + h·, ·i|p2 . 8 2

(6.49)

Proof. In view of Proposition 6.11.18, we need to check equality (6.49) only for any nonzero vector W ∈ p1 . Let us take W = e8 e9 . Then by formula (6.48), hW,W i = 8. On the other hand, in view of Lemma 7.11.9, W defines a unit Killing vector field W on S15 , and W x0 is orthogonal to the fiber of the Hopf fibration pro at the point x0 . Therefore, (W,W )t = 1. This implies equality (6.49) for W . t u Remark 6.11.20. It follows from Propositions 6.11.16 and 6.11.17 that one needs to multiply the coefficients in (6.49) by 2 when using the inner product (6.29) for the Lie algebra spin(9) ∼ = so(9) itself. Any Spin(9)-invariant metric on S15 = Spin(9)/Spin(7) is generated (up to homothety) by an inner product on p of the form (6.49) for some t > 0. Note that for t = 1 we get a metric of constant curvature 1 on S15 and for t = 1/4 we get a Spin(9)-normal homogeneous metric on S15 = Spin(9)/Spin(7). Proposition 6.11.21. The homogeneous Riemannian space (S15 = Spin(9)/Spin(7), ψt ) is Spin(9)-generalized normal homogeneous for all t ∈ [1/4, 1]. Proof. By Theorem 7.11.10, the Euclidean sphere S15 is Clifford–Wolf homogeneous with respect to the Lie group Spin(9) ⊂ SO(16), hence, it is Spin(9)generalized normal homogeneous. Now, it suffices to apply Corollary 6.11.19, Theorems 6.4.8, and Corollary 6.10.5. t u Lemma 6.11.22. The metric ψt with t > 1 is not Spin(9)-generalized normal homogeneous. Proof. Suppose that the metric ψt is Spin(9)-generalized normal homogeneous on S15 = Spin(9)/Spin(7). By Proposition 6.8.7 we know that for every X ∈ p1 , Y ∈ p2 the inequality x1 h[[Y, X], X]h , [[Y, X], X]h i ≥ (x2 − x1 )h[[Y, X], X]p2 , [[Y, X], X]p2 i holds, where x1 =

1 8

and x2 =

t 2

by Corollary 6.11.19.

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Now we consider X = e2 · e9 ∈ p1 and Y = e1 · e2 + e3 · e4 + e5 · e6 − e7 · e8 ∈ p2 (this is the vector −X1 on p. 227 in [451], see also [217]). It is easy to check that [[Y, X], X]] = −4e1 · e2 = Z −Y , where the vector Z = −3e1 · e2 + e3 · e4 + e5 · e6 − e7 · e8 = −3(e1 · e2 + e7 · e8 ) + (e3 · e4 + e7 · e8 ) + (e5 · e6 + e7 · e8 ) ∈ spin(7) (all vectors appearing here in brackets are basis vectors for h in the first line of basis vectors for h in [451, p. 226], see also [217]). Since h−Y, −Y i = 32 and hZ, Zi = 96, we get 1 4t−1 t u 8 · 96 ≥ 8 · 32, i.e. t ≤ 1. Lemma 6.11.23. The metric ψt with t < 1/4 is not Spin(9)-generalized normal homogeneous. Proof. The metric ψt is generated by the inner product (6.49). Since the pair (spin(7) ⊕ p2 , spin(7)) is the symmetric pair (so(8), so(7)) (that corresponds to a two-point homogeneous space S7 = SO(8)/SO(7)), then for any non-trivial X ∈ p2 we see that Zh (X) (see Lemma 6.7.9) is isomorphic to so(6). By Corollary 6.7.10 we get that X is a δ -vector and (X, [U, [U, X]]p ) + ([U, X]p , [U, X]p ) ≤ 0 for all U ∈ spin(9). Take any U ∈ p1 such that [U, X] 6= 0, then [U, X] ∈ p1 and t 1 0 ≥ (X, [U, [U, X]]p ) + ([U, X]p , [U, X]p ) = hX, [U, [U, X]]i + h[U, X]p , [U, X]p i 2 8 t 1 1 − 4t = − h[U, X], [U, X]i + h[U, X], [U, X]i = h[U, X], [U, X]i. 2 8 8 Since [U, X] 6= 0, we get t ≥ 1/4.

t u

From Proposition 6.11.21, Lemma 6.11.22, and Lemma 6.11.23 we obviously get Theorem 6.11.24. The homogeneous Riemannian space (S15 = Spin(9)/Spin(7), ψt ) is Spin(9)-generalized normal homogeneous if and only if t ∈ [1/4, 1]. This completes the proof of Theorem 6.11.1, the main result of the section.

6.12 The Chebyshev Norm on the Lie Algebra of the Isometry Group of a Compact Homogeneous Finsler Manifold The main results of this section (based on the publication [78]) could be formulated as follows: a homogeneous compact manifold with an intrinsic metric (M, ρ) is G-δ homogeneous with respect to a connected compact subgroup G of its motions if and only if (M, ρ) is Finsler (with a G-invariant norm ν), G is a Lie group, and the set N(M, G) of all Ad(G)-invariant norms on the Lie algebra g of the group G such that for a bi-invariant intrinsic metric ρ1 on G, defined by them, the canonical projection p : (G, ρ1 ) → (M, ρ) is a submetry, is not empty. Moreover, the Chebyshev norm k·k on g for (M, ρ) (see the formula (6.50) below) is the minimal norm from N(M, G). This naturally forces the problem of investigating the set N(M, G) (including the Chebyshev norms) and invariant norms on compact Lie algebras. The case when G

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is the identity component of the full isometry group of the space (M, ρ) is especially important. Let (M, ν) be a smooth connected Finsler manifold with a continuous norm ν on T M and the corresponding intrinsic metric ρ. We know that the full isometry group Isom(M) of the space (M, ρ) is a Lie group; moreover, the stabilizer (isotropy) subgroup of any point in M is a compactp subgroup of Isom(M) [173]. In the case of a Riemannian manifold (M, µ), ν(v) = µ(v, v) for v ∈ T M. Let (M, ν) be a compact homogeneous Finsler manifold with a compact full connected isometry group G (in particular, G is a Lie subgroup in Isom(M)), and H its (compact) isotropy subgroup (stabilizer) of some point x ∈ M. Then M is naturally identified with the quotient space G/H. Consider the Lie algebras h and g, h ⊂ g, of the Lie groups H and G. We identify the Lie algebra g with the Lie algebra RG of right invariant vector fields on the Lie group G, which in turn are identified by means of the differential of the natural projection p : G → G/H = M with the Lie algebra of vector fields on M, generating 1-parameter isometry groups of the space (M, ν). The tangent space Mx to M at the point x is naturally identified with the vector quotient space g/h, while the differential T p e of the smooth map p at the point e ∈ G is identified with the canonical linear projection pr : g → g/h. Let us also consider an Ad(G)-invariant norm (which we will call the Chebyshev norm) k · k on g, defined by the formula kXk = max ν(X(x)). x∈M

(6.50)

Theorem 6.12.1. In the notation as above, a function d : G × G → R, defined by the equality d(g, h) = max ρ(g(x), h(x)), (6.51) x∈M

is a bi-invariant metric on G, compatible with the compact-open topology. Moreover, the metric d generates a bi-invariant intrinsic metric D on G, which coincides with the intrinsic Finsler metric on G, defined by the Ad(G)-invariant Chebyshev norm k · k on g. This theorem and its proof are simplified versions of Theorem 6.4.3 for Riemannian manifolds and its proof. Using the notation as above, it is not difficult to obtain a formula for convenient calculation of the Chebyshev norm. Proposition 6.12.2. The following formula for the Chebyshev norm of the element X ∈ g holds: kXk = max ν(x)(pr(Ad(a)(X))). (6.52) a∈G

In the case of a homogeneous Riemannian space (M, µ), let us consider some Ad(H)-invariant inner product (·, ·) and Ad(G)-invariant inner product h·, ·i on g, as well as the h·, ·i-orthogonal direct sum g = h ⊕ p, where the module p and µ(x) can be identified with g/h and the restriction of (·, ·) to p respectively. Denote by | · | a norm on p, generated by the inner product (·, ·). Then the restrictions of (·, ·)

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and h·, ·i to an arbitrary Ad(H)-invariant and Ad(H)-irreducible submodule q ⊂ p are proportional to each other (sometimes it is convenient to take as h·, ·i a nondegenerate Ad(G)-invariant quadratic form on g, which is not an inner product). Then the formula (6.52) takes the form kXk = max | (Ad(a)(X))p |, a∈G

(6.53)

where Yp means the p-component of the vector Y ∈ g (i.e. Yp is the h·, ·i-orthogonal projection of Y to p). Let G be any compact connected subgroup of the full isometry group Isom(M, ρ) of a compact metric space (M, ρ) and d be the bi-invariant metric (6.51) on G. One can easily prove Proposition 6.12.3. Let (M, ρ) be a homogeneous compact metric space, G be a transitive subgroup of the group Isom(M, ρ) with the corresponding bi-invariant metric d, and x ∈ M be any point. Then the canonical projection p : (G, d) → (M, ρ), defined by the formula p (g) = g(x), does not increase distances. Moreover, (M, ρ) is G-δ -homogeneous if and only if p : (G, d) → (M, ρ) is a submetry. Theorem 6.12.4. Let (M, ρ) be a compact G-δ -homogeneous space with intrinsic metric, where G is some closed subgroup of the group (Isom(M, ρ), d). Then there exists some transitive on M, arcwise connected normal subgroup G1 in G such that the restriction of the metric d to G1 × G1 generates a (finite) bi-invariant intrinsic metric D = din on G1 , (G1 , D) being metrically complete and locally compact, while p : (G1 , D) → (M, ρ) is a submetry. Proof. Let us define the set G1 = {g ∈ G : din (e, g) < ∞}. It is known that (G1 , din ) is an arcwise connected space with intrinsic metric (Definition 1.5.4). The function din is bi-invariant on G × G in view of the bi-invariance of the metric d and satisfies the triangle inequality. This immediately implies that (G1 , D = din ) is a normal subgroup of the group G with the bi-invariant intrinsic metric. To prove that the space (G1 , D) is locally compact and metrically complete, it suffices to show that any closed ball B(g, r), r > 0, in (G1 , D) is compact. Let gn be any sequence of elements in B(g, r). Then there is a path cn (t), 0 ≤ t ≤ 1, in (G1 , D) with parameter t, proportional to arc length, of length < r + n1 , joining the points g and gn . Moreover, the lengths of any part of this path, calculated in (G1 , D) and (G, d), coincide. The group (Isom(M), d) is compact in view of the compactness of the space (Isom(M), d); the group (G, d) is compact as a closed subgroup of the compact group (Isom(M), d). In view of the compactness of (G, d) and Theorems 4, 5 in [12, p. 62], there is a subsequence n(k) such that the paths cn(k) uniformly converge in (G, d), when k → +∞, to some path c(t), 0 ≤ t ≤ 1; moreover, the length of the path c does not exceed the lower limit of the lengths of the paths cn(k) (and, consequently, r). It is clear then that the image of the path c lies in B(g, r) ⊂ (G1 , D) and gn(k) → c(1), which is what was required. It remains to prove the last assertion, because it would imply that the group G1 is transitive on M. Let g ∈ G1 , p (g) = x, r > 0. Obviously, p (BD (g, r)) ⊂ B(x, r)

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in view of Proposition 6.12.3 and the inequality d ≤ din . We will prove that for any point y ∈ M, where ρ(x, y) = s > 0, there exists an element h ∈ G such that p (h) = y and a path in (G, d) of length s, joining the elements g and h. It is clear then that din (g, h) = d(g, h) = s = ρ(p (g) = x, p (h) = y), and the converse inclusion B(x, r) ⊂ p (BD (g, r)) would be proved. Since (M, ρ) is a compact space with intrinsic metric, there is (the shortest) path x = x(t), 0 ≤ t ≤ s, in (M, ρ) with the ends x and y of length s, parameterized by arc length. Put n = 2k , where k is any natural number. According to Proposition 6.12.3, one can find consequently elements gn,i ∈ G, i = 0, 1, . . . , n, such that   i 1 gn,0 = g, p(gn,i ) = x s , and d(gn,i+1 , gn,i ) = s, i = 0, . . . , n − 1. n n Then, since the mapping p : (G, d) → (M, ρ) does not increase distances, taking into account the triangle inequality for d,      i j |i − j| d(gn,i , gn, j ) = ρ x s ,x s = s n n n for all i, j ∈ {0, 1, . . . , n}. Using the compactness of the group (G, d) and the diagonal Cantor process, one can find a subsequence n(l) such that every sequence of i(l) the form gn(l),i(l) , where n(l) is a fixed dyadic number r ∈ [0, 1] for sufficiently large l, converges to a unique point g(rs). Then d(g(rs), g(r0 s)) = |r − r0 |s for all dyadic numbers r, r0 ∈ [0, 1]. Using the compactness of the group (G, d), one can uniquely extend by continuity the value g(t) to all real numbers t ∈ [0, s]. Set h = g(s). It is clear that g(0) = g, p(g(t)) = x(t) for all t ∈ [0, s], in particular, p(h) = y; d(g(t), g(t 0 )) = |t − t 0 | for all t,t 0 ∈ [0, s]. Hence, g(t), t ∈ [0, s], is the required path. t u Question 6.12.5. Is (G1 , D) a compact group? Proposition 6.12.6. Any continuous bijective homomorphism of a locally compact topological group satisfying the second countability axiom onto a group with leftinvariant complete metric is a homeomorphism, i.e. an isomorphism of topological groups. Proof. Let q : G → (H, dist) be the mentioned homomorphism and U an open neighborhood of the unit in G with compact closure U. It is stated that q(U) contains some nonempty open subset V in (H, dist). Otherwise the set q(U) is closed (as the image in a Hausdorff space of a compact set under a continuous mapping) and nowhere dense in (H, dist). In view of the Lindel¨of theorem, the T 2 -space G can be covered by an at most countable family of sets gnU, and so of sets gnU. Then the complete metric space (H, dist) is covered by no more than a countable family of nowhere dense subsets q(gnU), i.e. it is of the first category. This contradicts the Baire–Hausdorff theorem, which states that a nonempty complete metric space is of the second category. By Exercise 18 in [120, P. 44], q is an isomorphism of topological groups. t u

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Theorem 6.12.7. If, under the conditions of Theorem 6.12.4, (M, ρ) is a manifold, then (G1 , D) is a Lie group with a bi-invariant (intrinsic) Finslerian metric and (M, ρ) is a manifold with an (intrinsic) G1 -invariant Finslerian metric. Proof. Under the above conditions, the group (Isom(M), d) is a compact Lie group. By the Cartan theorem [6], its closed subgroup G with the topology induced from Isom(M) is also a compact Lie group. The identity map id : (G1 , D) → (G1 , d) ⊂ G does not increase distances and, hence, is continuous. Then the subgroup G1 ⊂ G, being a continuous image of the arcwise connected group (G1 , D), is also arcwise connected. Hence by the Kuranishi– Yamabe theorem 2.1.33, (G1 , τ) is a Lie subgroup of the Lie group G, where a base of the topology τ consists of arcwise connected components of open sets of the topology in G1 induced from G. It is clear that the Lie group (G1 , τ) is arcwise connected and the map Id : (G1 , D) → (G1 , τ) is also a continuous homomorphism. The Lie group (G1 , τ) admits a left-invariant Riemannian intrinsic metric dist, which is compatible with τ. In view of the local compactness of (G1 , τ), the metric dist is complete. It was established in the proof of Theorem 6.12.4 that every closed ball in (G1 , D) is compact. Thus, (G1 , D) is a T 2 -space. So, the homomorphism id : (G1 , D) → (G1 , dist) satisfies all conditions of Proposition 6.12.6; then it is an isomorphism of topological groups, and (G1 , D) is a Lie group with bi-invariant intrinsic metric. By [58], any bi-invariant intrinsic metric on a Lie group, in particular D on G1 , is Finslerian. This means that there is a norm F on T G1 , which is invariant with respect to differentials of all left and right translations on the Lie group G1 , such that D(g, h), where g, h ∈ G1 , is equal to the infimum of lengths of piecewise continuously differentiable paths in (G1 , F), joining the points g and h. The proved statements imply that M and the submetry p : (G1 , D) → (M, ρ) (see Theorem 6.12.4) can be naturally identified with the homogeneous space G1 /H, where H ⊂ G1 is the (compact) stabilizer of the point x ∈ M, and the canonical smooth projection p : (G1 , τ) → G1 /H respectively. Let g1 = (G1 )e , h = He be the Lie algebras of the Lie group (G1 , τ) and its compact Lie subgroup H. Apply the previous notation, replacing G and g with G1 and g1 respectively. Since p is a submetry for the intrinsic metrics D and ρ, and, moreover, the Finslerian metric D is defined by the norm F on T G1 , it is not difficult to show that ρ is also a Finslerian metric, defined by a unique G1 -invariant norm ν on T (G1 /H) with the condition that pr : (g1 , F(e)) → (g1 /h, ν(x)) is a linear submetry of finite-dimensional normed vector spaces. t u A direct corollary of Theorems 6.12.4, 6.12.7, and 6.12.1 is the following Theorem 6.12.8. Under the conditions and notations of Theorem 6.12.7 and its proof, the group G1 is the identity component of the compact Lie group G, and so it is a compact Lie group itself. Moreover, F(e) = k · k. An example of a δ -homogeneous manifold is any Lie group G with a bi-invariant intrinsic (necessarily Finslerian) metric, because left and right translations of G displace all elements of the group by one and the same distance. Let us now give several characterizations of compact δ -homogeneous manifolds with an intrinsic metric.

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Theorem 6.12.9. A compact homogeneous manifold (G/H = M, ρ) with G-invariant intrinsic metric, connected transitive Lie group G, and stabilizer H is G-δ homogeneous if and only if one of the following two equivalent conditions is satisfied: 1) (M, ρ) is G-normal in the generalized sense; 2) (M, ρ) is isometric to a homogeneous Finslerian manifold (M, ν), and for every tangent vector v ∈ Mx at any point x in M, there is a vector field X on M, generating a 1-parameter subgroup in G, such that X(x) = v, ν(X(x)) = maxy∈M ν(X(y)). Further, (M, ρ) is a (normal) homogeneous Riemannian manifold if it is G-normal in the usual sense. Proof. The necessity of conditions 1) and 2) follows from previous theorems in Section 6.12. The sufficiency of condition 1) is proved in Corollary 6.4.2. It is not difficult to see that condition 2) is equivalent to the statement that the projection pr : (g, k · k) → (g/h, ν(x)) is a linear submetry of normed vector spaces. One can deduce from here, analogously to the proof of the last assertion in Theorem 6.12.4, condition 1). If the space is normal, then we get the linear submetry pr : (g, F(e)) → (g/h, ν(x)) with a Euclidean norm F(e). Then ν(x) is also a Euclidean norm, and (M, ν) is a (normal) homogeneous Riemannian manifold. t u Analogously to Theorem 6.12.9, one can establish the following theorem, which was proved earlier for the Riemannian case (see Theorem 6.4.8). Theorem 6.12.10. A compact manifold (G/H, ρ) with a G-invariant intrinsic metric is G-δ -homogeneous for a Lie group G if and only if ρ is induced by some G-invariant norm ν on G/H and there is an Ad(G)-invariant centrally symmetric (relative to zero) convex body B in g such that P(B) = {v ∈ p | ν(v) ≤ 1}, where P : g → p is the h·, ·i-orthogonal projection. As B one can take C = {w ∈ g | ||w|| ≤ 1}, where k·k is the Ad(G)-invariant (possibly non-Euclidean) Chebyshev norm (6.52) on g. Example 6.12.11. Suppose that (G/H, ρ) in the statement of Theorem 6.12.10 is a normal homogeneous Riemannian manifold with respect to a bi-invariant Riemannian metric on G, induced by an Ad(G)-invariant inner product h·, ·i on g. Then as p e B we can also consider Ce = {w ∈ g | hw, wi ≤ 1}. In this case, P(C) = P(C). It follows from results in this section that a compact G-δ -homogeneous manifold (M, ρ) with an intrinsic metric is G0 -δ -homogeneous for the connected component of the unit G0 in the group G, in particular I(M)0 -δ -homogeneous. Therefore, as G one can choose a connected transitive isometry Lie group of the manifold (M, ρ). As will be shown by further examples of the Chebyshev norm, the set N(M, G) in the beginning of this section can have more than one element. Therefore, the following problem is very interesting:

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Problem 6.12.12. Study the set N(M, G) and its properties (among them the properties of the Chebyshev norms on g) for G-(δ -)homogeneous Finsler (in particular, Riemannian) manifolds. Especially important is the case G = I(M)0 . On the grounds of the non-expandability of submetries and Theorem 6.12.10, we get the following result. Proposition 6.12.13. If | · | ∈ N(M, G), then k · k ≤ | · |. If | · |1 , | · |3 ∈ N(M, G), and | · |1 ≤ | · |2 ≤ | · |3 for some Ad(G)-invariant norm | · |2 on g, then | · |2 ∈ N(M, G). The convex linear combination of the unit balls of norms in N(M, G), as well as the convex hull of their union, give the unit ball of some norm in N(M, G). Many results on homogeneous Riemannian manifolds can be extended to homogeneous Busemann G-spaces (geodesic spaces), briefly, G-spaces [132], [53], [54], [55], [62], [69]. We need to omit here other sources. A G-space M is a locally compact, complete intrinsic metric space. Then by the Cohn-Vossen theorem 1.5.5 it is finitely compact, i.e. any closed ball in M is compact, and any two points in M can be joined by a shortest arc (path) with length equal to the distance between these points. The last property means that M is a geodesic space in the sense of M. Gromov. Other requirements on M model the properties of geodesics in “regular” Finsler spaces. These are local conditions of extension of shortest arcs and uniqueness of such extension [132]. These conditions imply that any shortest arc in M can be extended to a unique geodesic, i.e. locally isometric mapping of R to M. Notice that a “Finsler space” in the sense of Busemann is any C1 -smooth manifold M with a continuous norm on the tangent bundle T M, without any additional condition. It is unknown whether a general G-space has a finite topological dimension (any G-space satisfies the second countability axiom in which case all the usual definitions of dimension coincide). In [52] it is proved that any G-space which has a point x, such that all closed balls sufficiently close to x are convex, is finite-dimensional. But there are homogeneous two-dimensional G-spaces with no convex closed ball of positive radius [236]. Nowadays it is known only that in general, G-spaces of dimension n ≤ 4 are topological manifolds. This was proved by Busemann himself for n = 1, 2 in [132], by B. Krakus for n = 3 in 1968, and by P. Thurston for n = 4 in 1996. P. D. Andreev proved that any G-space, which locally has nonpositive curvature in Busemann’s sense, is a manifold [23]. The mentioned curvature condition means that in any “small” triangle, with three shortest arcs as its sides and three vertices as intersections of corresponding two sides, the distance between the mid-points of any two sides of the triangle is less than or equal to half of the length of the third side. Notice that any sufficiently small closed ball in such G-space is convex. J. Szenthe proved in 1963 that the isometry group of any G-space, satisfying such convexity condition, is a Lie group. The isometry group of any compact G-space is a Lie group [132].

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It is presently unknown whether the isometry group of a general noncompact Gspace is a Lie group. But if a G-space is homogeneous then this is true [53, 54]. Any homogeneous G-space of dimension ≤ 3 is isometric to a homogeneous Finslerian space supplied with its intrinsic metric. It must be true for any dimension but there is no proof of this statement. On the other hand, this is true for any symmetric G-space M, i.e. G-space such that for any point x ∈ M there exists an involutive isometry with a discrete set of fixed points including x. Then M is homogeneous; as a smooth homogeneous space of its Lie isometry group, it can be identified with a symmetric Riemannian manifold M1 ; Isom(M) ⊂ Isom(M1 ); and the metric on M coincides with an Isom(M)-invariant intrinsic Finslerian metric. There are also necessary and sufficient conditions for a Finslerian norm on T M which gives a symmetric G-space [55, 69]. Thus, the Cartan theory of Riemannian symmetric spaces extends in many respects to G-spaces. In particular, the universal covering space of any locally symmetric G-space is symmetric [55, 69], any symmetric G-space M is geodesic orbit and as sets, geodesics in M coincide with geodesics in M1 . Moreover, if a G-space is two-point homogeneous then it is isometric to a Riemannian symmetric space of rank one. If M is a noncompact G-space, then this statement is true even in the more general situation when the isometry group of M is transitive on the set of its geodesics [62]. In the compact case this statement has been proved by Busemann [132]. Notice in this connection that in the paper [428] Z. I. Szabo gave a short topological proof of the symmetry of two-point homogeneous Riemannian spaces. Before that, the proofs applied classifications of Lie groups acting transitively on spheres and of symmetric Riemannian spaces.

6.13 Calculations of the Chebyshev Norms for Some Manifolds

This section is motivated by previous results about generalized normal homogeneous Riemannian manifolds and based on the publication [78]. Since the explicit calculation of the Chebyshev norm for concrete homogeneous Riemannian manifolds is quite complicated, it is natural to solve this problem for some classes of “simple” manifolds. In Subsections 6.13.2, 6.13.3, and 6.13.4 the Chebyshev norms for the Euclidean spheres, Berger’s spheres, and invariant Riemannian metrics on the space SO(5)/U(2) are calculated. In the course of the presentation, the connections between invariant norms on the compact Lie algebras and the spectral properties of matrices from different representations of Lie algebras become clear (see [222, 323, 321, 322, 352] and the bibliography in the latter three papers for the spectral properties).

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6.13.1 On the L¨owner–John Ellipsoids To study the properties of Chebyshev norms, the notion of the L¨owner–John ellipsoid can be useful. Recall that the L¨owner–John ellipsoid (also called the L¨owner ellipsoid) for a given convex compact body U ⊂ Rn is an ellipsoid of maximal volume contained in U. It is well known that there exists a unique ellipsoid with this property [38, 274, 324]. Below we consider some well-known properties of the L¨owner–John ellipsoids for the unit balls in finite-dimensional normed vector spaces. Theorem 6.13.1 ([274, 324]). Let X = (Rn , | · |) be a normed vector space and (·, ·) be an inner product, defined by the L¨owner–John ellipsoid for the unit ball U in the space X. Then there are a natural number s, n ≤ s ≤ n(n + 1)/2, vectors xi ∈ X, and positive real numbers λi , 1 ≤ i ≤ s, such that |xi |2 = (xi , xi ) = 1 for all i = 1, . . . , s; s

x = ∑ λi (x, xi )xi for all x ∈ X. i=1

s

Moreover, ∑ λi = n and the following inequality holds: i=1

|x|2 ≤ (x, x) ≤ n|x|2 for all x ∈ X. As an addition to this theorem, we provide one more result from [274]. Theorem 6.13.2 ([274, 324]). Let us consider a norm | · |1 in Rn such that (in the notation of the previous theorem) for everyp1 ≤ i ≤ s, we have the equality |xi |1 = 1 and, for all x ∈ Rn , the inequality |x|1 ≤ (x, x). Then the L¨owner–John ellipsoid for p the unit ball of norm | · |1 coincides with the unit ball of the Euclidean norm (x, x) (and with the L¨owner–John ellipsoid for the unit ball of the norm | · |). The consideration of Example 6.12.11 leads to the following natural questions. Question 6.13.3. Let (G/H, µ) be a homogeneous Riemannian manifold which is normal with respect to a bi-invariant Riemannian metric on G defined by an Ad(G)invariant inner product h·, ·i on the Lie algebra g. Let t be a Cartan subalgebra in g, U be a closed unit ball of the Chebyshev norm k · k in t, and E be the L¨owner–John ellipsoid for U. Is it true that E = {x ∈ t | hx, xi ≤ 1}? Question 6.13.4. Suppose that a W -invariant inner product (·, ·) on t (W is the Weyl group for t) is defined by the L¨owner–John ellipsoid inscribed into the convex set {v ∈ t, kvk ≤ 1}. Is it true that for a (unique) extension of (·, ·) to an Ad(G)-invariant inner product on g and a bi-invariant Riemannian metric η on G corresponding to the inner product (·, ·), the canonical projection p : (G, η) → (G/H, µ) is a Riemannian submersion? Note that the answers to both posed questions are negative in the general case, as follows from examples to be considered in Subsection 6.13.3.

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6.13.2 The Chebyshev Norm for Euclidean Spheres Consider the sphere Sm−1 with the canonical metric µ of constant sectional curvature 1. The full connected isometry group for this sphere is the group G = SO(m). Consider an embedding H = SO(m−1) ⊂ SO(m) defined as follows: A 7→ diag(1, A) for A ∈ SO(m − 1). Then Sm−1 is naturally identified with SO(m)/SO(m − 1), and the metric µ under consideration is normal (up to homothety) with respect to a biinvariant metric on SO(m) defined by the inner product hX,Y i = −1/2 trace(XY ) for X,Y ∈ so(m). Put g = so(m), h = so(m − 1) (according to the embedding under consideration), and let p be the orthogonal complement to h in g with respect to h·, ·i. Recall that the algebra so(m) is naturally identified with the Lie algebra of Killing fields on Sm−1 . Moreover, the Ad(G)-invariant Chebyshev norm k · k on g = so(m) is defined by the Riemannian metric µ on M = Sm−1 as follows: p kXk = max µ(x)(X(x), X(x)) x∈M

for a Killing vector field X on (M, µ). We are going to describe this norm explicitly. Theorem 6.13.5. The Chebyshev norm on the Lie algebra so(m), m ≥ 2, corresponding to the Euclidean sphere Sm−1 = SO(m)/SO(m − 1), can be calculated by the formula kXk = max |λi |, 1≤i≤m

where λi , 1 ≤ i ≤ m, are the eigenvalues of the matrix X. Proof. Consider any X ∈ so(m), then according to (6.53), we get kXk2 = max hAd(A)(X)|p , Ad(A)(X)|p i. A∈SO(m)

Let the (i, j)-th entry of the matrices A and X be ai j and xi j respectively; then the (i, j)-th entry of the matrix A−1 is a ji . Consider Xe = Ad(A)(X) = AXA−1 . Direct calculations show that the entries of the latter matrix are xei j = ∑ aik a jl xkl . k,l

Now it is easy to see that e p , X| e pi hAd(A)(X)|p , Ad(A)(X)|p i = hX| = ∑(e x1 j )2 = j

∑

a1k a jl a1p a jq xkl x pq .

j,k,l,p,q

Since ∑ a jl a jq = δlq (the Kronecker symbol), j

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6 Generalized Normal Homogeneous Manifolds With Intrinsic Metrics

hAd(A)(X)| p , Ad(A)(X)| p i =

∑ a1k a1p xkl x pl = ∑ a1k a1p vkp , k,l,p

k,p

where vi j means the (i, j)-th entry of the matrix V = XX t = −X 2 . Let now m = 2n or m = 2n + 1, where n ≥ 1. Consider the vectors Fi = E(2i−1)(2i) − E(2i)(2i−1) for i = 1, ..., n. Here Ei j means an (m × m)-matrix with zero entries except for the (i, j)-th entry, which is 1. The linear span of the vectors Fi is a standard Cartan subalgebra t in so(m) [6]. Now consider a vector of the type X = u1 F1 + u2 F2 + · · · + un Fn and calculate its Chebyshev norm. Note that in this case the matrix V = −X 2 is diagonal: V = diag(u21 , u21 , u22 , u22 , . . . , u2n , u2n , (0)), therefore, hAd(A)(X)| p , Ad(A)(X)| p i = ∑ a1k a1p vkp = ∑ a21k vkk , k,p

k

where v(2i−1)(2i−1) = v(2i)(2i) = u2i for 1 ≤ i ≤ n. Since ∑ a21k = 1, and for every set k

of numbers a1k with such a property there is a matrix A ∈ SO(m) whose first row consists of these numbers, we get kXk2 = max hAd(A)(X)|p , Ad(A)(X)|p i A∈SO(m)

= max

max {u2i } ∑ a21k vkk = 1≤i≤n

A∈SO(m) k

for the vector X = u1 F1 + u2 F2 + · · · + un Fn . Consequently, kXk = max |ui |. 1≤i≤n

In force of the Ad(SO(m))-invariance of the Chebyshev norm, this formula is valid for all X ∈ so(m). t u Corollary 6.13.6. The unit ball of the Chebyshev norm on the standard Cartan subalgebra t of the Lie algebras so(2n) and so(2n + 1) is the cube K = {u1 F1 + u2 F2 + · · · + un Fn | max |ui | ≤ 1}, 1≤i≤n

and the unit sphere is the surface of this cube. According to obvious symmetries of the cube, its L¨owner–John ellipsoid is the set (the ball) n

B = {u1 F1 + u2 F2 + · · · + un Fn | ∑ u2i ≤ 1}. i=1

This set coincides with the set

6.13 Calculations of the Chebyshev Norms for Some Manifolds

395

{X ∈ t | hX, Xi ≤ 1}. Thus, an inner product corresponding to the L¨owner–John ellipsoid, on so(m), generates a bi-invariant metric on SO(m) such that the natural projection of SO(m) to Sm−1 = SO(m)/SO(m − 1) is a Riemannian submersion (see Questions 6.13.3 and 6.13.4). Note pthat for m = 2 and m = 3 the Chebyshev norm coincides with the Euclidean norm −1/2 trace(X 2 ). This is not the case for m ≥ 4. Lemma 6.13.7. Suppose that the absolute value of every eigenvalue of the matrix Q ∈ so(m) does not exceed 1. Then for every entry of this matrix the inequality |qi j | ≤ 1 holds. Proof. Clearly, the matrix −Q2 is symmetric, all its eigenvalues are real, and they do not exceed 1. Then the matrix I + Q2 is nonnegative in the operator sense. In particular, all of its diagonal entries are nonnegative. Therefore, for any i we get the inequality m

1≥

∑ q2i j ,

j=1

t u

which implies the assertion of the lemma.

Proposition 6.13.8. For m ≥ 4, the unit sphere of the Chebyshev norm on so(m) is not smooth. Proof. An arbitrary linear functional on so(m) has the form l(X) = trace(X · A) for some matrix A ∈ so(m). Now consider the vector Y = F1 + F2 (here we have used the condition m ≥ 4). Consider also the functionals l1 (X) = trace(X · F1 ) and l2 (X) = trace(X · F2 ). Denote by U and S respectively the unit ball and the unit sphere of the Chebyshev norm under consideration. It is clear that trace(X · F1 ) = −2x12 ,

trace(X · F2 ) = −2x34 .

By Lemma 6.13.7 we get the inequalities l1 (X) ≥ −2 and l2 (X) ≥ −2 for any X from the unit ball U. On the other hand, the equalities l1 (Y ) = l2 (Y ) = −2 are obvious. Since the functionals under consideration are linearly independent, we get two supporting hyperplanes to the ball U at the point Y ∈ S. Therefore, the sphere S is not smooth at the point Y . t u Remark 6.13.9. The above example shows that, in general, the unit sphere of the norm vector space (g, k · k) is not smooth. It is also clear that the closed unit ball in this example is not strictly convex.

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6 Generalized Normal Homogeneous Manifolds With Intrinsic Metrics

6.13.3 The Chebyshev Norm for the Berger Spheres Transitive and effective actions of (connected) Lie groups on spheres are given in Table 4.1. All generalized normal homogeneous (δ -homogeneous) metrics on spheres are classified in Section 6.11. Now we consider more closely a description of U(n)invariant metrics on S2n−1 , following [501]. The space of such metrics (up to homothety) is one-dimensional. All these metrics can be obtained by using the Hopf fibration S1 → S2n−1 → CPn−1 . Let us fix the canonical metric can of constant sectional curvature 1 on S2n−1 , and for every A > 0 consider the metric gA on S2n−1 which is obtained by multiplying by A the metric can along the fibers S1 of the above Hopf fibration. The family of metrics gA for A > 0 coincides (up to homothety) with the family of all U(n)invariant metrics on S2n−1 . The full connected isometry group of the metric gA for A 6= 1 is the group U(n). Note √ that the metric gA has positive sectional curvature if and only if A ∈ normal (naturally reductive) with respect to U(n) if (0, 2/ 3). The metric  q gA is  q q n n n and only if A ∈ 0, 2(n−1) (respectively, A 6= 2(n−1) ). For A = 2(n−1) the q n metric gA is SU(n)-normal. Therefore, if A > 2(n−1) and A 6= 1 then the metric gA is not normal with respect to any isometry group. √ Thus, for A ∈ (1, 2/ 3) the metric gA is naturally reductive (with respect to U(n)), weakly symmetric, and has positive sectional curvature. Let us consider the homogeneous space G/H = U(2)/S1 = (SU(2) × S1 )/S1 , where the isotropy group is embedded as follows: H = S1 = diag(S1 ) ⊂ S1 × S1 ⊂ SU(2) × S1 , where the embedding of the second multiple S1 in the product S1 × S1 is identical, and the embedding of the first multiple is defined by some fixed embedding i : S1 → SU(2). Since all circles (maximal tori) in the group SU(2) are pairwise conjugate via the action of inner automorphisms of the group SU(2), in fact we get a unique homogeneous space (up to homogeneous spaces isomorphism) diffeomorphic to the three-dimensional sphere S3 . In the Lie algebra g = su(2)⊕R, we choose vectors e1 , . . . , e4 such that ei ∈ su(2) for 1 ≤ i ≤ 3, e4 ∈ R, h = LH = Lin{e3 + e4 },

[e1 , e2 ] = e3 ,

[e2 , e3 ] = e1 ,

[e3 , e1 ] = e2 .

Let us fix on g an Ad(G)-invariant inner product h·, ·i, such that the vectors ei , i = 1, . . . , 4, form an orthonormal basis with respect to this inner product. For a real α 6= 0, consider on g a nondegenerate Ad(G)-invariant quadratic form (·, ·)α = h·, ·i|su(2) + αh·, ·i|R .

6.13 Calculations of the Chebyshev Norms for Some Manifolds

397

Now consider the orthogonal complement p to h in g with respect to (·, ·)α . For α ∈ (−∞, −1) ∪ (0, ∞), the restriction of (·, ·)α to p defines a G-naturally reductive metric ρα on the space G/H under consideration, and this metric is G-normal for positive α. Note that G is the full connected isometry group for the corresponding homogeneous Riemannian manifolds (see e.g. [501]). Remark 6.13.10. Invariant metrics generated by the (possibly, indefinite) inner products (·, ·)α for various (·, ·)α exhaust all U(2)-invariant Riemannian metrics on S3 (up to homothety), except for bi-invariant metrics (that have a larger isometry group). It is also known that the metric ρα is proportional to a U(2)-invariant metric α gA on S3 , where A2 = 1+α [501]. Recall that the metric gA has positive sectional √ curvature if and only if A ∈ (0, 2/ 3). The Riemannian manifolds (S3 , gA ) for such values of A are called Berger’s spheres. The values A > 1 correspond to negative α. Further we describe the Chebyshev norm k · kα generated by (G/H, ρα ) on g. According to (6.53), for an element V ∈ g we get kV kα = max | Ad(a)(V )p |α , a∈G

where | · |α is the Euclidean norm generated on p by the inner product (·, ·)α . Note that any orientation-preserving orthogonal (with respect to the inner product h·, ·i) transformation of the Lie algebra su(2) is its inner automorphism. According to Theorem 2.7.2, it suffices to get the values of the Chebyshev norm for vectors of some Cartan subalgebra of g. Let us fix k = Lin{e3 , e4 }, a Cartan subalgebra in g. Put V = xe3 + ye4 ∈ k. Theorem 6.13.11. For the Chebyshev norm of the vector V = xe3 + ye4 ∈ k we have the equality r α (|x| + |y|) kV kα = α +1 for α ∈ (−∞, −1) and the equality (q kV kα =

α (|x| + |y|) for |αy| ≥ |x|, p α+1 x2 + αy2 for |αy| ≤ |x|,

for α ∈ (0, ∞). In order to prove this theorem, we need the following auxiliary result. α Lemma 6.13.12. Consider f : R3 → R, f (x, y, z) = x2 + y2 + α+1 (s − z)2 for some real s and for α ∈ (−∞, −1)∪(0, ∞), let M(t, s) be the maximal value of this function under the constraint x2 + y2 + z2 = t 2 for a given real number t. Then

M(t, s) = for α ∈ (−∞, −1) and

α (|t| + |s|)2 α +1

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6 Generalized Normal Homogeneous Manifolds With Intrinsic Metrics

 M(t, s) =

α 2 α+1 (|t| + |s|) t 2 + αs2

for |αs| ≥ |t|, for |αs| ≤ |t|,

for α ∈ (0, ∞). Proof. It is clear that M(t, s) = max ϕ(z), |z|≤|t|

where ϕ(z) = t 2 − z2 +

α 1 2 2α α 2 (s − z)2 = − z − sz + t 2 + s . α +1 α +1 α +1 α +1

The function ϕ(z) has a unique critical point z0 = −αs. Note that ϕ(−|t|) =

α (|t| + s)2 , α +1

ϕ(|t|) =

α (|t| − s)2 , α +1

max{ϕ(−|t|), ϕ(|t|)} =

ϕ(z0 ) = t 2 + αs2 ,

α (|t| + |s|)2 . α +1

If α < −1, then obviously M(t, s) = max{ϕ(−|t|), ϕ(|t|)} =

α (|t| + |s|)2 , α +1

since the function ϕ(z) is convex in this case. If α > 0, then the function ϕ(z) is concave and attains its maximal value on R at the point z0 . Therefore, it attains a maximal value on the interval [−|t|, |t|] either at the critical point z0 (if |z0 | ≤ |t|) or at one of the ends of the interval (if |z0 | ≥ |t|). Considering these variants we get the assertion of the lemma. t u Proof (of Theorem 6.13.11). Consider the vector Ve := Ad(a)(V ) = l1 e1 + l2 e2 + l3 e3 + l4 e4 for some a ∈ U(2). Clearly, l4 = y and l12 + l22 + l32 = x2 . Further, since the vectors e1 ,

e2 ,

ee3 := − p

α 1 e3 + p e4 α(α + 1) α(α + 1)

form an orthonormal basis in p with respect to (·, ·)α , we have |Vep |2 = l12 + l22 +

α (q − l3 )2 . α +1

It is easy to see that for any triple of real numbers (l1 , l2 , l3 ) under the constraint l12 + l22 + l32 = x2 , it is possible to find a ∈ U(2) such that Ad(a)(V ) = ±l1 e1 + l2 e2 + l3 e3 + ye4 . Using the above calculations and Lemma 6.13.12, we get the theorem.

t u

6.13 Calculations of the Chebyshev Norms for Some Manifolds

399

Let Uα be the unit ball of the norm k · kα on the Cartan subalgebra t. For α > 0 consider also the ellipse Eα = {V = xe3 + ye4 ∈ t | x2 + αy2 ≤ 1}, i.e. the set of vectors in t with length ≤ 1 with respect to the Euclidean norm induced by the inner product (·, ·)α . Moreover, the sets r   α Kα = V = xe3 + ye4 ∈ t | (|x| + |y|) ≤ 1 α +1 and

  α +1 2 2 Bα = V = xe3 + ye4 ∈ t | x + y ≤ . 2α

will be useful for our considerations. Note that Uα ⊂ Kα (moreover, Uα = Kα for α < −1), and for α > 0 the inclusions Eα ⊂ Uα ,

Bα ⊂ Kα .

hold. Moreover, the ball Bα is the L¨owner–John ellipsoid for the square Kα . Proposition 6.13.13. If α > 1, then Bα is the L¨owner–John ellipsoid for Uα . Proof. It is easy to see that for α > 1 the inclusion Bα ⊂ Uα holds. Indeed, the coordinates of the intersection points ∂ (Eα ) ∩ ∂ (Bα ) satisfy the inequality |αy| ≥ |x|. Using the definition of Uα , it is easy to get the above inclusion. Further, Bα is the L¨owner–John ellipsoid for Uα , since Bα is the L¨owner–John ellipsoid for a larger set Kα . t u Remark 6.13.14. This proposition shows that in general the answer to Question 6.13.3 is negative. An arbitrary Ad(G)-invariant inner product on g has the form (·, ·)β ,γ = β h·, ·i|su(2) + γh·, ·i|R for some positive constants β and γ. We denote by ρ¯ β ,γ the corresponding biinvariant Riemannian metric on G. Let us fix some α > 0 and consider the natural projection πα,β ,γ : (G, ρ¯ β ,γ ) → (G/H, ρα ). The following theorem holds. Theorem 6.13.15. The projection πα,β ,γ is a submetry if and only if β = 1 and γ = α. Proof. The assertion that for β = 1 and γ = α the map πα,β ,γ is a submetry, is obvious. Suppose that πα,β ,γ is a submetry. Then the induced map of tangent spaces

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6 Generalized Normal Homogeneous Manifolds With Intrinsic Metrics

πeα,β ,γ : (g, (·, ·)β ,γ ) → (p = g/h, (·, ·)α ) is also a submetry. Note that under the projection π : g → p = g/h the q vector V = x1 e1 + x2 e2 + α x3 e3 + x4 e4 moves to the vector π(V ) = x1 e1 + x2 e2 + α+1 (x4 − x3 )e e3 ; moreover, as was mentioned above, the vectors e1 ,

e2 ,

ee3 = − p

α 1 e3 + p e4 α(α + 1) α(α + 1)

form an orthonormal basis with respect to (·, ·)α in p. The vector e1 is fixed under the projection π, but π −1 (e1 ) = e1 +h. Since πeα,β ,γ is a submetry, the equality β = 1 necessarily holds. Now we show that γ = α. There is a vector V ∈ g such that π(V ) = ee3 , and (V,V )1,γ ≤ 1. It is clear that V = te3 + se4 for some real t and s, and the conditions r α 2 2 t + γs ≤ 1, (s − t) = 1 α +1 q must hold. Therefore, t = s − α+1 α and r 2

(1 + γ)s − 2

α +1 1 s + ≤ 0. α α

The discriminant of the quadric trinomial in the last inequality must be nonnegative, which implies that α ≥ γ. Further, any vector V = te3 + se4 with the condition t 2 + γs2 = 1 must be projected to the vector π(V ) such that (π(V ), π(V ))α ≤ 1, i.e. the inequality α 2 √ γ and s = − √ 1 , then we get α+1 (s − t) ≤ 1 must hold. Consider t = γ(γ+1)

γ(γ+1)

α γ+1 α+1 γ

≤ 1, which is equivalent to the inequality α ≤ γ. Thus, if πα,β ,γ is a submetry, then β = 1 and γ = α. t u Let us turn back to the case α > 1. Consider an Ad(G)-invariant inner product on g such that Bα (the L¨owner–John ellipsoid for Uα ) is the unit ball for this inner product, q on k. It is easy to see that this inner product coincides with (·, ·)η,η for 2α η = α+1 , and ρ¯ η,η is a bi-invariant metric, generated by this inner product, on G. On the other hand, as was shown above, the natural projection

πα,η,η : (G, ρ¯ η,η ) → (G/H, ρα ) is not a submetry. Remark 6.13.16. This argument shows that in general the answer to Question 6.13.4 is negative. Finally, we shall prove the following assertion (see Remark 6.13.10).

6.13 Calculations of the Chebyshev Norms for Some Manifolds

401

Proposition 6.13.17. The homogeneous Riemannian manifold (G/H, ρα ) is not δ homogeneous for α < −1. Proof. As was shown above, for α < −1, the vector V = xe3 + ye4 ∈ k satisfies the equality r α kV kα = (|x| + |y|). α +1 4

Now, if W = ∑ xi ei then, as is easy to check, i=1

r kW kα =

α α +1

q

x12 + x22 + x32 + |x4 |

 .

If (G/H, ρα ) is δ -homogeneous, then the projection πeα : (g, k · kα ) → (p = g/h, | · |α ) must be a submetry. As was noted, under the natural projection π : g → p = g/h the vector q W = x1 e1 + x2 e2 + x3 e3 + x4 e4 moves to the vector π(W ) = x1 e1 + α x2 e2 + α+1 (x4 − x3 )e e3 and π −1 (e1 ) = e1 + h. Now it suffices to note that |e1 |α = p (e1 , e1 )α = 1. On the other hand, r α min ke1 + ukα = ke1 kα = >1 u∈h α +1

for α < −1; hence, πeα is not a submetry and (G/H, ρα ) is not δ -homogeneous for α < −1. t u

6.13.4 The Chebyshev Norms for Invariant Metrics on SO(5)/U(2) As was noted repeatedly, the SO(5)-invariant metric µx1 ,x2 on the homogeneous space M = SO(5)/U(2) is generalized normal homogeneous but is not normal homogeneous if and only if x1 < x2 < 2x1 . Below we shall compute the Chebyshev norms on the Lie algebra so(5), which correspond to the homogeneous Riemannian manifolds (M = SO(5)/U(2), µx1 ,x2 ) for 2x1 6= x2 (for 2x1 = x2 the full isometry group of (M = SO(5)/U(2), µx1 ,x2 ) is locally isomorphic to the group SO(6)). There is a description of (M = SO(5)/U(2), µx1 ,x2 ) in Example 4.12.14. In this case U(2) ⊂ SO(4) ⊂ SO(5), and the pairs (SO(5), SO(4)), (SO(4),U(2)) are irreducible symmetric. Consider the Ad(SO(5))-invariant inner product hA, Bi = √ −1/2 trace(A · B) for A, B ∈ so(5). The matrices A + −1B ∈ u(2), where     0c ad A= , B= −c 0 d b

402

6 Generalized Normal Homogeneous Manifolds With Intrinsic Metrics

are embedded in so(4) via √

A + −1B 7→



AB −B A



in order to get the symmetric pair (so(4), u(2)). We also use the standard embedding A 7→ diag(A, 0) of the Lie algebra so(4) in so(5). Let us consider the natural h·, ·iorthogonal decomposition: g = so(5) = so(4) ⊕ p1 = u(2) ⊕ p2 ⊕ p1 ,

p = p1 ⊕ p2 ,

where      



 0 c a d 0  −c 0 d b 0    −a −d 0 c 0  u(2) = Z =   ;     −d −b −c 0 0    0 0 0 0 0    0 0 0 0 k      0 0 0 0 l    0 0 0 0 m p1 = X =   ;    0 0 0 0 n    −k −l −m −n 0    0 e 0 f 0      −e 0 − f 0 0     0 f 0 −e 0  p2 = Y =   ;   −f 0 e 0 0    0 0 0 0 0

a, b, c, d ∈ R

k, l, m, n ∈ R

                

,

,

    

e, f ∈ R

     

,

    

the modules p1 , p2 are Ad(U(2))-invariant and Ad(U(2))-irreducible. It is clear that hX, Xi = k2 + l 2 + m2 + n2 for X ∈ p1 and hY,Y i = 2e2 + 2 f 2 for Y ∈ p2 . Now consider on SO(5)/U(2) the SO(5)-invariant Riemannian metric µ = µx1 ,x2 generated by the inner product (·, ·) = x1 h·, ·i|p1 + x2 h·, ·i|p2 on p for some positive numbers x1 and x2 . We shall deal with the case x2 6= 2x1 . First we prove two auxiliary results using a special notation. Let Ei, j be a (5 × 5)-matrix with zero entries except for the (i, j)-th entry, which is 1. For any 1 ≤ i < j ≤ 5 put Fi, j = Ei, j − E j,i . Consider the subspace q = R · F1,5 ⊕ R · (F1,4 − F2,3 ) ⊂ p = p1 ⊕ p2 . Proposition 6.13.18. For any vector V ∈ p, there exists an a ∈ H = U(2) such that Ad(a)(V ) ∈ q.

6.13 Calculations of the Chebyshev Norms for Some Manifolds

403

Proof. Let V = X + Y , where X ∈ p1 and Y ∈ p2 . The group Ad(U(2)) acts transitively on the unit sphere in p1 . Therefore we can suppose that X = bF1,5 for some b ∈ R. We have Y = c1 (F1,2 − F3,4 ) + c2 (F1,4 − F2,3 ) for some real numbers c1 and c2 . Note that [F2,4 , X] = 0. Thus, X is fixed under Ad(a), where a = exp(tF2,4 ). On the other hand, Ad(a)(Y ) = ce1 (F1,2 − F3,4 ) + ce2 (F1,4 − F2,3 ) ∈ p2 , where ce1 = c1 cos(t) + c2 sin(t), ce2 = c2 cos(t) − c1 sin(t). For an appropriate t ∈ R we get ce1 = 0. Hence, Ad(a)(V ) = bF1,5 + ce2 (F1,4 − F2,3 ) ∈ q. t u Proposition 6.13.19. A vector W = X +Y + Z, where X +Y ∈ q, X ∈ p1 , Y ∈ p2 , and Z ∈ h = u(2), is a nontrivial geodesic vector on (SO(5)/U(2), µ), x2 6= x1 , x2 6= 2x1 if and only if one of the following conditions holds: 1 1) W = bF1,5 + xx21 cF1,4 + x2 −2x x1 cF2,3 for some b 6= 0, c 6= 0; 2) W = d(F1,4 − F2,3 ) + a1 (F1,2 + F3,4 ) + a2 (F1,4 + F2,3 ) + a3 (F1,3 − F2,4 ) for some d 6= 0, a1 , a2 , a3 ∈ R; 3) W = eF1,5 + f F2,4 for some e 6= 0 and f ∈ R.

Proof. Let us consider W = X + Y + Z, where X = bF1,5 ∈ p1 , Y = c(F1,4 − F2,3 ), Z = b1 (F1,2 + F3,4 ) + b2 (F1,4 + F2,3 ) + b3 F1,3 + b4 F2,4 . By Proposition 6.8.1, the vector W is geodesic if and only if the next two equalities hold: [Z,Y ] = 0,

[X,Y ] = x1 /(x2 − x1 )[X, Z].

Direct computations show that [X,Y ] = bcF4,5 , [Z,Y ] = c(b3 + b4 )(F1,2 − F3,4 ), [X, Z] = b(b1 F2,5 + b3 F3,5 + b2 F4,5 ). If b 6= 0 and c 6= 0, then b1 = b3 = b4 = 0 1 and b2 = x2x−x c. If b = 0 and c 6= 0, then b4 = −b3 . If b 6= 0 and c = 0, then 1 b1 = b2 = b3 = 0. The proposition is proved. t u Denote by k · kx1 ,x2 the Chebyshev norm, corresponding to the homogeneous Riemannian manifold (M = SO(5)/U(2), µx1 ,x2 ), on the Lie algebra so(5). Fix any W ∈ so(5), then the matrix −W 2 is symmetric and has the eigenvalues 0, λ 2 , λ 2 , µ 2 , µ 2 , where µ ≥ λ ≥ 0 (it is clear that λ and µ are the absolute values of the corresponding eigenvalues of the matrix W ). Since the norm k · kx1 ,x2 is Ad(SO(5))-invariant, its value on the vector W depends only on λ and µ as is easy to see. Remark 6.13.20. We will repeatedly use, without additional comments, the values λ and µ, introduced above, in the rest of this subsection. Consider O(W ), an orbit of W in so(5) with respect to the action of the ade ∈ O(W ) such that joint group Ad(SO(5)). Since this orbit is compact, there is a W e e (W |p , W |p ) ≥ (V |p ,V |p ) for every V ∈ O(W ). According to the definition of the Chebyshev norm, we get that q e |p , W e |p ) = kW kx ,x . (W 1 2 e ∈ O(W ) with such a property are called δ -vectors. It is clear that the Vectors W set of δ -vectors is invariant with respect to the action of the subgroup Ad(U(2)) ⊂

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6 Generalized Normal Homogeneous Manifolds With Intrinsic Metrics

Ad(SO(5)). In order to find such vectors we need Lemma 6.13.21, which is an obvious corollary of Propositions 6.13.18 and 6.13.19. Lemma 6.13.21. Suppose that x2 6= 2x1 and x2 6= x1 . Then for any W ∈ so(5) there is a δ -vector V ∈ O(W ) such that its p-component Vp lies in the space q. If V is such a vector, then it may have only one of the following three forms: 1 1) V = bF1,5 + xx21 cF1,4 + x2 −2x x1 cF2,3 , where b 6= 0, c 6= 0; 2) V = d(F1,4 − F2,3 ) + a1 (F1,2 + F3,4 ) + a2 (F1,4 + F2,3 ) + a3 (F1,3 − F2,4 ), where d 6= 0, a1 , a2 , a3 ∈ R; 3) V = eF1,5 + f F2,4 , where e 6= 0 and f ∈ R.

Thus, we get an algorithm to calculate the Chebyshev norm of W ∈ so(5). Define a subset of matrices Mλ ,µ ⊂ so(5) as follows: Mλ ,µ = {V ∈ so(5) | −V 2 has the eigenvalues 0, λ 2 , λ 2 , µ 2 , µ 2 }. Then kW kx1 ,x2 = max{(V |p ,V |p ) | V ∈ Mλ ,µ }. Moreover, Lemma 6.13.21 permits us to deal only with vectors V of the form 1), 2), or 3) in its statement. The following result could be easily checked by direct calculations. Lemma 6.13.22. 2 1 1) Let V = bF1,5 + xx21 cF1,4 + x2 −2x x1 cF2,3 , where b, c ∈ R. Then (Vp ,Vp ) = x1 b + 2 2 2x2 c , and the matrix −V has the eigenvalues

0,

c2 (2x1 − x2 )2 , x12

c2 (2x1 − x2 )2 , x12

b2 x12 + c2 x22 , x12

b2 x12 + c2 x22 . x12

2) Let V = d(F1,4 − F2,3 ) + a1 (F1,2 + F3,4 ) + a2 (F1,4 + F2,3 ) + a3 (F1,3 − F2,4 ), where d 6= 0, a1 , a2 , a3 ∈ R. Then (Vp ,Vp ) = 2x2 d 2 , and the matrix −V 2 has the eigenvalues 0, (|d| − s)2 , (|d| − s)2 , (|d| + s)2 , (|d| + s)2 , q where s = a21 + a22 + a23 . 3) Let V = eF1,5 + f F2,4 , where e 6= 0 and f ∈ R. Then (Vp ,Vp ) = x1 e2 , and the matrix −V 2 has the eigenvalues 0,

e2 ,

e2 ,

f 2,

f 2.

Using this lemma, it is easy to get an explicit form of the Chebyshev norm. Let us first prove some auxiliary results.

6.13 Calculations of the Chebyshev Norms for Some Manifolds

405

Lemma 6.13.23. Let x2 6= 2x1 and x2 6= x1 . Then kW kx21 ,x2 = max{I1 , I2 , I3 }, where ( I1 =

x2 x1 µ 2 + 2xx11−x λ 2 for µ ≥ 2 0 for µ
x2√, then the inequality I2 ≥ (≤) I3 is equivalent to the inequality λ ≥ (≤) 2x1xx22 −x2 µ. Proof. It is clear that 2I2 − 2I3 = x2 λ 2 + 2x2 λ µ + (x2 − 2x1 )µ 2 . Taking into ac√ count that the polynomial x2t 2 + 2x2t + (x2 − 2x1 ) has the roots 2x1xx22 −x2 and √ − 2x1 x2 −x2 x2

t u

< 0, we get both assertions of the lemma.

Theorem 6.13.25. If 2x1 < x2 , then for the Chebyshev norm of the vector W the equality r x2 kW kx1 ,x2 = (λ + µ) 2 holds. If 2x1 > x2 > x1 , then for the Chebyshev norm of the vector W the equality q  x1 µ 2 + x1 x2 λ 2 for µ ≥ x2 λ , 2x1 −x2 2x1 −x2 q kW kx1 ,x2 = x2 2  for µ ≤ 2x1x−x λ 2 (λ + µ) 2 holds. If x1 ≥ x2 , then for the Chebyshev norm of the vector W we get the equality r x1 x2 kW kx1 ,x2 = x1 µ 2 + λ 2. 2x1 − x2 Proof. Note that x1 µ 2 +

x1 x2 x2 2x1 − x2 λ 2 − (λ + µ)2 = 2x1 − x2 2 2

 µ−

x2 λ 2x1 − x2

2 .

Therefore, for 2x1 < x2 the inequality I1 ≤ I2 holds. Moreover, in this case 2I2 = x2 (λ + µ)2 ≥ 2x1 µ 2 = I3 . Consequently, the first assertion of the theorem is proved. 2 Under the condition 2x1 > x2 > x1 the inequality 2x1x−x > 1 holds. It is clear that 2 x2 for µ ≥ 2x1 −x2 λ the inequalities 2x1 − x2 I1 − I2 = 2 hold. If µ ≤

x2 2x1 −x2 λ ,



x2 λ− µ 2x1 − x2

then, as it is easy to see,

2 ≥ 0 and I1 ≥ I3

6.13 Calculations of the Chebyshev Norms for Some Manifolds

λ≥

2x1 − x2 µ≥ x2

√

407

2x1 x2 − x2 µ. x2

According to Lemma 6.13.24, I2 ≥ I3 in this case, which proves the second assertion of the theorem. It suffices to prove the third assertion under the assumption x2 6= x1 , and to get the corresponding result for x2 = x1 by passing to the limit. Note that for x1 > x2 the 2 inequality 0 < 2x1x−x < 1 holds. Therefore, there is no number µ ≥ λ ≥ 0 under the 2 x2 condition µ < 2x1 −x2 λ . Further, as in the previous case, we get I1 ≥ I2 and I1 ≥ I3 for any µ ≥ λ ≥ 0, which completes the proof of the theorem. t u Note especially that only for x2 = x1 , the Chebyshev norm on so(5) is Euclidean (in this case it is generated by the Ad(SO(5))-invariant inner product x1 h·, ·i on so(5)). It is possible to consider the restriction of the Chebyshev norm k · kx1 ,x2 to a Cartan subalgebra t of the Lie algebra so(5). As such a subalgebra we consider t = {V = aF1,2 + bF3,4 | a, b ∈ R}. Taking into account that the eigenvalues of the matrix −V 2 = −(aF1,2 + bF3,4 )2 are 0, a2 , b2 (the last two eigenvalues have multiplicity 2), we immediately get from Theorem 6.13.25 the following result. Corollary 6.13.26. For 2x1 < x2 , the equality r
x2 kV kx1 ,x2 = |a| + |b| . 2 holds. For 2x1 > x2 > x1 , we have the equality q x2  x1 max{a2 , b2 } + 2xx11−x min{a2 , b2 }  2   x  2   for max{|a|, |b|} ≥ 2x1 −x2 min{|a|, |b|}, kV kx1 ,x2 = q   x2    2 (|a| + |b|)   2 for max{|a|, |b|} ≤ 2x1x−x min{|a|, |b|}. 2 For x1 ≥ x2 , the equality r kV kx1 ,x2 =

x1 max{a2 , b2 } +

x1 x2 min{a2 , b2 } 2x1 − x2

holds. We also note that Theorems 6.13.25 and 6.4.8 lead to a proof of the generalized normality of Riemannian manifolds (M = SO(5)/U(2), µx1 ,x2 ) for x1 < x2 < 2x1 , which differs from the original one in [73].

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6 Generalized Normal Homogeneous Manifolds With Intrinsic Metrics

6.14 Almost Normal Homogeneous Riemannian Manifolds In this section, we consider a class of compact homogeneous Riemannian manifolds (M = G/H, µ) almost normal with respect to a transitive isometry group G for which by definition there exists a G-left-invariant and an H-right-invariant inner product ν on G such that the canonical projection p : (G, ν) → (G/H, µ) is a Riemannian submersion and the norm | · | of the inner product ν is at least the biinvariant Chebyshev norm on G defined by the space (M, µ). Below we consider results that show relations of this class of Riemannian manifolds with some other wellknown classes. The presentation is based on the paper [65] by V. N. Berestovskii. All Riemannian manifolds mentioned in this section are assumed to be connected, compact, and homogeneous. In addition, we impose the minimal condition under which a natural generalization of Question 6.13.3 makes sense; namely, the bi-invariant Chebyshev norm on the transitive isometry group of the manifold under consideration does not exceed the norm on the group defined by some left-invariant inner product (Riemannian metric) for which the canonical projection onto the manifold is a Riemannian submersion. Such Riemannian manifolds are called almost normal homogeneous (see Definition 6.14.1 below) according to [65]. A natural generalization of the above-mentioned Question 6.13.3 is whether the unit ball of an invariant inner product on a transitive Lie group is a L¨owner–John ellipsoid (see Subsection 6.13.1 and [38, 274, 321]), i.e. an inscribed solid ellipsoid of maximal volume for the unit ball of the Chebyshev norm in the case of a homogeneous almost normal Riemannian manifold; the same question can be asked for the restrictions of the inner product and the Chebyshev norm to a Cartan subgroup (maximal torus) of the Lie group. Below we prove that the answers to these questions are affirmative in the case of normal homogeneous Riemannian manifolds with simple transitive Lie group of motions. The results of Section 6.12 imply that the answer to the version of the question for a Cartan subgroup is in general negative for normal homogeneous Riemannian manifolds with non-simple transitive isometry group; a counterexample is given by the Berger spheres with some normal Riemannian metrics. Definition 6.14.1. A Riemannian manifold (M, µ) is called almost G-normal homogeneous if G is a connected compact Lie subgroup in the full connected isometry group, transitive on M with the isotropy subgroup H at some point x0 ∈ M, and there exists a G-left-invariant and H-right-invariant Riemannian metric ν on G such that the canonical projection p : (G, ν) → (M = G/H, µ) is a Riemannian submersion and p | · | := ν(· , ·) ≥ k · k, (6.54) where k·k is the Chebyshev norm on the tangent bundle T G with respect to the Riemannian metric µ. A manifold (M, µ) is called almost normal homogeneous if it is almost G-normal homogeneous for some Lie group G. Remark 6.14.2. By the left G-invariance of ν and k · k, (6.54) is equivalent to the inequality | · |e ≥ k · ke , and the last inequality is equivalent to the inclusion

6.14 Almost Normal Homogeneous Riemannian Manifolds

409

C := B | · | (0, 1) ⊂ B := B k · k (0, 1) of the closed unit balls centered at the origin 0 ∈ g = Ge corresponding to these norms. Below we shall use the Ad(H)-invariant decomposition g = h ⊕ p into a direct sum orthogonal with respect to the inner product νe on g = Ge , where h = He , and the corresponding projection pr : g → p. The map p p : (G, ν) → (G/H, µ) is a Riemannianpsubmersion if and only if | pr(ξ )|e = µ(d pe (ξ ), d pe (ξ )) for any ξ ∈ g. Hence µ(d pe (ξ ), d pe (ξ )) = |ξ |e if ξ ∈ p. Then Proposition 6.12.2 implies that kξ k = max | pr(Ad(g)(ξ ))|e , ξ ∈ g = RG. (6.55) g∈G

Proposition 6.14.3. Suppose that G is a connected compact transitive Lie group of motions of a Riemannian manifold (M, µ) with stabilizer H at some point x0 ∈ M and ν is a bi-invariant Riemannian metric on G such that the canonical projection p : (G, ν) → (M = G/H, µ) is a Riemannian submersion. Then (6.54) holds. Proof. The proposition is a straightforward consequence of the bi-invariance of the norms | · |, k · k and (6.55). Corollary 6.14.4. Every (connected compact) G-normal homogeneous Riemannian manifold is almost G-normal homogeneous. t u Theorem 6.14.5 ([65]). An almost G-normal homogeneous Riemannian manifold (M, µ) is G-δ -homogeneous (or , equivalently, G-generalized normal homogeneous) and naturally reductive with respect to the Lie group G and the subspace p. Proof. By what has been said above, for proving the first assertion, it suffices to show that the linear projection

pr : g, k · k → p, | · |   maps the unit ball B := w ∈ g : kwk ≤ 1 onto the unit ball C1 := v ∈ p : |v| ≤ 1 . From (6.55) it follows that pr does not increase the norm. Therefore, pr(B) ⊂ C1 . On the other hand, Remark 6.14.2 and the fact that p : (G, ν) → (G/H, µ) is a Riemannian submersion imply that pr(B) ⊃ B ∩ p ⊃ B| · | (0, 1) ∩ p = C1 . Thus, pr(B) = C1 , as required. The proof of the first assertion implies that B1 ∩p = C1 , i.e. every vector ξ ∈ p is a δ -vector. This observation and (6.55) imply that, for a right-invariant vector field X with X(e) = ξ , the Killing vector field X on (M = G/H, µ) satisfying the equality X ◦ p = T p ◦ X attains its maximum length with respect to | · | at the point x0 = H. Consequently, the orbit of x0 under the one-parameter motion group in (M, µ) generated by X is a geodesic according to 3) in Theorem 3.1.7. In other words, every

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6 Generalized Normal Homogeneous Manifolds With Intrinsic Metrics

vector ξ ∈ p is also a geodesic vector (see Section 5.2). Consequently, a homogeneous Riemannian manifold (M, µ) is naturally reductive with respect to the isometry group G and the reductive subspace p (see Subsection 4.10, Theorem 5.2.6, and Remark 5.2.7). t u Question 6.14.6. Is every almost normal homogeneous Riemannian manifold normal homogenous? Theorem 6.14.7 ([65]). Suppose that G is a connected compact simple transitive isometry group of a Riemannian manifold (M, µ) with the isotropy subgroup H at some point x0 ∈ M, ν is a bi-invariant Riemannian metric on G such that the canonical projection p : (G, ν) → (M = G/H, µ) is a Riemannian submersion, and g = Ge . Then the unit ball C in the Euclidean space (g, νe ) is
the (solid) L¨owner–John ellipsoid of the unit ball B of the normed space g, k · ke , where k · k is the bi-invariant Chebyshev norm on the tangent bundle T G. Proof. Observe first that, by Proposition 6.14.3 and Remark 6.14.2, we have the inclusion C ⊂ B. Moreover, the proof of Theorem 6.14.5 implies that C ∩ p = B ∩ p.

(6.56)

Suppose that the theorem fails. Then the L¨owner–John ellipsoid C0 of B does not coincide with C. By the uniqueness of the L¨owner–John ellipsoid and the Ad(G)invariance and central symmetry of B with respect to zero, C0 is also Ad(G)-invariant and centrally symmetric with respect to zero. But by the simplicity of G, the Ad(G)invariant inner product on g is unique up to proportionality; hence, C0 = λC, where λ > 1. This contradicts (6.56) and the inclusion C0 ⊂ B. t u Question 6.14.8. Does Theorem 6.14.7 hold without the condition of simplicity on G? Theorem 6.14.9 ([65]). Under the conditions of Theorem 6.14.7, let t ⊂ g be a Cartan subalgebra. Then, in the notation of Theorem 6.14.7, C ∩t is the (solid) L¨owner– John ellipsoid of the set B ∩ t. Proof. By the Ad(G)invariance of B and C, the sets C ∩ t ⊂ B ∩ t are invariant under the Weyl group W = W (t) ⊂ Ad(G) of the subalgebra t (of the Lie algebra of some maximal torus in G). Let D be the L¨owner–John ellipsoid for B ∩ t. By the uniqueness of the L¨owner–John ellipsoid and the W -invariance and the central symmetry of B ∩ t with respect to zero, the ellipsoid D is also W -invariant and centrally symmetric with respect to zero. Then D is the unit ball with respect to a W -invariant inner product (· , ·) on t. By Vinberg’s Theorem 2.7.2, every W -invariant norm on t extends to a unique Ad(G)-invariant norm on g; the same assertion holds for an inner product by Theorem 2.7.5. In particular, the
inner product (· , ·) respectively, the restriction of the Chebyshev norm k · k to t extends to a unique Ad(G)-invariant
inner product (· , ·)0 on g respectively, a unique Ad(G)-invariant norm k · k0 on g . Clearly, k · k0 = k · k. Moreover, the orbit of every element v ∈ g with respect

6.14 Almost Normal Homogeneous Riemannian Manifolds

411

0 0 to Ad(G) intersects t (see Theorem 2.5.22).
Therefore, C ⊂ B, where C is the 0 unit ball of the Euclidean space g, (· , ·) . Since G is simple, an Ad(G)-invariant inner product on g is unique up to proportionality; hence, C0 = λC, where λ ≤ 1 by Theorem 6.14.7 and D = C0 ∩ t = λ (C ∩ t). Consequently, λ = 1 because D is the L¨owner–John ellipsoid of B ∩ t. The theorem is proved. t u

Remark 6.14.10. Theorem 6.14.9 in general fails for a nonsimple Lie group G, for example, for some G-normal Berger spheres S3 = U(2)/S1 with nonsemisimple Lie group
G = U(2) and the diagonal embedding of the subgroup S1 in S1 × S1 ⊂ U(2) , see Section 6.12. Remark 6.14.11. In Subsection 6.13.2, Theorem 6.14.9 was obtained for all Euclidean normal spheres Sn = SO(n + 1)/SO(n), n ≥ 1, after calculating the Chebyshev norms. For n 6= 1, 3, the Lie groups SO(n + 1) are simple, so that Theorem 6.14.9 is applicable. Recall also that every transitive isometry group of a connected compact homogeneous Riemannian manifold of positive Euler characteristic indecomposable into a direct metric product is simple (cf. Theorem 4.12.3).

Chapter 7

Clifford–Wolf Homogeneous Riemannian Manifolds

Abstract In this chapter, using connections between Clifford–Wolf isometries and Killing vector fields of constant length on a given Riemannian manifold, we classify simply connected Clifford–Wolf homogeneous Riemannian manifolds. We also get the classification of complete simply connected Riemannian manifolds with the Killing property defined and studied previously by J. E. D’Atri and H. K. Nickerson. The next goal of the chapter is to study properties of Clifford–Killing spaces, that is, real vector spaces of Killing vector fields of constant length, on odd-dimensional round spheres, and discuss numerous connections between these spaces and various classical objects. Finally, we consider some results related to restrictively Clifford– Wolf homogeneous Riemannian manifolds. One of the main goals of this chapter is to get the complete isometric classification of simply connected Clifford–Wolf homogeneous Riemannian manifolds (cf. Definitions 6.1.1 and 6.1.2) obtained in [76, 80]. It is not difficult to see that Euclidean spaces, odd-dimensional round spheres, and Lie groups with bi-invariant Riemannian metrics, as well as direct metric products of Clifford–Wolf homogeneous Riemannian spaces are also Clifford–Wolf homogeneous. The main result of this chapter states that the opposite statement is also true. More exactly, Theorem 7.0.1 (V. N. Berestovskii–Yu. G. Nikonorov, [80]). A simply connected Riemannian manifold is Clifford–Wolf homogeneous if and only if it is isometric to a direct metric product of a Euclidean space, odd-dimensional spheres of constant sectional curvature, and simply connected compact simple Lie groups with bi-invariant Riemannian metrics (some of these factors may be absent). The rest of this chapter is concerned with the investigation of different interconnections between the class of Clifford–Wolf homogeneous Riemannian manifolds and other naturally defined classes. In particular, based on [80] and [83], we study properties of Clifford–Killing spaces, that is, real vector spaces of Killing vector fields of constant length, on odd-dimensional round spheres, and discuss numerous connections between these spaces and various classical objects. The final section contains a classification of restrictively Clifford–Wolf homogeneous Riemannian manifolds. © Springer Nature Switzerland AG 2020 V. Berestovskii and Y. Nikonorov, Riemannian Manifolds and Homogeneous Geodesics, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-56658-6_7

413

414

7 Clifford–Wolf Homogeneous Riemannian Manifolds

7.1 Preliminary Results H. Freudenthal classified in [216] all individual Clifford–Wolf translations on symmetric spaces. Notice that several classical Riemannian manifolds possesses a 1parameter group of Clifford–Wolf translations. For instance, it is known that among irreducible compact simply connected symmetric spaces only odd-dimensional spheres, the spaces SU(2m)/Sp(m), m ≥ 2, and simple compact Lie groups, supplied with some bi-invariant Riemannian metric, admit 1-parameter groups of Clifford–Wolf translations [476]. Let us consider some examples. Obviously, every Euclidean space En is Clifford– Wolf homogeneous. Since En can be treated as a (commutative) additive vector group with a bi-invariant inner product, the following example can be considered as a generalization. Example 7.1.1. Let G be a Lie group supplied with a bi-invariant Riemannian metric ρ. In this case both the group of left translations L(G) and the group of right translations R(G) consist of Clifford–Wolf isometries of (G, ρ). Therefore, (G, ρ) is Clifford–Wolf homogeneous. Note also that in [54] the following result has been proved: A Riemannian manifold (M, g) admits a transitive group Γ of Clifford– Wolf translations if and only if it is isometric to some Lie group G supplied with a bi-invariant Riemannian metric. Example 7.1.2. Every odd-dimensional round sphere S2n−1 is Clifford–Wolf homogeneous. Indeed, n

S2n−1 = {ξ = (z1 , . . . , zn ) ∈ Cn :

∑ |zk |2 = 1}. k=1

Then the formula γ(t)(ξ ) = eit ξ defines a 1-parameter group of Clifford–Wolf translations on S2n−1 with all orbits as geodesic circles. Now, since S2n−1 is homogeneous and isotropic, any its geodesic circle is an orbit of a 1-parameter group of Clifford– Wolf translations, and so S2n−1 is Clifford–Wolf homogeneous. Note that S1 and S3 can be treated as the Lie groups SO(2) and SU(2) with bi-invariant Riemannian metrics. Note also that any direct metric product of Clifford–Wolf homogeneous Riemannian manifolds is Clifford–Wolf homogeneous itself. On the other hand, the condition for a Riemannian manifold to be Clifford–Wolf homogeneous is quite strong. Therefore, one should hope to get a complete classification of such manifolds. In what follows, we shall also need the following Definition 7.1.3. An intrinsic metric space (X, d) is called strongly Clifford–Wolf homogeneous if for every two points x, y ∈ X there is a 1-parameter group γ(t), t ∈ R, of Clifford–Wolf translations of the space (X, d) such that for sufficiently small |t|, γ(t) shifts all points of (X, d) by distance |t|, and γ(s)(x) = y, where d(x, y) = s.

7.2 Once More about Killing Vector Fields of Constant Length

415

It is clear that any strongly Clifford–Wolf homogeneous intrinsic metric space is Clifford–Wolf homogeneous and a Clifford–Wolf homogeneous intrinsic metric space is restrictively Clifford–Wolf homogeneous (Definition 6.2.1). The following question is natural: Question 7.1.4. Are the above three classes pairwise distinct (in particular, in the case of Riemannian manifolds)? Theorems 6.2.9 and 4.8.1 imply the following result. Theorem 7.1.5. If a connected restrictively Clifford–Wolf homogeneous Riemannian manifold is symmetric, then it is strongly Clifford–Wolf homogeneous.

7.2 Once More about Killing Vector Fields of Constant Length Now, we are going to prove the following theorem, which plays a key role in our study of Clifford–Wolf homogeneous Riemannian manifolds and their generalizations. Theorem 7.2.1. For any Killing vector field of constant length Z on a Riemannian manifold (M, g), we have (∇Z R) (·, Z) Z ≡ 0. Proof. It suffices to prove that g((∇Z R)(X, Z)Z,Y ) = 0 for all smooth vector fields X and Y on M. From Proposition 3.3.5 we know that g(∇X Z, ∇Y Z) = g(R(X, Z)Z,Y ). Therefore, Z · g(∇X Z, ∇Y Z) = Z · g(R(X, Z)Z,Y ). Further, by Proposition 3.3.5 we conclude that Z · g(∇X Z, ∇Y Z) = g(∇Z ∇X Z, ∇Y Z) + g(∇X Z, ∇Z ∇Y Z) = g(R(Y, Z)Z, ∇Z X) + g(R(X, Z)Z, ∇Z Y ). On the other hand, Z · g(R(X, Z)Z,Y ) = g((∇Z R)(X, Z)Z,Y ) + g(R(∇Z X, Z)Z,Y ) + g(R(X, Z)Z, ∇Z Y ) = g((∇Z R)(X, Z)Z,Y ) + g(R(Y, Z)Z, ∇Z X) + g(R(X, Z)Z, ∇Z Y ). Combining the equations above, we get g((∇Z R)(X, Z)Z,Y ) = 0, which proves the theorem. t u Note that the condition (∇Z R)(·, Z)Z = 0 means that for any geodesic γ, that is an integral curve of the field Z, a covariant derivative of any normal Jacobi field along γ is also a normal Jacobi field (see Section 2.33 in [105]).

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7 Clifford–Wolf Homogeneous Riemannian Manifolds

7.3 The Proof of the Main Result We will need the following useful proposition. Proposition 7.3.1 (Proposition 2.35 in [105]). If the Levi-Civita derivative of the curvature tensor R of a Riemannian manifold (M, g) satisfies the condition (∇X R)(·, X)X = 0 for all X ∈ T M, then (M, g) is locally symmetric. Proof. Let us prove that along any geodesic γ, parameterized by arc length, the symmetric linear map X 7→ R(X, γ 0 )γ 0 of Mγ has parallel eigenvectors and its eigenvalues are constant. Indeed, if E(s) is a parallel vector field along the geodesic then R(E(s), γ 0 (s))γ 0 (s) is clearly a parallel vector field due to (∇X R)(·, X)X = 0, X ∈ T M. Let Ei , i = 2, . . . , n := dim M, be (n−1) orthogonal eigenvectors relative to eigenvalues λi , i = 2, . . . , n, of R(E(s0 ), γ 0 (s0 ))γ 0 (s0 ) at a point γ(so ) of γ. If Ei (s), i = 2, . . . , n, are parallel vector fields along γ with initial conditions Ei (s0 ) = Ei , then λi Ei (s) and R(E(s), γ 0 (s))γ 0 (s) are parallel vector fields with the same initial conditions, so that they are equal along γ. Now, notice that if Y (s) = ∑ni=2 fi (s)Ei is a normal Jacobi field along the geodesic s 7→ γ(s) in (M, g), then the functions fi (s) satisfy the equations 00

f i (s) + λi fi (s) = 0,

i = 2, ..., n.

Fix a point γ(s0 ) on γ. The norm of the normal Jacobi field J(s) along γ such that J(s0 ) = 0 and J 0 (s0 ) = X0 , where X0 is any normal tangent vector in Mγ (s0 ) , depends only on the distance between γ(s0 ) and γ(s) (i. e., |s − s0 |). Let x 0 be a point in M, and let σ be the geodesic symmetry at x 0 defined in a normal neighborhood V of x 0 , where the exponential map Expx 0 is a diffeomorphism. For x ∈ V , let γ be the unique geodesic from x 0 to m with γ(0) = x 0 and γ(s) = x.
Then σ (x) = Expx 0 − Exp −1 (x) = γ(−s). x0 By Proposition 1.4.19, for every tangent vector X in Mx we have kTx σ (X)k = k − J(−s)k = kJ(s)k = kXk, where we realize X as the value at s of a normal Jacobi field along γ such that J(s0 ) = 0 and J 0 (s0 ) = X0 for some X0 in Mx 0 . The map σ is an isometry, thus (M, g) is locally symmetric. t u Now, using Theorem 7.2.1, we can prove the following theorem. Theorem 7.3.2. Every restrictively Clifford–Wolf homogeneous Riemannian manifold (M, g) is locally symmetric. Proof. According to Proposition 7.3.1, in order to prove that (M, g) is locally symmetric, it suffices to prove (∇X R)(·, X)X = 0 for any X ∈ Mx at a fixed point x ∈ M. By Theorem 6.2.9, there is a non-trivial Killing field Z of constant length on (M, g) such that Z(x) = X. By Theorem 7.2.1, we get (∇Z R)(·, Z)Z = 0 at every point of M. In particular,

7.3 The Proof of the Main Result

417

(∇X R)(·, X)X = (∇Z(x) R)(·, Z(x))Z(x) = 0. The theorem is proved.

t u

Proposition 7.3.3 (Theorem 5.5.1 in [476]). Let M be a simply connected compact irreducible symmetric space that is not isometric to a Lie group supplied with a bi-invariant Riemannian metric. If M admits a non-trivial Killing vector field X of constant length, then either M is an odd-dimensional sphere S2n−1 for n ≥ 3, or M = SU(2n)/Sp(n), n ≥ 3. Proof. The assertion of the proposition follows from Theorem 5.5.1 in [476], where it is proved that among the irreducible compact simply connected symmetric spaces only odd-dimensional spheres, spaces SU(2m)/Sp(m), m ≥ 2, and compact simple Lie groups with bi-invariant Riemannian metrics, admit 1-parameter groups of Clifford–Wolf translations. Now we give a somewhat different proof of Proposition 7.3.3. Let G be the identity component of the full isometry group of M. Consider a 1-parameter isometry group µ(t), t ∈ R, generated by X. This group consists of Clifford–Wolf translations of M according to Theorem 4.8.1. By Lemma 4.8.2, Zµ , the centralizer of the flow µ in G, acts transitively on M. It is clear that the identity component K = K(Zµ ) of Zµ is a Lie group, which acts transitively on M and has a non-discrete center (this center contains µ(t), t ∈ R). Note that M = G/H is a homogeneous space, where H is the isotropy subgroup at some point x ∈ M. Moreover, due to the assumption of the theorem and the exact homotopy sequence from the proof of Lemma 4.15.7, G and H are connected (recall that M is simply connected) and G is a simple compact Lie group. In [375, Theorem 4.1], A. L. Onishchik classified all connected proper subgroups K of the group G that act transitively on the homogeneous space G/H, where G is a simple compact connected Lie group and H is its connected closed subgroup. If in this situation K has a non-discrete center, then this theorem implies that either G/H is an odd-dimensional sphere, or G/H = SU(2n)/Sp(n), n ≥ 3. Since the center of K(Zµ ) is non-discrete, it proves the proposition. t u Proposition 7.3.4. Let M be a symmetric space SU(2n)/Sp(n), where n ≥ 3. Then every√Killing vector field of constant length on M has the form Ad(s)(tU), where U = −1 diag(1, 1, . . . , 1, −(2n − 1)) ∈ su(2n), s ∈ SU(2n), t ∈ R. Moreover, M is not restrictively Clifford–Wolf homogeneous. Proof. Let us reformulate the calculations on page 89 of [476]. According to Theorem 4.1 in [375], there is a unique (up to a conjugation in SU(2n)) connected subgroup with non-discrete center K ⊂ SU(2n) that acts transitively on the homogeneous space M = SU(2n)/Sp(n) (n ≥ 3). This is the group SU(2n − 1) × S1 , where 1 SU(2n √− 1) is embedded in SU(2n) via A → diag(A, 1) and S = exp(tU), t ∈ R, U = −1 diag(1, 1, . . . , 1, −(2n − 1)) ∈ su(2n). If SU(2n−1)×S1 is the centralizer of a Killing field V ∈ su(2n), then clearly V = tU for a suitable t ∈ R. It is clear also that any such Killing field has constant length

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on M, since it lies in the center of the Lie algebra of the group acting transitively on M. This proves the first assertion of the proposition. Note that dim(SU(2n)/Sp(n)) = (n − 1)(2n + 1). On the other hand, we can easily calculate the dimension of the set of Killing fields of constant length on M. Indeed, this set is Ad(SU(2n))(tU) (the orbit of tU ∈ su(2n) under the adjoint action of the group SU(2n)), t ∈ R. For any fixed t this orbit is SU(2n)/S(U(2n−1)·U(1)), and dim(SU(2n)/S(U(2n − 1) ·U(1))) = 4n − 2. Since dim(M) = (n − 1)(2n + 1) > (4n − 2) + 1 for n ≥ 3, M is not restrictively Clifford–Wolf homogeneous. Otherwise, by Theorem 6.2.9, for any x ∈ M and U ∈ Mx , there is a Killing vector field of constant length X on M such that X(x) = U, which is impossible by the previous calculation of dimensions. t u Let us consider the following useful result. Theorem 7.3.5. Let M be a simply connected (restrictively) Clifford–Wolf homogeneous Riemannian manifold and M = M0 × M1 × · · · × Mk be its de Rham decomposition, where M0 is a Euclidean space, and the other Mi , 1 ≤ i ≤ k, are simply connected compact Riemannian manifolds. Then every Mi is a (restrictively) Clifford–Wolf homogeneous Riemannian manifold. Moreover, any isometry f of M is a Clifford–Wolf translation if and only if it is a product of some Clifford–Wolf translations fi on Mi . Proof. By Lemma 6.3.4 (see also Corollary 3.1.4 in [476]), every Clifford–Wolf translation f on M is uniquely represented in the form f = ( f1 , . . . , fk ), where f j , j = 1, . . . , k, are Clifford–Wolf translations on M j . This observation proves the second assertions of the theorem. Obviously, the first assertion is a consequence of the second one. t u Now we can prove the main result of this chapter. Proof (of Theorem 7.0.1). Let M be a simply connected Clifford–Wolf homogeneous Riemannian manifold. By Lemma 6.1.3, M is complete. Therefore, according to Theorems 7.3.2 and 4.7.8, M is a symmetric space. Let us consider the de Rham decomposition M = M0 × M1 × · · · × Mk , where M0 is a Euclidean space, and the other Mi , 1 ≤ i ≤ k, are simply connected compact irreducible symmetric spaces. By Theorem 7.3.5 (see also Corollary 3.1.4 in [476] or Lemma 6.3.4), every Clifford–Wolf translation f on M is uniquely represented in the form f = ( f1 , . . . , fk ), where f j , j = 1, . . . , k, are Clifford–Wolf translations on M j . Hence, every Mi , 0 ≤ i ≤ k, is Clifford–Wolf homogeneous. To prove the necessity it remains to apply Propositions 7.3.3 and 7.3.4. The sufficiency follows from the Examples 7.1.1, 7.1.2 and the fact that a direct metric product of simply connected Clifford–Wolf homogeneous spaces is itself simply connected and Clifford–Wolf homogeneous. t u

7.4 Clifford–Killing Spaces

419

7.4 Clifford–Killing Spaces This section is devoted to a deeper study of Clifford–Killing spaces. The following proposition is evident. Proposition 7.4.1. A collection {X1 , . . . , Xl } of Killing vector fields on a connected Riemannian manifold (M, g) constitutes a basis of a finite-dimensional vector space CKl (over R) of Killing vector fields of constant length if and only if the vector fields X1 , . . . , Xl are linearly independent and all inner products g(Xi , X j ); i, j = 1, . . . , l, are constant. Under this CKl admits an orthonormal basis of (unit) Killing vector fields. We shall call such a space CKl , l ≥ 1, a Clifford–Killing space or simply CKspace. Below we give a simple method to check the condition g(X,Y ) = const for given Killing vector fields X,Y . Lemma 7.4.2. Let X and Y be Killing fields on a Riemannian manifold (M, g). Then a point x ∈ M is a critical point of the function x 7→ gx (X,Y ) if and only if 1 ∇X Y (x) = −∇Y X (x) = [X,Y ] (x). 2 Proof. Since ∇X Y − ∇Y X = [X,Y ], it suffices to prove that x is a critical point of g(X,Y ) if and only if ∇X Y (x) + ∇Y X (x) = 0. For any Killing vector field W and arbitrary vector fields U and V on (M, g), we have the equality g(∇U W,V ) + g(U, ∇V W ) = 0 (cf. 1) in Proposition 3.1.6). Since X and Y are Killing vector fields, for any smooth vector field Z we get 0 = Z · g(X,Y ) = g(∇Z X,Y ) + g(X, ∇Z Y ) = −g(Z, ∇Y X) − g(∇X Y, Z) = −g(Z, ∇Y X + ∇X Y ), t u

which proves the lemma.

Corollary 7.4.3. Let X and Y be Killing vector fields on a connected Riemannian manifold (M, g). Then g(X,Y ) = const if and only if 1 ∇X Y = −∇Y X = [X,Y ]. 2 In particular, a Killing vector field X has constant length if and only if ∇X X = 0. Definition 7.4.4. Let V and W be some CK-spaces on (M, g). We say that V is (properly) equivalent to W if there exists a (preserving orientation) isometry f of (M, g) onto itself such that d f (V ) = W . The following question is very interesting. Problem 7.4.5. Classify all homogeneous Riemannian manifolds that admit nontrivial CK-spaces. For any such manifold, classify, up to a (proper) equivalence, all (in particular, all maximal by inclusion) possible CK-spaces.

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This difficult question has been considered in full generality firstly in [80]. All consequent sections are related to this question. We shall see that it is closely connected with some impressive classical and recent results.

7.5 Riemannian Manifolds with the Killing Property In the paper [168], J. E. D’Atri and H. K. Nickerson studied Riemannian manifolds with the Killing property. Definition 7.5.1 ([168]). A Riemannian manifold (M, g) is said to have the Killing property if for each point x ∈ M there exists an orthonormal frame {X1 , . . . , Xn } of vector fields in a neighborhood of x such that each Xi , i = 1, . . . , n, is a Killing vector field (local infinitesimal isometry). Such a frame is called a Killing frame. Remark 7.5.2. We shall call also the property from Definition 7.5.1 the “local Killing property”, while a Riemannian manifold which has a globally defined Killing frame will be considered as a “manifold with the global Killing property”. Note that a global Killing frame on (M, g) defines an absolute parallelism in the sense of Cartan and Schouten [146]. In fact, even in the indefinite metric setting it defines an absolute parallelism consistent with the Riemannian structure; see [481] and [482]. Obviously, every Lie group supplied with a bi-invariant Riemannian metric has the (global) Killing property. Note that a generalization of the Killing property is the divergence property, which we shall not treat here (for details, see [168]). Proposition 7.5.3 ([168]). Every Riemannian manifold with the local Killing property is locally symmetric. Proof. We prove the proposition by a method different from the original one in [168]. By definition, for any point x in a given manifold (M, g) there is a Killing frame {X1 , . . . , Xn } in some neighborhood U of the point x. Since g(Xi , X j ) = δi j , for any real constants ai , the local vector field a1 X1 + a2 X2 + · · · + an Xn is a local Killing field of constant length. As a corollary, for any vector v ∈ Mx there is a Killing field Z of constant length in U such that Z(x) = v. Then, by the proof of Theorem 7.2.1, which in fact doesn’t require a global character of vector fields, we get that (∇Z R)(·, Z)Z (x) = 0 or (∇v R)(·, v) v = 0. By Proposition 7.3.1, (M, g) is locally symmetric. t u Remark 7.5.4. Another proof of the above proposition is given in [481]. The manifold (M, g) has a (locally defined) consistent absolute parallelism defined by a local Killing frame, so it is locally symmetric by Theorem 4.14 in [481]. It is well known that every (not necessarily complete) locally symmetric Riemannian manifold is locally isometric to a symmetric space (cf. e.g. [256]). Therefore, local properties of manifolds with the Killing property can be obtained from the study of complete (simply connected) Riemannian manifolds.

7.5 Riemannian Manifolds with the Killing Property

421

Proposition 7.5.5. Every simply connected complete Riemannian manifold (M, g) with the local Killing property is a symmetric space and has the global Killing property. Every simply connected complete Riemannian manifold (M, g) with the global Killing property is strongly Clifford–Wolf homogeneous. Proof. Since (M, g) is locally symmetric (according to Proposition 7.5.3), complete and simply connected, it is a symmetric space (see e.g. [256]). In particular, (M, g) is real analytic. For any point x in a given manifold (M, g), there is a Killing frame {X1 , . . . , Xn } in some neighborhood U of the point x. We state now that the above (local) Killing frame uniquely extends to a global Killing frame in (M, g). Indeed, every locally defined Killing vector field on (M, g) is a restriction of a globally defined Killing vector field (for instance, by Theorems 1 and 2 in [370], each local Killing vector field in any simply connected real-analytic Riemannian manifold admits a unique extension to a Killing vector field on the whole manifold). Let us prove the second assertion. As a result of the above discussion, there is a global Killing frame {X1 , . . . , Xn } on (M, g). This means that g(Xi , X j ) = δi j for every 1 ≤ i, j ≤ n. Therefore, any linear combination of vector fields X1 , . . . , Xn over R is a Killing vector field of constant length. By the completeness and Theorem 6.2.9, we get that (M, g) is restrictively Clifford–Wolf homogeneous. Now it is enough to apply the first assertion of the proposition and Theorem 7.1.5. t u The authors of [168] tried but failed to classify simply connected complete Riemannian manifolds with the Killing property (see page 407 right before section 5 in [168]). The following theorem completely solves this problem. Theorem 7.5.6 ([80]). A simply connected complete Riemannian manifold (M, g) has the Killing property if and only if it is isometric to a direct metric product of a Euclidean space, compact simply connected simple Lie groups with bi-invariant metrics, and round spheres S7 (some mentioned factors could be absent). Remark 7.5.7. The assertion of Theorem 7.5.6 could also be derived from the first assertion of Proposition 7.5.5 and Theorem 9.1 in [482]. Proof. The sufficiency of this statement follows from the well-known fact that every mentioned factor has the Killing property (see also the next section, where the round sphere S7 is discussed in more detail). Let us prove the necessity. As a corollary of Proposition 7.5.5 and Theorem 7.0.1, (M, g) must have the form as in Theorem 7.0.1. But we may leave only S7 as factors among odd-dimensional spheres for the following reason. It is easy to see that every factor of the corresponding product also has the Killing property (see Theorem 4.1 in [168]). It is well known (see also Corollary 7.6.5 below) that only S3 and S7 have the Killing property among (simply connected) odd-dimensional round spheres. But the round sphere S3 can be considered as a compact simply connected simple Lie group SU(2) supplied with a bi-invariant Riemannian metric. Therefore, we can exclude S3 . t u

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7.6 Results by A. Hurwitz and J. Radon A. Hurwitz has posed the following question. Problem 7.6.1. For a given natural number m, find the maximal natural number p = ρ(m) such that there is a bilinear real vector-function z = (z1 , . . . , zm ) = z(x, y) of real vectors x = (x1 , . . . , x p ) and y = (y1 , . . . , ym ), which satisfies the equation

x12 + · · · + x2p y21 + · · · + y2m = z21 + · · · + z2m (7.1) for all x ∈ R p and y ∈ Rm . Using some equivalent formulations, J. Radon in [393] and A. Hurwitz in [267] independently obtained the following answer. Theorem 7.6.2. If m = 24α+β m0 , where β = 0, 1, 2, 3, α is a non-negative integer, and m0 is odd, then ρ(m) = 8α + 2β (this mapping m 7→ ρ(m) is called the Radon– Hurwitz function). Let us consider some equivalent formulations and corollaries, following in some respects J. Radon. It is clear that any bilinear function z = z(x, y) can be represented in the form ! p

z = z(x, y) =

∑ x j (A j y) =

j=1

p

∑ x jA j

(7.2)

y,

j=1

where A j are (m × m)-matrices and one considers z and y as vectors-columns. Putting xi = δ ji , i = 1, . . . , p for a fixed j from {1, . . . , p} and using equation (7.1), one can easily see that 1) every A j must be an orthogonal matrix. On the other hand, the same equation shows that 2) for any fixed y ∈ Rm , A j y, j = 1, . . . , p must be mutually orthogonal vectors in Rm with lengths, all equal to |y|. As a corollary, we must have p ≤ m. Finally, the equation (7.1) and the last form of equation (7.2) show that   3) for any unit vector x ∈ R p , the matrix ∑ pj=1 x j A j must be orthogonal. The last statement explains the title of the paper [393] and gives an equivalent form of the question 7.6.1, which was considered by J. Radon. Now, it is clear that if we change every matrix A j in the bilinear form (7.2) to B j = A j A, where A is a fixed orthogonal (m×m)-matrix, then we get another bilinear form, which also satisfies the equation (7.1). If we take A = A−1 p , and change the notation B j to A j , we get the following form of (7.2): ! p−1

z = z(x, y) =

p−1

∑ x j (A j y) + x p (Iy) = ∑ x j A j + x p I

j=1

j=1

y.

(7.3)

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423

Now, applying the properties 1) and 2) for the obtained bilinear form (7.3), we get that all matrices A j , j = 1, . . . p − 1, in (7.3) must be both orthogonal and skewsymmetric, since A j y ⊥ y = Iy for all y ∈ Rm (it implies in particular that only p ≥ 2 is possible when m is even). Now, from Theorem 7.6.2, the properties 1) and 2), and the last statement, we easily obtain the following Theorem 7.6.3. A nontrivial collection of vector fields on Sm−1 consists of mutually orthogonal unit Killing vector field on Sm−1 if and only if it can be presented in the form X j (y) = A j y, y ∈ Sm−1 , j = 1, . . . , p − 1, where (m × m)-matrices A j , which are both orthogonal and skew-symmetric (thus m is even), are taken from the equation (7.3), defining a bilinear form satisfying the equation (7.1). The maximal number of such fields is equal to ρ(m) − 1, see Theorem 7.6.2. Theorem 7.6.3 implies Theorem 7.6.4. The maximal dimension l of Clifford–Killing spaces CKl on Sm−1 is equal to ρ(m) − 1. Corollary 7.6.5. The maximal dimension l of Clifford–Killing spaces CKl on Sm−1 is equal to m − 1 > 0 if and only if m ∈ {2, 4, 8}. As a corollary, S1 , S3 , and S7 are all round spheres with the Killing property. Note that the last result is related to the existence of algebras of complex, quaternion, and Cayley numbers. Later on, Eckmann, Lee, and Wong reproved the Hurwitz–Radon Theorem in [199], [319], and [486]. The methods of Radon and Hurwitz yield complicated schemes for the actual construction of the forms (7.3) for p = ρ(m), these methods have been simplified by Adams, Lax, and Phillips [8], as well as by Zvengrowski [504], by Balabaev [36], and by Ognikyan [371]. The purpose of the paper [486] is to study maximal sets of mutually Cliffordparallel (n − 1)-planes in the real elliptic (2n − 1)-spaces RP2n−1 . This problem is equivalent to the search for maximal sets of mutually isoclinic n-planes in Euclidean 2n-spaces, which in its turn is found to be connected with the Hurwitz–Radon problem on the composition of quadratic forms. L. K. Hua encountered the Hurwitz matrix equations (7.4), (7.5) in his geometries of matrices (see the book [456] by Wan), but was apparently unaware of the works of Hurwitz and others, he obtained a wrong conclusion concerning the largest number of the matrices which form a solution of the Hurwitz equations in [264, Theorem 35]. E. Stiefel introduced his famous characteristic classes in [422] to study the problem of the maximal number of continuous linearly independent (at all points) tangent vector fields on closed orientable smooth manifolds. Let us mention here only the following Sufficient Stiefel condition from [422]: If there exist q + 1 real quadratic matrices A j , 0 ≤ j ≤ q, of order n such that for any set of real numbers λ j , not all zero, the matrix ∑ j λ j A j is non-singular, then there exist q linearly independent tangent vector fields on the projective (n − 1)-space and (n − 1)-sphere. It should be noted also that J. F. Adams has proved that there are no ρ(m) continuous orthonormal (or, equivalently, linear independent) tangent vector fields on the sphere Sm−1 [5] (see also Theorem 13.10 of Chapter 15 in [268]).

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7.7 Clifford–Killing Spaces on Spheres and Clifford Algebras and Modules In this section we continue the study of Clifford–Killing spaces on spheres Sm−1 , supplied with their canonical Riemannian metrics of constant sectional curvature 1. As noted before, those spaces are Clifford–Wolf homogeneous when m is even. The material of Sections 7.7, 7.8 and 7.9 is closely related to that of [477] and [478], especially in connection with the Hurwitz equations and Helgason spheres. Theorem 7.7.1. Real (m×m)-matrices u1 , u2 , . . . , ul , l ≥ 1, define (pairwise orthogonal at every point of Sm−1 ) unit Killing vector fields Ui (x) := ui x, x ∈ Sm−1 , on Sm−1 if and only if ui ∈ O(m) ∩ so(m), u2i = −I, i = 1, . . . , l, (7.4) ui u j + u j ui = 0,

i 6= j.

(7.5)

In this case m is even and ui ∈ SO(m). Proof. The vector fields Ui are unit Killing vector fields on Sm−1 if and only if they can be presented as follows: Ui (x) = ui x, x ∈ Sm−1 , where ui ∈ O(m) ∩ so(m) (see Theorem 7.6.3). This implies that m is even, (x, y) = (ui x, ui y) = −(u2i x, y), and u2i = −I (here (·, ·) means the standard inner product in Rm ). It is clear that under the first condition in (7.4), the vector fields Ui and U j on Sm−1 are orthogonal if and only if ((ui + u j )x, (ui + u j )x) = (ui x, ui x) + (u j x, u j x),

x ∈ Sm−1 ,

which is equivalent to the identity (ui u j x, x) ≡ 0, x ∈ Rm , or ui u j ∈ so(m), or −(ui u j ) = (ui u j )t = utj uti = (−u j )(−ui ) = u j ui , i.e. (7.5). A matrix ui ∈ O(m) is skew symmetric if and only if ui is orthogonally similar to a matrix u = diag(C, . . . ,C), where C ∈ O(2) is skew symmetric. This implies that ui ∈ SO(m). t u Remark 7.7.2. According to Theorem 7.7.1, later on we may assume that m is even and m = 2n. Theorem 7.7.1 naturally leads to the notion of associative algebras over the field R, with generators e1 , . . . , el , such that e2i = −1, ei e j + e j ei = 0, i 6= j (and any other relation in the algebra is some corollary of the indicated relations). Such algebra Cll is called the Clifford algebra (with respect to the negative definite quadratic form −(y, y) in Rk and orthonormal basis {e1 , . . . , el } in (Rk , (·, ·))). These algebras include the algebra Cl1 = C of complex numbers and the algebra Cl2 = H of quaternions. The others Clifford algebras can be described as follows: Cl3 = H ⊕ H, Cl4 = H ⊗R R(2), where the algebra R(2) is generated by the following symmetric

7.7 Clifford–Killing Spaces on Spheres and Clifford Algebras and Modules

425

(2 × 2)-matrices: diag{1, −1} and the permutation matrix of vectors in the canonical basis {e1 , e2 } for R2 . After this one can apply “the periodicity law” Clk+4 = Clk ⊗R Cl4 . For more details, see [268, 223]. Let us consider Ll,2n , the algebra of linear operators (on R2n ) that is generated by the operators (the matrices) ui , i = 1, . . . , l ≥ 1. Theorem 7.7.1 implies that Ll,2n is a homomorphic image of the Clifford algebra Cll . It is easy to see that the kernel of the natural homomorphism c : Cll → Ll,2n is a two-sided ideal of Cll . The above description of Clifford algebras shows that Cll , l 6= 4k + 3, contains no proper two-sided ideal, while the Clifford algebra Cll = Cll−1 ⊕Cll−1 contains exactly two proper two-sided ideals C1 and C2 that are both isomorphic to Cll−1 if l = 4k + 3. Thus, in any case Ll,2n is isomorphic either to Cll , or to Cll−1 (if l 6= 4k + 3, then necessarily Ll,2n is isomorphic to Cll ). Now we consider an example where Ll,2n is not isomorphic to Cll . Example 7.7.3. Let us consider a Lie algebra CK3 ⊂ so(2n), where 2n = 4k. In this case CK3 is the linear span of the vectors u1 , u2 , u3 := u1 u2 . Clearly, the algebra L3,2n is not isomorphic to Cl3 . It is easy to calculate the dimensions of both of these associative algebras. The dimension of Clm , considered as a vector space over R, is 2m [268]. In particular, dim(Cl3 ) = 8. On the other hand, L3,2n = Lin{1, u1 , u2 , u3 }, dim(L3,2n ) = 4, since u1 u3 = u1 u1 u2 = −u2 ,

u2 u3 = u2 u1 u2 = u1 .

Notice that in this case L3,2n is isomorphic to the quaternion algebra H = Cl2 . The above-mentioned homomorphism c : Cll → Ll,2n defines a representation of the Clifford algebra Cll in R2n , so the last vector space together with the representation c becomes a Clifford module (over Cll ). We saw that any Clifford–Killing space CKl ⊂ so(2n) defines the structure of a Clifford module on R2n over Cll . It is very important that in this way one can get every Clifford module. We need some information on the classification of Clifford modules over Cll [268]: a) If l 6= 4k + 3, then there exists (up to equivalence) precisely one irreducible Clifford module µl over Cll with the representation cl . Every Clifford module νl over Cll is isomorphic to the m-fold direct sum of µl , that is, νl ∼ = ⊕m µl .

(7.6)

b) If l = 4k + 3, then there exist (up to an equivalence) precisely two nonequivalent irreducible Clifford modules µl,1 , µl,2 over Cll with representations c1 = cl−1 ◦ π1 and c2 = cl−1 ◦ π2 , where πi is the natural projection of Cll onto the maximal ideal (Cl )i ∼ = Cl−1 , i = 1, 2. The modules µl,1 and µl, 2 have the same dimension and every Clifford module νl over Cll is isomorphic to νl ∼ = ⊕m1 µl,1 ⊕ ⊕m2 µl,2

(7.7)

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for some non-negative integers m1 and m2 . It is clear that the representation of Cll , corresponding to the module νl , is exact if and only if both numbers m1 and m2 are positive. The dimension n0 of µl or µl,1 , µl,2 is equal to n0 = 24α+β , if 8α +2β −1 −1 < l ≤ 8α + 2β − 1, where α is a non-negative integer and β = 0, 1, 2, 3. In some sense, this is the function, inverse to the Radon–Hurwitz function ρ(m), from Theorem 7.6.2. The previous discussion implies Theorem 7.7.4. The sphere S2n−1 admits a Clifford–Killing space CKl if and only if 1 ≤ l ≤ ρ(2n) − 1. In this case n0 = n0 (l) divides 2n. All Clifford–Killing spaces CKl for S2n−1 are pairwise equivalent (all spaces CKl ⊂ so(2n) are equivalent with respect to O(2n)) if and only if l 6= 4k + 3. If l = 4k + 3, then there exist exactly [n/n0 (l)] + 1 non-equivalent classes of Clifford–Killing spaces CKl for S2n−1 . In particular, all Clifford–Killing spaces CKρ(2n)−1 for S2n−1 are pairwise equivalent if and only if 2n = 24α+β n0 , where α is a non-negative integer, n0 is odd, and β = 1 or β = 3. If β = 0 or β = 2 in the above notation, then there exist exactly [n0 /2] + 1 non-equivalent classes of Clifford–Killing spaces CKρ(2n)−1 for S2n−1 . This theorem together with Theorem 7.7.6 below give the exact number of (proper) equivalence classes for Clifford–Killing spaces on S2n−1 . Proposition 7.7.5. The set of unit Killing vector fields on the sphere S2n−1 represents by itself a union of two disjoint orbits with respect to the adjoint action of the group SO(2n), and one orbit with respect to the adjoint action of the group O(2n). Proof. By Theorem 7.7.1, an arbitrary unit Killing vector field on the sphere S2n−1 is defined by a matrix U from SO(2n) ∩ so(2n) with the condition U 2 = −I. Thus there is a matrix A(U) ∈ O(2n) such that A(U)UA(U)−1 = diag(C, . . . ,C), where C is a fixed matrix C ∈ SO(2) ∩ so(2). Moreover, if A0 (U) is another such matrix, then A(U)[A0 (U)]−1 ∈ SO(2n). This implies that every two matrices U,V ∈ SO(2n) ∩ so(2n) are equivalent in O(2n), and equivalent in SO(2n) if and only if A(U)A(V )−1 ∈ SO(2n). This finishes the proof. t u Theorem 7.7.6. If 2n ≡ 2 (mod 4), then any two spaces of the type CKl (necessarily l = 1) are SO(2n)-equivalent. If 2n ≡ 0 (mod 4), then any O(2n)-equivalence class of spaces CKl ⊂ so(2n) contains exactly two SO(2n)-equivalence classes. Proof. We shall use the vectors U and V as in Proposition 7.7.5, where A(U) ∈ SO(2n) and A(V ) ∈ O(2n) \ SO(2n). Let us suppose that 2n ≡ 2 (mod 4). Then l = 1 and any space CK1 is spanned onto some unit Killing vector field X. Note that X is equivalent with −X ∈ CK1 by means of an orthogonal matrix with determinant −1, thus either X or −X is equivalent to the vector U in SO(2n). Now, let us suppose that 2n ≡ 0 (mod 4). Consider any space CKl which is spanned on unit Killing vector fields U1 , . . . ,Ul . If l = 1, then X1 is obviously equivalent to −X1 in SO(2n). If l > 1, then any two unit Killing vector fields in CKl can be continuously deformed (in the set CKl ) to another one. Therefore, all unit Killing

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vector fields in CKl are simultaneously equivalent to (only one of) U or V in SO(2n), for example, to U. Now, let us consider B ∈ O(2n) \ SO(2n) and the space CKl0 , which is equivalent to CKl by B. Since neither U nor −U is equivalent to V in SO(2n), any unit Killing vector in CKl0 is equivalent to the vector V , which is not equivalent in SO(2n) to unit Killing vectors in CKl . Thus the spaces CKl and CKl0 are not equivalent to each other in SO(2n). On the other hand, since SO(2n) is an index 2 subgroup of O(2n), it is clear that any O(2n)-equivalence class contains at most two SO(2n)-equivalence classes. t u It should be noted that subspaces CKl of Lie algebras so(2n) play an important role in various mathematical theories. For example, Theorem 7.7.4 and the assertions on p. 23 in [104] imply that there is a bijection between the O(2n)-classes of CKl ⊂ so(2n) (for all possible pairs (2n, k) of this type) and the isometry classes of generalized Heisenberg groups first studied by A. Kaplan in [280] (see also Section 5.3). These are special two-step nilpotent groups admitting one-dimensional solvable Einstein extensions that are the well-known Damek–Ricci spaces [167]. Note that Damek–Ricci spaces are harmonic Riemannian manifolds and most of them are not symmetric. We refer the reader to [104, 198, 231, 46, 166, 252] and references therein for the deep theory of generalized Heisenberg groups and Damek– Ricci spaces. Note also that there are some useful generalizations of subspaces of the type CKl in Lie algebras so(2n). One of them is the notion of uniform subspaces of so(2n) [231]. Such subspaces are used for producing new Einstein solvmanifolds with two-step nilpotent nilradical (see [231] and [285] for details).

7.8 Clifford–Killing Spaces for S2n−1 and Unit Radon Spheres in O(2n) Now, we supply the Lie algebra so(2n) with the following Ad(SO(2n))-invariant inner product: 1 (U,V ) = − trace(UV ). (7.8) 2n The Lie group SO(2n), supplied with the corresponding bi-invariant inner Riemannian metric ρ, is a symmetric space. This metric uniquely extends to a bi-invariant “metric” ρ on O(2n) (ρ(x, y) = +∞ if and only if x, y lie in different connected components). Note that (X, X) = 1 for every unit Killing field X on the sphere S2n−1 supplied with the canonical metric of constant curvature 1, and the Killing form B of so(2n) is connected with the form (7.8) by the formula B(U,V ) = 2(n − 1) trace(UV ) = −4n(n − 1) (U,V ). Let us recall that the forms trace(UV ) and B(U,V ) on so(2n) are forms associated with the identical and the adjoint representations of so(2n) respectively [181].

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Definition 7.8.1. Let A1 , . . . , A p , 1 ≤ p ≤ ρ(2n), be matrices from O(2n), defining p the bilinear form (7.2). Then the set of all matrices of the form ∑i=1 xi Ai , where x = (x1 , . . . , x p ) is a unit vector in R p , is called a Radon unit sphere for the form (7.2) or simply a Radon unit sphere, and is denoted by RS p−1|2n . It follows from Section 7.6 and Theorem 7.7.1 that always RS p−1|2n ∈ O(2n), and ∈ SO(2n) if and only if at least one of the matrices A1 , . . . , A p is in SO(2n). Moreover, RS p−1|2n A is also a Radon unit sphere for every A ∈ O(2n), and left and right translations are isometries in (SO(2n), ρ). Thus, from a geometric point of view, we can consider only the Radon unit sphere defined by a form (7.3). As was mentioned in Section 7.6, in this case A p = I, and the linear span of A1 , . . . , A p−1 is the Clifford–Killing space CKl ⊂ so(2n), where l = p − 1. The goal of this section is the following RS p−1|2n

Theorem 7.8.2. In the notation of Definition 7.8.1, the map p

x = (x1 , . . . , xl+1=p ) ∈ Sl ⊂ Rl+1 →

∑ x j (A j ) ∈ (O(2n), ρ)

j=1

preserves distances and has the image RSl|2n . As a corollary, any Radon sphere RSl|2n , defined by a form (7.2), is a totally geodesic submanifold in (O(2n), ρ), which is isometric to the standard unit sphere Sl ⊂ Rl+1 with unit sectional curvature. If A p = I, it is equal to exp(CKl ), where CKl is the linear span of A1 , . . . , Al=p−1 from the formula (7.3). Conversely, the image exp(CKl ) ⊂ SO(2n) of an arbitrary Clifford–Killing space CKl ⊂ so(2n) is a Radon unit sphere RSl|2n for some form (7.3). Proof. First we shall prove the third assertion. It was proved in Section 7.6 and Theorem 7.7.1 that all A1 , . . . , A p−1=l are elements of SO(2n) ∩ so(2n) that define mutually orthogonal unit Killing vector fields on S2n−1 . Consider any C ∈ RSl|2n ⊂ SO(2n), that is C = ∑li=1 xi Ai + xl+1 I, where x = (x1 , . . . , xl+1 ) is a unit vector in 2 6= 1. Then the matrix Rl+1 . First, let us suppose that xl+1 A := q

1 ∑li=1 xi2

l

∑ xi Ai

i=1

is in SO(2n) ∩ so(2n) ∩ CKl and defines a unit Killing vector field on S2n−1 . Obviously, the vector C can be represented in the form C = (cos r)I + (sin r)A, r ∈ R, where cos r = xl+1 . Now for any t, s ∈ R we have A2 = −I by Theorem 7.7.1 and so    (cost)I + (sint)A (cos s)I + (sin s)A = (cos(t + s))I + (sin(t + s))A. This means that the set of matrices (cost)I + (sint)A, t ∈ R, constitutes a 1parameter subgroup in SO(2n) with the tangent vector A, and C = exp(rA) ∈ exp(CKl ). If xl+1 = 1 or xl+1 = −1, then for any matrix A ∈ CKl , defining a unit

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Killing vector field, we have that exp(tA) ∈ RSl|2n for all t ∈ R, and exp(0A) = C or exp(πA) = C by the above arguments. So, we have proved the required equality RSl|2n = exp(CKl ). This implies in particular that exp(CKl ) topologically is Sl . According to Section 7.4.1, any subspace CKl=p−1 on S2n−1 has a basis of l mutually orthogonal unit Killing vector fields, which are defined by some matrices A1 , . . . A p−1 in SO(2n) ∩ so(2n) by Theorem 7.7.1. (Let us note also that by the same Theorem, Ai A j = −A j Ai , if i 6= j, and so (Ai , A j ) = 0 by the formula (7.8).) Now, it is clear that formula (7.3) defines the required bilinear form, and the last assertion follows from the third one. By the last two assertions, RSl|2n = exp CKl , where CKl is the linear span of A1 , . . . , Al , where l = p − 1. Now, if A ∈ CKl ∩ SO(2n), then by Theorem 7.7.1, A2 = −I and the formula (7.8) gives us that (A, A) = 1. This fact, the Ad(SO(2n))invariance of the inner product (7.8), and the discussions in the first part of the proof imply that exp(tA), t ∈ R, is a geodesic circle in SO(2n) of length 2π, entirely lying in RSl|2n . Let us suppose that B and C are two different points in RSl|2n , defined by unit vectors b and c in Rl+1 . Then there is a unit vector d ∈ Rl+1 , which is orthogonal to b, such that c = (cos r)b + (sin r)d for some r ∈ [0, π]. The vector d defines the corresponding element D ∈ RSl|2n . Matrices A01 = B, A02=p = D define a bilinear form z(x, y) with necessary properties by the formula (7.2), and the corresponding Radon sphere RS1,2n ⊂ O(2n), containing the points A, D,C. By the previous results, the right translation by D−1 , which is an isometry on (O(2n), ρ), transforms this Radon sphere to another one of the form exp(CK1 ), which is a geodesic circle in (SO(2n), ρ) of length 2π. This implies that the curve c(t) := (cost)B + (sint)D, t ∈ [O, r], joining the points B and C in RSk|2n , is a shortest geodesic in (O(2n), ρ), parameterized by the arc-length. Thus we have proved the second assertion. The first assertion follows from the second one and the bi-invariance of the “metric” ρ on O(2n). t u

7.9 Triple Lie Systems in so(2n) and Totally Geodesic Spheres in SO(2n) Recall that a linear subspace a of a Lie algebra g is called a Lie triple system if [a, [a, a]] ⊂ a. We shall need the following Lemma 7.9.1. If a is a Lie triple system of a Lie algebra g, then h := [a, a] and k := h + a are subalgebras of g and h ∩ a is an ideal of k. Proof. From [a, [a, a]] ⊂ a and the Jacobi identity we get [h, h] ⊂ h and [h, a] ⊂ a, that proves the first assertion of the lemma. Now, it is easy to see that [a, h ∩ a] ⊂ h ∩ a and [h, h ∩ a] ⊂ h ∩ a, which proves the second assertion. t u

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Lemma 7.9.2. Let G be a compact semisimple Lie group supplied with a bi-invariant Riemannian metric ρ. Suppose that a is a Lie triple system in g, the Lie algebra of G. Then M := exp(a) is a totally geodesic submanifold of G, in particular, M is a symmetric space. If the universal Riemannian covering of M is irreducible, then one of the following two conditions holds: 1) a is a simple Lie subalgebra of g and M is a simple compact Lie group with a bi-invariant Riemannian metric. 2) The Lie algebra h := [a, a] satisfies the relation h ∩ a = 0, and (h ⊕ a, h) is a symmetric pair corresponding to M. In particular, exp(h⊕a) is the full connected isometry group of M. If in this case the Lie algebra h ⊕ a is not simple, then M is a simple compact Lie group with a bi-invariant Riemannian metric. Proof. Here we consider exp(a) and exp(h ⊕ a) in the Lie theoretical sense. On the other hand, since ρ is bi-invariant, they could also be treated in the Riemannian sense. Let (·, ·) be a Ad(G)-invariant inner product on g that generates ρ. Let us consider the standard representation of (G, ρ) as a symmetric space: G × G/ diag(G). Consider e g = g ⊕ g, the Lie algebra of G × G, and ek = diag(g) ⊂ e g,

e p = {(X, −X) | X ∈ g}.

Then e g = ek ⊕ e p is a Cartan decomposition for the symmetric space G × G/ diag(G). Now consider e a = {(X, −X) | X ∈ a} ⊂ e p. It is clear that e a is a Lie triple system in e g, [e a,e a] = h ⊕ h ⊂ ek. Using the standard theory of Lie triple systems in symmetric spaces (see e.g. [256]), we conclude that the pair ([e a,e a] ⊕ e a,e a) is symmetric and corresponds to the symmetric space M1 := exp(e a). It is clear that M1 (supplied with the Riemannian metric induced by the inner product 12 (·, ·) + 12 (·, ·) on g ⊕ g) is isometric to M. In this case it is sufficient to consider an isometry i : M → M1 defined as follows: i(exp(tX)) = (exp(tX), exp(−tX)), where t ∈ R, X ∈ a. This proves the first assertion of the lemma. Let us suppose that the universal covering of M = M1 is an irreducible symmetric space. Then the Lie algebra [e a,e a] ⊕ e a, being the Lie algebra of the full isometry group of M1 , is either simple or a direct sum of two copies of some simple Lie algebra. Recall that u := h ∩ a is an ideal in the Lie algebra h + a by Lemma 7.9.1. Let us consider the maps π1 : [e a,e a] ⊕ e a → h + a,

π2 : [e a,e a] ⊕ e a → h + a,

defined as follows. Consider Z = (Y,Y ) + (X, −X), where X ∈ a and Y ∈ [a, a], then we put π1 (Z) = Y + X and π2 (Z) = Y − X. It is easy to see that π1 and π2 are Lie algebra epimorphisms. The kernel of π1 is u1 = {(0, X) | X ∈ u} and the kernel of π2 is u2 = {(X, 0) | X ∈ u}. In particular, we get that u ⊕ u = u1 ⊕ u2 is an ideal in the Lie algebra [e a,e a] ⊕ e a. Suppose that the Lie algebra [e a,e a] ⊕ e a is isomorphic to s ⊕ s, where s is a simple Lie algebra. If u is not trivial, then [e a,e a] ⊕ e a coincides with its ideal u ⊕ u = u1 ⊕ u2 (this ideal could not be proper in this case). Moreover, u = a = [a, a] = h is a simple

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Lie algebra (since u1 is the kernel of π1 , then h + a is isomorphic to u2 ∼ u = h ∩ a). In this case M is a connected simple compact Lie group supplied with a bi-invariant Riemannian metric, and we get Condition 1) of the lemma. If u is trivial, then π1 is an isomorphism. Therefore, h + a = h ⊕ a is isomorphic to s ⊕ s, and h is isomorphic to diag(s) (since diag(s) is a unique proper subalgebra in s ⊕ s). Hence M is a simple compact Lie group (with the Lie algebra isomorphic to s) with a bi-invariant Riemannian metric, and we get Condition 2) of the lemma with non-simple h ⊕ a. Now, if the Lie algebra [e a,e a] ⊕ e a is simple, then u is trivial and π1 is an isomorphism. Therefore, h + a = h ⊕ a = π1 ([e a,e a] ⊕ e a) is a simple Lie algebra that is the Lie algebra of the full isometry group of M. Moreover, by the definition of π1 , we obtain π1 ([e a,e a]) = h. Therefore, Condition 2) of the lemma with simple h ⊕ a holds. t u Theorem 7.9.3. Let CKl be a Clifford–Killing subspace of dimension l in the Lie algebra so(2n). Then the following assertions hold: 1) CKl is a Lie triple system in the Lie algebra so(2n). 2) For every CK1 , the image exp(CK1 ) is a closed geodesic of length 2π in (SO(2n), ρ). If l ≥ 2, then the image exp(CKl ) is a totally geodesic sphere Sl of constant sectional curvature 1 in (SO(2n), ρ). 3) If CKl is a Lie subalgebra of so(2n) then either l = 1, or l = 3. 4) Every CK1 is a commutative Lie subalgebra of so(2n), the image exp(CK1 ) consist of Clifford–Wolf translations of S2n−1 , and the exp(CK1 )-orbits constitute a totally geodesic foliation of equidistant (great) circles in the sphere S2n−1 . 5) If CK3 is a Lie subalgebra of so(2n) then CK3 = [CK3 ,CK3 ] is isomorphic to so(3) ∼ su(2), and exp(CK3 ) = S3 is the group SU(2) supplied with a biinvariant Riemannian metric. Moreover, exp(CK3 ) = SU(2) consists of Clifford– Wolf translations of S2n−1 and, consequently, the exp(CK3 )-orbits constitute a totally geodesic foliation of equidistant (great) 3-spheres in the sphere S2n−1 . 6) If CKl is not a Lie subalgebra of so(2n) then CKl ∩ [CKl ,CKl ] is trivial, subspaces CKl ⊕ [CKl ,CKl ] and [CKl ,CKl ] are Lie subalgebras of so(2n) such that the pair (CKl ⊕ [CKl ,CKl ], [CKl ,CKl ]) is the symmetric pair (so(l + 1), so(l)) and exp(CKl + [CKl ,CKl ]) is isomorphic to SO(l + 1), the full connected isometry group of exp(CKl ) = Sl . Proof. Let CKl be the linear span of the unit Killing vector fields U1 , . . . ,Ul . By Theorem 7.7.1 we get that Ui2 = −I and UiU j +U jUi = 0 for i 6= j. Let us show that [CKl , [CKl ,CKl ]] ⊂ CKl . If the indices i, j, k are pairwise distinct, then [Ui ,U j ] = 2UiU j and [Uk , [Ui ,U j ]] = 2UkUiU j − 2UiU jUk = −2UiUkU j + 2UiUkU j = 0. On the other hand, [Ui , [Ui ,U j ]] = 2UiUiU j − 2UiU jUi = −2U j + 2UiUiU j = −4U j ∈ CKl . Therefore, CKl is a Lie triple system in so(2n), which proves the first assertion of the theorem.

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The second assertion immediately follows from Theorem 7.8.2. If CKl is a Lie algebra, then necessarily l = 1 or l = 3, which proves the third assertion of the theorem. The fourth assertion immediately follows from Theorem 4.8.1. If CK3 is a Lie subalgebra of so(2n), then CK3 is isomorphic to so(3) ∼ su(2). In this case exp(CK3 ) = SU(2) = S3 with a metric of constant sectional curvature 1. Obviously, every X ∈ CK3 is a Killing vector field of constant length on S2n−1 . By Theorem 4.8.1, we get that exp(CK3 ) = SU(2) consists of Clifford–Wolf translations of S2n−1 . Now, it easy to see that exp(CK3 )-orbits constitute a totally geodesic foliation of equidistant (great) 3-spheres in the sphere S2n−1 . This proves the fifth assertion of the theorem. By Lemma 7.9.2, CKl +[CKl ,CKl ] is a Lie subalgebra of so(2n) and the subgroup exp(CKl +[CKl ,CKl ]) of SO(2n) is a connected isometry group of M := exp(CKl ) = Sl . If l = 3 and CK3 is not a Lie subalgebra of so(2n), then (by Lemma 7.9.2) the Lie algebra CK3 + [CK3 ,CK3 ] = CK3 ⊕ [CK3 ,CK3 ] is isomorphic to so(4) ∼ so(3)⊕so(3) and exp(CK3 ⊕[CK3 ,CK3 ]) = SO(4). Now, suppose that l 6= 1, 3. Then the sphere Sl = exp(CKl ) is not a Lie subgroup of SO(2n). By Lemma 7.9.2, we get that CKl ∩ [CKl ,CKl ] is trivial, the Lie algebra CKl ⊕ [CKl ,CKl ] is so(l + 1) and exp(CKl ⊕ [CKl ,CKl ]) = SO(l + 1). The theorem is completely proved. t u According to Theorems 7.8.2 and 7.9.3, the study of subspaces CKl is related to the study of totally geodesic spheres in SO(2n). Note that there are well-known totally geodesic spheres in SO(2n), Helgason spheres. In [255], S. Helgason proved that every compact irreducible Riemannian symmetric space M with the maximal sectional curvature κ contains totally geodesic submanifolds of constant curvature κ. Any two such submanifolds of the same dimension are equivalent under the full connected isometry group of M. The maximal dimension of any such submanifold is 1 + m(δ ), where m(δ ) is the multiplicity of the highest restricted root δ . Moreover, if M is not a real projective space, then such submanifolds of dimension 1 + m(δ ) are actually spheres. In the case when M is a simple compact Lie group with a bi-invariant Riemannian metric, m(δ ) = 2. Therefore, the maximal dimension of submanifolds as above is 3. Note that M = (SO(2n), ρ), n ≥ 2, is not a real projective space, therefore, there are 3-dimensional totally geodesic Helgason spheres in (SO(2n), ρ). It is easy to give a description of these spheres (see [255]). First we recall some well-known facts (see e.g. H of Chapter 8 in [106]). Let Ei j be a (2n × 2n)-matrix with zero entries except for the (i, j)-th entry, which is 1. Consider the matrices Fi = E(2i)(2i−1) − E(2i−1)(2i) for 1 ≤ i ≤ n. These matrices define a basis of a standard Cartan subalgebra t in n

so(2n). Hence every X in t has the form X = ∑ λi Fi . Every root (with respect to t) i=1

of so(2n) has the form λi ± λ j , i 6= j. Note that all these roots have the same length. Let Vλi ±λ j be the (two-dimensional) root space of the root λi ± λ j in so(2n). Put Uλi ±λ j = R · (Fi ± Fj ) ⊕ Vλi ±λ j . In this notation, exp(Uλi ±λ j ) is a Helgason sphere in (SO(2n), ρ) (for details, see the proof of Theorem 1.2 in [255]). Moreover, by

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Theorem 1.1 in [255] any two Helgason spheres in (SO(2n), ρ) are equivalent under the full connected isometry group of (SO(2n), ρ). Therefore, every Helgason sphere in (SO(2n), ρ) is conjugate in SO(2n) either to the sphere exp(Uλ1 −λ2 ), or to the sphere exp(Uλ1 +λ2 ).

Proposition 7.9.4. The spheres S3 = exp(CK3 ) in Assertion 5 of Theorem 7.9.3 are Helgason spheres of constant curvature 1 in (SO(4), ρ). Every Clifford–Killing space CK3 in so(4) is an ideal of so(4). ∼ so(3) ⊕ so(3). There are only two non-proportional roots Proof. Note that so(4) = for the standard Cartan algebra t: λ1 + λ2 and λ1 − λ2 . It is easy to see that Uλ1 +λ2 and Uλ1 −λ2 are pairwise commuting Lie algebras isomorphic to so(3) ∼ su(2). In particular, exp(Uλ1 ±λ2 ) are spheres of the type exp(CK3 ) in (SO(4), ρ) (see Assertion 5 in Theorem 7.9.3). Note also, that Uλ1 +λ2 and Uλ1 −λ2 can be naturally identified with the Lie algebras of left and right shifts on S3 = SU(2). On the other hand, (this follows from the previous discussion) exp(Uλ1 ±λ2 ) are Helgason spheres (SO(4), ρ). By Assertion 2 in Theorem 7.9.3 (or by Theorem 7.8.2) these spheres have constant curvature 1. This proves the first assertion of the proposition. Now, let CK3 be an arbitrary Clifford–Killing space of so(4). By Theorem 7.8.2, exp(CK3 ) is a sphere S3 of constant curvature 1. Since its sectional curvature coincides with the sectional curvature of the Helgason spheres exp(Uλ1 ±λ2 ), exp(CK3 ) should be a Helgason sphere too. From the description of Helgason spheres right before the statement of the proposition, we get that exp(CK3 ) is conjugate in SO(4) either to exp(Uλ1 −λ2 ), or to exp(Uλ1 +λ2 ). In particular, CK3 is conjugate in SO(4) either to Uλ1 −λ2 , or to Uλ1 +λ2 . Since Uλ1 ±λ2 are ideals in so(4), this proves the second assertion of the proposition. t u

Proposition 7.9.5. Every Helgason sphere in (SO(2n), ρ), n ≥ 2, has constant sectional curvature n/2. In particular, for n ≥ 3 all Helgason spheres are distinct from the spheres in Theorem 7.8.2.

Proof. For n = 2 the assertion of the proposition follows from Proposition 7.9.4. Now, let us consider the case n ≥ 3. The subgroup H = diag(SO(4), 1, . . . , 1) ⊂ SO(2n) with a Riemannian bi-invariant metric ρ1 induced by ρ, is a totally geodesic submanifold in (SO(2n), ρ). On the other hand, all roots of h = Lie(H) are roots of so(2n). From the above description of Helgason spheres we get that every Helgason sphere in (H, ρ1 ) is also a Helgason sphere in (SO(2n), ρ). It is easy to see that (H, 2n ρ1 ) is isometric to (SO(4), ρ 0 ), where ρ 0 is a bi-invariant Riemannian metric, generated by the inner product (7.8) for n = 2. Since all Helgason spheres in (SO(4), ρ 0 ) have constant curvature 1 by Proposition 7.9.4, every Helgason sphere in (H, ρ1 ) has constant curvature n/2. t u

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7.10 Lie Algebras in Clifford–Killing Spaces for S2n−1 Here we discuss some results related to Lie algebras contained in Clifford–Killing subspaces CKl of so(2n). Proposition 7.10.1. Let X,Y be linearly independent Killing vector fields on S2n−1 with constant inner products can(X, X), can(Y,Y ), can(X,Y ). Then the Lie bracket [X,Y ] is a non-trivial Killing vector field of constant length on Sn−1 . If X,Y are unit orthogonal Killing vector fields on S2n−1 , then the triple of vector fields 1 {X,Y, Z := [X,Y ]} 2 constitutes an orthonormal basis of the Lie algebra CK3 of vector fields on S2n−1 with relations [X,Y ] = 2Z, [Z, X] = 2Y , [Y, Z] = 2X. Consequently, Assertion 5) of Theorem 7.9.3 holds. Proof. According to Corollary 7.4.3, we get 12 [X,Y ] = ∇X Y = −∇Y X. Then, the formula can(∇X Y, ∇X Y ) = can(R(X,Y )Y, X) holds by Proposition 3.3.5. Since can(X,Y ) and the sectional curvature of (Sn−1 , can) are constant, the expression can(R(X,Y )Y, X) is a positive constant. Consequently, [X,Y ] is a (non-trivial) Killing field of constant length, which proves the first assertion of the proposition. The orthonormality of the triple of Killing vector fields {X,Y, Z} on S2n−1 follows from the orthonormality of the pair {X,Y }, the proof of the first part of Proposition, and the relations 0 = Y can(X, X) = − can([X,Y ], X), 0 = X can(Y,Y ) = can([X,Y ],Y ). The first commutation relation follows from the definition. By Theorem 7.7.1, we get the equality: 1 [Z, X] = {(XY −Y X)X − X(XY −Y X)} 2 1 1 = {XY X −Y XX − XXY + XY X} = (2Y + 2Y ) = 2Y. 2 2 The third commutation relation can be proved analogously. Then the linear span CK3 of vectors X,Y, Z is a Lie algebra. Now, it suffices to apply Theorem 7.9.3. t u Proposition 7.10.2. If l ≥ 4, then the space CKl contains no Lie subalgebra of dimension ≥ 2. Proof. This proposition immediately follows from Theorem 7.9.3. Indeed, in this case the intersection CKl ∩ [CKl ,CKl ] is trivial, while CKl contains no two-dimensional commutative Lie subalgebra, because Sl is a CROSS. t u Corollary 7.10.3. A sphere S2n−1 admits a space CKρ(2n)−1 , which is a Lie algebra, if and only if ρ(2n) = 2 or ρ(2n) = 4, i.e. when n is not a multiple of 4. Proof. If 8 divides 2n, then by Theorem 7.6.4, the dimension of the space CKρ(2n)−1 is more than 4, and by Proposition 7.10.2, this space cannot be a Lie algebra.

7.10 Lie Algebras in Clifford–Killing Spaces for S2n−1

435

If 4 divides 2n, but 8 doesn’t divide 2n, then ρ(2n) − 1 = 3 and by Proposition 7.10.1, one can take the Lie algebra with the basis X,Y, Z, indicated there as a space CK3 . Finally, if 4 doesn’t divide 2n, then ρ(2n) − 1 = 1, and any space CK1 is a Lie algebra. t u Propositions 7.10.1, 7.10.2, and Theorem 7.6.4 immediately imply Corollary 7.10.4. If 8 divides 2n, then there is a space CK3 that is a Lie algebra (isomorphic to su(2)), which is not contained in any space CKρ(2n)−1 . From Propositions 7.10.1 and 7.10.2, we get Corollary 7.10.5. 1) If 4 divides 2n, then there is a space CK3 ⊂ so(2n), which is a Lie algebra, isomorphic to so(3). 2) If 8 divides 2n, then there exists a space CK3 ⊂ so(2n), which is not a Lie algebra. Example 7.10.6. Here we consider two examples of the spaces CK3 , where the first (respectively, the second) one is (respectively, is not) a subalgebra of so(2n). In the Lie algebra so(4), consider the vectors       0 0 0 −1 0 0 −1 0 0 −1 0 0  0 0 −1 0  0 0 0 1 1 0 0 0      U1 =   0 1 0 0  , U2 =  1 0 0 0  , U3 =  0 0 0 −1  . 1 0 0 0 0 −1 0 0 0 0 1 0 It is easy to check that U12 = U22 = U32 = −I, U1U2 = −U3 , U2U1 = U3 , U1U3 = U2 , U3U1 = −U2 , U2U3 = −U1 , U3U2 = U1 . Therefore, the linear span of the vectors Ui , 1 ≤ i ≤ 3, in so(4) is a Lie subalgebra of the type CK3 . Now, consider in so(8) the vectors e1 = diag(U1 , −U1 ), U

e2 = diag(U2 , −U2 ), U

e3 = diag(U3 , −U3 ). U

e2 = U e2 = U e 2 = −I, It is easy to check that U 1 2 3 e1U e2 = diag(−U3 , −U3 ), U e1U e3 = diag(U2 ,U2 ), U e2U e3 = diag(−U1 , −U1 ), U e2U e1 = diag(U3 ,U3 ), U e3U e1 = diag(−U2 , −U2 ), U e3U e2 = diag(U1 ,U1 ). U ei , 1 ≤ i ≤ 3, in so(8) is a subspace of Therefore, the linear span of the vectors U the type CK3 , but is not a Lie subalgebra. It is easy to check that the Lie algebra CK3 + [CK3 ,CK3 ] is isomorphic to so(3) ⊕ so(3).

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7 Clifford–Wolf Homogeneous Riemannian Manifolds

7.11 Killing Vector Fields of Constant Length and G-Clifford–Wolf Homogeneous Metrics on Round Spheres In the previous sections of this book, there are various results, connected with characterizations of Killing vector fields of constant length on Riemannian manifolds. The set of such fields on some classical Riemannian manifolds has a rather complicated structure. Here we shall discuss the corresponding questions for the Euclidean spheres Sn . Recall that even-dimensional spheres do not admit non-trivial Killing fields of constant length, because they have positive Euler characteristic. The presentation in this section is based on the paper [83].

7.11.1 Descriptions of Killing Vector Fields of Constant Length and G-Clifford–Wolf Homogeneous Metrics Note that, by Theorem 7.7.1, a Killing vector field U ∈ so(m) on the round sphere Sm−1 has unit length if and only if U ∈ O(m) ∩ so(m) and U 2 = −I. Here, we consider some generalizations of this result. Recall that there are several connected compact Lie groups acting transitively on spheres (see Examples 4.2.6 and 4.11.4). Apart from the standard representation of Sn as the homogeneous space SO(n + 1)/SO(n), we have also S2n+1 = SU(n + 1)/SU(n),

S2n+1 = U(n + 1)/U(n),

S4n+3 = Sp(n + 1)Sp(1)/Sp(n)Sp(1), S15 = Spin(9)/Spin(7),

S4n+3 = Sp(n + 1)/Sp(n),

S4n+3 = Sp(n + 1)U(1)/Sp(n)U(1),

S7 = Spin(7)/G2 ,

S6 = G2 /SU(3).

We are going to consider unit Killing fields on round spheres, closely related to some of these homogeneous spaces. We shall need a unified notation. By F we mean one of the following three classical fields: R of real numbers, C of complex numbers, and H of quaternions with the usual inclusions R ⊂ C ⊂ H. Since H contains both R and C as subfields, it will sometimes be convenient to call elements of all the mentioned fields quaternions. By Fn we denote an n-dimensional vector space over the field F (right in the case F = H); we understand vectors as column vectors. Now we assume that the vector space Fn+1 is supplied with the standard inner (scalar) product (v, w) = ℜ(vT w), generating the norm k · k. The Lie group of all linear transformations of Fn+1 over F, preserving the inner product (·, ·), is denoted by UF (n + 1). The elements of UF (n + 1) are represented by T ((n + 1) × (n + 1))-matrices A over F with the property AA = AA∗ = Idn+1 , where Idn+1 is the unit diagonal ((n + 1) × (n + 1))-matrix; A ∈ UF (n + 1) acts on Fn+1 in the usual way: if x ∈ Fn+1 , then A(x) = Ax. The Lie algebra uF (n + 1) of the Lie T group UF (n + 1) consists of matrices U satisfying the equation U +U = U +U ∗ =

7.11 Killing Vector Fields of Constant Length and G-Clifford–Wolf Homogeneous . . .

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0. Thus UR (n + 1) = O(n + 1), UC (n + 1) = U(n + 1), UH (n + 1) = Sp(n + 1), and uR (n + 1) = so(n + 1), uC (n + 1) = u(n + 1), uH (n + 1) = sp(n + 1). It is clear that the Lie group UF (n + 1) acts transitively and isometrically on the unit sphere Sd(F)(n+1)−1 = {x ∈ Fn+1 | kxk = 1}, where d(F) is the dimension of F over R. Here and later, elements of the Lie algebra uF (n + 1) are identified with the Killing vector fields on Sd(F)(n+1)−1 . The following proposition is well known (cf. Proposition 7.7.5). Proposition 7.11.1. The following assertions are equivalent: 1) A Killing vector field U ∈ uF (n + 1) has constant length C ≥ 0 on Sd(F)(n+1)−1 . 2) U 2 = −C2 Id. If F = R or F = C then 1) and 2) are equivalent to √ 3) All eigenvalues of U have the form ± C i. Moreover, the following assertions hold: 4) U ∈ uF (n + 1) is a unit Killing vector field on Sd(F)(n+1)−1 if and only if U ∈ uF (n + 1) ∩UF (n + 1).

(7.9)

5) If F = R or U ∈ su(n + 1), then the last condition in 4) is equivalent to U ∈ so(n + 1) ∩ SO(n + 1) or U ∈ su(n + 1) ∩ SU(n + 1) respectively, where n + 1 must be even in both cases. Proof. The value of the Killing vector field U ∈ uF (n + 1) at a point x ∈ Sd(F)(n+1)−1 is U(x) = Ux, and its length at this point is equal to kUxk. It is clear that U has constant length C on Sd(F)(n+1)−1 if and only if kUxk = Ckxk for all x ∈ Fn+1 . This is equivalent to the equalities T

C2 (x, x) = (Ux,Ux) = ℜ(xU Ux) = −(x,U 2 x) = −(U 2 x, x), where U 2 is a self-adjoint (symmetric) linear operator. It is clear that the above equalities are equivalent to the equality U 2 = −C2 Id. So the assertions 1) and 2) are equivalent. 3) We can assume that U is non-zero. Since U is skew-Hermitian, 2) means that the matrix V := √1C U is orthogonal or unitary. Hence, all eigenvalues of V have unit norm. On the other hand, V is also skew-symmetric or skew-Hermitian and all eigenvalues of V should be purely imaginary. Therefore, in these cases 1), 2), and 3) are pairwise equivalent. 4) Assertion 2) for C = 1 has the form Id = −U 2 = UU ∗ (since U ∗ = −U ∈ uF (n + 1)), which is equivalent to (7.9). 5) By assertions 2) and 3), in both these cases U is a unit Killing vector field on Sd(F)(n+1)−1 if and only if all eigenvalues of U have the form ±i, and additionally the sum of all these eigenvalues is equal to zero. This proves all statements in 5). t u

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7 Clifford–Wolf Homogeneous Riemannian Manifolds

Note that the last assertion in Proposition 7.11.1 also shows that non-zero Killing vector fields U ∈ so(n + 1) (respectively U ∈ su(n + 1)) of constant length on Sn (respectively, on S2n+1 ) can exist only for odd n. Thus later we shall consider only odddimensional spheres S2n+1 . Note that the set of Killing vector fields U ∈ so(2(n+1)) of constant length on odd-dimensional spheres S2n+1 is very extensive. Theorem 7.11.2. Any unit sphere Sd(F)(n+1)−1 , n ≥ 1, is UF (n + 1)-Clifford–Wolf homogeneous (in addition, for F = R, n + 1 should be even ). Proof. According to Theorems 4.8.1 and 6.2.9, it suffices to prove that for any tand(F)(n+1)−1 gent vector v ∈ Sx0 , where x0 ∈ Sd(F)(n+1)−1 , there exists a Killing vecd(F)(n+1)−1 tor field K(x) = Ux, x ∈ S , where U ∈ uF (n + 1), of constant length on Sd(F)(n+1)−1 , such that Ux0 = v. Since the Lie group G = UF (n + 1) acts transitively on Sd(F)(n+1)−1 by isometries, one can assume that x0 = (1, 0, . . . , 0)T . Then the Lie group H = UF (n) is the isotropy subgroup of the Lie group G at the point x0 , and one can consider Sd(F)(n+1)−1 as the homogeneous manifold (G/H = UF (n + 1)/UF (n), µ) with a suitable invariant Riemannian metric µ. One gets the Ad(UF (n))-invariant h·, ·i-orthogonal decomposition uF (n + 1) = p ⊕ uF (n) = p1 ⊕ p2 ⊕ uF (n) = p1 ⊕ uF (1) ⊕ uF (n),

(7.10)

0 −uT u 0nn

(7.11)

where  p1 =  p2 =



 T

u1 0n 0n 0nn

,u ∈ F

n

 ,

 , u1 ∈ ℑ(F) .

(7.12)

The lower indices above indicate the size of the zero block-matrices. Note that p2 = {0} and u1 = {0} for F = R. In view of the transitivity of the Lie group UF (n + 1) on Sd(F)(n+1)−1 , the tangent d(F)(n+1)−1 vector space Sx0 of Sd(F)(n+1)−1 at the point x0 is realized in the form Ux0 , U ∈ uF (n + 1). In addition, the correspondence d(F)(n+1)−1

U ∈ p → Ux0 = (u1 , u)T ∈ Sx0

d(F)(n+1)−1

defines an isomorphism of the vector space p onto the vector space Sx0 . Our goal is to find, for every vector-matrix X + Y ∈ p1 ⊕ p2 , a vector-matrix Z ∈ uF (n) of the form    0 0Tn Z= ,Unn ∈ uF (n) , (7.13) 0n Unn such that the Killing vector field K(x) := Ux, x ∈ Sd(F)(n+1)−1 on Sd(F)(n+1)−1 for the matrix U := X + Y + Z has constant length. Below we shall find such fields in different cases, applying Proposition 7.11.1.

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If u 6= 0, we first consider the case 1) u = (u2 , 0, . . . , 0)T ∈ Fn , u2 > 0. a) If F =R, then u1 = 0, and it suffices to take as U a block-diagonal matrix with 0 −u2 the blocks on the diagonal. u2 0 b) If F 6= R, it suffices to take as Unn a diagonal (n × n)-matrix with purely imaginary quaternions in F on the diagonal such that the element in the left upper corner is equal to −u1 , and the squared moduli of the others are equal to u22 − u21 . 2) Let us suppose that u ∈ Fn , |u| > 0. The group UF (n) acts transitively on every sphere in Fn with zero center. Therefore, there exists an element g ∈ UF (n) such that g(|u|, 0, . . . , 0)T = u. In this case, instead of Unn above one needs to take the matrix 0 = Ad(g)(U ), where U is the same as above with u := |u|. Unn nn nn 2 The proof for the case F = R is finished. 3) u = 0, i.e. X = 0. In this case, it suffices to take as Unn (which determines the vector Z by formula (7.13)) a diagonal (n × n)-matrix with arbitrary purely imaginary quaternions in F on the diagonal, whose squared moduli are equal to −u21 . t u The following result completes the previous theorem. Proposition 7.11.3. Any round Euclidean sphere S2n+1 , n ≥ 1, is SU(n+1)-Clifford– Wolf homogeneous for odd n. Every round Euclidean sphere S4n+3 , n ≥ 0, is Sp(n + 1)Sp(1)-Clifford–Wolf homogeneous and Sp(n + 1)U(1)-Clifford–Wolf homogeneous. The round sphere S7 is Spin(7)-Clifford–Wolf homogeneous. Proof. The proposition follows from the inclusions Sp(n + 1) ⊂ SU(2(n + 1)), Sp(n+1) ⊂ Sp(n+1)Sp(1), Sp(n+1) ⊂ Sp(n+1)U(1), Sp(2) ⊂ SU(4) ⊂ Spin(7), and Theorem 7.11.2. Indeed, if a Riemannian manifold is G-Clifford–Wolf homogeneous, then it is obviously G1 -Clifford–Wolf homogeneous for any isometry Lie group G1 such that G ⊂ G1 . t u Note that S6 admits no non-trivial Killing field of constant length. Hence, in order to get a complete picture, it suffices to check S15 = Spin(9)/Spin(7) for the property of being Spin(9)-Clifford–Wolf homogeneous. The affirmative result is obtained in Theorem 7.11.10 of Subsection 7.11.3.

7.11.2 The Spaces of Unit Killing Fields on Spheres Let g be the Lie algebra of a Lie group G acting transitively on some sphere Sn and let UKV F(g, n) be the set of all unit Killing vector fields, lying in g (see e.g. 4) and 5) of Proposition 7.11.1). The set UKV F(g, n), supplied with the induced topology from g, becomes a topological space. It is interesting to find the topological structure of these spaces. Obviously, UKV F(g, n) = 0/ for any g with even n and UKV F(so(2), 1) has exactly two points. In this and in the last sections, we shall study the spaces UKV F(g, n) for spheres Sn with odd n ≥ 3 and some connected transitive Lie groups G on Sn .

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7 Clifford–Wolf Homogeneous Riemannian Manifolds

Proposition 7.11.4. The space O(2n)/U(n) (with the symmetric space SO(2n)/U(n) as a connected component) can be considered as the space UKV F(so(2n), 2n − 1) for n ≥ 1. Proof. It is well known that any complex structure on a finite-dimensional vector space naturally defines some orientation. It follows from 1) and 2) in Proposition 7.11.1 that the set UKV F(so(2n), 2n − 1) can be identified with the set of all complex structures on R2n , compatible with the standard Euclidean structure of R2n (except, possibly, its standard orientation). As it was stated in [106], the symmetric space SO(2n)/U(n) is the space of all complex structures on R2n , compatible with the standard Euclidean structure. Since there are two orientations on any finite dimensional vector space, the last two statements imply the proposition. t u The spheres S2n+1 are Clifford–Wolf homogeneous even with respect to U(n + 1) and the spheres S4n+3 are Clifford–Wolf homogeneous with respect to SU(2(n + 1)) and Sp(n + 1) (see Proposition 7.11.3 and Theorem 7.11.2). We are going to describe the spaces UKV F(u(n + 1), 2n + 1),

UKV F(su(n + 1), 2n + 1),

UKV F(sp(n + 1), 4n + 3).

Let G be one of the groups U(n + 1), SU(2(k + 1)) and Sp(n + 1). Then the Lie group G acts by conjugation on the space L of all unit Killing vector fields from g on corresponding sphere. Therefore, L is a disjoint union of some orbits of Ad(G). Proposition 7.11.5. The K¨ahler symmetric space Sp(n + 1)/U(n + 1) can be interpreted as the space UKV F(sp(n + 1), 4n + 3), where sp(n + 1) ⊂ so(4(n + 1)). Proof. It follows from Proposition 7.11.1 and Example 8.116 in [106] that any matrix U ∈ UKV F(sp(n + 1), 4n + 3) is in an Ad(Sp(n + 1))-orbit of a matrix of the e = diag(i, i, . . . , i). Hence, by the same Example 8.116 in [106], type U
UKV F(sp(n + 1), 4n + 3) = Ad(Sp(n + 1)) diag(i, i, . . . , i) = Sp(n + 1)/U(n + 1). t u Remark 7.11.6. One can give another proof of Proposition 7.11.5 using Proposition 7.11.1 and the fact that Sp(n + 1)/U(n + 1) can be identified with the set of all complex structures in Hn+1 (see e.g. Table 10.125 in [106]). Proposition 7.11.7. The space UKV F(u(n + 1), 2n + 1), u(n + 1) ⊂ so(2(n + 1)) can be naturally identified with the following disjoint union of the complex Grassmannians n+1 [

SU(n + 1)/S(U(l) ×U(n + 1 − l)).

(7.14)

i=0

The complex Grassmannian SU(2(k + 1))/S(U(k + 1) × U(k + 1)) can be interpreted as the space UKV F(su(2(k +1)), 4k +3), where su(2(k +1)) ⊂ so(4(k +1)).

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Proof. The following facts are well known. Any skew-Hermitian complex linear endomorphism of Cn+1 (in other words an element U ∈ u(n + 1)) admits an orthonormal basis b consisting of eigenvectors of U. For any orthonormal basis b in Cn+1 there is a unique element T ∈ U(n + 1) which transfers b to the standard (orthonormal) basis e of Cn+1 ; multiplying, if necessary, a vector of b by −1, one can assume that T ∈ SU(n + 1). This observation and 3) of Proposition 7.11.1 imply that any matrix U ∈ UKV F(u(n + 1), 2n + 1) is in an Ad(SU(n + 1))-orbit of a matrix e = diag(i, . . . , i , −i, . . . , −i ), where l = 0, 1, . . . , n + 1. Then we get the of the type U | {z } | {z } l

n+1−l

first assertion of the proposition, since S(U(l) × U(n + 1 − l)) is the centralizer of e in SU(n + 1). U Now, by the previous argument, any matrix U ∈ UKV F(su(2(k + 1)), 4k + 3) e above with n + 1 = 2(k + 1) and is in an Ad(SU(2(k + 1)))-orbit of the matrix U e l = k + 1 since trace(U) = trace(U) = 0, and we get the second assertion of the proposition. t u Remark 7.11.8. Note that for any given vector v ∈ Sx2n+1 , one can choose a unit 0 Killing vector field in the proof of Theorem 7.11.2 from any principal orbit (i.e. of maximal dimension) among the orbits in (7.14), projecting to v. If n + 1 = 2(k + 1), then a unique principal orbit is exactly the complex Grassmannian from Proposition 7.11.7.

7.11.3 Unit Killing Fields on the Sphere S15 = Spin(9)/Spin(7) In this section, we continue to study the action of Spin(9) on the sphere S15 , which was started in Subsection 6.11.3. In what follows we use the results and the notations from the latter subsection. In general, the exposition is based on the paper [83]. Lemma 7.11.9. An element U ∈ spin(9) ⊂ so(16) is a unit Killing vector field on S15 if and only if it can be represented as a product U = v · w of orthonormal vectors in (R9 , (·, ·)). Proof. By Lemma 6.11.15, U = v · w for orthonormal vectors in (Rn , (·, ·)) if and only if U 2 = −1. We identify U with its image θ (U) ∈ so(16). Since φ : Cl 8 ∼ = R(16) is an isomorphism and (I9 )−1 (spin(9)) ⊂ Cl 8 , the previous equality is equivalent to the equality U 2 = − Id, which in turn by Proposition 7.11.1 is equivalent to the statement that U is a unit Killing vector field on S15 . t u Theorem 7.11.10. The Euclidean sphere S15 is Clifford–Wolf homogeneous with respect to the Lie group Spin(9) ⊂ SO(16). Proof. According to Theorems 4.8.1 and 6.2.9, it suffices to prove that for every tangent vector u ∈ Sx150 there is a Killing vector field U ∈ spin(9) of constant length on S15 such that U(x0 ) = u. We can identify Sx150 with p. Let p : g → p be the corresponding linear projection.

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7 Clifford–Wolf Homogeneous Riemannian Manifolds

For any unit vector v ∈ R8 let v⊥ be the orthogonal complement to v in R8 . Then we get the 7-dimensional linear subspace Vv = {v · u, u ∈ v⊥ } ⊂ spin(8) ⊂ spin(9). Analogously we define the 8-dimensional subspace Ww ⊂ spin(9) for any unit vector w in R9 . It is clear that Vv and Ww consist of simple bivectors. Therefore by Lemma 7.11.9, the linear space Vv consists of Killing vector fields in so(16) of constant length on S15 , in other words, Vv is some 7-dimensional Clifford–Killing space in so(16) [80]. Analogously, Ww is some 8-dimensional Clifford–Killing space in so(16). It is clear that the kernel of the map p : Vv → p2

(7.15)

is zero. Otherwise the intersection h ∩Vv has a non-zero Killing vector field of constant length on S15 , which is impossible. Since p2 and Vv are both 7-dimensional, we get a linear isomorphism p := pv in (7.15). So, for any vector u ∈ p2 and any v ∈ S7 ⊂ R8 there is a Killing vector field K(v, u) ∈ Vv such that p(K(v, u)) = u. In fact, p1 = We9 (see e.g. [377, 451]), so it is itself a Clifford–Killing space in so(16). Now let W ∈ p be any vector. Then W = W1 + W2 , where Wi ∈ pi , i = 1, 2. We can assume that W1 6= 0 and W2 6= 0. Then W1 = e9 · v = w · (−|v|)e9 ∈ Ww , where w = v/|v| for some v ∈ R8 . By the previous consideration, there is a vector Z2 = w · u ∈ Vw ⊂ Ww such that p(Z2 ) = W2 . Then the element W1 + Z2 ∈ Ww ⊂ spin(9) is a Killing vector field of constant length on S15 such that p(W1 + Z2 ) = W . t u Remark 7.11.11. Theorem 7.11.10 was first proved with the help of a computer (see the discussion in [83]), presenting an explicit expression for Killing vector fields from spin(9) of constant length on S15 , projecting to any given tangent vector in Sx150 , under the additional requirement that there is only one nonzero vector component in p2 . This is sufficient, since Spin(7) acts transitively on spheres with zero center in p2 . For vectors in p1 , only the vectors themselves were obtained. It used an explicit expression for the embedding monomorphism θ : spin(9) → so(16), provided by T. Friedrich in his paper [217]. (Another expression for θ is given in [272] in terms of the so-called Clifford cross-section η : S15 → V916 , see pp. 3–4 in [272].) The obtained expressions for Killing vector fields turned out to be rather complicated but verifiable by hand with some difficulties. To get the required Killing vector field U ∈ spin(9) ⊂ so(16) of constant length on S15 one needs to solve the matrix equation U 2 = −s2 Id for a desired skew-symmetric (16 × 16)-matrix U. This gives (14 · 15)/2 + 14 = 119 scalar equations. It is also not difficult to trace on a computer the condition for a skew-symmetric matrix U to be a δ -vector at the point (1, 0, . . . , 0) of any round Euclidean sphere. Remark 7.11.12. Note that the Clifford–Killing space p1 in so(16) has maximal possible dimension 8; also 15 is the minimal dimension for Euclidean spheres having Clifford–Killing spaces of dimension more than 7, see Theorem 7.7.4.

7.11 Killing Vector Fields of Constant Length and G-Clifford–Wolf Homogeneous . . .

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Proposition 7.11.13. The Grassmannian G+ (9, 2) = SO(9)/(SO(2) × SO(7)) of oriented real 2-planes in R9 can be interpreted as the space UKV F(spin(9), 15) of all Killing vector fields of unit length on S15 , lying in the Lie algebra spin(9) ⊂ so(16). Proof. It follows from Lemmas 6.11.15 and 7.11.9 that an element U ∈ spin(9) ⊂ so(16) gives a unit Killing vector field on S15 if and only if it can be presented in the form U = v · w, where v, w are orthonormal vectors in (R9 , (·, ·)). Now one can check easily, directly or using Proposition 6.11.12, that two products v1 · w1 and v2 · w2 for pairs of orthonormal vectors in (R9 , (·, ·)) coincide if and only if these pairs define one and the same oriented 2-plane in R9 . This finishes the proof of the proposition. t u Now we want to describe the space of unit Killing vector fields on S15 , lying in the Lie algebra spin(9), whose image under above projection p : spin(9) → p is situated in p2 (respectively, p1 ). Proposition 7.11.14. The Grassmannian G+ (8, 2) = SO(8)/(SO(2) × SO(6)) of oriented real 2-planes in R8 can be interpreted as the space of all Killing vector fields of unit length on S15 , lying in the Lie subalgebra spin(8) ⊂ spin(9) ⊂ so(16). This can also be considered as the space of all unit Killing vector fields on S15 , lying in spin(9) and projecting under p into p2 . Proof. The first statement is proved in the same way as Proposition 7.11.13. The second statement follows from the first statement and the relations p(spin(8)) = p2 , p−1 (p2 ) ⊂ spin(8). t u The following proposition gives a scheme to search all (unit) Killing vector fields on S15 , lying in spin(9) and projecting into p2 (which actually always lie in spin(8) by Proposition 7.11.14), applied in the proof of Theorem 7.11.10. Proposition 7.11.15. Let S7 and S6 be the unit spheres respectively in (R8 , (·, ·)) and (p2 , 12 h·, ·i|p2 ). Then there is the following sequence of real-analytic maps S7 × S6

(Id ◦p1 )×K

−→

q

incl

p

V28 −→ G+ (8, 2) −→ spin(8) −→ S6 .

(7.16)

Here V28 = SO(8)/SO(6) is the homogeneous Stiefel manifold consisting of all orthonormal 2-frames in R8 [268], q is the canonical projection, incl is a natural inclusion map given by Proposition 7.11.14, and K is the map from the proof of Theorem 7.11.10. The first map (Id ◦p1 ) × K associates the pair (v, K(v, u)) ∈ V28 to a pair (v, u) of unit vectors in S7 × S6 ; it is a diffeomorphism. Moreover, for any point (v, u) ∈ S7 × S6 , incl(q(((Id ◦p1 ) × K)(v, u))) is a unit Killing vector field on S15 , lying in spin(8), which is projected to u by p. Any unit Killing vector field on S15 , lying in spin(9) and projecting to u ∈ S6 , has this form. Proof. It is clear that there exists the inverse map f to (Id ◦p1 ) × K, which is defined by the formula f (v, w) = (v, p(incl(q(v, w)))). Obviously, this map is real-analytic.

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7 Clifford–Wolf Homogeneous Riemannian Manifolds

It is enough to prove that f is a diffeomorphism. First we define another diffeomorphism F : V28 → S7 × S6 . As a corollary of classical results of Hurwitz–Radon, there is a 7-dimensional Clifford–Killing space CK7 on S7 (see Theorems 7.6.2 and 7.7.4) with some orthonormal basis {Y1 , . . . ,Y7 }. For any pair (v, w) ∈ V28 , w is a tangent vector to S7 at the point v ∈ S7 . So it can be presented in the form w = s1Y1 (v) + · · · + s7Y7 (v), where s21 + · · · + s27 = 1, thus we can identify S = (s1 , . . . , s7 ) with a point in S6 . By the definition, F(v, w) = (v, S) ∈ S7 × S6 . It is clear that F is a diffeomorphism. Now we see that the first component p1 ( f (F −1 (v, S))) = v of f ◦ F −1 is identical to v for a fixed S, while its second component p2 ( f (F −1 (v, S))) = p2 ( f (v, w)) isometrically depends on w ∈ v⊥ (hence, on S) for a fixed v, because v · w remains in the fixed Clifford–Killing space Vv ⊂ spin(8), while the map (7.15) is non-degenerate linear. Hence, the differential D( f ◦ F −1 ) of f ◦ F −1 is non-degenerate at any point (v, S) ∈ S7 × S9 . Therefore, the differential D f of f is non-degenerate at any point (v, w) ∈ V28 . By the inverse function theorem, the maps f and (Id ◦p1 ) × K are mutually inverse real-analytic diffeomorphisms. The last assertion follows from Proposition 7.11.14. The statement before it follows from the fact that f and (Id ◦p1 ) × K are mutually inverse maps. t u We get an analogous proposition, presenting the scheme to search for (unit) Killing vector fields on S15 , lying in spin(9) and projecting into p \ (p1 ∪ p2 ), similarly to the second part in the proof of Theorem 7.11.10. Let S14 and S17 be the unit spheres in (p, (·, ·)1 ) and (p1 , 81 h·, ·i|p1 ) respectively, S6 is the same as in Proposition 7.11.15. It is known that S14 is the join S17 ∗S6 [220]. This means that S14√ is the image of a continuous map J : S17 × S6 × [0, 1] → S14 , where J(x, y, s) = sx + 1 − s2 y. In addition, J is a real-analytic homeomorphism of S17 × S6 × (0, 1) onto S14 \ (S17 ∪ S6 ). Proposition 7.11.16. Let S7 be the same as in Proposition 7.11.15. Then there is the following sequence of real-analytic maps J −1

g×Id × Id

I×Id

S14 \ (S17 ∪ S6 ) −→ S17 × S6 × (0, 1) −→ S7 × S6 × (0, 1) −→ Id ×J

Q

incl

p

V28 × (0, 1) −→ V29 −→ G+ (9, 2) −→ spin(9) −→ S14 \ (S17 ∪ S6 ). Here g maps an element v · e9 ∈ S17 to v ∈ S7 , I = (Id ◦p1 ) × X, (Id ×J)(v, w, s) := (v, J(e9 , w, s)), Q is the canonical projection, incl is a natural inclusion map given by Proposition 7.11.13. Moreover, the composition f of all maps but the last one (in the above diagram), applied to any vector u ∈ S14 \ (S17 ∪ S6 ), gives some unit Killing vector field on S15 , lying in spin(9), which is projected under p to the vector u. Proof. This proposition easily follows from Proposition 7.11.15.

t u

Proposition 7.11.17. The image A of the set S14 \ (S17 ∪ S6 ) in G+ (9, 2) under the map f from Proposition 7.11.16 is open and connected in G+ (9, 2). Its closure is equal to G+ (9, 2) and its boundary consists of two disjoint connected components, G+ (8, 2) and p1 . A nonzero vector u ∈ p is a projection under p of a unique Killing vector field of constant length on S15 , lying in spin(9), if and only if u ∈ / p2 .

7.11 Killing Vector Fields of Constant Length and G-Clifford–Wolf Homogeneous . . .

445

Proof. The dimensions of S14 and G+ (9, 2) = SO(9)/(SO(2) × SO(7)) are both equal to 14. The composition of all maps in the diagram from Proposition 7.11.16 is identical on the open subset S14 \ (S17 ∪ S6 ) ⊂ S14 ; all maps in Proposition 7.11.16 are real-analytic. Then the set A is open in G+ (9, 2). The boundary of the set A in G+ (9, 2) consists of two closed connected components. The first, which we denote by C, is p1 , of dimension 8. Another one, which we denote by D, consists of unit Killing fields on S15 , projecting to S6 ⊂ p2 . Therefore, by Propositions 7.11.15 and 7.11.14, D ⊂ G+ (8, 2). Thus the topological dimension of D is no more than the dimension of G+ (8, 2) = SO(8)/(SO(2) × SO(6)), which is equal to 12. So C ∪ D cannot divide G+ (9, 2). Let us suppose that the closure of A in G+ (9, 2) is not equal to G+ (9, 2). Then there is a point X, lying in the nonempty open subset G+ (9, 2)\(A∪C ∪D). Since C ∪D does not divide G+ (9, 2), and G+ (9, 2) is arcwise connected, there is an arc in the open subset G+ (9, 2) \ (C ∪ D), joining the point X with a given point Y ∈ A. But this is impossible, because A is connected and open in G+ (9, 2), while A ∪C ∪ D is closed in G+ (9, 2). Therefore, we get D = G+ (8, 2). The last statement follows from the previous ones. t u Remark 7.11.18. Since π2 (SO(9)/(SO(2) × SO(7))) = π1 (SO(2)) = Z, the Grassmannian G+ (9, 2) is not homeomorphic to S14 , see the book [220] by Rokhlin and Fuks. Corollary 7.11.19. The space of all unit Killing vector fields on S15 , lying in the Lie algebra spin(9) ⊂ so(16) and projecting under p into p1 , is the space p1 itself. In addition, p is identical on p1 . Corollary 7.11.20. Proposition 7.11.16 gives all unit Killing vector fields on S15 from spin(9), projecting to p \ (p1 ∪ p2 ). Thus Propositions 7.11.15, 7.11.16, and Corollary 7.11.19 altogether present the way to get all unit Killing vector fields on S15 from spin(9), projecting to any given point in S14 ⊂ p. It follows from Propositions 7.11.4, 7.11.5, 7.11.7, and 7.11.13 that the symmetric spaces O(2n)/U(n), Sp(n + 1)/U(n + 1), SU(2(n + 1))/S(U(n + 1) ×U(n + 1)), and G+ (9, 2) can be considered as real-analytic closed submanifolds in so(2n), sp(n + 1), su(2(n + 1)), and spin(9) ⊂ so(16), respectively. Clearly, they do not intersect the corresponding isotropy Lie subalgebras so(2n + 1), sp(n), s(u(n + 1) ⊕ u(n + 1)), and spin(7). Similar statements hold for each individual orbit in (7.14). Moreover, by Theorems 7.11.2 and 7.11.10 together with Proposition 7.11.3, the natural linear projections map these symmetric spaces, or their union (7.14), respectively onto the unit spheres S2(n−1) , S4n+2 , S4n+2 , S14 , and S2n in the tangent spaces to S2n−1 , S4n+3 , S4n+3 , S15 , and S2n+1 at their initial points. Indeed, it is possible to reconstruct the search for all required unit Killing vector fields on S15 from G+ (9, 2), and the space G+ (9, 2) itself, knowing only the map f from Proposition 7.11.16. There is a unique continuous extension of f to S14 \ S6 , which we denote by f , but there is no such extension to all S14 . Consider the continuous map φ := f ◦ J : S17 × S6 × (0, 1] → G+ (9, 2).

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7 Clifford–Wolf Homogeneous Riemannian Manifolds

It has a unique continuous extension to all S17 × S6 × [0, 1], which we also denote by φ . Now the algorithm to search for all required unit Killing vector fields on S15 is expressed by the equalities φ (S17 × S6 × [0, 1]) = G+ (9, 2);

φ (S17 × S6 × {0}) = G+ (8, 2);

p ◦ φ = J.

Since φ is a surjective, continuous, and closed map, we also get the space G+ (9, 2) as the quotient space of S17 × S6 × [0, 1] with respect to φ . Let us take s = 21 and O := J(S17 × S6 × [0, s)), which is an open tubular (π/4)neighborhood of S6 in S14 . Then the formula ω(J(v, w,t)) = φ (v, w, 2t − 1),

1/2 ≤ t ≤ 1,

defines a surjective, continuous, and closed map of the complement CO for the neighborhood O of S6 in S14 onto G+ (9, 2). So G+ (9, 2) is the quotient space of CO, and ω is a homeomorphism on CO \ B, where B is the common boundary of O and CO. Change ω|B by the surjective continuous map q ◦ F −1 ◦ h : B → G+ (8, 2), where h is the canonical homeomorphism of B onto S17 × S6 , and q and F are taken from Proposition 7.11.15 and its proof. Then the space G+ (9, 2) is a result of gluing CO with G+ (8, 2) by the map q ◦ F −1 ◦ h. One can prove that there exists some realanalytic map c : G+ (8, 2) → S6 such that c ◦ q ◦ F −1 = p2 ,

where

p2 : S 7 × S 6 → S 6

(7.17)

is the projection to the second factor. In addition, if r : S6 → RP6 is the canonical projection, then r ◦ c : G+ (8, 2) → RP6 is a real-analytic fibration with the fiber CP3 . Unlike (7.17) now we only have the more complicated (although analogous) formula (p ◦ ω)(J(v, w, s)) = A(v) ◦ p2 , where A(v) ∈ SO(7), for the restriction p : G+ (8, 2) → S6 of the above linear projection p : spin(8) → p2 . We don’t know whether the real-analytic map r ◦ p : G+ (8, 2) → RP6 is a fibration with the fiber CP3 .

7.12 Restrictively Clifford–Wolf Homogeneous Riemannian Manifolds In this section we classify all restrictively Clifford–Wolf homogeneous Riemannian manifolds. The results of this section were obtained in [81]. With the goal of locally using in this section, let us state the following corollary of Theorems 6.2.9, 6.3.6, 7.3.2, and 7.1.5.

7.12 Restrictively Clifford–Wolf Homogeneous Riemannian Manifolds

447

Theorem 7.12.1. A Riemannian manifold (M, g) is restrictively Clifford–Wolf homogeneous if and only if it is metrically complete and every geodesic γ in (M, g) is an integral curve of some Killing vector field of constant length on (M, g). Any Riemannian covering of a restrictively Clifford–Wolf homogeneous Riemannian manifold is itself restrictively Clifford–Wolf homogeneous. The universal Riemannian covering of a restrictively Clifford–Wolf homogeneous Riemannian manifold is symmetric and strongly Clifford–Wolf homogeneous. J. A. Wolf obtained in [476] the following result. Theorem 7.12.2. Let Γ be the sliding group of the universal Riemannian covering e → M of a complete connected locally symmetric Riemannian manifold M. p:M e is a symmetric space, and M is homogeneous if and only if Γ is a group of Then M e Clifford–Wolf translations on M. e and the Notice that M and Γ are naturally identified with the quotient space Γ \M fundamental group π1 (M) respectively. J. A. Wolf obtained in [476] (see also [483]) e the following refinement of Theorem 7.12.2 for special spaces M. e is a compact simple simply connected Lie group with a biTheorem 7.12.3. If M e is homogeneous if and only if Γ is invariant Riemannian metric, then M = Γ \M e conjugate in the full isometry group of M to a group of left shifts by elements of e and M is symmetric if and only if B is a central some discrete subgroup B in M, e subgroup in M. Proposition 7.12.4. A direct metric product M = M1 × · · · × Mk of indecomposable Riemannian manifolds is homogeneous and locally symmetric if and only if any of its factors is homogeneous and locally symmetric. In addition, a vector field on M is a Killing vector field of constant length on M if and only if it is a product of Killing vector fields of constant length on factors; M is restrictively Clifford–Wolf homogeneous if and only if every factor satisfies this property. Proof. The sufficiency in the first statement is evident. Suppose that M is homogeneous and locally symmetric. Corollary 8.10 of Chapter VII in [291] states that any totally geodesic submanifold of a homogeneous Riemannian manifold M is itself homogeneous. In particular, any factor Mi , i = 1, . . . , k, is homogeneous. Since locally symmetric Riemannian manifolds are characterized by the equality ∇R = 0, where ∇ is the Levi-Civita connection and R is the curvature tensor [256], then any fi , factor Mi is locally symmetric. Then we get by Theorem 7.12.2 that Mi = Γi \M f where Γi is a group of Clifford–Wolf translations on Mi , i = 1, . . . , k. The sufficiency in the second statement is also evident. Assume that X is a Killing vector field of constant length on M. Theorem 4.8.8 implies that X is p-connected e where with a unique Γ -invariant Killing vector field Y of constant length on M, e → Γ \M f1 × · · · × M fk = M, p:M

Γ = Γ1 × · · · × Γk ,

e is symmetric, since it is simply connected and is the natural projection. Now M locally symmetric (see [256]). Then, according to Theorem 4.8.1, the 1-parameter

448

7 Clifford–Wolf Homogeneous Riemannian Manifolds

motion group γe(t), t ∈ R, generated by the Killing vector field Y of constant length e consists of Clifford–Wolf translations on M. e According to Corollary 3.1.4 in on M, e [476], any Clifford–Wolf translation f on M is uniquely presented in the form f = fj . Then γe(t) = ( f1 , . . . , fk ) where f j , j = 1, . . . , k, are Clifford–Wolf translations on M fi , i = 1, ..., k. This γe1 (t) × · · · × γek (t), where γei (t) are Clifford–Wolf translations on M observation, Proposition 3.4.5, and Theorem 4.8.8 imply that we get a direct metric product Y = Y1 × · · · × Yk of Γi -invariant Killing vector fields of constant length on fi , i = 1, . . . , k, that are pi -connected with the Killing vector fields Xi of the factors M fi → Γi \M fi , and constant length on Mi with respect to the natural projections pi : M X = X1 × · · · × Xk . The last assertion of the proposition follows from the previous two assertions and Theorem 7.12.1. t u Proposition 7.12.5. For any compact homogeneous locally symmetric Riemannian manifold M, there exists a finite-sheeted Riemannian covering mapping p f : M → M f on a compact homogeneous locally symmetric Riemannian manifold M f , which has the form of the direct metric product M f = T m × M1 × · · · × Mk , where T m is fi , i = 1, . . . , k, with some finite groups Γi of a (locally Euclidean) torus, Mi = Γi \M Clifford–Wolf translations on simply connected compact indecomposable Riemanfi . There is a one-to-one correspondence between the set nian symmetric manifolds M of all Killing vector fields of constant length on M and the set of all Killing vector fields of constant length on M f such that each vector field X on M corresponds to a unique vector field Z on M f with the property d(p f )(X) = Z, and conversely. Moreover, a vector field Z on M f is a Killing vector field of constant length if and only if it is a direct metric product Z0 × Z1 × . . . Zk of Killing vector fields of constant length on T m and M j , j = 1, . . . k,. e0 × M e+ ) with a group Γ of Clifford–Wolf Proof. By Theorem 7.12.2, M = Γ \(M e where M e0 = translations of some simply connected Riemannian symmetric space M, e+ = M f1 × · · · × M fk is a direct metric product with indecomposable comEm and M pact simply connected Riemannian symmetric factors. According to Corollary 3.1.4 from [476], any element f ∈ Γ is uniquely presented in a form f = ( f0 , f1 , . . . , fk ), fj . It is clear that the where f j , j = 0, 1, . . . , k, are Clifford–Wolf translations on M correspondence f ∈ Γ → f j , j = 0, 1, . . . , k, defines a homomorphism of the group fj , and Γ ⊂ Γ0 ×Γ1 × · · · ×Γk . Γ onto the group Γj of Clifford–Wolf translations on M The last inclusion defines a finite-sheeted Riemannian covering mapping f1 × · · · × Γk \M fk := T m × M1 × · · · × Mk . p f : M → M f := Γ0 \E m × Γ1 \M e is a Killing vector field Furthermore, By Proposition 7.12.4, a vector field Y on M of constant length if and only if it is a direct product Y0 × Y1 × · · · × Yk of Killing e j , j = 0, 1, . . . , k. This observation implies that a vector fields of constant length on M e is invariant with respect to all elements Killing vector field Y of constant length on M of Γ if and only if Y is invariant with respect to all elements in Γ0 × Γ1 × · · · × Γk . Thus, with the help of Theorem 4.8.8, we get the second statement of the proposition. The last statement follows from Proposition 7.12.4. t u

7.12 Restrictively Clifford–Wolf Homogeneous Riemannian Manifolds

449

Theorem 7.12.6. A Riemannian manifold M is restrictively Clifford–Wolf homogee1 × · · · × M ek ), where Γ is a neous if and only if it is isometric to a space Γ \(Em × M subgroup of a group Γ0 ×Γ1 × · · · ×Γk , Γ0 is a discrete group of parallel translations ei is either a round sphere of odd dimension di , and Γi is a subgroup of on Em , M SO(di + 1) of the form 1), or 2), or 3) from Theorem 4.8.18; or a compact simply connected simple Lie group with a bi-invariant Riemannian metric, and Γi is a finite ei . group of left translations on M Proof. Let us prove the necessity. If M is restrictively CW-homogeneous then by e is CW-homogeneous. It follows Theorem 7.12.1, its universal covering space M e1 × from Theorems 7.0.1 and 7.12.2, that M is isometric to some space Γ \(E m × M e e · · · × Mk ), where the spaces Mi are indicated in the statement of Theorem 7.0.1, e It is possible to find and Γ is a discrete group of Clifford–Wolf translations on M. ei as in Proposition the groups Γj , j = 0, 1, . . . , k, of Clifford–Wolf translations on M 7.12.5. Then Γ ⊂ Γ0 × Γ1 × · · · × Γk , while Theorem 7.0.1 and Proposition 7.12.5 imply that M is restrictively Clifford–Wolf homogeneous if and only if the space f1 × · · · × Γk \M fk is restrictively Clifford–Wolf homogeneous. M f := Γ0 \E m × Γ1 \M In turn, Proposition 7.12.4 implies that this is equivalent to the statement that all f1 , . . . ,Γk \M fk are restrictively CW-homogeneous. This means spaces Γ0 \E m , Γ1 \M that Γ0 is a discrete group of parallel translations on Em , and Γ0 \Em is even CWhomogeneous. ei is a compact simply connected simple Lie group with a bi-invariant RieIf M mannian metric, then by Theorem 7.12.3, we may suppose that Γi is a finite group ei . It is clear that for any such group Γi , every left-invariant vector of left shifts on M e is a Killing vector field of constant length, and Γi \M e is restrictively field Y on M Clifford–Wolf homogeneous according to Theorems 4.8.8 and 7.12.1. ei is a round odd-dimensional sphere, and Mi = Γi \M fi is Further we assume that M one of the spaces described in Theorem 4.8.18. In the first two cases of Theorem 4.8.18, the space Mi is a round sphere or the real projective space of the odd dimension di . Consequently, it is symmetric and restrictively CW-homogeneous by Theorems 7.0.1, 4.8.8, and 7.12.1. ei ⊂ Cni , the multiIn the third case of Theorem 4.8.18, for the round sphere M plication by {diag(e2πi(1/q) , ..., e2πi(1/q) )} ⊂ U(ni ) generates a group Γi , as well as the standard complex algebra C of R-linear operators on R2ni = Cni . That is, the centralizer of Γi in O(2ni ) coincides with U(ni ) (see, for example, Examples 22 and ei is restrictively CW-homogeneous 23 in Section 1.2 of the book [377]). Thus Γi \M by Theorem 7.12.1 and Theorem 6.2.9. In the fourth case it is clear that the centralizer of Γi in SO(di + 1) contains a group Sp(li ) (acting on column vectors by left multiplication). On the other hand, any binary dihedral or binary polyhedral group is not contained in the complex field of the real quaternion algebra H by Theorem 2.6.7 in [483]. This means that a minimal algebra over R, containing all elements of such a group, is the algebra H. Now, using again Examples 22 and 23 in Section 1.2 of the book [377], we get that the centralizer of Γi in SO(di + 1) is the group Sp(li ). The case li = 2 actually fi is the compact simply connected simple Lie corresponds to the situation when M group SU(2) = Sp(1); it was considered above in a more general context.

450

7 Clifford–Wolf Homogeneous Riemannian Manifolds

Notice that the above argument also contains the proof of the sufficiency in the theorem. t u e be a round unit sphere S2n−1 of dimension 2n − 1 ≥ 3, Proposition 7.12.7. Let M n e M ⊂ C , d + 1 = 2n, and Γ = {diag(e2πi(p/q) , ..., e2πi(p/q) )} ⊂ U(n), 0 ≤ p < q e is with p an integer and q ≥ 3 a given natural number. Then the space M = Γ \M restrictively Clifford–Wolf homogeneous but is not Clifford–Wolf homogeneous. Proof. The first statement was proved in Theorem 7.12.6. Let us prove the second one. The group Γ is contained in the group C := {diag(e2πit , ..., e2πit) ) |t ∈ R} ⊂ U(n). The inclusions {e} ⊂ Γ ⊂ C induce the Riemannian submersions p1 : S2n−1 → M, p2 : M → CPn , and their composition p = p2 ◦ p1 : S2n−1 → CPn , which is the Hopf fibration. The injectivity radius and the diameter of the space CPn are equal to π/2. Consider an arbitrary number r, where π/3 < r < π/2. Since p2 is the Riemannian submersion, there exist two points x1 , x2 ∈ M such that ρM (x1 , x2 ) = r. We state that there is no CW-translation f on M such that f (x1 ) = x2 . In fact, suppose that such translation f exists. Then f is induced by an isometry fe of S2n−1 normalizing Γ (see the proof of Theorem 2.4.17 in [483]). It is obvious that fe normalizes the group C. Consequently, fe and f induce an isometry g of CPn (assuming that p2 ( f (x)) = g(y) for all points y ∈ CPn and x ∈ p−1 2 (y)). Hence ρCPn (y, g(y)) ≤ r for every point y ∈ CPn and there is a unique smooth vector field X on CPn such that ExpCPn (X(y)) = g(y) for all points y ∈ CPn because r < π/2. Since χ(CPn ) > 0, there exists a point −1 z ∈ CPn such that X(z) = 0. This means that f (x) ⊂ p−1 2 (z) for any point x ∈ p2 (z). Now it is clear that the equality ρM (x, f (x)) = r is impossible because the Γ -orbit of the point x is π/q-dense on the circle p−1 (z) ⊂ S2n−1 and π/q ≤ π/3 < r. This contradiction proves the proposition. t u Remark 7.12.8. In the case n = 2 the previous proposition gives an example of a restrictively Clifford–Wolf homogeneous but not Clifford–Wolf homogeneous quotient space of the Lie group SU(2)/D, where C := {γ(t),t ∈ [0, 2π)} is a maximal torus in SU(2) and D := {γ(2pπ/q)} with natural q ≥ 3 and integer p such that 0 ≤ p < q. e with biProposition 7.12.9. For any compact simply connected simple Lie group M e such invariant Riemannian metric, there is a finite commutative subgroup F ⊂ M e that M = M/F is not Clifford–Wolf homogeneous, but is restrictively Clifford–Wolf homogeneous. e r be a positive real number, which Proof. Let T be an arbitrary maximal torus in M, e e is less than half of the injectivity radius of M, and the quotient space N = M/T is supplied with the quotient metric. One can take any r/2-dense finite subgroup in e normalizing the T as F. Then it is possible to show that any element in Isom(M), e group Γ of right shifts of the space M by elements from F, also normalizes the group

7.12 Restrictively Clifford–Wolf Homogeneous Riemannian Manifolds

451

C of right shifts by elements from T . As a corollary, using the proof of Theorem e 2.7.13 in [483], one can see that any isometry of M is induced by an isometry of M, which normalizes Γ , and, consequently, C. Now, consider the natural Riemannian submersion p : M → N. Note that N has positive Euler characteristic by Theorem 4.3.1. Now, as in the the proof of Proposition 7.12.7, one can show that there are two elements x1 , x2 ∈ M with the condition ρM (x1 , x2 ) = 2r, such that there is no Clifford-Wolf translation on M moving x1 to x2 . t u

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List of Tables

2.1 2.2

Simple Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 Properties of simple Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

4.1 4.2

Transitive actions on spheres and invariant metrics . . . . . . . . . . . . . . . . 211 The pairs (g, h) defining generalized Wallach spaces G/H with simple G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 δ -pinchings of the pair for the Wallach spaces . . . . . . . . . . . . . . . . . . . 236 δ -pinchings of pairs for odd-dimensional manifolds . . . . . . . . . . . . . . . 237

4.3 4.4 5.1 5.2

Invariant and geodesic orbit Riemannian metrics on spheres . . . . . . . . 300 Riemannian GO-spaces G/K fibered over irreducible symmetric spaces G/H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321

6.1 6.2

Normal homogeneous metrics on spheres . . . . . . . . . . . . . . . . . . . . . . . 368 Generalized normal homogeneous metrics on spheres . . . . . . . . . . . . . 369

© Springer Nature Switzerland AG 2020 V. Berestovskii and Y. Nikonorov, Riemannian Manifolds and Homogeneous Geodesics, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-56658-6

473

Symbol Index

AX Y , 66 Ad, 82 Ad(g), 81 ad(X), 82 Aut(g), 90 B(x, r), xiv B(X,Y ), 89 C, 50 cg (k), 88 C(g), 87 C∞ (M), 3 Cl n , 378 Conj, 50 Cx , 48 ∆ = ∆ (g), 106 δ (M, g), 71 Der(g), 90, 91 d f , 10 diam(M), 65 d(x, y), 36 Exp, 33 exp, 79 Expx , 33 F = F (D), 14 F ∞ (D), 14 F ∞ ( f ), 28

F ∞ (M), 8 Γi kj , 20 gk , 88 gl , 88 GL+ (n, R), 80 gl(n, R), 80 gl(V ), 80 GL(V ), 80 I(g), 81 Isom0 (M, g), 167 Isom(M, g), 167 J, 34 K(σ ), 25 (2) Kθ (K1 , K2 ), 365, 366 (p)

Kθ (K1 , K2 ), 364 L0 , 52 l0 (x), 52 LG, 76 lg , 75 LX , 16 M (n, D), 73 M(n, R), 80 M¨ob(n), 243 Mp , 3

© Springer Nature Switzerland AG 2020 V. Berestovskii and Y. Nikonorov, Riemannian Manifolds and Homogeneous Geodesics, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-56658-6

475

476

∇, 17 ∇X A, 17 ng (k), 88 nilrad, 89 O(n), 84 Ox , 87 R, 21 rad, 89 Radinj, 34 Radinj(M), 52 RG, 76 rg , 75 Ric, 24 ric, 25 rk G, 104 rk g, 104 r(M), 57 r(x), 57 S, 72 s, 50 s0 , 52

Symbol Index

s0 (x), 52 S1 M, 50 sc, 25 SL(n, R), 84 Sn , 2 SO(n), 84 Su M, 154 S(x, r), xiv T , 17 t, 103 θi , 106 T M, 7 T m , 98 Tφ, 8 U (X,Y ), 124, 229 U(x, r), xiv U(X,Y ), 91, 229 W , 106 [X,Y ], 11

Subject Index

A action completely reducible, 205 action of a Lie group, 86 almost effective, 86 almost free, 86 effective, 86 free, 86 left, 86 right, 86 simply transitive, 87 transitive, 87 adjoint representation of a Lie algebra, 82 of a Lie group, 82 Aloff–Wallach space, 220–221, 235 arc-wise connected subgroup, 84 atlas, 1 smooth, 1 automorphism of a Lie algebra, 90 B bilinear symmetric form invariant, 97 C canonical connection, 185 of the first kind, 185 of the second kind, 185

Cartan decomposition, 189 Cartan subalgebra, 103 center of a Lie algebra, 87 central symmetry, 188 centralizer of a Lie subalgebra, 88 chart, 1 Chebyshev norm, 337, 384–386, 389, 391–408 Christoffel symbol, 20 Clifford algebra, 378, 424 representation, 425 Clifford–Killing space, 424–435 Clifford module, 425 Clifford translation, see Clifford–Wolf translation Clifford–Wolf homogeneity radius, 330 Clifford–Wolf translation, 128–132, 329 commutator of vector fields, 11 components of a vector in a chart, 7 conjugate point, 40 first, 40 conjugate set of a point, 50 conjugate set of a Riemannian manifold, 50 conjugate value, 40 first, 40

© Springer Nature Switzerland AG 2020 V. Berestovskii and Y. Nikonorov, Riemannian Manifolds and Homogeneous Geodesics, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-56658-6

477

478

convexity radius, 57 convex subset, 56 coordinate system Riemannian normal, 34 covariant derivative (= linear connection), 17 covering, 62 covering manifold, 62 regular, 62 universal, 62 covering mapping, 62 curvature tensor, 21 of a linear connection, 21 of a Riemannian manifold, 21 covariant, 23 cusp of a hyperbolic manifold, 161 cut locus, 48 cut point, 48 cut set of a Riemannian manifold, 50 D Damek–Ricci space, 427 D’Atri space, 271 deck transformation, 62 δ -homogeneity radius, 330 δ -pinching, 71, 234, 354 of a pair, 236, 237 δ -vector, 346–351, 353, 356, 359–361, 409 δ (x)-translation, 329 derivation of a Lie algebra, 90 diffeomorphism, 2 differential of a function, 10 differential of a map, 8 differential structure, 2 discrete subgroup of a Lie group, 156–164 distribution, 14 completely integrable, 14 involutive, 14 dual 2-means, 364 dual p-means, 364 Dynkin diagram, 110

Subject Index

E Einstein manifold, 25 end of a manifold, 163 Euler characteristics, 177 exponential map of a Lie group, 79 of a Riemannian manifold, 33 F Finsler manifold homogeneous compact, 385–390 first variation of arc length, 29 flag manifold, 216, 280 flow of a vector field, 12 foliation, 14 full connected isometry group of a Riemannian manifold, 167 full isometry group of a Riemannian manifold, 167 fundamental group, 62 G gauge group, 173 G-Clifford–Wolf homogeneity radius, 330 G-δ -homogeneity radius, 330 generalized Heisenberg group, 280, 427 generalized Wallach space, 221–224 geodesic, 30 closed, 188 homogeneous, 257 geodesic flow, 32 geodesic graph, 275 geodesic loop, 188 geodesic orbit manifold, 270, 333 geodesic orbit metric, 271 geodesic orbit space, 271 proper, 306–310 geodesic variation, 30 geodesic vector, 258, 350, 358, 359, 410 geodesic vector field, 32

Subject Index

GO-manifold, see geodesic orbit manifold GO-metric, see geodesic orbit metric GO-space, see geodesic orbit space group of deck transformations, 62 H Helgason sphere, 432, 433 homogeneous G-space, 171 homogeneous Riemannian space generalized G-normal, 330 G-normal, 330 naturally reductive, 276 restrictively Clifford–Wolf, 330 restrictively G-Clifford–Wolf, 330 strongly Clifford–Wolf, 414 homogeneous space, 171 Finsler, 209 isotropy irreducible, 208 non-reductive, 176 reductive, 174–176 strongly isotropy irreducible, 208 I ideal of a Lie algebra, 87 immersion, 8 ineffective kernel, 86 infinitesimal translation, 125 injectivity radius, 34, 51, 52 integral manifold, 14 intrinsic metric, 38 of a Riemannian manifold, 36 intrinsic metric space generalized G-normal homogeneous, 330 G-normal homogeneous, 330 restrictively Clifford–Wolf homogeneous, 330 restrictively δ -homogeneous, 330 restrictively G-Clifford–Wolf homogeneous, 330 restrictively G-δ -homogeneous, 330 strongly Clifford–Wolf homogeneous, 414

479

invariant affine connection on a reductive homogeneous space, 184–186 involutive automorphism, 189 isometric action of S1 , 144 almost free, 131 free, 131 isometry of a Riemannian manifold, 118 isometry group of a Riemannian manifold, 167 isomorphic modules, 210 isotropy subgroup, 86, 167, 172 Iwasawa group, 321 J Jacobi field, 30 Jacobi identity, 11 K Killing equation, 120 Killing form, 89 Killing vector field, 168 of constant length, 125–132, 250–256, 288, 415, 436–446 quasiregular, 131, 146–152 regular, 131, 146 L Ledger–Obata space, 225–228, 317–319 left translation, 75 length of a smooth curve, 26 of a smooth path, 26 Levi-Civita connection, 19 Levi factor, 89, 179 Levi subalgebra, 89, 179 Levi subgroup, 179 Lie algebra, 11 abelian (= commutative), 78 compact, 97 Heisenberg type, 280 H-type, 280–281 k-step nilpotent, 88

480

k-step solvable, 89 nilpotent, 88 of a Lie group, 77 semisimple, 87 simple, 87 simply laced, 111 solvable, 89 Lie bracket of vector fields, 11 Lie derivative, 16 Lie group, 75 H-type, 280 nilpotent, 88 semisimple, 87 simple, 87 solvable, 89 Lie group epimorphism, 76 Lie group inner automorphism, 81 Lie group isomorphism, 76 Lie group monomorphism, 76 Lie subalgebra compatible with a decomposition, 294 Lie subgroup, 81 compatible with a decomposition, 294 virtual, 81 Lie triple system, 429–432 linear connection (= covariant derivative), 16 linear isotropy group, 86 linear operator equivariant, 210 locally Euclidean space, 197–200 locally symmetric space, 190, 416–417 L¨owner ellipsoid, see L¨owner–John ellipsoid L¨owner–John ellipsoid, 392, 395, 399, 400, 408, 410 M manifold non-orientable, 2 orientable, 2 oriented, 2

Subject Index

simply connected, 62 smooth, 2 topological, 1 metric endomorphism (= metric operator), 210, 292 metric Lie algebra, 246 solvable, 247 metric line, 73 metric operator, 211 metric space Clifford–Wolf homogeneous, 329 δ -homogeneous, 329 G-Clifford–Wolf homogeneous, 329 G-δ -homogeneous, 329 with intrinsic metric, 38 metric tensor (= Riemannian metric), 18 M¨obius group, 243 M-space, 325 N natural connection without torsion, 185 neighborhood star-shaped, 39 nilmanifold, 72 nilradical of a Lie algebra, 89 normalizer of a Lie subalgebra, 88 of a subgroup, 172 O O’Neill tensor, 66 one-parameter subgroup of a Lie group, 78 origin of a vector, 7 P parabolic isometry of a hyperbolic manifold, 161 parametrization, 26 by the arc length, 27 proportionally to the arc length, 27 periodicity function

Subject Index

of a flow, 137 principal isotropy algebra, 315, 316 group, 315 product of smooth manifolds, 3 proper action, 168 proper mapping, 49, 168 R radical of a Lie algebra, 89 Radon–Hurwitz function, 422, 426 Radon sphere, 428 reductive complement, 174, 178, 184 decomposition, 174, 178, 184 regular vector, 104 relative equilibria, 258 representation completely reducible, 85, 175 faithful, 81 irreducible, 81 of a Lie algebra, 81 of a Lie group, 81 Ricci curvature, 25 Ricci tensor, 24 of a linear connection, 24 of a Riemannian manifold, 24 Riemannian homogeneous space, see homogeneous Riemannian space Riemannian manifold almost G-normal homogeneous, 408 almost normal homogeneous, 408–411 Clifford–Wolf homogeneous, 413–418 G-homogeneous, 186 homogeneous, 186 Killing, 207 naturally reductive, 218 normal, 207 standard, 207 locally symmetric, 190

481

restrictively Clifford–Wolf homogeneous, 415, 446–451 restrictively δ -homogeneous, 330 restrictively G-δ -homogeneous, 330 strongly Clifford–Wolf homogeneous, 415, 447 weakly symmetric, 277 with global Killing property, 420–421 with Killing property, 420 with local Killing property, 420–421 Riemannian metric, 18 bi-invariant, 94 Einstein left-invariant, 322–324 homogeneous naturally reductive, 206 standard, 207 left-invariant, 91 naturally reductive, 323 right-invariant, 94 Riemannian space G-homogeneous, 186 homogeneous, 186 naturally reductive, 206–208 right translation, 75 root long, 110 negative, 108 of a Cartan subalgebra, 107 of a Lie algebra, 106 of a Lie group, 106 positive, 108 short, 110 simple, 110 root system, 107 abstract, 109–112 simple, 110 simply laced, 111 S Sasaki metric, 154 scalar curvature, 25

482

sectional curvature, 25 smooth curve, 26 oriented, 26 regular, 26 smooth path, 26 regular, 26 special decomposition, 310–313 spherical bundle, 50, 154 spherical space forms homogeneous, 200–204 stabilizer, 86, 167, 172 stationary subalgebra of points in general position, 315 stationary subgroup, 86 strictly convex subset, 56 strongly convex subset, 57 subalgebra of a Lie algebra compact, 179 compactly embedded, 179 subgroup of a Lie group compactly embedded, 179 submanifold, 9 smooth open, 2 totally geodesic, 343–345, 353, 362, 363 virtual, 8 submersion, 9 Riemannian, 66 submetry, 70, 330, 336, 339, 346, 384, 386, 388, 399 symmetric space, 417 Riemannian, 188, 191–194 T tangent bundle, 7

Subject Index

manifold, 7 vector, 3 vector space, 3 tensor field, 14 of degree (0, m), 14 of degree (1, m), 14 three-locally-symmetric space (= generalized Wallach space), 222 topological group, 76 locally compact, 76 torus, 78 totally geodesic subspace, 295 transvection on a symmetric space, 192, 276 U unit spherical bundle, 50 V variation of a path, 29 with fixed end points, 29 vector regular, 251 vector field, 7 Killing, 119 left-invariant, 76 parallel, 134, 136, 137 right-invariant, 76 vertical, 9 vector field along a curve, 30 parallel, 30 W Wallach space, 234, 327 weakly symmetric space, 277–279 Weyl group of a Cartan subalgebra, 106 of a Lie group, 106