Real Hypersurfaces in Hermitian Symmetric Spaces 9783110689839, 9783110689785

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Jürgen Berndt, Young Jin Suh Real Hypersurfaces in Hermitian Symmetric Spaces

Advances in Analysis and Geometry

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Editor in Chief Jie Xiao, Memorial University, Canada Editorial Board Der-Chen Chang, Georgetown University, USA Goong Chen, Texas A&M University, USA Andrea Colesanti, University of Florence, Italy Robert McCann, University of Toronto, Canada De-Qi Zhang, National University of Singapore, Singapore Kehe Zhu, University at Albany, USA

Volume 5

Jürgen Berndt, Young Jin Suh

Real Hypersurfaces in Hermitian Symmetric Spaces |

Mathematics Subject Classification 2020 53-02, 53C15, 53C35, 53C40, 53C55, 53D15 Authors Prof. Jürgen Berndt King’s College London Strand Campus London WC2R 2LS United Kingdom [email protected]

Prof. Young Jin Suh Kyungpook National University Research Institute of Real & Complex Manifolds Daegu 41566 Republic of Korea [email protected]

ISBN 978-3-11-068978-5 e-ISBN (PDF) 978-3-11-068983-9 e-ISBN (EPUB) 978-3-11-068991-4 ISSN 2511-0438 Library of Congress Control Number: 2022930118 Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2022 Walter de Gruyter GmbH, Berlin/Boston Typesetting: VTeX UAB, Lithuania Printing and binding: CPI books GmbH, Leck www.degruyter.com

Preface This monograph originates from research collaborations of the two authors over a period of more than 20 years and covers aspects of the geometry of real hypersurfaces in Hermitian symmetric spaces. The research has been motivated by an attempt to generalize classical topics about the geometry of real hypersurfaces in complex projective spaces to other Kähler manifolds. We initially focused on complex quadrics and complex 2-plane Grassmannians, and their non-compact dual manifolds, the complex hyperbolic quadrics and the complex hyperbolic 2-plane Grassmannians. More recent research deals with a more general approach in Hermitian symmetric spaces of compact type. The focus in this monograph is on the research results obtained by the two authors and their collaborators, but related work is also included. The exhaustive bibliography at the end of the monograph enables the reader to find relevant literature with more details. The second author was supported by Grant Proj. No. NRF-2018-R1D1A1B-05040381 from the National Research Foundation of Korea. We would like to thank Dr. Hyunjin Lee for providing technical support. Dr. Lee was supported by Grant Proj. No. NRF-2019-R1I1A1A-01050300 from the National Research Foundation of Korea. Jürgen Berndt Young Jin Suh

https://doi.org/10.1515/9783110689839-201

Introduction Real hypersurfaces in complex projective spaces and in complex hyperbolic spaces have been of interest to geometers since many years. A thorough introduction and overview to this topic can be found in the excellent monograph [33] by Thomas E. Cecil and Patrick J. Ryan. In this monograph we extend some of this research to other Hermitian symmetric spaces, with particular focus on the complex quadric Qn = SOn+2 /SOn SO2 (n ≥ 3), ∗ the complex hyperbolic quadric Qn = SOn,2 /SOn SO2 (n ≥ 3), the complex 2-plane Grassmannian G2 (ℂk+2 ) = SU k+2 /S(Uk U2 ) (k ≥ 3) and the complex hyperbolic 2-plane Grassmannian G2∗ (ℂk+2 ) = SU k,2 /S(Uk U2 ) (k ≥ 3). The reader is expected to have prior knowledge of basic concepts of Riemannian geometry and submanifolds. In Chapter 1 we review concepts from Riemannian geometry that are most relevant to the contents of this monograph. This includes a brief introduction to Kähler geometry, to Riemannian symmetric spaces and to Hermitian symmetric spaces. We will not provide proofs but refer the reader to suitable literature. We discuss in detail the construction of complex quadrics, complex hyperbolic quadrics, complex 2-plane Grassmannians and complex hyperbolic 2-plane Grassmannians. The complex quadrics and complex hyperbolic quadrics are equipped with two geometric structures, namely a Kähler structure and a circle bundle of real structures. The complex 2-plane Grassmannians and complex hyperbolic 2-plane Grassmannians are also equipped with two geometric structures, namely a Kähler structure and a quaternionic Kähler structure. We will explain how these geometric structures are constructed, as they are important for the investigations of real hypersurfaces in these spaces. In Chapter 2 we present relevant basic concepts from submanifold geometry. This includes the fundamental equations of submanifold geometry. Important classes of submanifolds will be discussed, such as totally geodesic submanifolds and curvatureadapted submanifolds. We will then describe a fundamental technique, based on Jacobi fields, for investigating the geometry of focal sets of hypersurfaces and of tubes around submanifolds. In the final part of this chapter we will encounter homogeneous hypersurfaces. In Chapter 3 we discuss the geometry of real hypersurfaces in Kähler manifolds. The Kähler structure of a Kähler manifold induces a so-called almost contact metric structure on a real hypersurface in the manifold. We discuss basic properties of almost contact metric structures and what they tell us about the geometry of real hypersurfaces in Kähler manifolds. One important ingredient of an almost contact metric structure is a unit vector field on the real hypersurface that is known as the Reeb vector field. When the Reeb vector field is a principal curvature vector of the real hypersurface everywhere, then the real hypersurface is called a Hopf hypersurface. Hopf hypersurfaces are important submanifolds in Kähler geometry. We will derive some https://doi.org/10.1515/9783110689839-202

VIII | Introduction fundamental equations for Hopf hypersurfaces involving the shape operator and curvature quantities. We will then refine these equations for contact hypersurfaces and for real hypersurfaces with isometric Reeb flow. We will also review some important classification results for real hypersurfaces in complex projective spaces and in complex hyperbolic spaces, which to some extent motivated the research in this monograph. In the next four chapters we investigate the geometry of real hypersurfaces in complex 2-plane Grassmannians (Chapter 4), in complex hyperbolic 2-plane Grassmannians (Chapter 5), in complex quadrics (Chapter 6) and in complex hyperbolic quadrics (Chapter 7). In each of these chapters we will first derive some basic equations for real hypersurfaces involving the almost contact metric structure, the shape operator and curvature quantities. These equations will be fundamental for investigating the geometry of real hypersurfaces in these spaces. We will then discuss their totally geodesic submanifolds and their homogeneous real hypersurfaces. For some of the homogeneous real hypersurfaces we will investigate their geometry in more detail and derive some interesting geometric properties. We will then investigate in how far these homogeneous real hypersurfaces can be characterized by some of their geometric properties. In the final Chapter 8 we will generalize some of this to Hermitian symmetric spaces of compact type. For this we will first explain the general structure theory of Hermitian symmetric spaces of compact type, which involves structure theory of certain semisimple real Lie algebras. We then apply the general structure theory to investigate real hypersurfaces with isometric Reeb flow and contact hypersurfaces.

Contents Preface | V Introduction | VII 1 1.1 1.1.1 1.1.2 1.1.3 1.1.4 1.1.5 1.1.6 1.1.7 1.2 1.2.1 1.2.2 1.2.3 1.2.4 1.2.5 1.2.6 1.2.7 1.3 1.3.1 1.3.2 1.3.3 1.3.4 1.3.5 1.4 1.4.1 1.4.2 1.4.3 1.4.4 1.4.5

Riemannian geometry | 1 Riemannian manifolds | 1 Manifolds, maps and vector fields | 1 Riemannian metrics and isometries | 2 The Riemannian connection | 3 Geodesics and the exponential map | 5 Curvature | 6 Jacobi fields | 8 Killing vector fields | 9 Kähler manifolds | 10 Kähler structures | 10 Curvature | 11 Complex Euclidean spaces | 13 Complex projective spaces | 13 Complex hyperbolic spaces | 14 Complex space forms | 15 Quaternionic Kähler manifolds | 15 Riemannian symmetric spaces | 16 Riemannian locally symmetric spaces | 16 Cartan decomposition and Riemannian symmetric pairs | 17 Curvature, geodesics and Jacobi fields | 18 Semisimple Riemannian symmetric spaces, rank and duality | 20 Classification of Riemannian symmetric spaces | 21 Hermitian symmetric spaces | 23 Definition and classification | 23 Complex quadrics | 24 Complex hyperbolic quadrics | 27 Complex 2-plane Grassmannians | 31 Complex hyperbolic 2-plane Grassmannians | 34

2 2.1 2.2 2.3 2.4 2.5

Submanifolds of Riemannian manifolds | 37 Fundamental equations of submanifold theory | 37 Totally geodesic submanifolds | 39 Curvature-adapted submanifolds | 40 M-Jacobi fields | 40 Equidistant hypersurfaces and focal sets | 41

X | Contents 2.6 2.7

Tubes | 45 Homogeneous hypersurfaces | 48

3 3.1 3.2 3.3 3.4 3.5 3.6

Real hypersurfaces in Kähler manifolds | 53 Submanifolds | 53 Real hypersurfaces | 55 Hopf hypersurfaces | 59 Real hypersurfaces with isometric Reeb flow | 65 Contact hypersurfaces | 71 Real hypersurfaces in complex space forms | 84

4 4.1 4.2 4.3 4.4 4.5

Real hypersurfaces in complex 2-plane Grassmannians | 89 Basic equations for real hypersurfaces | 89 Totally geodesic submanifolds | 97 Homogeneous real hypersurfaces | 99 Hopf hypersurfaces | 105 Real hypersurfaces whose maximal quaternionic subbundle is invariant under the shape operator | 109 Real hypersurfaces with isometric Reeb flow | 122 Contact hypersurfaces | 129 The Ricci tensor of Hopf hypersurfaces | 132 The normal Jacobi operator of Hopf hypersurfaces | 143 The structure Jacobi operator of Hopf hypersurfaces | 148

4.6 4.7 4.8 4.9 4.10 5 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10

Real hypersurfaces in complex hyperbolic 2-plane Grassmannians | 153 Basic equations for real hypersurfaces | 153 Totally geodesic submanifolds | 158 Homogeneous real hypersurfaces | 158 Horospheres | 163 Tubes around totally geodesic submanifolds | 171 Hopf hypersurfaces | 179 Real hypersurfaces whose maximal quaternionic subbundle is invariant under the shape operator | 183 Real hypersurfaces with isometric Reeb flow | 204 Contact hypersurfaces | 212 Pseudo-Einstein real hypersurfaces | 215

6 6.1 6.2 6.3 6.4

Real hypersurfaces in complex quadrics | 219 Basic equations for real hypersurfaces | 219 Totally geodesic submanifolds | 229 Homogeneous real hypersurfaces | 232 Hopf hypersurfaces | 240

Contents | XI

6.5 6.6 6.7 6.8

Real hypersurfaces with isometric Reeb flow | 244 Contact hypersurfaces | 252 Real hypersurfaces with Reeb parallel shape operator | 259 The Ricci tensor of Hopf hypersurfaces | 265

7 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9

Real hypersurfaces in complex hyperbolic quadrics | 281 Basic equations for real hypersurfaces | 281 Totally geodesic submanifolds | 286 Homogeneous real hypersurfaces | 287 Horospheres | 291 Tubes around totally geodesic submanifolds | 297 Hopf hypersurfaces | 305 Real hypersurfaces with isometric Reeb flow | 309 Contact hypersurfaces | 318 Further results | 326

8 8.1 8.2 8.3

Real hypersurfaces in Hermitian symmetric spaces | 331 Structure theory of compact Hermitian symmetric spaces | 331 Real hypersurfaces with isometric Reeb flow | 340 Contact hypersurfaces | 353

Bibliography | 363 Index | 369

1 Riemannian geometry In this chapter we present a brief introduction to Riemannian geometry, Kähler geometry, Riemannian symmetric spaces and Hermitian symmetric spaces. Our selection of the material presented here is guided by what is most relevant in our context. For more thorough introductions to these topics we recommend the reader to consult other literature. For Riemannian geometry there are many introductory textbooks, for example by Chavel [34], do Carmo [39], Gallot, Hulin and Lafontaine [44], Kobayashi and Nomizu [59], O’Neill [78] and Sakai [87]. More specialized results for Kähler geometry can be found for example in Ballmann [4] and Moroianu [73]. For Riemannian symmetric spaces two standard references are Helgason [45] and Loos [68], but another good introduction can be found in Wolf [122]. For Riemannian symmetric spaces of non-compact type see also Eberlein [41]. An introduction to Hermitian symmetric spaces can be found in Helgason [45, Chapter VIII].

1.1 Riemannian manifolds 1.1.1 Manifolds, maps and vector fields Let M be a finite-dimensional smooth manifold and m = dim(M). By smooth we always mean C ∞ . Manifolds are always assumed to satisfy the second countability axiom and hence they are paracompact. For each p ∈ M we denote by Tp M the tangent space of M at p. The tangent bundle of M is denoted by TM. For 0 ≠ v ∈ Tp M we denote by ℝv the real span of v, that is, ℝv = {av ∈ Tp M : a ∈ ℝ}. We denote by ℱ (M) the algebra of smooth real-valued functions f : M → ℝ on M and by Λ1 (M) the ℱ (M)-module of smooth 1-forms on M. For f ∈ ℱ (M) and p ∈ M we denote by dp f : Tp M → ℝ,

X 󳨃→ dp f (X)

the differential of f at p and by df ∈ Λ1 (M) the corresponding 1-form. We sometimes write Xp f for dp f (X) and Xf for df (X). A vector field on M is a smooth map X : M → TM with π ∘ X = idM , where π : TM → M is the canonical projection. We denote by X(M) the set of smooth vector fields on M. If x1 , . . . , xm are local coordinates of M on an open neighborhood U of M, then the restriction X|U of X ∈ X(M) to U can be uniquely expressed as m

X|U = ∑ X i i=1

https://doi.org/10.1515/9783110689839-001

𝜕 . 𝜕xi

2 | 1 Riemannian geometry As usual, we can identify a vector field with a derivation on the algebra ℱ (M). This allows us to define the Lie bracket [X, Y] ∈ X(M) of vector fields X, Y ∈ X(M) by [X, Y]f = X(Yf ) − Y(Xf )

for all f ∈ ℱ (M).

If x1 , . . . , xm are local coordinates on M and m

X = ∑ Xi i=1

m

Y = ∑ Yi

𝜕 , 𝜕xi

i=1

𝜕 , 𝜕x i

then m

[X, Y]f = ∑ (X j i,j=1

𝜕X i 𝜕f 𝜕Y i − Yj j ) j . j 𝜕x 𝜕x 𝜕x

Thus m

∑(X j j=1

i 𝜕Y i j 𝜕X − Y ) 𝜕xj 𝜕xj

(i = 1, . . . , m)

are the coefficients of the Lie bracket [X, Y] with respect to the coordinate vector fields 𝜕 , . . . , 𝜕x𝜕m . For X, Y, Z ∈ X(M) the Jacobi identity 𝜕x 1 [[X, Y], Z] + [[Y, Z], X] + [[Z, X], Y] = 0 holds, which means that X(M) is a real Lie algebra. We may also regard X(M) as a module over the algebra ℱ (M). If f ∈ ℱ (M) and X ∈ X(M), then fX defined by (fX)p = f (p)Xp for all p ∈ M is a vector field on M and [fX, gY] = fg[X, Y] + f (Xg)Y − g(Yf )X for all f , g ∈ ℱ (M). If M, N are smooth manifolds and F : M → N is a smooth map, we denote by dp F : Tp M → Tf (p) N the differential of F at p. For a tensor field A of type (0, s) (s ≥ 1) on N, we can define a tensor field F ∗ A of type (0, s) on M by (F ∗ A)(v1 , . . . , vs ) = A(dp F(v1 ), . . . , dp F(vs )) for all v1 , . . . , vs ∈ Tp M and p ∈ M. The tensor field F ∗ A is called the pullback of A by F. 1.1.2 Riemannian metrics and isometries A positive definite smooth symmetric tensor field g on M of type (0, 2) is called a Riemannian metric on M. A Riemannian metric is a family (gp )p∈M of inner products on the

1.1 Riemannian manifolds | 3

tangent spaces Tp M so that g(X, Y) is smooth for all X, Y ∈ X(M). A smooth manifold M together with a Riemannian metric g is called a Riemannian manifold. Let V be an m-dimensional real vector space equipped with an inner product ⟨⋅, ⋅⟩. For p ∈ V denote by πp : Tp V → V the canonical identification. For u, v ∈ Tp V we define gp (u, v) = ⟨πp (u), πp (v)⟩. Then g defines a Riemannian metric on V. If V = ℝm , then the above metric is called the canonical Riemannian metric on ℝm . If x1 , . . . , x m are the standard coordinates on ℝm , then 𝜕x𝜕 1 , . . . , 𝜕x𝜕m is an orthonormal basis of Tp ℝm at each point p ∈ ℝm with respect to the canonical Riemannian metric on ℝm . Let (M1 , g1 ) and (M2 , g2 ) be Riemannian manifolds and M = M1 × M2 . Denote by πi : M → Mi the canonical projections. Then g = π1∗ g1 + π2∗ g2 is a Riemannian metric on M, the so-called product Riemannian metric on M1 × M2 . This construction can be extended in an obvious way to the product of a finite number of Riemannian manifolds. The symmetries of a Riemannian manifold are the so-called isometries. Let (M, gM ) and (N, gN ) be Riemannian manifolds. A diffeomorphism f : M → N is called an isometry if f ∗ gN = gM . Let (M, g) be a Riemannian manifold. We denote by I(M) or I(M, g) the set of all isometries from (M, g) onto itself. This set is a subgroup of the group of all diffeomorphisms of M and is a Lie group with respect to the compact open topology. The group I(M) is called the isometry group of M and acts as a Lie transformation group on M. We denote by I o (M) or I o (M, g) the identity component of I(M), that is, the connected component of I(M) containing the identity transformation idM of M. Isometries of Riemannian manifolds are rigid in the following sense. Theorem 1.1.1. Let M be a connected Riemannian manifold and f , g ∈ I(M). If there exists a point p ∈ M with f (p) = g(p) and dp f = dp g, then f = g. Another important feature of local isometries is that they can be extended to global isometries in certain situations. Theorem 1.1.2. Let M be a connected, simply connected, complete, real analytic Riemannian manifold. Then every local isometry of M can be extended to a global isometry of M.

1.1.3 The Riemannian connection There is no canonical way to differentiate smooth vector fields on a smooth manifold. The theory of studying the various possibilities for such a differentiation process is

4 | 1 Riemannian geometry called theory of linear connections, or covariant derivatives. A linear connection on a smooth manifold M is an ℝ-bilinear map ∇ : X(M) × X(M) → X(M),

(X, Y) 󳨃→ ∇X Y

satisfying (i) ∇fX Y = f ∇X Y for all f ∈ ℱ (M), (ii) ∇X fY = f ∇X Y + df (X)Y for all f ∈ ℱ (M). The vector field ∇X Y is also known as the covariant derivative of the vector field Y in direction of the vector field X. In the context of Riemannian geometry, we like to have linear connections that are compatible with the Riemannian metric. A linear connection ∇ satisfying Zg(X, Y) = g(∇Z X, Y) + g(X, ∇Z Y) for all X, Y, Z ∈ X(M) is called a metric linear connection. There are many metric linear connections on a Riemannian manifold. However, using the concept of torsion of a connection, we can single out a particular important one. A linear connection ∇ is called torsion-free if it satisfies ∇X Y − ∇Y X − [X, Y] = 0 for all X, Y ∈ X(M). A fundamental result of Riemannian geometry states the following. Theorem 1.1.3. On each Riemannian manifold there exists a unique torsion-free metric linear connection. This connection is called the Riemannian connection or Levi-Civita connection of the Riemannian manifold M. Unless otherwise stated, ∇ usually denotes the Riemannian connection of a Riemannian manifold. From the above properties we can explicitly calculate the Riemannian connection using the Koszul formula: 2g(∇X Y, Z) = Xg(Y, Z) + Yg(Z, X) − Zg(X, Y)

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

Let c : I → M be a smooth curve in M defined on an interval I ⊆ ℝ. We denote the set of smooth vector fields along c by Xc (M). The Riemannian connection ∇ on M induces a differentiation process for vector fields along c. The derivative of X ∈ Xc (M) is denoted by X ′ and is characterized by the following properties: (i) (X + Y)′ = X ′ + Y ′ for all X, Y ∈ Xc (M), (ii) (fX)′ = f ′ X + fX ′ for all X ∈ Xc (M) and all smooth functions f : I → ℝ, (iii) (X ∘ c)′ = ∇ċ X for all X ∈ X(M).

1.1 Riemannian manifolds | 5

We sometimes write ∇ċ ċ instead of ċ′ . Since ∇ is a metric connection, we have g(X, Y)′ = g(X ′ , Y) + g(X, Y ′ )

for all X, Y ∈ Xc (M).

A vector field X ∈ Xc (M) is called parallel if X ′ = 0. General theory of ordinary differential equations implies that for every t0 ∈ I and every v ∈ Tc(t0 ) M there exists exactly one parallel vector field Bv along c with Bv (t0 ) = v. For each t ∈ I there is then a well-defined linear isometry Pc (t) : Tc(t0 ) M → Tc(t) M,

v 󳨃→ Bv (t),

which is called the parallel transport along c. The covariant derivative and parallel transport along c are related by X ′ (t) =

d 󵄨󵄨󵄨󵄨 −1 󵄨 (P (t + s)) X(t + s). ds 󵄨󵄨󵄨s=0 c

1.1.4 Geodesics and the exponential map A smooth curve γ : I → M for which the tangent vector field γ̇ is parallel, that is, ∇γ̇ γ̇ = γ̇ ′ = 0, is called a geodesic. Geodesics play a similar role in Riemannian geometry as lines in Euclidean geometry. In particular, geodesics are locally distance-minimizing curves. General theory of ordinary differential equations implies the following existence and uniqueness result for geodesics. Theorem 1.1.4. Let M be a Riemannian manifold. For every point p ∈ M and every tangent vector v ∈ Tp M there exists a unique geodesic γv : Iv → M with 0 ∈ Iv , γv (0) = p, ̇ γ̇v (0) = v such that for any other geodesic γ : I → M with 0 ∈ I, γ(0) = p and γ(0) =v we have I ⊆ Iv . The geodesic γv in Theorem 1.1.4 is called the maximal geodesic in M through p and tangent to v. The Hopf–Rinow theorem states that a Riemannian manifold is complete if and only if Iv = ℝ for all v ∈ TM. For p ∈ M we define T̃ p M = {v ∈ Tp M : 1 ∈ Iv } and ̃ = {v ∈ TM : 1 ∈ Iv } = ⋃ T̃ TM p M. p∈M

6 | 1 Riemannian geometry ̃ = TM if and only if M is complete. The expoBy the Hopf–Rinow theorem we have TM nential map of M is the map ̃ → M, exp : TM

v 󳨃→ γv (1).

For p ∈ M the exponential map expp is the restriction of exp to T̃ p M. There exists an open neighborhood U of 0 ∈ Tp M such that the restriction expp |U of expp to U is a diffeomorphism into M.

1.1.5 Curvature The most important concept in Riemannian geometry is curvature. There are different types of curvature and all of them can be deduced from the Riemannian curvature tensor R(X, Y)Z = ∇X ∇Y Z − ∇Y ∇X Z − ∇[X,Y] Z. The Riemannian curvature tensor satisfies the algebraic curvature identities g(R(X, Y)Z, W) = −g(R(Y, X)Z, W) = −g(R(X, Y)W, Z) = g(R(Z, W)X, Y).

(1.1)

Moreover, R satisfies the first Bianchi identity R(X, Y)Z + R(Y, Z)X + R(Z, X)Y = 0

(1.2)

and the second Bianchi identity (∇X R)(Y, Z)W + (∇Y R)(Z, X)W + (∇Z R)(X, Y)W = 0. By contracting the curvature tensor we get the Ricci tensor of M: ric(X, Y) = tr(Z 󳨃→ R(Z, X)Y). From the algebraic curvature identities we see that the Ricci tensor is symmetric, that is, ric(X, Y) = ric(Y, X). The corresponding self-adjoint tensor field Ric of type (1, 1), given by ric(X, Y) = g(Ric(X), Y),

1.1 Riemannian manifolds | 7

is also known as the Ricci tensor of M. A Riemannian manifold for which there exists a real number c ∈ ℝ such that the Ricci tensor satisfies ric(X, Y) = cg(X, Y) for all X, Y ∈ X(M) is called an Einstein manifold. By contracting the Ricci tensor we obtain the scalar curvature s = tr(Ric) of M. A Riemannian manifold (M, g) is said to have harmonic curvature if the divergence of the Riemannian curvature tensor R of M vanishes. If dim(M) ≥ 3, then M has harmonic curvature if and only if the Ricci tensor of M is a Codazzi tensor, that is, if (∇X Ric)Y = (∇Y Ric)X

(1.3)

holds for all X, Y ∈ X(M). This implies that every connected Riemannian manifold with harmonic curvature has constant scalar curvature. If dim(M) ≥ 4, then M has harmonic curvature if and only if M has harmonic Weyl tensor and constant scalar curvature. The Riemannian manifold (M, g) is said to have harmonic Weyl tensor if the divergence of the Weyl conformal tensor of (M, g) vanishes. If n = dim(M) ≥ 4, then M s has harmonic Weyl tensor if and only if Ric − 2n−2 id is a Codazzi tensor, that is, if (∇X (Ric −

s s id))Y = (∇Y (Ric − id))X 2n − 2 2n − 2

holds for all X, Y ∈ X(M). A Riemannian manifold (M, g) is said to be a Ricci soliton if there exist a vector field V ∈ X(M) and a real number c ∈ ℝ such that the Ricci tensor Ricg of (M, g) satisfies 1 Ricg = cg − ℒV g, 2 where ℒV g is the Lie derivative of the Riemannian metric g with respect to V. The vector field V arising in the above definition is said to be the potential field of the Ricci soliton. Ricci solitons are generalizations of Einstein manifolds. A Ricci soliton is said to be shrinking if c > 0, steady if c = 0, and expanding if c < 0. A Ricci soliton (M, g0 ) is a self-similar solution to the Ricci flow equation 𝜕t gt = −2Ricgt .

8 | 1 Riemannian geometry Let G2 (TM) be the Grassmann bundle over M consisting of all 2-dimensional linear subspaces V of Tp M, p ∈ M. Let V ∈ G2 (TM), choose a basis u, v of V, and define K(V) =

g(R(u, v)v, u) . g(u, u)g(v, v) − g(u, v)2

One can show that the definition of K(V) is independent of the choice of the basis u, v. In particular, if u, v are orthonormal, we have K(V) = g(R(u, v)v, u). The map K : G2 (TM) → ℝ ,

V 󳨃→ K(V)

is the sectional curvature function of M, and K(V) is the sectional curvature of M with respect to V. Since there exists an open neighborhood U of 0 in Tp M such that expp |U is a diffeomorphism into M, the image expp (U) is a surface in M. The sectional curvature K(V) coincides with Gaussian curvature of this surface at the point p. A Riemannian manifold M is said to have constant sectional curvature if the sectional curvature function is constant. If M is connected and dim(M) ≥ 3, the second Bianchi identity and Schur’s lemma imply that if the sectional curvature function depends only on the point p, then M has constant sectional curvature. Theorem 1.1.5. A Riemannian manifold (M, g) has constant sectional curvature κ ∈ ℝ if and only if the Riemannian curvature tensor R of (M, g) satisfies R(X, Y)Z = κ(g(Y, Z)X − g(X, Z)Y) for all X, Y, Z ∈ X(M). A connected, simply connected, complete Riemannian manifold with constant sectional curvature is also called a space form or real space form. The standard models for real space forms are the spheres Sm (κ) (m ≥ 2, κ > 0), the Euclidean spaces ℝm (m ≥ 1, κ = 0) and the real hyperbolic spaces ℝH m (κ) (m ≥ 2, κ < 0), all of them equipped with their canonical Riemannian metrics (up to rescaling). We often write Sm for Sm (1) and ℝH m for ℝH m (−1). 1.1.6 Jacobi fields Let γ : I → M be a geodesic. A vector field Y ∈ Xγ (M) is called a Jacobi field along γ if it satisfies the linear second order ordinary differential equation Y ′′ + R(Y, γ)̇ γ̇ = 0. General theory of ordinary differential equations implies that the Jacobi fields along a geodesic form a 2m-dimensional real vector space, where m = dim(M). Moreover,

1.1 Riemannian manifolds | 9

every Jacobi field is uniquely determined by the initial values Y(t0 ) and Y ′ (t0 ) at some point t0 ∈ I. Jacobi fields arise geometrically as infinitesimal variational vector fields of geodesic variations. The self-adjoint tensor field Rγ = R(⋅, γ)̇ γ̇ is called the Jacobi operator along the geodesic γ. More generally, if v ∈ TM or X is a vector field on M, then Rv = R(⋅, v)v and RX = R(⋅, X)X is called the Jacobi operator of M with respect to v and X, respectively. Jacobi operators contain important information about the geometry of Riemannian manifolds. Jacobi fields can be used to calculate the differential of the exponential map. Let p ∈ M and expp be the exponential map of M at p. For v ∈ T̃ p M we identify Tv (Tp M) with Tp M in the canonical way. Then we have dv expp (w) = Yw (1) for each w ∈ Tp M, where Yw is the Jacobi field along γv with initial values Yw (0) = 0 and Yw′ (0) = w. 1.1.7 Killing vector fields For a vector field X ∈ X(M) on a Riemannian manifold M we denote by ΦX : D → M the flow of M, where D ⊂ ℝ × M is the flow domain of X. For each t ∈ ℝ we define ΦXt : Dt = {p ∈ M : (t, p) ∈ D} → M ,

p 󳨃→ ΦX (t, p),

which, if Dt ≠ 0, is a local diffeomorphism on M generated by the flow of X. A vector field X is complete if its flow domain D is equal to ℝ × M. A vector field X on a Riemannian manifold M is called a Killing vector field if the local diffeomorphisms ΦXt : Dt → M are isometries. This just means that the Lie derivative ℒX g of the Riemannian metric g of M with respect to X vanishes. A useful characterization of Killing vector fields is that a vector field X ∈ X(M) is a Killing vector field if and only if its covariant derivative ∇X is a skew-symmetric tensor field on M. A Killing vector field is completely determined by its value and its covariant derivative at any given point. In particular, a Killing vector field X for which Xp = 0 and (∇X)p = 0 at some point p ∈ M vanishes at each point of M. For a complete Killing vector field X on a Riemannian manifold M, the corresponding one-parameter group (ΦXt )t∈ℝ consists of isometries of M. Conversely, suppose we have a one-parameter group Φt of isometries on a Riemannian manifold M. Then Xp =

d 󵄨󵄨󵄨󵄨 󵄨 (t 󳨃→ Φt (p)) dt 󵄨󵄨󵄨t=0

10 | 1 Riemannian geometry defines a complete Killing vector field X on M with ΦXt = Φt for all t ∈ ℝ. If X is a Killing vector field on M and Xp = 0, then t 󳨃→ dp ΦXt is a curve in SO(Tp M) and d 󵄨󵄨󵄨󵄨 X 󵄨 (t 󳨃→ dp Φt ) = (∇X)p ∈ so(Tp M), dt 󵄨󵄨󵄨t=0 where so(Tp M) is the Lie algebra of skew-symmetric transformations of Tp M.

1.2 Kähler manifolds 1.2.1 Kähler structures An almost complex structure on a smooth manifold M is a tensor field J of type (1, 1) on M satisfying J 2 = −idTM . An almost complex manifold (M, J) is a smooth manifold M equipped with an almost complex structure J. Each tangent space of an almost complex manifold is isomorphic to a complex vector space, which implies that the dimension m = dim(M) of an almost complex manifold M is an even number. If we write m = 2n, then n is the complex dimension dimℂ (M) of M. Let (M, J) be an almost complex manifold and p ∈ M. For 0 ≠ v ∈ Tp M we denote by ℂv = {av + bJv ∈ Tp M : a, b ∈ ℝ} the complex span of v. A Hermitian metric on an almost complex manifold (M, J) is a Riemannian metric g for which the almost complex structure J on M is orthogonal, that is, g(JX, JY) = g(X, Y) for all X, Y ∈ X(M). An orthogonal almost complex structure on a Riemannian manifold is called an almost Hermitian structure. A subspace V of Tp M is called complex if JV = V, and it is called totally real if g(JV, V) = {0}. Every complex manifold M has a canonical almost complex structure. In fact, if z = (z1 , . . . , zn ) with zν = xν + iyν are local complex coordinates on M, we can define an almost complex structure J on M by J

𝜕 𝜕 = , 𝜕xν 𝜕yν

J

𝜕 𝜕 =− . 𝜕yν 𝜕xν

These local almost complex structures are compatible on the intersection of any two coordinate neighborhoods and hence induce an almost complex structure, which is called the induced complex structure on M. An almost complex structure J on a smooth manifold M is integrable if M can be equipped with the structure of a complex

1.2 Kähler manifolds |

11

manifold so that J is the induced complex structure. A famous result by Newlander– Nirenberg says that the almost complex structure J of an almost complex manifold M is integrable if and only if [X, Y] + J[JX, Y] + J[X, JY] − [JX, JY] = 0 holds for all X, Y ∈ X(M). A Hermitian manifold is an almost Hermitian manifold with an integrable almost complex structure. The almost Hermitian structure of a Hermitian manifold is called a Hermitian structure. The 2-form ω on a Hermitian manifold (M, g, J) defined by ω(X, Y) = g(JX, Y) for all X, Y ∈ X(M) is called the Kähler form of M. A Kähler manifold is a Hermitian manifold whose Kähler form is closed. In this situation the Hermitian structure is called a Kähler structure and the Hermitian metric is called a Kähler metric. A Hermitian manifold (M, g, J) is a Kähler manifold if and only if its Hermitian structure J is parallel with respect to the Riemannian connection ∇ of (M, g), that is, if ∇J = 0. The latter condition characterizes the Kähler manifolds among all Hermitian manifolds by the geometric property that parallel translation along curves commutes with the Hermitian structure J.

1.2.2 Curvature Let (M, g, J) be a Kähler manifold and R the Riemannian curvature tensor of (M, g). Since ∇J = 0, we immediately see that R(X, Y)JZ = JR(X, Y)Z

(1.4)

holds for all X, Y, Z ∈ X(M). From the curvature identities (1.1) and (1.4) we also get g(R(X, Y)Z, W) = g(R(JX, JY)Z, W) = g(R(X, Y)JZ, JW)

(1.5)

for all X, Y, Z, W ∈ X(M). Let G2J (TM) be the Grassmann bundle over M consisting of all 2-dimensional J-invariant linear subspaces V of Tp M, p ∈ M. Thus every V ∈ G2J (TM) is a complex line in the corresponding tangent space of M. The restriction of the sectional curvature function K to G2J (TM) is called the holomorphic sectional curvature function on M and K(V) is called the holomorphic sectional curvature of M with respect to V ∈ G2J (TM). A Kähler manifold M is said to have constant holomorphic sectional curvature if the holomorphic sectional curvature function is constant.

12 | 1 Riemannian geometry Theorem 1.2.1. A Kähler manifold (M, g, J) has constant holomorphic sectional curvature c ∈ ℝ if and only if its Riemannian curvature tensor R satisfies R(X, Y)Z =

c (g(Y, Z)X − g(X, Z)Y 4 + g(JY, Z)JX − g(JX, Z)JY − 2g(JX, Y)JZ)

(1.6)

for all X, Y, Z ∈ X(M). For distinct planes V1 , V2 ∈ G2J (Tp M), p ∈ M, the holomorphic bisectional curvature is defined by H(V1 , V2 ) = g(R(v1 , Jv1 )Jv2 , v2 ), where v1 ∈ V1 and v2 ∈ V2 are unit vectors. It is straightforward to show that the definition of the holomorphic bisectional curvature H(V1 , V2 ) is independent of the choice of unit vectors v1 ∈ V1 and v2 ∈ V2 . The notion of holomorphic bisectional curvature stems from the fact that H(V1 , V2 ) = g(R(v1 , Jv1 )Jv2 , v2 )

= g(R(v1 , Jv1 )Jv1 , v1 ) + g(R(v2 , Jv2 )Jv2 , v2 ) = K(V1 ) + K(V2 ).

The above curvature identities for Kähler manifolds lead to some additional identities for the Ricci tensor. Let Z1 , . . . , Zn be a local orthonormal complex frame field on M, that is, Z1 , JZ1 , . . . , Zn , JZn is a local orthonormal frame field on M. Then, using (1.5), (1.1) and (1.2), we calculate n

n

ν=1 n

ν=1 n

ric(X, Y) = ∑ g(R(Zν , X)Y, Zν ) + ∑ g(R(JZν , X)Y, JZν ) = ∑ g(R(Zν , X)JY, JZν ) − ∑ g(R(JZν , X)JY, Zν ) ν=1 n

ν=1 n

= ∑ g(R(X, Zν )JZν , JY) + ∑ g(R(JZν , X)Zν , JY) ν=1

ν=1

n

n

= − ∑ g(R(Zν , JZν )X, JY) = ∑ g(R(Zν , JZν )JX, Y). ν=1

ν=1

Thus n

Ric(X) = ∑ R(Zν , JZν )JX. ν=1

In particular, this implies Ric(JX) = JRic(X)

(1.7)

1.2 Kähler manifolds |

13

and hence ric(JX, JY) = ric(X, Y). 1.2.3 Complex Euclidean spaces The complex vector space ℂn (n ∈ ℕ) is in a canonical way an n-dimensional complex manifold. For p ∈ ℂn denote by πp : Tp ℂn → ℂn the canonical isomorphism. We define a Riemannian metric g on ℂn by gp (u, v) = ⟨πp (u), πp (v)⟩ for all u, v ∈ Tp ℂn and p ∈ ℂn , where ⟨⋅, ⋅⟩ is the real part of the standard Hermitian inner product on ℂn , that is, n

⟨a, b⟩ = Re( ∑ aν b̄ ν ) ν=1

(a, b ∈ ℂn ).

The metric g is called the canonical Riemannian metric on ℂn . The complex structure J on ℂn is given by the equation πp (Ju) = iπp (u). It is easy to verify that (ℂn , g, J) is a Kähler manifold. In fact, (ℂn , g, J) is a Kähler manifold with constant holomorphic sectional curvature 0. The Kähler manifold (ℂn , g, J) is known as the n-dimensional complex Euclidean space.

1.2.4 Complex projective spaces Let n ∈ ℕ. We define an equivalence relation ∼ on ℂn+1 \{0} by z1 ∼ z2 if and only if there exists λ ∈ ℂ \ {0} so that z2 = z1 λ. We denote the quotient space (ℂn+1 \ {0})/ ∼ by ℂP n . By construction, the points in ℂP n are in one-to-one correspondence with the complex lines through 0 ∈ ℂn+1 . We equip ℂP n with the quotient topology with respect to the canonical projection τ : ℂn+1 \ {0} → ℂP n . Then ℂP n is a compact Hausdorff space and τ is a continuous map. There exists a unique complex manifold structure on ℂP n so that τ is a holomorphic submersion. In this way ℂP n becomes an n-dimensional complex manifold (ℂP n , J). For z ∈ ℂn+1 \ {0} we also write [z] = τ(z) ∈ ℂP n . Let S2n+1 be the unit sphere in ℂn+1 and denote by π the restriction of τ to S2n+1 . We consider S2n+1 with the Riemannian metric induced from ℂn+1 , which is the standard metric on S2n+1 turning it into a real space form with constant sectional curvature 1. The map π : S2n+1 → ℂP n is a surjective submersion whose fibers are 1-dimensional circles. There exists a unique Riemannian metric g on ℂP n so that π becomes a Riemannian submersion. In this setup, the map π : S2n+1 → ℂP n is known as the Hopf

14 | 1 Riemannian geometry map from S2n+1 onto ℂP n and the Riemannian metric g is known as the Fubini– Study metric on ℂP n . The manifold (ℂP n , g, J) is a Kähler manifold and called the n-dimensional complex projective space. The complex projective space (ℂP n , g, J) is a Kähler manifold with constant holomorphic sectional curvature 4. By a suitable rescaling of the Fubini–Study metric we obtain a Riemannian metric gc on ℂP n so that ℂP n (c) = (ℂP n , gc , J) is a complex space form with constant holomorphic sectional curvature c > 0. Note that ℂP 1 (c) is isometric to the sphere S2 (c) with constant sectional curvature c.

1.2.5 Complex hyperbolic spaces The standard construction of the complex hyperbolic space involves semi-Riemannian geometry. As we do not use semi-Riemannian geometry anywhere else in this monograph, we refer the reader to [78] for details on this topic. Let n ∈ ℕ. We define on ℂn+1 the symmetric bilinear form n+1

β(a, b) = Re(−a1 b̄ 1 + ∑ aν b̄ ν ) ν=2

(a, b ∈ ℂn+1 ).

Since β is invariant under the canonical action of S1 = {eit ∈ ℂ : t ∈ ℝ}, the set ℂH n = {[z] ∈ ℂP n : z ∈ ℂn+1 \ {0}, β(z, z) < 0} is an open subset of ℂP n and therefore an n-dimensional complex manifold (ℂH n , J) with respect to the induced complex manifold structure. The points in ℂH n are in oneto-one correspondence with the complex lines through 0 ∈ ℂn+1 on which β is negative definite. Let gβ be the semi-Riemannian metric on ℂn+1 induced by β. Then H 2n+1 = {z ∈ ℂn+1 : β(z, z) < 0} is a Lorentz hypersurface of (ℂn+1 , gβ ), the so-called anti-de Sitter space. The canonical projection π : H 2n+1 → ℂH n , z 󳨃→ [z] is a surjective submersion with 1-dimensional totally geodesic timelike fibers. There exists a unique Riemannian metric g on ℂH n so that π becomes a semi-Riemannian submersion. The map π : H 2n+1 → ℂH n is known as the Hopf map from H 2n+1 onto ℂH n and the Riemannian metric g is known as the Fubini–Study metric on ℂH n . The manifold (ℂH n , g, J) is a Kähler manifold and called the n-dimensional complex hyperbolic space. The complex hyperbolic space (ℂH n , g, J) is a Kähler manifold with constant holomorphic sectional curvature −4. By a suitable rescaling of the Fubini–Study metric we obtain a Riemannian metric gc on ℂH n so that ℂH n (c) = (ℂH n , gc , J) is a Kähler manifold with constant holomorphic sectional

1.2 Kähler manifolds |

15

curvature c < 0. Note that ℂH 1 (c) is isometric to the real hyperbolic plane ℝH 2 (c) of constant sectional curvature c.

1.2.6 Complex space forms A connected, simply connected, complete Kähler manifold with constant holomorphic sectional curvature is called a complex space form. Theorem 1.2.2. Let M be a connected Kähler manifold with constant holomorphic sectional curvature c ∈ ℝ and n = dimℂ (M). Then M is an open part of the complex Euclidean space ℂn (if c = 0), the complex projective space ℂP n (c) (if c > 0) or the complex hyperbolic space ℂH n (c) (if c < 0). 1.2.7 Quaternionic Kähler manifolds A quaternionic Kähler structure on a Riemannian manifold M is a rank 3 vector subbundle J of the endomorphism bundle End(TM) over M with the following properties: (i) for each p in M there exist an open neighborhood U of p in M and sections J1 , J2 , J3 of J over U so that Jν is an almost Hermitian structure on U and Jν Jν+1 = Jν+2 = −Jν+1 Jν

(ν = 1, 2, 3, index modulo 3);

(ii) J is a parallel subbundle of End(TM), that is, if J is a section in J and X ∈ X(M), then ∇X J is also a section in J. Each triple J1 , J2 , J3 of the above kind is called a canonical local basis of J, or, if restricted to the tangent space Tp M of M at p, a canonical basis of Jp . It follows from (ii) that for any canonical local basis J1 , J2 , J3 of J there exist three local 1-forms q1 , q2 , q3 such that ∇X Jν = qν+2 (X)Jν+1 − qν+1 (X)Jν+2 for all vector fields X ∈ X(M). A quaternionic Kähler manifold (M, g, J) is a Riemannian manifold (M, g) equipped with a quaternionic Kähler structure J. The canonical bases of a quaternionic Kähler structure turn the tangent spaces of a quaternionic Kähler manifold into quaternionic vector spaces. Therefore, the dimension of a quaternionic Kähler manifold M is 4n for some n ∈ ℕ, that is, dimℍ (M) = n. Let M be a quaternionic Kähler manifold and p ∈ M. For 0 ≠ v ∈ Tp M we define ℍv = {av + bJ1 v + cJ2 v + dJ3 v ∈ Tp M : a, b, c, d ∈ ℝ},

16 | 1 Riemannian geometry Jv = {bJ1 v + cJ2 v + dJ3 v ∈ Tp M : b, c, d ∈ ℝ}, where J1 , J2 , J3 is a canonical basis of Jp . The choice of canonical basis does not matter for the definitions of ℍv and Jv. A subspace V of Tp M is said to be (i) totally real if g(JV, V) = {0} for all J ∈ Jp ; (ii) totally complex if there exists a canonical basis J1 , J2 , J3 of Jp such that J1 V = V, g(J2 V, V) = {0} and g(J3 V, V) = {0}; (iii) quaternionic if JV ⊆ V for all J ∈ Jp . The standard examples of such subspaces are ℝk ⊂ ℍn (totally real), ℂk ⊂ ℍn (totally complex) and ℍk ⊆ ℍn (quaternionic). A 4n-dimensional connected Riemannian manifold M can be equipped with a quaternionic Kähler structure if and only if its holonomy group is contained in Spn Sp1 . Some standard examples of quaternionic Kähler manifolds are the quaternionic Euclidean space ℍn , the quaternionic projective space ℍP n = Spn+1 /Spn Sp1 (n ≥ 2) and the quaternionic hyperbolic space ℍH n = Sp1,n /Spn Sp1 (n ≥ 2).

1.3 Riemannian symmetric spaces 1.3.1 Riemannian locally symmetric spaces Let M be a Riemannian manifold, p ∈ M and ϵ ∈ ℝ+ so that Uϵ (0) = {v ∈ Tp M : gp (v, v) < ϵ} is contained in T̃ p M. Then, for sufficiently small ϵ, Bϵ (p) = expp (Uϵ (0)) is the open ball in M consisting of all points in M with distance less than ϵ to p. The map sp : Bϵ (p) → Bϵ (p),

expp (v) 󳨃→ expp (−v)

reflects in p the geodesics of M through p. The map sp is called a local geodesic symmetry of M at p. A connected Riemannian manifold is called a Riemannian locally symmetric space if for each point p in M there exists an open ball Bϵ (p) such that the local geodesic symmetry sp : Bϵ (p) → Bϵ (p) is an isometry. A connected Riemannian manifold is called a Riemannian symmetric space if at each point p ∈ M such a local geodesic symmetry extends to a global isometry sp ∈ I(M). This is equivalent to saying that there exists an involutive isometry sp ∈ I(M) such that p is an isolated fixed point of sp . In such a case sp is called the symmetry or geodesic symmetry of M in p.

1.3 Riemannian symmetric spaces | 17

Let M be a Riemannian homogeneous space and assume that there exists a symmetry sp ∈ I(M) at a point p ∈ M. Let q ∈ M and g ∈ I(M) with g(p) = q. Then sq = g ∘ sp ∘ g −1 is a symmetry of M at q. Therefore, to prove that a Riemannian homogeneous space is symmetric, it suffices to construct a symmetry at one point. In view of this we can easily find examples of Riemannian symmetric spaces. The Euclidean space ℝm is symmetric with symmetry s0 : ℝm → ℝm ,

p 󳨃→ −p

at the origin 0 ∈ ℝm . The map so : Sm → Sm ,

(p1 , . . . , pm , pm+1 ) 󳨃→ (−p1 , . . . , −pm , pm+1 )

is a symmetry of the unit sphere Sm ⊂ ℝm+1 at o = (0, . . . , 0, 1). Let G be a connected compact Lie group. Any Ad(G)-invariant inner product on g extends to a bi-invariant Riemannian metric on G. With respect to such a Riemannian metric, the inversion se : G → G ,

g 󳨃→ g −1

is a symmetry of G at the identity e of G. Thus, every connected compact Lie group with a bi-invariant Riemannian metric is a Riemannian symmetric space. If M is a connected, complete, Riemannian locally symmetric space, then its Riemannian universal covering is a Riemannian symmetric space. Using the symmetries, one can show that every Riemannian symmetric space is a homogeneous Riemannian manifold. Note that there are complete Riemannian locally symmetric spaces that are not symmetric, even not homogeneous. For example, let M be a compact Riemann surface with genus ≥ 2 and equipped with a Riemannian metric of constant sectional curvature −1. It is known that the isometry group of M is finite, so M is not homogeneous and therefore also not symmetric. On the other hand, M is locally isometric to the real hyperbolic plane ℝH 2 and hence locally symmetric. 1.3.2 Cartan decomposition and Riemannian symmetric pairs Let G be a connected Lie group and s a non-trivial involutive automorphism of G, and denote by e ∈ G the identity of G. We denote by Gs ⊂ G the set of fixed points of s and by Gso the identity component of Gs . Let K be a closed subgroup of G with Gso ⊆ K ⊆ Gs . We denote by g and k the Lie algebra of G and K, respectively. We identify in the canonical way the Lie algebra of a Lie group with the tangent space of the Lie group at the identity, that is, g ≅ Te G. Then the differential σ = de s is an involutive automorphism of g and k = {X ∈ g : σ(X) = X}.

18 | 1 Riemannian geometry The linear subspace p = {X ∈ g : σ(X) = −X} of g is called the standard complement of k in g. Then we have g = k ⊕ p (direct sum of vector spaces) and [k, p] ⊆ p ,

[p, p] ⊆ k.

This particular decomposition of g is called the Cartan decomposition or standard decomposition of g with respect to σ. In this situation, the pair (G, K) is called a Riemannian symmetric pair if AdG (K) is a compact subgroup of GL(g) and p is equipped with an AdG (K)-invariant inner product ⟨⋅, ⋅⟩. Here AdG : G → GL(g) denotes the adjoint representation of G. Suppose (G, K) is a Riemannian symmetric pair and s the corresponding involutive automorphism of G. The inner product ⟨⋅, ⋅⟩ on p determines a G-invariant Riemannian metric on the homogeneous space M = G/K and the map M→M,

gK 󳨃→ s(g)K

is a symmetry of M at o = eK ∈ M. Thus, M is a Riemannian symmetric space. Conversely, assume that M is a Riemannian symmetric space. Let G = I o (M), o ∈ M, so be the symmetry of M at o and K = Go be the isotropy group of G at o. Then s:G→G,

g 󳨃→ so gso

is an involutive automorphism of G with Gso ⊆ K ⊆ Gs , and the inner product on the standard complement p ≅ To M of k in g is AdG (K)-invariant. In this way, the Riemannian symmetric space M determines a Riemannian symmetric pair (G, K). This Riemannian symmetric pair is effective, that is, every normal subgroup of G contained in K is trivial. As described here, there is a one-to-one correspondence between Riemannian symmetric spaces and effective Riemannian symmetric pairs. 1.3.3 Curvature, geodesics and Jacobi fields The fundamental relation between Riemannian locally symmetric spaces and curvature is the following surprising fact. Theorem 1.3.1. A Riemannian manifold (M, g) is locally symmetric if and only if the Riemannian curvature tensor R of (M, g) is parallel with respect to the Riemannian connection ∇ of (M, g), that is, if ∇R = 0.

1.3 Riemannian symmetric spaces | 19

Let M be a Riemannian symmetric space, o ∈ M, G = I o (M), K the isotropy group of G at o and g = k ⊕ p the corresponding Cartan decomposition of g. This gives us a realization of M as a homogeneous space G/K. Denote by π : G → G/K,

g 󳨃→ gK

the canonical projection. If we restrict the differential de π : g → TeK G/K ≅ To M to p, we get an isomorphism p → To M. We will use this isomorphism to identify tangent vectors in To M with elements in p. Let v ∈ To M ≅ p. The maximal geodesic γv in M with γv (0) = o and γ̇v (0) = v is given by γv (t) = Exp(tv)(o), where Exp : g → G is the Lie exponential map. The curve Exp(tv) is a one-parameter group of isometries of M. Let X v ∈ X(M) be the corresponding Killing vector field on v M and ΦX be the flow of X v . Then the parallel transport Pγv (t) along γv from o = γv (0) to γv (t) coincides with the differential v

do ΦXt : To M → Tγv (t) M. We now describe the Jacobi fields along the geodesic γv . First of all, the Jacobi operator Rv : To M → To M,

u 󳨃→ R(u, v)v

is self-adjoint. We denote by Spec(Rv ) the eigenvalues of Rv and by Eκ the eigenspace of κ ∈ Spec(Rv ). For κ ∈ ℝ we define the functions 1 sin(√κt) { { { √κ sinκ (t) = {t { { 1 { √−κ sinh(√−κt)

cos(√κt) { { { cosκ (t) = {1 { { {cosh(√−κt)

if κ > 0, if κ = 0,

if κ < 0,

if κ > 0,

if κ = 0, if κ < 0,

1 { √κ tan(√κt) { sinκ (t) { tanκ (t) = = t { cosκ (t) { { 1 { √−κ tanh(√−κt)

if κ > 0, if κ = 0,

if κ < 0,

20 | 1 Riemannian geometry {√κ cot(√κt) { cosκ (t) { = { 1t cotκ (t) = sinκ (t) { { {√−κ coth(√−κt)

if κ > 0, if κ = 0,

if κ < 0.

Note that sin′κ = cosκ and cos′κ = −κ sinκ . Since ∇R = 0, the eigenvalues of the Jacobi operator Rγv are constant along γ and the eigenspaces of Rγv are invariant under parallel transport along γv . It follows that every Jacobi field along γv can be written as a linear combination of the so-called basic Jacobi fields along γv given by Yu (t) = cosκ (t)Pγv (t)(u) ,

Zu (t) = sinκ (t)Pγv (t)(u),

where κ ∈ Spec(Rv ) and u ∈ Eκ . The vector field Yu is the Jacobi field along γv with initial values Yu (0) = u and Yu′ (0) = 0, and the vector field Zu is the Jacobi field along γv with initial values Zu (0) = 0 and Zu′ (0) = u. Note that we always have 0 ∈ Spec(Rv ) because Rv v = 0. Moreover, the basic Jacobi fields Yv and Zv are Yv (t) = γ̇v (t) and Zv (t) = t γ̇v (t). The Riemannian curvature tensor Ro of M at o can also be described by the simple formula Ro (X, Y)Z = −[[X, Y], Z] for all X, Y, Z ∈ p ≅ To M. This is the starting point for calculating curvature quantities using algebraic tools, but in many situations this leads to very complicated algebraic expressions.

1.3.4 Semisimple Riemannian symmetric spaces, rank and duality Let M be a Riemannian symmetric space and M̃ its Riemannian universal covering ̃ where the Euclidean space. Let M̃ 0 × M̃ 1 × ⋅ ⋅ ⋅ × M̃ k be the de Rham decomposition of M, ̃ factor M0 is isometric to a Euclidean space of dimension ≥ 0. Each M̃ i , i > 0, is a simply connected, irreducible, Riemannian symmetric space. A semisimple Riemannian symmetric space is a Riemannian symmetric space for which M̃ 0 has dimension 0. This notion is due to the fact that I o (M) is a semisimple Lie group if M̃ 0 is trivial. A Riemannian symmetric space M is said to be of compact type if M is semisimple and compact, and it is said to be of non-compact type if M is semisimple and non-compact. Riemannian symmetric spaces of non-compact type are always simply connected. An s-representation is the isotropy representation of a simply connected, semisimple, Riemannian symmetric space M = G/K with G = I o (M). The rank of a Riemannian symmetric space M = G/K is the dimension of a maximal Abelian subspace a of p in a Cartan decomposition g = k ⊕ p of the Lie algebra

1.3 Riemannian symmetric spaces | 21

g of G = I o (M). The rank corresponds to the maximal possible dimension of a flat totally geodesic submanifold of M. Every tangent vector v ∈ p ≅ To M is contained in at least one maximal Abelian subspace a of p. A tangent vector v ∈ p ≅ To M is said to be regular if there exists exactly one maximal Abelian subspace a of p containing v. Otherwise, v is said to be singular. It is clear from the definition that every non-zero tangent vector of a Riemannian symmetric space of rank 1 is regular. Thus, the concept of singular tangent vectors is of relevance only for higher rank symmetric spaces. Let (G, K) be a Riemannian symmetric pair so that G/K is a simply connected Riemannian symmetric space of compact type or of non-compact type, respectively. Consider the complexification gℂ = g + ig of g and the Cartan decomposition g = k ⊕ p of g. Then g∗ = k ⊕ ip is a real subalgebra of gℂ with respect to the induced Lie algebra structure. Let G∗ be the real Lie subgroup of Gℂ with Lie algebra g∗ . Then G∗ /K is a simply connected Riemannian symmetric space of non-compact type or of compact type, respectively, with Cartan decomposition g∗ = k ⊕ ip. This feature is known as duality between Riemannian symmetric spaces of compact type and of non-compact type. Duality establishes a one-to-one correspondence between simply connected Riemannian symmetric spaces of compact type and of non-compact type.

1.3.5 Classification of Riemannian symmetric spaces Every simply connected Riemannian symmetric space decomposes into the Riemannian product of a Euclidean space and some simply connected, irreducible, Riemannian symmetric spaces. Thus, the classification problem for simply connected Riemannian symmetric spaces reduces to the classification of simply connected, irreducible Riemannian symmetric spaces. Any such space is either of compact type or of non-compact type. The concept of duality enables us to reduce the classification problem to those of non-compact type. The crucial step for deriving the latter classification is to show that every non-compact irreducible Riemannian symmetric space is of the form M = G/K with some non-compact real simple Lie group G with trivial center and K a maximal compact subgroup of G. If the complexification of g is simple as a complex Lie algebra, then M is said to be of type III, otherwise M is said to be of type IV. The corresponding compact, irreducible, Riemannian symmetric spaces are said to be of types I and II, respectively. The complete list of simply connected, irreducible, Riemannian symmetric spaces is given in Tables 1.1 and 1.2. In the tables we denote by E6 , E7 , E8 , F4 , G2 the connected, simply connected, compact, real Lie group with Lie algebra e6 , e7 , e8 , f4 , g2 , respectively. The symmetric space SOp+q /SOp SOq is the Grassmann manifold of all p-dimensional oriented linear subspaces of ℝp+q and will often be denoted by Gp+ (ℝp+q ). The Grassmann manifold G2+ (ℝ4 ) is isometric to the Riemannian product S2 × S2 and hence reducible. So, strictly speaking, this special case has to be excluded from Table 1.1. Disregarding the orientation of the p-planes, we have a natural 2-fold

22 | 1 Riemannian geometry Table 1.1: Riemannian symmetric spaces of types I and III. Type I (compact)

Type III (non-compact)

Dimension

Rank

SOp+q /SOp SOq SU p+q /S(Up Uq ) Spp+q /Spp Spq SU r+1 /SOr+1 SO4r /U2r SO4r+2 /U2r+1 Spr /Ur SU 2r+2 /Spr+1 E6 /Sp4 E6 /SU 6 Sp1 E6 /Spin10 U1 E6 /F4 E7 /SU 8 E7 /SO12 Sp1 E7 /E6 U1 E8 /SO16 E8 /E7 Sp1 F4 /Sp3 Sp1 F4 /Spin9 G2 /SO4

SOop,q /SOp SOq

pq 2pq 4pq 1 r(r + 3) 2 2r(2r − 1) 2r(2r + 1) r(r + 1) r(2r + 3) 42 40 32 26 70 64 54 128 112 28 16 8

min{p, q} min{p, q} min{p, q} r r r r r 6 4 2 2 7 4 3 8 4 4 1 2

SU p,q /S(Up Uq ) Spp,q /Spp Spq SLr+1 (ℝ)/SOr+1 SO∗4r /U2r SO∗4r+2 /U2r+1 Spr (ℝ)/Ur SU ∗2r+2 /Spr+1 E66 /Sp4 E62 /SU 6 Sp1 E6−14 /Spin10 U1 E6−26 /F4 E77 /SU 8 E7−5 /SO12 Sp1 E7−25 /E6 U1 E88 /SO16 E8−24 /E7 Sp1 F44 /Sp3 Sp1 F4−20 /Spin9 G22 /SO4

Table 1.2: Riemannian symmetric spaces of types II and IV. Type II (compact)

Type IV (non-compact)

Dimension

Rank

SU r+1 Spin2r+1 Spr Spin2r E6 E7 E8 F4 G2

SLr+1 (ℂ)/SU r+1 SO2r+1 (ℂ)/SO2r+1 Spr (ℂ)/Spr SO2r (ℂ)/SO2r E6ℂ /E6 E7ℂ /E7 E8ℂ /E8 F4ℂ /F4 G2ℂ /G2

r(r + 2) r(2r + 1) r(2r + 1) r(2r − 1) 78 133 248 52 14

r r r r 6 7 8 4 2

covering map Gp+ (ℝp+q ) → Gp (ℝp+q ) onto the Grassmann manifold Gp (ℝp+q ) of all p-dimensional linear subspaces of ℝp+q , which can be written as the homogeneous space SOp+q /S(Op Oq ). Similarly, the symmetric space SU p+q /S(Up Uq ) is the Grassmann manifold of all p-dimensional complex linear subspaces of ℂp+q and will be denoted by Gp (ℂp+q ).

1.4 Hermitian symmetric spaces | 23

Eventually, the symmetric space Spp+q /Spp Spq is the Grassmann manifold of all p-dimensional quaternionic linear subspaces of ℍp+q and will be denoted by Gp (ℍp+q ). The Grassmann manifold G1+ (ℝ1+q ) is the q-dimensional sphere Sq . The Grassmann manifold G1 (ℝ1+q ) (resp. G1 (ℂ1+q ) or G1 (ℍ1+q )) is the q-dimensional real (resp. complex or quaternionic) projective space ℝP q (resp. ℂP q or ℍP q ). The dual space of the sphere Sq is the real hyperbolic space ℝH q . The dual space of the complex projective space ℂP q (resp. the quaternionic projective space ℍP q ) is the complex hyperbolic space ℂH q (resp. the quaternionic hyperbolic space ℍH q ). In small dimensions, certain symmetric spaces are isometric to each other (with a suitable normalization of the Riemannian metric): S2 = ℂP 1 = SU 2 /SO2 = SO4 /U2 = Sp1 /U1 ,

S5 = SU 4 /Sp2 ,

G2+ (ℝ6 ) = G2 (ℂ4 ),

ℂP 3 = SO6 /U3 ,

G2+ (ℝ8 ) = SO8 /U4 ,

S4 = ℍP 1 ,

G2+ (ℝ5 ) = Sp2 /U2 ,

G3+ (ℝ6 ) = SU 4 /SO4 .

In the non-compact case, there are isometries between the corresponding dual symmetric spaces. Since Spin2 is isomorphic to U1 and Spin4 is isomorphic to the product SU 2 × SU 2 we have to assume r ≥ 3 for Spin2r in Table 1.2. In small dimensions there are the following additional isomorphisms: Spin3 = SU 2 = Sp1 ,

Spin5 = Sp2 ,

Spin6 = SU 4 .

In the non-compact case, there are isometries between the corresponding dual symmetric spaces.

1.4 Hermitian symmetric spaces 1.4.1 Definition and classification A good introduction to Hermitian symmetric spaces is the paper [121] by Wolf. A Hermitian symmetric space is a Riemannian symmetric space that is equipped with a Kähler structure for which the geodesic symmetries are holomorphic maps. The simplest example of a Hermitian symmetric space is the complex Euclidean space ℂn . For semisimple Riemannian symmetric spaces one can easily decide whether or not it is Hermitian. In fact, let (G, K) be the Riemannian symmetric pair of an irreducible Riemannian semisimple symmetric space M. Then the center of K is either discrete or 1-dimensional. The irreducible semisimple Hermitian symmetric spaces are precisely those for which the center of K is 1-dimensional. From Tables 1.1 and 1.2 we can

24 | 1 Riemannian geometry Table 1.3: Irreducible Hermitian symmetric spaces. Type I (compact)

Type III (non-compact)

Dimension

Rank

Comments

SO2+n /SO2 SOn SU r+k /S(Ur Uk ) SO2k /Uk Spr /Ur E6 /Spin10 U1 E7 /E6 U1

SOo2,n /SO2 SOn

2n 2rk k(k − 1) r(r + 1) 32 54

2 r [ 2k ] r 2 3

n≥3 k≥r≥1 k≥2 r≥1

SU r,k /S(Ur Uk ) SO∗2k /Uk Spr (ℝ)/Ur E6−14 /Spin10 U1 E7−25 /E6 U1

then deduce the classification of irreducible Hermitian symmetric spaces, which we summarize in Table 1.3. The Hermitian symmetric space SO2+n /SO2 SOn is the real Grassmann manifold G2+ (ℝ2+n ) of oriented 2-dimensional subspaces of ℝ2+n . The Hermitian symmetric space SU r+k /S(Ur Uk ) is the complex Grassmann manifold Gr (ℂr+k ) of r-dimensional complex subspaces of ℂr+k . In low dimensions and for the compact type we have the following isometries: S2 ≅ SO3 /SO2 ≅ ℂP 1 ≅ G1 (ℂ2 ) ≅ SU 2 /S(U1 U1 ) ≅ SO4 /U2 ≅ Sp1 /U1 ,

ℂP 3 ≅ G1 (ℂ4 ) ≅ SU 4 /S(U1 U3 ) ≅ SO6 /U3 ,

G2+ (ℝ5 ) ≅ SO5 /SO2 SO3 ≅ Sp2 /U2 ,

G2+ (ℝ6 ) ≅ SO6 /SO2 SO4 ≅ G2 (ℂ4 ) ≅ SU 4 /S(U2 U2 ),

G2+ (ℝ8 ) ≅ SO8 /SO2 SO6 ≅ SO8 /U4 .

Using duality, we obtain isometries between the dual Hermitian symmetric spaces of non-compact type. Every semisimple Hermitian symmetric space is simply connected and hence decomposes into the Riemannian product of irreducible Hermitian symmetric spaces.

1.4.2 Complex quadrics 2 Let n be a positive integer. The homogeneous quadratic equation z12 + ⋅ ⋅ ⋅ + zn+2 = 0 on n+2 ℂ defines a complex hypersurface 2 Qn = {[z] ∈ ℂP n+1 : z12 + ⋅ ⋅ ⋅ + zn+2 = 0}

of the (n + 1)-dimensional complex projective space ℂP n+1 . This complex hypersurface is generally known as the n-dimensional complex quadric in ℂP n+1 . The complex structure J on ℂP n+1 naturally induces a complex structure on Qn , which we will denote by J as well. We equip Qn with the Riemannian metric g induced from the Fubini–Study

1.4 Hermitian symmetric spaces | 25

metric on ℂP n+1 with constant holomorphic sectional curvature 4. The 1-dimensional complex quadric Q1 is isometric to the 2-dimensional sphere S2 (4) with constant sectional curvature 4. The 2-dimensional complex quadric Q2 is isometric to the Riemannian product S2 (4)×S2 (4). For n ≥ 2, the triple (Qn , g, J) is a Hermitian symmetric space of rank 2 and its maximal sectional curvature is equal to 4. The complex projective space ℂP n+1 can be realized as the Hermitian symmetric space SU 2+n /S(U1 Un+1 ). The special unitary group SU 2+n acts on ℂP n+1 by isometries and contains in a canonical way the special orthogonal group SO2+n . The action of SO2+n on ℂP n+1 is of cohomogeneity 1, that is, the codimension of a principal orbit of the action has codimension 1. The action has exactly two singular orbits. Let e1 , . . . , en+2 be the standard Hermitian orthonormal basis of ℂn+2 . The orbit SOn+2 ⋅ [e1 ] of SO2+n through [e1 ] ∈ ℂP n+1 is the real projective space ℝP n+1 in ℂP n+1 . The second singular orbit is the orbit of SO2+n through [ √12 (e1 + ie2 )] ∈ ℂP n+2 . This orbit coincides with the complex quadric Qn ⊂ ℂP n+1 as defined above. The isotropy group of SO2+n at [ √12 (e1 + ie2 )] is SO2 SOn , which provides a realization of SO2+n /SO2 SOn

as the complex quadric Qn in ℂP n+1 . Another geometric realization arises from the fact that SO2+n acts transitively on the Grassmann manifold G2+ (ℝn+2 ) of all oriented 2-dimensional linear subspaces of ℝn+2 . The isotropy group of SO2+n at the oriented 2-plane ℝ2 ⊕ {0} ⊂ ℝ2 ⊕ ℝn ≅ ℝn+2 is SO2 SOn . We thus have three different models in this situation: SO2+n /SO2 SOn ≅ Qn ≅ G2+ (ℝn+2 ). Using the Hopf map S2n+3 → ℂP n+1 , we can identify for each [z] ∈ ℂP n+1 the tangent space T[z] ℂP n+1 with the orthogonal complement ℂn+2 ⊖ [z] of the complex line [z] = ℂz in ℂn+2 . For [z] ∈ Qn the tangent space T[z] Qn can then be identified ̄ of the 2-dimensional canonically with the orthogonal complement ℂn+2 ⊖ ([z] ⊕ [z]) complex subspace [z] ⊕ [z]̄ = ℂz ⊕ ℂz̄ in ℂn+2 : ̄ T[z] Qn = ℂn+2 ⊖ ([z] ⊕ [z]).

Note that z̄ is a unit normal vector of Qn in ℂP n+1 at the point [z]. To discuss further details of the geometry of the complex quadric, we use some concepts from submanifold geometry which will be discussed in more detail in Section 2. We denote by Az̄ the shape operator of Qn in ℂP n+1 with respect to z.̄ Then we have Az̄ v = v for all v ∈ T[z] Qn , that is, the shape operator Az̄ coincides with complex conjugation in ℂn+2 restricted to T[z] Qn . The shape operator Az̄ is an antilinear involution on the complex vector space T[z] Qn inducing the decomposition T[z] Qn = V(Az̄ ) ⊕ JV(Az̄ ), where V(Az̄ ) = ℝn+2 ∩ T[z] Qn is the (+1)-eigenspace and JV(Az̄ ) = iℝn+2 ∩ T[z] Qn is the (−1)-eigenspace of Az̄ . Geometrically this means that the shape operator Az̄ defines

26 | 1 Riemannian geometry a real structure on the complex vector space T[z] Qn . Recall that a real structure on a complex vector space V is by definition an antilinear involution C : V → V, which can be considered as complex conjugation. Since the normal space ν[z] Qn of Qn in ℂP n+1 at [z] is a 1-dimensional complex subspace of T[z] ℂP n+1 , every normal vector in ν[z] Qn can be written as λz̄ with some λ ∈ ℂ. The family (Aλz̄ )λ∈ℂ of shape operators of Qn defines a rank 2 vector subbundle A of the endomorphism bundle End(TQn ). Since the second fundamental form of the embedding Qn ⊂ ℂP n+1 is parallel (see, e. g., [90]), A is a parallel subbundle of End(TQn ). For λ ∈ S1 ⊂ ℂ we again get a real structure Aλz̄ on T[z] Qn and we have V(Aλ2 z̄ ) = λV(Az̄ ). We thus have an S1 -subbundle of A consisting of real structures on the tangent spaces of Qn . To avoid notational confusion with the symbol A that were are using in general for shape operators, we are going to use the symbol C to denote a real structure in A. The Gauss equation for the complex hypersurface Qn ⊂ ℂP n+1 implies that the Riemannian curvature tensor R of Qn can be expressed in terms of the Riemannian metric g, the complex structure J and a generic real structure C in A. By contraction we then also obtain the Ricci tensor of Qn . Theorem 1.4.1 ([84]). The Riemannian curvature tensor R of the complex quadric Qn = SO2+n /SO2 SOn , n ≥ 3, satisfies R(X, Y)Z = g(Y, Z)X − g(X, Z)Y + g(JY, Z)JX − g(JX, Z)JY − 2g(JX, Y)JZ + g(CY, Z)CX − g(CX, Z)CY + g(JCY, Z)JCX − g(JCX, Z)JCY

for all X, Y, Z ∈ X(Qn ), where C is an arbitrary real structure in A. The Ricci tensor Ric of Qn satisfies Ric(X) = 2nX for all X ∈ X(Qn ). Note that the complex structure J anticommutes with each endomorphism C ∈ A, that is, CJ = −JC. From the explicit expression of the curvature tensor it is straightforward to find the singular tangent vectors of Qn , n ≥ 2. There are exactly two types of singular tangent vectors v ∈ T[z] Qn , which can be characterized as follows: (i) If there exists a real structure C ∈ A[z] such that v ∈ V(C), then v is singular. Such a singular tangent vector is called A-principal. (ii) If there exist a real structure C ∈ A[z] and orthonormal vectors u, w ∈ V(C) such v = √12 (u + Jw), then v is singular. Such a singular tangent vector is called that ||v|| A-isotropic.

1.4 Hermitian symmetric spaces | 27

Basic complex linear algebra shows that for every unit tangent vector v ∈ T[z] Qn there exist a real structure C ∈ A[z] and orthonormal vectors u, w ∈ V(C) such that v = cos(t)u + sin(t)Jw for some t ∈ [0, π4 ]. The singular tangent vectors correspond to the values t = 0 and t = π4 . Let us briefly summarize how we constructed the Riemannian structure on Qn . We start with the unit sphere S2n+3 together with its standard Riemannian metric of constant sectional curvature 1. The Hopf map S2n+3 → ℂP n+1 then induces the Fubini– Study metric on ℂP n+1 with constant holomorphic sectional curvature 4. This metric induces on the complex quadric Qn a Riemannian metric whose sectional curvature lies in the interval [0, 4]. The above expression for the Riemannian curvature tensor of Qn implies that the maximal value 4 of the sectional curvature is obtained precisely for the 2-planes of the form V = ℂv, where v is an A-isotropic singular tangent vector of the quadric. More general, if v is a singular tangent vector of Qn , the Jacobi operator Rv = R(⋅, v)v has the following eigenvalues and eigenspaces. Lemma 1.4.2. Let v ∈ T[z] Qn be a singular unit tangent vector. Then v is either A-principal or A-isotropic. (i) If v is A-principal, say v ∈ V(C) with real structure C ∈ A, then the eigenvalues and eigenspaces of the Jacobi operator Rv are 0 2

ℝv ⊕ (JV(C) ⊖ ℝJv), ℝJv ⊕ (V(C) ⊖ ℝv).

(ii) If v is A-isotropic, say v = √12 (u + Jw) with orthonormal vectors u, w ∈ V(C) for some real structure C ∈ A, then the eigenvalues and eigenspaces of the Jacobi operator Rv are 0

ℝv ⊕ ℂCv,

1 T[z] Qn ⊖ (ℂv ⊕ ℂCv),

4 ℝJv. 1.4.3 Complex hyperbolic quadrics

The complex hyperbolic quadric Qn = SOo2,n /SO2 SOn is the non-compact dual symmetric space of the complex quadric Qn = SO2+n /SO2 SOn . The 1-dimensional complex hy∗ perbolic quadric Q1 is isometric to the 2-dimensional real hyperbolic space ℝH 2 (−4) ∗ with constant sectional curvature −4. The 2-dimensional complex quadric Q2 is isometric to the Riemannian product ℝH 2 (−4) × ℝH 2 (−4). For n ≥ 3, the complex hyper∗

28 | 1 Riemannian geometry bolic quadric Qn is an irreducible Hermitian symmetric space of non-compact type ∗ with rk(Qn ) = 2. We saw in Section 1.4.2 that the complex quadric Qn can be realized as a homogeneous complex hypersurface in the complex projective space ℂP n+1 . One might think that there is a dual model for that. However, the complex hyperbolic quadric ∗ Qn cannot be realized as a complex hypersurface of the complex hyperbolic space ℂH n+1 . In fact, Smyth [91] proved that every homogeneous complex hypersurface in ℂH n+1 is a totally geodesic complex hyperbolic hyperplane ℂH n . Another result by Smyth [91] states that any complex hypersurface of ℂH n+1 for which the square of the shape operator has constant eigenvalues (counted with multiplicities) is totally ∗ geodesic. This also excludes the possibility of a model for Qn as a complex hypersurface of ℂH n+1 with the analogous property for the shape operator. We therefore need a different approach in this situation. We will use the symmetric ∗ space model Qn = SOo2,n /SO2 SOn for our approach (see also [58]). Let G = SOo2,n and ∗ K = SO2 SOn be the isotropy group of G at o ∈ Qn . The Lie algebra g = so2,n of G is given by ∗

g = {Y ∈ gl2+n (ℝ) : Y ⊤ I2,n + I2,n Y = 0} with 0 ) ∈ gln+2 (ℝ). In

−I2 0

I2,n = ( It follows that we can write A g = {( ⊤ X

X ) : A ∈ so2 , B ∈ son , X ∈ M2,n (ℝ)} , B

where M2,n (ℝ) is the real vector space of 2×n matrices with real coefficients. The Cartan involution on g is σ(Y) = −Y ⊤ and therefore the Cartan decomposition is g = k ⊕ p with k = {Y ∈ g : σ(Y) = Y} = {(

A 0

0 ) : A ∈ so2 , B ∈ son } ≅ so2 ⊕ son , B

0 p = {Y ∈ g : σ(Y) = −Y} = {( ⊤ X

X ∗ ) : X ∈ M2,n (ℝ)} ≅ To Qn . 0

We will identify X ∈ M2,n (ℝ) ≅ To Qn with the matrix ( X0⊤ X0 ) ∈ p. If B is the Killing form of g, then ∗

0 ⟨X, Y⟩ = −B (( ⊤ X

X 0 ) , σ (( ⊤ 0 Y

Y ))) = 2ntr(Y ⊤ X) 0

1.4 Hermitian symmetric spaces | 29

is an inner product on p. This inner product is clearly Ad(K)-invariant and induces ∗ ∗ an Ad(G)-invariant Riemannian metric on Qn for which Qn becomes a Riemannian symmetric space. The matrix 0 j = (−1

0

1 0

0 In

)∈K

lies in the center of K. We can define an orthogonal complex structure J on p by JX = Ad(j)X for all X ∈ p. Because j is in the center of K, the orthogonal complex structure J ∗ on p is Ad(K)-invariant and induces an Ad(G)-invariant Kähler structure on Qn , which n∗ we also denote by J. This turns Q into a Hermitian symmetric space. Similar to the complex quadric, there is another important geometric structure on the complex hyperbolic quadric besides the Riemannian metric and the Kähler structure, namely an S1 -bundle of real structures (or conjugations). The situation here is ∗ different though from that of the complex quadric in that for Qn the real structures cannot be realized as the shape operator of a complex hypersurface in a complex space form. Let 1 c0 = (0

0 −1 0

0 In

).

Note that c0 ∈ ̸ K = SO2 SOn , but c0 ∈ O2 SOn , that is, c0 is in the isotropy group at o of ∗ the full isometry group of Qn . The adjoint transformation Ad(c0 ) leaves p invariant ∗ and C0 = Ad(c0 )|p is an antilinear involution on p ≅ To Qn satisfying C0 J + JC0 = 0. In ∗ other words, C0 is a real structure on To Qn . The involution C0 commutes with Ad(g) for all g ∈ SOn ⊂ K but not for all g ∈ K. More precisely, for g = (g1 , g2 ) ∈ K with − sin(t) g1 ∈ SO2 and g2 ∈ SOn , say g1 = ( cos(t) sin(t) cos(t) ) with t ∈ ℝ, so that Ad(g1 ) corresponds to multiplication with the complex number μ = eit , we have C0 ∘ Ad(g) = μ−2 Ad(g) ∘ C0 . This equation shows that the object which is Ad(K)-invariant and therefore geometrically relevant is not the real structure C0 by itself, but rather the “circle of real structures” {λC0 : λ ∈ S1 }. This set is Ad(K)-invariant and therefore generates an Ad(G)-invariant S1 -subbundle ∗ A0 of the endomorphism bundle End(TQn ), consisting of real structures (or conjun∗ gations) on the tangent spaces of Q . This S1 -bundle naturally extends to an Ad(G)-

30 | 1 Riemannian geometry invariant vector subbundle A of End(TQn ), which is parallel with respect to the in∗ duced connection on End(TQn ). We are mainly interested though in the circle of real structures in A. For any real structure C ∈ A the tangent line to the fiber of A through ∗ C is spanned by JC. For every p ∈ Qn and real structure C ∈ Ap we have an orthogonal decomposition ∗

Tp Qn = V(C) ⊕ JV(C) ∗

into two totally real subspaces of Tp Qn . Here V(C) and JV(C) are the (+1)- and (−1)eigenspace of C, respectively. As for the complex quadric, the Riemannian metric g, the Kähler structure J and ∗ a real structure C on Qn can be used to give an explicit expression of the Riemannian ∗ curvature tensor R of Qn . For simplicity, we normalize here the Riemannian metric ∗ on Qn so that the minimum of the sectional curvature is equal to −4. ∗

Theorem 1.4.3. The Riemannian curvature tensor R of the complex hyperbolic quadric ∗ Qn = SO2,n /SO2 SOn , n ≥ 3, satisfies R(X, Y)Z = −g(Y, Z)X + g(X, Z)Y − g(JY, Z)JX + g(JX, Z)JY + 2g(JX, Y)JZ − g(CY, Z)CX + g(CX, Z)CY − g(JCY, Z)JCX + g(JCX, Z)JCY

for all X, Y, Z ∈ X(Qn ), where C is an arbitrary real structure in A. The Ricci tensor Ric ∗ of Qn satisfies ∗

Ric(X) = −2nX for all X ∈ X(Qn ). ∗

Note that this expression for R is just the negative of the corresponding expression for the curvature tensor in the complex quadric, which is a fact coming from duality between symmetric spaces of compact type and of non-compact type. From the explicit expression of the curvature tensor it is straightforward to find ∗ the singular tangent vectors of Qn , n ≥ 2. There are exactly two types of singular n∗ tangent vectors v ∈ Tp Q , which can be characterized as follows: (i) If there exists a real structure C ∈ Ap such that v ∈ V(C), then v is singular. Such a singular tangent vector is called A-principal. (ii) If there exist a real structure C ∈ Ap and orthonormal vectors u, w ∈ V(C) such v = √12 (u + Jw), then v is singular. Such a singular tangent vector is called that ||v|| A-isotropic. Basic complex linear algebra shows that for every unit tangent vector v ∈ Tp Qn there exist a real structure C ∈ Ap and orthonormal vectors u, w ∈ V(C) such that ∗

v = cos(t)u + sin(t)Jw

1.4 Hermitian symmetric spaces | 31

for some t ∈ [0, π4 ]. The singular tangent vectors correspond to the values t = 0 and t = π4 . ∗ The above expression for the Riemannian curvature tensor of Qn implies that the minimal value −4 of the sectional curvature is obtained precisely for the 2-planes of the form V = ℂv, where v is an A-isotropic singular tangent vector of the quadric. More ∗ generally, if v is a singular tangent vector of Qn , the Jacobi operator Rv = R(⋅, v)v has the following eigenvalues and eigenspaces. Lemma 1.4.4. Let v ∈ Tp Qn be a singular unit tangent vector. Then v is either A-principal or A-isotropic. (i) If v is A-principal, say v ∈ V(C) with real structure C ∈ Ap , then the eigenvalues and eigenspaces of the Jacobi operator Rv are ∗

0

ℝv ⊕ (JV(C) ⊖ ℝJv),

−2

ℝJv ⊕ (V(C) ⊖ ℝv).

(ii) If v is A-isotropic, say v = √12 (u + Jw) with orthonormal vectors u, w ∈ V(C) for some real structure C ∈ Ap , then the eigenvalues and eigenspaces of the Jacobi operator Rv are 0

ℝv ⊕ ℂCv,

−1

Tp Qn ⊖ (ℂv ⊕ ℂCv),

−4

ℝJv.

∗

1.4.4 Complex 2-plane Grassmannians We denote by Gr (ℂr+k ) the set of all r-dimensional complex subspaces of ℂr+k , where r ≥ 1 and k ≥ 1. The special unitary group G = SU r+k acts transitively on Gr (ℂr+k ) and the isotropy group K at ℂr ⊕ {0} ⊂ ℂr ⊕ ℂk = ℂr+k is K = S(Ur Uk ). Thus Gr (ℂr+k ) can be identified with the homogeneous space G/K, which we equip with the unique analytic structure for which the natural action of G on Gr (ℂr+k ) becomes analytic. Since Gr (ℂr+k ) ≅ Gk (ℂr+k ), we can assume r ≤ k. Denote by g and k the Lie algebra of G and K, respectively, and by p the orthogonal complement of k in p with respect to the Killing form B of g. Then g = k ⊕ p is an Ad(K)invariant reductive decomposition of g. We put o = eK and identify To Gr (ℂr+k ) with p in the usual manner. Since B is negative definite on g, its negative restricted to p × p defines an inner product on p. By Ad(K)-invariance of B, this inner product can be extended to a G-invariant Riemannian metric gB on Gr (ℂr+k ), the so-called Killing metric on Gr (ℂr+k ). In this way Gr (ℂr+k ) becomes a Riemannian symmetric space. The maximum value of the sectional curvature of the Killing metric on Gr (ℂr+k ) is the square length of the highest root in a restricted root space decomposition of sur+k . As this

32 | 1 Riemannian geometry value changes with the dimension r + k, we rescale the Killing metric by a constant factor so that the maximum value of the sectional curvature is 4 for r = 1 and 8 for r ≥ 2. For r = 1 we then obtain the k-dimensional complex projective space ℂP k with the Fubini–Study metric that we introduced in Section 1.2.4. The Lie algebra k has the direct sum decomposition k = sur ⊕ suk ⊕ ℝ, where ℝ is the center of k. Since k is the holonomy algebra of Gr (ℂr+k ), the center ℝ induces a Kähler structure J on Gr (ℂr+k ). This turns Gr (ℂr+k ) into a Hermitian symmetric space. We have rk(Gr (ℂr+k )) = r, m = dim(Gr (ℂr+k )) = 2rk, and n = dimℂ (Gr (ℂr+k )) = rk. If r = 2, the normal subalgebra su2 of k induces, via the adjoint representation, a quaternionic Kähler structure J on G2 (ℂ2+k ). Thus, the complex 2-plane Grassmannian G2 (ℂ2+k ) is equipped with both a Kähler structure J and a quaternionic Kähler structure J. Lemma 1.4.5. Let M = G2 (ℂ2+k ), k ≥ 2, with its Kähler structure J and its quaternionic Kähler structure J. Let J1 be a (local) almost Hermitian structure in J. Then we have JJ1 = J1 J,

(JJ1 )2 = id,

tr(JJ1 ) = 0.

The complex Grassmann manifold G2 (ℂ3 ) is isometric to the complex projective plane ℂP 2 (8) with constant holomorphic sectional curvature 8. Note that the isomorphism Spin6 ≅ SU 4 leads to an isometry between G2 (ℂ4 ) and the real Grassmann manifold G2+ (ℝ6 ) of oriented 2-dimensional linear subspaces of ℝ6 , which is isometric to the complex quadric Q4 = SO6 /SO2 SO4 (with a suitably rescaled metric by a constant factor). Theorem 1.4.6 ([6]). The Riemannian curvature tensor R of the complex 2-plane Grassmannian G2 (ℂ2+k ) = SU 2+k /S(U2 Uk ), k ≥ 3, satisfies R(X, Y)Z = g(Y, Z)X − g(X, Z)Y + g(JY, Z)JX − g(JX, Z)JY − 2g(JX, Y)JZ 3

+ ∑ (g(Jν Y, Z)Jν X − g(Jν X, Z)Jν Y − 2g(Jν X, Y)Jν Z ν=1

+ g(Jν JY, Z)Jν JX − g(Jν JX, Z)Jν JY) for all X, Y, Z ∈ X(G2 (ℂ2+k )), where J1 , J2 , J3 is any canonical local basis of J. The Ricci tensor Ric of G2 (ℂ2+k ) satisfies Ric(X) = 4(k + 2)X. Proof. (Outline, see [6] for details.) We start with the unit sphere S4k+7 ⊂ ℍk+2 , equipped with the standard metric of constant sectional curvature 1. Then consider the Hopf map S4k+7 → ℍP k+1 onto the quaternionic projective space ℍP k+1 . There is a unique Riemannian structure on ℍP k+1 so that the Hopf map becomes a Riemannian submersion, with sectional curvature K on ℍP k+1 bounded by 1 ≤ K ≤ 4.

1.4 Hermitian symmetric spaces | 33

Next, consider the standard totally geodesic embedding of the complex projective space ℂP k+1 into ℍP k+1 . The focal set of ℂP k+1 in ℍP k+1 is an embedding of the homogeneous space SU 2+k /SU 2 SU k into ℍP k+1 as a submanifold with codimension 3. In fact, the isometric action of SU k+2 ⊂ Spk+2 = I o (ℍP k+1 ) is of cohomogeneity 1 and ℂP k+1 and SU 2+k /SU 2 SU k are the two singular orbits of this action. Using Jacobi field theory it is straightforward to calculate the shape operator of SU 2+k /SU 2 SU k in ℍP k+1 . Using the Gauss equation we can then work out the Riemannian curvature tensor of SU 2+k /SU 2 SU k with respect to the Riemannian metric induced from ℍP k+1 . The homogeneous space SU 2+k /SU 2 SU k is a circle bundle over SU 2+k /S(U2 Uk ) = G2 (ℂ2+k ) and is equipped with a Sasakian structure. This is a special case of the standard circle bundle G/H → G/K with K = U1 H over any Hermitian symmetric space G/K of compact type. We equip SU 2+k /S(U2 Uk ) = G2 (ℂ2+k ) with the Riemannian metric so that the canonical projection from SU 2+k /SU 2 SU k onto SU 2+k /S(U2 Uk ) becomes a Riemannian submersion. The Riemannian structure on G2 (ℂ2+k ) obtained in this way coincides, up to rescaling, with the one obtained from the Killing form of su2+k . The quaternionic Kähler structure on G2 (ℂ2+k ) is induced from the restriction of the quaternionic Kähler structure on ℍP k+1 to the maximal quaternionic subbundle of the tangent bundle of SU 2+k /SU 2 SU k . The Kähler structure on G2 (ℂ2+k ) is induced from the Sasakian structure on SU 2+k /SU 2 SU k . Then, using the well-developed theory of Riemannian submersions, we can work out explicitly the Riemannian curvature tensor of G2 (ℂ2+k ) = SU 2+k /S(U2 Uk ). The expression for the Ricci tensor follows by contracting R. From the explicit expression of the curvature tensor we can find the singular tangent vectors of G2 (ℂ2+k ), k ≥ 3. There are exactly two types of singular tangent vectors v ∈ Tp G2 (ℂ2+k ), characterized by the geometric properties Jv ⊥ Jv and Jv ∈ Jv. If v is a singular tangent vector of G2 (ℂ2+k ), the Jacobi operator Rv = R(⋅, v)v has the following eigenvalues and eigenspaces. Lemma 1.4.7. Let v ∈ G2 (ℂ2+k ), n ≥ 3, be a singular unit tangent vector at p ∈ G2 (ℂ2+k ). Then either Jv ⊥ Jv or Jv ∈ Jv. (i) If Jv ⊥ Jv, then the eigenvalues and eigenspaces of the Jacobi operator Rv are 0

ℝv ⊕ JJv,

1 Tp G2 (ℂ2+k ) ⊖ (ℍv ⊕ ℍJv),

4 ℝJv ⊕ Jv.

(ii) If Jv ∈ Jv, say Jv = J1 v with an almost Hermitian structure J1 ∈ Jp , then the eigenvalues and eigenspaces of the Jacobi operator Rv are 0

ℝv ⊕ {u ∈ Tp G2 (ℂ2+k ) : u ⊥ ℍv, Ju = −J1 u},

34 | 1 Riemannian geometry 2 (ℍv ⊖ ℂv) ⊕ {u ∈ Tp G2 (ℂ2+k ) : u ⊥ ℍv, Ju = J1 u},

8 ℝJv.

It follows from Lemma 1.4.7 that the maximum value 8 of the sectional curvature of G2 (ℂ2+k ) is obtained exactly for the 2-dimensional planes ℂv, where v is a singular tangent vector of type Jv ∈ Jv. 1.4.5 Complex hyperbolic 2-plane Grassmannians The non-compact dual symmetric space of the complex Grassmann manifold Gr (ℂr+k ) = SU r+k /S(Ur Uk ) is the complex hyperbolic Grassmann manifold Gr∗ (ℂr+k ) = SU r,k /S(Ur Uk ). Here we consider ℂr+k being equipped with the indefinite Hermitian form r

r+k

ν=1

ν=r+1

⟨a, b⟩ = − ∑ aν b̄ ν + ∑ aν b̄ ν

(a, b ∈ ℂr+k ).

Then Gr∗ (ℂr+k ) is the set of all r-dimensional linear subspaces of ℂr+k on which the restriction of ⟨⋅, ⋅⟩ is negative definite. The indefinite special unitary group G = SU r,k acts transitively on Gr∗ (ℂr+k ) and the isotropy group K at ℂr ⊕ {0} ⊂ ℂr ⊕ ℂk = ℂr+k is K = S(Ur Uk ). Thus Gr∗ (ℂr+k ) can be identified with the homogeneous space G/K, which we equip with the unique analytic structure for which the natural action of G on Gr∗ (ℂr+k ) becomes analytic. Since Gr∗ (ℂr+k ) ≅ Gk∗ (ℂr+k ), we can assume r ≤ k. Denote by g and k the Lie algebras of G and K, respectively, and by p the orthogonal complement of k in p with respect to the Killing form B of g. Then g = k ⊕ p is an Ad(K)invariant reductive decomposition of g. We put o = eK and identify To Gr∗ (ℂr+k ) with p in the usual manner. Since B is positive definite on p, it defines an inner product on p. By Ad(K)-invariance of B, this inner product can be extended to a G-invariant Riemannian metric gB on Gr∗ (ℂr+k ), the so-called Killing metric on Gr∗ (ℂr+k ). In this way Gr∗ (ℂr+k ) becomes a Riemannian symmetric space. We rescale the Killing metric by a constant factor so that the minimum value of the sectional curvature is −4 for r = 1 and −8 for r ≥ 2. For r = 1 we then obtain the k-dimensional complex hyperbolic space ℂH k with the Fubini–Study metric that we introduced in Section 1.2.5. The Lie algebra k has the direct sum decomposition k = sur ⊕ suk ⊕ ℝ, where ℝ is the center of k. Since k is the holonomy algebra of Gr∗ (ℂr+k ), the center ℝ induces a Kähler structure J on Gr∗ (ℂr+k ). This turns Gr∗ (ℂr+k ) into a Hermitian symmetric space. We have rk(Gr∗ (ℂr+k )) = r, m = dim(Gr∗ (ℂr+k )) = 2rk and n = dimℂ (Gr∗ (ℂr+k )) = rk. If r = 2, the normal subalgebra su2 of k induces, via the adjoint representation, a quaternionic Kähler structure J on G2∗ (ℂ2+k ). Thus, the complex hyperbolic 2-plane Grassmannian G2∗ (ℂ2+k ) is equipped with both a Kähler structure J and a quaternionic Kähler structure J.

1.4 Hermitian symmetric spaces | 35

Lemma 1.4.8. Let M = G2∗ (ℂ2+k ), k ≥ 2, with its Kähler structure J and its quaternionic Kähler structure J. Let J1 be a (local) almost Hermitian structure in J. Then we have JJ1 = J1 J,

(JJ1 )2 = id,

tr(JJ1 ) = 0.

The complex hyperbolic Grassmann manifold G2∗ (ℂ3 ) is isometric to the complex hyperbolic plane ℂH 2 (−8) with constant holomorphic sectional curvature −8. Note that the isomorphism Spin2,4 ≅ SU 2,2 leads to an isometry between G2∗ (ℂ4 ) and the ∗ complex hyperbolic quadric Q4 = SOo2,4 /SO2 SO4 (with a suitably rescaled metric by a constant factor). Theorem 1.4.9. The Riemannian curvature tensor R of the complex hyperbolic Grassmann manifold G2∗ (ℂ2+k ) = SU 2,k /S(U2 Uk ), k ≥ 3, satisfies R(X, Y)Z = −g(Y, Z)X + g(X, Z)Y − g(JY, Z)JX + g(JX, Z)JY + 2g(JX, Y)JZ 3

− ∑ (g(Jν Y, Z)Jν X − g(Jν X, Z)Jν Y − 2g(Jν X, Y)Jν Z ν=1

+ g(Jν JY, Z)Jν JX − g(Jν JX, Z)Jν JY) for all X, Y, Z ∈ X(G2∗ (ℂ2+k )), where J1 , J2 , J3 is any canonical local basis of J. The Ricci tensor Ric of G2∗ (ℂ2+k ) satisfies Ric(X) = −4(k + 2)X. Proof. This follows from Theorem 1.4.6, since the Riemannian curvature tensors of dual Riemannian symmetric spaces differ only by sign. From the explicit expression of the curvature tensor we can find the singular tangent vectors of G2∗ (ℂ2+k ), k ≥ 3. There are exactly two types of singular tangent vectors v ∈ Tp G2∗ (ℂ2+k ), characterized by the geometric properties Jv ⊥ Jv and Jv ∈ Jv. If v is a singular tangent vector of G2∗ (ℂ2+k ), the Jacobi operator Rv = R(⋅, v)v has the following eigenvalues and eigenspaces. Lemma 1.4.10. Let v ∈ G2∗ (ℂ2+k ), n ≥ 3, be a singular unit tangent vector at p ∈ G2∗ (ℂ2+k ). Then either Jv ⊥ Jv or Jv ∈ Jv. (i) If Jv ⊥ Jv, then the eigenvalues and eigenspaces of the Jacobi operator Rv are 0

ℝv ⊕ JJv,

−1 Tp G2∗ (ℂ2+k ) ⊖ (ℍv ⊕ ℍJv), −4 ℝJv ⊕ Jv.

36 | 1 Riemannian geometry (ii) If Jv ∈ Jv, say Jv = J1 v with an almost Hermitian structure J1 ∈ Jp , then the eigenvalues and eigenspaces of the Jacobi operator Rv are 0 ℝv ⊕ {u ∈ Tp G2∗ (ℂ2+k ) : u ⊥ ℍv, Ju = −J1 u},

−2 (ℍv ⊖ ℂv) ⊕ {u ∈ Tp G2∗ (ℂ2+k ) : u ⊥ ℍv, Ju = J1 u}, −8 ℝJv. It follows from Lemma 1.4.10 that the minimum value −8 of the sectional curvature of G2∗ (ℂ2+k ) is obtained exactly for the 2-dimensional planes ℂv, where v is a singular tangent vector of type Jv ∈ Jv.

2 Submanifolds of Riemannian manifolds In this chapter we present basic material about submanifolds of Riemannian manifolds.

2.1 Fundamental equations of submanifold theory Let M̄ be a Riemannian manifold with Riemannian metric g,̄ Riemannian curvature tensor R̄ and Riemannian connection ∇.̄ ̄ The codimension of M is defined as Let M be a submanifold of M. ̄ − dim(M). codim(M) = dim(M) The tangent bundle and the normal bundle of M will be denoted by TM and νM, rē p ∈ M, we denote by v⊤ and v⊥ the orthogonal spectively. For a tangent vector v ∈ Tp M, projection of v onto Tp M and νp M, respectively. By restricting the Riemannian metric ḡ to tangent vectors of M we obtain a Riemannian metric g on M. We denote by R the Riemannian curvature tensor of (M, g) and by ∇ the Riemannian connection of (M, g). We denote by Γ(νM) the space of sections in the normal bundle νM of M. Note that a section ξ ∈ Γ(νM) is a normal vector field on M, that is, a vector field on M so that ξp ∈ νp M for all p ∈ M. The following fundamental equations of submanifold geometry encode significant ̄ information about the geometry of the submanifold M in the Riemannian manifold M. Theorem 2.1.1 (Fundamental equations of submanifold geometry). Let M be a submanifold of a Riemannian manifold M.̄ Then the following equations hold for all tangent vector fields X, Y, Z, W ∈ X(M) and normal vector fields ζ , η ∈ Γ(νM): Gauss formula: ∇̄X Y = ∇X Y + α(X, Y), Weingarten formula: ∇̄X ζ = −Aζ X + ∇X⊥ ζ , Gauss equation: ̄ Y)Z, W) = g(R(X, Y)Z, W) − g(α(Y, ̄ ̄ g(̄ R(X, Z), α(X, W)) + g(α(X, Z), α(Y, W)), Codazzi equation: ̄ Y)Z) = (∇⊥ α)(Y, Z) − (∇⊥ α)(X, Z), (R(X, X Y ⊥

https://doi.org/10.1515/9783110689839-002

38 | 2 Submanifolds of Riemannian manifolds Ricci equation: ̄ Y)ζ , η) = g(R ̄ ⊥ (X, Y)ζ , η) − g((Aζ Aη − Aη Aζ )X, Y). g(̄ R(X, We have to add a few remarks about the objects in these equations. The νM-valued (1, 2)-tensor field α on M, which appears in the Gauss formula, is the second fundamental form of M. It is defined as the normal component of ∇̄X Y, so α(X, Y) = (∇̄X Y)⊥ . Since ∇ and ∇̄ are torsion-free connections, α is symmetric, that is, α(X, Y) = α(Y, X). The symbol A in the Weingarten formula denotes the shape operator of M. The second fundamental form and the shape operator are related by ̄ g(α(X, Y), ζ ) = g(Aζ X, Y). For each normal vector ζ ∈ νp M, the shape operator Aζ is a self-adjoint endomorphism of the tangent space Tp M. Its eigenvalues are called the principal curvatures of M at p with respect to ζ and the corresponding eigenspaces are called the principal curvature spaces of Tp M with respect to ζ . The vector field ∇X⊥ ζ in the Weingarten formula is the normal component of ∇̄X ζ , that is, ∇X⊥ ζ = (∇̄X ζ )⊥ , and ∇⊥ is the normal covariant derivative of M. The νM-valued tensor field ∇⊥ α of type (3, 1) on M, which appears in the Codazzi equation, is the covariant derivative of α and defined by (∇X⊥ α)(Y, Z) = ∇X⊥ α(Y, Z) − α(∇X Y, Z) − α(Y, ∇X Z). Finally, the normal curvature tensor R⊥ , which appears in the Ricci equation, is the curvature tensor of the normal bundle of M defined by ⊥ R⊥ (X, Y)ζ = ∇X⊥ ∇Y⊥ ζ − ∇Y⊥ ∇X⊥ ζ − ∇[X,Y] ζ.

Two submanifolds M and M ′ of M̄ are said to be congruent if there exists an isom̄ with f (M) = M ′ . etry f ∈ I(M)

2.2 Totally geodesic submanifolds | 39

2.2 Totally geodesic submanifolds Definition 2.2.1. A submanifold M of a Riemannian manifold M̄ is said to be totally geodesic if the second fundamental α of M vanishes, or equivalently, if the shape operator A is trivial. If α = 0, then the Gauss formula implies that ∇̄γ̇ γ̇ = ∇γ̇ γ̇ = 0 for every geodesic γ ̄ in M. Thus every geodesic in M is also a geodesic in M. ̄ Then the Conversely, assume that every geodesic in M is also a geodesic in M. Gauss formula implies that α(γ,̇ γ)̇ = 0 for every geodesic γ in M. This implies that α(v, v) = 0 for all v ∈ TM. Since α is symmetric, we have α(v + w, v + w) = α(v, v) + α(w, w) + 2α(v, w) and hence α(v, w) = 0 for all v, w ∈ Tp M, p ∈ M. Thus α = 0 and we have proved the following. Proposition 2.2.2. A submanifold M of a Riemannian manifold M̄ is totally geodesic if and only if every geodesic in M is a geodesic in M.̄ It was proved by Murphy and Wilhelm [75] that generic Riemannian manifolds do not admit totally geodesic submanifolds apart from the obvious ones: points, geodesics, open subsets. The general existence problem was settled by Élie Cartan (see, e. g., [9, Theorem 10.3.3] for a proof). Theorem 2.2.3 (Élie Cartan). Let M̄ be a Riemannian manifold, p ∈ M̄ and V be a linear subspace of Tp M.̄ There exists a totally geodesic submanifold M of M̄ with p ∈ M and Tp M = V if and only if there exists a real number ϵ ∈ ℝ+ such that for every geodesic ̇ ̇ γ : [0, 1] → M̄ with γ(0) = p, γ(0) ∈ V and ‖γ(0)‖ < ϵ the Riemannian curvature tensor ̄ of M at γ(1) preserves the parallel translate of V along γ from p to γ(1). Assume that there exists a totally geodesic submanifold M of M̄ with p ∈ M and ̄ ̄ Tp M = V. Since the exponential map (expM̄ )p : T̃ p M → M maps straight line segments ̃ ̄ there is an open neighborhood U of 0 in Tp M̄ through 0 ∈ Tp M̄ to geodesics in M,

such that (expM̄ )p maps U ∩ V diffeomorphically onto some open neighborhood of p in M. This implies that M is uniquely determined near p and that every totally geodesic submanifold of M̄ containing p and tangent to V is contained as an open part in a maximal one with this property. This feature is known as rigidity of totally geodesic submanifolds. ̄ every subspace V of Tp M̄ is the tangent space of a totally In a real space form M, geodesic submanifold. Theorem 2.2.4. Let M̄ be a real space form, p ∈ M̄ and V be a linear subspace of Tp M̄ ̄ Then there exists a totally geodesic submanifold M of M̄ with 2 ≤ dim(V) < dim(M).

40 | 2 Submanifolds of Riemannian manifolds with p ∈ M and Tp M = V. If M is maximal, then M is isometric to an m-dimensional real space form with the same sectional curvature as M.̄

2.3 Curvature-adapted submanifolds Let M be a submanifold of a Riemannian manifold M̄ and ζ ∈ νp M be a normal vector ̄ ζ )ζ : Tp M̄ → Tp M. ̄ If R̄ ζ (Tp M) ⊆ Tp M, then of M. Consider the Jacobi operator R̄ ζ = R(⋅, ̄ the restriction Kζ of Rζ to Tp M is a self-adjoint endomorphism of Tp M, the so-called normal Jacobi operator of M with respect to ζ . The family K = (Kζ )ζ ∈νM is called the normal Jacobi operator of M. Definition 2.3.1. A submanifold M of a Riemannian manifold M̄ is curvature-adapted if for every normal vector ζ ∈ νp M, p ∈ M, the following two conditions are satisfied: (i) R̄ ζ (Tp M) ⊆ Tp M; (ii) the normal Jacobi operator Kζ and the shape operator Aζ of M are simultaneously diagonalizable, that is, Kζ Aζ = Aζ Kζ . Remark 2.3.2. (i) Since R̄ λζ = λ2 R̄ ζ for all λ > 0, it suffices to check conditions (i) and (ii) in Definition 2.3.1 only for unit normal vectors. (ii) Since R̄ ζ ζ = 0 and R̄ ζ is self-adjoint, condition (i) in Definition 2.3.1 is always satisfied if codim(M) = 1. Curvature-adapted submanifolds were introduced in [24]. They are a very useful class of submanifolds in the context of focal sets and tubes, as we will see in Sections 2.5 and 2.6. If M̄ is a real space form with constant sectional curvature κ, then R̄ ζ (Tp M) ⊆ Tp M is always satisfied and Kζ = κid for all unit normal vectors ζ ∈ νM. It follows that every submanifold of a real space form is curvature-adapted. The concept of curvatureadapted submanifolds is more restrictive though in other Riemannian manifolds (see, e. g., Proposition 3.3.3).

2.4 M-Jacobi fields Focal sets and tubes will play an important role in our investigations. The general methodology for investigating the geometry of focal sets and tubes is based on Jacobi field theory. In this section we will provide the necessary background for this. ̄ p ∈ M and ζ ∈ νp M be a unit normal vector. Consider Let M be a submanifold of M, the maximal geodesic γ = γζ : I → M̄ that is defined by the initial values γ(0) = p and

2.5 Equidistant hypersurfaces and focal sets | 41

̇ γ(0) = ζ . Let V(s, t) = γs (t) be a smooth geodesic variation of γ = γ0 with c(s) = γs (0) ∈ M and ζ (s) = γ̇s (0) ∈ νc(s) M for all s. Geometrically, V is a geodesic variation of γ by geodesics in M̄ intersecting M perpendicularly along the curve c in M. The Jacobi field Y along γ induced by this geodesic variation is determined by the initial values Y(0) =

d 󵄨󵄨󵄨󵄨 d 󵄨󵄨󵄨󵄨 d 󵄨󵄨󵄨󵄨 ̇ ∈ Tp M 󵄨󵄨 V(s, 0) = 󵄨󵄨 γs (0) = 󵄨 c(s) = c(0) ds 󵄨󵄨s=0 ds 󵄨󵄨s=0 ds 󵄨󵄨󵄨s=0

and, using the Weingarten formula, d 󵄨󵄨󵄨󵄨 d 󵄨󵄨󵄨󵄨 󵄨󵄨 V(s, t) = ∇̄c(0) 󵄨 V(s, t) = ∇̄c(0) ̇ ̇ γ̇s ds 󵄨󵄨s=0 dt 󵄨󵄨󵄨t=0 ⊥ ̄ = ∇̄c(0) ̇ ζ = ∇Y(0) ζ = −Aζ (0) Y(0) + ∇Y(0) ζ .

Y ′ (0) = ∇̄γ(0) ̇

Thus, the initial values of Y satisfy Y(0) ∈ Tp M

and Y ′ (0) + Aζ Y(0) ∈ νp M.

A Jacobi field Y along γ whose initial values satisfy these two conditions is called an M-Jacobi field. These particular Jacobi fields correspond to geodesic variations of geodesics intersecting M perpendicularly. Since a Jacobi field Y along γ is uniquely determined by its values Y(0) and Y ′ (0), we see that M-Jacobi fields along γ form ̄ ̄ an m-dimensional linear subspace of the 2m-dimensional vector space of all Jacobi ̄ Obviously, Y(t) = t γ(t) ̇ is an M-Jacobi field along γ. fields along γ, where m̄ = dim(M). Since this particular Jacobi field is of little relevance, we define J(M, γ) as the (m̄ − 1)dimensional vector space consisting of all M-Jacobi fields along γ which are perpeṅ dicular to the M-Jacobi field t 󳨃→ t γ(t), that is, ̄ : Y ′′ + R̄ γ Y = 0, Y(0) ∈ Tp M, J(M, γζ ) = {Y ∈ Xγζ (M) ζ Y ′ (0) = −Aζ Y(0) + η with η ∈ νp M ⊖ ℝζ }.

2.5 Equidistant hypersurfaces and focal sets ̄ so m = dim(M) = dim(M) ̄ − 1 = m̄ − 1. Assume that Let M be a hypersurface in M, ̄ ζ ∈ Γ(νM) is a unit normal vector field on M. Let r ∈ ℝ+ and assume that rζp ∈ T̃ p M for ̄ all p ∈ M. This is always satisfied if M is complete. We now investigate the equidistant displacement of M in direction ζ at distance r. For this we define the smooth map Φr : M → M̄ ,

p 󳨃→ expM̄ (rζp ),

̃ ̄ The smooth map Φr parametrizes where expM̄ : T M̄ → M̄ is the exponential map of M. the parallel displacement Mr = Φr (M) of M in direction ζ at distance r. The set Mr is an

42 | 2 Submanifolds of Riemannian manifolds immersed submanifold of M̄ if and only if Φr has constant rank m. If Φr has constant rank less than m, then locally Φr is a submersion onto a smooth submanifold of M̄ whose dimension is equal to the rank of Φr . Let p ∈ M and γ = γζp : t 󳨃→ expM̄ (tζp ). In this situation the m-dimensional vector space J(M, γ) of M-Jacobi fields along γ is ̄ : Y ′′ + R̄ γ Y = 0, Y(0) ∈ Tp M, Y ′ (0) = −Aζ Y(0)}. J(M, γ) = {Y ∈ Xγ (M) p ̇ Let Y ∈ J(M, γ) and c be a smooth curve in M with c(0) = p and c(0) = Y(0). Then V(s, t) = exp(tζc(s) ) is a smooth geodesic variation of γ consisting of geodesics of M̄ intersecting M perpendicularly with corresponding M-Jacobi field Y. For the differential dp Φr of Φr at p we calculate ̇ dp Φr (Y(0)) = dp Φr (c(0)) =

d 󵄨󵄨󵄨󵄨 󵄨 exp(rζc(s) ) = Y(r). ds 󵄨󵄨󵄨s=0

Thus rk(dp Φr ) < m if and only if there exists a non-zero M-Jacobi field Y ∈ J(M, γ) with Y(r) = 0. In this case Φr (p) is called a focal point of M along γ and the dimension of the kernel of dp Φr is called the multiplicity of the focal point. If Φr (p) is a focal point of M along γ, its multiplicity is the dimension of the linear subspace of J(M, γ) consisting of all M-Jacobi fields Y ∈ J(M, γ) with Y(r) = 0. The geometric interpretation of Jacobi fields in terms of geodesic variations implies that Φr (p) is a focal point of M along γ if and only if there exists a non-trivial geodesic variation of γ all of whose geodesics intersect M orthogonally and meet infinitesimally close at Φr (p). If there exists a positive integer k such that Φr (q) is a focal point of M along γζq with multiplicity k for all q in some sufficiently small open neighborhood U of p, then Φr |U parametrizes an (m − k)̄ Such a submanifold F is called a (local) dimensional embedded submanifold F of M. ̄ If Φr (p) is not a focal point of M along γ, then rk(dp Φr ) = m focal manifold of M in M. in some open neighborhood U of p. Then, if U is sufficiently small, Φr |U parametrizes ̄ which is called a (local) equidistant hypersurface to an embedded hypersurface in M, ̄ In both cases, the vector γ(r) ̇ is a unit normal vector of the focal manifold resp. M in M. equidistant hypersurface at Φr (p). Our next aim is to calculate the shape operator of the focal manifold resp. equidiṡ tant hypersurface with respect to γ(r). We denote the focal manifold resp. equidistant hypersurface by Mr (in general this is only a part of the original Mr ). Let p ∈ M and Y, c, V be as above. Then cr = Φr ∘ c is a curve in Mr with ċr (0) = ̇ dp Φr (c(0)) = Y(r). We define a unit normal vector field ηr on Mr along cr by ηr (s) = γ̇ζc(s) (r) and denote by Ar the shape operator of Mr . Then, using the Weingarten for-

2.5 Equidistant hypersurfaces and focal sets | 43

mula, we obtain d 󵄨󵄨󵄨 ⊥ Y ′ (r) = ∇̄cṙ (0) 󵄨󵄨󵄨 V(s, t) = ∇̄cṙ (0) ηr = η′r (0) = −Arηr (0) Y(r) + (η′r (0)) , 󵄨 dt 󵄨t=r where (⋅)⊥ denotes the orthogonal projection onto νγ(r) Mr . Thus, if Y is an M-Jacobi field along γ in J(M, γ), the shape operator Ar of Mr satisfies ′ Arγ(r) ̇ Y(r) = −(Y (r)) , ⊤

where (⋅)⊤ denotes the orthogonal projection onto Tγ(r) Mr . If, in particular, Mr is a hȳ then Y ′ (r) is tangent to Mr because Y is perpendicular to γ̇ and the persurface in M, ̇ normal space of Mr at γ(r) is spanned by γ(r). We summarize this in the following theorem. Theorem 2.5.1. Let M be a hypersurface in a Riemannian manifold M,̄ r ∈ ℝ+ , ζ a unit normal vector field on M and Mr the equidistant displacement of M in direction ζ at distance r. Suppose that Mr is a submanifold of M.̄ Let p ∈ M and γ be the geodesic in ̇ M̄ with γ(0) = p and γ(0) = ζp . Then Tγ(r) Mr = {Y(r) : Y ∈ J(M, γ)}. ̇ ∈ νγ(r) Mr is given by The shape operator Ar of Mr with respect to γ(r) ′ Arγ(r) ̇ Y(r) = −(Y (r)) , ⊤

where (Y ′ (r))⊤ is the component of Y ′ (r) tangent to Mr . If, in particular, Mr has codimension 1, then Y ′ (r) is tangent to Mr . If Mr has codimension 1, there is an efficient way to describe the shape operator A . In this situation, we denote by γ ⊥ the parallel subbundle of the tangent bundle of ̄ and put ̇ in Tγ(t) M, M̄ along γ that is defined by the orthogonal complements of ℝγ(t) r

̄ ̄ ̇ γ|̇ γ⊥ . R̄ ⊥ γ = Rγ |γ ⊥ = R(⋅, γ) Let D be the End(γ ⊥ )-valued tensor field along γ given by D′′ + R̄ ⊥ γ ∘D=0,

D(0) = idTp M ,

D′ (0) = −Aζp .

If v ∈ Tp M and Bv is the parallel vector field along γ with Bv (0) = v, then Y = DBv is the Jacobi field along γ with initial values Y(0) = v and Y ′ (0) = −Aζp v. Thus γ(r) is a focal ̄ then D(r) point of M along γ if and only if D(r) is singular. If Mr is a hypersurface in M, is regular, and we obtain

r ′ ′ ′ Arγ(r) ̇ D(r)Bv (r) = Aγ(r) ̇ Y(r) = −Y (r) = −(DBv ) (r) = −D (r)Bv (r).

44 | 2 Submanifolds of Riemannian manifolds ̇ satisfies the equation Therefore, the shape operator Ar of Mr with respect to γ(r) Arγ(r) = −D′ (r) ∘ D−1 (r). ̇ Now assume in addition that M̄ is a Riemannian locally symmetric space. Then ̄ ̄ ∇R = 0 and hence the eigenvalues of R̄ ⊥ γ are constant along γ and the corresponding eigenspaces are invariant under parallel translation along γ. Therefore, if M is curvature-adapted, we can calculate explicitly the principal curvatures and principal curvature spaces of Mr . To describe them explicitly, we define what we call basic M-Jacobi fields along γ. A non-zero M-Jacobi field Y ∈ J(M, γ) is called basic if Y(0) is an eigenvector of the normal Jacobi operator Kζ and of the shape operator Aζ . To be more specific, we denote by Spec(Kζ ) and Spec(Aζ ) the set of eigenvalues of Kζ and Aζ , respectively. If κ, λ ∈ ℝ we define Eκ = {v ∈ Tp M : Kζ v = κv}, Tλ = {v ∈ Tp M : Aζ v = λv}. The basic M-Jacobi fields along γ are the M-Jacobi fields of the form Yv (t) = (cosκ (t) − λ sinκ (t))Bv (t), where Bv (t) is the parallel vector field along γ with Bv (0) = v and v ∈ Eκ ∩ Tλ ≠ {0}. From Theorem 2.5.1 we then obtain the following. Corollary 2.5.2. Let M be a curvature-adapted hypersurface in a Riemannian locally symmetric space M,̄ r ∈ ℝ+ and ζ a unit normal vector field of M. Let Mr be the equidistant displacement of M in direction ζ at distance r. For p ∈ M we put γp = γζp . (i) Let p ∈ M. Then γp (r) is a focal point of M along γp if and only if there exist κ ∈ Spec(Kζp ) and λ ∈ Spec(Aζp ) such that Eκ ∩ Tλ ≠ {0} and λ = cotκ (r). If γp (r) is a focal point of M along γp , then its multiplicity μp is μp =

∑

κ∈Spec(Kζp )

dim(Eκ ∩ Tcotκ (r) ).

(ii) If Mr is a hypersurface in M,̄ then the principal curvatures of Mr at γp (r) with respect to γ̇p (r) are κ sinκ (r) + λ cosκ (r) , cosκ (r) − λ sinκ (r) where κ ∈ Spec(Kζp ) and λ ∈ Spec(Aζp ) with Eκ ∩ Tλ ≠ {0}. The corresponding principal curvature spaces are the parallel translates of Eκ ∩ Tλ along γp from p = γp (0) to γp (r).

2.6 Tubes | 45

(iii) If Mr is a focal manifold of M in M,̄ then the principal curvatures of Mr at γp (r) with respect to γ̇p (r) are κ sinκ (r) + λ cosκ (r) , cosκ (r) − λ sinκ (r) where κ ∈ Spec(Kζp ) and λ ∈ Spec(Aζp ) with λ ≠ cotκ (r) and Eκ ∩ Tλ ≠ {0}. The corresponding principal curvature spaces are the parallel translates of Eκ ∩Tλ along γp from p = γp (0) to γp (r).

2.6 Tubes ̄ Let M be a submanifold of a Riemannian manifold M̄ with m = dim(M) ≥ 0 and m−m = 1 codim(M) ≥ 2. Let r ∈ ℝ+ and consider the unit normal bundle ν M = {ζ ∈ νM : ‖ζ ‖ = 1} of M. The tube of radius r around M is defined as Mr = Φr (M), where Φr : ν1 M → M,̄

ζ 󳨃→ expM̄ (rζ ).

̃ ̄ which For this to be well-defined we require that {tζ ∈ νM : ζ ∈ ν1 M, t ∈ [0, r]} ⊂ T M, ̄ is always satisfied if M is complete. We are interested in the case when Mr is a hypersurface. For this we need that dζ Φr : Tζ ν1 M → TexpM̄ (rζ ) M̄ has maximal rank m̄ − 1 for all ζ ∈ ν1 M. Let p ∈ M and ζ ∈ νp1 M. There is a canonical isomorphism Tζ ν1 M → Tp M ⊕ (νp M ⊖ ℝζ ),

u 󳨃→ (dζ τ(u), Vζ−1 (u)),

where τ : ν1 M → M is the canonical projection and Vζ : νp M → Tζ νp M is the canonical vector space isomorphism (also known as the vertical lift) given by Vζ (η) = ċζη (0) with cζη (t) = ζ + tη. The differential of Φr can then be computed using M-Jacobi fields by dζ Φr (u) = Y(r), where u ∈ Tζ ν1 M and Y ∈ J(M, γζ ) is the M-Jacobi field along γζ : t 󳨃→ expM̄ (tζ ) with initial values Y(0) = dζ τ(u) and Y ′ (0) = Vζ−1 (u) − Aζ dζ τ(u). Thus Mr is an immersed hypersurface if and only if Y(r) ≠ 0 for all 0 ≠ Y ∈ J(M, γζ ) and ζ ∈ ν1 M. In this case we have TΦr (ζ ) Mr = {Y(r) : Y ∈ J(M, γζ )}. We now define a vector field ηr along Φr by ηrζ = γ̇ζ (r). The vector ηrζ ∈ TΦr (ζ ) M̄ is a unit normal vector of Mr at Φr (ζ ), and thus ηr is a unit normal vector field along Φr .

46 | 2 Submanifolds of Riemannian manifolds If Y ∈ J(M, γζ ) is the M-Jacobi field along γζ : t 󳨃→ expM̄ (tζ ) with initial values Y(0) = dζ τ(u) and Y ′ (0) = Vζ−1 (u) − Aζ dζ τ(u), then Y ′ (r) = ∇̄u ηr . Thus, if Y is an M-Jacobi field along γζ in J(M, γζ ), the shape operator Ar of Mr satisfies Arγζ̇ (r) Y(r) = −Y ′ (r). We summarize this in the following theorem. Theorem 2.6.1. Let M be a submanifold of a Riemannian manifold M̄ and r ∈ ℝ+ . Assume that the tube Mr = Φr (M) of radius r ∈ ℝ+ around M is a hypersurface in M.̄ Let p ∈ M and ζ be a unit normal vector of M at p. Let γ be the geodesic in M̄ with γ(0) = p ̇ and γ(0) = ζ . Then Tγ(r) Mr = {Y(r) : Y ∈ J(M, γ)} ̇ ∈ νγ(r) Mr is given by and the shape operator Ar of Mr with respect to γ(r) ′ Arγ(r) ̇ Y(r) = −Y (r).

Similar to what we wrote after Theorem 2.5.1, we can describe the shape operator Ar in a simple manner. We again denote by γ ⊥ the parallel subbundle of the tangent ̇ in Tγ(t) M̄ bundle of M̄ along γ that is defined by the orthogonal complements of ℝγ(t) and put ̄ ̄ ̇ γ|̇ γ⊥ . R̄ ⊥ γ = Rγ |γ ⊥ = R(⋅, γ) We decompose Tp M̄ ⊖ ℝζ orthogonally into Tp M̄ ⊖ ℝζ = Tp M ⊕ (νp M ⊖ ℝζ ).

(2.1)

Let D be the End(γ ⊥ )-valued tensor field along γ given by D′′ + R̄ ⊥ γ ∘D=0,

D(0) = (

idTp M 0

0 ), 0

−Aζ D′ (0) = ( p 0

0 ), idνp M⊖ℝζ

where the decomposition of the matrices is with respect to decomposition (2.1). If v ∈ Tp M and Bv is the parallel vector field along γ with Bv (0) = v, then Y = DBv is the M-Jacobi field along γ with initial values Y(0) = v and Y ′ (0) = −Aζp v. If v ∈ νp M ⊖ ℝζ and Bv is the parallel vector field along γ with Bv (0) = v, then Y = DBv is the M-Jacobi field along γ with initial values Y(0) = 0 and Y ′ (0) = v. Then we obtain r ′ ′ ′ Arγ(r) ̇ D(r)Bv (r) = Aγ(r) ̇ Y(r) = −Y (r) = −(DBv ) (r) = −D (r)Bv (r).

2.6 Tubes | 47

̇ satisfies the equation Therefore, the shape operator Ar of Mr with respect to γ(r) Arγ(r) = −D′ (r) ∘ D−1 (r). ̇ We now assume that M̄ is a Riemannian locally symmetric space. Then ∇̄ R̄ = 0 and hence the eigenvalues of R̄ ⊥ γ are constant along γ and the corresponding eigenspaces are invariant under parallel translation along γ. Therefore, if M is curvature-adapted, we can calculate explicitly the principal curvatures and principal curvature spaces of Mr . To describe them explicitly, we consider the basic M-Jacobi fields along γ in this context. A non-zero M-Jacobi field Y ∈ J(M, γ) is called basic if: (i) Y(0) ∈ Tp M is an eigenvector of the normal Jacobi operator Kζ and of the shape operator Aζ ; or (ii) Y(0) ∈ νp M ⊖ ℝζ is an eigenvector of the Jacobi operator R̄ ζ . We denote by Spec(Kζ ) and Spec(Aζ ) the set of eigenvalues of Kζ and Aζ , respectively, ̄ and by Spec(R̄ ⊥ ζ ) the set of eigenvalues of Rζ |νp M⊖ℝζ . If κ, λ ∈ ℝ we define Eκ = {v ∈ Tp M : Kζ v = κv}, Tλ = {v ∈ Tp M : Aζ v = λv}, Vκ = {v ∈ νp M ⊖ ℝζ : R̄ ζ v = κv}. The basic M-Jacobi fields along γ are the M-Jacobi fields of the form (cosκ (t) − λ sinκ (t))Bv (t), Yv (t) = { sinκ (t)Bv (t),

if v ∈ Eκ ∩ Tλ ≠ {0}, if v ∈ Vκ ≠ {0},

where Bv (t) is the parallel vector field along γ with Bv (0) = v. From Theorem 2.6.1 we then obtain the following. Corollary 2.6.2. Let M be a curvature-adapted submanifold of a Riemannian locally symmetric space M̄ and r ∈ ℝ+ . Assume that the tube Mr = Φr (M) of radius r ∈ ℝ+ around M is a hypersurface in M.̄ Let p ∈ M and ζ be unit normal vector of M at p. Let γ ̇ be the geodesic in M̄ with γ(0) = p and γ(0) = ζ . Then Mr is curvature-adapted and the ̇ are principal curvatures of Mr at γ(r) with respect to γ(r) κ sinκ (r)+λ cosκ (r)

{ cosκ (r)−λ sinκ (r) − cotκ (r),

,

where κ ∈ Spec(Kζp ), λ ∈ Spec(Aζp ) with Eκ ∩ Tλ ≠ {0}, where κ ∈ Spec(R̄ ⊥ ) with Vκ ≠ {0}. ζ

The corresponding principal curvature spaces are the parallel translates of Eκ ∩ Tλ and Vκ , respectively, along γ from p = γ(0) to γ(r). Example 2.6.3. Let M̄ be a real space form with constant sectional curvature κ and ̄ Then the tube Mr with radius r assume that M = {p} consists of a single point p ∈ M.

48 | 2 Submanifolds of Riemannian manifolds ̄ where we require r < π for κ > 0. From around M is a geodesic hypersphere in M, √κ Corollary 2.6.2 we obtain that Mr has only one principal curvature, namely −√κ cot(√κr), { { { 1 − cotκ (r) = {− r , { { {−√−κ coth(√−κr),

if κ > 0, if κ = 0, if κ < 0.

This is the well-known principal curvature of a geodesic hypersphere with radius r with respect to the outward unit normal vector field in a real space form of constant sectional curvature κ. Example 2.6.4. Let M̄ be an n-dimensional complex space form M̄ n (c) with constant holomorphic sectional curvature c ≠ 0 and assume that M = {p} consists of a single ̄ The tube Mr with radius r around M is a geodesic hypersphere in M, ̄ point p ∈ M. π ̄ where we require r < √c for c > 0. Let ζ ∈ νp M = Tp M be a unit tangent vector of M̄ c at p. From (1.6) we easily see that the eigenvalues of R̄ ⊥ ζ are c and 4 with corrersponding eigenspaces ℝJζ and Tp M̄ ⊖ ℂζ . From Corollary 2.6.2 we obtain that Mr has two principal curvatures at each point, namely

−√c cot(√cr), − cotc (r) = { −√−c coth(√−cr),

if c > 0, if c < 0

and − 1 √c cot( 21 √cr), − cot c (r) = { 21 4 − 2 √−c coth( 21 √−cr),

if c > 0, if c < 0.

The multiplicity of the principal curvature − cotc (r) is equal to 1 and the corresponding principal curvature spaces are Jνq Mr , q ∈ Mr . The multiplicity of the principal curvature − cot c (r) is equal to 2(n − 1) and the corresponding principal curvature spaces are 4 the maximal holomorphic subspaces of Tq Mr , q ∈ Mr .

2.7 Homogeneous hypersurfaces A submanifold M of a Riemannian manifold M̄ is called a homogeneous submanifold ̄ with f (p) = q and of M̄ if for any two points p, q ∈ M there exists an isometry f ∈ I(M) ̄ f (M) = M. If codim(M) = 1, then M is also called a homogeneous hypersurface of M. ̄ If M is a homogeneous submanifold of M, then ̄ : f (M) = M} GM = {f ∈ I(M)

2.7 Homogeneous hypersurfaces |

49

acts transitively on M and therefore M is necessarily a Riemannian homogeneous space. Then also the identity component of GM acts transitively on M. Important examples of homogeneous hypersurfaces are principal orbits of cō The homogeneity one actions. Assume that G is a connected closed subgroup of I(M). ̄ cohomogeneity of the action of G on M is the codimension of a principal orbit of the action. Therefore principal orbits of cohomogeneity one actions are homogeneous hypersurfaces. In this monograph homogeneous hypersurfaces will play an important role as they provide “model spaces” for hypersurface with “nice” geometric properties. We start by presenting the classifications of homogeneous hypersurfaces in real space forms, and we will discuss their classifications in Hermitian symmetric spaces later. Since isometries preserve Riemannian connections, we immediately get the following. Proposition 2.7.1. Every homogeneous hypersurface in a Riemannian manifold has constant principal curvatures. In a space of constant sectional curvature a hypersurface has constant principal curvatures if and only if it is isoparametric. An isoparametric hypersurface is a level hypersurface of an isoparametric function. This is a smooth function f : M̄ → ℝ so that ‖df ‖2 = a(f ) and Δf = b(f ) for some smooth function a and some continuous function b on ℝ. The geometric meaning of the first condition is that the level sets of f are equidistant. The second condition then means that the level hypersurfaces have constant mean curvature. Isoparametric functions and hypersurfaces in ℝm+1 were classified by Levi-Civita [66] for m = 2 and by Segre [89] in the general case. It is easy to see that each of these isoparametric hypersurfaces is homogeneous. This establishes the classification of homogeneous hypersurfaces in Euclidean spaces. Theorem 2.7.2. A hypersurface M in ℝm+1 , m ≥ 2, is homogeneous if and only if M is congruent to: (i) the sphere Sm (r) with radius r ∈ ℝ+ in ℝm+1 , (ii) the hyperplane ℝm in ℝm+1 or (iii) the tube of radius r ∈ ℝ+ around the k-dimensional subspace ℝk of ℝm+1 for some k ∈ {1, . . . , m − 1}. The isometry group of ℝm+1 is the semidirect product I(ℝm+1 ) = Om+1 ⋉ ℝm+1 , where ℝm+1 acts on itself by translations. The spheres Sm (r) are the principal orbits of the action of SOm+1 ⊂ Om+1 , and the hyperplane ℝm is a principal orbit of the action of the subgroup ℝm × {0} in the translation group ℝm+1 . The tubes of radius r ∈ ℝ+ around ℝk in ℝm+1 are the principal orbits of the action of SOm−k+1 ⋉ ℝk , where we decompose ℝm+1 = ℝk ⊕ ℝm−k+1 and SOm−k+1 acts on the second factor.

50 | 2 Submanifolds of Riemannian manifolds In a series of papers [26–29], Cartan made an attempt to classify the isoparametric hypersurfaces in real hyperbolic spaces and the spheres. He succeeded in the case of the hyperbolic spaces and obtained various results in the case of the sphere. As concerns the hyperbolic spaces, the crucial step in the classification is a formula which is derived from the equations of Gauss and Codazzi and describes a relation among the principal curvatures. This formula implies that the number of distinct principal curvatures is at most 2. If there is just one principal curvature, then the hypersurface is umbilical and hence a horosphere, a geodesic hypersphere, a totally geodesic real hyperbolic hyperplane or an equidistant hypersurface to it. If there are two distinct principal curvatures, one can use focal set theory to deduce that the hypersurface is a tube around a totally geodesic real hyperbolic subspace. All these spaces are in fact homogeneous, which implies the following. Theorem 2.7.3. A hypersurface M in ℝH m+1 , m ≥ 2, is homogeneous if and only if M is congruent to: (i) a geodesic hypersphere with radius r ∈ ℝ+ in ℝH m+1 , (ii) a horosphere in ℝH m+1 , (iii) the totally geodesic hyperplane ℝH m in ℝH m+1 or an equidistant hypersurface in it or (iv) the tube of radius r ∈ ℝ+ around the k-dimensional real hyperbolic subspace ℝH k of ℝH m+1 for some k ∈ {1, . . . , m − 1}. As subgroups of the identity component SOo1,m+1 of the isometry group of ℝH m+1 one may choose (i) the isotropy group SOm+1 ; (ii) the nilpotent subgroup N in an Iwasawa decomposition KAN of SOo1,m+1 , which is isomorphic to the Abelian Lie group ℝm ; (iii) SOo1,m ; or (iv) SOo1,k × SOm+1−k . For spheres Cartan’s formula does not provide sufficient information to determine the possible number of distinct principal curvatures. Only later it was proved by Münzner [74], using sphere bundles and methods from algebraic topology, that the number g of distinct principal curvatures of an isoparametric hypersurface in Sm+1 equals 1, 2, 3, 4 or 6. Already Cartan classified the isoparametric hypersurfaces with at most three distinct principal curvatures. They all turn out to be homogeneous. Surprisingly, for g = 4 there are inhomogeneous isoparametric hypersurfaces. The first such examples were discovered by Ozeki and Takeuchi [80]; later Ferus, Karcher and Münzner [42] constructed further series of examples by using representations of Clifford algebras. It was shown by Abresch [1] that the case g = 6 occurs only in S7 and S13 . Dorfmeister and Neher [40] proved that in S7 an isoparametric hypersurface with g = 6 must be homogeneous, and Miyaoka [70] proved the analogous result for S13 . Now, for the special case of homogeneous hypersurfaces, the following result by Hsiang and Lawson [46] settles also the remaining case g = 4.

2.7 Homogeneous hypersurfaces | 51

Theorem 2.7.4. A hypersurface in Sm+1 is homogeneous if and only if it is (congruent to) a principal orbit of the isotropy representation of a semisimple Riemannian symmetric space G/K with dim(G/K) = m + 2 and rk(G/K) = 2. We can therefore read off the classification of homogeneous hypersurfaces in spheres from the classification of compact, simply connected, Riemannian symmetric spaces. It is remarkable that homogeneous hypersurfaces in spheres correspond to certain Riemannian symmetric spaces via s-representations. A beautiful survey paper about isoparametric hypersurfaces was written by QuoShin Chi [37], who also settled the case g = 4 for inhomogeneous isoparametric hypersurfaces.

3 Real hypersurfaces in Kähler manifolds In this chapter we present some basic material about submanifolds and real hypersurfaces in Kähler manifolds and, as a special case, in Hermitian symmetric spaces.

3.1 Submanifolds Let M̄ be a Kähler manifold with Kähler structure J and Kähler metric g. Let M be a ̄ We will denote the induced Riemannian metric on M also by g. We submanifold of M. ̄ put m = dimℝ (M) and n = dimℂ (M). Definition 3.1.1. A submanifold M of a Kähler manifold M̄ is said to be: (i) complex (or invariant) if JTp M ⊆ Tp M

for all p ∈ M;

(ii) totally real (or anti-invariant) if JTp M ⊆ νp M

for all p ∈ M;

Jνp M ⊆ Tp M

for all p ∈ M.

(iii) anti-holomorphic if

Note that JTp M ⊆ Tp M is equivalent to Jνp M ⊆ νp M. If M is a submanifold of M̄ and u ∈ Tp M, p ∈ M, then Ju decomposes orthogonally into the tangential component Pu ∈ Tp M and the normal component Fu ∈ νp M: Ju = Pu + Fu.

(3.1)

This construction induces a skew-symmetric tensor field P on M and a νM-valued tensor field F on M. A submanifold M of M̄ is complex if and only if F = 0. A submanifold M of M̄ is totally real if and only if P = 0. If ζ ∈ νp M, p ∈ M, then Jζ decomposes orthogonally into the tangential component tζ ∈ Tp M and the normal component fζ ∈ νp M: Jζ = tζ + fζ . A submanifold M of M̄ is complex if and only if t = 0. A submanifold M of M̄ is antiholomorphic if and only if f = 0. For u, v ∈ Tp M and ζ , ρ ∈ νp M we have g(Pu, v) + g(u, Pv) = 0 ,

g(Fu, ζ ) + g(u, tζ ) = 0 ,

https://doi.org/10.1515/9783110689839-003

g(fζ , ρ) + g(ζ , fρ) = 0.

54 | 3 Real hypersurfaces in Kähler manifolds From −u = J 2 u = J(Pu + Fu) = P 2 u + FPu + tFu + fFu we obtain P 2 + tF = −idTM

and FP + fF = 0.

From −ζ = J 2 ζ = J(tζ + fζ ) = Ptζ + Ftζ + tfζ + f 2 ζ we obtain f 2 + Ft = −idνM

and Pt + tf = 0.

For the covariant derivatives, we can use the Gauss and Weingarten formulae. First, we have 0 = (∇̄X J)Y = ∇̄X JY − J ∇̄X Y = ∇̄X (PY + FY) − J∇X Y − Jh(X, Y)

= ∇X PY + h(X, PY) − AFY X + ∇X⊥ FY

− P∇X Y − F∇X Y − th(X, Y) − fh(X, Y)

= (∇X P)Y − AFY X − th(X, Y) + (∇X⊥ F)Y + h(X, PY) − fh(X, Y) for all X, Y ∈ X(M). The tangential and normal component of this equation gives (∇X P)Y = AFY X + th(X, Y),

(∇X⊥ F)Y

= fh(X, Y) − h(X, PY).

Next, we have 0 = (∇̄X J)ζ = ∇̄X Jζ − J ∇̄X ζ = ∇̄X (tζ + fζ ) + JAζ X − J∇⊥ ζ X

= ∇X tζ + h(X, tζ ) − Afζ X + ∇X⊥ fζ + PAζ X + FAζ X − t∇X⊥ ζ − f ∇X⊥ ζ = (∇X t)ζ − Afζ X + PAζ X + (∇X⊥ f )ζ + h(X, tζ ) + FAζ X for all X ∈ X(M). The tangential and normal component of this equation gives (∇X t)ζ = Afζ X − PAζ X, (∇X⊥ f )ζ = −FAζ X − h(X, tζ ).

(3.2)

3.2 Real hypersurfaces |

55

Since F = 0 and t = 0 for a complex submanifold, (3.2) implies the following. Lemma 3.1.2. Let M be a complex submanifold of a Kähler manifold M.̄ Then ∇P = 0 and AJζ X = JAζ X

(3.3)

for all X ∈ X(M) and ζ ∈ Γ(νM). The equation ∇P = 0 implies that the restriction of the Kähler structure of M̄ to a complex submanifold M induces a Kähler structure on M. Corollary 3.1.3. Let M be a complex submanifold of a Kähler manifold M.̄ Then M is a minimal submanifold of M.̄ Proof. Let p ∈ M, u, v ∈ Tp M and ζ ∈ νp M. From (3.3) we have g(Jh(u, v), ζ ) = −g(h(u, v), Jζ ) = −g(AJζ u, v) = −g(JAζ u, v) = g(Aζ u, Jv) = g(h(u, Jv), ζ ). This implies Jh(u, v) = h(u, Jv) and thus h(Ju, Jv) = Jh(Ju, v) = Jh(v, Ju) = J 2 h(v, u) = −h(v, u) = −h(u, v). Now choose a complex orthonormal basis e1 , . . . , ek of the tangent space Tp M, where 2k = m = dimℝ (M). Then k

tr(h) = ∑ (h(eν , eν ) + h(Jeν , Jeν )) = 0, ν=1

which means that M is a minimal submanifold.

3.2 Real hypersurfaces Let M̄ be a Kähler manifold with Kähler structure J and Kähler metric g. Let M be a real ̄ We will denote the induced Riemannian metric on M also by g. We hypersurface in M. ̄ and m = dimℝ (M) = 2n − 1. put n = dimℂ (M) Let ζ be a (local) unit normal vector field on M. We denote by A = Aζ the shape operator of M with respect to ζ and by K = Kζ = R̄ ζ |TM the normal Jacobi operator of M with respect to ζ . Definition 3.2.1. The unit vector field ξ = −Jζ

56 | 3 Real hypersurfaces in Kähler manifolds is called the Reeb vector field on M. The flow of the Reeb vector field ξ is called the Reeb flow on M. We define a 1-form η on M by η(X) = g(X, ξ ) for all X ∈ X(M) and a skew-symmetric tensor field ϕ on M by ϕX = JX − η(X)ζ for all X ∈ X(M). The skew-symmetric tensor field ϕ is called the structure tensor field on M. The quadruple (ϕ, ξ , η, g) is called the induced almost contact metric structure on M. The subbundle 𝒞 = ker(η) = TM ∩ J(TM)

of TM is called the maximal holomorphic subbundle of the tangent bundle TM. We denote by Γ(𝒞 ) the set of all vector fields X on M with values in 𝒞 , that is, Γ(𝒞 ) = {X ∈ X(M) : Xp ∈ 𝒞p for all p ∈ M} = {X ∈ X(M) : η(X) = 0}.

Note that we always have η(ξ ) = 1 and ϕξ = 0. Using the Kähler property ∇J̄ = 0 and the Weingarten formula, we obtain 0 = (∇̄X J)ζ = ∇̄X Jζ − J ∇̄X ζ = −∇̄X ξ + JAX for all X ∈ X(M). The tangential component of this equation induces the important relation ∇X ξ = ϕAX

(3.4)

for all X ∈ X(M). With the notation from (3.1), we have PX = ϕX and FX = η(X)ζ for all X ∈ X(M). Equation (3.2) therefore implies (∇X ϕ)Y = η(Y)AX − g(AX, Y)ξ

(3.5)

3.2 Real hypersurfaces |

57

for all X, Y ∈ X(M). Using the definition for the exterior derivative, equation (3.4) and the fact that ∇ is torsion-free, we obtain dη(X, Y) = d(η(Y))(X) − d(η(X))(Y) − η([X, Y]) = Xg(Y, ξ ) − Yg(X, ξ ) − g([X, Y], ξ )

= g(∇X Y, ξ ) + g(Y, ∇X ξ ) − g(∇Y X, ξ ) − g(X, ∇Y ξ ) − g([X, Y], ξ )

= g(Y, ϕAX) − g(X, ϕAY) and hence

dη(X, Y) = g((Aϕ + ϕA)X, Y).

(3.6)

Definition 3.2.2. Let M be a real hypersurface in a Kähler manifold M̄ with induced almost contact metric structure (ϕ, ξ , η, g). The fundamental 2-form ω on M is defined by ω(X, Y) = g(ϕX, Y) for all X, Y ∈ X(M). Proposition 3.2.3. The fundamental 2-form ω on a real hypersurface in a Kähler manifold is closed, that is, dω = 0. Proof. By definition, we have dω(X, Y, Z) = d(ω(Y, Z))(X) + d(ω(Z, X))(Y) + d(ω(X, Y))(Z) − ω([X, Y], Z) − ω([Y, Z], X) − ω([Z, X], Y)

= g(∇X ϕY, Z) + g(ϕY, ∇X Z) + g(∇Y ϕZ, X) + g(ϕZ, ∇Y X) + g(∇Z ϕX, Y) + g(ϕX, ∇Z Y)

− g(ϕ[X, Y], Z) − g(ϕ[Y, Z], X) − g(ϕ[Z, X], Y)

= g((∇X ϕ)Y, Z) + g((∇Y ϕ)Z, X) + g((∇Z ϕ)X, Y)

for all X, Y ∈ X(M). Inserting the expression for ∇ϕ as in (3.5) into the previous equation gives dω = 0. The following lemma contains a useful equation involving the normal Jacobi operator K and the structure tensor field ϕ. Lemma 3.2.4. Let M be a real hypersurface in a Kähler manifold M.̄ Then we have ̄ Y)ξ , ζ ) = −g((Kϕ + ϕK)X, Y) g(R(X, for all X, Y ∈ X(M).

58 | 3 Real hypersurfaces in Kähler manifolds Proof. Using the algebraic Bianchi identity and other curvature identities we obtain ̄ Y)ξ , ζ ) = −g(R(Y, ̄ ̄ , X)Y, ζ ) g(R(X, ξ )X, ζ ) − g(R(ξ ̄ ̄ , JX)Y, ζ ) = −g(R(JY, Jξ )X, ζ ) − g(R(Jξ ̄ ̄ , JX)Y, ζ ) = g(R(JY, ζ )ζ , X) − g(R(ζ ̄ ζ )ζ , JY) − g(R(Y, ̄ = g(R(X, ζ )ζ , JX) = g(KX, JY) − g(KY, JX)

= g(KX, ϕY) − g(KY, ϕX) = −g((Kϕ + ϕK)X, Y).

Finally, we consider a special case of the Ricci soliton equation. Lemma 3.2.5. Let M be a real hypersurface in a Kähler manifold M.̄ Assume that (M, g) is a Ricci soliton with potential field ξ , that is, there exists c ∈ ℝ such that 1 Ric = cg − ℒξ g. 2 Then we have 1 Ric(ϕX) + ϕRic(X) = 2cϕX + (η(X)Aξ − η(AX)ξ ) 2 for all X ∈ X(M). Proof. For X, Y ∈ X(M) we have (ℒξ g)(X, Y) = ℒξ g(X, Y) − g(ℒξ X, Y) − g(X, ℒξ Y) = g(∇ξ X, Y) + g(X, ∇ξ Y) − g([ξ , X], Y) − g(X, [ξ , Y]) = g(∇X ξ , Y) + g(X, ∇Y ξ )

= g(ϕAX, Y) + g(X, ϕAY) = g((ϕA − Aϕ)X, Y). Now assume that Ric = cg − 21 ℒξ g. Then 1 Ric(ϕX) = cϕX − (ϕA − Aϕ)ϕX 2 1 1 = cϕX − ϕAϕX + Aϕ2 X 2 2 1 1 1 = cϕX − ϕAϕX − AX + η(X)Aξ 2 2 2 and 1 ϕRic(X) = cϕX − ϕ(ϕA − Aϕ)X 2

3.3 Hopf hypersurfaces | 59

1 1 = cϕX − ϕ2 AX + ϕAϕX 2 2 1 1 1 = cϕX + AX − η(AX)ξ + ϕAϕX. 2 2 2 Adding up the previous two equations gives the assertion.

3.3 Hopf hypersurfaces Definition 3.3.1. Let M be a real hypersurface in a Kähler manifold M̄ with induced almost contact metric structure (ϕ, ξ , η, g). Then M is called a Hopf hypersurface if the Reeb flow on M is a geodesic flow, that is, if the integral curves of the Reeb vector field ξ are geodesics in M. This definition is motivated by the special case of the unit sphere S2n−1 in ℂn . The Reeb vector field on S2n−1 generates the 1-dimensional fibers of the Hopf fibration induced by the Riemannian submersion S2n−1 → ℂP n−1 . The fibers are totally geodesic circles in S2n−1 and are the images of the integral curves of the Reeb vector field on S2n−1 . Proposition 3.3.2. Let M be a real hypersurface in a Kähler manifold M̄ with induced almost contact metric structure (ϕ, ξ , η, g). The following statements are equivalent: ̄ (i) M is a Hopf hypersurface in M; (ii) ∇ξ ξ = 0; (iii) the Reeb vector field ξ is a principal curvature vector of M at every point; (iv) the maximal holomorphic subbundle 𝒞 of TM is invariant under the shape operator A of M. Proof. Let p ∈ M and c : I → M be an integral curve of the Reeb vector field ξ with c(0) = p. Then we have ′ ′ ∇ξp ξ = ∇c(0) ̇ ξ = (ξ ∘ c) (0) = ċ (0).

If M is a Hopf hypersurface, then we have ċ′ (0) = 0 by definition and therefore ∇ξp ξ = 0. Since this holds at any point p ∈ M, we obtain ∇ξ ξ = 0. Conversely, if ∇ξ ξ = 0, then ċ′ = ∇ċ ξ = ∇ξ ∘c ξ = 0 for any integral curve c of ξ . Thus any integral curve of ξ is a geodesic in M and hence M is a Hopf hypersurface. This establishes the equivalence of (i) and (ii). By definition, the kernel ker(ϕ) of the structure tensor field ϕ is spanned by the Reeb vector field, ker(ϕ) = ℝξ .

60 | 3 Real hypersurfaces in Kähler manifolds From (3.4) we therefore see that ∇ξ ξ = 0 if and only if Aξ ∈ ℝξ , which shows that (ii) and (iii) are equivalent. We have the orthogonal decomposition TM = 𝒞 ⊕ ℝξ . Since the shape operator A is self-adjoint, the equivalence of (iii) and (iv) is obvious. We shall now see that in complex space forms the notions of Hopf hypersurface and of curvature-adapted hypersurface coincide. We first look at the concept of curvature-adapted submanifolds of complex space forms. Proposition 3.3.3. Let M be a submanifold of a complex space form M̄ with constant holomorphic sectional curvature c ≠ 0. Then M is curvature-adapted if and only if: (i) M is a complex submanifold of M̄ or (ii) M is an anti-holomorphic submanifold of M̄ and for every unit normal vector ζ ∈ νp M the vector Jζ is a principal curvature vector of M with respect to ζ . Proof. We recall that M is curvature-adapted if for every normal vector ζ ∈ νp M, p ∈ M, the following two conditions are satisfied: (i) R̄ ζ (Tp M) ⊂ Tp M; (ii) the normal Jacobi operator Kζ = R̄ ζ |Tp M and the shape operator Aζ of M satisfy Kζ Aζ = Aζ Kζ . Obviously, we need to check these two conditions only for unit normal vectors. Let ζ ∈ νp M be a unit normal vector of M. It follows from (1.6) that the eigenvalues of the Jacobi operator R̄ ζ are 0, c and 4c with corresponding eigenspaces ℝζ , ℝJζ and Tp M̄ ⊖ ℂζ . Therefore R̄ ζ leaves Tp M invariant if and only if Jζ ∈ νp M or Jζ ∈ Tp M. By continuity, this implies that R̄ ζ (Tp M) ⊂ Tp M holds for all unit normal vectors ζ if and only if M is a complex submanifold or M is an anti-holomorphic submanifold. If M is a complex submanifold, then Kζ = 4c idTp M and therefore Kζ Aζ = Aζ Kζ . If M is an anti-holomorphic submanifold, then Kζ Aζ = Aζ Kζ holds if and only if Jζ is a principal curvature vector of M with respect to ζ . This finishes the proof. For real hypersurfaces this immediately implies the following. Corollary 3.3.4. Let M be a real hypersurface in a complex space form M̄ with constant holomorphic sectional curvature c ≠ 0. Then M is curvature-adapted if and only if M is a Hopf hypersurface.

3.3 Hopf hypersurfaces | 61

̄ then the princiDefinition 3.3.5. If M is a Hopf hypersurface in a Kähler manifold M, pal curvature function α = g(Aξ , ξ ) is called the Hopf principal curvature function on M. Proposition 3.3.6. Let M be a Hopf hypersurface in a Kähler manifold M.̄ The differential dα of the Hopf principal curvature function α satisfies dα(X) = dα(ξ )η(X) − η(KϕX)

(3.7)

for all X ∈ X(M). Equivalently, the gradient gradM α of α is gradM α = dα(ξ )ξ + ϕKξ .

(3.8)

Proof. Using Lemma 3.2.4 and the Codazzi equation we get − g((Kϕ + ϕK)X, Y) ̄ Y)ξ , ζ ) = g(R(X,

= g((∇X A)Y − (∇Y A)X, ξ ) = g((∇X A)ξ , Y) − g((∇Y A)ξ , X)

= dα(X)η(Y) − dα(Y)η(X) + αg((Aϕ + ϕA)X, Y) − 2g(AϕAX, Y)

(3.9)

for all X, Y ∈ X(M). Inserting Y = ξ and using Aξ = αξ and ϕξ = 0, we obtain dα(X) = dα(ξ )η(X) − g(KϕX, ξ ) = dα(ξ )η(X) − η(KϕX). This readily implies the equation for the gradient of α. Proposition 3.3.6 immediately implies the following corollary. Corollary 3.3.7. Let M be a Hopf hypersurface in a Kähler manifold M.̄ Then the Hopf principal curvature function α is constant if and only if the two following properties hold: (i) α is constant along the Reeb flow on M, that is, dα(ξ ) = 0; (ii) the Reeb vector field ξ is an eigenvector of the normal Jacobi operator K everywhere, that is, Kξ = κξ , where κ is the holomorphic sectional curvature function of M̄ with respect to the complex lines spanned by the normal vectors of M. The following result provides a useful relation between the structure tensor field ϕ, the shape operator A and the normal Jacobi operator K of Hopf hypersurfaces.

62 | 3 Real hypersurfaces in Kähler manifolds Proposition 3.3.8. Let M be a Hopf hypersurface in a Kähler manifold M.̄ Then we have g((α(Aϕ + ϕA) − 2AϕA)X, Y)

= η(KϕX)η(Y) − η(KϕY)η(X) − g((Kϕ + ϕK)X, Y)

for all X, Y ∈ X(M). Proof. This follows by inserting dα(X) and dα(Y) as in (3.7) into equation (3.9). As a consequence we derive useful equations for the principal curvatures of Hopf hypersurfaces in Kähler manifolds. Corollary 3.3.9. Let M be a Hopf hypersurface in a Kähler manifold M.̄ Let X, Y ∈ Γ(𝒞 ) with AX = λX and AY = μY. Then we have (α(λ + μ) − 2λμ)g(ϕX, Y) = −g((Kϕ + ϕK)X, Y). Proof. The equation follows immediately from Proposition 3.3.8 when inserting X, Y ∈

𝒞 with AX = λX and AY = μY.

Corollary 3.3.10. Let M be a Hopf hypersurface in a Kähler manifold M.̄ Let X ∈ Γ(𝒞 ) with AX = λX. Then we have (α − 2λ)AϕX + αλϕX = η(KϕX)ξ − KϕX − ϕKX = −(KϕX)𝒞 − ϕKX, where (KϕX)𝒞 is the orthogonal projection of KϕX onto 𝒞 . Proof. The equation follows immediately from Proposition 3.3.8 when inserting X ∈ Γ(𝒞 ) with AX = λX. For Ricci solitons, we obtain as a special case of Lemma 3.2.5 the following. Lemma 3.3.11. Let M be a Hopf hypersurface in a Kähler manifold M.̄ Assume that (M, g) is a Ricci soliton with potential field ξ , that is, there exists c ∈ ℝ such that 1 Ric = cg − ℒξ g. 2 Then we have Ric(ϕX) + ϕRic(X) = 2cϕX for all X ∈ X(M). In Hermitian symmetric spaces we have ∇̄ R̄ = 0. Using this fact we can develop Proposition 3.3.8 further.

3.3 Hopf hypersurfaces | 63

Proposition 3.3.12. Let M be a Hopf hypersurface in a Hermitian symmetric space M.̄ Then dα(ξ )g((Aϕ + ϕA)X, Y)

= η(X)η(KAY) − η(Y)η(KAX) − αη(X)η(KY) + αη(Y)η(KX) + 3g((AϕKϕ − ϕKϕA)X, Y) + g((KA − AK)X, Y)

holds for all X, Y ∈ X(M). Proof. From (3.8) we have gradM α = dα(ξ )ξ + ϕKξ = dα(ξ )ξ + ϕR̄ ζ ξ . The Hessian of α is given by hessM α(X, Y) = g(∇X gradM α, Y) = Xdα(Y) − dα(∇X Y). As the Hessian of a function is symmetric, we have 0 = hessM α(X, Y) − hessM α(Y, X) = Xdα(Y) − Ydα(X) − dα([X, Y]). Since M̄ is a Hermitian symmetric space, we have ∇̄ R̄ = 0 and ∇J̄ = 0. Using these identities we first calculate Xdα(Y) by Xdα(Y) = X(dα(ξ )η(Y) − g(R̄ ζ ξ , JY)) = Xdα(ξ )η(Y) + dα(ξ )Xη(Y) − Xg(R̄ ζ ξ , JY)

= hessM α(X, ξ )η(Y) + dα(∇X ξ )η(Y) + dα(ξ )η(∇X Y) + dα(ξ )g(Y, ∇X ξ ) − g(∇̄X R̄ ζ ξ , JY) − g(R̄ ζ ξ , ∇̄X JY)

= hessM α(X, ξ )η(Y) + dα(ϕAX)η(Y) + dα(ξ )η(∇X Y) + dα(ξ )g(Y, ϕAX) ̄ , ∇̄X ζ )ζ , JY) − g(R(ξ ̄ , ζ )∇̄X ζ , JY) − g(R(̄ ∇̄X ξ , ζ )ζ , JY) − g(R(ξ ̄ , ζ )ζ , ∇̄X JY) − g(R(ξ

= hessM α(X, ξ )η(Y) + dα(ϕAX)η(Y) + dα(ξ )η(∇X Y) + dα(ξ )g(Y, ϕAX) ̄ , ζ )∇̄X ξ , Y) − g(R(ξ ̄ , ζ )ζ , J ∇̄X Y) − 2g(R(̄ ∇̄X ξ , ζ )ζ , JY) − g(R(ξ

= hessM α(X, ξ )η(Y) + dα(ϕAX)η(Y) + dα(ξ )η(∇X Y) + dα(ξ )g(Y, ϕAX) ̄ ̄ , ζ )JAX, Y) − g(R(ξ ̄ , ζ )ζ , J ∇̄X Y). − 2g(R(JAX, ζ )ζ , JY) − g(R(ξ

This implies 0 = Xdα(Y) − Ydα(X) − dα([X, Y])

= hessM α(X, ξ )η(Y) + dα(ϕAX)η(Y) + dα(ξ )η(∇X Y) + dα(ξ )g(Y, ϕAX)

64 | 3 Real hypersurfaces in Kähler manifolds ̄ ̄ , ζ )JAX, Y) − g(R(ξ ̄ , ζ )ζ , J ∇̄X Y) − 2g(R(JAX, ζ )ζ , JY) − g(R(ξ

− hessM α(Y, ξ )η(X) − dα(ϕAY)η(X) − dα(ξ )η(∇Y X) − dα(ξ )g(X, ϕAY) ̄ ̄ , ζ )JAY, X) + g(R(ξ ̄ , ζ )ζ , J ∇̄Y X) + 2g(R(ϕAY, ζ )ζ , JX) + g(R(ξ ̄ , ζ )ζ , J[X, Y]) − dα(ξ )η([X, Y]) + g(R(ξ

= hessM α(X, ξ )η(Y) + dα(ϕAX)η(Y) + dα(ξ )g(Y, ϕAX) ̄ ̄ , ζ )JAX, Y) − 2g(R(JAX, ζ )ζ , JY) − g(R(ξ

− hessM α(Y, ξ )η(X) − dα(ϕAY)η(X) − dα(ξ )g(X, ϕAY) ̄ ̄ , ζ )JAY, X) + 2g(R(JAY, ζ )ζ , JX) + g(R(ξ

= hessM α(X, ξ )η(Y) − hessM α(Y, ξ )η(X) + dα(ϕAX)η(Y) − dα(ϕAY)η(X) + dα(ξ )g((Aϕ + ϕA)X, Y) ̄ ̄ , ζ )JAX, Y) − 2g(R(JAX, ζ )ζ , JY) − g(R(ξ

̄ ̄ , ζ )JAY, X). + 2g(R(JAY, ζ )ζ , JX) + g(R(ξ

Inserting Y = ξ gives hessM α(X, ξ ) = hessM α(ξ , ξ )η(X) − dα(ϕAX) + g(R̄ ζ ξ , AX) − αg(R̄ ζ ξ , X). Inserting this and the analogous equation for hessM α(Y, ξ ) into the previous equation leads to dα(ξ )g((Aϕ + ϕA)X, Y) = η(X)g(R̄ ζ ξ , AY) − η(Y)g(R̄ ζ ξ , AX) ̄ , ζ )Jξ , Y) + αη(Y)g(R(ξ ̄ , ζ )Jξ , X) − αη(X)g(R(ξ ̄ ̄ , ζ )JAX, Y) + 2g(R(JAX, ζ )ζ , JY) + g(R(ξ ̄ ̄ , ζ )JAY, X). − 2g(R(JAY, ζ )ζ , JX) − g(R(ξ

Using again the algebraic Bianchi identity, we can rewrite ̄ , ζ )JAX, Y) = g(R(JAX, ̄ ̄ g(R(ξ ζ )ζ , JY) + g(R(Y, ζ )ζ , AX). Inserting this equation and the corresponding one with X and Y interchanged into the previous equation leads to the equation in the assertion. As a consequence we derive a useful equation for the principal curvatures of Hopf hypersurfaces in Hermitian symmetric spaces. Corollary 3.3.13. Let M be a Hopf hypersurface in a Hermitian symmetric space M.̄ Let X, Y ∈ Γ(𝒞 ) with AX = λX and AY = μY. Then we have (λ + μ)dα(ξ )g(ϕX, Y) = 3(λ − μ)g(KϕX, ϕY) + (λ − μ)g(KX, Y).

3.4 Real hypersurfaces with isometric Reeb flow

| 65

Proof. The equation follows immediately when inserting X, Y ∈ Γ(𝒞 ) with AX = λX and AY = μY into Proposition 3.3.12.

3.4 Real hypersurfaces with isometric Reeb flow In this section we investigate real hypersurfaces in Kähler manifolds whose Reeb flow is an isometric flow. Proposition 3.4.1. Let M be a real hypersurface in a Kähler manifold M.̄ Then the following statements are equivalent: (i) the Reeb flow on M is an isometric flow (or equivalently, the Reeb vector field ξ is a Killing vector field); (ii) the covariant derivative ∇ξ of the Reeb vector field ξ on M is a skew-symmetric tensor field; (iii) the shape operator A of M and the structure tensor field ϕ on M commute, that is, Aϕ = ϕA; (iv) M is a Hopf hypersurface and the principal curvature spaces in 𝒞 are complex subspaces, that is, Aξ = αξ and AϕX = λϕX for all X ∈ Γ(𝒞 ) with AX = λX. Proof. The Reeb flow on M is an isometric flow if and only if ξ is a Killing vector field. The equivalence of (i) and (ii) uses a well-known fact for Killing vector fields on Riemannian manifolds (see Section 1.1.7). Furthermore, we have 0 = g(∇X ξ , Y) + g(X, ∇Y ξ ) = −g(∇̄X Jζ , Y) − g(X, ∇̄Y Jζ ) = −g(J ∇̄X ζ , Y) − g(X, J ∇̄Y ζ ) = g(JAX, Y) − g(X, JAY) = g(ϕAX, Y) − g(X, ϕAY) = g((ϕA − Aϕ)X, Y)

for all X, Y ∈ X(M). This shows the equivalence of (ii) and (iii). The equivalence of (iii) and (iv) is straightforward. Proposition 3.4.2. Let M be a real hypersurface with isometric Reeb flow in a Kähler manifold M.̄ Then we have (A − αI)AϕX = KϕX − η(KϕX)ξ = (KϕX)𝒞 for all X ∈ X(M), where (KϕX)𝒞 is the orthogonal projection of KϕX onto 𝒞 . Proof. Since M has isometric Reeb flow, we have Aϕ = ϕA by Proposition 3.4.1. Differentiating Aϕ = ϕA covariantly, we obtain (∇X A)ϕY + A(∇X ϕ)Y = (∇X ϕ)AY + ϕ(∇X A)Y

66 | 3 Real hypersurfaces in Kähler manifolds for all X, Y ∈ X(M). Using (3.5), this leads to (∇X A)ϕY − ϕ(∇X A)Y = αg(AX, Y)ξ − g(AX, AY)ξ + αη(Y)AX − η(Y)A2 X. Taking inner product with a vector field Z ∈ X(M) gives g((∇X A)Y, ϕZ) + g((∇X A)Z, ϕY)

= αη(Z)g(AX, Y) − η(Z)g(AX, AY) + αη(Y)g(AX, Z) − η(Y)g(AX, AZ),

which implies g((∇X A)Y, ϕZ) + g((∇X A)Z, ϕY) + g((∇Y A)Z, ϕX) + g((∇Y A)X, ϕZ) − g((∇Z A)X, ϕY) − g((∇Z A)Y, ϕX)

= 2αη(Z)g(AX, Y) − 2η(Z)g(AX, AY). The left-hand side of this equation can be rewritten as 2g((∇X A)Y, ϕZ) − g((∇X A)Y − (∇Y A)X, ϕZ)

+ g((∇Y A)Z − (∇Z A)Y, ϕX) − g((∇Z A)X − (∇X A)Z, ϕY),

and using the Codazzi equation this implies 2g((∇X A)Y, ϕZ) = 2αη(Z)g(AX, Y) − 2η(Z)g(AX, AY) ̄ Y)ϕZ, ζ ) − g(R(Y, ̄ ̄ X)ϕY, ζ ). + g(R(X, Z)ϕX, ζ ) + g(R(Z, Replacing Z by ϕZ, the left-hand side of the previous equation becomes −2g((∇X A)Y, Z) + 2η(Z)g((∇X A)Y, ξ ). Since g((∇X A)Y, ξ ) = g((∇X A)ξ , Y) = g(∇X (αξ ), Y) − g(A∇X ξ , Y), we see that 2dα(X)η(Y)η(Z) + 2αη(Z)g(ϕAX, Y) − 2η(Z)g(AϕAX, Y) − 2g((∇X A)Y, Z) becomes the left-hand side of the previous equation when we replace Z by ϕZ. Replacing Z by ϕZ also on the right-hand side we get 2dα(X)η(Y)η(Z) + 2αη(Z)g(ϕAX, Y) − 2η(Z)g(AϕAX, Y) − 2g((∇X A)Y, Z) ̄ Y)ϕ2 Z, ζ ) − g(R(Y, ̄ ̄ = g(R(X, ϕZ)ϕX, ζ ) + g(R(ϕZ, X)ϕY, ζ )

3.4 Real hypersurfaces with isometric Reeb flow

| 67

̄ Y)ζ , ϕ2 Z) − g(R(ϕX, ̄ ̄ = −g(R(X, ζ )Y, ϕZ) − g(R(ϕY, ζ )X, ϕZ) ̄ ̄ = g(R(X, Y)ζ , Z) − η(Z)g(R(X, Y)ζ , ξ ) ̄ ̄ − g(R(ϕX, ζ )Y, ϕZ) − g(R(ϕY, ζ )X, ϕZ),

or equivalently, 2g((∇X A)Y, Z) = 2dα(X)η(Y)η(Z) + 2αη(Z)g(ϕAX, Y) − 2η(Z)g(AϕAX, Y) ̄ Y)ζ , Z) + η(Z)g(R(X, ̄ Y)ζ , ξ ) − g(R(X, ̄ ̄ + g(R(ϕX, ζ )Y, ϕZ) + g(R(ϕY, ζ )X, ϕZ).

Since JZ = ϕZ + η(Z)ζ , this implies 2g((∇X A)Y, Z) = (2dα(X)η(Y) + 2αg(ϕAX, Y) − 2g(AϕAX, Y) ̄ Y)ξ , ζ ) − g(R(ϕX, ̄ ̄ − g(R(X, ζ )Y, ζ ) − g(R(ϕY, ζ )X, ζ ))η(Z) ̄ Y)ζ , Z) − g(J R(ϕX, ̄ ̄ − g(R(X, ζ )Y, Z) − g(J R(ϕY, ζ )X, Z).

Since M̄ is a Kähler manifold, we have ̄ ̄ g(J R(ϕX, ζ )Y, Z) = g(R(ϕX, ζ )JY, Z) ̄ ̄ = g(R(ϕX, ζ )ϕY, Z) + η(Y)g(R(ϕX, ζ )ζ , Z). The previous equation then becomes 2g((∇X A)Y, Z) = (2dα(X)η(Y) + 2αg(ϕAX, Y) − 2g(AϕAX, Y) ̄ Y)ξ , ζ ) − g(R(ϕX, ̄ ̄ − g(R(X, ζ )Y, ζ ) − g(R(ϕY, ζ )X, ζ ))η(Z) ̄ ̄ − η(Y)g(R(ϕX, ζ )ζ , Z) − η(X)g(R(ϕY, ζ )ζ , Z) ̄ Y)ζ , Z) − g(R(ϕX, ̄ ̄ − g(R(X, ζ )ϕY, Z) − g(R(ϕY, ζ )ϕX, Z).

Inserting Y = ξ and using basic curvature identities for Kähler manifolds, we obtain ̄ ξ )ξ , ζ ) + g(R(ϕX, ̄ 2g((∇X A)ξ , Z) = (2dα(X) − g(R(X, ζ )ζ , ξ ))η(Z) ̄ ̄ ξ )ζ , Z) − g(R(ϕX, ζ )ζ , Z) − g(R(X,

̄ ̄ = (2dα(X) + g(R(JX, ζ )ζ , ξ ) + g(R(ϕX, ζ )ζ , ξ ))η(Z) ̄ ̄ − g(R(ϕX, ζ )ζ , Z) − g(R(JX, ζ )ζ , Z)

̄ ̄ = (2dα(X) + 2g(R(ϕX, ζ )ζ , ξ ))η(Z) − 2g(R(ϕX, ζ )ζ , Z). On the other hand, we have (∇X A)ξ = dα(X)ξ + α∇X ξ − A∇X ξ = dα(X)ξ + αAϕX − A2 ϕX.

68 | 3 Real hypersurfaces in Kähler manifolds Inserting this into the previous equation implies ̄ ̄ αg(AϕX, Z) − g(A2 ϕX, Z) = g(R(ϕX, ζ )ζ , ξ )η(Z) − g(R(ϕX, ζ )ζ , Z), which implies the assertion since K = R̄ ζ |TM . Combining the equations in Proposition 3.3.8 and Proposition 3.4.2 leads to the following. Corollary 3.4.3. Let M be a real hypersurface with isometric Reeb flow in a Kähler manifold M.̄ Then we have g((Kϕ − ϕK)X, Y) = η(KϕX)η(Y) + η(KϕY)η(X) for all X, Y ∈ X(M), or equivalently, (Kϕ − ϕK)X = η(KϕX)ξ − η(X)ϕKξ for all X ∈ X(M). As a consequence, we obtain useful equations for the principal curvatures of real hypersurfaces with isometric Reeb flow in Kähler manifolds. Corollary 3.4.4. Let M be a real hypersurface with isometric Reeb flow in a Kähler manifold M.̄ Let X, Y ∈ Γ(𝒞 ) with AX = λX and AY = μY. Then we have (α(λ + μ) − 2λμ)g(ϕX, Y) = −2g(KϕX, Y). Proof. The equation follows immediately from Corollary 3.4.3 and Corollary 3.3.9. Corollary 3.4.5. Let M be a real hypersurface with isometric Reeb flow in a Kähler manifold M.̄ Let X ∈ Γ(𝒞 ) with AX = λX. Then we have KX = λ(λ − α)X + η(KX)ξ . Proof. From Corollary 3.3.10 and Proposition 3.4.2, using the fact that Aϕ = ϕA, we obtain λ(λ − α)ϕX = ϕKX. Applying ϕ to both sides of the equation leads to the assertion. The next result provides useful information about the eigenspaces of the normal Jacobi operator if the Reeb flow is an isometric flow. Proposition 3.4.6. Let M be a real hypersurface with isometric Reeb flow in a Kähler manifold M.̄ Then we have g(KX, Y) = g(KϕX, ϕY) for all X, Y ∈ Γ(𝒞 ).

3.4 Real hypersurfaces with isometric Reeb flow

| 69

Proof. By linearity, it suffices to prove this for vector fields X ∈ Γ(𝒞 ) with AX = λX. Then we also have AϕX = ϕAX = λϕX. From Corollary 3.4.5 we then get g(KX, Y) = λ(λ − α)g(X, Y) = λ(λ − α)g(ϕX, ϕY) = g(KϕX, ϕY) for all Y ∈ Γ(𝒞 ), which proves the assertion. We now restrict to the case that the Kähler manifold M̄ is a Hermitian symmetric space. In this situation we can use the equation ∇̄ R̄ = 0. From Corollary 3.3.13 and Proposition 3.4.6 we immediately get the following. Corollary 3.4.7. Let M be a real hypersurface with isometric Reeb flow in a Hermitian symmetric space M.̄ Let X, Y ∈ Γ(𝒞 ) with AX = λX and AY = μY. Then we have (λ + μ)dα(ξ )g(ϕX, Y) = 4(λ − μ)g(KX, Y). Proposition 3.4.8. Let M be a real hypersurface with isometric Reeb flow in an irreducible Hermitian symmetric space M.̄ Then we have dα(ξ ) = 0. Proof. Assume that dα(ξ ) ≠ 0 at some point p ∈ M. Then, by continuity, dα(ξ ) ≠ 0 on an open neighborhood U of p in M. The following calculations are valid on U. Choosing Y = ϕX in Corollary 3.4.7, so μ = λ, we see that AX = 0

for all X ∈ Γ(𝒞 ).

For X, Y, Z ∈ Γ(𝒞 ) we therefore get g((∇X A)Y, Z) = g(∇X AY, Z) − g(A∇X Y, Z) = 0 and hence ̄ Y)Z, ζ ) = 0 g(̄ R(X, by the Codazzi equation. This also implies ̄ Y)Z, ξ ) = g(̄ R(X, ̄ Y)JZ, Jξ ) = g(̄ R(X, ̄ Y)ϕZ, ζ ) = 0. g(̄ R(X, Altogether this implies that R(̄ 𝒞 , 𝒞 )𝒞 ⊆ 𝒞 , that is, 𝒞q is a curvature-invariant subspace of Tq M̄ at each point q ∈ U. It follows from Theorem 2.2.3 that there exists a totally geodesic submanifold Σ of M̄ with Tp Σ = 𝒞p . Since 𝒞 is J-invariant, Σ is a totally ̄ It follows that the index i(M) ̄ of M̄ is at most geodesic complex hypersurface in M. ̄ ̄ It 2, where i(M) is the minimal codimension of the totally geodesic submanifold of M. was proved in [13] that the only irreducible Hermitian symmetric spaces with index ≤ 2 are the complex projective space ℂP n , the complex quadric Qn and their non-compact

70 | 3 Real hypersurfaces in Kähler manifolds dual symmetric spaces ℂH n and Qn , which all are known to admit totally geodesic complex hypersurfaces. From Proposition 3.4.6 and Corollary 3.3.9 we obtain ∗

g(KX, Y) = g(KϕX, ϕY) = −g(ϕKX, ϕY) = −g(KX, Y) for all X, Y ∈ Γ(𝒞 ), which implies g(KX, Y) = 0 for all X, Y ∈ Γ(𝒞 ) and hence KX ∈ ℝξ

for all X ∈ Γ(𝒞 ).

̄ However, it follows from (1.6) This implies dim(ker(K)) ≥ 2n − 2, where n = dimℂ (M). that for ℂP n (c) (resp. ℂH n (c)) this is not possible since the eigenvalues of K are 4c and c (resp. 4c and c) with multiplicity 2n − 2 and 1, respectively. For the complex quadric Qn , it follows from Lemma 1.4.2 that dim(ker(K)) ≤ n. Since Q2 is not irreducible, we have n ≥ 3 and therefore dim(ker(K)) ≥ 2n−2 is not possible. By duality, the analogous statement holds for the non-compact dual symmetric space of the complex quadric. Altogether it follows that the assumption dα(ξ ) ≠ 0 leads to a contradiction. Thus we must have dα(ξ ) = 0. From Corollary 3.3.9, Corollary 3.4.7 and Proposition 3.4.8 we obtain the following. Corollary 3.4.9. Let M be a real hypersurface with isometric Reeb flow in an irreducible Hermitian symmetric space M.̄ Let X, Y ∈ Γ(𝒞 ) with AX = λX and AY = μY. Then we have (λ − μ)g(KX, Y) = 0

(3.10)

and, if ‖X‖ = 1, λ2 − αλ − g(KX, X) = 0. The Hopf principal curvature function α and the normal Jacobi operator are related by the following result. Proposition 3.4.10. Let M be a real hypersurface with isometric Reeb flow in an irreducible Hermitian symmetric space M.̄ Then the following statements are equivalent: (i) the Hopf principal curvature function α is constant; (ii) the Reeb vector field ξ is an eigenvector of the normal Jacobi operator K everywhere. Proof. From Proposition 3.4.8 and (3.7) we obtain dα(X) = −η(KϕX)

3.5 Contact hypersurfaces | 71

for all X ∈ X(M). Thus α is constant if and only if KϕX ∈ Γ(𝒞 ) for all X ∈ X(M). However, we have KϕX ∈ Γ(𝒞 ) if and only if 0 = g(KϕX, ξ ) = g(ϕX, Kξ ) for all X ∈ X(M), which is equivalent to Kξ ∈ ℝξ . This finishes the proof. The following result provides a geometric characterization, in terms of totally geodesic submanifolds, of the property that ξ is an eigenvector of the normal Jacobi operator K. Proposition 3.4.11. Let M̄ be a Hermitian symmetric space, o ∈ M̄ and ζ ∈ To M.̄ Then the following statements are equivalent: ̄ ζ )ζ ; (i) Jζ is an eigenvector of R̄ ζ = R(⋅, (ii) there exists a totally geodesic complex submanifold Σ of M̄ with o ∈ Σ and To Σ = ℂζ . ̄ ζ )ζ , say R̄ ζ Jζ = κJζ . Let G Proof. We first assume that Jζ is an eigenvector of R̄ ζ = R(⋅, be the identity component of the isometry group of M̄ and K the isotropy group of G at o. Denote by g = k ⊕ p the corresponding Cartan decomposition of the Lie algebra g of G. With the usual identification To M̄ ≅ p, the curvature tensor R̄ of M̄ can be written ̄ Y)Z = −[[X, Y], Z]. The assumption gives us [[Jζ , ζ ], ζ ] = −κJζ , and since J is as R(X, parallel this implies ̄ , Jζ )Jζ = −J R(ζ ̄ , Jζ )ζ = J R(Jζ ̄ , ζ )ζ = −κζ . [[ζ , Jζ ], Jζ ] = −R(ζ It follows that ℂζ is a Lie triple system in p, which implies (ii). ̄ The converse is obvious since Σ is complex and totally geodesic in M. Proposition 3.4.12. Let M be a real hypersurface with isometric Reeb flow in an irreducible Hermitian symmetric space M.̄ If the Hopf principal curvature function α is constant, then M is curvature-adapted. Proof. From Proposition 3.4.10 we see that the Reeb vector field ξ is an eigenvector of the normal Jacobi operator K everywhere. Equation (3.10) then implies that M is curvature-adapted.

3.5 Contact hypersurfaces A contact manifold is an odd-dimensional smooth manifold M together with a 1-form η on M satisfying η ∧ (dη)k ≠ 0,

72 | 3 Real hypersurfaces in Kähler manifolds where dimℝ (M) = 2k + 1. Such a 1-form η is called a contact form. The kernel of η defines a hyperplane distribution 𝒞 on M, the so-called contact distribution. The contact condition η ∧ (dη)k ≠ 0 means that the maximal possible dimension of a submanifold of M all of whose tangent spaces are contained in 𝒞 is equal to k. The contact condition therefore can be seen as a measure for maximal non-integrability of 𝒞 . Contact manifolds are objects studied in contact geometry, the odd-dimensional analog of symplectic geometry. The natural question in our context is: Given a real hypersurface in a Kähler manifold M̄ with almost contact metric structure (ϕ, ξ , η, g), when is η a contact form? We begin with a standard example in this context. Example 3.5.1. Let S2n−1 (r) be the sphere with radius r ∈ ℝ+ in the complex Euclidean space ℂn (see Section 1.2.3) and consider the almost contact metric structure (ϕ, ξ , η, g) on S2n−1 (r). The Reeb vector field ξ on S2n−1 (r) is defined by πp (ξp ) = − 1r ip, where πp : Tp ℂn → ℂn is the canonical isomorphism. A straightforward calculation shows that dη(X, Y) = − 2r ω(X, Y), where ω is the fundamental 2-form on S2n−1 (r). Since ω has rank 2(n − 1) on the kernel 𝒞 of η, it follows that η ∧ (dη)n−1 ≠ 0. Geometrically this means that the maximal holomorphic subbundle 𝒞 of TS2n−1 (r) is as far as possible away from being integrable. The maximal dimension of a submanifold of S2n−1 (r) whose tangent spaces are all contained in 𝒞 is equal to n − 1. Definition 3.5.2. A real hypersurface M in a Kähler manifold M̄ with induced almost contact metric structure (ϕ, ξ , η, g) is said to be a contact hypersurface if there exists an everywhere non-zero smooth function f : M → ℝ such that dη = 2fω, where ω is the fundamental 2-form on M. Remark 3.5.3. If dη = 2fω, then η ∧ dηn−1 = (2f )n−1 (η ∧ ωn−1 ) ≠ 0 everywhere. Thus every contact hypersurface in a Kähler manifold is a contact manifold. Proposition 3.5.4. Let M be a connected real hypersurface in a Kähler manifold M̄ and ̄ > 2. Assume that there exists an everywhere non-zero smooth function n = dimℂ (M) f : M → ℝ such that dη = 2fω. Then f is constant. Proof. We take the exterior derivative of the equation dη = 2fω and use Proposition 3.2.3 to get 0 = d2 η = 2df ∧ ω, or equivalently, 0 = df (X)g(ϕY, Z) + df (Y)g(ϕZ, X) + df (Z)g(ϕX, Y)

(3.11)

for all X, Y, Z ∈ X(M). Inserting X = ξ , Y ∈ Γ(𝒞 ) with ‖Y‖ = 1 and Z = ϕY gives df (ξ ) = 0. Let X ∈ Γ(𝒞 ). Since n > 2, we can choose a (local) unit vector field Y ∈ Γ(𝒞 ) which is perpendicular to both X and ϕX. Inserting X, Y and Z = ϕY into equation (3.11) gives df (X) = 0. Altogether, since M is connected, this implies that f is constant.

3.5 Contact hypersurfaces | 73

The corresponding statement for n = 2 does not hold. Before discussing this, we prove the following useful characterization of contact hypersurfaces. Proposition 3.5.5. Let M be a real hypersurface in a Kähler manifold M.̄ Then M is a contact hypersurface if and only if there exists an everywhere non-zero smooth function f on M such that Aϕ + ϕA = 2fϕ,

(3.12)

where A is the shape operator of M. Proof. The equation dη = 2fω means that dη(X, Y) = 2fg(ϕX, Y) for all X, Y ∈ X(M). If M is a contact hypersurface, then dη(X, Y) = 2fg(ϕX, Y), which implies Aϕ + ϕA = 2fϕ by (3.6). Conversely, if Aϕ + ϕA = 2fϕ, then dη(X, Y) = 2fg(ϕX, Y) = 2fω(X, Y), which implies that M is a contact hypersurface. Proposition 3.5.6. Let M be a contact hypersurface in a Kähler manifold M.̄ Then the following statements hold: (i) M is a Hopf hypersurface; (ii) if X ∈ Γ(𝒞 ) with AX = λX, then ϕX ∈ Γ(𝒞 ) satisfies AϕX = (2f − λ)ϕX: AX = λX

󳨐⇒

AϕX = (2f − λ)ϕX;

(iii) the mean curvature of M is given by tr(A) = α + 2(n − 1)f ,

(3.13)

where α = g(Aξ , ξ ) is the Hopf principal curvature function. Proof. Since ϕξ = 0, equation (3.12) implies ϕAξ = 0. Therefore Aξ ∈ ker(ϕ) = ℝξ , which implies that M is a Hopf hypersurface by Proposition 3.3.2. Let X ∈ Γ(𝒞 ) with AX = λX. Then equation (3.12) implies AϕX = (2f − λ)ϕX. By adding up the principal curvatures we obtain tr(A) = α + 2(n − 1)f . We now come back to contact hypersurfaces in 2-dimensional Kähler manifolds. Proposition 3.5.7. Let M be a real hypersurface in a Kähler manifold M̄ with n = ̄ = 2. Then M is a contact hypersurface if and only if M is a Hopf hypersurface dimℂ (M) and tr(A) ≠ α everywhere. Proof. The “only if” part follows from Proposition 3.5.6. Assume that M is a Hopf hypersurface and tr(A) ≠ α everywhere. Then we have Aξ = αξ and A𝒞 ⊆ 𝒞 . Let X ∈ Γ(𝒞 )

74 | 3 Real hypersurfaces in Kähler manifolds with AX = λX. Since the rank of 𝒞 is equal to 2, the vector field ϕX ∈ Γ(𝒞 ) must be a principal curvature vector of M everywhere, say AϕX = μϕX. Then we have AϕX + ϕAX = (μ + λ)ϕX and Aϕ(ϕX) + ϕA(ϕX) = −(λ + μ)X = (λ + μ)ϕ(ϕX). This shows that the equation Aϕ + ϕA = 2fϕ holds with 2f = (λ + μ) = tr(A) − α nonzero everywhere by assumption. It follows from Proposition 3.5.5 that M is a contact hypersurface. Example 3.5.8. Let M̄ be the complex projective plane ℂP 2 equipped with the Fubini– Study metric of constant holomorphic sectional curvature 4. Let Σ be a complex curve in ℂP 2 , that is, Σ is a 1-dimensional complex submanifold of ℂP 2 . Then, at least locally and for small radii, the tubes around Σ are well-defined real hypersurfaces in ℂP 2 . Since any complex curve in ℂP 2 is curvature-adapted, we can calculate the principal curvatures of the tubes using Corollary 2.6.2. Any such tube is a Hopf hypersurface with constant Hopf principal curvature α = −2 cot(2r), where r is the radius. Generically the mean curvature of such a tube is different from −2 cot(2r), and therefore is a contact hypersurface in ℂP 2 by Proposition 3.5.7. We continue our discussion of contact hypersurfaces in Kähler manifolds. For p ∈ M and X ∈ Tp M̄ we denote by X𝒞 the orthogonal projection of X onto 𝒞 . Proposition 3.5.9. Let M be a contact hypersurface in a Kähler manifold M.̄ Then we have ̄ , ζ )ϕX) 2(A2 − 2fA + αf )X = −(R(ξ 𝒞 for all X ∈ Γ(𝒞 ). In particular, for all X ∈ Γ(𝒞 ) with AX = λX we have ̄ , ζ )ϕX) . 2(λ2 − 2λf + αf )X = −(R(ξ 𝒞 Proof. Since M is a contact hypersurface, we know from Proposition 3.5.6 that M is a Hopf hypersurface and hence Aξ = αξ . From Proposition 3.5.5 we also know that Aϕ + ϕA = 2fϕ. Lemma 3.2.4 and Proposition 3.3.8 then imply ̄ Y)ξ , ζ ) η(KϕX)η(Y) − η(KϕY)η(X) + g(R(X,

= η(KϕX)η(Y) − η(KϕY)η(X) − g((Kϕ + ϕK)X, Y) = g((α(Aϕ + ϕA) − 2AϕA)X, Y)

= g((2αfϕ − 2A(2fϕ − Aϕ))X, Y) = g(2(A2 − 2fA + αf )ϕX, Y)

3.5 Contact hypersurfaces | 75

for all X, Y ∈ X(M). Choosing X ∈ Γ(𝒞 ) gives ̄ Y)ξ , ζ ). g(2(A2 − 2fA + αf )ϕX, Y) = η(KϕX)η(Y) + g(R(X, Replacing X by ϕX implies ̄ g(2(A2 − 2fA + αf )X, Y) = η(KX)η(Y) − g(R(ϕX, Y)ξ , ζ ) ̄ , ζ )ϕX, Y). = η(KX)η(Y) − g(R(ξ Inserting Y = ξ gives 0 = 0 and hence no information. Inserting Y ∈ Γ(𝒞 ) gives ̄ , ζ )ϕX, Y). g(2(A2 − 2fA + αf )X, Y) = −g(R(ξ This finishes the proof. Proposition 3.5.10. Let M be a contact hypersurface in a Kähler manifold M.̄ Then the Hopf principal curvature function α is constant if and only if the Reeb vector field ξ is an eigenvector of the normal Jacobi operator K everywhere. Proof. First assume that ξ is an eigenvector of the normal Jacobi operator K everywhere. From Proposition 3.3.6 we then have gradM α = dα(ξ )ξ .

(3.14)

Therefore the Hessian hessM α of α is given by (hessM α)(X, Y) = g(∇X gradM α, Y) = d(dα(ξ ))(X)η(Y) + dα(ξ )g(ϕAX, Y). As hessM α is a symmetric bilinear form, the last equation implies 0 = dα(ξ )g((Aϕ + ϕA)X, Y) = 2fdα(ξ )g(ϕX, Y) for all X, Y ∈ Γ(𝒞 ). Since f is non-zero everywhere, this implies dα(ξ ) = 0 and hence α is constant by (3.14). Conversely, assume that α is constant. From Proposition 3.3.6 we get ϕKξ = 0 and hence Kξ ∈ ker(ϕ) = ℝξ . Proposition 3.5.11. Let M be a contact hypersurface in a Kähler manifold M.̄ Then we have ‖A‖2 = tr(A2 ) = α2 + 2(n − 1)f (2f − α) − ricM (ζ , ζ ) + η(Kξ ), ̄

where ricM is the Ricci tensor of M.̄ ̄

76 | 3 Real hypersurfaces in Kähler manifolds Proof. We choose a local orthonormal frame field of M̄ along M of the form E1 , E2 = JE1 , . . . , E2n−3 , E2n−2 = JE2n−3 , E2n−1 = ξ , E2n = Jξ = ζ . ̄ Since M̄ is Kähler, its (1, 1)-Ricci tensor RicM can by calculated by n

n

ν=1

ν=1

̄ 2ν−1 , JE2ν−1 )JX = ∑ R(E ̄ 2ν , JE2ν )JX RicM (X) = ∑ R(E ̄

along M (see (1.7)). Using Propositions 3.5.9 and 3.5.6 we get 2n−2

tr(A2 ) = α2 + ∑ g(A2 Eν , Eν ) ν=1

2n−2

1 ̄ , ζ )JEν , Eν )) = α2 + ∑ (2fg(AEν , Eν ) − αfg(Eν , Eν ) − g(R(ξ 2 ν=1 = α2 + 2f (tr(A) − α) − 2(n − 1)αf −

1 2n−2 ̄ ν , JEν )Jζ , ζ ) ∑ g(R(E 2 ν=1

̄ , Jζ )Jζ , ζ )) = α2 + 4(n − 1)f 2 − 2(n − 1)αf − (g(RicM (ζ ), ζ ) − g(R(ζ ̄

= α2 + 2(n − 1)f (2f − α) − ricM (ζ , ζ ) + g(Kξ , ξ ), ̄

which proves the assertion. Proposition 3.5.12. Let M be a contact hypersurface in a Kähler manifold M̄ and n = ̄ > 2. Then M has constant mean curvature if and only if the Reeb vector field ξ dimℂ (M) is an eigenvector of the normal Jacobi operator K everywhere. Proof. It follows from (3.13) and Proposition 3.5.4 that M has constant mean curvature if and only if the Hopf principal curvature function is constant. The assertion then follows from Proposition 3.5.10. From Proposition 3.5.10 and Proposition 3.5.12 we immediately get the following. Theorem 3.5.13. Let M be a contact hypersurface in a Kähler manifold M̄ and ̄ > 2. Then the following statements are equivalent: dimℂ (M) (i) the Hopf principal curvature function α is constant; (ii) M has constant mean curvature; (iii) the Reeb vector field ξ is an eigenvector of the normal Jacobi operator K everywhere. We now restrict to the case that the Kähler manifold M̄ is a Hermitian symmetric space. In this situation we can use the equation ∇̄ R̄ = 0.

3.5 Contact hypersurfaces | 77

Proposition 3.5.14. Let M be a contact hypersurface in a Hermitian symmetric space M̄ ̄ > 2. Then we have and dimℂ (M) 2g(((∇X A)A + A(∇X A) − 2f (∇X A) + dα(X)f )Y, Z) ̄ = −2g(R(AX, ξ )JY, Z) + g(AX, Y)η(KϕZ) + g(AX, Z)η(KϕY) − η(∇X Y)η(KZ) − η(∇X Z)η(KY)

for all X ∈ X(M) and Y, Z ∈ Γ(𝒞 ). Proof. According to Proposition 3.5.9 we have ̄ , ζ )JY, Z) 2g((A2 − 2fA + αf )Y, Z) = g(R(Jζ for all vector fields Y, Z ∈ Γ(𝒞 ). Differentiating with respect to X ∈ X(M) leads to 2g(∇X (A2 − 2fA + αf )Y, Z) + 2g((A2 − 2fA + αf )Y, ∇X Z) ̄ , ζ )JY, Z) + g(R(Jζ ̄ , ζ )JY, ∇̄X Z). = g(∇̄X R(Jζ We have 2g(∇X (A2 − 2fA + αf )Y, Z)

= 2g((∇X (A2 − 2fA + αf ))Y, Z) + 2g((A2 − 2fA + αf )∇X Y, Z)

= 2g(((∇X A)A + A(∇X A) − 2f (∇X A) + dα(X)f )Y, Z) + 2g((A2 − 2fA + αf )∇X Y, Z)

and, using (1.5) and the facts that ∇̄ R̄ = 0 and ∇J̄ = 0, ̄ , ζ )JY, Z) g(∇̄X R(Jζ ̄ ̄ , AX)JY, Z) + g(R(Jζ ̄ , ζ )J ∇̄X Y, Z) = −g(R(JAX, ζ )JY, Z) − g(R(Jζ ̄ ̄ , ζ )JZ, ∇̄X Y). = 2g(R(AX, Jζ )JY, Z) + g(R(Jζ

Altogether, 2g(((∇X A)A + A(∇X A) − 2f (∇X A) + dα(X)f )Y, Z)

+ 2g(A2 ∇X Y, Z) − 4fg(A∇X Y, Z) + 2αfg(∇X Y, Z)

+ 2g(A2 Y, ∇X Z) − 4fg(AY, ∇X Z) + 2αfg(Y, ∇X Z) ̄ ̄ , ζ )JZ, ∇̄X Y) + g(R(Jζ ̄ , ζ )JY, ∇̄X Z). = 2g(R(AX, Jζ )JY, Z) + g(R(Jζ According to the Gauss formula we have ∇̄X Y = ∇X Y + g(AX, Y)ζ .

78 | 3 Real hypersurfaces in Kähler manifolds Furthermore, ∇X Y = (∇X Y)𝒞 + g(∇X Y, Jζ )Jζ = (∇X Y)𝒞 − η(∇X Y)Jζ . Inserting these two equations and the analogous ones when we replace Y by Z into the previous one leads to the equation in the assertion. By taking the trace of the equation in Proposition 3.5.14 we obtain the following. Proposition 3.5.15. Let M be a contact hypersurface in an irreducible Hermitian sym̄ > 2. Then we have metric space M̄ and n = dimℂ (M) d(tr(A2 ))(X) − 2fdα(ξ )η(X) + (2n − 1)fdα(X) = 2η(KϕAX) − 2fη(KϕX) for all X ∈ X(M). In particular, d(tr(A2 ))(ξ ) + (2n − 3)fdα(ξ ) = 0. Proof. We first evaluate tr(∇X A) at a point p ∈ M. Let E1 , . . . , E2n−1 be a local orthonormal frame field of M. We can assume that E2k (p) = JE2k−1 (p) ∈ 𝒞p for k ∈ {1, . . . , n − 1} and E2n−1 (p) = Jζp . We may also assume that ∇Xp Ei = 0. Using (3.13) and (3.7) we obtain 2n−1

2n−1

i=1

i=1

tr(∇Xp A) = ∑ g((∇X A)Ei , Ei )|p = ∑ g(∇X (AEi ), Ei )|p 2n−1

= ∑ Xp g(AEi , Ei ) = dp (tr(A))(Xp ) = dp α(Xp ) i=1

= dp α(ξp )η(Xp ) − η(KϕXp ). By varying with p we get tr(∇X A) = d(tr(A))(X) = dα(X) = dα(ξ )η(X) − η(KϕX). Next, we have 2n−1

2tr((∇X A)A)|p = 2 ∑ g((∇X A)AEi , Ei )|p i=1

2n−1

2n−1

= 2 ∑ g(∇X (A2 Ei ), Ei )|p − 2 ∑ g(∇X (AEi ), AEi )|p i=1

2n−1

2n−1

i=1

i=1

i=1

= 2 ∑ Xp g(A2 Ei , Ei ) − ∑ Xp g(AEi , AEi ) = dp (tr(A2 ))(Xp ) = dp (‖A‖2 )(Xp ).

3.5 Contact hypersurfaces | 79

Thus, taking the trace of the equation in Proposition 3.5.14 leads to d(tr(A2 ))(X) − 2fdα(ξ )η(X) + 2fη(KϕX) + (2n − 1)dα(X)f 2n−1

̄ = η(KϕAX) − ∑ g(R(AX, ξ )JEi , Ei ) i=1

at p and thus, by varying with p, everywhere. At the point p we calculate 2n−1

̄ ξ )JEi , Ei ) ∑ g(R(AX, i=1

2n

̄ ̄ = ∑ g(R(AX, ξ )JEi , Ei ) − g(R(AX, ξ )Jζ , ζ ) i=1 2n

̄ i , JEi )ξ , AX) − g(R(JAX, ̄ = ∑ g(R(E Jξ )Jζ , ζ ) i=1 n

n

i=1 n

i=1 n

̄ 2i−1 , JE2i−1 )ξ , AX) + ∑ g(R(E ̄ 2i , JE2i )ξ , AX) − g(R(JAX, ̄ = ∑ g(R(E ζ )Jζ , ζ ) ̄ 2i−1 , JE2i−1 )ξ , AX) − ∑ g(R(JE ̄ 2i−1 , E2i−1 )ξ , AX) − g(R̄ ζ ξ , JAX) = ∑ g(R(E i=1

i=1

n

̄ 2i−1 , JE2i−1 )Jζ , AX) − g(Kξ , ϕAX) = −2 ∑ g(R(E i=1

= −2ricM (ζ , AX) − η(KϕAX) ̄

= −η(KϕAX), where for the second last equality we use (1.7) and for the last equality we use that M̄ is irreducible so that it is an Einstein manifold. By varying with p we obtain that the equation holds at any point. Inserting this into the previous equation gives d(tr(A2 ))(X) − 2fdα(ξ )η(X) + (2n − 1)dα(X)f = 2η(KϕAX) − 2fη(KϕX). Inserting X = ξ leads to d(tr(A2 ))(ξ ) + (2n − 3)fdα(ξ ) = 0. This finishes the proof. Proposition 3.5.16. Let M be a contact hypersurface in an irreducible Hermitian sym̄ > 2. Then we have metric space M̄ and n = dimℂ (M) dα(ξ ) = 0.

80 | 3 Real hypersurfaces in Kähler manifolds Proof. According to Proposition 3.5.11 we have tr(A2 ) = α2 + 2(n − 1)f (2f − α) − ricM (ζ , ζ ) + η(Kξ ). ̄

Since M̄ is an Einstein manifold, the function ricM (ζ , ζ ) is constant. Moreover, since ∇̄ R̄ = 0, we have ̄

̄ , ζ )ζ , ξ ) ξg(R̄ ζ ξ , ξ ) = ξg(R(ξ ̄ , ∇̄ξ ζ )ζ , ξ ) = g(R(̄ ∇̄ξ ξ , ζ )ζ , ξ ) + g(R(ξ ̄ , ζ )∇̄ξ ζ , ξ ) + g(R(ξ ̄ , ζ )ζ , ∇̄ξ ξ ). + g(R(ξ From the Gauss formula we get ∇̄ξ ξ = ∇ξ ξ + g(Aξ , ξ )ζ = ϕAξ + αζ = αζ , and the Weingarten formula implies ∇̄ξ ζ = Aξ = αξ . Inserting this into the previous equation yields ξη(Kξ ) = ξg(R̄ ζ ξ , ξ ) = 0. Altogether we thus get d(tr(A2 ))(ξ ) = 2αdα(ξ ) − 2(n − 1)fdα(ξ ). On the other hand, by Proposition 3.5.15 we have d(tr(A2 ))(ξ ) = −(2n − 3)fdα(ξ ). Comparing the previous two equations leads to 0 = 2αdα(ξ ) − fdα(ξ ) = (2α − f )dα(ξ ). Since f is constant by Proposition 3.5.4, this implies the assertion. Proposition 3.5.17. Let M be a contact hypersurface in an irreducible Hermitian sym̄ > 2. If X ∈ Γ(𝒞 ) with AX = λX, then metric space M̄ and n = dimℂ (M) (2(α − λ) − f )dα(X) = 0. Proof. Assume that AX = λX with X ∈ Γ(𝒞 ). From Proposition 3.5.15 and (3.7) we get d(tr(A2 ))(X) + (2n − 1)fdα(X) = 2(λ − f )η(KϕX) = 2(f − λ)dα(X).

3.5 Contact hypersurfaces | 81

On the other hand, from Proposition 3.5.11 we have d(tr(A2 ))(X) = 2(α − (n − 1)f )dα(X) + Xη(Kξ ). The last term can be calculated as follows: ̄ , ζ )ζ , ξ ) Xη(Kξ ) = Xg(R̄ ζ ξ , ξ ) = Xg(R(ξ ̄ , ∇̄X ζ )ζ , ξ ) + g(R(ξ ̄ , ζ )∇̄X ζ , ξ ) + g(R(ξ ̄ , ζ )ζ , ∇̄X ξ ) = g(R(̄ ∇̄X ξ , ζ )ζ , ξ ) + g(R(ξ ̄ ̄ , AX)ζ , ξ ) − g(R(ξ ̄ , ζ )AX, ξ ) + g(R(ξ ̄ , ζ )ζ , ϕAX) = g(R(ϕAX, ζ )ζ , ξ ) − g(R(ξ

̄ ̄ , X)ζ , ξ ) − g(R(ξ ̄ , ζ )X, ξ ) + g(R(ξ ̄ , ζ )ζ , ϕX)) = λ(g(R(ϕX, ζ )ζ , ξ ) − g(R(ξ ̄ ̄ , JX)Jζ , Jξ ) − g(R(Jξ ̄ , Jζ )JX, Jξ ) + g(R(ξ ̄ , ζ )ζ , JX)) = λ(g(R(JX, ζ )ζ , ξ ) − g(R(Jξ ̄ ̄ , JX)ξ , ζ ) + g(R(ζ ̄ , ξ )JX, ζ ) + g(R(ξ ̄ , ζ )ζ , JX)) = λ(g(R(JX, ζ )ζ , ξ ) + g(R(ζ ̄ , ζ )ζ , JX) = 4λg(Kξ , ϕX) = 4λη(KϕX) = −4λdα(X), = 4λg(R(ξ

where for the last equality we use Proposition 3.5.16 and (3.7). Inserting this into the previous equation gives d(tr(A2 ))(X) = 2(α − (n − 1)f )dα(X) − 4λdα(X). Comparing the two equations for d(tr(A2 ))(X) finishes the proof. Proposition 3.5.18. Let M be a contact hypersurface in an irreducible Hermitian sym̄ > 2. Then metric space M̄ and n = dimℂ (M) 0 = 2fg(KϕX, X) + g(AϕKX, X) + g(ϕKAX, X) for all X ∈ Γ(𝒞 ) with ‖X‖ = 1. Proof. Since M is a contact hypersurface, we have Aϕ + ϕA = 2fϕ by Proposition 3.5.5. Inserting this into the equation in Proposition 3.3.12 and using the fact that dα(ξ ) = 0 by Proposition 3.5.16 gives 0 = η(X)η(KAY) − η(Y)η(KAX) − αη(X)η(KY) + αη(Y)η(KX) + 3g((AϕKϕ − ϕKϕA)X, Y) + g((KA − AK)X, Y)

for all X, Y ∈ X(M). For X, Y ∈ Γ(𝒞 ) this simplifies to 0 = 3g((AϕKϕ − ϕKϕA)X, Y) + g((KA − AK)X, Y). In particular, if Y = ϕX = JX and ‖X‖ = 1, this gives 0 = 3g((AϕKϕ − ϕKϕA)X, ϕX) + g((KA − AK)X, ϕX)

= 3g(AϕKϕX, ϕX) − 3g(ϕKϕAX, ϕX) + g(KAX, ϕX) − g(AKX, ϕX)

82 | 3 Real hypersurfaces in Kähler manifolds = −3g(KϕX, ϕAϕX) − 3g(KϕAX, X) − g(ϕKAX, X) + g(ϕAKX, X). Since ϕAϕX = ϕ(2fϕX − ϕAX) = −2fX + AX, this implies 0 = 6fg(KϕX, X) + 3g(AϕKX, X) + 2g(ϕKAX, X) + g(ϕAKX, X). Inserting ϕAX = 2fϕX − AϕX into the last term of the previous equation gives 0 = 4fg(KϕX, X) + 2g(AϕKX, X) + 2g(ϕKAX, X). This finishes the proof. Theorem 3.5.19. Let M be a contact hypersurface in an irreducible Hermitian symmetric ̄ > 2. Then the following statements hold: space M,̄ n = dimℂ (M) (i) The Hopf principal curvature function α is constant. (ii) The contact hypersurface M has constant mean curvature. (iii) The Reeb vector field ξ is an eigenvector of the normal Jacobi operator K everywhere. Proof. From (3.7) and Proposition 3.5.16 we have dα(X) = −η(KϕX) = −g(Kξ , ϕX) = −g(R̄ ζ ξ , JX) for all X ∈ X(M). For the Hessian hessM α of α we thus get hessM α(X, Y) = Xdα(Y) − dα(∇X Y) = −Xg(R̄ ζ ξ , JY) + g(R̄ ζ ξ , J∇X Y) for all X, Y ∈ X(M). Since g is a metric connection, using the Gauss formula, we obtain hessM α(X, Y) = −g(∇̄X R̄ ζ ξ , JY) − g(R̄ ζ ξ , ∇̄X JY) + g(R̄ ζ ξ , J ∇̄X Y) + g(AX, Y)g(R̄ ζ ξ , ξ ). Then, using the assumption that ∇̄ R̄ = 0 (since M̄ is a symmetric space), ∇J̄ = 0 (since M̄ is a Kähler manifold) and the Weingarten formula, we get ̄ , ∇̄X ζ )ζ , JY) hessM α(X, Y) = −g(R(̄ ∇̄X ξ , ζ )ζ , JY) − g(R(ξ ̄ , ζ )∇̄X ζ , JY) + g(AX, Y)g(R̄ ζ ξ , ξ ) − g(R(ξ ̄ ∇̄X ζ , ζ )ζ , JY) − g(R(ξ ̄ , ∇̄X ζ )ζ , JY) = g(R(J ̄ , ζ )∇̄X ζ , JY) + g(AX, Y)g(R̄ ζ ξ , ξ ) − g(R(ξ ̄ ̄ , AX)ζ , JY) = −g(R(JAX, ζ )ζ , JY) + g(R(ξ ̄ , ζ )AX, JY) + g(AX, Y)g(R̄ ζ ξ , ξ ). + g(R(ξ

3.5 Contact hypersurfaces | 83

Since hessM α is symmetric and A is self-adjoint, this implies 0 = hessM α(Y, X) − hessM α(X, Y) ̄ ̄ , AX)ζ , JY) − g(R(ξ ̄ , ζ )AX, JY) = g(R(JAX, ζ )ζ , JY) − g(R(ξ

̄ ̄ , AY)ζ , JX) + g(R(ξ ̄ , ζ )AY, JX). − g(R(JAY, ζ )ζ , JX) + g(R(ξ

Putting Y = ξ gives ̄ , ζ )AX, ζ ) + αg(R(ξ ̄ , ζ )ξ , JX). 0 = −g(R(ξ Using some basic algebraic curvature identities, this leads to ̄ , ζ )AX, ζ ) + αg(R(ξ ̄ , ζ )ξ , JX) 0 = −g(R(ξ ̄ , ζ )ζ , AX) − αg(R(ζ ̄ , ξ )ξ , JX) = g(R(ξ ̄ , Jξ )Jξ , JJX) = g(R̄ ζ ξ , AX) − αg(R(Jζ ̄ , ζ )ζ , X) = g(R̄ ζ ξ , AX) − αg(R(ξ = g(R̄ ζ ξ , AX) − αg(R̄ ζ ξ , X). Thus, if X ∈ Γ(𝒞 ) with AX = λX, then 0 = (λ − α)g(R̄ ζ ξ , X) = (λ − α)g(Kξ , X).

(3.15)

Therefore, if the principal curvature function α has multiplicity 1 everywhere, then ξ is an eigenvector of the normal Jacobi operator K everywhere. We now need to understand what happens if the multiplicity of α is greater than 1 somewhere. According to (3.15) there exists (locally) a unit vector field Z ∈ Γ(𝒞 ) with AZ = αZ so that Kξ = bZ + cξ with some continuous functions b and c. In particular, this gives dα(X) = −g(Kξ , JX) = −g(bZ + cξ , JX) = −bg(Z, JX) = bg(JZ, X), and hence gradM α = bJZ. From Proposition 3.5.17 we have (2(α − λ) − f )dα(X) = 0 for X ∈ Γ(𝒞 ) with AX = λX. Inserting X = JZ, we have λ = 2f − α by Proposition 3.5.6 and hence 0 = (4(α − f ) − f )b = (4α − 5f )b.

84 | 3 Real hypersurfaces in Kähler manifolds Let U = {p ∈ M : b(p) ≠ 0}. If U = 0, then b = 0 on M and therefore ξ is an eigenvector of K everywhere, which implies that α is constant on M by Theorem 3.5.13. Now assume that U ≠ 0. Since b = g(Kξ , Z) is a continuous function, U is an open subset of M and α = 5f /4 is constant on U by Proposition 3.5.4. Thus, by continuity of α, α is constant on the closure Ū of U. If Ū = M we conclude that α is constant on M. Assume that M \ Ū ≠ 0. Then ξ is an eigenvector of K on M \ U,̄ which implies that α is locally constant on the open set M \ U,̄ using again Theorem 3.5.13. Since M is connected, we can then conclude that α is constant on M also in case M \ Ū ≠ 0. The assertion then follows by using Theorem 3.5.13. Proposition 3.5.20. Let M be a contact hypersurface in an irreducible Hermitian sym̄ > 2. Then M is curvature-adapted. metric space M̄ and n = dimℂ (M) Proof. We have to show that the shape operator A and the normal Jacobi operator K are simultaneously diagonalizable. Since Aξ = αξ and ξ is an eigenvector of K by Theorem 3.5.19, we can restrict to the maximal holomorphic subbundle 𝒞 . Let X, Y ∈ Γ(𝒞 ) with AX = λX and AY = μY. Since α is constant by Theorem 3.5.19, we obtain from Corollary 3.3.13 that 0 = (λ − μ)(3g(KϕX, ϕY) + g(KX, Y)). From Proposition 3.5.6 we know that AϕX = (2f −λ)ϕX and AϕY = (2f −μ)ϕY. Applying the previous argument to ϕX, ϕY instead of X, Y, we obtain 0 = (λ − μ)(3g(KX, Y) + g(KϕX, ϕY)). The previous two equations imply 0 = (λ − μ)g(KX, Y). This shows that K preserves the principal curvature spaces in 𝒞 , which proves the assertion.

3.6 Real hypersurfaces in complex space forms In this section we briefly discuss some classification results for real hypersurfaces in complex projective spaces and in complex hyperbolic spaces that are related to our investigations in the subsequent chapters. As we mentioned in the introduction, many details can be found in the monograph [33]. Homogeneous hypersurfaces have interesting geometric properties and it is natural to classify them and to investigate their geometry. The homogeneous hypersurfaces in complex projective spaces were classified by Takagi [116]. The classification is closely related to the classification of homogeneous hypersurfaces in spheres, which

3.6 Real hypersurfaces in complex space forms | 85

we presented in Theorem 2.7.4. Recall that a hypersurface in the sphere Sm+1 is homogeneous if and only if it is (congruent to) a principal orbit of the isotropy representation of a semisimple Riemannian symmetric space G/K with dim(G/K) = m + 2 and rk(G/K) = 2. Now assume that G/K is also Hermitian symmetric. Then m = 2n is even and the orbits of the 1-dimensional center of K in the sphere S2n+1 ⊂ To G/K ≅ ℂn+1 are the fibers of the Hopf map S2n+1 → ℂP n . The action of the isotropy group K on S2n+1 therefore descends to an action of K on ℂP n . This leads to the classification of homogeneous hypersurfaces in ℂP n . Theorem 3.6.1. A real hypersurface in ℂP n is homogeneous if and only if it is (congruent to) the projection via the Hopf map S2n+1 → ℂP n of a principal orbit of the isotropy representation of a semisimple Hermitian symmetric space G/K with dim(G/K) = 2n + 2 and rk(G/K) = 2. Explicitly, and more geometrically, this gives the following list of homogeneous real hypersurfaces in ℂP n . Theorem 3.6.2 ([116]). A real hypersurface in ℂP n is homogeneous if and only if it is congruent to one of the following hypersurfaces: (A) a tube around the totally geodesic embedding of the complex projective space ℂP k into ℂP n for some k ∈ {0, . . . , n − 1}, (B) a tube around the standard embedding of the complex quadric Qn−1 into ℂP n , (C) a tube around the Segre embedding of ℂP 1 × ℂP k into ℂP 2k+1 , (D) a tube around the Plücker embedding of the complex Grassmann manifold G2 (ℂ5 ) into ℂP 9 , (E) a tube around the half spin embedding of the Hermitian symmetric space SO10 /U5 into ℂP 15 . The classification of homogeneous real hypersurfaces in complex hyperbolic spaces is more involved. The classification of homogeneous Hopf hypersurfaces in ℂH n follows from the classification of Hopf hypersurfaces with constant principal curvatures in ℂH n and is as follows. Theorem 3.6.3 ([5]). A Hopf hypersurface in ℂH n is homogeneous if and only if it is congruent to one of the following hypersurfaces: (A) a tube around the totally geodesic embedding of the complex hyperbolic space ℂH k into ℂH n for some k ∈ {0, . . . , n − 1}, (B) a tube around the totally geodesic embedding of the real hyperbolic space ℝH n into ℂH n , (C) a horosphere in ℂH n . It follows from Takagi’s list in Theorem 3.6.2 that every homogeneous real hypersurface in the complex projective space ℂP n is a Hopf hypersurface. Lohnherr and Reckziegel [67] discovered a homogeneous ruled real hypersurface in ℂH n which is

86 | 3 Real hypersurfaces in Kähler manifolds not a Hopf hypersurface. This homogeneous ruled real hypersurface can be described in Lie theoretic terms by using an Iwasawa decomposition of SU 1,n (see [7]). Following this Lie theoretic approach more rigorously, a complete classification of homogeneous real hypersurfaces in ℂH n was finally obtained by Berndt and Tamaru in [22]. We omit a detailed description here since it involves Lie theoretic concepts that are not relevant for subsequent investigations. Real hypersurfaces with isometric Reeb flow in complex projective spaces were studied by Okumura [77]. Okumura considered the condition Aϕ = ϕA. Using the Hopf map π : S2n+1 → ℂP n from the Riemannian sphere S2n+1 onto ℂP n , Okumura first showed that π −1 (M) is a hypersurface with parallel second fundamental form in S2n+1 . Using known classification results for such hypersurfaces he then concluded that M satisfies Aϕ = ϕA if and only if M is locally congruent to a tube around a totally geodesic complex projective subspace ℂP k ⊂ ℂP n for some k ∈ {0, . . . , n − 1}. Using Proposition 3.4.1 we obtain the following. Theorem 3.6.4 ([77]). Let M be a connected real hypersurface in the complex projective space ℂP n . Then M has isometric Reeb flow if and only if M is an open part of a tube around a totally geodesic complex projective subspace ℂP k ⊂ ℂP n for some k ∈ {0, . . . , n − 1}. Real hypersurfaces with isometric Reeb flow in complex hyperbolic spaces were classified by Montiel and Romero [72]. They used a similar approach as Okumura, considering the Hopf map π : H 2n+1 → ℂH n from anti-de Sitter space H 2n+1 onto ℂH n , and investigated the geometry of π −1 (M). Since anti-de Sitter space is a Lorentzian manifold, the calculations are more involved. The conclusion is the following. Theorem 3.6.5 ([72]). Let M be a connected real hypersurface in the complex hyperbolic space ℂH n . Then M has isometric Reeb flow if and only if M is either a horosphere in ℂH n or an open part of a tube around a totally geodesic complex hyperbolic subspace ℂH k ⊂ ℂH n for some k ∈ {0, . . . , n − 1}. The first systematic study of contact hypersurfaces in Kähler manifolds was carried out by Masafumi Okumura [76] in 1966. Okumura proved the very useful characterization Aϕ + ϕA = 2fϕ of contact hypersurfaces in Proposition 3.5.5. This immediately implies that every contact hypersurface in a Kähler manifold is a Hopf hypersurface. He also proved that f is constant in complex space forms of dimension ≥ 3. Okumura constructed the following examples of contact hypersurfaces: (i) The Riemannian product of ℝn and a sphere Sn−1 (r) ⊂ ℝn is a contact hypersurface of ℝn × ℝn ≅ ℂn . These examples are precisely the tubes around ℝn ⊂ ℂn . (ii) Every totally umbilical hypersurface with non-zero mean curvature in a Kähler manifold is a contact hypersurface. In particular, the sphere S2n−1 (r) is a contact hypersurface of ℂn .

3.6 Real hypersurfaces in complex space forms |

87

He also proved that every complete, simply connected contact hypersurface in the complex Euclidean space ℂn is congruent to either a sphere or a tube around ℝn in ℂn . More generally, Okumura considered contact hypersurfaces in complex space forms and proved that any such hypersurface has at most three distinct principal curvatures and all of them are constant. Using Okumura’s [76] result about the constancy of the principal curvatures and Takagi’s [117, 118] classification of real hypersurfaces in complex projective spaces with at most three distinct constant principal curvatures, Kon [61] classified contact hypersurfaces in complex projective spaces. Theorem 3.6.6 ([61]). Let M be a connected real hypersurface in the complex projective space ℂP n , n ≥ 3. Then M is a contact hypersurface if and only if M is an open part of a geodesic hypersphere in ℂP n or of a tube around the totally geodesic real projective space ℝP n in ℂP n . Contact hypersurfaces in complex hyperbolic space ℂH n , n ≥ 3, were classified by Vernon. In [119], Vernon first derives some necessary conditions for the shape operator of M and then applies rigidity results from [120] to derive the classification of contact hypersurfaces in ℂH n . Theorem 3.6.7 ([119]). Let M be a connected real hypersurface in the complex hyperbolic space ℂH n , n ≥ 3. Then M is a contact hypersurface if and only if M is an open part of a horosphere in ℂH n , of a geodesic hypersphere in ℂH n , of a tube around the totally geodesic real hyperbolic space ℝH n in ℂH n or of a tube around the totally geodesic complex hyperbolic space ℂH n−1 in ℂH n . Real hypersurfaces with isometric Reeb flow and contact hypersurfaces in some other Hermitian symmetric spaces will be discussed thoroughly in the subsequent chapters.

4 Real hypersurfaces in complex 2-plane Grassmannians In this chapter we investigate real hypersurfaces in the complex Grassmann manifold G2 (ℂ2+k ) = SU 2+k /S(U2 Uk ), k ≥ 3. We denote by g the Riemannian metric, by ∇̄ the Riemannian connection, by R̄ the Riemannian curvature tensor and by ric the Ricci tensor of G2 (ℂ2+k ).

4.1 Basic equations for real hypersurfaces Let M be a real hypersurface in G2 (ℂ2+k ) with unit normal vector field ζ . We denote by g the Riemannian metric, by ∇ the Riemannian connection, by R the Riemannian curvature tensor and by A the shape operator of M with respect to the unit normal vector field ζ . The Kähler structure J on G2 (ℂ2+k ) induces an almost contact metric structure (ϕ, ξ , η, g) on M. The vector field ξ = −Jζ is the Reeb vector field on M. The maximal holomorphic subbundle of TM is denoted by 𝒞 and the orthogonal complement of 𝒞 in TM is denoted by 𝒞 ⊥ . Thus we have TM = 𝒞 ⊕ 𝒞 ⊥ ,

⊥

𝒞 = ℝξ = ℝJζ .

The quaternionic Kähler structure J on G2 (ℂ2+k ) induces further structures on M. Let J1 , J2 , J2 be a local canonical basis of J. Then each of the three almost Hermitian structures Jν induces an almost contact metric structure (ϕν , ξν , ην , g) on M. The maximal quaternionic subbundle of TM is denoted by 𝒟 and the orthogonal complement of 𝒟 in TM is denoted by 𝒟⊥ . Thus we have TM = 𝒟 ⊕ 𝒟⊥ ,

⊥

𝒟 = Jζ .

We define ℂζ = ℝζ ⊕ ℝJζ = ℝζ ⊕ ℝξ , ℍζ = ℝζ ⊕ Jζ , ℍℂζ = ℍζ + ℍJζ = ℍζ + ℍξ . From the equation JJν = Jν J we obtain ϕξν + η(ξν )ζ = Jξν = −JJν ζ = −Jν Jζ = Jν ξ = ϕν ξ + ην (ξ )ζ . The tangential and normal component of this equation gives ϕξν = ϕν ξ https://doi.org/10.1515/9783110689839-004

90 | 4 Real hypersurfaces in complex 2-plane Grassmannians and η(ξν ) = ην (ξ ), respectively. Next, we have JJν X = J(ϕν X + ην (X)ζ ) = ϕϕν X + η(ϕν X)ζ − ην (X)ξ ,

Jν JX = Jν (ϕX + η(X)ζ ) = ϕν ϕX + ην (ϕX)ζ − η(X)ξν . Since JJν = Jν J, the tangential component gives ϕϕν X − ην (X)ξ = ϕν ϕX − η(X)ξν , and the normal component gives η(ϕν X) = ην (ϕX). From the equations Jν Jν+1 = Jν+2 = −Jν+1 Jν we obtain ϕν ξν+1 = Jν ξν+1 = −Jν Jν+1 ζ = −Jν+2 ζ = ξν+2 and ϕν+1 ξν = Jν+1 ξν = −Jν+1 Jν ζ = Jν+2 ζ = −ξν+2 . Next, we have

Jν Jν+1 X = Jν (ϕν+1 X + ην+1 (X)ζ ) = ϕν ϕν+1 X + ην (ϕν+1 X)ζ − ην+1 (X)ξν ,

Jν+1 Jν X = Jν+1 (ϕν X + ην (X)ζ ) = ϕν+1 ϕν X + ην+1 (ϕν X)ζ − ην (X)ξν+1 . Since Jν Jν+1 = −Jν+1 Jν , the tangential component gives ϕν ϕν+1 X − ην+1 (X)ξν = −ϕν+1 ϕν X + ην (X)ξν+1 , and the normal component gives ην (ϕν+1 X) = −ην+1 (ϕν X). Also, we have Jν+2 X = ϕν+2 X + ην+2 (X)ζ . Since Jν Jν+1 = Jν+2 , the tangential component gives ϕν ϕν+1 X − ην+1 (X)ξν = ϕν+2 X

4.1 Basic equations for real hypersurfaces | 91

and the normal component gives ην (ϕν+1 X) = ην+2 (X). Since the Kähler structure J is parallel, we have 0 = (∇̄X J)ζ = ∇̄X Jζ − J ∇̄X ζ = ∇X Jζ + g(AX, Jζ )ζ − J ∇̄X ζ

= −∇X ξ − η(AX)ζ + JAX = −∇X ξ − η(AX)ζ + ϕAX + η(AX)ζ = −∇X ξ + ϕAX.

Since the quaternionic Kähler structure J on G2 (ℂ2+k ) is a parallel subbundle of End(TG2 (ℂ2+k )), it follows that for any canonical local basis J1 , J2 , J3 of J there exist three local 1-forms q1 , q2 , q3 such that ∇̄X Jν = qν+2 (X)Jν+1 − qν+1 (X)Jν+2 . This implies (∇̄X Jν )ζ = −qν+2 (X)ξν+1 + qν+1 (X)ξν+2 . On the other hand, we have (∇̄X Jν )ζ = ∇̄X Jν ζ − Jν ∇̄X ζ = ∇X Jν ζ + g(AX, Jν ζ )ζ − Jν ∇̄X ζ

= −∇X ξν − ην (AX)ζ + Jν AX = −∇X ξν − ην (AX)ζ + ϕν AX + ην (AX)ζ = −∇X ξν + ϕν AX.

Comparing the previous two equations gives ∇X ξν = qν+2 (X)ξν+1 − qν+1 (X)ξν+2 + ϕν AX. For the covariant derivatives we obtain (∇X η)Y = g(ϕAX, Y) and (∇X ην )Y = qν+2 (X)ην+1 (Y) − qν+1 (X)ην+2 (Y) + g(ϕν AX, Y). Moreover, (∇X ϕ)Y = ∇X ϕY − ϕ∇X Y = ∇̄X ϕY − g(AX, ϕY)ζ − ϕ∇X Y = ∇̄X JY − ∇̄X (η(Y)ζ ) + g(ϕAX, Y)ζ − ϕ∇X Y

92 | 4 Real hypersurfaces in complex 2-plane Grassmannians = J ∇̄X Y − η(∇X Y)ζ − g(Y, ∇X ξ )ζ − η(Y)∇̄X ζ + g(∇X ξ , Y)ζ − ϕ∇X Y = J ∇̄X Y − η(∇X Y)ζ + η(Y)AX − J∇X Y + η(∇X Y)ζ = η(Y)AX + J ∇̄X Y − J∇X Y = η(Y)AX + g(AX, Y)Jζ = η(Y)AX − g(AX, Y)ξ

and (∇X ϕν )Y = ∇X ϕν Y − ϕν ∇X Y = ∇̄X ϕν Y − g(AX, ϕν Y)ζ − ϕν ∇X Y = ∇̄X Jν Y − ∇̄X (ην (Y)ζ ) + g(ϕν AX, Y)ζ − ϕν ∇X Y = Jν ∇̄X Y + qν+2 (X)Jν+1 Y − qν+1 (X)Jν+2 Y

− ην (∇X Y)ζ − g(Y, ∇X ξν )ζ − ην (Y)∇̄X ζ + g(ϕν AX, Y)ζ − ϕν ∇X Y = Jν ∇̄X Y + qν+2 (X)Jν+1 Y − qν+1 (X)Jν+2 Y − ην (∇X Y)ζ − qν+2 (X)ην+1 (Y)ζ + qν+1 (X)ην+2 (Y)ζ

− g(Y, ϕν AX)ζ + ην (Y)AX + g(ϕν AX, Y)ζ − Jν ∇X Y + ην (∇X Y)ζ

= qν+2 (X)ϕν+1 Y − qν+1 (X)ϕν+2 Y + ην (Y)AX − g(AX, Y)ξν . We summarize this in the following lemma.

Lemma 4.1.1 (Basic structure equations). Let M be a real hypersurface in G2 (ℂ2+k ). The almost contact metric structures (ϕ, ξ , η, g) and (ϕν , ξν , ην , g) (ν = 1, 2, 3) satisfy the following relations (with indices modulo 3): ϕξν = ϕν ξ ,

ϕν ξν+1 = ξν+2 = −ϕν+1 ξν , η(ξν ) = ην (ξ ),

η(ϕν X) = ην (ϕX),

ην (ϕν+1 X) = −ην+1 (ϕν X),

(4.1)

ϕϕν X − ϕν ϕX = ην (X)ξ − η(X)ξν ,

(4.2)

∇X ξ = ϕAX,

(4.3)

ϕν ϕν+1 X − ην+1 (X)ξν = ϕν+2 X = −ϕν+1 ϕν X + ην (X)ξν+1 , ∇X ξν = qν+2 (X)ξν+1 − qν+1 (X)ξν+2 + ϕν AX,

(∇X η)Y = g(ϕAX, Y),

(∇X ην )Y = qν+2 (X)ην+1 (Y) − qν+1 (X)ην+2 (Y) + g(ϕν AX, Y),

(∇X ϕ)Y = η(Y)AX − g(AX, Y)ξ ,

(4.4)

4.1 Basic equations for real hypersurfaces | 93

(∇X ϕν )Y = qν+2 (X)ϕν+1 Y − qν+1 (X)ϕν+2 Y + ην (Y)AX − g(AX, Y)ξν .

The fundamental equations of submanifold geometry (Theorem 2.1.1) can be rewritten using the explicit expression of the curvature tensor of G2 (ℂ2+k ) as given in Theorem 1.4.6. Theorem 4.1.2 (Fundamental structure equations). Let M be a real hypersurface in G2 (ℂ2+k ). Then the following equations hold for all X, Y, Z ∈ X(M): Gauss formula: ∇̄X Y = ∇X Y + g(AX, Y)ζ , Weingarten formula: ∇̄X ζ = −AX, Gauss equation: R(X, Y)Z = g(Y, Z)X − g(X, Z)Y + g(ϕY, Z)ϕX − g(ϕX, Z)ϕY − 2g(ϕX, Y)ϕZ 3

+ ∑ (g(ϕν Y, Z)ϕν X − g(ϕν X, Z)ϕν Y − 2g(ϕν X, Y)ϕν Z) ν=1 3

+ ∑ (g(ϕν ϕY, Z)ϕν ϕX − g(ϕν ϕX, Z)ϕν ϕY) ν=1 3

+ ∑ ην (Z)(η(X)ϕν ϕY − η(Y)ϕν ϕX) ν=1 3

+ ∑ (η(Y)g(ϕν ϕX, Z) − η(X)g(ϕν ϕY, Z))ξν ν=1

+ g(AY, Z)AX − g(AX, Z)AY,

(4.5)

Codazzi equation: (∇X A)Y − (∇Y A)X = η(X)ϕY − η(Y)ϕX − 2g(ϕX, Y)ξ 3

+ ∑ (ην (X)ϕν Y − ην (Y)ϕν X − 2g(ϕν X, Y)ξν ) ν=1 3

+ ∑ (ην (ϕX)ϕν ϕY − ην (ϕY)ϕν ϕX) ν=1 3

+ ∑ (η(X)ην (ϕY) − η(Y)ην (ϕX))ξν . ν=1

(4.6)

94 | 4 Real hypersurfaces in complex 2-plane Grassmannians We now turn our attention to the normal Jacobi operator K = R̄ ζ . Theorem 4.1.3. Let M be a real hypersurface in G2 (ℂ2+k ). The normal Jacobi operator K of M satisfies 3

KX = X + 3η(X)ξ + ∑ (3ην (X)ξν + ην (ξ )η(X)ξν + ην (ϕX)ϕν ξ − ην (ξ )ϕν ϕX) ν=1

for all X ∈ X(M). Proof. From Theorem 1.4.6 we get ̄ ζ )ζ = g(ζ , ζ )X − g(X, ζ )ζ + g(Jζ , ζ )JX − g(JX, ζ )Jζ − 2g(JX, ζ )Jζ KX = R(X, 3

+ ∑ (g(Jν ζ , ζ )Jν X − g(Jν X, ζ )Jν ζ − 2g(Jν X, ζ )Jν ζ ν=1

+ g(Jν Jζ , ζ )Jν JX − g(Jν JX, ζ )Jν Jζ ) for all X ∈ X(M). We have g(ζ , ζ ) = 1,

g(X, ζ ) = g(Jζ , ζ ) = g(Jν ζ , ζ ) = 0,

g(JX, ζ )Jζ = −g(X, Jζ )Jζ = −g(X, ξ )ξ = −η(X)ξ ,

g(Jν X, ζ )Jν ζ = −g(X, Jν ζ )Jν ζ = −g(X, ξν )ξν = −ην (X)ξν ,

g(Jν Jζ , ζ ) = −g(Jζ , Jν ζ ) = −g(ξ , ξν ) = −ην (ξ ) = −η(ξν ),

g(Jν JX, ζ )Jν Jζ = g(JJν X, ζ )Jν Jζ = −g(Jν X, Jζ )Jν Jζ = −η(ϕν X)Jν ξ . Inserting this into the previous equation leads to 3

KX = X + 3η(X)ξ + ∑ (3ην (X)ξν − ην (ξ )Jν JX + η(ϕν X)Jν ξ ). ν=1

Next, we have Jν JX = ϕν ϕX + ην (ϕX)ζ − η(X)ξν , Jν ξ = ϕν ξ + ην (ξ )ζ .

Inserting this into the previous equation and using (4.1) gives 3

KX = X + 3η(X)ξ + ∑ (3ην (X)ξν − ην (ξ )ϕν ϕX + ην (ξ )η(X)ξν + ην (ϕX)ϕν ξ ). ν=1

This finishes the proof.

(4.7)

4.1 Basic equations for real hypersurfaces | 95

Proposition 4.1.4. Let M be a real hypersurface in G2 (ℂ2+k ), k ≥ 3, and p ∈ M. The following statements are equivalent: (i) Kξp = κξp for some κ ∈ ℝ, (ii) ξp ∈ 𝒟p or ξp ∈ 𝒟p⊥ ,

(iii) the unit normal vector ζp is a singular tangent vector of G2 (ℂ2+k ). Proof. Inserting X = ξ into (4.7) gives Kξ = 4ξ + 4η1 (ξ )ξ1 + 4η2 (ξ )ξ2 + 4η3 (ξ )ξ3 . This implies that ξp is an eigenvector of K if and only if ξp ∈ 𝒟p or ξp ∈ 𝒟p⊥ . We have ξp ∈ 𝒟p if and only if Jζp ⊥ Jζp and ξp ∈ 𝒟p⊥ if and only if Jζp ∈ Jζp . The assertion then

follows from the description of the singular tangent vectors of G2 (ℂ2+k ) in Section 1.4.4. Theorem 4.1.5. Let M be a real hypersurface in G2 (ℂ2+k ). The structure Jacobi operator Rξ of M satisfies

3

3

ν=1

ν=1

Rξ X = X − η(X)ξ − 3 ∑ η(ϕν X)ϕν ξ − ∑ ην (ξ )ϕν ϕX 3

+ ∑ (η(X)η(ξν ) − ην (X))ξν + η(Aξ )AX − η(AX)Aξ ν=1

for all X ∈ X(M). Proof. Inserting Y = Z = ξ into the Gauss equation (4.5) gives R(X, ξ )ξ = g(ξ , ξ )X − g(X, ξ )ξ + g(ϕξ , ξ )ϕX − g(ϕX, ξ )ϕξ − 2g(ϕX, ξ )ϕξ 3

+ ∑ (g(ϕν ξ , ξ )ϕν X − g(ϕν X, ξ )ϕν ξ − 2g(ϕν X, ξ )ϕν ξ ) ν=1 3

+ ∑ (g(ϕν ϕξ , ξ )ϕν ϕX − g(ϕν ϕX, ξ )ϕν ϕξ ) ν=1 3

+ ∑ ην (ξ )(η(X)ϕν ϕξ − η(ξ )ϕν ϕX) ν=1 3

+ ∑ (η(ξ )g(ϕν ϕX, ξ ) − η(X)g(ϕν ϕξ , ξ ))ξν ν=1

+ g(Aξ , ξ )AX − g(AX, ξ )Aξ .

(4.8)

96 | 4 Real hypersurfaces in complex 2-plane Grassmannians We have g(ξ , ξ ) = η(ξ ) = 1, g(X, ξ ) = η(X), ϕξ = 0, g(ϕν ξ , ξ ) = 0. Thus the previous equation simplifies to 3

3

3

ν=1

ν=1

ν=1

R(X, ξ )ξ = X − η(X)ξ − 3 ∑ η(ϕν X)ϕν ξ − ∑ ην (ξ )ϕν ϕX + ∑ η(ϕν ϕX)ξν + η(Aξ )AX − η(AX)Aξ . From (4.2) we have ϕν ϕX = ϕϕν X − ην (X)ξ + η(X)ξν and hence η(ϕν ϕX) = η(X)η(ξν ) − ην (X). Inserting this into the previous equation leads to (4.8). For the Ricci tensor of M we obtain the following. Theorem 4.1.6. Let M be a real hypersurface in G2 (ℂ2+k ). The Ricci tensor of M satisfies ric(X, Y) = 4(k + 2)g(X, Y) − g(KX, Y) + tr(A)g(AX, Y) − g(A2 X, Y),

(4.9)

or equivalently, Ric(X) = 4(k + 2)X − KX + tr(A)AX − A2 X, for all X, Y ∈ X(M). Proof. Let E1 , . . . , E4k be a local orthonormal frame field of G2 (ℂ2+k ) so that E4k = ζ . The Gauss equation implies 4k−1

ric(X, Y) = ∑ g(R(X, Eν )Eν , Y) ν=1

4k−1

̄ Eν )Eν , Y) = ∑ g(R(X, ν=1

4k−1

4k−1

ν=1

ν=1

+ ∑ g(AEν , Eν )g(AX, Y) − ∑ g(AX, Eν )g(AEν , Y) for all X, Y ∈ X(M). Using the expression for the Ricci tensor of G2 (ℂ2+k ) in Theorem 1.4.6, we get 4k−1

4k

ν=1

ν=1

̄ Eν )Eν , Y) = ∑ g(R(X, ̄ Eν )Eν , Y) − g(R(X, ̄ E4k )E4k , Y) ∑ g(R(X,

4.2 Totally geodesic submanifolds | 97

̄ ζ )ζ , Y) = ric(X, Y) − g(R(X,

= 4(k + 2)g(X, Y) − g(KX, Y). We also have 4k−1

∑ g(AEν , Eν )g(AX, Y) = tr(A)g(AX, Y)

ν=1

and 4k−1

4k−1

ν=1

ν=1

∑ g(AX, Eν )g(AEν , Y) = ∑ g(AX, Eν )g(AY, Eν ) = g(AX, AY) = g(A2 X, Y).

Altogether this implies the assertion. Contracting equation (4.9) implies the following. Corollary 4.1.7. Let M be a real hypersurface in G2 (ℂ2+k ). The scalar curvature s of M satisfies s = 4(k + 2)(4k − 2) + tr(A)2 − tr(A2 ).

4.2 Totally geodesic submanifolds Tubes around certain totally geodesic submanifolds will play an eminent role in our studies of real hypersurfaces. For this reason we briefly revisit here the classification of totally geodesic submanifolds in complex 2-plane Grassmannians. The totally geodesic submanifolds in G2 (ℂ2+k ) were classified by Klein in [57]. We present here the maximal totally geodesic submanifolds, as the tubes around some of them are important for our investigations. Maximal totally geodesic submanifolds in G2 (ℂ2+k ) were already classified in [36], but with one omission that was discovered later by Klein in [57]. We describe now the standard embeddings for these maximal totally geodesic submanifolds. The linear embedding ℂ1+k → ℂ2+k ,

(z1 , . . . , z1+k ) 󳨃→ (z1 , . . . , z1+k , 0)

induces a totally geodesic submanifold G2 (ℂ1+k ) ⊂ G2 (ℂ2+k ). This submanifold is complex (with respect to J) and quaternionic (with respect to J). Consider again the linear embedding ℂ1+k → ℂ2+k ,

(z1 , . . . , z1+k ) 󳨃→ (z1 , . . . , z1+k , 0)

98 | 4 Real hypersurfaces in complex 2-plane Grassmannians and the complex projective space ℂP k induced from ℂ1+k . Every line [z] ∈ ℂP k defines a 2-plane V[z] = ℂz ⊕ ℂ(0, . . . , 0, 1). The embedding ℂP k → G2 (ℂ2+k ),

[z] 󳨃→ V[z]

realizes ℂP k as a maximal totally geodesic submanifold of G2 (ℂ2+k ). This submanifold is complex (with respect to J) and totally complex (with respect to J). Decompose ℂ2+k into ℂ2+k = ℂ1+a ⊕ ℂ1+b with a + b = k and 0 < a < k and consider the corresponding complex projective spaces ℂP a and ℂP b . The embedding ℂP a × ℂP b → G2 (ℂ2+k ),

([z], [w]) 󳨃→ [z] ⊕ [w] = ℂz ⊕ ℂw

realizes ℂP a × ℂP b as a maximal totally geodesic submanifold of G2 (ℂ2+k ). This submanifold is complex (with respect to J) and totally complex (with respect to J). Consider the linear embedding ℝ2+k → ℂ2+k ,

(x1 , . . . , x2+k ) 󳨃→ (x1 , . . . , x2+k )

and the real 2-plane Grassmannian G2 (ℝ2+k ) induced from ℝ2+k . The embedding G2 (ℝ2+k ) → G2 (ℂ2+k ),

ℝx ⊕ ℝy 󳨃→ ℂx ⊕ ℂy

realizes G2 (ℝ2+k ) as a maximal totally geodesic submanifold of G2 (ℂ2+k ). This submanifold is totally real (with respect to J) and totally complex (with respect to J). Assume that k is even, say k = 2l. Then we have a natural identification ℂ2+2l ≅ ℍ1+l ≅ ℂ1+l ⊕ ℂ1+l j, where we write quaternionic vectors as z + wj with z, w ∈ ℂ1+l . Consider the quaternionic projective space ℍP l induced from ℍ1+l . The embedding ℍP l → G2 (ℂ2+2l ),

[z + wj] 󳨃→ ℂz ⊕ ℂwj

realizes ℍP l as a maximal totally geodesic submanifold of G2 (ℂ2+2l ). This submanifold is totally real (with respect to J) and quaternionic (with respect to J). The standard representation of Sp3 on ℍ3 ≅ ℂ6 induces a representation of Sp3 on Λ3 ℂ6 . This 20-dimensional representation decomposes into two irreducible representations ℂ6 ⊕ ℂ14 . The representation on ℂ6 is equivalent to the standard representation. The restriction to SU 3 of the Sp3 -representation on ℂ14 leaves a 6-dimensional complex linear subspace V 6 ≅ ℂ6 in ℂ14 invariant. The induced action of SU 3 on G2 (V 6 ) ≅ G2 (ℂ6 ) has exactly one totally geodesic orbit. This orbit is isometric to a complex projective space ℂP 2 (c) (with a suitable c > 0) and a maximal totally geodesic submanifold of G2 (ℂ6 ) which is neither totally real nor complex (with respect to J) and neither totally real nor totally complex nor quaternionic (with respect to J). We call this the non-standard totally geodesic embedding of ℂP 2 into G2 (ℂ6 ).

4.3 Homogeneous real hypersurfaces | 99

Theorem 4.2.1 ([36, 57]). Let Σ be a maximal totally geodesic submanifold of the complex 2-plane Grassmannian G2 (ℂ2+k ) = SU 2+k /S(U2 Uk ), k ≥ 3. Then Σ is congruent to one of the following maximal totally geodesic submanifolds: (i) the complex and quaternionic totally geodesic embedding of G2 (ℂ1+k ) into G2 (ℂ2+k ); (ii) the complex and totally complex totally geodesic embedding of ℂP k into G2 (ℂ2+k ); (iii) the complex and totally complex totally geodesic embedding of ℂP a × ℂP k−a into G2 (ℂ2+k ), where a ∈ {1, . . . , k − 1}; (iv) the totally real and totally complex totally geodesic embedding of G2 (ℝ2+k ) into G2 (ℂ2+k ); (v) (only for k = 2l even) the totally real and quaternionic totally geodesic embedding of ℍP l into G2 (ℂ2+2l ); (vi) (only for k = 4) the non-standard totally geodesic embedding of ℂP 2 into G2 (ℂ6 ).

4.3 Homogeneous real hypersurfaces The homogeneous real hypersurfaces in G2 (ℂ2+k ) were classified by Kollross in [60]. In fact, Kollross classified in [60] the cohomogeneity one actions on irreducible Riemannian symmetric spaces of compact type up to orbit equivalence. The principal orbits of cohomogeneity one actions are homogeneous hypersurfaces. Conversely, any homogeneous hypersurface arises by definition as an orbit of a cohomogeneity one action. Kollross proved that any cohomogeneity one action on G2 (ℂ2+k ) is orbit equivalent to the action of S(U1+k U1 ) or of Sp1+l (only for k = 2l even) on G2 (ℂ2+k ). Any cohomogeneity one action on G2 (ℂ2+k ) has exactly two singular orbits. Geometrically the two singular orbits are focal sets of each other. The principal orbits can then be realized geometrically as the tubes around either of the two singular orbits. The singular orbits of the action of S(U1+k U1 ) are both totally geodesic. One of them is congruent to the totally geodesic G2 (ℂ1+k ) ⊂ G2 (ℂ2+k ), the other one is congruent to the totally geodesic ℂP k ⊂ G2 (ℂ2+k ). The principal orbits are the tubes around G2 (ℂ1+k ) ⊂ G2 (ℂ2+k ), or equivalently, the tubes around ℂP k ⊂ G2 (ℂ2+k ). With our normalization of the Riemannian on G2 (ℂ2+k ), the distance between the two focal sets is π π and therefore the tubes are well-defined for the radii 0 < r < 2√ . 2√2 2 Assume that k = 2l is even. One of the singular orbits of the action of Sp1+l is congruent to a totally geodesic ℍP l ⊂ G2 (ℂ2+2l ). The other singular orbit is a complex homogeneous real hypersurface in G2 (ℂ2+2l ) isometric to Sp1+l /Spl−1 U2 . The principal orbits are the tubes around ℍP l ⊂ G2 (ℂ2+2l ), or equivalently, the tubes around Sp1+l /Spl−1 U2 ⊂ G2 (ℂ2+2l ). With our normalization of the Riemannian metric on G2 (ℂ2+2l ), the distance between the two focal sets is π4 and therefore the tubes are well-defined for the radii 0 < r < π4 .

100 | 4 Real hypersurfaces in complex 2-plane Grassmannians Theorem 4.3.1 ([60]). Let M be a homogeneous real hypersurface in the complex 2-plane Grassmannian G2 (ℂ2+k ) = SU 2+k /S(U2 Uk ), k ≥ 3. Then M is congruent to one of the following homogeneous real hypersurfaces: π around the totally geodesic G2 (ℂ1+k ) ⊂ G2 (ℂ2+k ); (A) the tube with radius 0 < r < 2√ 2 (B) (only for k = 2l even) the tube with radius 0 < r < π4 around the totally geodesic ℍP l ⊂ G2 (ℂ2+2l ). We will now investigate the geometry of these homogeneous real hypersurfaces. We π start with the tubes around G2 (ℂ1+k ) ⊂ G2 (ℂ2+k ). Let 0 < r < 2√ and Σ = G2 (ℂ1+k ). 2 We can assume that o ∈ Σ. We have dim(Σ) = 4(k − 1) and hence codim(Σ) = 4. Let ζ ∈ νo Σ be a unit normal vector. Since Σ is complex (with respect to J) and quaternionic (with respect to J), we see that the normal space νo Σ is complex (with respect to J) and quaternionic (with respect to J). Thus there exists an almost Hermitian structure J1 ∈ J such that Jζ = J1 ζ . It follows that ζ is a singular tangent vector of G2 (ℂ2+k ) of type Jζ ∈ Jζ . From Lemma 1.4.7 we see that the Jacobi operator R̄ ζ leaves νo Σ and To Σ invariant. Therefore Σ is a curvature-adapted submanifold of G2 (ℂ2+k ). We can therefore apply Corollary 2.6.2 to calculate the principal curvatures and corresponding principal curvature spaces of the tube Σr with radius r around Σ. Using the notations of Section 2.6, we have Spec(Kζ ) = {0, 2}, Spec(Aζ ) = {0} and Spec(R̄ ⊥ ζ ) = {2, 8} and corresponding eigenspaces E0 = {u ∈ To Σ : Ju = −J1 u}, E2 = {u ∈ To Σ : Ju = J1 u},

T0 = To Σ,

V2 = ℍζ ⊖ ℂζ ,

V8 = ℝJζ .

Let γζ be the geodesic in G2 (ℂ2+k ) with γζ (0) and γ̇ζ (0) = ζ . It follows from Corollary 2.6.2 that the principal curvatures of Σr at γζ (r) with respect to −γ̇ζ (r) are μ = 0, λ = −2 tan2 (r) = −√2 tan(√2r), β = cot2 (r) = √2 cot(√2r) and α = cot8 (r) = 2√2 cot(2√2r). The corresponding principal curvature spaces are the parallel translates of E0 , E2 , V2 and V8 along γζ from 0 = γζ (0) to p = γζ (r). Since Σr is homogeneous, these conclusions hold at any point. Thus we have proved the following. Theorem 4.3.2. Let M be the tube with radius 0 < r
0. Then there exists some r ∈ (0, π4 ) so that β = 2 cot(2r). Then 0 = 4 + αβ gives α = −2 tan(2r). The solutions of x2 − βx − 1 = 0 are then λ = cot(r) and μ = − tan(r). The statement about the principal curvature spaces follows from Lemma 4.5.7 and Lemma 4.5.9. We now come to the main result of Case 1. Theorem 4.5.11. Let M be a connected Hopf hypersurface in G2 (ℂ2+k ), k ≥ 3, with A𝒟 ⊆ 𝒟 and assume that the Reeb vector field ξ is tangent to 𝒟 everywhere. Then the quaternionic dimension k of G2 (ℂ2+k ) is even, say k = 2l, and M is an open part of the tube with radius r ∈ (0, π4 ) around a totally geodesic quaternionic projective space ℍP l in G2 (ℂ2+2l ). Proof. From Proposition 4.5.10 we know that k = 2l. By assumption, the normal vector field ζ of M is singular and of type Jζ ⊥ Jζ everywhere. Using Lemma 1.4.7 we see that the eigenvalues of the normal Jacobi operator Kζ are 0, 1 and 4 with corresponding eigenspaces Jξ , (ℍℂζ )⊥ and ℝξ ⊕ 𝒟⊥ , respectively. From Proposition 4.5.10 we get Jξ = Tγ ,

(ℍℂζ )⊥ = Tλ ⊕ Tμ ,

ℝξ ⊕ 𝒟⊥ = Tα ⊕ Tβ .

It follows that M is curvature-adapted and we can apply Corollary 2.5.2. For p ∈ M denote by cp the geodesic in G2 (ℂ2+2l ) with cp (0) = p and ċp (0) = ζp and by Φr the smooth map Φr : M → G2 (ℂ2+2l ),

p 󳨃→ cp (r).

Geometrically, Φr is the displacement of M at distance r in the direction of the normal vector field ζ . From Corollary 2.5.2(i) we obtain that cp (r) is a focal point of M along cp and, locally, Φr is a submersion onto a 4l-dimensional submanifold P of G2 (ℂ2+2l ). Moreover, the tangent space of P at Φr (p) is obtained by parallel translation of (Tα ⊕ Tγ ⊕ Tμ )(p) =(ℍξ ⊕ Tμ )(p), which is a quaternionic (with respect to J) and totally real (with respect to J) subspace of Tp G2 (ℂ2+2l ). Since both J and J are parallel along cp , also TΦr (p) P is a quaternionic and totally real subspace of

4.5 Real hypersurfaces with invariant maximal quaternionic subbundle | 117

TΦr (p) G2 (ℂ2+2l ). Thus P is a quaternionic and totally real submanifold of G2 (ℂ2+2l ). Since P is quaternionic, it is totally geodesic in G2 (ℂ2+2l ) (see [3]). The only quaternionic totally geodesic submanifolds of G2 (ℂ2+2l ) of half dimension are open parts of G2 (ℂ2+l ) and ℍP l (see [6]). However, only ℍP l is embedded in G2 (ℂ2+2l ) as a totally real submanifold. So we conclude that P is an open part of a totally geodesic ℍP l in G2 (ℂ2+2l ). Rigidity of totally geodesic submanifolds finally implies that M is an open part of the tube with radius r around a totally geodesic ℍP l in G2 (ℂ2+2l ). Theorem 4.5.11 has the following interesting consequence. Theorem 4.5.12. Let M be a connected Hopf hypersurface in G2 (ℂ2+k ), k ≥ 3. Then the following two statements are equivalent: (i) the Reeb vector field ξ is tangent to the maximal quaternionic subbundle 𝒟 of TM everywhere (or equivalently, the normal vector field ζ is singular of type Jζ ⊥ Jζ everywhere); (ii) the quaternionic dimension k of G2 (ℂ2+k ) is even, say k = 2l, and M is an open part of the tube with radius r ∈ (0, π4 ) around a totally geodesic quaternionic projective space ℍP l in G2 (ℂ2+2l ). Proof. According to Theorem 4.3.4, (ii) implies (i). Conversely, if the Reeb vector field ξ is tangent to the maximal quaternionic subbundle 𝒟 of TM everywhere, then Proposition 4.4.3 implies that A𝒟 ⊆ 𝒟, and we can apply Theorem 4.5.11. Case 2: The Reeb vector field ξ is tangent to 𝒟 ⊥ everywhere We now assume ξ ∈ Γ(𝒟⊥ ). Then the unit normal vector field ζ is a singular tangent vector field on G2 (ℂ2+k ) of type Jζ ∈ Jζ everywhere. There exists an almost Hermitian structure J1 ∈ J such that Jζ = J1 ζ . Then we have ξ = ξ1 ,

α = β1 ,

ϕξ2 = −ξ3 ,

ϕξ3 = ξ2 ,

ϕ𝒟 ⊆ 𝒟.

In the following the indices ν and μ will be either 2 or 3 and distinct from each other. Lemma 4.5.13. We have 2β2 β3 − α(β2 + β3 ) − 4 = 0. Proof. Inserting X = ξν into equation (4.12) yields (2βν − α)Aϕξν = (4 + αβν )ϕξν . Since 2βν − α = 0 implies 0 = 4 + αβν = 4 + 2βν2 , which is impossible, we get 2βν − α ≠ 0

and

Aϕξν =

4 + αβν ϕξ . 2βν − α ν

118 | 4 Real hypersurfaces in complex 2-plane Grassmannians Using Aξμ = βμ ξμ , ϕξ2 = −ξ3 and ϕξ3 = ξ2 we obtain βμ =

4 + αβν , 2βν − α

which proves the assertion. Lemma 4.5.14. If X ∈ Γ(𝒟) with AX = λX, then 2λ ≠ βν

and

Aϕν X = λν ϕν X

with

λν =

2 + λβν . 2λ − βν

Proof. Let X ∈ Γ(𝒟) with AX = λX. The 𝒟-component of the equation in Lemma 4.5.1 is 0 = (2λ − βν )Aϕν X − (2 + λβν )ϕν X. If 2λ = βν , then 0 = 2 + λβν = 2 + 2λ2 , which is impossible. The assertion now follows easily. Lemma 4.5.15. If X ∈ Γ(𝒟) with AX = λX, then either ϕX = ϕ1 X

and 2λ2 λ3 − α(λ2 + λ3 ) = 0

or ϕX = −ϕ1 X

and 2λ2 λ3 − α(λ2 + λ3 ) − 4 = 0.

Proof. Let X ∈ Γ(𝒟) with AX = λX. We replace X by ϕ2 X in the equation in Lemma 4.5.1 with index 1. Then the 𝒟-component of the resulting equation is 0 = (2λ2 λ3 − α(λ2 + λ3 ) − 2)ϕ3 X − 2ϕ2 ϕX. Applying ϕ2 to this equation yields 0 = (2λ2 λ3 − α(λ2 + λ3 ) − 2)ϕ1 X + 2ϕX, which implies the assertion. Lemma 4.5.16. If X ∈ Γ(𝒟) with AX = λX and ϕX = ϕ1 X, then 2λ ≠ α

and

Aϕ1 X =

4 + λα ϕ X. 2λ − α 1

Proof. Let X ∈ Γ(𝒟) with AX = λX and ϕX = ϕ1 X. Then the 𝒟-component of the equation in Lemma 4.5.1 with index 1 is (2λ − α)Aϕ1 X = (4 + λα)ϕ1 X.

4.5 Real hypersurfaces with invariant maximal quaternionic subbundle | 119

If 2λ = α, then 0 = 4 + λα = 4 + 2λ2 , which is impossible, and the assertion now follows. Lemma 4.5.17. We have β2 = β3 . Proof. From Lemma 4.5.14 we derive Aϕ1 X = Aϕ2 ϕ3 X = (λ3 )2 ϕ2 ϕ3 X =

λ(4 + β2 β3 ) + 2(β2 − β3 ) ϕX (4 + β2 β3 ) − 2λ(β2 − β3 ) 1

(4.23)

and Aϕ1 X = −Aϕ3 ϕ2 X = −(λ2 )3 ϕ3 ϕ2 X =

λ(4 + β2 β3 ) − 2(β2 − β3 ) ϕ X. (4 + β2 β3 ) + 2λ(β2 − β3 ) 1

Comparing these two equations leads to 4 + β2 β3 = 0

or

β2 = β3 .

We first assume that 4 + β2 β3 = 0. Then Lemma 4.5.13 shows that α ≠ 0 and β2 , β3 are the two solutions of the quadratic equation αx2 + 12x − 4α = 0. If X satisfies ϕX = −ϕ1 X, then ϕ2 X satisfies ϕϕ2 X = ϕ1 ϕ2 X. Thus we may choose X so that ϕX = ϕ1 X. From Lemma 4.5.16 and equation (4.23) we obtain 4 + λα 1 ϕ X. − ϕ1 X = Aϕ1 X = λ 2λ − α 1 Therefore, λ is a solution of the quadratic equation αx2 + 6x − α = 0. It follows that 2λ is a solution of αx2 + 12x − 4α = 0 and must therefore coincide with β2 or β3 , which is a contradiction to Lemma 4.5.14. It follows that 4 + β2 β3 ≠ 0 and hence β2 = β3 . According to Lemma 4.5.17 we can define β = β2 = β3 . Lemma 4.5.18. The restriction of the shape operator A to 𝒟 has exactly two distinct eigenvalues λ ≠ 0 and μ = 0 with the same multiplicities 2k − 2. The corresponding eigenspaces Tλ and Tμ satisfy Tλ = {v ∈ 𝒟 : Jv = J1 v}

and

Tμ = {v ∈ 𝒟 : Jv = −J1 v}.

120 | 4 Real hypersurfaces in complex 2-plane Grassmannians Moreover, λ and β are the distinct solutions of the quadratic equation x2 − αx − 2 = 0

(4.24)

and satisfy λβ = −2. Proof. Lemma 4.5.13 implies that β is a solution of the quadratic equation (4.24). Let X ∈ Γ(𝒟) with AX = λX. Equation (4.23) shows that Aϕ1 X = λϕ1 X. So the eigenspaces of A|𝒟 are ϕ1 -invariant, which implies λ2 = λ3 for all eigenvalues λ of A|𝒟 . Moreover, if X satisfies ϕX = ϕ1 X, then Lemma 4.5.16 implies that λ is also a solution of equation (4.24). We put μ = λ2 = λ3 . According to Lemma 4.5.15, μ satisfies μ(μ−α) = 0 and hence μ ∈ {0, α}. If λ = β, then Lemma 4.5.14 implies that λ is a solution of x 2 − μx + 2 = 0, which is impossible for μ = 0 and contradicts λ2 − αλ − 2 = 0 for μ = α. Therefore, λ and β are the distinct solutions of the quadratic equation (4.24), that is, 1 λ, β ∈ { (α ± √α2 + 8)}, 2

λ ≠ β.

So λβ = −2, and using again Lemma 4.5.14 proves that μ = 0. This finishes the proof. Proposition 4.5.19. Let M be a connected Hopf hypersurface in G2 (ℂ2+k ) with A𝒟 ⊆ 𝒟 and assume that the Reeb vector field ξ is tangent to 𝒟⊥ everywhere. Let J1 ∈ J be π the almost Hermitian structure with Jζ = J1 ζ . Then M has three (if r = 4√ ) or four 2 (otherwise) distinct constant principal curvatures α = 2√2 cot(2√2r),

β = √2 cot(√2r),

λ = −√2 tan(√2r),

π with some r ∈ (0, 2√ ). The corresponding multiplicities are 2

m(α) = 1,

m(β) = 2,

m(λ) = 2(k − 1) = m(μ),

and the corresponding principal curvature spaces are Tα = ℝξ = 𝒞 ⊥ ,

Tβ = 𝒟⊥ ⊖ 𝒞 ⊥ ,

Tλ = {v ∈ 𝒟 : Jv = J1 v},

Tμ = {v ∈ 𝒟 : Jv = −J1 v}.

Proof. Since ξ = ξ1 ∈ Γ(𝒟⊥ ), equation (4.11) gives gradM α = dα(ξ )ξ . As in the proof for Proposition 4.5.10 we can show that 0 = dα(ξ )g((Aϕ + ϕA)X, Y)

μ=0

4.5 Real hypersurfaces with invariant maximal quaternionic subbundle | 121

for all X, Y ∈ X(M). Suppose that Aϕ + ϕA = 0 at some point p ∈ M. Then, at p, we use the ϕ1 -invariance of Tλ to derive λϕX = λϕ1 X = Aϕ1 X = AϕX = −ϕAX = −λϕX. This implies λ = 0, which contradicts Lemma 4.5.18. Thus Aϕ + ϕA ≠ 0 everywhere, which implies dα(ξ ) = 0 everywhere. We conclude that α is constant on M and Lemma 4.5.18 implies that M has constant principal curvatures. From λβ = −2 we see that λ and β have different sign. We may choose the unit normal vector field ζ in π ). We then use such a way that β is positive, say β = √2 cot(√2r) with some r ∈ (0, 2√ 2 2 Lemma 4.5.18 to compute α = 2√2 cot(2√2r) and λ = − = −√2 tan(√2r). This finishes β

the proof of the proposition.

We now come to the main result of Case 2. Theorem 4.5.20. Let M be a connected Hopf hypersurface in G2 (ℂ2+k ), k ≥ 3, with A𝒟 ⊆ 𝒟 and assume that the Reeb vector field ξ is tangent to 𝒟⊥ everywhere. Then M is an open π part of the tube with radius r ∈ (0, 2√ ) around a totally geodesic G2 (ℂ1+k ) in G2 (ℂ2+k ). 2 Proof. By assumption, the normal vector field ζ of M is singular and of type Jζ ∈ Jζ everywhere, say Jζ = J1 ζ with an almost Hermitian structure J1 ∈ J. Using Lemma 1.4.7 we see that the eigenvalues of the normal Jacobi operator Kζ are 0, 2 and 8 with corresponding eigenspaces {v ∈ 𝒟 : Jv = −J1 v},

(𝒟⊥ ⊖ 𝒞 ⊥ ) ⊕ {v ∈ 𝒟 : Jv = J1 v},

ℝξ = 𝒞 ⊥ ,

respectively. From Proposition 4.5.19 we get {v ∈ 𝒟 : Jv = −J1 v} = Tμ , (𝒟 ⊖ 𝒞 ) ⊕ {v ∈ 𝒟 : Jv = J1 v} = Tβ ⊕ Tλ , ⊥

⊥

ℝξ = 𝒞 ⊥ = Tα . It follows that M is curvature-adapted and we can apply Corollary 2.5.2. For p ∈ M denote by cp the geodesic in G2 (ℂ2+k ) with cp (0) = p and ċp (0) = ζp and by Φr the smooth map Φr : M → G2 (ℂ2+k ),

p 󳨃→ cp (r).

Geometrically, Φr is the displacement of M at distance r in the direction of the normal vector field ζ . From Corollary 2.5.2(i) we obtain that cp (r) is a focal point of M along cp and, locally, Φr is a submersion onto a (4k − 4)-dimensional submanifold P of G2 (ℂ2+k ). Moreover, the tangent space of P at Φr (p) is obtained by parallel translation of (Tλ ⊕ Tμ )(p) = 𝒟(p), which is a quaternionic subspace of Tp G2 (ℂ2+k ). Since J is parallel along cp , also TΦr (p) P is a quaternionic subspace of TΦr (p) G2 (ℂ2+k ). Thus P is

122 | 4 Real hypersurfaces in complex 2-plane Grassmannians a quaternionic submanifold of G2 (ℂ2+k ). Since P is quaternionic, it is totally geodesic in G2 (ℂ2+k ) (see [3]). Every totally geodesic quaternionic hypersurface in G2 (ℂ2+k ) is congruent to an open part of a totally geodesic G2 (ℂ1+k ) in G2 (ℂ2+k ) (see [6]). So we conclude that P is an open part of a totally geodesic G2 (ℂ1+k ) in G2 (ℂ2+k ). Rigidity of totally geodesic submanifolds finally implies that M is an open part of the tube with radius r around a totally geodesic G2 (ℂ1+k ) in G2 (ℂ2+k ). This finishes the discussions of the two cases and we now come to the main result of this section. Theorem 4.5.21 ([14]). Let M be a connected real hypersurface in G2 (ℂ2+k ), k ≥ 3. Then the following statements are equivalent: (i) the maximal complex subbundle 𝒞 and the maximal quaternionic subbundle 𝒟 of the tangent bundle of M are invariant under the shape operator A of M; (ii) M is congruent to an open part of one of the following homogeneous real hypersurfaces: π (A) a tube with radius r ∈ (0, 2√ ) around the complex and quaternionic totally 2

geodesic embedding of G2 (ℂ1+k ) into G2 (ℂ2+k ), (B) (only if k = 2l is even) a tube with radius r ∈ (0, π4 ) around the real and quaternionic totally geodesic embedding of the quaternionic projective space ℍP l into G2 (ℂ2+2l ).

Proof. The result follows from Proposition 4.3.3, Proposition 4.3.5, Theorem 4.5.11 and Theorem 4.5.20. From Theorem 4.3.1 we obtain the following interesting consequence. Corollary 4.5.22. Let M be a connected real hypersurface in G2 (ℂ2+k ), k ≥ 3. Then the maximal complex subbundle 𝒞 and the maximal quaternionic subbundle 𝒟 of the tangent bundle of M are invariant under the shape operator A of M if and only if M is an open part of a homogeneous real hypersurface in G2 (ℂ2+k ).

4.6 Real hypersurfaces with isometric Reeb flow In this section we investigate and classify real hypersurfaces with isometric Reeb flow in G2 (ℂ2+k ), k ≥ 3. We already studied real hypersurfaces with isometric Reeb flow in Kähler manifolds in Section 3.4. In particular, from Proposition 3.4.1 we already know that any such hypersurface is a Hopf hypersurface and satisfies Aϕ = ϕA.

4.6 Real hypersurfaces with isometric Reeb flow

| 123

Proposition 4.6.1. Let M be a real hypersurface with isometric Reeb flow in G2 (ℂ2+k ), k ≥ 3. Then we have αAϕX − A2 ϕX + ϕX 3

= ∑ (4ην (ϕX)ην (ξ )ξ − ην (ξ )ϕν X + ην (X)ϕν ξ − 3ην (ϕX)ξν ) ν=1

(4.25)

for all X ∈ X(M). Proof. Since M has isometric Reeb flow, we have Aϕ − ϕA = 0 by Proposition 3.4.1. Differentiating this equation covariantly we obtain (∇X A)ϕY + η(Y)A2 X − αg(AX, Y)ξ − ϕ(∇X A)Y − η(AY)AX + g(AX, AY)ξ = 0 for all X, Y ∈ X(M). Taking inner product with Z ∈ X(M) gives g((∇X A)Y, ϕZ) + g((∇X A)Z, ϕY)

= αη(Y)g(AX, Z) + αη(Z)g(AX, Y) − η(Y)g(AX, AZ) − η(Z)g(AX, AY),

which implies g((∇X A)Y, ϕZ) + g((∇X A)Z, ϕY) + g((∇Y A)Z, ϕX)

+ g((∇Y A)X, ϕZ) − g((∇Z A)X, ϕY) − g((∇Z A)Y, ϕX)

= 2αη(Z)g(AX, Y) − 2η(Z)g(AX, AY).

The left-hand side of this equation can be rewritten as 2g((∇X A)Y, ϕZ) − g((∇X A)Y − (∇Y A)X, ϕZ)

+ g((∇Y A)Z − (∇Z A)Y, ϕX) − g((∇Z A)X − (∇X A)Z, ϕY),

and using the Codazzi equation this implies g((∇X A)Y, ϕZ)

= αη(Z)g(AX, Y) − η(Z)g(AX, AY) − η(Y)g(X, Z) + η(Z)g(X, Y) 3

+ ∑ (ην (Y)g(ϕν ϕX, Z) − ην (Z)g(ϕν ϕX, Y) ν=1

+ ην (ϕY)g(ϕν X, Z) − ην (ϕZ)g(ϕν X, Y) + 2ην (ϕX)g(ϕν Y, Z)). Replacing Z by ϕZ, the left-hand side of the previous equation becomes −g((∇X A)Y, Z) + η(Z)g((∇X A)Y, ξ ).

124 | 4 Real hypersurfaces in complex 2-plane Grassmannians Since g((∇X A)Y, ξ ) = g((∇X A)ξ , Y) = g(∇X (αξ ), Y) − g(A∇X ξ , Y), we see that dα(X)η(Y)η(Z) + αη(Z)g(ϕAX, Y) − η(Z)g(AϕAX, Y) − g((∇X A)Y, Z) becomes the left-hand side of the previous equation when we replace Z by ϕZ. Replacing Z by ϕZ also on the right-hand side we eventually get 3

(∇X A)Y = −η(Y)ϕX − ∑ (ην (Y)ϕν X − ην (ϕY)ϕϕν X − 2ην (ϕX)ϕϕν Y) ν=1

+ (dα(X)η(Y) + αg(AϕX, Y) − g(A2 ϕX, Y))ξ 3

+ ( ∑ (ην (ϕX)ην (Y) + ην (ξ )g(ϕν X, Y)))ξ 3

ν=1

3

− ∑ g(ϕν X, Y)ξν + ∑ (η(X)ην (Y) − g(ϕν ϕX, Y))ϕν ξ . ν=1

ν=1

Inserting Y = ξ then yields 3

(∇X A)ξ = −ϕX − ∑ (ην (ξ )ϕν X − 2ην (ϕX)ϕϕν ξ ) ν=1

3

+ (dα(X) + ∑ (ην (ϕX)ην (ξ ) + ην (ξ )g(ϕν X, ξ )))ξ 3

ν=1

3

− ∑ g(ϕν X, ξ )ξν + ∑ (η(X)ην (ξ ) − g(ϕν ϕX, ξ ))ϕν ξ . ν=1

ν=1

On the other hand, we have (∇X A)ξ = dα(X)ξ + α∇X ξ − A∇X ξ = dα(X)ξ + αϕAX − A2 ϕX. Comparing these two expressions for (∇X A)ξ implies equation (4.25). An important step towards the classification of real hypersurfaces with isometric Reeb flow in G2 (ℂ2+k ) is the following. Proposition 4.6.2. Let M be a real hypersurface with isometric Reeb flow in G2 (ℂ2+k ), k ≥ 3. Then the maximal quaternionic subbundle 𝒟 of TM is contained in the maximal complex subbundle 𝒞 of TM. In particular, the Reeb vector field ξ is perpendicular to 𝒟 everywhere, or equivalently, the unit normal vector field ζ of M is singular of type Jζ ∈ Jζ everywhere.

4.6 Real hypersurfaces with isometric Reeb flow

| 125

Proof. Let X ∈ Γ(𝒟). From equation (4.25) we get αg(AϕX, ϕX) − g(A2 ϕX, ϕX) + g(ϕX, ϕX) 3

= ∑ (ην (X)2 − 3ην (ϕX)2 − ην (ξ )g(ϕν X, ϕX)), ν=1

and equation (4.10) implies αg(AϕX, ϕX) − g(A2 ϕX, ϕX) + g(ϕX, ϕX) 3

= − ∑ (ην (X)2 + ην (ϕX)2 + ην (ξ )g(ϕν X, ϕX)). ν=1

Comparing the previous two equations shows that 3

3

ν=1

ν=1

∑ ην (X)2 = ∑ ην (ϕX)2

for all X ∈ Γ(𝒟). Consequently, ϕX ∈ Γ(𝒞 ∩𝒟) for all X ∈ Γ(𝒞 ∩𝒟). This shows that 𝒞 ∩𝒟 is ϕ-invariant. Since ξ is perpendicular to 𝒞 and hence also to 𝒞 ∩ 𝒟, the subbundle 𝒞 ∩ 𝒟 has even rank (pointwise). As rk(𝒞 ∩ 𝒟) = rk(𝒞 ) + rk(𝒟) − rk(𝒞 + 𝒟) and rk(𝒞 ) = 4k − 2, rk(𝒟) = 4k − 4 and rk(𝒞 + 𝒟) ∈ {4k − 2, 4k − 1}, we therefore must have rk(𝒞 + 𝒟) = 4k − 2. This means that 𝒟 ⊂ 𝒞 , which shows that ξ is perpendicular to 𝒟 everywhere. Thus there exists an almost Hermitian structure J1 ∈ J so that ξ = ξ1 , which is equivalent to Jζ = J1 ζ , which shows that ζ is singular everywhere. We will now prove that the principal curvatures of a real hypersurface in G2 (ℂ2+k ) with isometric Reeb flow are constant. Proposition 4.6.3. Let M be a connected real hypersurface with isometric Reeb flow in G2 (ℂ2+k ), k ≥ 3. We choose a canonical local basis in J so that ξ1 = ξ (see Proposition 4.6.2). Then the Hopf principal curvature function α is constant and for all X ∈ Γ(𝒞 ) with AX = λX one of the following two statements holds: (i) λ(λ − α) = 0, X ∈ Γ(𝒟) and ϕX = −ϕ1 X; (ii) λ2 − αλ − 2 = 0 and ϕX𝒟 = ϕ1 X𝒟 , where X𝒟 denotes the orthogonal projection of X onto 𝒟. In particular, all principal curvatures of M are constant. Proof. Since ξ = ξ1 , equation (4.11) reduces to gradM α = dα(ξ )ξ ,

126 | 4 Real hypersurfaces in complex 2-plane Grassmannians which implies ∇X gradM α = d(dα(ξ ))(X)ξ + dα(ξ )ϕAX. The symmetry of the Hessian of α and the identity Aϕ = ϕA imply that 2dα(ξ )g(AϕX, Y) = d(dα(ξ ))(X)η(Y) − d(dα(ξ ))(Y)η(X) for all X, Y ∈ X(M). Putting X = ξ implies d(dα(ξ ))(Y) = d(dα(ξ ))(ξ )η(Y) for all Y ∈ X(M), and inserting this and the corresponding expression for d(dα(ξ ))(X) into the previous equation implies dα(ξ )g(AϕX, Y) = 0 for all X, Y ∈ X(M). Thus either dα(ξ ) = 0, which implies gradM α = 0 and hence α is constant, or Aϕ = 0. But the latter equation implies that A restricted to 𝒞 vanishes. In particular, inserting X = ξ2 into equation (4.25) yields ξ3 = 0, which is impossible. Thus α is constant. Now let X ∈ Γ(𝒞 ) with AX = λX. First, we have AϕX = ϕAX = λϕX. Thus, from equation (4.12) we get (λ2 − αλ − 1)ϕX − ϕ1 X = 2η3 (X)ξ2 − 2η2 (X)ξ3 . Replacing X by ϕX we obtain (λ2 − αλ − 1)X + ϕ1 ϕX = 2η2 (X)ξ2 + 2η3 (X)ξ3 . We now decompose X into X = X𝒟 + η2 (X)ξ2 + η3 (X)ξ3 and insert this expression into the previous equation, which leads to 0 = (λ2 − αλ − 1)X𝒟 + ϕ1 ϕX𝒟 + (λ2 − αλ − 2)η2 (X)ξ2 + (λ2 − αλ − 2)η3 (X)ξ3 . It is clear that ϕ1 , and hence also ϕ, leaves 𝒟 invariant. Thus ϕ1 ϕX𝒟 ∈ Γ(𝒟), and therefore the previous equation splits into three equations: 0 = (λ2 − αλ − 1)X𝒟 + ϕ1 ϕX𝒟 , 0 = (λ2 − αλ − 2)η2 (X)ξ2 ,

0 = (λ2 − αλ − 2)η3 (X)ξ3 .

4.6 Real hypersurfaces with isometric Reeb flow

| 127

If λ2 − αλ − 2 = 0, then the first equation implies ϕ1 ϕX𝒟 = −X𝒟 . On the other hand, if λ2 − αλ − 2 ≠ 0, then the last two equations imply X ∈ Γ(𝒟). The first equation shows that ϕ1 ϕX𝒟 and X𝒟 are proportional. In fact, since both ϕ1 and ϕ act orthogonally on 𝒟, we must have ϕ1 ϕX𝒟 = ±X𝒟 . If ϕ1 ϕX𝒟 = −X𝒟 , then the first equation yields λ2 − αλ − 2 = 0, which is a contradiction. Hence we must have ϕ1 ϕX𝒟 = X𝒟 , and the first equation implies λ2 − αλ = 0. This shows that either (i) or (ii) holds. The constancy of the principal curvatures then follows from the constancy of α. We can now state and prove our main classification result for this section. Theorem 4.6.4 ([15]). Let M be a connected real hypersurface in G2 (ℂ2+k ), k ≥ 3. Then M has isometric Reeb flow if and only if M is congruent to an open part of the tube with π radius r ∈ (0, 2√ ) around a totally geodesic G2 (ℂ1+k ) in G2 (ℂ2+k ). 2 Proof. We know from Proposition 4.3.3 that every tube around a totally geodesic G2 (ℂ1+k ) in G2 (ℂ2+k ) has isometric Reeb flow. Conversely, let M be a connected real hypersurface in G2 (ℂ2+k ) with isometric Reeb flow. From Proposition 4.6.3 we see that M has constant principal curvatures, and the number of different principal curvatures is at most four. We choose a real number r π such that with 0 < r < 2√ 2 α = 2√2 cot(2√2r). Then, from Proposition 4.6.3, the other possible principal curvatures are μ = 0,

β = √2 cot(√2r),

λ = −√2 tan(√2r).

Note that β and λ are the two different solutions of the quadratic equation x2 − αx − 2 = 0. For ρ ∈ {α, μ, β, λ} we define Tρ = {v ∈ 𝒞 : Av = ρv}. Then we have 𝒞 = Tα ⊕ Tμ ⊕ Tβ ⊕ Tλ

and, if Tρ is non-trivial, Tρ is the subbundle of TM consisting of all principal curvature vectors of M with respect to ρ which are tangent to 𝒞 . According to Proposition 3.4.1, each Tρ is a complex subbundle of TM.

128 | 4 Real hypersurfaces in complex 2-plane Grassmannians For p ∈ M we denote by cp the geodesic in G2 (ℂ2+k ) with cp (0) = p and ċp (0) = ζp and by Φr the smooth map Φr : M → G2 (ℂ2+k ),

p 󳨃→ cp (r).

Geometrically, Φr is the displacement of M at distance r in direction of the normal vector field ζ . For each p ∈ M the differential dp Φr of Φr at p can be computed using Jacobi fields by means of dp Φr (v) = Zv (r) (see Section 2.5). Here, v ∈ Tp M and Zv is the Jacobi field along cp with initial values Zv (0) = v and Zv′ (0) = −Av. Since Jζ = −ξ = −ξ1 = J1 ζ , we see that ζ is a singular tangent vector of type Jζ ∈ Jζ at each point. Using the explicit description of the Jacobi operator R̄ ζ for the case Jζ ∈ Jζ in Lemma 1.4.7 and of the shape operator A of M in Proposition 4.6.3, we see that Tα ⊕ Tμ is contained in the 0-eigenspace of R̄ ζ and Tβ ⊕ Tλ in the 2-eigenspace of R̄ ζ . For the Jacobi fields Zv along cp we thus get the expressions (cos(2√2t) − 2√α 2 sin(2√2t))Bv (t), if v ∈ ℝξ , { { { Zv (t) = {(cos(√2t) − √ρ sin(√2t))Bv (t), if v ∈ Tρ and ρ ∈ {β, λ}, 2 { { if v ∈ Tρ and ρ ∈ {α, μ}, {(1 − ρt)Bv (t), where Bv denotes the parallel vector field along cp with Bv (0) = v. This shows that the kernel of dΦr is ℝξ ⊕ Tβ and that Φr is of constant rank dim(Tα ⊕ Tλ ⊕ Tμ ). So, locally, Φr is a submersion onto a submanifold P of G2 (ℂ2+k ). Moreover, the tangent space of P at Φr (p) is obtained by parallel translation of (Tα ⊕ Tλ ⊕ Tμ )(p) (see Theorem 2.5.1), which is a complex subspace of Tp G2 (ℂ2+k ). Since J is parallel along cp , also TΦr (p) P is a complex subspace of TΦr (p) G2 (ℂ2+k ). Thus P is a complex submanifold of G2 (ℂ2+k ). It is clear that ηp = ċp (r) is a unit normal vector of P at Φr (p). The shape operator APηp of P with respect to ηp is given by APηp Zv (r) = −Zv′ (r) (see Theorem 2.5.1). From this we immediately get that for each ρ ∈ {α, λ, μ} the parallel translate of Tρ (p) along cp from p to Φr (p) is a principal curvature space of P with respect to ρ, provided that Tρ (p) is non-trivial. Moreover, the corresponding principal curvature is

2√2 cot(2√2r) 1−2√2 cot(2√2r)r

for ρ = α and 0 for ρ ∈ {μ, λ}. Since any complex submani-

π fold of a Kähler manifold is minimal this implies that either Tα is trivial or r = 4√ (in 2 which case the corresponding principal curvature becomes zero). The vectors of the form ηq , q ∈ Φ−1 r ({Φr (p)}), form an open subset of the unit sphere in the normal space

4.7 Contact hypersurfaces | 129

of P at Φr (p). Since APηq vanishes for all ηq it now follows that P is totally geodesic in

G2 (ℂ2+k ). Rigidity of totally geodesic submanifolds now implies that the entire submanifold M is an open part of the tube with radius r around a connected, complete, totally geodesic, complex submanifold P of G2 (ℂ2+k ). The vector space {v ∈ Tp G2 (ℂ2+k ) : Jv = −J1 v} is the +1-eigenspace of the selfadjoint endomorphism JJ1 = J1 J at p. Using the fact that (JJ1 )2 = I and tr(JJ1 ) = 0, we easily see that this eigenspace is a complex vector space of complex dimension k. Thus Tα (p) ⊕ Tμ (p) = {v ∈ 𝒟(p) : Jv = −J1 v} is a complex vector space of complex dimension k − 1, and the above arguments show that the parallel translate of this complex vector space along cp from p to Φr (p) lies in TΦr (p) P. Since both J and J are parallel along cp , it follows that TΦr (p) P contains a (k − 1)-dimensional complex subspace of the form {v : Jv = J1′ v} for some fixed almost Hermitian structure J1′ ∈ J. Any such subspace is invariant under the curvature tensor R̄ and hence there exists a connected, complete, totally geodesic, complex submanifold Σ of G2 (ℂ2+k ) with Φr (p) ∈ Σ and TΦr (p) Σ equal to that curvature-invariant complex subspace. Since that curvature-invariant subspace consists entirely of singular tangent vectors of this special type, we see that Σ is a totally geodesic ℂP k−1 . We thus conclude that P contains a totally geodesic ℂP k−1 . The classification of totally geodesic submanifolds in G2 (ℂ2+k ) (see [57]) shows that the only totally geodesic submanifolds in G2 (ℂ2+k ) containing a totally geodesic ℂP k−1 are ℂP k−1 , ℂP k , ℂP k−1 × ℂP 1 and G2 (ℂ1+k ). The normal spaces of both ℂP k−1 and ℂP k−1 × ℂP 1 contain regular tangent vectors of G2 (ℂ2+k ). Emanating along the geodesic in direction of such a regular tangent vector would give a normal vector to M which is a regular. But this contradicts the fact that the normal vector field ζ of M is singular everywhere. Thus P is either ℂP k or G2 (ℂ1+k ). Since ℂP k is the focal set of G2 (ℂ1+k ) in G2 (ℂ2+k ), we finally see that M is an open part of the tube with radius r around a totally geodesic G2 (ℂ1+k ) in G2 (ℂ2+k ). This finishes the proof.

4.7 Contact hypersurfaces In this section we study contact hypersurfaces in the complex 2-plane Grassmannian G2 (ℂ2+k ), k ≥ 3. We already investigated contact hypersurfaces in Kähler manifolds in Section 3.5. We also proved in Proposition 4.3.5 that the tube with radius 0 < r < π4 around the totally geodesic ℍP l ⊂ G2 (ℂ2+2l ), l ≥ 2, is a contact hypersurface. A natural question is whether there are other contact hypersurfaces in G2 (ℂ2+k ). We will see that there are no other contact hypersurfaces.

130 | 4 Real hypersurfaces in complex 2-plane Grassmannians Let M be a connected contact hypersurface in G2 (ℂ2+k ), k ≥ 3. From Proposition 3.5.4 and Proposition 3.5.5 we see that there exists a constant f ≠ 0 such that Aϕ + ϕA = 2fϕ. Moreover, from Proposition 3.5.6 we know that M is a Hopf hypersurface and the mean curvature of M is given by tr(A) = α + (4k − 2)f , where α = g(Aξ , ξ ) is the Hopf principal curvature function. We also know that for all X ∈ Γ(𝒞 ) we have AX = λX

󳨐⇒

AϕX = (2f − λ)ϕX.

From Proposition 3.5.9 we have the useful equation ̄ , ζ )ϕX) 2(A2 − 2fA + αf )X = −(R(ξ 𝒞

(4.26)

for all X ∈ Γ(𝒞 ). Using the explicit expression for the Riemannian curvature tensor R̄ of G2 (ℂ2+k ) in Theorem 1.4.6 we compute (note that ϕX = JX for X ∈ Γ(𝒞 )) ̄ , ζ )ϕX = g(ζ , ϕX)ξ − g(ξ , ϕX)ζ + g(Jζ , ϕX)Jξ − g(Jξ , ϕX)Jζ − 2g(Jξ , ζ )JϕX R(ξ 3

+ ∑ (g(Jν ζ , ϕX)Jν ξ − g(Jν ξ , ϕX)Jν ζ − 2g(Jν ξ , ζ )Jν ϕX ν=1

+ g(Jν Jζ , ϕX)Jν Jξ − g(Jν Jξ , ϕX)Jν Jζ ) 3

= 2X + 2 ∑ (−g(ξν , ϕX)Jν ξ + g(Jν ξ , ϕX)ξν − g(ξ , ξν )Jν ϕX) ν=1 3

= 2X − 2 ∑ (ην (ϕX)Jν ξ + η(ϕν ϕX)ξν + ην (ξ )Jν ϕX). ν=1

The 𝒞 -component of Jν ξ is ϕν ξ , the 𝒞 -component of ξν is ξν − η(ξν )ξ and the 𝒞 -component of Jν ϕX is ϕν ϕX − η(ϕν ϕX)ξ . Altogether, we get 3

̄ , ζ )ϕX) = 2X − 2 ∑ (ην (ϕX)ϕν ξ + η(ϕν ϕX)ξν − η(ϕν ϕX)η(ξν )ξ (R(ξ 𝒞 ν=1

+ ην (ξ )ϕν ϕX − ην (ξ )η(ϕν ϕX)ξ ) 3

= 2X − 2 ∑ (ην (ϕX)ϕν ξ + η(ϕν ϕX)ξν ν=1

+ ην (ξ )ϕν ϕX − 2η(ϕν ϕX)η(ξν )ξ ).

4.7 Contact hypersurfaces | 131

We have η(ϕν ϕX) = ην (ϕ2 X) = −ην (X) for X ∈ Γ(𝒞 ) and hence 3

̄ , ζ )ϕX) = 2X − 2 ∑ (ην (ϕX)ϕν ξ − ην (X)ξν (R(ξ 𝒞 ν=1

+ ην (ξ )ϕν ϕX + 2ην (X)η(ξν )ξ ). Combining this with equation (4.26) gives the following. Proposition 4.7.1. Let M be a connected contact hypersurface in G2 (ℂ2+k ), k ≥ 3. Then we have (A2 − 2fA + αf + 1)X 3

= ∑ (ην (ϕX)ϕν ξ − ην (X)ξν + ην (ξ )ϕν ϕX + 2ην (X)η(ξν )ξ ) ν=1

(4.27)

for all X ∈ Γ(𝒞 ). Proposition 4.7.2. Let M be a connected contact hypersurface in G2 (ℂ2+k ), k ≥ 3. Then ξ is tangent to the maximal quaternionic subbundle 𝒟 everywhere. Proof. From Theorem 3.5.19 we know that the Reeb vector field ξ is an eigenvector of the normal Jacobi operator K everywhere. From Proposition 4.1.4 we then see that ξ is either tangent to 𝒟 or tangent to 𝒟⊥ everywhere. Let us assume that ξ is tangent to 𝒟⊥ everywhere. With a suitable choice of local canonical basis we can assume that ξ = ξ1 (locally). Then, using the basic structure equations, equation (4.27) becomes (A2 − 2fA + αf + 1)X = ϕ1 ϕX − 2η2 (X)ξ2 − 2η3 (X)ξ3 for all X ∈ Γ(𝒞 ). Let 0 ≠ X ∈ Γ(𝒞 ) with AX = λX. Then (4.28) implies (λ2 − 2fλ + αf + 1)X = ϕ1 ϕX − 2η2 (X)ξ2 − 2η3 (X)ξ3 . We decompose X into X = X𝒟 + η2 (X)ξ2 + η3 (X)ξ3 , where X𝒟 is the orthogonal projection of X onto 𝒟. Then we have ϕ1 ϕX = ϕ1 ϕX𝒟 + η2 (X)ξ2 + η3 (X)ξ3 , and inserting this into the previous equation leads to the equation 0 = (λ2 − 2fλ + αf + 1)X𝒟 − ϕ1 ϕX𝒟 .

(4.28)

132 | 4 Real hypersurfaces in complex 2-plane Grassmannians Since ξ is tangent to 𝒟⊥ everywhere, we have ϕ1 ϕ = J1 J on 𝒟. Since the trace of J1 J is zero and ξ = ξ1 , this trace is zero when restricting J1 J to 𝒟. Therefore, since J1 J is a multiple of the identity on 𝒟, this multiple must be zero, and hence J1 J = 0 on 𝒟, which is a contradiction. It follows that ξ is tangent to 𝒟 everywhere. We now state the main result of this section, which was proved by Suh in [93] under the additional assumption that the contact hypersurface has constant mean curvature. This assumption now turns out to be redundant by Theorem 3.5.19. Theorem 4.7.3. Let M be a connected real hypersurface in G2 (ℂ2+k ), k ≥ 3. Then M is a contact hypersurface if and only if the quaternionic dimension k of G2 (ℂ2+k ) is even, say k = 2l, and M is an open part of the tube with radius r ∈ (0, π4 ) around a totally geodesic quaternionic projective space ℍP l in G2 (ℂ2+2l ). Proof. Let M be a connected contact hypersurface in G2 (ℂ2+k ). Then M is a Hopf hypersurface by Proposition 3.5.6. By Theorem 3.5.19, the Reeb vector field ξ is an eigenvector of the normal Jacobi operator K everywhere. From Proposition 4.1.4 we see that ξ is tangent to 𝒟 or to 𝒟⊥ everywhere. Then Proposition 4.7.2 tells us that ξ is tangent to 𝒟 everywhere. It then follows from Theorem 4.5.12 that the quaternionic dimension k of G2 (ℂ2+k ) is even, say k = 2l, and M is an open part of the tube with radius r ∈ (0, π4 ) around a totally geodesic quaternionic projective space ℍP l in G2 (ℂ2+2l ). Conversely, let M be an open part of the tube with radius r ∈ (0, π4 ) around a totally geodesic quaternionic projective space ℍP l in G2 (ℂ2+2l ). Then M is a contact hypersurface with constant mean curvature by Proposition 4.3.5.

4.8 The Ricci tensor of Hopf hypersurfaces In this section we investigate properties of the Ricci tensor of Hopf hypersurfaces in G2 (ℂ2+k ), k ≥ 3. Let M be a real hypersurface in G2 (ℂ2+k ) and Ric the Ricci tensor of M. We already computed the Ricci tensor in Theorem 4.1.6. More precisely, we have Ric(X) = 4(k + 2)X − KX + hAX − A2 X for all X ∈ X(M), where K is the normal Jacobi operator of M and h = tr(A). We computed the normal Jacobi operator of real hypersurfaces in G2 (ℂ2+k ) in Theorem 4.1.3, namely 3

KX = X + 3η(X)ξ + ∑ (3ην (X)ξν + ην (ξ )η(X)ξν + ην (ϕX)ϕν ξ − ην (ξ )ϕν ϕX). ν=1

4.8 The Ricci tensor of Hopf hypersurfaces | 133

Altogether, this gives Ric(X) = (4k + 7)X − 3η(X)ξ + hAX − A2 X 3

− ∑ (3ην (X)ξν + ην (ξ )η(X)ξν + ην (ϕX)ϕν ξ − ην (ξ )ϕν ϕX) ν=1

for all X ∈ X(M). The covariant derivative of the Ricci tensor is defined by (∇X Ric)Y = ∇X Ric(Y) − Ric(∇X Y). From (4.29) we then get (∇X Ric)Y = −3∇X (η(Y)ξ ) + ∇X (hAY) − ∇X (A2 Y) 3

− ∑ (3∇X (ην (Y)ξν ) + ∇X (ην (ξ )η(Y)ξν ) ν=1

+ ∇X (ην (ϕY)ϕν ξ ) − ∇X (ην (ξ )ϕν ϕY)) + 3η(∇X Y)ξ − hA∇X Y + A2 ∇X Y 3

+ ∑ (3ην (∇X Y)ξν + ην (ξ )η(∇X Y)ξν ν=1

+ ην (ϕ∇X Y)ϕν ξ − ην (ξ )ϕν ϕ∇X Y)

= −3((∇X η)Y)ξ − 3η(Y)∇X ξ + dh(X)AY + h(∇X A)Y − (∇X A2 )Y 3

− ∑ (3((∇X ην )Y)ξν + 3ην (Y)∇X ξν ν=1

+ ην (∇X ξ )η(Y)ξν + η(∇X ξν )η(Y)ξν

+ ην (ξ )((∇X η)Y)ξν + ην (ξ )η(Y)∇X ξν

+ ην ((∇X ϕ)Y)ϕν ξ + g(ϕY, ∇X ξν )ϕν ξ + ην (ϕY)(∇X ϕ)ξν + ην (ϕY)ϕ∇X ξν − ην (∇X ξ )ϕν ϕY − η(∇X ξν )ϕν ϕY

− ην (ξ )(∇X ϕν )ϕY − ην (ξ )ϕν (∇X ϕ)Y). Using the basic structure equations this leads to (∇X Ric)Y

= −3g(ϕAX, Y)ξ − 3η(Y)ϕAX + dh(X)AY + h(∇X A)Y − (∇X A2 )Y 3

− ∑ (3qν+2 (X)ην+1 (Y)ξν − 3qν+1 (X)ην+2 (Y)ξν + 3g(ϕν AX, Y)ξν ν=1

+ 3ην (Y)qν+2 (X)ξν+1 − 3ην (Y)qν+1 (X)ξν+2 + 3ην (Y)ϕν AX

(4.29)

134 | 4 Real hypersurfaces in complex 2-plane Grassmannians + ην (ϕAX)η(Y)ξν

+ qν+2 (X)ην+1 (ξ )η(Y)ξν − qν+1 (X)ην+2 (ξ )η(Y)ξν + η(ϕν AX)η(Y)ξν + ην (ξ )g(ϕAX, Y)ξν + ην (ξ )η(Y)ϕν AX

+ ην (ξ )η(Y)qν+2 (X)ξν+1 − ην (ξ )η(Y)qν+1 (X)ξν+2 + η(Y)ην (AX)ϕν ξ − g(AX, Y)ην (ξ )ϕν ξ

+ qν+2 (X)ην+1 (ϕY)ϕν ξ − qν+1 (X)ην+2 (ϕY)ϕν ξ + g(ϕY, ϕν AX)ϕν ξ + ην (ϕY)ην (ξ )AX − ην (ϕY)ην (AX)ξ

+ ην (ϕY)qν+2 (X)ϕν+1 ξ − ην (ϕY)qν+1 (X)ϕν+2 ξ + ην (ϕY)ϕϕν AX

− ην (ϕAX)ϕν ϕY − η(ϕν AX)ϕν ϕY

− qν+2 (X)ην+1 (ξ )ϕν ϕY + qν+1 (X)ην+2 (ξ )ϕν ϕY

− ην (ξ )qν+2 (X)ϕν+1 ϕY + ην (ξ )qν+1 (X)ϕν+2 ϕY

− ην (ξ )ην (ϕY)AX + ην (ξ )g(AX, ϕY)ξν

− ην (ξ )η(Y)ϕν AX + ην (ξ )g(AX, Y)ϕν ξ ).

Reorganizing this equation eventually leads to the following explicit expression for the covariant derivative (∇X Ric)Y of the Ricci tensor of M: (∇X Ric)Y

= −3g(ϕAX, Y)ξ − 3η(Y)ϕAX + dh(X)AY + h(∇X A)Y − (∇X A2 )Y 3

− ∑ (3g(ϕν AX, Y)ξν + 2ην (ϕAX)η(Y)ξν ν=1

− η(AX)ην (ϕY)ξν + η(AX)ην (Y)ϕν ξ − g(ϕν ϕAX, Y)ϕν ξ

+ 3ην (Y)ϕν AX + ην (ϕY)ϕν ϕAX − 2ην (ϕAX)ϕν ϕY).

(4.30)

We now assume that M is a Hopf hypersurface, so Aξ = αξ . From (4.11) we have 3

dα(X) = g(gradM α, X) = dα(ξ )g(ξ , X) + 4 ∑ ην (ξ )g(ϕν ξ , X) ν=1

3

= dα(ξ )η(X) − 4 ∑ ην (ξ )ην (ϕX). ν=1

Using (4.31) we obtain (∇X A)ξ = ∇X Aξ − A∇X ξ = dα(X)ξ + α∇X ξ − A∇X ξ 3

= dα(ξ )η(X)ξ − 4 ∑ ην (ξ )ην (ϕX)ξ + αϕAX − AϕAX ν=1

(4.31)

4.8 The Ricci tensor of Hopf hypersurfaces | 135

and (∇X A2 )ξ = (∇X A)Aξ + A(∇X A)ξ = α(∇X A)ξ + A(∇X A)ξ 3

= αdα(ξ )η(X)ξ − 4α ∑ ην (ξ )ην (ϕX)ξ + α2 ϕAX − αAϕAX ν=1

3

+ αdα(ξ )η(X)ξ − 4α ∑ ην (ξ )ην (ϕX)ξ + αAϕAX − A2 ϕAX ν=1

3

= 2αdα(ξ )η(X)ξ − 8α ∑ ην (ξ )ην (ϕX)ξ + α2 ϕAX − A2 ϕAX. ν=1

We now assume that the Ricci tensor is parallel, that is, ∇Ric = 0. Inserting Y = ξ into equation (4.30) then gives 0 = − 3ϕAX + dh(X)Aξ + h(∇X A)ξ − (∇X A2 )ξ 3

− ∑ (5ην (ϕAX)ξν + ην (AX)ϕν ξ + 3ην (ξ )ϕν AX) ν=1

= − 3ϕAX + αdh(X)ξ

3

+ hdα(ξ )η(X)ξ − 4h ∑ ην (ξ )ην (ϕX)ξ + hαϕAX − hAϕAX ν=1

3

− 2αdα(ξ )η(X)ξ + 8α ∑ ην (ξ )ην (ϕX)ξ − α2 ϕAX + A2 ϕAX 3

ν=1

− ∑ (5ην (ϕAX)ξν + ην (AX)ϕν ξ + 3ην (ξ )ϕν AX) ν=1 2

= − (α − hα + 3)ϕAX − hAϕAX + A2 ϕAX

3

+ αdh(X)ξ + (h − 2α)dα(ξ )η(X)ξ − 4(h − 2α) ∑ ην (ξ )ην (ϕX)ξ ν=1

3

− ∑ (5ην (ϕAX)ξν + ην (AX)ϕν ξ + 3ην (ξ )ϕν AX). ν=1

(4.32)

Inserting X = ξ implies 3

0 = (αdh(ξ ) + (h − 2α)dα(ξ )η(ξ ))ξ − 4α ∑ ην (ξ )ϕν ξ . ν=1

If α ≠ 0, this equation implies ην (ξ ) = 0 for all ν ∈ {1, 2, 3} and hence ξ ∈ Γ(𝒟). From Theorem 4.5.12 we then conclude that the quaternionic dimension k of G2 (ℂ2+k ) is even, say k = 2l, and M is an open part of the tube with radius r ∈ (0, π4 ) around a totally geodesic quaternionic projective space ℍP l in G2 (ℂ2+2l ). From Proposition 4.3.5(i)

136 | 4 Real hypersurfaces in complex 2-plane Grassmannians we know that h = (8l − 2) cot(2r) − 2 tan(2r). On the other hand, from Theorem 4.3.4 we have Aξν = βξν with β = 2 cot(2r). Inserting X = ξν into equation (4.32) and using the fact that Aϕξν = 0 by Theorem 4.3.4 yields 0 = −(α2 − hα + 4)βϕξν . Since β = 2 cot(2r) ≠ 0 for r ∈ (0, π4 ) and α = −2 tan(2r), this implies 0 = α2 − hα + 4 = 4 tan2 (2r) + 2 tan(2r)((8l − 2) cot(2r) − 2 tan(2r)) + 4 = 16l, which is a contradiction. We therefore must have α = 0. Then (4.31) implies 3

0 = ∑ ην (ξ )ην (ϕX). ν=1

Replacing X by ϕX leads to 3

3

0 = ∑ ην (ξ )ην (ϕ2 X) = ∑ ην (ξ )ην (−X + η(X)ξ ) ν=1

ν=1

3

3

ν=1

ν=1

= − ∑ ην (ξ )ην (X) + ∑ ην (ξ )2 η(X). It follows that 3

0 = ∑ ην (ξ )2 η(X) ν=1

for all X ∈ Γ(𝒟). If η(X) ≠ 0 for some X ∈ Γ(𝒟), then we have 0 = η1 (ξ )2 + η2 (ξ )2 + η3 (ξ )2 and hence ξ ∈ Γ(𝒟). From Theorem 4.5.12 we see that the quaternionic dimension k of G2 (ℂ2+k ) is even, say k = 2l, and M is an open part of the tube with radius r ∈ (0, π4 ) around a totally geodesic quaternionic projective space ℍP l in G2 (ℂ2+2l ). As we have seen above, these real hypersurfaces do not have parallel Ricci tensor. It follows that η(X) = 0 for all X ∈ Γ(𝒟), which implies 𝒟 ⊂ 𝒞 and hence ξ ∈ Γ(𝒟⊥ ). Thus we can choose a local canonical basis J1 , J2 , J3 so that ξ = ξ1 (locally). Then we have η1 (ϕX) = g(ϕX, ξ1 ) = g(ϕX, ξ ) = −g(X, ϕX) = 0,

η2 (ϕX) = g(ϕX, ξ2 ) = −g(X, ϕξ2 ) = −g(X, ϕ2 ξ ) = −g(X, ϕ2 ξ1 )

4.8 The Ricci tensor of Hopf hypersurfaces | 137

= g(X, ξ3 ) = η3 (X),

η3 (ϕX) = g(ϕX, ξ3 ) = −g(X, ϕξ3 ) = −g(X, ϕ3 ξ ) = −g(X, ϕ3 ξ1 ) = −g(X, ξ2 ) = −η2 (X)

for all X ∈ X(M). Moreover, from (4.2) we get ϕϕ1 = ϕ1 ϕ. From (4.10) we obtain −g(AϕAX, Y) + g(ϕX, Y) + g(ϕ1 X, Y) = 2η2 (X)η3 (Y) − 2η2 (Y)η3 (X), or equivalently, AϕAX = ϕX + ϕ1 X + 2η3 (X)ξ2 − 2η2 (X)ξ3 . This implies A2 ϕAX = AϕX + Aϕ1 X + 2η3 (X)Aξ2 − 2η2 (X)Aξ3 . On the other hand, (4.32) becomes (note that α = 0 now) 0 = −3ϕAX − hAϕAX + A2 ϕAX + 3ϕ1 AX − 4η3 (AX)ξ2 + 4η2 (AX)ξ3 = −3ϕAX + 3ϕ1 AX − 4η3 (AX)ξ2 + 4η2 (AX)ξ3 − hϕX − hϕ1 X − 2hη3 (X)ξ2 + 2hη2 (X)ξ3

+ AϕX + Aϕ1 X + 2η3 (X)Aξ2 − 2η2 (X)Aξ3 .

Inserting X = ξ2 gives 0 = −3ϕAξ2 + 3ϕ1 Aξ2 − 4η3 (Aξ2 )ξ2 + 4η2 (Aξ2 )ξ3 + 2hξ3 − 2Aξ3 , and inserting X = ξ3 gives 0 = −3ϕAξ3 + 3ϕ1 Aξ3 − 4η3 (Aξ3 )ξ2 + 4η2 (Aξ3 )ξ3 − 2hξ2 + 2Aξ2 . Taking inner product of the previous two equations with Y ∈ Γ(𝒟) then leads to 0 = 3g(Aξ2 , ϕY) − 3g(Aξ2 , ϕ1 Y) − 2g(Aξ3 , Y) and 0 = 3g(Aξ3 , ϕY) − 3g(Aξ3 , ϕ1 Y) + 2g(Aξ2 , Y).

138 | 4 Real hypersurfaces in complex 2-plane Grassmannians Now, 2g(Aξ3 , Y) = 3g(Aξ2 , ϕY) − 3g(Aξ2 , ϕ1 Y) implies 2g(Aξ3 , ϕY) = 3g(Aξ2 , ϕ2 Y) − 3g(Aξ2 , ϕ1 ϕY) = −3g(Aξ2 , Y) − 3g(Aξ2 , ϕ1 ϕY) and 2g(Aξ3 , ϕ1 Y) = 3g(Aξ2 , ϕϕ1 Y) − 3g(Aξ2 , ϕ21 Y) = 3g(Aξ2 , ϕϕ1 Y) + 3g(Aξ2 , Y). Then we obtain 0 = 6g(Aξ3 , ϕY) − 6g(Aξ3 , ϕ1 Y) + 4g(Aξ2 , Y)

= −9g(Aξ2 , Y) − 9g(Aξ2 , ϕ1 ϕY) − 9g(Aξ2 , ϕϕ1 Y) − 9g(Aξ2 , Y) + 4g(Aξ2 , Y) = −14g(Aξ2 , Y) − 18g(Aξ2 , ϕ1 ϕY).

Analogously, 2g(Aξ2 , Y) = −3g(Aξ3 , ϕY) + 3g(Aξ3 , ϕ1 Y) implies 2g(Aξ2 , ϕY) = −3g(Aξ3 , ϕ2 Y) + 3g(Aξ3 , ϕ1 ϕY) = 3g(Aξ3 , Y) + 3g(Aξ3 , ϕ1 ϕY) and 2g(Aξ2 , ϕ1 Y) = −3g(Aξ3 , ϕϕ1 Y) + 3g(Aξ3 , ϕ21 Y) = −3g(Aξ3 , ϕϕ1 Y) − 3g(Aξ3 , Y). Then we obtain 0 = 6g(Aξ2 , ϕY) − 6g(Aξ2 , ϕ1 Y) − 4g(Aξ3 , Y)

= 9g(Aξ3 , Y) + 9g(Aξ3 , ϕ1 ϕY) + 9g(Aξ3 , ϕϕ1 Y) + 9g(Aξ3 , Y) − 4g(Aξ3 , Y) = 14g(Aξ3 , Y) + 18g(Aξ3 , ϕ1 ϕY).

Altogether we have proved 7g(Aξμ , Y) = −9g(Aξμ , ϕ1 ϕY)

for μ ∈ {2, 3} and Y ∈ Γ(𝒟).

(4.33)

On the other hand, taking the covariant derivative of ξ = ξ1 , we get ϕAX = ∇X ξ = ∇X ξ1 = q3 (X)ξ2 − q2 (X)ξ3 + ϕ1 AX.

(4.34)

4.8 The Ricci tensor of Hopf hypersurfaces | 139

Then, by taking the inner product of (4.34) with ξ2 and ξ3 , respectively, we obtain q2 (X) = 2g(AX, ξ2 )

and q3 (X) = 2g(AX, ξ3 ).

From this, (4.34) becomes ϕAX = 2g(AX, ξ3 )ξ2 − 2g(AX, ξ2 )ξ3 + ϕ1 AX.

(4.35)

Applying the structure tensor ϕ to (4.35) and using ϕ2 X = −X + η(X)ξ , it follows that ϕϕ1 AX = −AX + η(AX)ξ + 2g(AX, ξ2 )ξ2 + 2g(AX, ξ3 )ξ3 .

(4.36)

Substituting X = ξμ with μ ∈ {2, 3} in (4.36) and taking the inner product with Y ∈ Γ(𝒟) gives g(ϕϕ1 Aξμ , Y) = −g(Aξμ , Y). Thus, (4.33) becomes 2g(Aξμ , Y) = 0

for μ ∈ {2, 3} and Y ∈ Γ(𝒟),

which implies that the maximal quaternionic subbundle 𝒟 is invariant under the shape operator A of M, that is, A𝒟 ⊆ 𝒟. From Theorem 4.5.20 we see that M is an π ) around a totally geodesic G2 (ℂ1+k ) open part of the tube with radius r ∈ (0, 2√ 2

in G2 (ℂ2+k ) with vanishing Hopf principal curvature α = 0. However, according to Proposition 4.5.19, the Hopf principal curvature α = 2√2 cot(2√2r) never vanishes. Consequently, the case α = 0 is not possible. Altogether we have now proved the following. Theorem 4.8.1 ([96]). There are no Hopf hypersurfaces with parallel Ricci tensor in G2 (ℂ2+k ), k ≥ 3. Since every Einstein manifold has parallel Ricci tensor, Theorem 4.8.1 implies the following. Corollary 4.8.2. There are no Einstein Hopf hypersurfaces in G2 (ℂ2+k ), k ≥ 3. The Ricci tensor of M is said to be ϕ-invariant if Ric ϕ = ϕ Ric. It follows immediately from the definition that the Reeb vector field ξ of a real hypersurface with a ϕ-invariant Ricci tensor in G2 (ℂ2+k ) is an eigenvector of the Ricci tensor everywhere. Taking the covariant derivative of the identity Ric ϕ = ϕ Ric implies (∇X Ric)ϕY + Ric((∇X ϕ)Y) = (∇X ϕ)Ric(Y) + ϕ(∇X Ric)Y

140 | 4 Real hypersurfaces in complex 2-plane Grassmannians for all X, Y ∈ X(M). Using (4.30), this can be rearranged as follows: − 3g(ϕAX, ϕY)ξ + dh(X)AϕY + h(∇X A)ϕY − (∇X A2 )ϕY + η(Y)Ric(AX) − g(AX, Y)Ric(ξ ) 3

− ∑ (3g(ϕν AX, ϕY)ξν − η(AX)ην (ϕ2 Y)ξν ν=1

+ η(AX)ην (ϕY)ϕν ξ − g(ϕν ϕAX, ϕY)ϕν ξ

+ 3ην (ϕY)ϕν AX + ην (ϕ2 Y)ϕν ϕAX − 2ην (ϕAX)ϕν ϕ2 Y)

= η(Ric(Y))AX − g(AX, Ric(Y))ξ

− 3η(Y)ϕ2 AX + dh(X)ϕAY + hϕ(∇X A)Y − ϕ(∇X A2 )Y 3

− ∑ (3g(ϕν AX, Y)ϕξν + 2ην (ϕAX)η(Y)ϕξν ν=1

− η(AX)ην (ϕY)ϕξν + η(AX)ην (Y)ϕϕν ξ − g(ϕν ϕAX, Y)ϕϕν ξ + 3ην (Y)ϕϕν AX + ην (ϕY)ϕϕν ϕAX − 2ην (ϕAX)ϕϕν ϕY).

(4.37)

Inserting X = Y = ξ into (4.37) and assuming that M is Hopf, that is, Aξ = αξ , we obtain 3

hdα(ξ )ξ − 4α ∑ ην (ξ )ϕ2 ξν = 0. ν=1

From this, by taking inner product with ξ , we obtain hdα(ξ ) = 0 and thus 3

α ∑ ην (ξ )(ξν − ην (ξ )ξ ) = 0. ν=1

In order to show that ξ is tangent to 𝒟 or to 𝒟⊥ everywhere, we write ξ = X1 + X2 with X1 ∈ Γ(𝒟) and X2 ∈ Γ(𝒟⊥ ). It then follows that 3

3

α ∑ ην (ξ )(ξν − ην (ξ )X2 ) − α ∑ ην (ξ )ην (ξ )X1 = 0. ν=1 ν=1 ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟ ∈Γ(𝒟 ⊥ )

(4.38)

∈Γ(𝒟)

By taking the inner product of (4.38) with X1 , we obtain 3

α ∑ ην (ξ )2 = 0. ν=1

If α ≠ 0, this implies ην (ξ ) = 0 for all ν ∈ {1, 2, 3} and it follows that ξ ∈ Γ(𝒟). If α = 0, then ξ ∈ Γ(𝒟) or ξ ∈ Γ(𝒟⊥ ) by Corollary 4.4.2. We thus have proved the following.

4.8 The Ricci tensor of Hopf hypersurfaces | 141

Lemma 4.8.3. Let M be a Hopf hypersurface with ϕ-invariant Ricci tensor in the complex 2-plane Grassmannian G2 (ℂ2+k ), k ≥ 3. Then the Reeb vector field ξ is tangent to 𝒟 or to 𝒟⊥ everywhere. If ξ ∈ Γ(𝒟), we can apply Proposition 4.4.3 and Theorem 4.5.11. We conclude that the quaternionic dimension k of G2 (ℂ2+k ) is even, say k = 2l, and M is an open part of the tube with radius r ∈ (0, π4 ) around a totally geodesic quaternionic projective space ℍP l in G2 (ℂ2+2l ). However, an explicit computation shows that for none of these tubes the Ricci tensor is ϕ-invariant. If ξ ∈ Γ(𝒟⊥ ), it was shown in [94] that A𝒟 ⊆ 𝒟. We can then apply Theorem 4.5.20 to conclude that M is an open part of the tube with π radius r ∈ (0, 2√ ) around a totally geodesic G2 (ℂ1+k ) in G2 (ℂ2+k ). We proved in Propo2 sition 4.3.3 that any such tube has ϕ-invariant Ricci tensor. Altogether, we conclude the following. Theorem 4.8.4 ([94]). Let M be a connected Hopf hypersurface with ϕ-invariant Ricci π tensor in G2 (ℂ2+k ), k ≥ 3. Then M is an open part of the tube with radius r ∈ (0, 2√ ) 2 around a totally geodesic G2 (ℂ1+k ) in G2 (ℂ2+k ).

We now discuss Hopf hypersurfaces with harmonic curvature in G2 (ℂ2+k ), k ≥ 3. Recall from (1.3) that M has harmonic curvature if and only if (∇X Ric)Y = (∇Y Ric)X holds for all X, Y ∈ X(M). We already computed (∇X Ric)Y in (4.30). Using this expression, the identity (∇X Ric)Y = (∇Y Ric)X was investigated thoroughly in [98] under some additional assumptions, leading to the following partial classification. Theorem 4.8.5 ([98]). Let M be a Hopf hypersurface with harmonic curvature and constant mean curvature in G2 (ℂ2+k ), k ≥ 3. If Aϕ = ϕA holds on 𝒟⊥ , then M is an open π ) around a totally geodesic G2 (ℂ1+k ) in G2 (ℂ2+k ), part of the tube with radius r ∈ (0, 2√ 2 where r satisfies 3 cot2 (√2r) = 4(k − 1). If dim(M) ≥ 4, M has harmonic curvature if and only if M has harmonic Weyl tensor and constant scalar curvature. From Theorem 4.8.5 we therefore obtain the following. Corollary 4.8.6 ([98]). Let M be a Hopf hypersurface with harmonic Weyl tensor in G2 (ℂ2+k ), k ≥ 3. Assume that M has constant scalar curvature and constant mean curπ vature. If Aϕ = ϕA holds on 𝒟⊥ , then M is an open part of the tube with radius r ∈ (0, 2√ ) 2

around a totally geodesic G2 (ℂ1+k ) in G2 (ℂ2+k ), where r satisfies 3 cot2 (√2r) = 4(k − 1).

Let M be a Hopf hypersurface in G2 (ℂ2+k ). When the Ricci tensor Ric and the structure tensor ϕ satisfy Ric ∘ ϕ + ϕ ∘ Ric = 2cϕ,

c ∈ ℝ \ {0},

(4.39)

142 | 4 Real hypersurfaces in complex 2-plane Grassmannians then the Ricci tensor of M is said to be pseudo-ϕ-anti-invariant. If c = 0 in the previous equation, then the Ricci tensor of M is said to be ϕ-anti-invariant. Recall from Lemma 3.3.11 that (4.39) can be motivated by Ricci solitons. The equation can also be motivated from well-known results about Einstein and pseudo-Einstein real hypersurfaces in complex space forms. Such hypersurfaces were investigated and classified by Kon [61] for non-negative holomorphic sectional curvature and by Cecil and Ryan [31] for negative holomorphic sectional curvature. Any such hypersurface has a pseudo-ϕ-anti-invariant Ricci tensor (see Yano and Kon [123]). It can be easily verified that a tube of radius r ∈ (0, π4 ) around a totally geodesic quaternionic projective space ℍP l in G2 (ℂ2+2l ) has pseudo-ϕ-anti-invariant Ricci tensor (see Berndt and Suh [14] and Pérez, Suh and Watanabe [83]). The converse problem has been considered by Jeong and Suh in [51]. They proved the following. Theorem 4.8.7 ([51]). Let M be a Hopf hypersurface with pseudo-ϕ-anti-invariant Ricci tensor in G2 (ℂ2+k ), k ≥ 3. Then one of the following statements holds: (i) ξ ∈ Γ(𝒟), k = 2l is even and M is locally congruent to a tube with radius r ∈ (0, π4 ) around the totally geodesic quaternionic projective space ℍP l in G2 (ℂ2+2l ); (ii) ξ ∈ Γ(𝒟⊥ ) and c = 4k+2+ α2 (h−α), where α = g(Aξ , ξ ) is the Hopf principal curvature and h is the mean curvature of M. We do not know if there exists a Hopf hypersurface with pseudo-ϕ-anti-invariant Ricci tensor in G2 (ℂ2+k ) satisfying condition (i). One can easily verify that the tubes around a totally geodesic G2 (ℂ1+k ) in G2 (ℂ2+k ) do not satisfy condition (i). Let M be a Hopf hypersurface in G2 (ℂ2+k ) and assume that M is a Ricci soliton with potential field ξ , that is, 1 Ric = cg − ℒξ g. 2 From Lemma 3.3.11 we know that Ric(ϕX) + ϕRic(X) = 2cϕX, and thus M has pseudo-ϕ-anti-invariant Ricci tensor. It follows from Theorem 4.8.7 that ξ ∈ Γ(𝒟) or ξ ∈ Γ(𝒟⊥ ). In [51], Jeong and Suh proved that the constant c satisfies c = 4(k + 1) + α(h − α) if ξ ∈ Γ(𝒟) and c = 4k + α(h − α) if ξ ∈ Γ(𝒟⊥ ). If ξ ∈ Γ(𝒟⊥ ), we also have c = 4k + 2 + α2 (h − α) from Theorem 4.8.7(ii), which implies α(h − α) = 2c − 8k − 4. Inserting this into c = 4k + α(h − α) gives c = 4k + 2c − 8k − 4 = −4k − 4 + 2c and hence c = 4(k + 1) > 0. A basic computation also shows that the tubes in part (i) of Theorem 4.8.7 are not Ricci solitons with ξ as potential field. Thus we conclude the following.

4.9 The normal Jacobi operator of Hopf hypersurfaces | 143

Corollary 4.8.8 ([51]). Let M be a Hopf hypersurface in G2 (ℂ2+k ), k ≥ 3, and assume that M is a Ricci soliton with ξ as potential field. Then c = 4(k + 1) > 0 and thus (M, g, ξ , c) is a shrinking Ricci soliton. By the results in [38] we know that any shrinking Ricci soliton on a closed manifold has positive scalar curvature. From Corollary 4.8.8 we then obtain the following. Corollary 4.8.9 ([51]). Let M be a closed Hopf hypersurface in G2 (ℂ2+k ), k ≥ 3, and assume that M is a Ricci soliton with ξ as potential field. Then M has positive scalar curvature. Remark 4.8.10. We finish with a few remarks. (1) Pérez, Suh and Watanabe obtained in [83] a classification of pseudo-Einstein real hypersurfaces in G2 (ℂ2+k ). We say that M is a pseudo-Einstein real hypersurface in G2 (ℂ2+k ) if there exist a, b, c ∈ ℝ so that the Ricci tensor Ric of M satisfies 3

Ric(X) = aX + bη(X)ξ + c ∑ ην (X)ξν ν=1

for all X ∈ X(M). However, M is proper pseudo-Einstein when c ≠ 0, and it is not pseudo-ϕ-anti-invariant. (2) We saw in Theorem 4.8.1 that there are no Hopf hypersurfaces with parallel Ricci tensor in G2 (ℂ2+k ). Motivated by this non-existence result, we characterized tubes π ) around the totally geodesic G2 (ℂ1+k ) in G2 (ℂ2+k ) by the with radius r ∈ (0, 2√ 2 property ℒξ Ric = 0 (see [95]). (3) When we consider a ruled real hypersurface MR = Σ∗ × G2 (ℂ1+k ) in G2 (ℂ2+k ), the shape operator satisfies Aξ = βξ2 , Aξ2 = βξ and AX = 0 for any X orthogonal to ξ = ξ1 and ξ2 (see [48]). So the ruled real hypersurface MR is not Hopf. The mean curvature and the Hopf principal curvature vanish. There do not exist any Ricci solitons among ruled real hypersurfaces in G2 (ℂ2+k ).

4.9 The normal Jacobi operator of Hopf hypersurfaces Let M be a real hypersurface in G2 (ℂ2+k ) and K(:= R̄ ζ ) the normal Jacobi operator of M. We already computed the normal Jacobi operator in Theorem 4.1.3. More precisely, we have 3

KY = Y + 3η(Y)ξ + ∑ (3ην (Y)ξν + ην (ξ )η(Y)ξν + ην (ϕY)ϕν ξ − ην (ξ )ϕν ϕY) ν=1

(4.40)

for all Y ∈ X(M). The covariant derivative of the normal Jacobi operator is defined by (∇X K)Y = ∇X KY − K∇X Y

144 | 4 Real hypersurfaces in complex 2-plane Grassmannians for all X, Y ∈ X(M). From (4.40) we then obtain (∇X K)Y = 3(∇X η)(Y)ξ + 3η(Y)∇X ξ 3

+ 3 ∑ ((∇X ην )(Y)ξν + ην (Y)∇X ξν ) ν=1

3

− ∑ (d(ην (ξ ))(X)(ϕν ϕY − η(Y)ξν ) − (∇X ην )(ϕY)ϕν ξ ν=1

− ην ((∇X ϕ)Y)ϕν ξ − ην (ϕY)∇X ϕν ξ

+ ην (ξ ){(∇X ϕν ϕ)Y − (∇X η)(Y)ξν − η(Y)∇X ξν }).

We have d(ην (ξ ))(X) = g(∇X ξν , ξ ) + g(ξν , ∇X ξ )

= qν+2 (X)ην+1 (ξ ) − qν+1 (X)ην+2 (ξ ) + 2g(ϕν AX, ξ ),

and using the basic equations for real hypersurfaces in Section 4.1, a lengthy but straightforward computation leads to (∇X K)Y = 3g(ϕAX, Y)ξ + 3η(Y)ϕAX 3

+ 3 ∑ (g(ϕν AX, Y)ξν + ην (Y)ϕν AX) ν=1

3

+ ∑ (2ην (ϕAX)η(Y)ξν − 2ην (ϕAX)ϕν ϕY − ην (ξ )ην (ϕY)AX ν=1

+ ην (ξ )g(AX, ϕY)ξν − ην (ξ )η(Y)ϕν AX + ην (ξ )g(AX, Y)ϕν ξ + ην (ξ )g(ϕAX, Y)ξν + ην (ξ )η(Y)ϕν AX + g(ϕν AX, ϕY)ϕν ξ + η(Y)ην (AX)ϕν ξ − g(AX, Y)ην (ξ )ϕν ξ + ην (ϕY)ϕν ϕAX − ην (ϕY)g(AX, ξ )ξν + ην (ϕY)η(ξν )AX).

(4.41)

Let M be a Hopf hypersurface in G2 (ℂ2+k ) and assume that the normal Jacobi operator K of M is parallel. Then, putting X = Y = ξ in (4.41) and using our assumption that M is Hopf, the equation becomes 3

4α ∑ ην (ξν )ϕν ξ = 0. ν=1

From this it follows that α=0

or

3

∑ ην (ξ )ϕν ξ = 0.

ν=1

If α = 0, then we have ξ ∈ Γ(𝒟) or ξ ∈ Γ(𝒟⊥ ) by Corollary 4.4.2.

4.9 The normal Jacobi operator of Hopf hypersurfaces | 145

Assume that ∑3ν=1 ην (ξ )ϕν ξ = 0. We decompose ξ into ξ = η(X0 )X0 + η(ξ1 )ξ1 for some unit vector fields X0 ∈ Γ(𝒟) and ξ1 ∈ Γ(𝒟⊥ ). Then we get 0 = η(ξ1 )η(X0 )ϕ1 X0 . This gives that η(X0 ) = 0 or η(ξ1 ) = 0, which means that ξ ∈ Γ(𝒟⊥ ) or ξ ∈ Γ(𝒟). Summing up the above facts, we obtain the following. Lemma 4.9.1 ([47]). Let M be a Hopf hypersurface with parallel normal Jacobi operator in G2 (ℂ2+k ), k ≥ 3. Then the Reeb vector field ξ is either tangent to 𝒟 or to 𝒟⊥ everywhere. Using this lemma, we can prove the following non-existence result. Theorem 4.9.2 ([47]). There are no Hopf hypersurfaces with parallel normal Jacobi operator in G2 (ℂ2+k ), k ≥ 3. We now consider some commuting conditions for the normal Jacobi operator. Firstly, from (4.40), the condition of KA = AK is equivalent to 3η(AX)ξ − 3η(X)Aξ 3

= ∑ (3ην (X)Aξν − ην (ξ )Aϕν ϕX + η(X)ην (ξ )Aξν + ην (ϕX)Aϕν ξ ) ν=1

3

− ∑ (3ην (AX)ξν + ην (ξ )ϕν ϕAX − η(AX)ην (ξ )ξν − ην (ϕAX)ϕν ξ ) ν=1

for all X ∈ X(M). From this, we assert the following. Theorem 4.9.3 ([82]). Let M be a real hypersurface in G2 (ℂ2+k ), k ≥ 3, and assume that A𝒟 ⊆ 𝒟. Then M is a Hopf hypersurface if and only if KA = AK and ξ is tangent to 𝒟 or to 𝒟⊥ everywhere. By virtue of this result, if the normal Jacobi operator K commutes with the structure tensor field ϕ, that is, Kϕ = ϕK, we get the following result. Theorem 4.9.4 ([82]). Let M be real hypersurface in G2 (ℂ2+k ), k ≥ 3, and assume that A𝒟 ⊆ 𝒟. If Kϕ = ϕK and KA = AK, then M is locally congruent to an open part of the π ) around the totally geodesic G2 (ℂ1+k ) in G2 (ℂ2+k ). tube with radius r ∈ (0, 2√ 2 We now present a non-existence result for Hopf hypersurfaces in complex 2-plane Grassmannians whose normal Jacobi operator K is a Codazzi tensor, that is, (∇X K)Y = (∇Y K)X for all X, Y ∈ X(M).

146 | 4 Real hypersurfaces in complex 2-plane Grassmannians Theorem 4.9.5. There are no Hopf hypersurfaces in G2 (ℂ2+k ), k ≥ 3, whose normal Jacobi operator is a Codazzi tensor and for which the orthogonal projection of the Reeb vector field ξ onto 𝒟 or onto 𝒟⊥ is invariant under the shape operator. In fact, from (4.41) the equation (∇X K)Y = (∇Y K)X becomes 3g(ϕAX, Y)ξ + 3η(Y)ϕAX 3

+ 3 ∑ (g(ϕν AX, Y)ξν + ην (Y)ϕν AX) ν=1

3

+ ∑ (2ην (ϕAX)η(Y)ξν − 2ην (ϕAX)ϕν ϕY + g(ϕν AX, ϕY)ϕν ξ ν=1

+ η(Y)ην (AX)ϕν ξ + ην (ϕY)ϕν ϕAX − ην (ϕY)g(AX, ξ )ξν ) = 3g(ϕAY, X)ξ + 3η(X)ϕAY 3

+ 3 ∑ (g(ϕν AY, X)ξν + ην (X)ϕν AY) ν=1

3

+ ∑ (2ην (ϕAY)η(X)ξν − 2ην (ϕAY)ϕν ϕX + g(ϕν AY, ϕX)ϕν ξ ν=1

+ η(X)ην (AY)ϕν ξ + ην (ϕX)ϕν ϕAY − ην (ϕX)g(AY, ξ )ξν )

(4.42)

for all X, Y ∈ X(M). Then, using the assumption that M is Hopf and putting X = ξμ and Y = ξ into (4.42), we get 3

3ϕAξμ + ∑ (5g(ϕν Aξμ , ξ )ξν + 3ην (ξ )ϕν Aξμ + ην (Aξμ )ϕν ξ ) ν=1

3

= 3α ∑ (g(ϕν ξ , ξμ )ξν + ην (ξμ )ϕν ξ ) ν=1

3

+ α ∑ (g(ϕν ξ , ϕξμ )ϕν ξ + η(ξμ )ην (ξ )ϕν ξ − ην (ϕξμ )ξν ). ν=1

(4.43)

Using this formula, we can prove the following. Lemma 4.9.6. Let M be a Hopf hypersurface in G2 (ℂ2+k ), k ≥ 3, whose normal Jacobi operator is a Codazzi tensor. If the orthogonal projection of the Reeb vector field ξ onto 𝒟 or onto 𝒟⊥ is invariant under the shape operator of M, then ξ is tangent to 𝒟 or to 𝒟⊥ everywhere. Proof. When the Hopf principal curvature function α = g(Aξ , ξ ) vanishes on M, we see that the Reeb vector field ξ is tangent to 𝒟 or to 𝒟⊥ by virtue of Corollary 4.4.2. Thus we can assume that α ≠ 0.

4.9 The normal Jacobi operator of Hopf hypersurfaces | 147

Let us assume that ξ = η(X0 )X0 + η(ξ1 )ξ1 for some unit vector field X0 ∈ Γ(𝒟) such that η(X0 ) and η(ξ1 ) are non-zero. By putting μ = 1 in (4.43), we obtain 3

3ϕAξ1 + ∑ (5g(ϕν Aξ1 , ξ )ξν + 3ην (ξ )ϕν Aξ1 + ην (Aξ1 )ϕν ξ ) ν=1 3

= 3α ∑ (g(ϕν ξ , ξ1 )ξν + ην (ξ1 )ϕν ξ ) ν=1

3

+ α ∑ (g(ϕν ξ , ϕξ1 )ϕν ξ + η(ξ1 )ην (ξ )ϕν ξ − ην (ϕξ1 )ξν ). ν=1

(4.44)

By assumption, the orthogonal projection of the Reeb vector field ξ onto 𝒟 or onto 𝒟⊥ is invariant under the shape operator. Then it follows that AX0 = αX0

and

Aξ1 = αξ1 .

Substituting these formulas into (4.44), we obtain 0 = αη(ξ1 )η(X0 )ϕ1 X0 . Since α ≠ 0, η(X0 ) ≠ 0 and η(ξ1 ) ≠ 0, we obtain ϕ1 X0 = 0, which contradicts g(ϕ1 X0 , ϕ1 X0 ) = 1. Thus we must have η(X0 ) = 0 or η(ξ1 ) = 0, that is, the Reeb vector field ξ is tangent to 𝒟 or to 𝒟⊥ everywhere. By Lemma 4.9.6, let us consider the cases ξ ∈ Γ(𝒟⊥ ) and ξ ∈ Γ(𝒟), respectively. Then we obtain the following two lemmas. Lemma 4.9.7. Let M be a Hopf hypersurface in G2 (ℂ2+k ), k ≥ 3, whose normal Jacobi operator is a Codazzi tensor. If ξ ∈ Γ(𝒟⊥ ), then g(A𝒟, 𝒟⊥ ) = 0. Lemma 4.9.8. Let M be a Hopf hypersurface in G2 (ℂ2+k ), k ≥ 3, whose normal Jacobi operator is a Codazzi tensor. If ξ ∈ Γ(𝒟), then g(A𝒟, 𝒟⊥ ) = 0. From Lemma 4.9.7, together with Theorem 4.5.21, we know that any Hopf hypersurface in G2 (ℂk+2 ) whose normal Jacobi operator is a Codazzi tensor and with ξ ∈ Γ(𝒟⊥ ) π is locally congruent to a tube with radius r ∈ (0, 2√ ) around a totally geodesic G2 (ℂ1+k ) 2

in G2 (ℂ2+k ). So, let us check whether a tube around a totally geodesic G2 (ℂ1+k ) in G2 (ℂ2+k ) satisfies (∇X K)Y = (∇Y K)X. In order to do this, let us consider X ∈ Γ(Tλ ) with ‖X‖ = 1 in (4.43). Then it follows that 3λϕX + 3λϕ1 X = 0. Since ϕX = ϕ1 X, we get λϕX = 0. Thus we have λ = 0 and this cannot occur for π r ∈ (0, 2√ ). This makes a contradiction. 2

148 | 4 Real hypersurfaces in complex 2-plane Grassmannians From Lemma 4.9.8, applying Theorem 4.5.21, we know that any Hopf hypersurface in G2 (ℂ2+k ) whose normal Jacobi operator is a Codazzi tensor and with ξ ∈ Γ(𝒟) is congruent to a tube over a totally geodesic ℍP l in G2 (ℂ2+2l ). It is easy to verify that a tube around a totally geodesic ℍP l in G2 (ℂ2+2l ) does not satisfy (∇X K)Y = (∇Y K)X. So we also know that the normal Jacobi operator of a tube around a totally geodesic ℍpl in G2 (ℂ2+2l ) cannot be a Codazzi tensor. Then, using Lemmas 4.9.6, 4.9.7 and 4.9.8 together with the facts mentioned above, we can complete the proof of Theorem 4.9.5.

4.10 The structure Jacobi operator of Hopf hypersurfaces As we know, Jacobi operators on Riemannian manifolds contain important information about the geometry of Riemannian manifolds. A particular important Jacobi operator on a real hypersurface in a Kähler manifold is the Jacobi operator Rξ = R(⋅, ξ )ξ with respect to the Reeb vector field ξ . This Jacobi operator controls the geometry along the Reeb flow and is called structure Jacobi operator. In this section we study the structure Jacobi operator on real hypersurfaces of G2 (ℂ2+k ). Let M be a real hypersurface in G2 (ℂ2+k ). From the Gauss equation (4.5) we obtain the following expression for the structure Jacobi operator: Rξ X = X − η(X)ξ + η(Aξ )AX − η(AX)Aξ 3

+ ∑ (η(X)ην (ξ )ξν − ην (X)ξν − 3η(ϕν X)ϕν ξ − ην (ξ )ϕν ϕX). ν=1

(4.45)

Jeong, Pérez and Suh [50] investigated real hypersurfaces in G2 (ℂ2+k ) for which the structure Jacobi operator Rξ is parallel, that is, ∇X Rξ = 0 for all X ∈ X(M). By using (4.45), the covariant derivative of Rξ is given by (∇X Rξ )Y = ∇X (Rξ Y) − Rξ (∇X Y) = −g(ϕAX, Y)ξ − η(Y)ϕAX + η((∇X A)ξ )AY

+ 2η(AϕAX)AY + η(Aξ )(∇X A)Y − η((∇X A)Y)Aξ

− g(AY, ϕAX)Aξ − η(AY)(∇X A)ξ − η(AY)AϕAX 3

+ ∑ (2η(Y)ην (ϕAX)ξν − g(ϕν AX, Y)ξν ν=1

− ην (Y)ϕν AX − 3g(ϕν AX, ϕY)ϕν ξ

− 3η(Y)ην (AX)ϕν ξ − 3ην (ϕY)ϕν ϕAX

4.10 The structure Jacobi operator of Hopf hypersurfaces | 149

+ 3ην (ϕY)η(AX)ξν − 4ην (ξ )ην (ϕY)AX

+ 4ην (ξ )g(AX, Y)ϕν ξ − 2ην (ϕAX)ϕν ϕY)

(4.46)

for all X, Y ∈ X(M). Therefore, the structure Jacobi operator Rξ is parallel if and only if 0 = −g(ϕAX, Y)ξ − η(Y)ϕAX + η((∇X A)ξ )AY

+ 2η(AϕAX)AY + η(Aξ )(∇X A)Y − η(∇X A)Y)Aξ

− g(AY, ϕAX)Aξ − η(AY)(∇X A)ξ − η(AY)AϕAX 3

+ ∑ (2η(Y)ην (ϕAX)ξν − g(ϕν AX, Y)ξν ν=1

− ην (Y)ϕν AX − 3g(ϕν AX, ϕY)ϕν ξ

− 3η(Y)ην (AX)ϕν ξ − 3ην (ϕY)ϕν ϕAX

+ 3ην (ϕY)η(AX)ξν − 4ην (ξ )ην (ϕY)AX

+ 4ην (ξ )g(AX, Y)ϕν ξ − 2ην (ϕAX)ϕν ϕY).

A thorough investigation of this equation leads to the following non-existence result. Lemma 4.10.1 ([50]). There are no Hopf hypersurfaces with parallel structure Jacobi operator in G2 (ℂ2+k ), k ≥ 3, if the orthogonal projection of the Reeb vector field ξ onto 𝒟 or onto 𝒟⊥ is invariant under the shape operator. On the other hand, in [50] it was proved that for Hopf hypersurfaces the orthogonal projection of the Reeb vector field ξ onto 𝒟 or onto 𝒟⊥ is invariant under the shape operator if the Hopf principal curvature function α = g(Aξ , ξ ) of the Reeb vector field ξ is constant along the Reeb flow. So Lemma 4.10.1 leads to the following theorem. Theorem 4.10.2 ([50]). There are no Hopf hypersurfaces with parallel structure Jacobi operator in G2 (ℂ2+k ), k ≥ 3, if the Hopf principal curvature function α is constant along the Reeb flow. We now consider some commutativity conditions for the structure Jacobi operator. As we know, the complex 2-plane Grassmannian G2 (ℂ2+k ) has two different geometric structures, namely the Kähler structure J and the quaternionic Kähler structure J. These two structures deduce the structure tensor field ϕ on M and the local structure tensor fields ϕν (ν ∈ {1, 2, 3}). We consider the following commutativity conditions for the structure Jacobi operator: Rξ ϕ = ϕRξ and Rξ ϕν = ϕν Rξ

for ν ∈ {1, 2, 3}.

The following classification results were obtained in [115] and [52], respectively.

(4.47)

150 | 4 Real hypersurfaces in complex 2-plane Grassmannians Theorem 4.10.3 ([115]). Let M be a Hopf hypersurface in G2 (ℂ2+k ), k ≥ 3, with nonvanishing Hopf principal curvature function and satisfying (4.47). If the orthogonal projection of the Reeb vector field ξ onto 𝒟 is invariant under the shape operator of M, then π M is congruent to a tube with radius r ∈ (0, 2√ ) around the totally geodesic G2 (ℂ1+k ) in 2

G2 (ℂ2+k ).

Theorem 4.10.4 ([52]). There are no Hopf hypersurfaces in G2 (ℂ2+k ), k ≥ 3, with Rξ ϕν = ϕν Rξ for all ν ∈ {1, 2, 3} if the orthogonal projection of the Reeb vector field ξ onto 𝒟⊥ is invariant under the shape operator. Finally, we consider the condition that the structure Jacobi operator is a Codazzi tensor, that is, (∇X Rξ )Y = (∇Y Rξ )X for all X, Y ∈ X(M). If M is a Hopf hypersurface, the Codazzi condition and (4.46) lead to 0 = (∇X Rξ )Y − (∇Y Rξ )X = g((Aϕ + ϕA)Y, X)ξ + η(X)ϕAY − η(Y)ϕAX

+ η((∇X A)ξ )AY − η((∇Y A)ξ )AX + 2g(AϕAY, X)ξ

+ α((∇X A)Y − (∇Y A)X + η((∇Y A)X)ξ − η((∇X A)Y)ξ

+ η(X)(∇Y A)ξ − η(Y)(∇X A)ξ + η(X)AϕAY − η(Y)AϕAX) 3

+ ∑ (g((Aϕν + ϕν A)Y, X)ξν + ην (X)ϕν AY − ην (Y)ϕν AX ν=1

+ 2η(Y)ην (ϕAX)ξν − 2η(X)ην (ϕAY)ξν + 2ην (ϕAY)ϕν ϕX

− 2ην (ϕAX)ϕν ϕY + 3g(Aϕν ϕY, X)ϕν ξ − 3g(ϕϕν AY, X)ϕν ξ

+ 3η(X)ην (AY)ϕν ξ − 3η(Y)ην (AX)ϕν ξ + 3ην (ϕX)ϕν ϕAY − 3αην (ϕX)η(Y)ξν − 3ην (ϕY)ϕν ϕAX + 3αη(X)ην (ϕY)ξν

+ 4ην (ξ )ην (ϕX)AY − 4ην (ξ )ην (ϕY)AX).

Then, by taking Y = ξ and using the assumption that M is Hopf, it follows that 0 = −ϕAX − dα(ξ )AX − α(∇ξ A)X + 2αdα(ξ )η(X)ξ − αAϕAX 3

+ ∑ (ην (ϕAX)ξν − ην (ξ )ϕν AX − 3ην (AX)ϕν ξ ν=1

+ 4αην (X)ϕν ξ − 4αην (ϕX)ξν + 4αην (ξ )ην (ϕX)ξ ) for all X ∈ X(M), where we have used the formula (∇ξ A)ξ = dα(ξ )ξ derived from Aξ = αξ . From the above equation we can then assert the following.

4.10 The structure Jacobi operator of Hopf hypersurfaces | 151

Lemma 4.10.5 ([49]). Let M be a Hopf hypersurface in G2 (ℂ2+k ), k ≥ 3, whose structure Jacobi operator is a Codazzi tensor. If the orthogonal projection of the Reeb vector field ξ onto 𝒟 or onto 𝒟⊥ is invariant under the shape operator of M, then ξ is tangent to 𝒟 or to 𝒟⊥ everywhere. By using Lemma 4.10.5, we can then prove the following non-existence result. Theorem 4.10.6 ([49]). There are no Hopf hypersurfaces in G2 (ℂ2+k ), k ≥ 3, whose structure Jacobi operator is a Codazzi tensor if the orthogonal projection of the Reeb vector field ξ onto 𝒟 or onto 𝒟⊥ is invariant under the shape operator.

5 Real hypersurfaces in complex hyperbolic 2-plane Grassmannians In this chapter we investigate real hypersurfaces in the complex hyperbolic Grassmann manifold G2∗ (ℂ2+k ) = SU 2,k /S(U2 Uk ), k ≥ 3. We denote by g the Riemannian metric, by ∇̄ the Riemannian connection, by R̄ the Riemannian curvature tensor and by ric the Ricci tensor of G2∗ (ℂ2+k ). When the proofs of results are the same as for the complex Grassmann manifold G2 (ℂ2+k ), we refer to the corresponding proof in Chapter 4. Many proofs, however, require sign changes in parts of equations. For the sake of completeness, we will provide full proofs in this section in these cases.

5.1 Basic equations for real hypersurfaces Let M be a real hypersurface in G2∗ (ℂ2+k ) with unit normal vector field ζ . We denote by g the Riemannian metric, by ∇ the Riemannian connection, by R the Riemannian curvature tensor of M and by A the shape operator of M with respect to the unit normal vector field ζ . The Kähler structure J on G2∗ (ℂ2+k ) induces an almost contact metric structure (ϕ, ξ , η, g) on M. The vector field ξ = −Jζ is the Reeb vector field on M. The maximal holomorphic subbundle of TM is denoted by 𝒞 and the orthogonal complement of 𝒞 in TM is denoted by 𝒞 ⊥ . Thus we have TM = 𝒞 ⊕ 𝒞 ⊥ ,

⊥

𝒞 = ℝξ = ℝJζ .

The quaternionic Kähler structure J on G2∗ (ℂ2+k ) induces further structures on M. Let J1 , J2 , J2 be a local canonical basis of J. Then each of the three almost Hermitian structures Jν induces an almost contact metric structure (ϕν , ξν , ην , g) on M. The maximal quaternionic subbundle of TM is denoted by 𝒟 and the orthogonal complement of 𝒟 in TM is denoted by 𝒟⊥ . Thus we have TM = 𝒟 ⊕ 𝒟⊥ ,

⊥

𝒟 = Jζ .

We define ℂζ = ℝζ ⊕ ℝJζ = ℝζ ⊕ ℝξ , ℍζ = ℝζ ⊕ Jζ , ℍℂζ = ℍζ + ℍJζ = ℍζ + ℍξ . The proofs for the following basic structure equations are exactly the same as for the complex 2-plane Grassmannian in Section 4.1. https://doi.org/10.1515/9783110689839-005

154 | 5 Real hypersurfaces in complex hyperbolic 2-plane Grassmannians Lemma 5.1.1 (Basic structure equations). Let M be a real hypersurface in G2∗ (ℂ2+k ). The almost contact metric structures (ϕ, ξ , η, g) and (ϕν , ξν , ην , g) (ν = 1, 2, 3) satisfy the following relations (with indices modulo 3): ϕξν = ϕν ξ ,

ϕν ξν+1 = ξν+2 = −ϕν+1 ξν , η(ξν ) = ην (ξ ),

η(ϕν X) = ην (ϕX),

ην (ϕν+1 X) = −ην+1 (ϕν X),

ϕϕν X − ϕν ϕX = ην (X)ξ − η(X)ξν ,

(5.1)

∇X ξ = ϕAX,

(5.2)

ϕν ϕν+1 X − ην+1 (X)ξν = ϕν+2 X = −ϕν+1 ϕν X + ην (X)ξν+1 , ∇X ξν = qν+2 (X)ξν+1 − qν+1 (X)ξν+2 + ϕν AX,

(5.3)

(∇X η)Y = g(ϕAX, Y),

(∇X ην )Y = qν+2 (X)ην+1 (Y) − qν+1 (X)ην+2 (Y) + g(ϕν AX, Y),

(∇X ϕ)Y = η(Y)AX − g(AX, Y)ξ ,

(∇X ϕν )Y = qν+2 (X)ϕν+1 Y − qν+1 (X)ϕν+2 Y + ην (Y)AX − g(AX, Y)ξν .

The fundamental equations of submanifold geometry (Theorem 2.1.1) can be rewritten using the explicit expression of the curvature tensor of G2∗ (ℂ2+k ) as given in Theorem 1.4.9. Theorem 5.1.2 (Fundamental structure equations). Let M be a real hypersurface in G2∗ (ℂ2+k ). Then the following equations hold for all X, Y, Z ∈ X(M): Gauss formula: ∇̄X Y = ∇X Y + g(AX, Y)ζ , Weingarten formula: ∇̄X ζ = −AX, Gauss equation: R(X, Y)Z = g(X, Z)Y − g(Y, Z)X + g(ϕX, Z)ϕY − g(ϕY, Z)ϕX + 2g(ϕX, Y)ϕZ 3

− ∑ (g(ϕν Y, Z)ϕν X − g(ϕν X, Z)ϕν Y − 2g(ϕν X, Y)ϕν Z) ν=1

5.1 Basic equations for real hypersurfaces | 155 3

− ∑ (g(ϕν ϕY, Z)ϕν ϕX − g(ϕν ϕX, Z)ϕν ϕY) ν=1 3

− ∑ ην (Z)(η(X)ϕν ϕY − η(Y)ϕν ϕX) ν=1 3

− ∑ (η(Y)g(ϕν ϕX, Z) − η(X)g(ϕν ϕY, Z))ξν ν=1

+ g(AY, Z)AX − g(AX, Z)AY,

(5.4)

Codazzi equation: (∇X A)Y − (∇Y A)X = η(Y)ϕX − η(X)ϕY + 2g(ϕX, Y)ξ 3

− ∑ (ην (X)ϕν Y − ην (Y)ϕν X − 2g(ϕν X, Y)ξν ) ν=1 3

− ∑ (ην (ϕX)ϕν ϕY − ην (ϕY)ϕν ϕX) ν=1 3

− ∑ (η(X)ην (ϕY) − η(Y)ην (ϕX))ξν . ν=1

(5.5)

For the normal Jacobi operator K = R̄ ζ we obtain the following. Theorem 5.1.3. Let M be a real hypersurface in G2∗ (ℂ2+k ). The normal Jacobi operator K of M satisfies 3

KX = −X − 3η(X)ξ − ∑ (3ην (X)ξν + ην (ξ )η(X)ξν + ην (ϕX)ϕν ξ − ην (ξ )ϕν ϕX) ν=1

(5.6)

for all X ∈ X(M). Proof. The proof is entirely analogous to the proof of Theorem 4.1.3. It also follows directly from Theorem 4.1.3 since the Riemannian curvature tensor of G2∗ (ℂ2+k ) is the negative of the Riemannian curvature tensor of G2 (ℂ2+k ). Proposition 5.1.4. Let M be a real hypersurface in G2∗ (ℂ2+k ), k ≥ 3, and p ∈ M. The following statements are equivalent: (i) Kξp = κξp for some κ ∈ ℝ; (ii) ξp ∈ 𝒟p or ξp ∈ 𝒟p⊥ ; (iii) the unit normal vector ζp is a singular tangent vector of G2∗ (ℂ2+k ).

156 | 5 Real hypersurfaces in complex hyperbolic 2-plane Grassmannians Proof. Inserting X = ξ into (5.6) gives Kξ = −4ξ − 4η1 (ξ )ξ1 − 4η2 (ξ )ξ2 − 4η3 (ξ )ξ3 . This implies that ξp is an eigenvector of K if and only if ξp ∈ 𝒟p or ξp ∈ 𝒟p⊥ . We have ξp ∈ 𝒟p if and only if Jζp ⊥ Jζp and ξp ∈ 𝒟p⊥ if and only if Jζp ∈ Jζp . The assertion then follows from the description of the singular tangent vectors of G2∗ (ℂ2+k ) in Section 1.4.5.

Theorem 5.1.5. Let M be a real hypersurface in G2∗ (ℂ2+k ). The structure Jacobi operator Rξ of M satisfies 3

3

ν=1

ν=1

Rξ X = η(X)ξ − X + 3 ∑ η(ϕν X)ϕν ξ + ∑ ην (ξ )ϕν ϕX 3

− ∑ (η(X)η(ξν ) − ην (X))ξν + η(Aξ )AX − η(AX)Aξ ν=1

(5.7)

for all X ∈ X(M). Proof. Inserting Y = Z = ξ into the Gauss equation (5.4) gives R(X, ξ )ξ = g(X, ξ )ξ − g(ξ , ξ )X + −g(ϕX, ξ )ϕξ − g(ϕξ , ξ )ϕX + 2g(ϕX, ξ )ϕξ 3

− ∑ (g(ϕν ξ , ξ )ϕν X − g(ϕν X, ξ )ϕν ξ − 2g(ϕν X, ξ )ϕν ξ ) ν=1 3

− ∑ (g(ϕν ϕξ , ξ )ϕν ϕX − g(ϕν ϕX, ξ )ϕν ϕξ ) ν=1 3

− ∑ ην (ξ )(η(X)ϕν ϕξ − η(ξ )ϕν ϕX) ν=1 3

− ∑ (η(ξ )g(ϕν ϕX, ξ ) − η(X)g(ϕν ϕξ , ξ ))ξν ν=1

+ g(Aξ , ξ )AX − g(AX, ξ )Aξ . We have g(ξ , ξ ) = η(ξ ) = 1, g(X, ξ ) = η(X), ϕξ = 0, g(ϕν ξ , ξ ) = 0. Thus the previous equation simplifies to 3

3

3

ν=1

ν=1

ν=1

R(X, ξ )ξ = η(X)ξ − X + 3 ∑ η(ϕν X)ϕν ξ + ∑ ην (ξ )ϕν ϕX − ∑ η(ϕν ϕX)ξν + η(Aξ )AX − η(AX)Aξ . From (5.1) we have ϕν ϕX = ϕϕν X − ην (X)ξ + η(X)ξν

5.1 Basic equations for real hypersurfaces | 157

and hence η(ϕν ϕX) = η(X)η(ξν ) − ην (X). Inserting this into the previous equation leads to (5.7). For the Ricci tensor of M we obtain the following. Theorem 5.1.6. Let M be a real hypersurface in G2∗ (ℂ2+k ). The Ricci tensor of M satisfies ric(X, Y) = −4(k + 2)g(X, Y) − g(KX, Y) + tr(A)g(AX, Y) − g(A2 X, Y),

(5.8)

or equivalently, Ric(X) = −4(k + 2)X − KX + tr(A)AX − A2 X for all X ∈ X(M). Proof. Let E1 , . . . , E4k be a local orthonormal frame field of G2∗ (ℂ2+k ) with E4k = ζ . The Gauss equation implies 4k−1

ric(X, Y) = ∑ g(R(X, Eν )Eν , Y) ν=1

4k−1

̄ Eν )Eν , Y) = ∑ g(R(X, ν=1

4k−1

4k−1

ν=1

ν=1

+ ∑ g(AEν , Eν )g(AX, Y) − ∑ g(AX, Eν )g(AEν , Y) for all X, Y ∈ X(M). Using the expression for the Ricci tensor of G2∗ (ℂ2+k ) in Theorem 1.4.9, we get 4k−1

4k

ν=1

ν=1

̄ Eν )Eν , Y) − g(R(X, ̄ E4k )E4k , Y) ̄ Eν )Eν , Y) = ∑ g(R(X, ∑ g(R(X, ̄ ζ )Eζ , Y) = ric(X, Y) − g(R(X, = −4(k + 2)g(X, Y) − g(KX, Y).

We also have 4k−1

∑ g(AEν , Eν )g(AX, Y) = tr(A)g(AX, Y)

ν=1

158 | 5 Real hypersurfaces in complex hyperbolic 2-plane Grassmannians and 4k−1

4k−1

ν=1

ν=1

∑ g(AX, Eν )g(AEν , Y) = ∑ g(AX, Eν )g(AY, Eν ) = g(AX, AY) = g(A2 X, Y).

Altogether this implies the assertion. Contracting equation (5.8) implies the following. Corollary 5.1.7. Let M be a real hypersurface in G2∗ (ℂ2+k ). The scalar curvature s of M is given by s = −4(k + 2)(4k − 2) + tr(A)2 − tr(A2 ).

5.2 Totally geodesic submanifolds The concept of duality between Riemannian symmetric spaces of compact type and of non-compact type preserves Lie triple systems, and hence totally geodesic submanifolds. In Section 4.2 we discussed the maximal totally geodesic submanifolds of G2 (ℂ2+k ). By applying duality we obtain the following classification result. Theorem 5.2.1 ([36, 57]). Let Σ be a maximal totally geodesic submanifold of the complex hyperbolic 2-plane Grassmannian G2∗ (ℂ2+k ) = SU 2,k /S(U2 Uk ), k ≥ 3. Then Σ is congruent to one of the following maximal totally geodesic submanifolds: (i) the complex and quaternionic totally geodesic embedding of the complex hyperbolic 2-plane Grassmannian G2∗ (ℂ1+k ) into G2∗ (ℂ2+k ); (ii) the complex and totally complex totally geodesic embedding of the complex hyperbolic space ℂH k into G2∗ (ℂ2+k ); (iii) the complex and totally complex totally geodesic embedding of the Riemannian product ℂH a × ℂH k−a of complex hyperbolic spaces ℂH a and ℂH k−a into G2∗ (ℂ2+k ), where a ∈ {1, . . . , k − 1}; (iv) the totally real and totally complex totally geodesic embedding of the real hyperbolic 2-plane Grassmannian G2∗ (ℝ2+k ) into G2∗ (ℂ2+k ); (v) (only for k = 2l even) the totally real and quaternionic totally geodesic embedding of the quaternionic hyperbolic space ℍH l into G2 (ℂ2+2l ); (vi) (only for k = 4) the non-standard totally geodesic embedding of the complex hyperbolic plane ℂH 2 into G2∗ (ℂ6 ).

5.3 Homogeneous real hypersurfaces Ivan Solonenko [92] recently classified the cohomogeneity one actions on G2∗ (ℂ2+k ), k ≥ 3, up to orbit equivalence. This naturally leads to the classification of homogeneous real hypersurfaces in the complex hyperbolic 2-plane Grassmannian G2∗ (ℂ2+k ).

5.3 Homogeneous real hypersurfaces | 159

Geometrically, one can divide the homogeneous real hypersurfaces in G2∗ (ℂ2+k ) into two types: (I): There exists a closed subgroup H of SU 2,k such that the action of H on G2∗ (ℂ2+k ) is of cohomogeneity one and the orbits of H form a Riemannian foliation on G2∗ (ℂ2+k ). Every orbit is a homogeneous real hypersurface in G2∗ (ℂ2+k ). (II): There exists a closed subgroup H of SU 2,k such that the action of H on G2∗ (ℂ2+k ) is of cohomogeneity one and there exists exactly one singular orbit F. Every orbit of H different from F is a tube with some radius r ∈ ℝ+ around F and a homogeneous real hypersurface in G2∗ (ℂ2+k ). The homogeneous real hypersurfaces of type (I) were classified by Berndt and Tamaru in [20] in the more general context of irreducible Riemannian symmetric spaces of non-compact type. We now describe the construction of these homogeneous real hypersurfaces for G2∗ (ℂ2+k ). Let g be the Lie algebra of G = SU 2,k and k be the Lie algebra of the isotropy group K = S(U2 Uk ) of G at o ∈ G2∗ (ℂ2+k ). Let B be the Killing form of g and denote by p the orthogonal complement of k in g with respect to B. The resulting decomposition g = k ⊕ p is a Cartan decomposition of g. The Cartan involution θ ∈ Aut(g) on g is given by θ(X) = I2,k XI2,k

−I with I2,k = ( 2 0k,2

02,k ), Ik

where I2 and Ik are the identity (2 × 2)-matrix and (k × k)-matrix, respectively. Then ⟨X, Y⟩ = −B(X, θY) is a positive definite Ad(K)-invariant inner product on g. Its restriction to p induces a Riemannian metric g̃ on G2∗ (ℂ2+k ) = SU 2,k /S(U2 Uk ), which is also known as the Killing metric on G2∗ (ℂ2+k ). Note that g̃ = 8(k + 2)g, where g is the Riemannian metric on G2∗ (ℂ2+k ) that we introduced in Section 1.4.5. Hence the minimum 1 . of the sectional curvature of (G2∗ (ℂ2+k ), g)̃ is − k+2 The Lie algebra k decomposes orthogonally into k = su2 ⊕ suk ⊕ u1 , where u1 is the 1-dimensional center of k. The adjoint action of su2 on p induces the quaternionic Kähler structure J on G2∗ (ℂ2+k ) = SU 2,k /S(U2 Uk ), and the adjoint action of Z=

kiI 1 ( 2 2 + k 0k,2

02,k ) ∈ u1 −2iIk

(5.9)

induces the Kähler structure J on G2∗ (ℂ2+k ). We identify the tangent space To G2∗ (ℂ2+k ) of G2∗ (ℂ2+k ) at o with p in the usual way. Let a be a maximal Abelian subspace of p and a∗ be the dual vector space of a. For each α ∈ a∗ we define gα = {X ∈ g : ad(H)X = α(H)X for all H ∈ a}. If α ≠ 0 and gα ≠ {0}, then α is a restricted root and gα is a restricted root space. Let Σ ⊂ a∗ be the set of restricted roots. For each α ∈ Σ we define Hα ∈ a by α(H) = ⟨Hα , H⟩

160 | 5 Real hypersurfaces in complex hyperbolic 2-plane Grassmannians for all H ∈ a. The root spaces provide a restricted root space decomposition g = g0 ⊕ (⨁ gα ) α∈Σ

of g, where g0 = k0 ⊕ a and k0 ≅ uk−2 ⊕ u1 is the centralizer of a in k. The corresponding restricted root system is of type (BC)2 . We choose a set Λ = {α1 , α2 } of simple roots of Σ such that α1 is the longer root of the two simple roots and denote by Σ+ the resulting set of positive restricted roots. If we write, as usual, α1 = ϵ1 −ϵ2 and α2 = ϵ2 , the positive restricted roots are α1 = ϵ1 − ϵ2 ,

α2 = ϵ2 ,

α1 + α2 = ϵ1 ,

2α2 = 2ϵ2 ,

α1 + 2α2 = ϵ1 + ϵ2 ,

2α1 + 2α2 = 2ϵ1 .

The multiplicities of the restricted roots 2α2 and 2α1 + 2α2 are equal to 1, the multiplicities of α1 and α1 + 2α2 are equal to 2 and the multiplicities of α2 and α1 + α2 are equal to 2k − 4, respectively. We denote by C̄ + (Λ) the closed positive Weyl chamber in a that is determined by Λ. Note that C̄ + (Λ) is the closed cone in a bounded by the half-lines spanned by Hα1 +α2 and Hα1 +2α2 . We define a nilpotent subalgebra n of g by n = ⨁ gλ . λ∈Σ+

Then g = k ⊕ a ⊕ n is an Iwasawa decomposition of g, which induces a corresponding Iwasawa decomposition G = KAN of G. The subalgebra s = a ⊕ n of g is solvable, and the corresponding connected closed subgroup S = AN of G with Lie algebra s is solvable and simply connected and acts simply transitively on G2∗ (ℂ2+k ). In this way we can identify G2∗ (ℂ2+k ) with the solvable Lie group S = AN equipped with a suitable left-invariant Riemannian metric. Let ℓ be a linear line in a. Then the orthogonal complement sℓ = (a ⊕ n) ⊖ ℓ = (a ⊖ ℓ) ⊕ n of ℓ in a ⊕ n is a subalgebra of a ⊕ n of codimension 1. Let Sℓ be the connected Lie subgroup of AN with Lie algebra sℓ . Then the orbits of the action of Sℓ on G2∗ (ℂ2+k ) form a homogeneous foliation Fℓ on G2∗ (ℂ2+k ) of codimension 1. It was proved in [20] that any two leaves of the foliation Fℓ are isometrically congruent to each other. We denote by Mℓ the leaf containing the point o, that is, Mℓ = Sℓ ⋅ o.

5.3 Homogeneous real hypersurfaces | 161

Let i ∈ {1, 2}. For each unit vector ζ ∈ gαi the subspace sζ = a ⊕ (n ⊖ ℝζ ) is a subalgebra of a ⊕ n. Moreover, if ζ , η are two unit vectors in gαi , then there exists an isometry k in the centralizer of a in K o such that Ad(k)(sζ ) = sη (see [20], Lemma 4.1). This implies that for each i ∈ {1, 2} we obtain a congruence class of homogeneous foliations of codimension 1 on G2∗ (ℂ2+k ). More precisely, let αi ∈ Λ, ζ ∈ gαi be a unit vector and Sζ be the connected Lie subgroup of AN with Lie algebra sζ . Then the orbits of the action of Sζ on G2∗ (ℂ2+k ) form a homogeneous foliation Fζ of codimension 1 on G2∗ (ℂ2+k ). If η ∈ gαi is another unit vector, then the induced foliation Fη is isometrically congruent to Fζ under an isometry in the centralizer of a in K o . We denote by Fi a representative of this congruence class of homogeneous foliations of codimension 1 on G2∗ (ℂ2+k ), that is, Fi = Fζ for some unit vector ζ ∈ gαi . Analogously, we denote by Si the corresponding solvable subgroup of AN. The geometry of the leaves of the foliation Fi was investigated in [20]. We denote by Mi the orbit of Si containing o. Then Mi is a minimal homogeneous real hypersurface in G2∗ (ℂ2+k ). Any other two leaves of Fi are isometrically congruent if and only if they have the same distance to the minimal leaf Mi . The following result is a consequence of the main theorem in [20]. Theorem 5.3.1 ([20]). Let M be a homogeneous real hypersurface of type (I) in G2∗ (ℂ2+k ). Then M is isometrically congruent to Mℓ for some ℓ ∈ a or to M1 or M2 or one of its equidistant hypersurfaces. We now turn our attention to the homogeneous real hypersurfaces in G2∗ (ℂ2+k ) of type (II), which are precisely the homogeneous real hypersurfaces with a non-empty focal set. Any such homogeneous real hypersurface can be realized as a tube with radius r ∈ ℝ+ around its focal set. The homogeneous real hypersurfaces with a totally geodesic focal set were classified in [21]: Theorem 5.3.2 ([21]). Let M be a homogeneous real hypersurface in G2∗ (ℂ2+k ), k ≥ 2, with a totally geodesic focal set. Then M is isometrically congruent to one of the following homogeneous real hypersurfaces: (i) the tube with radius r ∈ ℝ+ around the complex and quaternionic totally geodesic embedding of the complex hyperbolic 2-plane Grassmannian G2∗ (ℂ1+k ) into G2∗ (ℂ2+k ); (ii) the tube with radius r ∈ ℝ+ around the complex and totally complex totally geodesic embedding of the complex hyperbolic space ℂH k into G2∗ (ℂ2+k ); (iii) (only if k = 2l is even) the tube with radius r ∈ ℝ+ around the totally real and quaternionic totally geodesic embedding of the quaternionic hyperbolic space ℍH l into G2∗ (ℂ2+2l ). The structure of homogeneous real hypersurfaces with a non-totally geodesic focal set was investigated in [23]. There it was shown that any such homogeneous real hypersurface can be obtained through one of two possible construction methods, namely the canonical extension method or the nilpotent construction method.

162 | 5 Real hypersurfaces in complex hyperbolic 2-plane Grassmannians We describe the first method now in more detail. The Lie algebra g = su2,k has two maximal parabolic subalgebras. For a more explicit description, we first define the reductive subalgebras and l2 = g−α2 ⊕ g0 ⊕ gα2 ≅ ℝ ⊕ su1,k−2

l1 = g−α1 ⊕ g0 ⊕ gα1 ≅ ℝ ⊕ so1,3 of g. Note that we have

g0 = a ⊕ k0 ≅ ℝ2 ⊕ (u1 ⊕ uk−2 ). We then define the nilpotent subalgebras n1 = n ⊖ gα1

and

n2 = n ⊖ gα2

q1 = l1 ⊕ n1

and

q2 = l2 ⊕ n2

of g. Then

are maximal parabolic subalgebras of g. Any parabolic subalgebra of g is conjugate to a subalgebra of q1 or q2 . We denote by L1 , L2 , N1 , N2 , Q1 , Q2 the closed subgroup of G with Lie algebra l1 , l2 , n1 , n2 , q1 , q2 , respectively. The orbit L1 ⋅ o is isometric to ℝ × ℝH 3 and the orbit L2 ⋅ o is isometric to ℝ × ℂH k−2 . Both orbits are totally geodesic submanifolds of G2∗ (ℂ2+k ). Let g1 = so1,3 and g2 = su1,k−2 , and let G1 ≅ SOo1,3 and G2 ≅ SU 1,k−2 be the corresponding closed subgroup of G with Lie algebra g1 , g2 , respectively. We define B1 = G1 ⋅ o = ℝH 3 and B2 = G2 ⋅ o = ℂH k−2 . The totally geodesic submanifolds B1 and B2 are boundary components of the maximal Satake compactification of G2∗ (ℂ2+k ). Now let Hi be a closed subgroup of Gi acting on Bi with cohomogeneity one so that Hi ⋅ o is a homogeneous hypersurface in Bi . Let hi be the Lie algebra of Hi . We then define the subalgebra h̃ i = hi ⊕ ni of qi and denote by H̃ i the corresponding closed subgroup of Qi . Then the orbit M = H̃ i ⋅ o is a homogeneous real hypersurface in G2∗ (ℂ2+k ). This construction method is called the canonical extension method. In our context it says that from every homogeneous hypersurface in ℝH 3 or in ℂH k−2 one can construct a homogeneous real hypersurface in G2∗ (ℂ2+k ) by canonical extension. The nilpotent construction method is more involved and we do not explain it explicitly here since it was recently proved by Solonenko [92] that the nilpotent construction method does not provide any new homogeneous real hypersurfaces in G2∗ (ℂ2+k ).

5.4 Horospheres | 163

Theorem 5.3.3 ([23, 92]). Every homogeneous real hypersurface in G2∗ (ℂ2+k ) with a non-totally geodesic focal set is obtained by canonical extension of a homogeneous hypersurface in the boundary component B1 = ℝH 3 or in the boundary component B2 = ℂH k−2 . The homogeneous hypersurfaces in real hyperbolic spaces and in complex hyperbolic spaces are all classified, which leads to a complete classification of homogeneous real hypersurfaces in G2∗ (ℂ2+k ) with non-totally geodesic focal sets that are obtained by the canonical extension method. For the classification of homogeneous hypersurfaces in real hyperbolic spaces we refer to Theorem 13.5.2 in [9]. The homogeneous real hypersurfaces in complex hyperbolic spaces were classified by Berndt and Tamaru in [22]. In the following two sections we will discuss the geometry of some of these homogeneous real hypersurfaces.

5.4 Horospheres The family Mℓ of homogeneous real hypersurfaces that we constructed in Section 5.3 contains the family of horospheres in G2∗ (ℂ2+k ). We will now discuss more thoroughly the geometry of these horospheres. Let Hℓ ∈ a be a unit vector, that is, ⟨Hℓ , Hℓ ⟩ = 1, such that ℓ = ℝHℓ . We recall from [20] that the shape operator AHℓ of (Mℓ , g)̃ with respect to the Hℓ is the adjoint transformation AHℓ = ad(Hℓ ) restricted to sℓ . It follows that the subspace a ⊖ ℓ is a principal curvature space of (Mℓ , g)̃ with corresponding principal curvature 0, and for each λ ∈ Σ+ the root space gλ is a principal curvature space of (Mℓ , g)̃ with corresponding principal curvature λ(Hℓ ). This implies that for the (constant) mean curvature μℓ of each leaf of (Mℓ , g)̃ we have μℓ =

1 ∑ (dim gλ )λ(Hℓ ). k − 1 λ∈Σ+

We now work this out in more detail. We denote by Mk1 ,k2 (ℂ) the real vector space of all (k1 × k2 )-matrices with complex coefficients and by 0k1 ,k2 the (k1 × k2 )-matrix with all coefficients equal to 0. For a = (a1 , a2 ) ∈ ℝ2 we put a1 0

Δ2,2 (a) = (

0 ). a2

164 | 5 Real hypersurfaces in complex hyperbolic 2-plane Grassmannians Then we have A g = {( ∗ C

C ) : A ∈ u2 , B ∈ uk , tr(A) + tr(B) = 0, C ∈ M2,k (ℂ)} , B

A k = {( 0k,2

02,k ) : A ∈ u2 , B ∈ uk , tr(A) + tr(B) = 0} , B

0 p = {( 2,2 C∗

C ) : C ∈ M2,k (ℂ)} , 0k,k

02,2 { { a = {(Δ2,2 (a) { { 0k−2,2

Δ2,2 (a) 02,2 0k−2,2

02,k−2 } } 02,k−2 ) : a ∈ ℝ2 } . } 0k−2,k−2 }

The two vectors 02,2 e1 = (Δ2,2 (1, 0) 0k−2,2

Δ2,2 (1, 0) 02,2 0k−2,2

02,k−2 02,k−2 ) , 0k−2,k−2

02,2 e2 = (Δ2,2 (0, 1) 0k−2,2

Δ2,2 (0, 1) 02,2 0k−2,2

02,k−2 02,k−2 ) 0k−2,k−2

form a basis for a. We denote by ϵ1 , ϵ2 ∈ a∗ the dual vectors of e1 , e2 . Then the root system Σ, the positive roots Σ+ and the simple roots Λ = {α1 , α2 } are given by Σ = {±ϵ1 ± ϵ2 , ±ϵ1 , ±ϵ2 , ±2ϵ1 , ±2ϵ2 },

Σ+ = {ϵ1 + ϵ2 , ϵ1 − ϵ2 , ϵ1 , ϵ2 , 2ϵ1 , 2ϵ2 }, α1 = ϵ1 − ϵ2 ,

α2 = ϵ2 .

For each λ ∈ Σ we define the corresponding restricted root space pλ in p by pλ = (gλ ⊕ g−λ ) ∩ p. Then we have p0 = a and 0 { { { { 0 { { { { ( { 0 { { {( ( 0 pϵ1 = {( ( v̄ { { ( 1 { { { { .. { { { . { { ̄ v {( k−2

0 0 0 0 0 .. . 0

0 0 0 0 0 .. . 0

0 0 0 0 0 .. . 0

v1 0 0 0 0 .. . 0

⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ .. . ⋅⋅⋅

vk−2 } } } } 0 } } } } } 0 ) } ) } } k−2 0 ) ) : v1 , . . . , vk−2 ∈ ℂ} ≅ ℂ , ) } } 0 ) } } } } .. } } } . } } 0 ) }

5.4 Horospheres | 165

0 { { { { 0 { { { { ( { 0 { {( { (0 pϵ2 = {( (0 { { ( { { { { .. { { { . { { {(0

p2ϵ1

p2ϵ2

pϵ1 −ϵ2

pϵ1 +ϵ2

0 { { { { 0 { { { { (−ix { { { {( ( = {( 0 (0 { { ( { { { { .. { { { . { { 0 {(

0 { { { { 0 { { { { ( { 0 { { {( (0 = {( (0 { { ( { { { { .. { { { . { { {(0

0 { { { { 0 { { { { (0 { { {( { ( = {( z̄ {(0 { ( { { { { .. { { { . { { {(0

0 { { { { 0 { { { { (0 { { { {( ( = {(−z̄ (0 { { ( { { { { .. { { { . { { 0 {(

0 0 0 0 v̄1 .. . v̄k−2

0 0 0 0 0 .. . 0

0 0 0 0 0 .. . 0

0 v1 0 0 0 .. . 0

0 0 0 0 0 .. . 0

ix 0 0 0 0 .. . 0

0 0 0 0 0 .. . 0

0 0 0 0 0 .. . 0

⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ .. . ⋅⋅⋅

0 0 0 −ix 0 .. . 0

0 0 0 0 0 .. . 0

0 ix 0 0 0 .. . 0

0 0 0 0 0 .. . 0

⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ .. . ⋅⋅⋅

0 0 z 0 0 .. . 0 0 0 z 0 0 .. . 0

0 z̄ 0 0 0 .. . 0 0 z̄ 0 0 0 .. . 0

z 0 0 0 0 .. . 0 −z 0 0 0 0 .. . 0

0 0 0 0 0 .. . 0

⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ .. . ⋅⋅⋅

⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ .. . ⋅⋅⋅ 0 0 0 0 0 .. . 0

⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ .. . ⋅⋅⋅

0 } } } } vk−2 } } } } } 0 ) } } ) } ) 0 ) : v , . . . , v ∈ ℂ ≅ ℂk−2 , 1 k−2 } } } 0 ) } ) } } } .. } } } . } } 0 ) }

0 } } } } 0 } } } } } 0) } } ) } ) 0) : x ∈ ℝ ≅ ℝ, } } ) } 0) } } } } .. } } } . } } 0) } 0 } } } } 0 } } } } } 0) } } ) } 0) ) : x ∈ ℝ} ≅ ℝ, } ) } 0) } } } } .. } } } . } } 0) }

0 } } } } 0 } } } } } 0) } } ) } ) 0) : z ∈ ℂ ≅ ℂ, } } } 0) } ) } } } .. } } } . } } 0) }

0 } } } } 0 } } } } } 0) } } ) } ) 0) : z ∈ ℂ ≅ ℂ. } ) } } 0) } } } } .. } } } . } } 0) }

166 | 5 Real hypersurfaces in complex hyperbolic 2-plane Grassmannians For t ∈ [0, π4 ] we define Ht = cos(t)e1 + sin(t)e2 ∈ a and denote by Mt the horosphere which coincides with the orbit SHt ⋅ o. Every horosphere in SU 2,k /S(U2 Uk ) is isometrically congruent to Mt for some t ∈ [0, π4 ], and two horospheres Mt1 and Mt2 are isometrically congruent if and only if t1 = t2 . The principal curvatures of (Mt , g)̃ with respect to Ht are 0 and λ(Ht ) (λ ∈ Σ+ ), and a ⊖ ℝHt and pλ are the corresponding principal curvature spaces. We now take into account our renormalization g̃ = 8(k + 2)g and compute the principal curvatures and corresponding eigenspaces and multiplicities of the horospheres Mt = (Mt , g), which we list in Table 5.1. Table 5.1: Principal curvatures of the horospheres Mt . Principal curvature

Eigenspace

Multiplicity

0 2√2 cos(t) 2√2 sin(t) √2(cos(t) − sin(t)) √2(cos(t) + sin(t)) √2 cos(t) √2 sin(t)

a ⊖ ℝHt p2ϵ1 p2ϵ2 pϵ1 −ϵ2 pϵ1 +ϵ2 pϵ1 pϵ2

1 1 1 2 2 2k − 4 2k − 4

Thus the number of distinct principal curvatures is 7 for k > 2 and 5 for k = 2 unless t ∈ {0, arctan( 21 ), π4 }. In these three cases we get Table 5.2. We now investigate the maximal complex subbundle 𝒞t of TMt . We recall that the complex structure J on p ≅ To G2∗ (ℂ2+k ) is given by JX = ad(Z)X for all X ∈ p, where Z is as in (5.9). In particular, we get JHt = iHt ∈ p2ϵ1 ⊕ p2ϵ2 . The maximal complex subbundle 𝒞t of Mt is invariant under the shape operator of Mt if and only if JHt is a principal curvature vector. Using the above tables and root space descriptions it is easy to see that JHt is a principal curvature vector of Mt if and only if t ∈ {0, π4 }. These two values for t correspond exactly to the boundary of the closed positive Weyl chamber C̄ + (Λ), and therefore to the two types of singular geodesics on G2∗ (ℂ2+k ). The quaternionic Kähler structure J on G2∗ (ℂ2+k ) is determined by the transformations ad(Q) on p with A Q ∈ {( 0k,2

02,k ) : A ∈ su2 } ⊂ k. 0k,k

5.4 Horospheres | 167 Table 5.2: Principal curvatures of the horospheres M0 , Marctan( 1 ) and M π . 2

t

Principal curvature

Eigenspace

Multiplicity

0

0 √2 2√2

ℝe2 ⊕ pϵ2 ⊕ p2ϵ2 pϵ1 ⊕ pϵ1 −ϵ2 ⊕ pϵ1 +ϵ2 p2ϵ1

2k − 2 2k 1

arctan( 21 )

0

ℝ(e1 − 2e2 )

1

π 4

√2 √5 2√2 √5 3√2 √5 4√2 √5

0 1 2

pϵ2 ⊕ pϵ1 −ϵ2

2k − 2

pϵ1 ⊕ p2ϵ2

2k − 3

pϵ1 +ϵ2

2

p2ϵ1

1

ℝ(e1 − e2 ) ⊕ pϵ1 −ϵ2 pϵ1 ⊕ pϵ2

3 4k − 8

p2ϵ1 ⊕ p2ϵ2 ⊕ pϵ1 +ϵ2

4

4

We now investigate the maximal quaternionic subbundle 𝒟t of TMt . For t = 0 we have H0 = e1 and JH0 = p2ϵ1 ⊕ (JH0 ∩ (pϵ1 −ϵ2 ⊕ pϵ1 +ϵ2 )). Using Table 5.2 we see that Je1 is invariant under the shape operator of M0 . This implies that the maximal quaternionic subbundle 𝒟0 of TM0 is invariant under the shape operator of M0 . Next, for t = π4 we have H π = √12 (e1 + e2 ). In this case we get 4

JH π = pϵ1 +ϵ2 ⊕ (JH π ∩ (p2ϵ1 ⊕ p2ϵ2 )), 4

4

which is contained in the 2-eigenspace of the shape operator according to Table 5.2. It follows that the maximal quaternionic subbundle 𝒟 π of TM π is invariant under the 4 4 shape operator of M π . Finally, for 0 < t < π4 we see that 4

JHt ⊂ p2ϵ1 ⊕ p2ϵ2 ⊕ pϵ1 −ϵ2 ⊕ pϵ1 +ϵ2 . We see from Tables 5.1 and 5.2 that the four root spaces we just listed correspond to distinct principal curvatures, and JHt is not equal to the sum of any three of them. We thus conclude that for 0 < t < π4 the maximal quaternionic subbundle of TMt is not invariant under the shape operator of Mt . We finally note that the angle between JHt and JHt is equal to 2t. Therefore the horospheres with a singular point at infinity are characterized by the geometric property that their normal vectors H satisfy JH ∈ JH or JH ⊥ JH. Since horospheres in G2∗ (ℂ2+k ) are homogeneous real hypersurfaces and isometries of G2∗ (ℂ2+k ) preserve angles as well as complex and quaternionic subspaces, it follows that the horospheres

168 | 5 Real hypersurfaces in complex hyperbolic 2-plane Grassmannians with a singular point at infinity can be characterized by the property that JH ∈ JH or JH ⊥ JH for some non-zero normal vector. Thus we have proved the following result. Theorem 5.4.1. Let M be a horosphere in G2∗ (ℂ2+k ), k ≥ 2. The following statements are equivalent: (i) the center of M is a singular point at infinity; (ii) A𝒞 ⊆ 𝒞 , that is, the shape operator of M preserves the maximal complex subbundle 𝒞 of TM; (iii) A𝒟 ⊆ 𝒟, that is, the shape operator of M preserves the maximal quaternionic subbundle 𝒟 of TM. We now study the geometry of the horospheres with singular points at infinity in more detail. From the above computations we obtain the following. Theorem 5.4.2. Let M be a horosphere in G2∗ (ℂ2+k ), k ≥ 2, with singular point at infinity of type Jζ ∈ Jζ . Then the unit normal vector field ζ of M is singular and satisfies Jζ = J1 ζ for some almost Hermitian structure J1 ∈ J (pointwise). Moreover, M has three distinct constant principal curvatures α = 2√2,

λ = √2,

μ=0

with corresponding principal curvature spaces Tα = ℝξ ,

Tλ = (𝒞 ⊖ 𝒟) ⊕ {v ∈ 𝒟 : Jv = J1 v},

Tμ = {v ∈ 𝒟 : Jv = −J1 v}.

The corresponding multiplicities of the principal curvatures are mα = 1,

mλ = 2k,

mμ = 2(k − 1).

From this we can deduce some geometric information about the horosphere in with singular point at infinity of type Jζ ∈ Jζ .

G2∗ (ℂ2+k )

Proposition 5.4.3. Let M be a horosphere in G2∗ (ℂ2+k ), k ≥ 2, with singular point at infinity of type Jζ ∈ Jζ . Then the following statements hold: (i) The mean curvature of M is equal to tr(A) = 2√2(k + 1). (ii) The horosphere M is a Hopf hypersurface, namely Aξ = 2√2ξ . (iii) The maximal holomorphic subbundle 𝒞 of TM and the maximal quaternionic subbundle 𝒟 of TM are invariant under the shape operator of M, that is, A𝒞 ⊆ 𝒞 and A𝒟 ⊆ 𝒟. (iv) The horosphere M is curvature-adapted, that is, AK = KA. (v) The horosphere M has isometric Reeb flow, or equivalently, Aϕ = ϕA.

5.4 Horospheres | 169

(vi) The Reeb vector field ξ is an eigenvector of the normal Jacobi operator K everywhere, namely Kξ = −8ξ . (vii) The normal Jacobi operator K and the structure tensor field ϕ commute, that is, Kϕ = ϕK. (viii) The Ricci tensor Ric and the structure tensor field ϕ commute, that is, Ric ϕ = ϕ Ric. Proof. (i): We can calculate the mean curvature of M using the principal curvatures and their multiplicities in Theorem 5.4.2. (ii): Since Tα = ℝξ , we have Aξ = αξ and thus M is a Hopf hypersurface. (iii): The maximal holomorphic subbundle 𝒞 of TM is invariant under the shape operator of M by (ii). For the maximal quaternionic subbundle 𝒟 of TM we have 𝒟 = (Tλ ⊖ (𝒞 ⊖ 𝒟)) ⊕ Tμ by Theorem 5.4.2, which implies that 𝒟 is invariant under the shape operator of M. (iv): It follows from Lemma 1.4.10(ii) and Theorem 5.4.2 that A and K have the same eigenspaces, and hence AK = KA. (v): From the description of the principal curvature spaces in Theorem 5.4.2 we see that Tλ and Tμ are ϕ-invariant. This implies Aϕ = ϕA and hence M has isometric Reeb flow by Proposition 3.4.1. (vi): By assumption, we have Jζ = J1 ζ . From Lemma 1.4.10(ii) we see that ξ = −Jζ is an eigenvector of K everywhere with corresponding eigenvalue −8. (vii): By assumption, we have Jζ = J1 ζ . From Lemma 1.4.10(ii) we see that the maximal holomorphic subbundle 𝒞 decomposes into 𝒞 = E0 ⊕ E−2 with E0 = {v ∈ 𝒟 : Jv = −J1 v},

E−2 = (𝒞 ⊖ 𝒟) ⊕ {v ∈ 𝒟 : Jv = J1 v}, where 0 and −2 are eigenvalues of K and E0 and E−2 are the corresponding eigenspaces. It is easy to see that both E0 and E−2 are ϕ-invariant. This shows that Kϕ = ϕK. (viii): This is an immediate consequence of Theorem 5.1.6 and parts (v) and (vii) of this proposition. For the horosphere with the other type of singular point at infinity we obtain the following. Theorem 5.4.4. Let M be a horosphere in G2∗ (ℂ2+k ), k ≥ 2, with singular point at infinity of type Jζ ⊥ Jζ . Then the unit normal vector field ζ of M is singular. Moreover, M has three distinct constant principal curvatures α = 2,

λ = 1,

μ=0

with corresponding principal curvature spaces Tα = 𝒞 ⊥ ⊕ 𝒟 ⊥ ,

Tλ = (ℍℂζ )⊥ ,

Tμ = J 𝒟⊥ = J𝒞 ⊥ .

170 | 5 Real hypersurfaces in complex hyperbolic 2-plane Grassmannians The corresponding multiplicities of the principal curvatures are mα = 4,

mλ = 4k − 8,

mμ = 3.

From this we can deduce some geometric information about the horosphere in G2∗ (ℂ2+k ) with singular point at infinity of type Jζ ⊥ Jζ . Proposition 5.4.5. Let M be a horosphere in G2∗ (ℂ2+k ), k ≥ 2, with singular point at infinity of type Jζ ⊥ Jζ . Then the following statements hold: (i) The mean curvature of M is equal to tr(A) = 4k. (ii) The horosphere M is a Hopf hypersurface, namely Aξ = 2ξ . (iii) The maximal holomorphic subbundle 𝒞 of TM and the maximal quaternionic subbundle 𝒟 of TM are invariant under the shape operator of M, that is, A𝒞 ⊆ 𝒞 and A𝒟 ⊆ 𝒟. (iv) The horosphere M is curvature-adapted, that is, AK = KA. (v) The horosphere M is a contact hypersurface and Aϕ + ϕA = 2ϕ. (vi) The Reeb vector field ξ is an eigenvector of the normal Jacobi operator K everywhere, namely Kξ = −4ξ . Proof. (i): We can calculate the mean curvature of M using the principal curvatures and their multiplicities in Theorem 5.4.4. (ii): Since ℝξ ⊂ Tα , we have Aξ = αξ and thus M is a Hopf hypersurface. (iii): The maximal holomorphic subbundle 𝒞 of TM is invariant under the shape operator of M by (ii). For the maximal quaternionic subbundle 𝒟 of TM we have 𝒟 = Tλ ⊕ Tμ by Theorem 5.4.4, which implies that 𝒟 is invariant under the shape operator of M. (iv): It follows from Lemma 1.4.10(i) and Theorem 5.4.4 that A and K have the same eigenspaces, and hence AK = KA. (v): From the description of the principal curvature spaces in Theorem 5.4.4 we compute: For X ∈ Γ(Tα ⊖ 𝒞 ⊥ ) we have ϕAX = 2ϕX

and AϕX = μϕX = 0.

For X ∈ Γ(Tμ ) we have ϕAX = μϕX = 0

and

AϕX = 2ϕX.

This implies (Aϕ + ϕA)X = 2 coth(2r)ϕX for X ∈ Γ((Tα ⊖ 𝒞 ⊥ ) ⊕ Tμ ). For X ∈ Γ(Tλ ) we have ϕAX = λϕX = ϕX

and AϕX = ϕX.

5.5 Tubes around totally geodesic submanifolds | 171

This implies (Aϕ + ϕA)X = 2ϕX for X ∈ Γ(Tλ ). We also have (Aϕ + ϕA)ξ = 0 = 2ϕξ . Altogether this implies Aϕ + ϕA = 2ϕ. From Proposition 3.5.5 we can conclude that M is a contact hypersurface. (vi): From Lemma 1.4.10(i) we see that ξ is an eigenvector of K everywhere, with corresponding eigenvalue −4.

5.5 Tubes around totally geodesic submanifolds We will now investigate the geometry of the tubes around totally geodesic submanifolds that we encountered in Theorem 5.3.2. We start with the tubes around G2∗ (ℂ1+k ) ⊂ G2∗ (ℂ2+k ). Let r ∈ ℝ+ and Σ = G2∗ (ℂ1+k ). We can assume that o ∈ Σ. We have dim(Σ) = 4(k − 1) and hence codim(Σ) = 4. Let ζ ∈ νo Σ be a unit normal vector. Since Σ is complex (with respect to J) and quaternionic (with respect to J), we see that the normal space νo Σ is complex (with respect to J) and quaternionic (with respect to J). Thus there exists an almost Hermitian structure J1 ∈ J such that Jζ = J1 ζ . It follows that ζ is a singular tangent vector of G2∗ (ℂ2+k ) of type Jζ ∈ Jζ . From Lemma 1.4.10 we see that the Jacobi operator R̄ ζ leaves νo Σ and To Σ invariant. Therefore Σ is a curvature-adapted submanifold of G2∗ (ℂ2+k ). We can therefore apply Corollary 2.6.2 to calculate the principal curvatures and corresponding principal curvature spaces of the tube Σr with radius r around Σ. Using the notations of Section 2.6, we have Spec(Kζ ) = {0, −2}, Spec(Aζ ) = {0}, Spec(R̄ ⊥ ζ ) = {−2, −8} and corresponding eigenspaces E0 = {v ∈ To Σ : Jv = −J1 v},

E−2 = {v ∈ To Σ : Jv = J1 v}, T0 = To Σ,

V−2 = ℍζ ⊖ ℂζ ,

V−8 = ℝJζ .

Let γζ be the geodesic in G2∗ (ℂ2+k ) with γζ (0) and γ̇ζ (0) = ζ . It follows from Corollary 2.6.2 that the principal curvatures of Σr at γζ (r) with respect to −γ̇ζ (r) are μ = 0, λ = 2 tan−2 (r) = √2 tanh(√2r), β = cot−2 (r) = √2 coth(√2r) and α = cot−8 (r) =

172 | 5 Real hypersurfaces in complex hyperbolic 2-plane Grassmannians 2√2 coth(2√2r). The corresponding principal curvature spaces are the parallel translates of E0 , E−2 , V−2 and V−8 along γζ from 0 = γζ (0) to p = γζ (r). Since Σr is homogeneous, these conclusions hold at any point of Σr . Thus we have proved the following theorem. Theorem 5.5.1. Let M be the tube with radius r ∈ ℝ+ around the complex and quaternionic totally geodesic embedding of the complex hyperbolic 2-plane Grassmannian G2∗ (ℂ1+k ) into G2∗ (ℂ2+k ). Then the unit normal vector field ζ of M is singular and satisfies Jζ = J1 ζ for some almost Hermitian structure J1 ∈ J (pointwise). Moreover, M has four distinct constant principal curvatures α = 2√2 coth(2√2r),

β = √2 coth(√2r),

λ = √2 tanh(√2r),

μ=0

with corresponding principal curvature spaces Tα = ℝξ ,

Tβ = 𝒟⊥ ⊖ ℝξ = ℍζ ⊖ ℂζ , Tλ = {v ∈ 𝒟 : Jv = J1 v},

Tμ = {v ∈ 𝒟 : Jv = −J1 v}. The corresponding multiplicities of the principal curvatures are mα = 1,

mβ = 2,

mλ = 2(k − 1) = mμ .

From this we can deduce some geometric information about the tubes around the totally geodesic G2∗ (ℂ1+k ) ⊂ G2∗ (ℂ2+k ). Proposition 5.5.2. Let M be the tube with radius r ∈ ℝ+ around the complex and quaternionic totally geodesic embedding of the complex hyperbolic 2-plane Grassmannian G2∗ (ℂ1+k ) into G2∗ (ℂ2+k ). Then the following statements hold: (i) The mean curvature of M is equal to tr(A) = 3√2 coth(√2r) + (2k − 1)√2 tanh(√2r) > 0. (ii) The tube M is a Hopf hypersurface, namely Aξ = 2√2 cot(2√2r)ξ . (iii) The maximal holomorphic subbundle 𝒞 of TM and the maximal quaternionic subbundle 𝒟 of TM are invariant under the shape operator of M, that is, A𝒞 ⊆ 𝒞 and A𝒟 ⊆ 𝒟. (iv) The tube M is curvature-adapted, that is, AK = KA. (v) The tube M has isometric Reeb flow, or equivalently, Aϕ = ϕA. (vi) The Reeb vector field ξ is an eigenvector of the normal Jacobi operator K everywhere, namely Kξ = −8ξ .

5.5 Tubes around totally geodesic submanifolds | 173

(vii) The normal Jacobi operator K and the structure tensor field ϕ commute, that is, Kϕ = ϕK. (viii) The Ricci tensor Ric and the structure tensor field ϕ commute, that is, Ric ϕ = ϕ Ric. Proof. (i): We can calculate the mean curvature of M using the expression of the principal curvatures in Theorem 5.5.1: tr(A) = 2√2 coth(2√2r) + 2√2 coth(√2r) + 2√2(k − 1) tanh(√2r) = √2(coth(√2r) + tanh(√2r)) + 2√2 coth(√2r) + 2√2(k − 1) tanh(√2r) = 3√2 coth(√2r) + (2k − 1)√2 tanh(√2r). (ii): Since Tα = ℝξ , we have Aξ = αξ and thus M is a Hopf hypersurface. (iii): The maximal holomorphic subbundle 𝒞 of TM is invariant under the shape operator of M by (ii). For the maximal quaternionic subbundle 𝒟 of TM we have 𝒟 = Tλ ⊕ Tμ by Theorem 5.5.1, which implies that 𝒟 is invariant under the shape operator of M. (iv): Since Σ = G2∗ (ℂ1+k ) is curvature-adapted in G2∗ (ℂ2+k ), the tubes around it are curvature-adapted by Corollary 2.6.2. (v): From the description of the principal curvature spaces in Theorem 5.5.1 we see that Tβ , Tλ and Tμ are ϕ-invariant. This implies Aϕ = ϕA and hence M has isometric Reeb flow by Proposition 3.4.1. (vi): From Theorem 5.5.1 we know that the unit normal vector ζ is a singular tangent vector of G2∗ (ℂ2+k ) of type Jζ = J1 ζ . From Lemma 1.4.10(ii) we see that ξ is an eigenvector of K everywhere with corresponding eigenvalue −8. (vii): From Theorem 5.5.1 we know that the unit normal vector ζ is a singular tangent vector of G2∗ (ℂ2+k ) of type Jζ = J1 ζ . From Lemma 1.4.10(ii) we see that the maximal holomorphic subbundle 𝒞 decomposes into 𝒞 = E0 ⊕ E−2 with E0 = {v ∈ 𝒟 : Jv = −J1 v},

E−2 = (𝒞 ⊖ 𝒟) ⊕ {v ∈ 𝒟 : Jv = J1 v}, where 0 and −2 are eigenvalues of K and E0 and E−2 are the corresponding eigenspaces. It is easy to see that both E0 and E−2 are ϕ-invariant. This shows that Kϕ = ϕK. (viii): This is an immediate consequence of Theorem 5.1.6 and parts (v) and (vii) of this theorem. We now consider the tubes around ℂH k ⊂ G2∗ (ℂ2+k ). Let r ∈ ℝ+ and Σ = ℂH k . We can assume that o ∈ Σ. We have dim(Σ) = 2k and hence codim(Σ) = 2k. Let ζ ∈ νo Σ be a unit normal vector. Since Σ is complex (with respect to J) and totally complex (with respect to J), we see that the normal space νo Σ is complex (with respect to J) and totally complex (with respect to J). Thus there exists an almost Hermitian structure J1 ∈ J such that Jζ = J1 ζ . It follows that ζ is a singular tangent vector of G2∗ (ℂ2+k ) of

174 | 5 Real hypersurfaces in complex hyperbolic 2-plane Grassmannians type Jζ ∈ Jζ . From Lemma 1.4.10 we see that the Jacobi operator R̄ ζ leaves νo Σ and To Σ invariant. Therefore Σ is a curvature-adapted submanifold of G2∗ (ℂ2+k ). We can therefore apply Corollary 2.6.2 to calculate the principal curvatures and corresponding principal curvature spaces of the tube Σr with radius r around Σ. Using the notations of Section 2.6, we have Spec(Kζ ) = {0, −2}, Spec(Aζ ) = {0}, Spec(R̄ ⊥ ζ ) = {−2, −8} and corresponding eigenspaces E0 = To Σ ⊖ (ℍζ ⊖ ℂζ ) = {v ∈ To G2∗ (ℂ2+k ) ⊖ ℍζ : Jv = −J1 v},

E−2 = ℍζ ⊖ ℂζ ,

T0 = To Σ = {v ∈ To G2∗ (ℂ2+k ) : Jv = −J1 v},

V−2 = νo Σ ⊖ ℂζ = {v ∈ To G2∗ (ℂ2+k ) ⊖ ℍζ : Jv = J1 v},

V−8 = ℝJζ .

Let γζ be the geodesic in G2∗ (ℂ2+k ) with γζ (0) and γ̇ζ (0) = ζ . It follows from Corollary 2.6.2 that the principal curvatures of Σr at γζ (r) with respect to −γ̇ζ (r) are μ = 0, λ = 2 tan−2 (r) = √2 tanh(√2r), β = cot−2 (r) = √2 coth(√2r), α = cot−8 (r) = 2√2 coth(2√2r). The corresponding principal curvature spaces are the parallel translates of E0 , E−2 , V−2 and V−8 along γζ from 0 = γζ (0) to p = γζ (r). Since Σr is homogeneous, these conclusions hold at any point of Σr . Thus we have proved the following theorem. Theorem 5.5.3. Let M be the tube with radius r ∈ ℝ+ around the complex and totally complex totally geodesic embedding of the complex hyperbolic space ℂH k into G2∗ (ℂ2+k ). Then the unit normal vector field ζ of M is singular and satisfies Jζ = J1 ζ for some almost Hermitian structure J1 ∈ J (pointwise). Moreover, M has four distinct constant principal curvatures α = 2√2 coth(2√2r),

β = √2 coth(√2r),

λ = √2 tanh(√2r),

μ=0

with corresponding principal curvature spaces Tα = ℝξ ,

Tλ = 𝒟⊥ ⊖ ℝξ = ℍζ ⊖ ℂζ ,

Tβ = {v ∈ 𝒟 : Jv = J1 v},

Tμ = {v ∈ 𝒟 : Jv = −J1 v}. The corresponding multiplicities of the principal curvatures are mα = 1,

mλ = 2,

mβ = 2(k − 1) = mμ .

From this we can deduce some geometric information about the tubes around the totally geodesic ℂH k ⊂ G2∗ (ℂ2+k ).

5.5 Tubes around totally geodesic submanifolds | 175

Proposition 5.5.4. Let M be the tube with radius r ∈ ℝ+ around the complex and totally complex totally geodesic embedding of the complex hyperbolic space ℂH k into G2∗ (ℂ2+k ). Then the following statements hold: (i) The mean curvature of M is equal to tr(A) = (2k − 1)√2 coth(√2r) + 3√2 tanh(√2r) > 0. (ii) The tube M is a Hopf hypersurface, namely Aξ = 2√2 cot(2√2r)ξ . (iii) The maximal holomorphic subbundle 𝒞 of TM and the maximal quaternionic subbundle 𝒟 of TM are invariant under the shape operator of M, that is, A𝒞 ⊆ 𝒞 and A𝒟 ⊆ 𝒟. (iv) The tube M is curvature-adapted, that is, AK = KA. (v) The tube M has isometric Reeb flow, or equivalently, Aϕ = ϕA. (vi) The Reeb vector field ξ is an eigenvector of the normal Jacobi operator K everywhere, namely Kξ = −8ξ . (vii) The normal Jacobi operator K and the structure tensor field ϕ commute, that is, Kϕ = ϕK. (viii) The Ricci tensor Ric and the structure tensor field ϕ commute, that is, Ric ϕ = ϕ Ric. Proof. (i): We can calculate the mean curvature of M using the expression of the principal curvatures in Theorem 5.5.3: tr(A) = 2√2 coth(2√2r) + 2(k − 1)√2 coth(√2r) + 2√2 tanh(√2r) = √2(coth(√2r) + tanh(√2r)) + 2(k − 1)√2 coth(√2r) + 2√2 tanh(√2r) = (2k − 1)√2 coth(√2r) + 3√2 tanh(√2r). (ii): Since Tα = ℝξ , we have Aξ = αξ and thus M is a Hopf hypersurface. (iii): The maximal holomorphic subbundle 𝒞 of TM is invariant under the shape operator of M by (ii). For the maximal quaternionic subbundle 𝒟 of TM we have 𝒟 = Tβ ⊕ Tμ by Theorem 5.5.3, which implies that 𝒟 is invariant under the shape operator of M. (iv): Since Σ = ℂH k is curvature-adapted in G2∗ (ℂ2+k ), the tubes around it are curvature-adapted by Corollary 2.6.2. (v): From the description of the principal curvature spaces in Theorem 5.5.3 we see that Tλ , Tβ and Tμ are ϕ-invariant. This implies Aϕ = ϕA and hence M has isometric Reeb flow by Proposition 3.4.1. (vi): From Theorem 5.5.3 we know that the unit normal vector ζ is a singular tangent vector of G2∗ (ℂ2+k ) of type Jζ = J1 ζ . From Lemma 1.4.10(ii) we see that ξ is an eigenvector of K everywhere with corresponding eigenvalue −8. (vii): From Theorem 5.5.1 we know that the unit normal vector ζ is a singular tangent vector of G2∗ (ℂ2+k ) of type Jζ = J1 ζ . From Lemma 1.4.10(ii) we see that the maximal

176 | 5 Real hypersurfaces in complex hyperbolic 2-plane Grassmannians holomorphic subbundle 𝒞 decomposes into 𝒞 = E0 ⊕ E−2 with E0 = {v ∈ 𝒟 : Jv = −J1 v},

E−2 = (𝒞 ⊖ 𝒟) ⊕ {v ∈ 𝒟 : Jv = J1 v}, where 0 and −2 are eigenvalues of K and E0 and E−2 are the corresponding eigenspaces. It is easy to see that both E0 and E−2 are ϕ-invariant. This shows that Kϕ = ϕK. (viii): This is an immediate consequence of Theorem 5.1.6 and parts (v) and (vii) of this theorem. We now investigate the geometry of the tubes around the totally geodesic quaternionic hyperbolic space ℍH l in G2∗ (ℂ2+2l ). Let r ∈ ℝ+ and Σ = ℍP l . We can assume that o ∈ Σ. We have dim(Σ) = 4l and hence codim(Σ) = 4l. Let ζ ∈ νo Σ be a unit normal vector. Since Σ is totally real (with respect to J) and quaternionic (with respect to J), we see that the normal space νo Σ is totally real (with respect to J) and quaternionic (with respect to J). More precisely, we have JTo Σ = νo Σ and both To Σ and νo Σ are invariant under J. It follows that ζ is a singular tangent vector of G2∗ (ℂ2+2l ) of type Jζ ⊥ Jζ . It follows from Lemma 1.4.10 that the Jacobi operator R̄ ζ leaves νo Σ and To Σ invariant. Therefore Σ is a curvature-adapted submanifold of G2∗ (ℂ2+2l ). We can therefore apply Corollary 2.6.2 to calculate the principal curvatures and corresponding principal curvature spaces of the tube Σr with radius r around Σ. Using the notations of Section 2.6, we have Spec(Kζ ) = {0, −1, −4}, Spec(Aζ ) = {0}, Spec(R̄ ⊥ ζ ) = {−1, −4} and corresponding eigenspaces E0 = JJζ ,

E−1 = To Σ ⊖ ℍJζ ,

E−4 = ℝJζ , T0 = To Σ,

V−1 = νo Σ ⊖ Jζ ,

V−4 = Jζ .

Let γζ be the geodesic in G2∗ (ℂ2+2l ) with γζ (0) and γ̇ζ (0) = ζ . It follows from Corollary 2.6.2 that the principal curvatures of Σr at γζ (r) with respect to −γ̇ζ (r) are γ = 0, μ = tan−1 (r) = tanh(r), α = tan−4 (r) = 2 tanh(2r), λ = cot−1 (r) = coth(r) and β = cot−4 (r) = 2 coth(2r). The corresponding principal curvature spaces are the parallel translates of E0 , E−1 , E−4 , V−1 and V−4 along γζ from 0 = γζ (0) to p = γζ (r). Since Σr is homogeneous, these conclusions hold at any point of Σr . We conclude the following. Theorem 5.5.5. Let M be the tube with radius r ∈ ℝ+ around the totally real and quaternionic totally geodesic embedding of the quaternionic hyperbolic space ℍH l into G2∗ (ℂ2+2l ). Then the unit normal vector field ζ of M is singular and satisfies Jζ ⊥ Jζ .

5.5 Tubes around totally geodesic submanifolds | 177

Moreover, M has five distinct constant principal curvatures α = 2 tanh(2r),

β = 2 coth(2r),

γ = 0,

λ = coth(r),

μ = tanh(r)

with corresponding principal curvature spaces Tα = ℝξ ,

Tβ = Jζ = ℝξ1 ⊕ ℝξ2 ⊕ ℝξ3 , Tγ = Jξ = ℝϕξ1 ⊕ ℝϕξ2 ⊕ ℝϕξ3 = ℝϕ1 ξ ⊕ ℝϕ2 ξ ⊕ ℝϕ3 ξ , Tλ = Pγζ (νo ℍH l ) ⊖ Jζ ,

Tμ = Pγζ (To ℍH l ) ⊖ ℍξ , where Pγζ denotes the parallel transport along γζ from ℍH l to M. The corresponding multiplicities of the principal curvatures are mα = 1,

mβ = 3 = mγ ,

mλ = 4(l − 1) = mμ .

Furthermore, we have JTβ = Tγ ,

JTλ = Tμ ,

JTλ = Tλ ,

JTμ = Tμ .

From this we can deduce some geometric information about the tubes around the totally geodesic ℍH l ⊂ G2∗ (ℂ2+2l ). Proposition 5.5.6. Let M be the tube with radius r ∈ ℝ+ around the totally real and quaternionic totally geodesic embedding of the quaternionic hyperbolic space ℍH l into G2∗ (ℂ2+2l ). Then the following statements hold: (i) The mean curvature of M is equal to tr(A) = 2 tanh(2r) + (8l − 2) coth(2r). (ii) The tube M is a Hopf hypersurface, namely Aξ = 2 tanh(2r)ξ . (iii) The maximal holomorphic subbundle 𝒞 of TM and the maximal quaternionic subbundle 𝒟 of TM are invariant under the shape operator of M, that is, A𝒞 ⊆ 𝒞 and A𝒟 ⊆ 𝒟. (iv) The tube M is curvature-adapted, that is, AK = KA. (v) The tube M is a contact hypersurface and Aϕ + ϕA = 2 coth(2r)ϕ. (vi) The Reeb vector field ξ is an eigenvector of the normal Jacobi operator K everywhere, namely Kξ = −4ξ . Proof. (i): We can calculate the mean curvature of M using the expression of the principal curvatures in Theorem 5.5.5. Using the identity 2 coth(2r) = coth(r) + tanh(r) we

178 | 5 Real hypersurfaces in complex hyperbolic 2-plane Grassmannians obtain tr(A) = 2 tanh(2r) + 6 coth(2r) + 4(l − 1) coth(r) + 4(l − 1) tanh(r) = 2 tanh(2r) + (8l − 2) coth(2r).

(ii): Since Tα = ℝξ , we have Aξ = αξ and thus M is a Hopf hypersurface. (iii): The maximal holomorphic subbundle 𝒞 of TM is invariant under the shape operator of M by (ii). Furthermore, we have 𝒟⊥ = Tβ by Theorem 5.5.5, which implies that 𝒟 is invariant under the shape operator of M. (iv): Since Σ = ℍH l is curvature-adapted in G2∗ (ℂ2+2l ), the tubes around it are curvature-adapted by Corollary 2.6.2. (v): From the description of the principal curvature spaces in Theorem 5.5.5 we know that ϕTβ = Tγ and ϕTλ = Tμ . For X ∈ Γ(Tβ ) we have ϕAX = βϕX = 2 coth(2r)ϕX

and

AϕX = γϕX = 0.

For X ∈ Γ(Tγ ) we have ϕAX = γϕX = 0

and AϕX = βϕX = 2 coth(2r)ϕX.

This implies (Aϕ + ϕA)X = 2 coth(2r)ϕX for X ∈ Γ(Tβ ⊕ Tγ ). For X ∈ Γ(Tλ ) we have ϕAX = λϕX = coth(r)ϕX

and AϕX = μϕX = tanh(r)ϕX.

For X ∈ Γ(Tμ ) we have ϕAX = μϕX = tanh(r)ϕX

and

AϕX = λϕX = coth(r)ϕX.

Since coth(r) + tanh(r) = 2 coth(2r), this implies (Aϕ + ϕA)X = 2 coth(2r)ϕX for X ∈ Γ(Tλ ⊕ Tμ ). We also have (Aϕ + ϕA)ξ = 0 = 2 coth(2r)ϕξ . Altogether this implies Aϕ + ϕA = 2 coth(2r)ϕ. From Proposition 3.5.5 we can conclude that M is a contact hypersurface. (vi): From Theorem 5.5.5 we know that the unit normal vector ζ is a singular tangent vector of G2∗ (ℂ2+2l ) of type Jζ ⊥ Jζ . From Lemma 1.4.10(i) we see that ξ is an eigenvector of K everywhere, with corresponding eigenvalue −4.

5.6 Hopf hypersurfaces | 179

5.6 Hopf hypersurfaces In this section we deduce some basic equations for Hopf hypersurfaces in G2∗ (ℂ2+k ). Let M be a Hopf hypersurface in G2∗ (ℂ2+k ). Then, by Proposition 3.3.2, the Reeb vector field ξ on M is a principal curvature vector of M everywhere. Thus we have Aξ = αξ with α = g(Aξ , ξ ). Equivalently, the maximal holomorphic subbundle 𝒞 of TM is invariant under A, that is, A𝒞 ⊆ 𝒞 . From the Codazzi equation (5.5) and the basic structure equations in Lemma 5.1.1 we obtain 3

2g(ϕX, Y) − 2 ∑ (ην (X)ην (ϕY) − ην (Y)ην (ϕX) − g(ϕν X, Y)ην (ξ )) ν=1

= g((∇X A)Y − (∇Y A)X, ξ )

= g((∇X A)ξ , Y) − g((∇Y A)ξ , X)

= g(∇X Aξ , Y) − g(A∇X ξ , Y) − g(∇Y Aξ , X) + g(A∇Y ξ , X) = g(∇X αξ , Y) − g(A∇X ξ , Y) − g(∇Y αξ , X) + g(A∇Y ξ , X) = dα(X)η(Y) + αg(ϕAX, Y) − g(AϕAXξ , Y)

− dα(Y)η(X) − αg(ϕAY, X) + g(AϕAY, X)

= dα(X)η(Y) − dα(Y)η(X) + αg((Aϕ + ϕA)X, Y) − 2g(AϕAX, Y) for all X, Y ∈ X(M). Inserting X = ξ implies 3

dα(Y) = dα(ξ )η(Y) + 4 ∑ ην (ξ )ην (ϕY). ν=1

Inserting this and the corresponding expression for dα(X) into the previous equation gives 3

2g(ϕX, Y) − 2 ∑ (ην (X)ην (ϕY) − ην (Y)ην (ϕX) − g(ϕν X, Y)ην (ξ )) 3

ν=1

= 4 ∑ (η(Y)ην (ϕX) − η(X)ην (ϕY))ην (ξ ) + αg((Aϕ + ϕA)X, Y) − 2g(AϕAX, Y). ν=1

From this we easily obtain the following. Lemma 5.6.1. Let M be a Hopf hypersurface in G2∗ (ℂ2+k ). Then we have αg((Aϕ + ϕA)X, Y) − 2g(AϕAX, Y) − 2g(ϕX, Y) 3

= 2 ∑ (ην (Y)ην (ϕX) − ην (X)ην (ϕY) + g(ϕν X, Y)ην (ξ ) ν=1

+ 2η(X)ην (ϕY)ην (ξ ) − 2η(Y)ην (ϕX)ην (ξ ))

(5.10)

180 | 5 Real hypersurfaces in complex hyperbolic 2-plane Grassmannians for all X, Y ∈ X(M) and 3

gradM α = dα(ξ )ξ − 4 ∑ ην (ξ )ϕν ξ . ν=1

(5.11)

Moreover, if X ∈ Γ(𝒞 ) with AX = λX, then (2λ − α)AϕX + (2 − αλ)ϕX 3

= 2 ∑ (2ην (ϕX)ην (ξ )ξ − ην (X)ϕν ξ − ην (ϕX)ξν − ην (ξ )ϕν X). ν=1

(5.12)

Corollary 5.6.2. Let M be a Hopf hypersurface in G2∗ (ℂ2+k ). Then the Hopf principal curvature function α is constant if and only if: (i) dα(ξ ) = 0 (or equivalently, α is constant along the Reeb flow lines on M) and (ii) ξ is tangent to 𝒟 or to 𝒟⊥ everywhere (or equivalently, the unit normal vector field ζ is a singular tangent vector of G2∗ (ℂ2+k ) everywhere). Proof. Assume that α is constant on M. Then gradM α = 0 and dα(ξ ) = 0, and from (5.11) we get 3

0 = ∑ ην (ξ )ϕν ξ . ν=1

We can choose a canonical local basis so that ξ = ξ𝒟 + η1 (ξ )ξ1 with ξ𝒟 ∈ Γ(𝒟) and η1 (ξ ) ≥ 0. Then η2 (ξ ) = 0 = η3 (ξ ) and hence 0 = η1 (ξ )ϕ1 ξ . If η1 (ξ ) = 0, then ξ = ξ𝒟 ∈ Γ(𝒟). If η1 (ξ ) > 0, then ϕ1 ξ = 0 and hence ξ = ξ1 , which implies ξ ∈ Γ(𝒟⊥ ). Conversely, assume that conditions (i) and (ii) are satisfied. If ξ is tangent to 𝒟 everywhere, then ην (ξ ) = 0 and we get gradM α = 0 from (5.11). If ξ is tangent to 𝒟⊥ everywhere, we can choose a local canonical basis so that ξ = ξ1 . Then η2 (ξ ) = 0 = η3 (ξ ) and hence gradM α = η1 (ξ )ϕ1 ξ = η1 (ξ )ϕ1 ξ1 = 0. We now consider Hopf hypersurfaces in G2∗ (ℂ2+k ) for which the Reeb vector field is tangent to 𝒟 everywhere. Proposition 5.6.3 ([99]). Let M be a Hopf hypersurface in G2∗ (ℂ2+k ). If the Reeb vector field ξ is tangent to the maximal quaternionic subbundle 𝒟 of TM everywhere (or equivalently, if the normal vector field ζ is singular of type Jζ ⊥ Jζ everywhere), then A𝒟 ⊆ 𝒟. Proof. Since ξ ∈ Γ(𝒟), we can decompose 𝒟 orthogonally into 𝒟 = 𝒟0 ⊕ ℍξ , where 𝒟0 is the quaternionic subbundle of 𝒟 perpendicular to ℍξ . By assumption, we have g(Aξν , ξ ) = g(ξν , Aξ ) = αg(ξν , ξ ) = 0.

(5.13)

5.6 Hopf hypersurfaces | 181

Next, using (5.2) and (5.3), we compute g(Aξν , ϕμ ξ ) = g(Aξν , ϕξμ ) = −g(ϕAξν , ξμ ) = −g(∇ξν ξ , ξμ ) = g(ξ , ∇ξν ξμ ) = g(ξ , ϕμ Aξν ) = −g(ϕμ ξ , Aξν ),

and hence g(Aξν , ϕμ ξ ) = 0.

(5.14)

From (5.13) and (5.14) we conclude that Aξν ∈ Γ(𝒟0 ) everywhere. Let X ∈ Γ(𝒟0 ). Then (5.10) implies αAϕX + αϕAX − 2AϕAX − 2ϕX = 0. Applying ϕ to this equation leads to αϕAϕX − αAX − 2ϕAϕAX + 2X = 0.

(5.15)

Taking inner product with ξν we obtain αg(AX, ξν ) = αg(ϕAϕX, ξν ) − 2g(ϕAϕAX, ξν ).

(5.16)

Using (5.2) and (5.3) we obtain g(ϕAϕX, ξν ) = g(∇ϕX ξ , ξν ) = −g(ξ , ∇ϕX ξν ) = −g(ξ , ϕν AϕX) = g(ϕν ξ , AϕX) = g(ϕξν , AϕX) = −g(ξν , ϕAϕX), and hence g(ϕAϕX, ξν ) = 0.

(5.17)

Using again (5.2) and (5.3) we obtain g(ϕAϕAX, ξν ) = g(∇ϕAX ξ , ξν ) = −g(ξ , ∇ϕAX ξν ) = −g(ξ , ϕν AϕAX) = g(ϕν ξ , AϕAX) = g(ϕξν , AϕAX) = −g(ξν , ϕAϕAX), and hence g(ϕAϕAX, ξν ) = 0.

(5.18)

From (5.16), (5.17) and (5.18) we conclude that αg(Aξν , X) = 0

(5.19)

182 | 5 Real hypersurfaces in complex hyperbolic 2-plane Grassmannians for all X ∈ Γ(𝒟0 ). If the closure of {p ∈ M : α(p) ≠ 0} is equal to M, the assertion follows from (5.13), (5.14) and (5.19). Assume that there exists an open subset of M on which α = 0. The following arguments are valid on this open subset of M. We continue with X ∈ Γ(𝒟0 ). From (5.15) we then have ϕAϕAX − X = 0 and hence AX = AϕAϕAX.

(5.20)

Taking inner product with ξν , using (5.2) and Aξ = 0, leads to g(AX, ξν ) = g(AϕAϕAX, ξν ) = −g(AϕAX, ϕAξν )

= g(AϕAX, −∇ξν ξ ) = g((∇ξν A)ϕAX, ξ ).

The last term can be investigated further using the Codazzi equation (5.5): (∇ξν A)ϕAX = (∇ϕAX A)ξν − η(ξν )ϕϕAX + η(ϕAX)ϕξν + 2g(ϕξν , ϕAX)ξ 3

− ∑ (ημ (ξν )ϕμ ϕAX − ημ (ϕAX)ϕμ ξν − 2g(ϕμ ξν , ϕAX)ξμ ) μ=1 3

− ∑ (ημ (ϕξν )ϕμ ϕϕAX − ημ (ϕϕAX)ϕμ ϕξν ) μ=1 3

− ∑ (η(ξν )ημ (ϕϕAX) − η(ϕAX)ημ (ϕξν ))ξμ μ=1

= (∇ϕAX A)ξν + 2g(ξν , AX)ξ − ϕν ϕAX 3

+ ∑ (ημ (ϕAX)ϕμ ξν + 2g(ϕμ ξν , ϕAX)ξμ − ημ (AX)ϕμ ϕξν ). μ=1

Taking inner product with ξ then leads to g((∇ξν A)ϕAX, ξ ) = g((∇ϕAX A)ξν , ξ ) + 4g(ξν , AX). Using the assumption that Aξ = 0, we have g((∇ϕAX A)ξν , ξ ) = g(∇ϕAX Aξν , ξ ) = −g(Aξν , ∇ϕAX ξ ) = −g(Aξν , ϕAϕAX). Inserting this into the previous equation gives g((∇ξν A)ϕAX, ξ ) = −g(Aξν , ϕAϕAX) + 4g(ξν , AX). Inserting this into (5.21) implies g(AϕAϕAX, ξν ) = 3g(AX, ξν ).

(5.21)

5.7 Real hypersurfaces with invariant maximal quaternionic subbundle | 183

Using (5.20) we then get g(Aξν , X) = 0 for all X ∈ Γ(𝒟0 ), which implies the assertion also in case α = 0.

5.7 Real hypersurfaces whose maximal quaternionic subbundle is invariant under the shape operator In this section we investigate real hypersurfaces in G2∗ (ℂ2+k ) whose maximal quaternionic subbundle 𝒟 is invariant under the shape operator A of M, that is, A𝒟 ⊆ 𝒟. Let M be a real hypersurface in G2∗ (ℂ2+k ) with A𝒟 ⊆ 𝒟. Then we also have A𝒟⊥ ⊆ 𝒟⊥ . By a suitable choice of the local canonical basis we can assume that ξ1 , ξ2 , ξ3 are principal curvature vectors of M everywhere, say Aξν = βν ξν . From the Codazzi equation (5.5) and the basic structure equations in Lemma 5.1.1 we obtain 2η(Y)ην (ϕX) − 2η(X)ην (ϕY) + 2g(ϕX, Y)ην (ξ ) + 2g(ϕν X, Y) − 2ην+1 (X)ην+2 (Y) + 2ην+1 (Y)ην+2 (X)

− 2ην+1 (ϕX)ην+2 (ϕY) + 2ην+1 (ϕY)ην+2 (ϕX)

= g((∇X A)Y − (∇Y A)X, ξν ) = g((∇X A)ξν , Y) − g((∇Y A)ξν , X) = g(∇X Aξν , Y) − g(A∇X ξν , Y) − g(∇Y Aξν , X) + g(A∇Y ξν , X)

= g(∇X βν ξν , Y) − g(A∇X ξν , Y) − g(∇Y βν ξν , X) + g(A∇Y ξν , X)

= dβν (X)ην (Y) − dβν (Y)ην (X) + βν g((Aϕν + ϕν A)X, Y) − 2g(Aϕν AX, Y) + (βν − βν+1 )(qν+2 (X)ην+1 (Y) − qν+2 (Y)ην+1 (X))

− (βν − βν+2 )(qν+1 (X)ην+2 (Y) − qν+1 (Y)ην+2 (X))

for all X, Y ∈ X(M). Inserting X = ξν into this equation gives dβν (Y) = dβν (ξν )ην (Y) + 4ην (ξ )ην (ϕY) − 2ην+1 (ξ )ην+1 (ϕY) − 2ην+2 (ξ )ην+2 (ϕY) + (βν − βν+1 )qν+2 (ξν )ην+1 (Y) − (βν − βν+2 )qν+1 (ξν )ην+2 (Y).

Inserting this and the corresponding expression for dβν (X) into the previous equation gives 2η(Y)ην (ϕX) − 2η(X)ην (ϕY) + 2g(ϕX, Y)ην (ξ ) + 2g(ϕν X, Y) − 2ην+1 (X)ην+2 (Y) + 2ην+1 (Y)ην+2 (X)

− 2ην+1 (ϕX)ην+2 (ϕY) + 2ην+1 (ϕY)ην+2 (ϕX)

= βν g((Aϕν + ϕν A)X, Y) − 2g(Aϕν AX, Y)

− 2ην+1 (ξ )(ην+1 (ϕX)ην (Y) − ην+1 (ϕY)ην (X))

− 2ην+2 (ξ )(ην+2 (ϕX)ην (Y) − ην+2 (ϕY)ην (X)) + 4ην (ξ )(ην (ϕX)ην (Y) − ην (ϕY)ην (X))

184 | 5 Real hypersurfaces in complex hyperbolic 2-plane Grassmannians + (βν − βν+1 )(qν+2 (ξν )(ην+1 (X)ην (Y) − ην+1 (Y)ην (X))

+ qν+2 (X)ην+1 (Y) − qν+2 (Y)ην+1 (X))

− (βν − βν+2 )(qν+1 (ξν )(ην+2 (X)ην (Y) − ην+2 (Y)ην (X))

+ qν+1 (X)ην+2 (Y) − qν+1 (Y)ην+2 (X)). From this we obtain the following.

Lemma 5.7.1. Let M be a real hypersurface in G2∗ (ℂ2+k ) with A-invariant maximal quaternionic subbundle 𝒟. If X ∈ Γ(𝒟) with AX = λX, then 0 = (2λ − βν )Aϕν X + (2 − λβν )ϕν X

+ (2ην+1 (ξ )ην+1 (ϕX) + 2ην+2 (ξ )ην+2 (ϕX) − 4ην (ξ )ην (ϕX))ξν

+ (βν − βν+1 )qν+2 (X)ξν+1 − (βν − βν+2 )qν+1 (X)ξν+2

+ 2ην (ξ )ϕX + 2ην (ϕX)ξ + 2η(X)ϕξν − 2ην+2 (ϕX)ϕξν+1 + 2ην+1 (ϕX)ϕξν+2 ,

where the canonical local basis is chosen so that Aξν = βν ξν . The ξν+1 -component of the equation in Lemma 5.7.1 gives 0 = (βν − βν+1 )qν+2 (X) + 4η(ξν )ην+1 (ϕX) + 2η(ξν+1 )ην (ϕX) − 2η(ξν+2 )η(X), whereas the ξν+2 -component leads to 0 = (βν − βν+2 )qν+1 (X) − 4η(ξν )ην+2 (ϕX) − 2η(ξν+2 )ην (ϕX) − 2η(ξν+1 )η(X). Both equations hold for any index. Raising the index of the second equation by one and then combining with the first equation yields 2η(ξν+2 )η(X) = η(ξν )ην+1 (ϕX) − η(ξν+1 )ην (ϕX). We thus have proved the following. Lemma 5.7.2. Let M be a real hypersurface in G2∗ (ℂ2+k ) with A-invariant maximal quaternionic subbundle 𝒟. Then we have 2η(ξν+2 )η(X) = η(ξν )ην+1 (ϕX) − η(ξν+1 )ην (ϕX) for all X ∈ Γ(𝒟). We now restrict to the special case of Hopf hypersurfaces. Let M be a Hopf hypersurface in G2∗ (ℂ2+k ) for which the maximal quaternionic subbundle 𝒟 is invariant under the shape operator A of M. We first prove the following crucial result.

5.7 Real hypersurfaces with invariant maximal quaternionic subbundle

| 185

Proposition 5.7.3. Let M be a Hopf hypersurface in G2∗ (ℂ2+k ) with A𝒟 ⊆ 𝒟. Then the Reeb vector field ξ is tangent to 𝒟 or to 𝒟⊥ everywhere (or equivalently, the normal vector field ζ is a singular tangent vector of G2∗ (ℂ2+k ) everywhere). Proof. We can write ξ = η(X)X + η(Z)Z with suitable unit vector fields X ∈ Γ(𝒟) and Z ∈ Γ(𝒟⊥ ). Since 𝒟 and 𝒟⊥ are invariant under A, we have AX ∈ Γ(𝒟) and AZ ∈ Γ(𝒟⊥ ). Thus αη(X)X + αη(Z)Z = αξ = Aξ = η(X)AX + η(Z)AZ implies η(X) = 0, η(Z) = 0 or (AX = αX and AZ = αZ). Suppose that the last possibility holds. Without loss of generality we may assume that the local canonical basis is chosen in such a way that Z = ξ3 . Then η(ξ1 ) = 0 = η(ξ2 ), and we get 2η(Z)η(X) = 2η(ξ3 )η(X) = η(ξ1 )η2 (ϕX) − η(ξ2 )η1 (ϕX) = 0 from Lemma 5.7.2. Therefore we necessarily have η(X) = 0 or η(Z) = 0, which means that ξ is either a vector field tangent to 𝒟⊥ or to 𝒟 everywhere. Case 1: The Reeb vector field ξ is tangent to 𝒟 everywhere We first assume ξ ∈ Γ(𝒟). Then the unit normal vector field ζ is a singular tangent vector of G2∗ (ℂ2+k ) of type Jζ ⊥ Jζ and the vector fields ξ , ξ1 , ξ2 , ξ3 , ϕξ1 , ϕξ2 , ϕξ3 are orthonormal. Moreover, we have Aξ = αξ and Aξν = βν ξν . Lemma 5.7.4. For each ν ∈ {1, 2, 3} we have Aϕξν = γν ϕξν and one of the following two cases holds: (i) αβν = 4 and γν = 0; 2 (ii) βν = α ≠ 0 and γν = α α−4 . Proof. Inserting X = ξν into (5.12) gives 0 = (2βν − α)Aϕξν + (4 − αβν )ϕξν . Inserting X = ξ into the equation in Lemma 5.7.1 gives 0 = (2α − βν )Aϕν ξ + (4 − αβν )ϕν ξ

+ (βν − βν+1 )qν+2 (ξ )ξν+1 − (βν − βν+2 )qν+1 (ξ )ξν+2 .

Since ϕν ξ = ϕξν is perpendicular to ξν+1 and ξν+2 , this implies 0 = (2α − βν )Aϕξν + (4 − αβν )ϕξν .

(5.22)

186 | 5 Real hypersurfaces in complex hyperbolic 2-plane Grassmannians Subtracting this equation from equation (5.22) gives 0 = (βν − α)Aϕξν . Thus we have βν = α or Aϕξν = 0. If βν = α, (5.22) gives 0 = αAϕξν + (4 − α2 )ϕξν , which implies α ≠ 0 and Aϕξν = γν ϕξν with γν =

α2 −4 . If Aϕξν α

= 0, (5.22) gives αβν = 4.

Lemma 5.7.5. For each ν ∈ {1, 2, 3} we have Aξν = βξν and Aϕξν = 0, where β is determined by αβ = 4. Proof. By evaluating the equation in Lemma 5.7.1 for X = ϕξν+1 and λ = γν+1 we get 2γν+1 γν+2 = βν (γν+1 + γν+2 )

(5.23)

for all ν ∈ {1, 2, 3}. Since βν ≠ 0 for all ν ∈ {1, 2, 3} according to Lemma 5.7.4, this implies that either γ1 = γ2 = γ3 = 0 or γν ≠ 0 for all ν ∈ {1, 2, 3}. Assume that γν ≠ 0 for all 2

ν ∈ {1, 2, 3}. Then, using Lemma 5.7.4, we have γν = α α−4 and βν = α for all ν ∈ {1, 2, 3}, and inserting this into (5.23) yields α2 = 4. However, α2 = 4 implies γν = 0, which contradicts our assumption that γν ≠ 0 for all ν ∈ {1, 2, 3}. We therefore must have γ1 = γ2 = γ3 = 0, and the assertion then follows from Lemma 5.7.4. We denote by ℍℂζ the real span of ζ , ξ , ξ1 , ξ2 , ξ3 , Jξ1 , Jξ2 , Jξ3 . It follows from Lemma 5.7.4 and the assumption that the orthogonal complement (ℍℂζ )⊥ of ℍℂζ in TM is invariant under the shape operator A. We now derive some equations for the principal curvatures of A restricted to (ℍℂζ )⊥ . Lemma 5.7.6. Let X ∈ Γ((ℍℂζ )⊥ ) with AX = λX. Then we have (2λ − α)AϕX = (αλ − 2)ϕX,

(αλ − 2)Aϕν X = (2λ − α)ϕν X.

(5.24) (5.25)

Proof. Equation (5.24) follows from equation (5.12), and equation (5.25) follows from Lemma 5.7.1 using the fact that αβ = 4 according to Lemma 5.7.5. If we assume 2λ − α = 0, we get αλ = 2 from (5.24) and therefore α2 = 4. Since αβ = 4 this implies α = β. For this reason we consider the two cases α ≠ β and α = β separately. We first assume that α ≠ β. Then we must have 2λ − α ≠ 0 and therefore also αλ − 2 ≠ 0 by (5.25). From (5.25) we then get Aϕν+1 X =

2λ − α ϕ X. αλ − 2 ν+1

5.7 Real hypersurfaces with invariant maximal quaternionic subbundle | 187

Applying (5.25) to ϕν+1 X instead of X we obtain Aϕν ϕν+1 X = λϕν ϕν+1 X. On the other hand, by (5.25) we also have Aϕν+2 X =

2λ − α ϕ X. αλ − 2 ν+2

Since ϕν ϕν+1 X = ϕν+2 X, the previous two equations imply that λ is a solution of the quadratic equation αx2 − 4x + α = 0. This shows that A restricted to (ℍℂζ )⊥ has at most two eigenvalues. Moreover, each solution of αx2 − 4x + α = 0 satisfies 2x − α = x, αx − 2 which means that the corresponding eigenspace is J-invariant. From (5.24) we see that αλ−2 ϕX is a principal curvature vector with principal curvature 2λ−α . If we assume that αλ−2 2λ−α

= λ, we get λ2 −αλ+1 = 0, which together with αλ2 −4λ+α = 0 leads to λ = 0 (which leads to α = 0 and contradicts αβ = 4) or α2 = 4 (which because of αβ = 4 leads to α = β αλ−2 ≠ λ. Altogether and contradicts the assumption α ≠ β). Therefore we must have 2λ−α ⊥ this shows that A restricted to (ℍℂζ ) has precisely two eigenvalues, namely the two distinct solutions λ1 and λ2 of αx2 − 4x + α = 0, and that J maps the two eigenspaces onto each other. It is easy to see that λ1 + λ2 =

4 = β ≠ 0. α

Since ξ ∈ Γ(𝒟), equation (5.11) gives gradM α = dα(ξ )ξ . So for the Hessian of α we get hessα (X, Y) = g(∇X gradM α, Y) = g(∇X dα(ξ )ξ , Y)

= d(dα(ξ ))(X)η(Y) + dα(ξ )g(∇X ξ , Y)

= d(dα(ξ ))(X)η(Y) + dα(ξ )g(ϕAX, Y) for all X, Y ∈ X(M). By symmetry of the Hessian this implies 0 = d(dα(ξ ))(X)η(Y) − d(dα(ξ ))(Y)η(X) + dα(ξ )g((Aϕ + ϕA)X, Y).

188 | 5 Real hypersurfaces in complex hyperbolic 2-plane Grassmannians For X = ξ this gives d(dα(ξ ))(Y) = d(dα(ξ ))(ξ )η(Y). Inserting this and the corresponding equation for X into the previous one yields 0 = dα(ξ )g((Aϕ + ϕA)X, Y) for all X, Y ∈ X(M). Suppose that Aϕ + ϕA vanishes at some point p of M. Then, at p, for v ∈ (ℍℂζ )⊥ with Av = λv we get Aϕv = −λϕv. But, since λ is a solution of αx 2 − 4x + α = 0 and α ≠ 0, −λ cannot be a solution of that equation as well, which contradicts the above statement that the principal curvatures on (ℍℂζ )⊥ are the two distinct solutions of αx2 − 4x + α = 0. Hence we must have Aϕ + ϕA ≠ 0 at each point of M. This implies dα(ξ ) = 0 and hence gradM α = 0, that is, α is constant on M. Since M is connected, we obtain that α is constant. From the equation αx2 − 4x + α = 0 we get α2 < 4, and hence we can write α = 2 tanh(2r) for some positive real number r and a suitable orientation of the normal vector. Writing α in this way, the two solutions of αx2 − 4x + α = 0 are λ = tanh(r) and μ = coth(r). From αβ = 4 we also get β = 2 coth(2r). We now assume that α = β. Since αβ = 4, we may assume that α = 2 (by a suitable orientation of the normal vector). Assume that there exists a principal curvature λ of A restricted to (ℍℂζ )⊥ such that λ ≠ 1. From (5.24) we then get AϕX =

αλ − 2 ϕX = ϕX, 2λ − α

and from (5.25) we obtain Aϕν X =

2λ − α ϕ X = ϕν X. αλ − 2 ν

Thus we have proved the following. Proposition 5.7.7. Let M be a connected Hopf hypersurface in G2∗ (ℂ2+k ) with A𝒟 ⊆ 𝒟 and assume that the Reeb vector field ξ is tangent to 𝒟 everywhere. Then one of the following three cases holds: (i) M has five (four for r = tanh−1 ( √13 ), in which case α = μ) distinct constant principal curvatures α = 2 tanh(2r),

β = 2 coth(2r),

γ = 0,

λ = tanh(r),

μ = coth(r),

5.7 Real hypersurfaces with invariant maximal quaternionic subbundle | 189

and the corresponding principal curvature spaces are Tα = 𝒞 ⊥ ,

Tβ = 𝒟⊥ ,

Tγ = J 𝒟⊥ = JTβ .

The principal curvature spaces Tλ and Tμ are invariant under J and are mapped onto each other by J. In particular, the quaternionic dimension of G2∗ (ℂ2+k ) must be even, say k = 2l. (ii) M has exactly three distinct constant principal curvatures α = 2 (= β),

γ = 0,

λ=1

with corresponding principal curvature spaces Tα = (𝒞 ∩ 𝒟)⊥ ,

Tγ = J 𝒟⊥ ,

Tλ = 𝒞 ∩ 𝒟 ∩ J 𝒟 .

(iii) M has at least four distinct principal curvatures, three of which are given by α = 2 (= β),

γ = 0,

λ=1

with corresponding principal curvature spaces Tα = (𝒞 ∩ 𝒟)⊥ ,

Tγ = J 𝒟⊥ ,

Tλ = 𝒞 ∩ 𝒟 ∩ J 𝒟 .

If μ is another (possibly non-constant) principal curvature function, then JTμ ⊂ Tλ and JTμ ⊂ Tλ . We now come to the main result for Case 1. Theorem 5.7.8. Let M be a connected Hopf hypersurface in G2∗ (ℂ2+k ), k ≥ 3, with A𝒟 ⊆ 𝒟 and assume that the Reeb vector field ξ is tangent to 𝒟 everywhere. Then one of the following three cases holds: (i) The quaternionic dimension k of G2∗ (ℂ2+k ) is even, say k = 2l, and M is an open part of a tube with radius r ∈ ℝ+ around the totally real and quaternionic totally geodesic embedding of the quaternionic hyperbolic space ℍH l in G2∗ (ℂ2+2l ). (ii) The Hopf hypersurface M is an open part of a horosphere in G2∗ (ℂ2+k ) whose center at infinity is singular and of type Jζ ⊥ Jζ . (iii) The Hopf hypersurface M has at least four distinct principal curvatures, three of which are given by α = β = 2,

γ = 0,

λ=1

with corresponding principal curvature spaces Tα = (𝒞 ∩ 𝒟)⊥ ,

Tγ = J 𝒟⊥ ,

Tλ = 𝒞 ∩ 𝒟 ∩ J 𝒟 .

190 | 5 Real hypersurfaces in complex hyperbolic 2-plane Grassmannians If μ is another (possibly non-constant) principal curvature function, then JTμ ⊂ Tλ and JTμ ⊂ Tλ . It is not known whether or not there exists a Hopf hypersurface in G2∗ (ℂ2+k ) whose principal curvatures and principal curvature spaces satisfy the relations in (iii). Proof. In view of Proposition 5.7.7, we need to investigate cases (i) and (ii). (i): By assumption, the normal vector field ζ of M is singular and of type Jζ ⊥ Jζ everywhere. Using Lemma 1.4.10 we see that the eigenvalues of the normal Jacobi operator K are 0, −1 and −4 with corresponding eigenspaces Jξ , (ℍℂζ )⊥ and ℝξ ⊕ 𝒟⊥ , respectively. From Proposition 5.7.7 we get Jξ = Tγ ,

(ℍℂζ )⊥ = Tλ ⊕ Tμ ,

ℝξ ⊕ 𝒟⊥ = Tα ⊕ Tβ .

It follows that M is curvature-adapted and we can apply Corollary 2.5.2. For p ∈ M denote by cp the geodesic in G2∗ (ℂ2+2l ) with cp (0) = p and ċp (0) = ζp and by Φr the smooth map Φr : M → G2∗ (ℂ2+2l ),

p 󳨃→ cp (r).

Geometrically, Φr is the displacement of M at distance r in the direction of the normal vector field ζ . From Corollary 2.5.2(i) we obtain that cp (r) is a focal point of M along cp and, locally, Φr is a submersion onto a 4l-dimensional submanifold P of G2∗ (ℂ2+2l ). Moreover, the tangent space of P at Φr (p) is obtained by parallel translation of (Tα ⊕ Tγ ⊕ Tμ )(p) =(ℍξ ⊕ Tμ )(p), which is a quaternionic (with respect to J) and totally real (with respect to J) subspace of Tp G2∗ (ℂ2+2l ). Since both J and J are parallel along cp , also TF(p) P is a quaternionic and totally real subspace of TΦr (p) G2∗ (ℂ2+2l ). Thus P is a quaternionic and totally real submanifold of G2∗ (ℂ2+2l ). Since P is quaternionic, it is totally geodesic in G2∗ (ℂ2+2l ) (see [3]). The only quaternionic totally geodesic submanifolds of G2∗ (ℂ2+2l ) of half dimension are open parts of G2∗ (ℂ2+l ) and ℍH l (see [6] and use duality). However, only ℍH l is embedded in G2∗ (ℂ2+2l ) as a totally real submanifold. So we conclude that P is an open part of a totally geodesic ℍH l in G2∗ (ℂ2+2l ). Rigidity of totally geodesic submanifolds finally implies that M is an open part of the tube with radius r ∈ ℝ+ around a totally geodesic ℍH l in G2∗ (ℂ2+2l ). (ii): As above, for p ∈ M we denote by cp the geodesic in G2∗ (ℂ2+k ) with cp (0) = p and ċp (0) = ζp and by Φr the smooth map Φr : M → G2∗ (ℂ2+k ),

p 󳨃→ cp (r).

For each p ∈ M the differential dp Φr of Φr at p can be computed using Jacobi fields by means of dp Φr (v) = Zv (r), where Zv is the Jacobi field along cp with initial value

5.7 Real hypersurfaces with invariant maximal quaternionic subbundle | 191

Zv (0) = v and Zv′ (0) = −Av (see Section 2.5). Using the explicit description of the Jacobi operator in Lemma 1.4.10(i) for the case Jζ ⊥ Jζ we get Bv (t), if v ∈ Tγ , { { { −t Zv (t) = {e Bv (t), if v ∈ Tλ , { { −2t {e Bv (t), if v ∈ Tα for all t ∈ ℝ, where Bv is the parallel vector field along cp with initial value Bv (0) = v. Now consider a geodesic variation in G2∗ (ℂ2+k ) consisting of geodesics cp . The corresponding Jacobi field is a linear combination of the three types of the Jacobi fields Zv listed above, and hence its length remains bounded when t → ∞. This shows that all geodesics cp in G2∗ (ℂ2+k ) are asymptotic to each other and hence determine a singular point z ∈ G2∗ (ℂ2+k )(∞) at infinity of type Jζ ⊥ Jζ . Therefore M is an integral manifold of the distribution on G2∗ (ℂ2+k ) given by the orthogonal complements of the tangent vectors of the geodesics in the asymptote class z. This distribution is integrable and the maximal leaves are the horospheres in G2∗ (ℂ2+k ) whose center at infinity is z. Uniqueness of integral manifolds of integrable distributions finally implies that M is an open part of a horosphere in G2∗ (ℂ2+k ) whose center is the singular point z at infinity. Theorem 5.7.8 has the following interesting consequence. Theorem 5.7.9 ([99]). Let M be a connected Hopf hypersurface in G2∗ (ℂ2+k ), k ≥ 3. Then the following two statements are equivalent: (a) The Reeb vector field ξ is tangent to the maximal quaternionic subbundle 𝒟 of TM everywhere (or equivalently, the normal vector field ζ is singular of type Jζ ⊥ Jζ everywhere). (b) One of the following three cases holds: (i) The quaternionic dimension k of G2∗ (ℂ2+k ) is even, say k = 2l, and M is an open part of a tube with radius r ∈ ℝ+ around the totally real and quaternionic totally geodesic embedding of the quaternionic hyperbolic space ℍH l in G2∗ (ℂ2+2l ). (ii) The Hopf hypersurface M is an open part of a horosphere in G2∗ (ℂ2+k ) whose center at infinity is singular and of type Jζ ⊥ Jζ . (iii) The Hopf hypersurface M has at least four distinct principal curvatures, three of which are given by α = 2,

γ = 0,

λ=1

with corresponding principal curvature spaces Tα = (𝒞 ∩ 𝒟)⊥ ,

Tγ = J 𝒟 ⊥ ,

Tλ = 𝒞 ∩ 𝒟 ∩ J 𝒟 .

If μ is another (possibly non-constant) principal curvature function, then JTμ ⊂ Tλ and JTμ ⊂ Tλ .

192 | 5 Real hypersurfaces in complex hyperbolic 2-plane Grassmannians Proof. Using Theorem 5.5.5 we easily see that (b) implies (a). Conversely, if the Reeb vector field ξ is tangent to the maximal quaternionic subbundle 𝒟 of TM everywhere, then Proposition 5.6.3 implies A𝒟 ⊆ 𝒟, and we can apply Theorem 5.7.8. Case 2: The Reeb vector field ξ is perpendicular to 𝒟 everywhere We now assume ξ ∈ Γ(𝒟⊥ ). Then the unit normal vector field ζ is a singular tangent vector of G2∗ (ℂ2+k ) of type Jζ ∈ Jζ . There exists an almost Hermitian structure J1 ∈ J so that Jζ = J1 ζ . We have Aξ = αξ and Aξν = βν ξν by assumption. Moreover, we have ξ = ξ1 ,

α = β1 ,

ϕξ2 = −ξ3 ,

ϕξ3 = ξ2 ,

ϕ𝒟 = 𝒟,

𝒟 ⊂ 𝒞.

By inserting X = ξ2 (or X = ξ3 ) into equation (5.12), we get 2β2 β3 − α(β2 + β3 ) + 4 = 0.

(5.26)

From equation (5.12) and the equation in Lemma 5.7.1 we immediately get the following. Lemma 5.7.10. Let X ∈ Γ(𝒟) with AX = λX. Then we have (2λ − α)AϕX = (αλ − 2)ϕX − 2ϕ1 X,

(2λ − α)Aϕ1 X = (αλ − 2)ϕ1 X − 2ϕX,

(2λ − βν )Aϕν X = (βν λ − 2)ϕν X,

ν ∈ {2, 3}.

(5.27) (5.28) (5.29)

By adding equations (5.27) and (5.28) we get (2λ − α)A(ϕ + ϕ1 )X = (αλ − 4)(ϕ + ϕ1 )X,

(5.30)

and by subtracting (5.28) from (5.27) we get (2λ − α)A(ϕ − ϕ1 )X = αλ(ϕ − ϕ1 )X.

(5.31)

Note that on 𝒟 we have (ϕϕ1 )2 = id𝒟 and tr(ϕϕ1 ) = 0. Let E+1 and E−1 be the eigenbundles of ϕϕ1 |𝒟 with respect to the eigenvalues +1 and −1, respectively. Then the maximal quaternionic subbundle 𝒟 of TM decomposes orthogonally into the Whitney sum 𝒟 = E+1 ⊕ E−1 , and the rank of both eigenbundles E±1 is equal to 2k − 2. We have X ∈ Γ(E+1 ) if and only if ϕX = −ϕ1 X and X ∈ Γ(E−1 ) if and only if ϕX = ϕ1 X. Lemma 5.7.11. Let X ∈ Γ(𝒟) with AX = λX. If 2λ = α, then λ = √2, α = 2√2 and X ∈ Γ(E−1 ). Proof. We can assume without loss of generality that α ≥ 0. From (5.30) and (5.31) we see that one of the following two statements holds:

5.7 Real hypersurfaces with invariant maximal quaternionic subbundle | 193

(i) λ = 0, α = 0 and X ∈ Γ(E+1 ); (ii) λ = √2, α = 2√2 and X ∈ Γ(E−1 ). We have to exclude case (i). Assume that λ = 0, α = 0 and X ∈ Γ(E+1 ). From (5.26) we get β2 β3 = −2, and therefore β2 and β3 are both non-zero. From (5.29) we get Aϕν X =

2 ϕ X, βν ν

ν ∈ {2, 3}.

By applying (5.29) for ν = 2 to ϕ3 X we obtain β − β3 2β 4 6 Aϕ1 X = ( − β2 )Aϕ2 ϕ3 X = ( 2 − 2)ϕ2 ϕ3 X = 2( 2 )ϕ1 X, β3 β3 β3 β3 and by applying (5.29) for ν = 3 to ϕ2 X we obtain 2β β − β2 4 6 Aϕ1 X = −( − β3 )Aϕ3 ϕ2 X = −( 3 − 2)ϕ3 ϕ2 X = 2( 3 )ϕ1 X. β2 β2 β2 β2 The previous two equations imply β2 = β3 , which contradicts β2 β3 = −2. It follows that case (i) cannot occur. We denote by Λ the set of all eigenvalues of A|𝒟 , and for each ρ ∈ Λ we denote by Tρ the corresponding eigenspace. We first assume that there exists λ ∈ Λ with 2λ = α. Then we have α = 2√2, λ = √2 and Tλ ⊆ E−1 according to Lemma 5.7.11. Since α = 2√2, (5.26) becomes 0 = β2 β3 − √2(β2 + β3 ) + 2 = (β2 − √2)(β3 − √2). Therefore we have β2 = √2 or β3 = √2. Without loss of generality we can assume that β2 = √2. From (5.29) we get Aϕ2 X = 0 for all X ∈ Γ(Tλ ). Applying (5.28) to ϕ2 X and using the fact that X ∈ Γ(E−1 ) we get Aϕ3 X = 0 for all X ∈ Γ(Tλ ). Thus we have shown that 0 ∈ Λ,

ϕ2 Tλ ⊆ T0 ,

ϕ3 Tλ ⊆ T0 .

Next, we apply (5.29) for ν = 2 to ϕ3 X, which yields Aϕ1 X = √2ϕ1 X, and applying (5.29) for ν = 3 to ϕ2 X gives β3 Aϕ1 X = 2ϕ1 X.

(5.32)

194 | 5 Real hypersurfaces in complex hyperbolic 2-plane Grassmannians Comparing the previous two equations shows that β3 = √2. Thus we have proved that β2 = β3 = √2 = λ,

ϕ1 Tλ ⊆ Tλ .

(5.33)

Now we choose ρ ∈ Λ \ {√2} and Y ∈ Γ(Tρ ). From (5.32) we know that Λ \ {√2} ≠ 0. From (5.29) and (5.33) we get (√2ρ − 1)Aϕν Y = (ρ − √2)ϕν Y,

ν ∈ {2, 3}.

Since ρ ≠ √2 this implies √2ρ ≠ 1 and ρ∗ =

ρ − √2 ∈ Λ, √2ρ − 1

ϕ2 Tρ ⊆ Tρ∗ ,

ϕ3 Tρ ⊆ Tρ∗ .

(5.34)

Note that (ρ∗ )∗ = ρ and 0∗ = √2 = λ. Finally, we apply (5.29) for ν = 2 to ϕ3 Y and obtain Aϕ1 Y = ρϕ1 Y, and therefore ϕ1 Tρ ⊆ Tρ .

(5.35)

From (5.28) and (5.35) we obtain (ρ − √2)ρϕ1 Y = (√2ρ − 1)ϕ1 Y − ϕY, and therefore ϕY = (−ρ2 + 2√2ρ − 1)ϕ1 Y. Since ϕY and ϕ1 Y have the same length, this implies Y ∈ Γ(E±1 ),

−ρ2 + 2√2ρ − 1 = ∓1.

The equation −ρ2 + 2√2ρ − 1 = 1 has ρ = √2 as a solution with multiplicity 2, and the equation −ρ2 + 2√2ρ − 1 = −1 has ρ = 0 and ρ = 2√2 as solutions. However, for √ √ ρ = 2√2 we would have 32 = ρ∗ ∈ Λ according to (5.34), but since 32 is not a solution 2 of −ρ + 2√2ρ − 1 = ∓1, we can dismiss the case ρ = 2√2. Altogether we have shown that Λ = {0, √2}. From (5.32)–(5.35) it is clear that T0 and T√2 have the same rank, and hence rk(T0 ) = rk(T√2 ) = 2k − 2. Since T√2 ⊆ E−1 and the rank of E−1 is 2k − 2, we get

5.7 Real hypersurfaces with invariant maximal quaternionic subbundle | 195

T1 = E−1 . From the two orthogonal decompositions 𝒟 = E+1 ⊕ E−1 = T0 ⊕ T√2

we also get T0 = E+1 . Thus we have proved the following. Proposition 5.7.12. Assume that there exists a principal curvature λ ∈ Λ with 2λ = α. Then M has three distinct constant principal curvatures 0, √2 and 2√2 with multiplicities 2k − 2, 2k and 1, respectively. The corresponding principal curvature spaces are E+1 , E−1 ⊕ (𝒞 ⊖ 𝒟) and 𝒞 ⊥ , respectively. We will now assume that 2λ ≠ α for all λ ∈ Λ. The linear maps 𝒟 → E+1 ,

X 󳨃→ (ϕ − ϕ1 )X

and 𝒟 → E−1 ,

X 󳨃→ (ϕ + ϕ1 )X

are epimorphisms, and according to (5.30) and (5.31) each of them maps principal curvature vectors in 𝒟 either to 0 or to a principal curvature vector in E+1 resp. E−1 . It follows that there exists a basis of principal curvature vectors in 𝒟 such that each vector in that basis is in E+1 or in E−1 . In other words, we have Tλ = (Tλ ∩ E+1 ) ⊕ (Tλ ∩ E−1 )

for all λ ∈ Λ.

From (5.27) and the ϕ-invariance of E±1 we get the following. Lemma 5.7.13. Let λ ∈ Λ. Then we have αλ ϕX ∈ Γ(E+1 ) for all X ∈ Γ(Tλ ∩ E+1 ), 2λ − α αλ − 4 AϕX = ϕX ∈ Γ(E−1 ) for all X ∈ Γ(Tλ ∩ E−1 ). 2λ − α AϕX =

(5.36)

This shows that the cardinality |Λ| of Λ satisfies |Λ| ≥ 2. From (5.29) we easily get the following. Lemma 5.7.14. Let λ ∈ Λ. If 2λ = βν , then (λ = 1 and βν = 2) or (λ = −1 and βν = −2). Moreover, if βν = 2, then 1 ∈ Λ and ϕν Tλ ⊆ T1 for all λ ∈ Λ \ {1}. Similarly, if βν = −2, then −1 ∈ Λ and ϕν Tλ ⊆ T−1 for all λ ∈ Λ \ {−1}. Lemma 5.7.15. Let λ ∈ Λ, ν ∈ {2, 3}, and assume that 2λ ≠ βν . Then we have λν =

βν λ − 2 ∈Λ 2λ − βν

and

ϕν Tλ ⊆ Tλν .

Moreover, if 2λ ≠ β2 and 2λ ≠ β3 , then one of the following two statements holds: (i) Tλ ⊆ E+1 and 2λ2 λ3 − α(λ2 + λ3 ) + 4 = 0; (ii) Tλ ⊆ E−1 and 2λ2 λ3 − α(λ2 + λ3 ) = 0.

(5.37)

196 | 5 Real hypersurfaces in complex hyperbolic 2-plane Grassmannians Proof. The first statement follows from (5.29). From (5.28) we obtain for X ∈ Γ(Tλ ) that 0 = (2λ2 − α)Aϕ1 ϕ2 X − (αλ2 − 2)ϕ1 ϕ2 X + 2ϕϕ2 X = (2λ2 − α)Aϕ3 X − (αλ2 − 2)ϕ3 X + 2ϕ2 ϕX

= (2λ2 − α)λ3 ϕ3 X − (αλ2 − 2)ϕ3 X + 2ϕ2 ϕX

= (2λ2 λ3 − α(λ2 + λ3 ) + 2)ϕ3 X + 2ϕ2 ϕX

= −(2λ2 λ3 − α(λ2 + λ3 ) + 2)ϕ2 ϕ1 X + 2ϕ2 ϕX.

This implies 0 = (2λ2 λ3 − α(λ2 + λ3 ) + 2)ϕ1 X − 2ϕX, from which the assertion easily follows. Lemma 5.7.16. Let λ, λ2 , λ3 ∈ Λ and assume that ϕν Tλ ⊆ Tλν for ν ∈ {2, 3}. If 2λ3 ≠ β2 and 2λ2 ≠ β3 , then at least one of the following three statements holds: (i) λ2 = 1; (ii) β2 β3 = 4; (iii) β2 = β3 . Proof. Let X ∈ Γ(Tλ ). From Lemma 5.7.15 we obtain (β β − 4)λ + 2(β3 − β2 ) β2 λ3 − 2 ϕϕ X= 2 3 ϕ X, 2λ3 − β2 2 3 (β2 β3 − 4) + 2(β3 − β2 )λ 1 β λ −2 (β β − 4)λ + 2(β2 − β3 ) Aϕ1 X = −Aϕ3 ϕ2 X = − 3 2 ϕ ϕX= 2 3 ϕ X. 2λ2 − β3 3 2 (β2 β3 − 4) + 2(β2 − β3 )λ 1

Aϕ1 X = Aϕ2 ϕ3 X =

Comparing these two equations leads to 0 = (λ2 − 1)(β2 β3 − 4)(β3 − β2 ), which implies the assertion. Lemma 5.7.17. If β22 ≠ 4 ≠ β32 , then β2 = β3 . Proof. If λ = 1 ∈ Λ, then Lemma 5.7.15 implies λ2 = λ3 = −1 and α < 0. If λ = −1 ∈ Λ, then Lemma 5.7.15 implies λ2 = λ3 = 1 and α > 0. Thus we have Λ ≠ {±1}, and it follows from Lemma 5.7.16 that β2 β3 = 4 or β2 = β3 . Let us assume that β2 β3 = 4. From (5.26) we obtain α(β2 + β3 ) = 12 and hence α ≠ 0. Moreover, from β2 β3 = 4 and (5.26) we see that β2 and β3 are the solutions of the quadratic equation αx2 − 12x + 4α = 0.

(5.38)

5.7 Real hypersurfaces with invariant maximal quaternionic subbundle | 197

From (5.37) we obtain λ2 λ3 = 1. If we choose λ ∈ Λ with Tλ ⊆ E+1 , Lemma 5.7.15(i) implies that λ2 and λ3 are the solutions of the quadratic equation αx2 − 6x + α = 0. It follows that both 2λ2 and 2λ3 are solutions of the quadratic equation (5.38), which means that β2 = 2λ2 or β2 = 2λ3 . In both cases we deduce β22 = 4 from Lemma 5.7.14, which is a contradiction to the assumption. Therefore we must have β2 β3 ≠ 4, and we conclude that β2 = β3 . Lemma 5.7.18. Assume that there exist λ ∈ Λ and ν ∈ {2, 3} such that 2λ = βν . Then λ = 1, β2 = β3 = 2, α = 3 and E−1 ⊆ Tλ . Proof. Without loss of generality we can assume that 2λ = β2 . Using Lemma 5.7.14 we can also assume that λ = 1 and β2 = 2 (by choosing a suitable orientation of the normal vector field). Inserting β2 = 2 into (5.26) gives (4 − α)β3 = 2α − 4. It follows from this equation that α ≠ 4 and β3 =

2α − 4 . 4−α

(5.39)

This implies β3 ≠ −2. We first assume that β3 ≠ 2 = β2 . Since |Λ| ≥ 2, there exists ρ ∈ Λ\{λ}, and any such ρ satisfies 2ρ ≠ β2 (since ρ ≠ λ and 2λ = β2 ) and 2ρ ≠ β3 (since β3 ≠ ±2 and because of Lemma 5.7.14). From Lemma 5.7.15 we see that ϕν Tρ ⊆ Tρν with ρ2 = 1 = λ and ρ3 =

β3 ρ − 2 ∈ Λ. 2ρ − β3

We have 2ρ2 = 2 = β2 ≠ β3 . Therefore, if 2ρ3 ≠ β2 , we deduce ρ = −1 from Lemma 5.7.16 (since 2 = β2 ≠ β3 and ρ ≠ λ). Otherwise, if 2ρ3 = β2 = 2 we get β3 ρ − 2 = ρ3 = 1 2ρ − β3 by Lemma 5.7.15, which is equivalent to (ρ + 1)β3 = 2ρ + 2. Since β3 ≠ 2 this implies ρ = −1 as well. Altogether we conclude that Λ = {±1}. We now α apply Lemma 5.7.13 to ρ = −1 ∈ Λ. Since 2+α ≠ 1 and α+4 ≠ 1, we must have either 2+α α α+4 = −1 or = −1. This implies α ∈ {−1, −3}. By applying Lemma 5.7.13 to ρ = 1 ∈ Λ, 2+α 2+α

198 | 5 Real hypersurfaces in complex hyperbolic 2-plane Grassmannians α ∈ Λ or α−4 ∈ Λ. However, for α ∈ {−1, −3} none of these numbers is we see that 2−α 2−α equal to ±1, which gives a contradiction. It follows that β3 ≠ 2 = β2 is not possible. Next, we assume β3 = β2 = 2. From (5.39) we then obtain α = 3. Assume that there exists ρ ∈ Λ \ {λ} such that Tρ ∩ E−1 ≠ {0} and let 0 ≠ X ∈ Γ(Tρ ∩ E−1 ). From (5.37) we obtain ϕ2 X ∈ Γ(Tλ ∩ E+1 ) and ϕ3 X ∈ Γ(Tλ ∩ E+1 ). Using (5.36) we then get

Aϕ3 X = Aϕ1 ϕ2 X = −Aϕϕ2 X = 3ϕϕ2 X = −3ϕ1 ϕ2 X = −3ϕ3 X, which contradicts ϕ3 X ∈ Γ(Tλ ). Thus we conclude that there exists no ρ ∈ Λ \ {λ} such that Tρ ∩ E−1 ≠ {0}, which means that E−1 ⊆ Tλ . We will now use the above results to derive some restrictions for the principal curvatures. Proposition 5.7.19. Let M be a connected real hypersurface in G2∗ (ℂ2+k ), k ≥ 2. Assume that the maximal complex subbundle 𝒞 of TM and the maximal quaternionic subbundle 𝒟 of TM are both invariant under the shape operator of M. If Jζ ∈ Jζ , then one of the following statements holds: (i) The real hypersurface M has exactly four distinct constant principal curvatures α = 2√2 coth(2√2r),

β = √2 coth(√2r),

λ1 = √2 tanh(√2r),

λ2 = 0,

and the corresponding principal curvature spaces are Tα = 𝒞 ⊥ ,

Tβ = 𝒞 ⊖ 𝒟,

Tλ1 = E−1 ,

Tλ2 = E+1 .

The principal curvature spaces Tλ1 and Tλ2 are complex (with respect to J) and totally complex (with respect to J). (ii) The real hypersurface M has exactly four distinct constant principal curvatures α = 2√2 coth(2√2r),

β = √2 tanh(√2r),

λ1 = √2 coth(√2r),

λ2 = 0,

and the corresponding principal curvature spaces are Tα = 𝒞 ⊥ ,

Tβ = 𝒞 ⊖ 𝒟,

Tλ1 = E−1 ,

Tλ2 = E+1 .

The principal curvature spaces Tλ1 and Tλ2 are complex (with respect to J) and totally complex (with respect to J). (iii) The real hypersurface M has exactly three distinct constant principal curvatures α = 2√2,

β = √2,

λ = 0,

with corresponding principal curvature spaces Tα = 𝒞 ⊥ ,

Tβ = (𝒞 ⊖ 𝒟) ⊕ E−1 ,

Tλ = E+1 .

5.7 Real hypersurfaces with invariant maximal quaternionic subbundle | 199

(iv) We have α = 3,

β2 = β3 = 2,

|Λ| ≥ 2,

λ = 1 ∈ Λ,

E−1 ⊆ Tλ .

Proof. If there exists a principal curvature λ ∈ Λ such that 2λ = βν for some ν ∈ {2, 3}, we get statement (iv) from Lemma 5.7.18. We now assume that 2λ ≠ βν for all λ ∈ Λ and all ν ∈ {2, 3}. We have to show that M satisfies (i), (ii) or (iii). If there exists a principal curvature λ ∈ Λ with 2λ = α, we get case (iii) from Proposition 5.7.12. Thus we can assume that 2λ ≠ α for all λ ∈ Λ. Since 2λ ≠ βν for all λ ∈ Λ, we obtain from Lemma 5.7.14 that βν2 ≠ 4 for ν ∈ {2, 3}. From Lemma 5.7.17 we obtain β2 = β3 . We denote by β the common value of β2 and β3 . Then (5.26) implies β2 − αβ + 2 = 0.

(5.40)

Note that if β is a solution of (5.40), then β2 is the other solution. In case β = √2, (5.40) has a root of multiplicity two. Let λ ∈ Λ and X ∈ Γ(Tλ ). Then, using Lemma 5.7.15, we see that for ν ∈ {2, 3} we have Aϕν X = λ∗ ϕν X with λ∗ =

βλ − 2 . 2λ − β

Note that (λ∗ )∗ = λ since β2 ≠ 4. Moreover, we have Tλ ⊆ E±1 and Tλ∗ ⊆ E−1

Tλ∗ ⊆ E+1

and λ∗2 − αλ∗ + 2 = 0,

and λ (λ − α) = 0, ∗

∗

if Tλ ⊆ E+1 ,

(5.41)

if Tλ ⊆ E−1 .

(5.42)

We choose λ ∈ Λ such that Tλ ⊆ E−1 . From (5.42) we then obtain λ∗ = 0 or λ∗ = α. Assume that λ∗ = α. Then we have α ∈ Λ and Tα ⊆ E+1 . From (5.40) and (5.41) we then get α∗ ∈ {β, β2 }. The equation α∗ = β is equivalent to β2 − αβ − 2 = 0, which contradicts

(5.40). The equation α∗ = β2 is equivalent to α(β2 − 4) = 0. Since β2 ≠ 4 this implies α = 0, which contradicts (5.40). Therefore λ∗ = α is impossible, and we conclude that λ∗ = 0 and hence λ = β2 . Altogether we conclude that Λ = {0, β2 }, T0 = E+1 and T 2 = E−1 . Since ξ = ξ1 , equation (5.11) gives

gradM α = dα(ξ )ξ . So for the Hessian of α we get hessα (X, Y) = g(∇X gradM α, Y) = g(∇X dα(ξ )ξ , Y)

= d(dα(ξ ))(X)η(Y) + dα(ξ )g(∇X ξ , Y)

= d(dα(ξ ))(X)η(Y) + dα(ξ )g(ϕAX, Y)

β

200 | 5 Real hypersurfaces in complex hyperbolic 2-plane Grassmannians for all X, Y ∈ X(M). By symmetry of the Hessian this implies 0 = d(dα(ξ ))(X)η(Y) − d(dα(ξ ))(Y)η(X) + dα(ξ )g((Aϕ + ϕA)X, Y). For X = ξ this gives d(dα(ξ ))(Y) = d(dα(ξ ))(ξ )η(Y). Inserting this and the corresponding equation for X into the previous one yields 0 = dα(ξ )g((Aϕ + ϕA)X, Y). Since E−1 is invariant under both ϕ and A, we get for all X ∈ Γ(E−1 ) that 0 = dα(ξ )g((Aϕ + ϕA)X, ϕX) =

2 dα(ξ )g(ϕX, ϕX), β

which implies dα(ξ ) = 0. Therefore gradM α = dα(ξ )ξ = 0, and since M is connected we conclude that α is constant. It follows easily from (5.40) that α2 ≥ 8. If α2 = 8, then β = α2 and hence 2λ1 − α = 0 for λ1 = β2 ∈ Λ, which contradicts our assumption that 2λ ≠ α for all λ ∈ Λ. Thus we have α2 > 8 and we can write α = 2√2 coth(2√2r) for some r ∈ ℝ+ (and possibly changing the orientation of the normal vector field). From (5.40) we then obtain β = √2 coth(√2r) or β = √2 tanh(√2r). If β = √2 coth(√2r), then λ1 = 2 = √2 tanh(√2r). If β β = √2 tanh(√2r), then λ1 = 2 = √2 coth(√2r). Altogether this shows that statement (i) or (ii) holds.

β

We now come to the main result of Case 2. Theorem 5.7.20. Let M be a connected Hopf hypersurface in G2∗ (ℂ2+k ), k ≥ 3, with A𝒟 ⊆ 𝒟 and assume that the Reeb vector field ξ is perpendicular to 𝒟 everywhere. Then there exists an almost Hermitian structure J1 ∈ J so that Jζ = J1 ζ and one of the following three cases holds: (i) The Hopf hypersurface M is congruent to an open part of the tube with radius r ∈ ℝ+ around the complex and quaternionic totally geodesic embedding of the complex hyperbolic 2-plane Grassmannian G2∗ (ℂ1+k ) into G2∗ (ℂ2+k ). (ii) The Hopf hypersurface M is congruent to an open part of the tube with radius r ∈ ℝ+ around the complex and totally complex totally geodesic embedding of the complex hyperbolic space ℂH k into G2∗ (ℂ2+k ). (iii) The Hopf hypersurface M is an open part of a horosphere in G2∗ (ℂ2+k ) whose center at infinity is singular and of type Jζ ∈ Jζ .

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Proof. In view of Proposition 5.7.19, we need to investigate four different cases. (i): By assumption, the normal vector field ζ of M is singular and of type Jζ ∈ Jζ everywhere, say Jζ = J1 ζ . Using Lemma 1.4.10 we see that the eigenvalues of the normal Jacobi operator Kζ are 0, −2 and −8 with corresponding eigenspaces E+1 , (𝒞 ⊖ 𝒟) ⊕ E−1 and ℝξ , respectively. From Proposition 5.7.19(i) we get E+1 = Tλ2 ,

(𝒞 ⊖ 𝒟) ⊕ E−1 = Tβ ⊕ Tλ1 ,

ℝξ = Tα .

It follows that M is curvature-adapted and we can apply Corollary 2.5.2. For p ∈ M denote by cp the geodesic in G2∗ (ℂ2+k ) with cp (0) = p and ċp (0) = ζp and by Φr the smooth map Φr : M → G2∗ (ℂ2+k ),

p 󳨃→ cp (r).

Geometrically, Φr is the displacement of M at distance r in the direction of the normal vector field ζ . From Corollary 2.5.2(i) we obtain that cp (r) is a focal point of M along cp and, locally, Φr is a submersion onto a (4k −4)-dimensional submanifold P of G2∗ (ℂ2+k ). Moreover, the tangent space of P at Φr (p) is obtained by parallel translation of (Tλ1 ⊕ Tλ2 )(p) = (E+1 ⊕ E−1 )(p) = 𝒟(p), which is a complex and quaternionic subspace of Tp G2∗ (ℂ2+k ). Since both J and J are parallel along cp , also TF(p) P is a complex and quaternionic subspace of TΦr (p) G2∗ (ℂ2+k ). Thus P is a complex and quaternionic submanifold of G2∗ (ℂ2+k ). Since P is quaternionic, it is totally geodesic in G2∗ (ℂ2+k ) (see [3]). The only quaternionic totally geodesic submanifolds of G2∗ (ℂ2+k ) of dimension 4k − 4 are open parts of G2∗ (ℂ1+k ) (see [6] and use duality). So we conclude that P is an open part of a totally geodesic G2∗ (ℂ1+k ) in G2∗ (ℂ2+k ). Rigidity of totally geodesic submanifolds finally implies that M is an open part of the tube with radius r around a totally geodesic G2∗ (ℂ1+k ) in G2∗ (ℂ2+2l ). (ii): By assumption, the normal vector field ζ of M is singular and of type Jζ ∈ Jζ everywhere, say Jζ = J1 ζ . Using Lemma 1.4.10 we see that the eigenvalues of the normal Jacobi operator Kζ are 0, −2 and −8 with corresponding eigenspaces E+1 , (𝒞 ⊖ 𝒟) ⊕ E−1 and ℝξ , respectively. From Proposition 5.7.19(i) we get E+1 = Tλ2 ,

(𝒞 ⊖ 𝒟) ⊕ E−1 = Tβ ⊕ Tλ1 ,

ℝξ = Tα .

It follows that M is curvature-adapted and we can apply Corollary 2.5.2. For p ∈ M denote by cp the geodesic in G2∗ (ℂ2+k ) with cp (0) = p and ċp (0) = ζp and by Φr the smooth map Φr : M → G2∗ (ℂ2+k ),

p 󳨃→ cp (r).

Geometrically, Φr is the displacement of M at distance r in the direction of the normal vector field ζ . From Corollary 2.5.2(i) we obtain that cp (r) is a focal point of M along

202 | 5 Real hypersurfaces in complex hyperbolic 2-plane Grassmannians cp and, locally, Φr is a submersion onto a 2k-dimensional submanifold P of G2∗ (ℂ2+k ). Moreover, the tangent space of P at FΦr (p) is obtained by parallel translation of (Tβ ⊕ Tλ2 )(p) = ((𝒞 ⊖ 𝒟) ⊕ E+1 )(p) = {v ∈ Tp M : Jv = −J1 v}, which is a complex (with respect to J) and totally complex subspace (with respect to J) of Tp G2∗ (ℂ2+k ). Since both J and J are parallel along cp , also TΦr (p) P is a complex and totally complex subspace of TΦr (p) G2∗ (ℂ2+k ). Thus P is a complex and totally complex submanifold of G2∗ (ℂ2+k ). From the classification of maximal totally geodesic submanifolds, as given in Theorem 5.2.1, if follows that P is an open part of the totally geodesic embedding of the complex hyperbolic space ℂH k into G2∗ (ℂ2+k ) or of the totally geodesic embedding of the Riemannian product ℂH a × ℂH k−a of complex hyperbolic spaces ℂH a and ℂH k−a into G2∗ (ℂ2+k ), where 0 < a < k. However, the tangent vectors to ℂH a are either in E+1 or in E−1 and the tangent vectors to ℂH k−a are then in the other eigenspace E−1 or in E+1 , respectively (see Section 17 in [6]). It follows that P is an open part of a totally geodesic ℂH k in G2∗ (ℂ2+k ). Rigidity of totally geodesic submanifolds finally implies that M is an open part of the tube with radius r around a totally geodesic ℂH k in G2∗ (ℂ2+2l ). (iii) As above, for p ∈ M we denote by cp the geodesic in G2∗ (ℂ2+k ) with cp (0) = p and ċp (0) = ζp and by Φr the smooth map Φr : M → G2∗ (ℂ2+k ),

p 󳨃→ cp (r).

For each p ∈ M the differential dp Φr of Φr at p can be computed using Jacobi fields by means of dp Φr (v) = Zv (r), where Zv is the Jacobi field along cp with initial value Zv (0) = v and Zv′ (0) = −Av (see Section 2.5). Using the explicit description of the Jacobi operator in Lemma 1.4.10(ii) for the case Jζ ∈ Jζ we get Bv (t), { { { Zv (t) = {e−√2t Bv (t), { { −2√2t Bv (t), {e

if v ∈ Tλ (p),

if v ∈ Tβ (p),

if v ∈ Tα (p)

for all t ∈ ℝ, where Bv is the parallel vector field along cp with initial value Bv (0) = v. Now consider a geodesic variation in G2∗ (ℂ2+k ) consisting of geodesics cp . The corresponding Jacobi field is a linear combination of the three types of the Jacobi fields Zv listed above, and hence its length remains bounded when t → ∞. This shows that all geodesics cp in G2∗ (ℂ2+k ) are asymptotic to each other and hence determine a singular point z ∈ G2∗ (ℂ2+k )(∞) at infinity of type Jζ ∈ Jζ . Therefore M is an integral manifold of the distribution on G2∗ (ℂ2+k ) given by the orthogonal complements of the tangent vectors of the geodesics in the asymptote class z. This distribution is integrable and the maximal leaves are the horospheres in G2∗ (ℂ2+k ) whose center at infinity is z. Unique-

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ness of integral manifolds of integrable distributions finally implies that M is an open part of a horosphere in G2∗ (ℂ2+k ) whose center is the singular point z at infinity. (iv): Let t ∈ ℝ+ such that √2 coth(√2t) = 2 = β = β2 = β3 . Then we have α = 3 = 2√2 coth(2√2t) and λ = 1 = √2 tanh(√2t). We define cp and Φr as above. Since M is a hypersurface, also Mr = Φr (M) is (locally) a hypersurface for sufficiently small r ∈ ℝ+ . The tangent vector ċp (r) is a unit normal vector of Mr at cp (r). Since ċp (0) = ζp is a singular tangent vector of type Jζ ∈ Jζ , every tangent vector of cp is singular and of type JX ∈ JX. The tangent space Tcp (r) Mr of Mr at cp (r) is obtained by parallel translation of Tp M along cp from cp (0) to cp (r). We denote by 𝒞r the maximal complex subbundle of TMr and by 𝒟r the maximal quaternionic subbundle of TMr . Let Ar be the shape operator of Mr with respect to ċp (r). Using Corollary 2.6.2, we obtain that the parallel translate of ξp along cp from p = cp (0) to cp (t) is a principal curvature vector of Mr with corresponding principal curvature αr = 2√2 coth(2√2(r + t)). It follows that (𝒞r )⊥ , and hence also 𝒞r , are invariant under the shape operator of Mr . Next, using again Corollary 2.6.2, we obtain that the parallel translate of v ∈ Tβ (p) along cp from p = cp (0) to cp (t) is a principal curvature vector of Mr with corresponding principal curvature βr = √2 coth(√2(r + t)). Since Tβ = 𝒞 ⊖ 𝒟 and both J and J are parallel, we conclude that 𝒞r ⊖ 𝒟r is invariant under Ar . Altogether this implies that 𝒟r is invariant under the shape operator of Mr . We thus have proved that Mr satisfies the assumptions of Proposition 5.7.19. It is easy to verify that the principal curvatures of Mr cannot satisfy (ii), (iii) or (iv) in Proposition 5.7.19, and hence must satisfy (i) in Proposition 5.7.19. Therefore Mr is an open part of a tube with radius r + t around a totally geodesic G2∗ (ℂ1+k ) in G2∗ (ℂ2+k ). This implies that M is an open part of a tube with radius t around a totally geodesic G2∗ (ℂ1+k ) in G2∗ (ℂ2+k ).

204 | 5 Real hypersurfaces in complex hyperbolic 2-plane Grassmannians This finishes the discussions for the two cases and we now come to the main result of this section. Theorem 5.7.21 ([16]). Let M be a connected real hypersurface in G2∗ (ℂ2+k ), k ≥ 3. Then the following statements are equivalent: (a) The maximal complex subbundle 𝒞 and the maximal quaternionic subbundle 𝒟 of the tangent bundle of M are invariant under the shape operator A of M; (b) M is congruent to an open part of one of the following hypersurfaces: (i) the tube with radius r ∈ ℝ+ around the complex and quaternionic totally geodesic embedding of G2∗ (ℂ1+k ) into G2∗ (ℂ2+k ), (ii) the tube with radius r ∈ ℝ+ around the complex and totally complex totally geodesic embedding of the complex hyperbolic space ℂH k into G2∗ (ℂ2+k ), (iii) (only if k = 2l is even) the tube with radius r ∈ ℝ+ around the real and quaternionic totally geodesic embedding of the quaternionic hyperbolic space ℍH l into G2∗ (ℂ2+2l ), (iv) a horosphere with singular point at infinity of type Jζ ∈ Jζ , (v) a horosphere with singular point at infinity of type Jζ ⊥ Jζ , (vi) The normal vector field ζ of M is singular of type Jζ ⊥ Jζ and M has at least four distinct principal curvatures, three of which are given by α = β = 2,

γ = 0,

λ = 1,

with corresponding principal curvature spaces Tα = (𝒞 ∩ 𝒟)⊥ ,

Tγ = J 𝒟 ⊥ ,

Tλ = 𝒞 ∩ 𝒟 ∩ J 𝒟 ;

if μ is another (possibly non-constant) principal curvature function, then JTμ ⊆ Tλ and JTμ ⊆ Tλ . Proof. The result follows from Proposition 5.4.3, Proposition 5.4.5, Proposition 5.5.2, Proposition 5.5.4, Proposition 5.5.6, Theorem 5.7.8 and Theorem 5.7.20. Note that case (ii) was overlooked in [16].

5.8 Real hypersurfaces with isometric Reeb flow In this section we investigate and classify real hypersurfaces with isometric Reeb flow in G2∗ (ℂ2+k ), k ≥ 3. We already studied real hypersurfaces with isometric Reeb flow in Kähler manifolds in Section 3.4. In particular, from Proposition 3.4.1 we already know that any such hypersurface is a Hopf hypersurface and satisfies Aϕ = ϕA.

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Proposition 5.8.1. Let M be a real hypersurface in G2∗ (ℂ2+k ) with isometric Reeb flow. Then we have αAϕX − A2 ϕX − ϕX 3

= ∑ (ην (ξ )ϕν X − 4ην (ϕX)ην (ξ )ξ − ην (X)ϕν ξ + 3ην (ϕX)ξν ) ν=1

(5.43)

for all X ∈ X(M). Proof. Since M has isometric Reeb flow, we have Aϕ − ϕA = 0 by Proposition 3.4.1. Differentiating this equation covariantly leads to (∇X A)ϕY + η(Y)A2 X − αg(AX, Y)ξ − ϕ(∇X A)Y − η(AY)AX + g(AX, AY)ξ = 0 for all X, Y ∈ X(M). Taking inner product with Z ∈ X(M) gives g((∇X A)Y, ϕZ) + g((∇X A)Z, ϕY)

= αη(Y)g(AX, Z) + αη(Z)g(AX, Y) − η(Y)g(AX, AZ) − η(Z)g(AX, AY),

which implies g((∇X A)Y, ϕZ) + g((∇X A)Z, ϕY) + g((∇Y A)Z, ϕX)

+ g((∇Y A)X, ϕZ) − g((∇Z A)X, ϕY) − g((∇Z A)Y, ϕX)

= 2αη(Z)g(AX, Y) − 2η(Z)g(AX, AY).

The left-hand side of this equation can be rewritten as 2g((∇X A)Y, ϕZ) − g((∇X A)Y − (∇Y A)X, ϕZ)

+ g((∇Y A)Z − (∇Z A)Y, ϕX) − g((∇Z A)X − (∇X A)Z, ϕY),

and using the Codazzi equation this implies g((∇X A)Y, ϕZ)

= 3αη(Z)g(AX, Y) − 3η(Z)g(AX, AY) + η(Y)g(X, Z) − η(Z)g(X, Y) 3

− ∑ (ην (Y)g(ϕν ϕX, Z) − ην (Z)g(ϕν ϕX, Y) ν=1

+ ην (ϕY)g(ϕν X, Z) − ην (ϕZ)g(ϕν X, Y) + 2ην (ϕX)g(ϕν Y, Z)). Replacing Z by ϕZ, the left-hand side of the previous equation becomes −g((∇X A)Y, Z) + η(Z)g((∇X A)Y, ξ ).

206 | 5 Real hypersurfaces in complex hyperbolic 2-plane Grassmannians Since g((∇X A)Y, ξ ) = g((∇X A)ξ , Y) = g(∇X (αξ ), Y) − g(A∇X ξ , Y), we see that dα(X)η(Y)η(Z) + αη(Z)g(ϕAX, Y) − η(Z)g(AϕAX, Y) − g((∇X A)Y, Z) becomes the left-hand side of the previous equation when we replace Z by ϕZ. Replacing Z by ϕZ also on the right-hand side we eventually get 3

(∇X A)Y = η(Y)ϕX + ∑ (ην (Y)ϕν X − ην (ϕY)ϕϕν X − 2ην (ϕX)ϕϕν Y) ν=1

+ (dα(X)η(Y) + αg(AϕX, Y) − g(A2 ϕX, Y))ξ 3

− ( ∑ (ην (ϕX)ην (Y) + ην (ξ )g(ϕν X, Y)))ξ 3

ν=1

3

+ ∑ g(ϕν X, Y)ξν − ∑ (η(X)ην (Y) − g(ϕν ϕX, Y))ϕν ξ . ν=1

ν=1

Inserting Y = ξ then yields 3

(∇X A)ξ = ϕX + ∑ (ην (ξ )ϕν X − 2ην (ϕX)ϕϕν ξ ) ν=1

3

+ (dα(X) − ∑ (ην (ϕX)ην (ξ ) + ην (ξ )g(ϕν X, ξ )))ξ 3

ν=1

3

+ ∑ g(ϕν X, ξ )ξν − ∑ (η(X)ην (ξ ) − g(ϕν ϕX, ξ ))ϕν ξ . ν=1

ν=1

On the other hand, we have (∇X A)ξ = dα(X)ξ + α∇X ξ − A∇X ξ = dα(X)ξ + αϕAX − A2 ϕX. Comparing these two expressions for (∇X A)ξ implies equation (5.43). An important step towards the classification of real hypersurfaces with isometric Reeb flow in G2∗ (ℂ2+k ) is the following. Proposition 5.8.2. Let M be a real hypersurface in G2∗ (ℂ2+k ) with isometric Reeb flow. Then the maximal quaternionic subbundle 𝒟 of TM is contained in the maximal complex subbundle 𝒞 of TM. In particular, the Reeb vector field ξ is perpendicular to 𝒟 everywhere, or equivalently, the unit normal vector field ζ of M is singular of type Jζ ∈ Jζ everywhere.

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Proof. Let X ∈ Γ(𝒟). From equation (5.43) we get αg(AϕX, ϕX) − g(A2 ϕX, ϕX) − g(ϕX, ϕX) 3

= ∑ (3ην (ϕX)2 − ην (X)2 + ην (ξ )g(ϕν X, ϕX)), ν=1

and equation (5.10) implies αg(AϕX, ϕX) − g(A2 ϕX, ϕX) − g(ϕX, ϕX) 3

= ∑ (ην (X)2 + ην (ϕX)2 + ην (ξ )g(ϕν X, ϕX)). ν=1

Comparing the previous two equations shows that 3

3

ν=1

ν=1

∑ ην (X)2 = ∑ ην (ϕX)2

for all X ∈ Γ(𝒟). Consequently, ϕX ∈ Γ(𝒞 ∩ 𝒟) for all X ∈ Γ(𝒞 ∩ 𝒟). This shows that 𝒞 ∩ 𝒟 is ϕ-invariant. Since ξ is perpendicular to 𝒞 , and hence also to 𝒞 ∩ 𝒟, the subspace 𝒞 ∩ 𝒟 has even dimension (pointwise). As dim(𝒞 ∩ 𝒟) = dim(𝒞 ) + dim(𝒟) − dim(𝒞 + 𝒟), dim(𝒞 ) = 4k − 2, dim(𝒟) = 4k − 4 and dim(𝒞 + 𝒟) ∈ {4k − 2, 4k − 1}, we must have dim(𝒞 + 𝒟) = 4k − 2. This means that 𝒟 ⊂ 𝒞 , which shows that ξ is perpendicular to 𝒟 everywhere. Thus there exists an almost Hermitian structure J1 ∈ J so that ξ = ξ1 , which is equivalent to Jζ = J1 ζ and shows that ζ is singular everywhere. We will now prove that the principal curvatures of a real hypersurface in G2∗ (ℂ2+k ) with isometric Reeb flow are constant. Proposition 5.8.3. Let M be a connected real hypersurface in G2∗ (ℂ2+k ) with isometric Reeb flow. We choose a canonical local basis in J so that ξ1 = ξ (see Proposition 5.8.2). Then the Hopf principal curvature function α is constant and for all X ∈ Γ(𝒞 ) with AX = λX one of the following two statements holds: (i) λ(λ − α) = 0, X ∈ Γ(𝒟) and ϕX = −ϕ1 X; (ii) λ2 − αλ + 2 = 0 and ϕX𝒟 = ϕ1 X𝒟 , where X𝒟 denotes the orthogonal projection of X onto 𝒟. In particular, all principal curvatures of M are constant. Proof. Since ξ = ξ1 , equation (5.11) reduces to gradM α = dα(ξ )ξ ,

208 | 5 Real hypersurfaces in complex hyperbolic 2-plane Grassmannians which implies ∇X gradM α = d(dα(ξ ))(X)ξ + dα(ξ )ϕAX for all X ∈ X(M). The symmetry of the Hessian of α and the identity Aϕ = ϕA imply 2dα(ξ )g(AϕX, Y) = d(dα(ξ ))(X)η(Y) − d(dα(ξ ))(Y)η(X) for all X, Y ∈ X(M). Putting X = ξ implies d(dα(ξ ))(Y) = d(dα(ξ ))(ξ )η(Y) for all Y ∈ X(M), and inserting this and the corresponding expression for d(dα(ξ ))(X) into the previous equation implies dα(ξ )g(AϕX, Y) = 0 for all X, Y ∈ X(M). Thus either dα(ξ ) = 0, which implies that gradM α = 0 and hence that α is constant, or Aϕ = 0. But the latter equation implies that A restricted to 𝒞 vanishes. In particular, inserting X = ξ2 into equation (5.43) yields ξ3 = 0, which is impossible. Thus α is constant. Now let X ∈ Γ(𝒞 ) with AX = λX. First, we have AϕX = ϕAX = λϕX. Thus, from equation (5.12) we get (λ2 − αλ + 1)ϕX + ϕ1 X = 2η2 (X)ξ3 − 2η3 (X)ξ2 . Replacing X by ϕX we obtain (λ2 − αλ + 1)X − ϕ1 ϕX = −2η2 (X)ξ2 − 2η3 (X)ξ3 . We now decompose X into X = X𝒟 + η2 (X)ξ2 + η3 (X)ξ3 and insert this expression into the previous equation, which leads to 0 = (λ2 − αλ + 1)X𝒟 − ϕ1 ϕX𝒟 + (λ2 − αλ + 2)η2 (X)ξ2 + (λ2 − αλ + 2)η3 (X)ξ3 . It is clear that ϕ1 , and hence also ϕ, leaves 𝒟 invariant. Thus we have ϕ1 ϕX𝒟 ∈ Γ(𝒟), and therefore the previous equation splits into three equations: 0 = (λ2 − αλ + 1)X𝒟 − ϕ1 ϕX𝒟 , 0 = (λ2 − αλ + 2)η2 (X)ξ2 ,

0 = (λ2 − αλ + 2)η3 (X)ξ3 .

If λ2 − αλ + 2 = 0, then the first equation implies ϕ1 ϕX𝒟 = −X𝒟 . On the other hand, if λ2 −αλ+2 ≠ 0, then the last two equations imply that X ∈ Γ(𝒟). The first equation shows

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that ϕ1 ϕX𝒟 and X𝒟 are proportional. In fact, since both ϕ1 and ϕ act orthogonally on 𝒟, we must have ϕ1 ϕX𝒟 = ±X𝒟 . If ϕ1 ϕX𝒟 = −X𝒟 , then the first equation yields λ2 − αλ + 2 = 0, which is a contradiction. Hence we must have ϕ1 ϕX𝒟 = X𝒟 , and the first equation implies λ2 − αλ = 0. This shows that either (i) or (ii) holds. The constancy of the principal curvatures then follows from the constancy of α. We can now state and prove the main result of this section. Theorem 5.8.4 ([97]). Let M be a connected real hypersurface in G2∗ (ℂ2+k ), k ≥ 3. Then M has isometric Reeb flow if and only if M is congruent to an open part of one of the following real hypersurfaces: (i) the tube with radius r ∈ ℝ+ around the complex and quaternionic totally geodesic embedding of the complex hyperbolic 2-plane Grassmannian G2∗ (ℂ1+k ) into G2∗ (ℂ2+k ), (ii) the tube with radius r ∈ ℝ+ around the complex and totally complex totally geodesic embedding of the complex hyperbolic space ℂH k into G2∗ (ℂ2+k ), (iii) the horosphere in G2∗ (ℂ2+k ) whose center at infinity is singular and of type Jζ ∈ Jζ . Proof. We know from Proposition 5.5.2, Proposition 5.5.4 and Proposition 5.4.3 that the real hypersurfaces in (i), (ii) and (iii) have isometric Reeb flow. Conversely, let M be a connected real hypersurface in G2∗ (ℂ2+k ) with isometric Reeb flow. From Proposition 5.8.3 we see that M has constant principal curvatures, and the number of different principal curvatures is at most four. Moreover, it is easy to see from Proposition 5.8.3 that the Hopf principal curvature α satisfies α ≠ 0. With a suitable orientation of the unit normal vector field ζ we can assume α > 0. We can also see from Proposition 5.8.3 that the quadratic equation x 2 − αx + 2 must have a solution, which implies α ≥ 2√2. Case 1: α = 2√2. Then the quadratic equation x2 − αx + 2 has only one solution, namely λ = √2. It follows from Proposition 5.8.3 that the corresponding eigenspaces are {v ∈ 𝒟 : ϕv = ϕ1 v} ⊕ (𝒟⊥ ⊖ 𝒞 ⊥ ). This shows in particular that 𝒟 is invariant by the shape operator A. It follows that M is a Hopf hypersurface with A𝒟 ⊆ 𝒟 and hence congruent to an open part of one of the real hypersurfaces listed in Theorem 5.7.21(b). The geometry of these real hypersurfaces was investigated in Sections 5.4 and 5.5 and it follows from these results that M is an open part of a horosphere in G2∗ (ℂ2+k ) whose center at infinity is singular and of type Jζ ∈ Jζ . Case 2: α > 2√2. Then there exists r ∈ ℝ+ so that α = 2√2 coth(2√2r).

210 | 5 Real hypersurfaces in complex hyperbolic 2-plane Grassmannians Then, from Proposition 5.8.3, the other possible principal curvatures are μ = 0,

β = √2 coth(√2r),

λ = √2 tanh(√2r).

Note that β and λ are the two different solutions of the quadratic equation x2 − αx + 2 = 0. For ρ ∈ {α, μ, β, λ} we define Tρ = {v ∈ 𝒞 : Av = ρv}. Then we have 𝒞 = Tα ⊕ Tμ ⊕ Tβ ⊕ Tλ

and, if Tρ is non-trivial, Tρ is the subbundle of TM consisting of all principal curvature vectors of M with respect to ρ and are tangent to 𝒞 . According to Proposition 3.4.1, each Tρ is a complex subbundle of TM. For p ∈ M we denote by cp the geodesic in G2∗ (ℂ2+k ) with cp (0) = p and ċp (0) = ζp and by Φr the smooth map Φr : M → G2∗ (ℂ2+k ),

p 󳨃→ cp (r).

Geometrically, Φr is the displacement of M at distance r in direction of the normal vector field ζ . For each p ∈ M the differential dp Φr of Φr at p can be computed using Jacobi fields by means of dp Φr (v) = Zv (r) (see Section 2.5). Here, Zv is the Jacobi field along cp with initial values Zv (0) = v and Zv′ (0) = −Av. Since Jζ = −ξ = −ξ1 = J1 ζ , we see that ζ is a singular tangent vector of type Jζ ∈ Jζ at each point. Using the explicit description of the Jacobi operator R̄ ζ for the case Jζ ∈ Jζ in Lemma 1.4.7 and of the shape operator A of M in Proposition 5.8.3 we see that Tα ⊕Tμ is contained in the 0-eigenspace of R̄ ζ and Tβ ⊕Tλ in the (−2)-eigenspace of R̄ ζ . For the Jacobi fields along cp we thus get the expressions (cosh(2√2t) − 2√α 2 sinh(2√2t))Bv (t), { { { Zv (t) = {(cosh(√2t) − √ρ sinh(√2t))Bv (t), 2 { { (1 − ρt)B (t), v {

if v ∈ ℝξp ,

if v ∈ Tρ (p) and ρ ∈ {β, λ},

if v ∈ Tρ (p) and ρ ∈ {α, μ},

where Bv (t) denotes the parallel vector field along cp (t) with Bv (0) = v. This shows that the kernel of dΦr is ℝξ ⊕Tβ and that Φr is of constant rank equal to rk(Tα ⊕ Tλ ⊕ Tμ ). So,

5.8 Real hypersurfaces with isometric Reeb flow

| 211

locally, Φr is a submersion onto a submanifold P of G2∗ (ℂ2+k ). Moreover, the tangent space of P at Φr (p) is obtained by parallel translation of (Tα ⊕ Tλ ⊕ Tμ )(p) (see Theorem 2.5.1), which is a complex subspace of Tp G2∗ (ℂ2+k ). Since J is parallel along cp , also TΦr (p) P is a complex subspace of TΦr (p) G2∗ (ℂ2+k ). Thus P is a complex submanifold of G2∗ (ℂ2+k ). It is clear that ηp = ċp (r) is a unit normal vector of P at Φr (p). The shape operator Bηp of P with respect to ηp is given by Bηp Zv (r) = −Zv′ (r) (see Theorem 2.5.1). From this we immediately get that for each ρ ∈ {α, λ, μ} the parallel translate of Tρ (p) along cp from p to Φr (p) is a principal curvature space of P with respect to ρ, provided that Tρ (p) is non-trivial. Moreover, the corresponding principal

2 2 coth(2 2r) for ρ = α and 0 for ρ ∈ {μ, λ}. Since any complex submancurvature is 1−2 √2 coth(2√2r)r ifold of a Kähler manifold is minimal this implies that Tα is trivial. The vectors of the form ηq , q ∈ Φ−1 r ({Φr (p)}), form an open subset of the unit sphere in the normal space of P at Φr (p). Since Bηq vanishes for all ηq it now follows that P is totally geodesic in √

√

G2∗ (ℂ2+k ). Rigidity of totally geodesic submanifolds now implies that the entire submanifold M is an open part of a tube with radius r around some connected, complete, totally geodesic, complex submanifold P of G2∗ (ℂ2+k ). The vector space {v ∈ Tp G2∗ (ℂ2+k ) : Jv = −J1 v} is the +1-eigenspace of the self-adjoint endomorphism JJ1 = J1 J. Using the fact that (JJ1 )2 = I and tr(JJ1 ) = 0 we easily see that this eigenspace is a complex vector space of complex dimension k. Thus Tα (p) ⊕ Tμ (p) = {v ∈ 𝒟(p) : Jv = −J1 v} is a complex vector space of complex dimension k − 1, and the above arguments show that the parallel translate of this complex vector space along cp from p to Φr (p) lies in TΦr (p) P. Since both J and J are parallel along cp , it follows that TΦr (p) P contains a (k − 1)-dimensional complex subspace of the form {v : Jv = J1′ v} for some fixed almost Hermitian structure J1′ ∈ J. Any such subspace is invariant under the curvature tensor R̄ and hence there exists a connected, complete, totally geodesic, complex submanifold Q of G2∗ (ℂ2+k ) with Φr (p) ∈ Q and TΦr (p) Q equal to that curvature-invariant complex subspace. Since that curvature-invariant subspace consists entirely of singular tangent vectors of this special type we see that Q is a totally geodesic ℂH k−1 . We thus conclude that P contains a totally geodesic ℂH k−1 . The classification of totally geodesic submanifolds in G2∗ (ℂ2+k ) shows that the only totally geodesic submanifolds in G2∗ (ℂ2+k ) containing a totally geodesic ℂH k−1 are ℂH k−1 , ℂH k , ℂH k−1 × ℂH 1 and

212 | 5 Real hypersurfaces in complex hyperbolic 2-plane Grassmannians G2∗ (ℂ1+k ). The normal spaces of both ℂH k−1 and ℂH k−1 × ℂH 1 contain regular tangent vectors of G2∗ (ℂ2+k ). Emanating along the geodesic in direction of such a regular tangent vector would give a normal vector to M which is a regular. But this contradicts the fact that the normal vector field ζ of M is singular everywhere. Thus P is either ℂH k or G2∗ (ℂ1+k ). It follows that M is congruent to an open part of the tube with radius r around a totally geodesic G2∗ (ℂ1+k ) in G2∗ (ℂ2+k ) or congruent to an open part of the tube with radius r around a totally geodesic ℂH k in G2∗ (ℂ2+k ). This finishes the proof.

5.9 Contact hypersurfaces In this section we study contact hypersurfaces in the complex hyperbolic 2-plane Grassmannian G2∗ (ℂ2+k ), k ≥ 3. We already investigated contact hypersurfaces in Kähler manifolds and in Hermitian symmetric spaces in Section 3.5. We also proved in Proposition 5.5.6 that the tube with radius r ∈ ℝ+ around the totally geodesic ℍH l ⊂ G2∗ (ℂ2+2l ), l ≥ 2, is a contact hypersurface. We also proved in Proposition 5.4.5 that a horosphere with singular point at infinity of type Jζ ⊥ Jζ is a contact hypersurface. A natural question is whether there are other contact hypersurfaces in G2∗ (ℂ2+k ). We will see that there are no other contact hypersurfaces. Let M be a connected contact hypersurface in G2∗ (ℂ2+k ), k ≥ 3. From Proposition 3.5.4 and Proposition 3.5.5 we see that there exists a constant f ≠ 0 such that Aϕ + ϕA = 2fϕ. Moreover, from Proposition 3.5.6 we know that M is a Hopf hypersurface and the mean curvature of M is given by tr(A) = α + (4k − 2)f , where α = g(Aξ , ξ ) is the Hopf principal curvature function. We also know that for all X ∈ Γ(𝒞 ) with AX = λX we have AϕX = (2f − λ)ϕX. From Proposition 3.5.9 we have the useful equation ̄ , ζ )ϕX) 2(A2 − 2fA + αf )X = −(R(ξ 𝒞

(5.44)

for all X ∈ Γ(𝒞 ). Using the explicit expression for the Riemannian curvature tensor R̄ of G2∗ (ℂ2+k ) in Theorem 1.4.9 we compute (note that ϕX = JX for X ∈ Γ(𝒞 )) ̄ , ζ )ϕX = −g(ζ , ϕX)ξ + g(ξ , ϕX)ζ − g(Jζ , ϕX)Jξ + g(Jξ , ϕX)Jζ + 2g(Jξ , ζ )JϕX R(ξ 3

− ∑ (g(Jν ζ , ϕX)Jν ξ − g(Jν ξ , ϕX)Jν ζ − 2g(Jν ξ , ζ )Jν ϕX ν=1

5.9 Contact hypersurfaces | 213

+ g(Jν Jζ , ϕX)Jν Jξ − g(Jν Jξ , ϕX)Jν Jζ ) 3

= −2X − 2 ∑ (−g(ξν , ϕX)Jν ξ + g(Jν ξ , ϕX)ξν − g(ξ , ξν )Jν ϕX) ν=1 3

= −2X + 2 ∑ (ην (ϕX)Jν ξ + η(ϕν ϕX)ξν + ην (ξ )Jν ϕX). ν=1

The 𝒞 -component of Jν ξ is ϕν ξ , the 𝒞 -component of ξν is ξν − η(ξν )ξ and the 𝒞 -component of Jν ϕX is ϕν ϕX − η(ϕν ϕX)ξ . Altogether, we get 3

̄ , ζ )ϕX) = −2X + 2 ∑ (ην (ϕX)ϕν ξ + η(ϕν ϕX)ξν − η(ϕν ϕX)η(ξν )ξ (R(ξ 𝒞 ν=1

+ ην (ξ )ϕν ϕX − ην (ξ )η(ϕν ϕX)ξ ) 3

= −2X + 2 ∑ (ην (ϕX)ϕν ξ + η(ϕν ϕX)ξν ν=1

+ ην (ξ )ϕν ϕX − 2η(ϕν ϕX)η(ξν )ξ ). We have η(ϕν ϕX) = ην (ϕ2 X) = −ην (X) for X ∈ Γ(𝒞 ) and hence 3

̄ , ζ )ϕX) = −2X + 2 ∑ (ην (ϕX)ϕν ξ − ην (X)ξν (R(ξ 𝒞 ν=1

+ ην (ξ )ϕν ϕX + 2ην (X)η(ξν )ξ ). Combining this with equation (5.44) gives the following. Proposition 5.9.1. Let M be a connected contact hypersurface in G2∗ (ℂ2+k ). Then we have (A2 − 2fA + αf − 1)X 3

= ∑ (ην (X)ξν − ην (ϕX)ϕν ξ − ην (ξ )ϕν ϕX − 2ην (X)η(ξν )ξ ) ν=1

(5.45)

for all X ∈ Γ(𝒞 ). Proposition 5.9.2. Let M be a connected contact hypersurface in G2∗ (ℂ2+k ). Then ξ is tangent to 𝒟 everywhere. Proof. From Theorem 3.5.19 we know that the Reeb vector field ξ is an eigenvector of the normal Jacobi operator K everywhere. From Proposition 4.1.4 we then see that ξ is either tangent to 𝒟 or tangent to 𝒟⊥ everywhere. Let us assume that ξ is tangent to 𝒟⊥ everywhere. With a suitable choice of local canonical basis we can assume that ξ = ξ1 (locally). Then, using the basic structure equations, equation (5.45) be-

214 | 5 Real hypersurfaces in complex hyperbolic 2-plane Grassmannians comes (A2 − 2fA + αf − 1)X = 2η2 (X)ξ2 + 2η3 (X)ξ3 − ϕ1 ϕX

(5.46)

for all X ∈ Γ(𝒞 ). Let 0 ≠ X ∈ Γ(𝒞 ) with AX = λX. Then (5.46) implies (λ2 − 2fλ + αf − 1)X = 2η2 (X)ξ2 + 2η3 (X)ξ3 − ϕ1 ϕX. We decompose X into X = X𝒟 +η2 (X)ξ2 +η3 (X)ξ3 , where X𝒟 is the orthogonal projection of X onto 𝒟. Then we have ϕ1 ϕX = ϕ1 ϕX𝒟 + η2 (X)ξ2 + η3 (X)ξ3 , and inserting this into the previous equation leads to the equation 0 = (λ2 − 2fλ + αf − 1)X𝒟 + ϕ1 ϕX𝒟 . Since ξ is tangent to 𝒟⊥ everywhere, we have ϕ1 ϕ = J1 J on 𝒟. Since the trace of J1 J is zero and ξ = ξ1 , this trace is zero when restricting J1 J to 𝒟. Therefore, since J1 J is a multiple of the identity on 𝒟, this multiple must be zero, and hence J1 J = 0 on 𝒟, which is a contradiction. It follows that ξ is tangent to 𝒟 everywhere. We now state the main result of this section, which was proved by the authors and Lee in [11] under the additional assumption that the contact hypersurface has constant mean curvature. This assumption now turns out to be redundant by Theorem 3.5.19. Theorem 5.9.3. Let M be a connected real hypersurface in G2∗ (ℂ2+k ), k ≥ 3. Then M is a contact hypersurface if and only if: (i) the quaternionic dimension k of G2∗ (ℂ2+k ) is even, say k = 2l, and M is an open part of the tube with radius r ∈ ℝ+ around a totally geodesic quaternionic hyperbolic space ℍH l in G2∗ (ℂ2+2l ); or (ii) M is an open part of a horosphere in G2∗ (ℂ2+k ) whose center at infinity is singular and of type Jζ ⊥ Jζ . Proof. The tube with radius r ∈ ℝ+ around a totally geodesic quaternionic hyperbolic space ℍH l in G2∗ (ℂ2+2l ) is a contact hypersurface by Proposition 5.5.6. A horosphere in G2∗ (ℂ2+k ) whose center at infinity is singular and of type Jζ ⊥ Jζ is a contact hypersurface by Proposition 5.4.5. Conversely, assume that M is a connected contact hypersurface. Then M is a Hopf hypersurface by Proposition 3.5.6. By Theorem 3.5.19, the Reeb vector field ξ is an eigenvector of the normal Jacobi operator K everywhere. From Proposition 5.1.4 we

5.10 Pseudo-Einstein real hypersurfaces | 215

see that ξ is tangent to 𝒟 or to 𝒟⊥ everywhere. Then Proposition 5.9.2 tells us that ξ is tangent to 𝒟 everywhere. We then apply Proposition 5.6.3 to see that the maximal quaternionic subbundle D of TM is invariant under the shape operator of M, that is, A𝒟 ⊆ 𝒟. From Theorem 5.7.8 we then obtain that M is an open part of the tube with radius r ∈ ℝ+ around a totally geodesic quaternionic hyperbolic space ℍH l in G2∗ (ℂ2+2l ) (if k = 2l is even) or of a horosphere in G2∗ (ℂ2+k ) whose center at infinity is singular and of type Jζ ⊥ Jζ , or M has at least four distinct principal curvatures, three of which are given by α = β = 2,

γ = 0,

λ=1

with corresponding principal curvature spaces Tα = (𝒞 ∩ 𝒟)⊥ ,

Tγ = J 𝒟⊥ ,

Tλ = 𝒞 ∩ 𝒟 ∩ J 𝒟 .

If μ is another (possibly non-constant) principal curvature function, then JTμ ⊂ Tλ and JTμ ⊂ Tλ . The last case is an exceptional case and we do not know if such a hypersurface exists. However, we do know the principal curvatures and the structure of the principal curvature spaces. Both 𝒟⊥ and ϕ𝒟⊥ are invariant under A with corresponding principal curvatures 2 and 0, respectively. This implies Aϕ + ϕA = 2ϕ on 𝒟 ⊕ 𝒟⊥ . There are two further principal curvatures 1 and μ ≠ 1 and ϕ maps the principal curvature space Tμ into the principal curvature space of 1. This implies Aϕ + ϕA = (μ + 1)ϕ ≠ 2ϕ on Tμ ⊕ ϕTμ . Thus, even if such a hypersurface exists, it cannot be a contact hypersurface.

5.10 Pseudo-Einstein real hypersurfaces Let M be a real hypersurface in G2∗ (ℂ2+k ) and Ric the Ricci tensor of M. We already computed the Ricci tensor in Theorem 5.1.6. More precisely, we have Ric(X) = −4(k + 2)X − KX + hAX − A2 X for all X ∈ X(M), where K is the normal Jacobi operator of M and h = tr(A). We computed the normal Jacobi operator of real hypersurfaces in G2∗ (ℂ2+k ) in Theorem 5.1.3, namely 3

KX = −X − 3η(X)ξ − ∑ (3ην (X)ξν + ην (ξ )η(X)ξν + ην (ϕX)ϕν ξ − ην (ξ )ϕν ϕX). ν=1

216 | 5 Real hypersurfaces in complex hyperbolic 2-plane Grassmannians Altogether, this gives Ric(X) = −(4k + 7)X + 3η(X)ξ + hAX − A2 X 3

+ ∑ (3ην (X)ξν + ην (ξ )η(X)ξν + ην (ϕX)ϕν ξ − ην (ξ )ϕν ϕX) ν=1

for all X ∈ X(M). Recall that a Riemannian manifold (M, g) is said to be an Einstein manifold if there exists a real number c ∈ ℝ such that the Ricci tensor of (M, g) satisfies Ric(X) = cX for all X ∈ X(M). Classically, Einstein hypersurfaces in real space forms have been studied by many differential geometers; see for instance Cartan [30] and Fialkow [43]. In complex space forms and in quaternionic space forms some geometers studied Einstein real hypersurfaces, Einstein complex hypersurfaces and more generally real hypersurfaces with parallel Ricci tensor; see Cecil and Ryan [32], Kimura [54, 55], Romero [85, 86] and Martínez and Pérez [69]. In this context, Kon [61] considered the concept of pseudo-Einstein real hypersurfaces in complex projective spaces. A real hypersurface M in complex projective space ℂP n is said to be pseudo-Einstein if there exist real numbers a, b ∈ ℝ such that the Ricci tensor of M satisfies Ric(X) = aX + bη(X)ξ for all X ∈ X(M). In [61], Kon has also given a complete classification of pseudoEinstein real hypersurfaces in ℂP n by using the work of Takagi [116] and proved that there do not exist any Einstein real hypersurfaces in ℂP n , n ≥ 3. The concept of pseudo-Einstein was generalized by Cecil and Ryan [32] by requiring that a, b are smooth functions on M rather than real numbers. By using the general theory of tubes, Cecil and Ryan [32] obtained a complete classification of such generalized pseudo-Einstein real hypersurfaces, and as a special case they also concluded that there do not exist any Einstein real hypersurfaces in ℂP n , n ≥ 3. On the other hand, Montíel [71] considered pseudo-Einstein real hypersurfaces in complex hyperbolic space ℂH n and derived a complete classification, which implied that there do not exist Einstein real hypersurfaces in ℂH n , n ≥ 3. For real hypersurfaces in quaternionic projective space ℍP n , the concept of pseudo-Einstein was considered by Martínez and Pérez [69]. In contrast to the complex case, there exist Einstein real hypersurfaces, and Pérez [81] proved that the only Einstein real hypersurfaces in ℍP n are (open parts of) the geodesic hyperspheres with 1 . radius r, where 0 < r < π2 and cot2 (r) = 2n

5.10 Pseudo-Einstein real hypersurfaces | 217

A real hypersurface M in the complex hyperbolic 2-plane Grassmannian G2∗ (ℂ2+k ) is said to be pseudo-Einstein if there exist real numbers a, b, c ∈ ℝ such that the Ricci tensor of M satisfies 3

Ric(X) = aX + bη(X)ξ + c ∑ ην (X)ξν ν=1

for all X ∈ X(M). Pérez, Suh and Watanabe [83] defined the concept of pseudo-Einstein real hypersurfaces in the dual compact Grassmannian G2 (ℂ2+k ) with the assumption that b and c are non-zero. In this case, the real hypersurface is said to be proper pseudoEinstein. In [83] a complete classification of proper pseudo-Einstein Hopf hypersurfaces in G2 (ℂ2+k ) was obtained. Motivated by these results, Suh [102] investigated pseudo-Einstein real hypersurfaces in G2∗ (ℂ2+k ) and obtained the following result. Theorem 5.10.1 ([102]). Let M be a pseudo-Einstein Hopf hypersurface in G2∗ (ℂ2+k ), k ≥ 3. Then M has four distinct constant principal curvatures 2, 0, λ = 1 and μ = q−4k+3 q such that p + q = 4(k − 2), where p and q denote the multiplicities of the principal curvatures λ and μ, respectively. In this case, M is a proper pseudo-Einstein real hypersurface with a = − 4k+5 , b = c = √32 . √2 It follows from Theorem 5.10.1 that there exist only proper pseudo-Einstein Hopf hypersurfaces in G2∗ (ℂ2+k ). From this we can conclude that there do not exist any Einstein Hopf hypersurfaces in G2∗ (ℂ2+k ), k ≥ 3. Now let us consider 𝒟⊥ -invariant Einstein real hypersurfaces in G2∗ (ℂ2+k ). Then from the 𝒟⊥ -invariant property we can prove that the Reeb vector field ξ is tangent to 𝒟 or to 𝒟⊥ everywhere. In each of these two cases we can deduce that M is Hopf. So, by using the classification of pseudo-Einstein Hopf hypersurfaces in Theorem 5.10.1, we obtain the following non-existence result. Theorem 5.10.2 ([102]). There are no 𝒟⊥ -invariant Einstein real hypersurfaces in G2∗ (ℂ2+k ), k ≥ 3.

6 Real hypersurfaces in complex quadrics In this chapter we investigate real hypersurfaces in the complex quadric Qn = SO2+n /SO2 SOn , n ≥ 3. We denote by g the Riemannian metric, by ∇̄ the Riemannian connection, by R̄ the Riemannian curvature tensor and by ric the Ricci tensor of Qn . We continue using other notations that we introduced in Section 1.4.2.

6.1 Basic equations for real hypersurfaces Let M be a real hypersurface in Qn with unit normal vector field ζ . We denote by g the Riemannian metric, by ∇ the Riemannian connection, by R the Riemannian curvature tensor and by A the shape operator of M with respect to the unit normal vector field ζ . The Kähler structure J on Qn induces an almost contact metric structure (ϕ, ξ , η, g) on M. The vector field ξ = −Jζ is the Reeb vector field on M. The maximal holomorphic subbundle of TM is denoted by 𝒞 and the orthogonal complement of 𝒞 in TM is denoted by 𝒞 ⊥ . Thus we have TM = 𝒞 ⊕ 𝒞 ⊥ ,

⊥

𝒞 = ℝξ = ℝJζ .

For a real structure C on Qn and X ∈ X(M) we decompose CX into its tangential and normal component, that is, CX = BX + ρ(X)ζ , where BX is the tangential component of CX and ρ(X) = g(CX, ζ ) = g(CX, Jξ ) = −g(JCX, ξ ) = g(CJX, ξ ) = g(JX, Cξ ). Since JX = ϕX + η(X)ζ and Cξ = Bξ + ρ(ξ )ζ , we also have ρ(X) = g(ϕX, Bξ ) + η(X)ρ(ξ ) = η(BϕX) + η(X)ρ(ξ ). We also define δ = g(ζ , Cζ ) = g(Jξ , CJξ ) = −g(Jξ , JCξ ) = −g(ξ , Cξ ) = −g(ξ , Bξ ) = −η(Bξ ). At each point p ∈ M we define 𝒬p = {v ∈ Tp M : Cv ∈ Tp M for all C ∈ Ap },

which is the maximal Ap -invariant subspace of Tp M. Lemma 6.1.1. Let M be a real hypersurface in Qn . The following statements are equivalent: https://doi.org/10.1515/9783110689839-006

220 | 6 Real hypersurfaces in complex quadrics (i) the normal vector ζp is an A-principal singular tangent vector of Qn ; (ii) the Reeb vector ξp is an A-principal singular tangent vector of Qn ; (iii) 𝒬p = 𝒞p ; (iv) there exists a real structure C ∈ Ap such that Cζp ∈ ℂνp M. Proof. We first assume that ζp is A-principal. By definition, this means that there exists a real structure C ∈ Ap such that ζp ∈ V(C), that is, Cζp = ζp . Then we have Cξp = −CJζp = JCζp = Jζp = −ξp . Thus ξp is A-principal with respect to the real structure −C. Conversely, assume that ξp is A-principal. Then there exists a real structure C ∈ Ap such that ξp ∈ V(C), that is, Cξp = ξp . We then have Cζp = CJξp = −JCξp = −Jξp = −ζp . Thus ζp is A-principal with respect to the real structure −C. Altogether we proved the equivalence of (i) and (ii). We assume again that ζp is A-principal, say Cζp = ζp . Then, as we just proved, we have Cξp = −ξp . It follows that C restricted to the plane ℂνp M is the orthogonal reflection in the line νp M. Since all endomorphisms in Ap differ just by scalar multiplication with a complex number, we see that ℂνp M is invariant under Ap . This implies that 𝒞p = Tp Qn ⊖ ℂνp M is invariant under Ap , and hence 𝒬p = 𝒞p . Conversely, assume that 𝒬p = 𝒞p . Then ℂνp M = Tp Qn ⊖ 𝒞p is invariant under Ap . Since the real dimension of ℂνp M is 2 and all real structures in Ap differ just by scalar multiplication with a unit complex number, there exists a real structure C ∈ Ap which fixes ζp . Thus ζp is A-principal. This proves the equivalence of (i) and (iii). The equivalence of (i) and (iv) follows from the fact that all real structures in Ap differ just by scalar multiplication with a unit complex number. Assume now that the normal vector ζp is not A-principal. Then there exists a real structure C ∈ Ap such that ζp = cos(t)Z1 + sin(t)JZ2 for some orthonormal vectors Z1 , Z2 ∈ V(C) and 0 < t ≤ π4 . Note that ζp is A-isotropic if and only if t = π4 . Then we get Cζp = cos(t)Z1 − sin(t)JZ2 , ξp = sin(t)Z2 − cos(t)JZ1 ,

Cξp = sin(t)Z2 + cos(t)JZ1 ,

6.1 Basic equations for real hypersurfaces | 221

and therefore 𝒬p = Tp Qn ⊖ ([Z1 ] ⊕ [Z2 ]) is strictly contained in 𝒞p . Moreover, from the above equations we obtain g(Cξp , ζp ) = 0, which implies Cξp = Bξp

and ρ(ξp ) = 0.

Furthermore, we have g(Bξp − η(Bξp )ξp , ζp ) = 0,

g(Bξp − η(Bξp )ξp , ξp ) = 0,

g(Bξp − η(Bξp )ξp , Bξp − η(Bξp )ξp ) = sin2 (2t), and therefore Up =

1 (Bξp − η(Bξp )ξp ) sin(2t)

is a unit vector in 𝒞p and 𝒞p = 𝒬p ⊕ ℂUp

(orthogonal sum).

If ζp is not A-principal at p, then ζ is not A-principal in an open neighborhood of p, and therefore U is a well-defined unit vector field on that open neighborhood. We summarize this in the following lemma. Lemma 6.1.2. Let M be a real hypersurface in Qn . Assume that ζp is not A-principal at p ∈ Qn . Then there exists an open neighborhood of p in M and a section C in A on that neighborhood consisting of real structures such that (i) Cξ = Bξ and ρ(ξ ) = 0, (ii) U = (Bξ − η(Bξ )ξ )/‖Bξ − η(Bξ )ξ ‖ is a unit vector field tangent to 𝒞 , (iii) 𝒞 = 𝒬 ⊕ ℂU. We will often assume that C is chosen in line with Lemma 6.1.2 in neighborhoods where the normal vector field ζ is not A-principal. The following lemma is also of interest in this context. If C is a real structure in A, then any real structure is of the form C ′ = cos(s)C + sin(s)JC with s ∈ [0, 2π). Lemma 6.1.3. Let M be a real hypersurface in Qn . Let C be a real structure in A and C ′ = cos(s)C + sin(s)JC.

222 | 6 Real hypersurfaces in complex quadrics The unit normal ζ can be written as ζ = cos(t)Z1 + sin(t)JZ2 with Z1 , Z2 ∈ V(C). Then we have g(C ′ ξ , ζ ) = − sin(s) cos(2t)

and

g(C ′ ξ , ξ ) = − cos(s) cos(2t).

In particular, g(C ′ ξ , ζ ) = 0

⇐⇒

C ′ = ±C

or

ζ is A-isotropic

g(C ′ ξ , ξ ) = 0

⇐⇒

C ′ = ±JC

or

ζ is A-isotropic.

and

Proof. Straightforward computations give ζ = cos(t)Z1 + sin(t)JZ2 ,

C ζ = cos(t)C ′ Z1 − sin(t)C ′ JZ2 ′

= cos(t)(cos(s)C + sin(s)JC)Z1 − sin(t)(cos(s)C + sin(s)JC)JZ2

= cos(t) cos(s)CZ1 + cos(t) sin(s)JCZ1 + sin(t) cos(s)JCZ2 − sin(t) sin(s)CZ2 = cos(t) cos(s)Z1 + cos(t) sin(s)JZ1 − sin(t) sin(s)Z2 + sin(t) cos(s)JZ2 ,

ξ = − cos(t)JZ1 + sin(t)Z2 ,

C ξ = − cos(t)C ′ JZ1 + sin(t)C ′ Z2 ′

= − cos(t)(cos(s)C + sin(s)JC)JZ1 + sin(t)(cos(s)C + sin(s)JC)Z2

= cos(t) cos(s)JCZ1 − cos(t) sin(s)CZ1 + sin(t) cos(s)CZ2 + sin(t) sin(s)JCZ2 = − cos(t) sin(s)Z1 + cos(t) cos(s)JZ1 + sin(t) cos(s)Z2 + sin(t) sin(s)JZ2 .

We then calculate g(C ′ ξ , ζ ) = − cos2 (t) sin(s) + sin(t)2 sin(s) = − sin(s) cos(2t),

g(C ′ ξ , ξ ) = − cos2 (t) cos(s) + sin(t)2 cos(s) = − cos(s) cos(2t). This finishes the proof. Corollary 6.1.4. Let M be a real hypersurface in Qn . If the unit normal vector field ζ is A-isotropic and C is a real structure on Qn , then we have Cξ = Bξ ,

ρ(ξ ) = 0,

η(Bξ ) = 0.

The analog for A-principal unit normal ζ is the following.

6.1 Basic equations for real hypersurfaces | 223

Corollary 6.1.5. Let M be a real hypersurface in Qn . Assume that the unit normal vector field ζ is A-principal and C is a real structure on Qn so that Cζ = ζ . Let C ′ = cos(s)C + sin(s)JC be any real structure in A. Then we have C ′ ξ = B′ ξ − sin(s)ζ

and

η(B′ ξ ) = − cos(s).

In particular, Cξ = Bξ ,

ρ(ξ ) = 0,

η(Bξ ) = −1.

We continue with some general equations. Let X, Y ∈ X(M). Since the Kähler structure J is parallel, we have 0 = (∇̄X J)ζ = ∇̄X Jζ − J ∇̄X ζ = ∇X Jζ + g(AX, Jζ )ζ − J ∇̄X ζ

= −∇X ξ − η(AX)ζ + JAX = −∇X ξ − η(AX)ζ + ϕAX + η(AX)ζ = −∇X ξ + ϕAX.

For the covariant derivatives of η and ϕ we obtain (∇X η)Y = g(ϕAX, Y) and (∇X ϕ)Y = ∇X ϕY − ϕ∇X Y = ∇̄X ϕY − g(AX, ϕY)ζ − ϕ∇X Y = ∇̄X JY − ∇̄X (η(Y)ζ ) + g(ϕAX, Y)ζ − ϕ∇X Y = J ∇̄X Y − η(∇X Y)ζ − g(Y, ∇X ξ )ζ − η(Y)∇̄X ζ + g(∇X ξ , Y)ζ − ϕ∇X Y = J ∇̄X Y − η(∇X Y)ζ + η(Y)AX − J∇X Y + η(∇X Y)ζ = η(Y)AX + J ∇̄X Y − J∇X Y

= η(Y)AX + g(AX, Y)Jζ

= η(Y)AX − g(AX, Y)ξ . We have g(BX, Y) = g(AX, Y) = g(X, AY) = g(X, BY) and hence B is a self-adjoint tensor field on M. From the equation 0 = CJ + JC

224 | 6 Real hypersurfaces in complex quadrics we obtain Cζ = CJξ = −JCξ = −J(Bξ + ρ(ξ )ζ ) = −JBξ − ρ(ξ )Jζ = −ϕBξ − η(Bξ )ζ + ρ(ξ )ξ .

It follows that δ = g(Cζ , ζ ) = −η(Bξ ), g(Cζ , ξ ) = ρ(ξ ),

g(Cζ , X) = −g(ϕBξ , X) = g(ϕX, Bξ )

for all X ∈ Γ(𝒞 ).

Moreover, we have 0 = CJX + JCX

= C(ϕX + η(X)ζ ) + J(BX + ρ(X)ζ ) = CϕX + η(X)Cζ + JBX + ρ(X)Jζ

= BϕX + ρ(ϕX)ζ − η(X)ϕBξ − η(X)η(Bξ )ζ + η(X)ρ(ξ )ξ + ϕBX + η(BX)ζ − ρ(X)ξ .

The ζ -component of this equation gives 0 = ρ(ϕX) − η(X)η(Bξ ) + η(BX) for all X ∈ X(M). Putting X = ξ gives no information, but for X ∈ Γ(𝒞 ) we obtain η(BX) = −ρ(ϕX). Taking the ξ -component of the above equation implies 0 = g(BϕX, ξ ) + η(X)ρ(ξ ) − ρ(X) for all X ∈ X(M). Putting X = ξ gives no information, but for X ∈ Γ(𝒞 ) we obtain ρ(X) = g(BϕX, ξ ). Finally, for the 𝒞 -component of the above equation, we take inner product with Y ∈ Γ(𝒞 ) and obtain 0 = g(BϕX, Y) − η(X)g(ϕBξ , Y) + g(ϕBX, Y). For X = ξ this gives no information, and for X, Y ∈ Γ(𝒞 ) this gives 0 = g(BϕX, Y) + g(ϕBX, Y) = g((Bϕ + ϕB)X, Y).

6.1 Basic equations for real hypersurfaces | 225

Since A is parallel and of rank 2, there exists a 1-form q on Qn so that (∇̄X C)Y = q(X)JCY

and (∇̄X JC)Y = −q(X)CY

for all X, Y ∈ X(Qn ) (see [90], Proposition 7). We have JCY = J(BY + ρ(Y)ζ ) = ϕBY + η(BY)ζ − ρ(Y)ξ . On the other hand, we have (∇̄X C)Y = ∇̄X CY − C ∇̄X Y = ∇̄X (BY + ρ(Y)ζ ) − C(∇X Y + g(AX, Y)ζ ) = ∇̄X BY + ∇̄X ρ(Y)ζ − C∇X Y − g(AX, Y)Cζ

= ∇X BY + g(AX, BY)ζ + dρ(Y)(X)ζ − ρ(Y)AX − B∇X Y − ρ(∇X Y)ζ

+ g(AX, Y)ϕBξ + g(AX, Y)η(Bξ )ζ − g(AX, Y)ρ(ξ )ξ

= (∇X B)Y − ρ(Y)AX + g(AX, Y)ϕBξ − g(AX, Y)ρ(ξ )ξ + g(AX, BY)ζ + dρ(Y)(X)ζ

− ρ(∇X Y)ζ + g(AX, Y)η(Bξ )ζ ,

(∇̄X C)Y = q(X)C ′ Y = q(X)B′ Y + q(X)q′ (Y)ζ . Comparing this with (∇̄X C)Y = q(X)(ϕBY + η(BY)ζ − ρ(Y)ξ ) implies q(X)(ϕBY − ρ(Y)ξ ) = (∇X B)Y − ρ(Y)AX + g(AX, Y)ϕBξ − g(AX, Y)ρ(ξ )ξ , q(X)η(BY) = g(AX, BY) + dρ(Y)(X) − ρ(∇X Y) + g(AX, Y)η(Bξ ).

We summarize this in the following lemma. Lemma 6.1.6 (Basic structure equations). Let M be a real hypersurface in Qn . The almost contact metric structure (ϕ, ξ , η, g), the real structure C and the induced structure (B, ρ) on M satisfy the following relations for X, Y ∈ X(M) (unless otherwise specified): 0 = ρ(ϕX) + η(BX) − η(X)η(Bξ ),

0 = g(BϕX, ξ ) + η(X)ρ(ξ ) − ρ(X),

0 = g(BϕX, Y) + g(ϕBX, Y) = g((Bϕ + ϕB)X, Y), δ = g(Cζ , ζ ) = −η(Bξ ),

X, Y ∈ Γ(𝒞 ),

226 | 6 Real hypersurfaces in complex quadrics g(Cζ , ξ ) = ρ(ξ ),

g(Cζ , X) = −g(ϕBξ , X) = g(ϕX, Bξ ), ∇X ξ = ϕAX,

X ∈ Γ(𝒞 ),

(∇X η)Y = g(ϕAX, Y),

(∇X ϕ)Y = η(Y)AX − g(AX, Y)ξ ,

(∇X B)Y = ρ(Y)AX + q(X)ϕBY − g(AX, Y)ϕBξ − q(X)ρ(Y)ξ + g(AX, Y)ρ(ξ )ξ ,

(6.1)

q(X)η(BY) = g(AX, BY) + dρ(Y)(X) − ρ(∇X Y) + g(AX, Y)η(Bξ ). The fundamental equations of submanifold geometry (Theorem 2.1.1) can be rewritten using the explicit expression of the curvature tensor of Qn as given in Theorem 1.4.1. Theorem 6.1.7 (Fundamental structure equations). Let M be a real hypersurface in Qn . Then the following equations hold for all X, Y, Z ∈ X(M): Gauss formula: ∇̄X Y = ∇X Y + g(AX, Y)ζ , Weingarten formula: ∇̄X ζ = −AX, Gauss equation: R(X, Y)Z = g(Y, Z)X − g(X, Z)Y + g(ϕY, Z)ϕX − g(ϕX, Z)ϕY − 2g(ϕX, Y)ϕZ + g(BY, Z)BX − g(BX, Z)BY

+ g(ϕBY, Z)ϕBX − g(ϕBX, Z)ϕBY − ρ(Y)η(Z)ϕBX + ρ(X)η(Z)ϕBY

− ρ(X)g(ϕBY, Z)ξ + ρ(Y)g(ϕBX, Z)ξ

+ g(AY, Z)AX − g(AX, Z)AY,

(6.2)

Codazzi equation: (∇X A)Y − (∇Y A)X = η(X)ϕY − η(Y)ϕX + ρ(X)BY − ρ(Y)BX + η(BX)ϕBY − η(BY)ϕBX

− 2g(ϕX, Y)ξ − η(BX)ρ(Y)ξ + η(BY)ρ(X)ξ . We now discuss the normal Jacobi operator K = R̄ ζ |TM .

(6.3)

6.1 Basic equations for real hypersurfaces | 227

Theorem 6.1.8. Let M be a real hypersurface in Qn . The normal Jacobi operator K of M satisfies KX = X − η(Bξ )BX − ρ(ξ )BϕX + ρ(ϕX)Bξ − η(Bξ )η(X)Bξ

+ ρ(ξ )η(X)ϕBξ + ρ(X)ϕBξ + 3η(X)ξ − ρ(X)ρ(ξ )ξ − ρ(ξ )2 η(X)ξ

(6.4)

for all X ∈ X(M). If we choose the real structure C so that ρ(ξ ) = 0, this simplifies to KX = X − η(Bξ )BX + ρ(ϕX)Bξ − η(Bξ )η(X)Bξ + ρ(X)ϕBξ + 3η(X)ξ . Proof. From Theorem 1.4.1 we get ̄ ζ )ζ = g(ζ , ζ )X − g(X, ζ )ζ + g(Jζ , ζ )JX − g(JX, ζ )Jζ − 2g(JX, ζ )Jζ KX = R(X,

+ g(Cζ , ζ )CX − g(CX, ζ )Cζ + g(JCζ , ζ )JCX − g(JCX, ζ )JCζ

for all X ∈ X(M). We have Jζ = −ξ and g(ζ , ζ ) = 1,

g(X, ζ ) = g(Jζ , ζ ) = 0,

g(JX, ζ ) = −g(X, Jζ ) = g(X, ξ ) = η(X),

g(Cζ , ζ ) = −η(Bξ ),

CX = BX + ρ(X)ζ ,

g(CX, ζ ) = ρ(X)g(ζ , ζ ) = ρ(X),

Cζ = −ϕBξ − η(Bξ )ζ + ρ(ξ )ξ ,

g(JCζ , ζ ) = −g(Cζ , Jζ ) = g(Cζ , ξ ) = ρ(ξ ),

g(JCX, ζ ) = −g(CX, Jζ ) = g(CX, ξ ) = g(BX, ξ ) = η(BX), JCX = −CJX = −CϕX − η(X)Cζ

= −BϕX − ρ(ϕX)ζ + η(X)ϕBξ + η(Bξ )η(X)ζ + ρ(ξ )η(X)ξ ,

g(JCX, ζ ) = −ρ(ϕX) + η(Bξ )η(X),

JCζ = −CJζ = Cξ = Bξ + ρ(ξ )ζ .

Inserting this into the previous equation leads to KX = X + 3η(X)ξ − η(Bξ )(BX + ρ(X)ζ ) − ρ(X)(−ϕBξ − η(Bξ )ζ + ρ(ξ )ξ )

+ ρ(ξ )(−BϕX − ρ(ϕX)ζ + η(X)ϕBξρ(ξ ) + η(Bξ )η(X)ζ − ρ(ξ )η(X)ξ ) − (−ρ(ϕX) + η(Bξ )η(X))(Bξ + ρ(ξ )ζ )

= X + 3η(X)ξ − η(Bξ )BX − η(Bξ )ρ(X)ζ

+ ρ(X)ϕBξ + ρ(X)η(Bξ )ζ − ρ(X)ρ(ξ )ξ

− ρ(ξ )BϕX − ρ(ξ )ρ(ϕX)ζ + ρ(ξ )η(X)ϕBξ + ρ(ξ )η(Bξ )η(X)ζ − ρ(ξ )2 η(X)ξ

(6.5)

228 | 6 Real hypersurfaces in complex quadrics + ρ(ϕX)Bξ − η(Bξ )η(X)Bξ + ρ(ϕX)ρ(ξ )ζ − η(Bξ )η(X)ρ(ξ )ζ

= X − η(Bξ )BX − ρ(ξ )BϕX + ρ(ϕX)Bξ − η(Bξ )η(X)Bξ

+ ρ(ξ )η(X)ϕBξ + ρ(X)ϕBξ + 3η(X)ξ − ρ(X)ρ(ξ )ξ − ρ(ξ )2 η(X)ξ .

It follows from Lemma 6.1.2 and Corollary 6.1.5 that we can always find a real structure C with ρ(ξ ) = 0. This finishes the proof. Proposition 6.1.9. Let M be a real hypersurface in Qn and p ∈ M. The following statements are equivalent: (i) Kξp = κξp for some κ ∈ ℝ; (ii) there exists a real structure C ∈ Ap such that the induced endomorphism B on Tp M satisfies Bξp ∈ 𝒞p or Bξp ∈ 𝒞p⊥ ; (iii) the unit normal vector ζp is a singular tangent vector of Qn . Proof. Inserting X = ξ into equation (6.5) implies Kξ = 4ξ − 2η(Bξ )Bξ . It follows that Kξp = κξp if and only if η(Bξp )Bξp ∈ ℝξp . The latter condition is equivalent to Bξp ∈ 𝒞p or Bξp ∈ 𝒞p⊥ . We have Bξp ∈ 𝒞p if and only if ζp is A-isotropic, and Bξp ∈ 𝒞p⊥ if and only if ζp is A-principal. The assertion then follows from the description of the singular tangent vectors of Qn in Section 1.4.2. We now compute an expression for the structure Jacobi operator. Theorem 6.1.10. Let M be a real hypersurface in Qn . The structure Jacobi operator Rξ = R(⋅, ξ )ξ of M satisfies Rξ X = X − η(X)ξ − ρ(ξ )ϕBX + ρ(X)ϕBξ + η(Bξ )BX − η(BX)Bξ + η(Aξ )AX − η(AX)Aξ

(6.6)

for all X ∈ X(M). Proof. Equation (6.6) follows by inserting Y = Z = ξ into the Gauss equation (6.2). For the Ricci tensor of M we obtain the following theorem. Theorem 6.1.11. Let M be a real hypersurface in Qn . The Ricci tensor of M satisfies ric(X, Y) = 2ng(X, Y) − g(KX, Y) + tr(A)g(AX, Y) − g(A2 X, Y),

(6.7)

or equivalently, Ric(X) = 2nX − KX + tr(A)AX − A2 X for all X ∈ X(M).

(6.8)

6.2 Totally geodesic submanifolds | 229

Proof. Let E1 , . . . , E2n be a local orthonormal frame field of Qn so that E2n = ζ . The Gauss equation implies 2n−1

ric(X, Y) = ∑ g(R(X, Eν )Eν , Y) ν=1

2n−1

̄ Eν )Eν , Y) = ∑ g(R(X, ν=1

2n−1

2n−1

ν=1

ν=1

+ ∑ g(AEν , Eν )g(AX, Y) − ∑ g(AX, Eν )g(AEν , Y) for all X, Y ∈ X(M). Using the expression for the Ricci tensor of Qn in Theorem 1.4.1, we get 2n−1

2n

ν=1

ν=1

̄ Eν )Eν , Y) − g(R(X, ̄ E2n )E2n , Y) ̄ Eν )Eν , Y) = ∑ g(R(X, ∑ g(R(X, ̄ ζ )ζ , Y) = ric(X, Y) − g(R(X, = 2ng(X, Y) − g(KX, Y).

We also have 2n−1

∑ g(AEν , Eν )g(AX, Y) = tr(A)g(AX, Y)

ν=1

and 2n−1

2n−1

ν=1

ν=1

∑ g(AX, Eν )g(AEν , Y) = ∑ g(AX, Eν )g(AY, Eν ) = g(AX, AY) = g(A2 X, Y).

Altogether this implies the assertion. Contracting equation (6.7) implies the following. Corollary 6.1.12. Let M be a real hypersurface in Qn . The scalar curvature s of M satisfies s = 4n(n − 1) + tr(A)2 − tr(A2 ).

6.2 Totally geodesic submanifolds Also in this chapter, tubes around certain totally geodesic submanifolds are important for our studies of real hypersurfaces. For this reason we briefly revisit here the classification of totally geodesic submanifolds in complex quadrics.

230 | 6 Real hypersurfaces in complex quadrics The totally geodesic submanifolds in the complex quadric Qn were investigated by Chen and Nagano in [35]. They obtained a classification, which later turned out to have omissions. A complete classification was then obtained by Klein in [56]. We present here the maximal totally geodesic submanifolds, as the tubes around some of them are relevant for our investigations. Maximal totally geodesic submanifolds in Qn were already classified in [35], but with one omission that was discovered by Klein in [56]. We describe now the standard embeddings for these maximal totally geodesic submanifolds. Recall that the complex quadric Qn can be identified with the real Grassmann manifold G2+ (ℝn+2 ) of oriented 2-planes in ℝn+2 . A standard map between these two different models of the same Riemannian symmetric space is Qn → G2+ (ℝn+2 ),

[z] = [x + iy] 󳨃→ ℝx ⊕ ℝy,

with the orientation of the 2-plane ℝx ⊕ ℝy given by the ordered pair (x, y). Here, we write z ∈ ℂn+2 as z = x + iy with x, y ∈ ℝn+2 . The quadric equation then becomes n+2

n+2

ν=1

ν=1

0 = ∑ zν2 = ∑ ((xν2 − yν2 ) + 2i(xν yν )), which in terms of x and y leads to ‖x‖ = ‖y‖ and

⟨x, y⟩ = 0.

The linear embedding ℂn+1 → ℂn+2 ,

(z1 , . . . , zn+1 ) 󳨃→ (z1 , . . . , zn+1 , 0)

induces a totally geodesic embedding of the complex projective space ℂP n into ℂP n+1 and a totally geodesic embedding of the complex quadric Qn−1 ⊂ ℂP n into Qn ⊂ ℂP n+1 , 2 Qn−1 = {[z] ∈ ℂP n+1 : z12 + ⋅ ⋅ ⋅ + zn+1 = 0, zn+2 = 0} = Qn ∩ ℂP n ⊂ Qn .

Equivalently, we can consider the linear embedding ℝn+1 → ℝn+2 ,

(x1 , . . . , xn+1 ) 󳨃→ (x1 , . . . , xn+1 , 0),

which induces a canonical totally geodesic embedding of G2+ (ℝn+1 ) into G2+ (ℝn+2 ). The complex hyperquadric Qn−1 ≅ G2+ (ℝn+1 ) in Qn ≅ G2+ (ℝn+2 ) is a complex submanifold of Qn . Consider the canonical isomorphism ℝn+2 ≅ ℝn+1 ⊕ ℝen+2

6.2 Totally geodesic submanifolds | 231

with en+2 = (0, . . . , 0, 1) and the unit sphere Sn in ℝn+1 . The map Sn → G2+ (ℝn+2 ),

x 󳨃→ ℝx ⊕ ℝen+2

is a total geodesic embedding of the unit sphere Sn into G2+ (ℝn+2 ) ≅ Qn . The sphere Sn is a totally real submanifold of Qn . Since dimℝ (Sn ) = n = dimℂ (Qn ), the sphere Sn is a real form of Qn . For this reason the complex quadric Qn is sometimes regarded as the complexification of the sphere Sn . Next, let a ∈ {1, . . . , n − 1} and consider the canonical isomorphism ℝn+2 ≅ ℝa+1 ⊕ ℝn−a+1 . Let Sa and Sn−a be the unit sphere in ℝa+1 and ℝn−a+1 , respectively. The map Sa × Sn−a → G2+ (ℝn+2 ),

(x, y) 󳨃→ ℝx ⊕ ℝy

is a totally geodesic immersion (more precisely, a 2-fold covering map onto its image) of the product Sa × Sn−a into G2+ (ℝn+2 ) ≅ Qn . The image (Sa × Sn−a )/ℤ2 is a totally real submanifold of Qn . Since dimℝ (Sa × Sn−a ) = n = dimℂ (Qn ), it is a real form of Qn . Assume that n is even, say n = 2m. Then we have a natural identification ℝn+2 ≅ 2m+2 ℝ ≅ ℂm+1 . Consider the complex projective space ℂP m induced from ℂm+1 . The tautological embedding ℂP m → G2+ (ℝ2m+2 ),

[z] 󳨃→ ℝz ⊕ ℝiz

realizes ℂP m as a totally geodesic submanifold of G2+ (ℝ2m+2 ). The complex projective space ℂP m is a complex submanifold of G2+ (ℝ2m+2 ) ≅ Q2m . Consider the irreducible representation of SO3 on the real vector space End+0 (ℝ3 ) of all orientation-preserving and trace-free endomorphisms of ℝ3 given by SO3 × End+0 (ℝ3 ) → End+0 (ℝ3 ),

(A, X) 󳨃→ AXA−1 .

We have a canonical isomorphism ℝ5 ≅ End+0 (ℝ3 ), which induces an isometric action of SO3 on G2+ (ℝ5 ). This action has exactly one totally geodesic singular orbit. This orbit is isometric to a 2-dimensional sphere S2 (with a suitable radius) and a maximal totally geodesic submanifold of G2+ (ℝ5 ) which is neither totally real nor complex. We call this the non-standard totally geodesic embedding of S2 into the complex quadric Q3 ≅ G2+ (ℝ5 ). Theorem 6.2.1 ([35, 56]). Let Σ be a maximal totally geodesic submanifold of the complex quadric Qn , n ≥ 3. Then Σ is congruent to one of the following maximal totally geodesic submanifolds: (i) the complex totally geodesic embedding of Qn−1 into Qn ; (ii) the totally real totally geodesic embedding of Sn into Qn ;

232 | 6 Real hypersurfaces in complex quadrics (iii) the totally real totally geodesic embedding of (Sa × Sn−a )/ℤ2 into Qn , where a ∈ {1, . . . , n − 1}; (iv) (only for n = 2m even) the complex totally geodesic embedding of ℂP m into Q2m ; (v) (only for n = 3) the non-standard totally geodesic embedding of S2 into Q3 .

6.3 Homogeneous real hypersurfaces The homogeneous real hypersurfaces in the complex quadric Qn , n ≥ 3, were classified by Kollross in [60]. Kollross classified the cohomogeneity one actions on irreducible Riemannian symmetric spaces of compact type up to orbit equivalence. The principal orbits of cohomogeneity one actions are homogeneous hypersurfaces. Conversely, any homogeneous hypersurface arises as an orbit of a cohomogeneity one action. Kollross proved that any cohomogeneity one action on Qn is orbit equivalent to the action of SOn+1 ⊂ SOn+2 on Qn , to the action of Um+1 ⊂ SO2m+2 on Q2m , to the action of Spm Sp1 ⊂ SO4m on Q4m−2 or to the action of Spin9 ⊂ SO16 on Q14 . Any cohomogeneity one action on Qn has exactly two singular orbits. Geometrically the two singular orbits are focal sets of each other. The principal orbits can then be realized geometrically as the tubes around either of the two singular orbits. The singular orbits of the action of SOn+1 are both totally geodesic. One of them is congruent to the totally geodesic Qn−1 ⊂ Qn , the other one is congruent to the totally geodesic Sn ⊂ Qn . The principal orbits are the tubes around Qn−1 ⊂ Qn , or equivalently, the tubes around Sn ⊂ Qn . With our normalization of the Riemannian metric π and therefore the tubes are wellon Qn , the distance between the two focal sets is 2√ 2 π defined for the radii 0 < r < 2√2 . The orbits can also be described as follows. Let en+2 =

(0, . . . , 0, 1) ∈ ℝn+2 and consider the canonical isomorphism ℝn+2 ≅ ℝn+1 ⊕ℝen+2 . To every unit vector v ∈ Sn ⊂ ℝn+1 we assign the 2-dimensional oriented plane ℝv ⊕ ℝen+2 . This describes a totally geodesic embedding of the unit sphere Sn = SOn+1 /SOn into G2+ (ℝn+2 ) ≅ Qn . For every V ∈ G2+ (ℝn+2 ) we measure the angle φ ∈ [0, π2 ] between V and ℝen+2 . Then V ∈ Sn if and only if φ = 0, that is, if en+2 ∈ V. The other singular orbit Qn−1 = SOn+1 /SO2 SOn−1 is given by all 2-planes which are perpendicular to en+2 , that is, φ = π2 . Each principal orbit consists of all 2-planes V with a fixed angle φ ∈ (0, π2 ) between V and ℝen+2 . As a homogeneous space, each principal orbit is isomorphic to the Stiefel manifold SOn+1 /SOn−1 . Assume that n = 2m is even. Both singular orbits of the action of Um+1 are congruent to a totally geodesic ℂP m ⊂ Q2m . The principal orbits are the tubes around ℂP m ⊂ Q2m . With our normalization of the Riemannian metric on Q2m , the distance between the two focal sets is π2 and therefore the tubes are well-defined for the radii 0 < r < π2 . A more explicit description of the orbits is as follows. Consider the canonical isomorphism ℝ2m+2 ≅ ℂm+1 . The complex projective space ℂP m is the set of all 2-dimensional linear subspaces V of ℂm+1 that are invariant under the complex structure of ℂm+1 , that is, all complex linear lines in ℂm+1 . The complex structure on ℂm+1

6.3 Homogeneous real hypersurfaces | 233

equips V with a natural orientation turning it into an oriented 2-dimensional real vector space. By mapping ℂz ∈ ℂP m to ℝz ⊕ ℝiz, with orientation given by the pair (z, iz), we get a totally geodesic embedding of ℂP m into G2+ (ℝ2m+2 ) ≅ Q2m . The second singular orbit of the Um+1 -action on Q2m consists of the complex linear lines in ℂm+1 with the opposite orientation. This is again a complex projective space ℂP m , but the embedding is given by ℂz 󳨃→ ℝz ⊕ ℝ(−iz), where the pair (z, −iz) describes the orientation of the real 2-plane ℝz ⊕ ℝ(−iz) = ℂz. The principal orbits of the Um+1 -action are characterized by the Kähler angle φ ∈ (0, π2 ] and the orientation unless φ = π2 . If φ = π2 , then V is a totally real subspace and Um+1 acts transitively on the set of oriented 2-dimensional totally real subspaces of ℂm+1 . If we consider the isomorphism ℂm+1 ≅ ℂ2 ⊕ ℂm−1 and the oriented 2-dimensional totally real subspace V = ℝ2 ⊂ ℂ2 and compute the isotropy group of Um+1 at V, we see that this principal orbit can be identified with the homogeneous space Um+1 /SO2 Um−1 . The other principal orbits are isomorphic (as homogeneous spaces) to this orbit. Assume that n = 4m − 2 and consider the natural isomorphism ℝ4m ≅ ℍm and the standard action of Spm Sp1 on ℍm . This action is transitive on the set of all totally complex oriented 2-dimensional subspaces of ℍm . Consider the canonical isomorphism ℍm ≅ ℍ ⊕ ℍm−1 and V = ℂ ⊂ ℍ. The isotropy group of Spm Sp1 at V is U1 Spm−1 SO2 . It follows that the set of all totally complex oriented 2-dimensional subspaces of ℍm is a 4m-dimensional singular orbit of the action of Spm Sp1 on G2+ (ℍm ) ≅ G2+ (ℝ4m ) ≅ Q4m−2 . The second singular orbit is given by the set of all totally real oriented 2-dimensional subspaces of ℍm . Consider the canonical isomorphism ℍm ≅ ℍ2 ⊕ ℍm−2 and V = ℝ2 ⊂ ℍ2 . The isotropy group of Spm Sp1 at V is SO2 Spm−2 Sp1 . It follows that the set of all totally real oriented 2-dimensional subspaces of ℍm is an (8m − 7)-dimensional singular orbit of the action of Spm Sp1 on G2+ (ℍm ) ≅ G2+ (ℝ4m ) ≅ Q4m−2 . The principal orbits can be characterized by the quaternionic Kähler angle (φ, π2 , π2 ) of 2-dimensional subspaces in ℍm (see [8] for the concept of quaternionic Kähler angle). The principal orbits have dimension 8m−5 and as homogeneous spaces are isomorphic to Spm Sp1 /SO2 Spm−2 SO2 . Finally, consider the spin representation of Spin9 on ℝ16 ≅ 𝕆2 , which induces an isometric action of Spin9 on G2+ (ℝ16 ) ≅ Q14 . Consider a canonical decomposition ℝ16 ≅ ℝ8 ⊕ ℝ8 , or equivalently, 𝕆2 ≅ 𝕆 ⊕ 𝕆, where the two 8-dimensional spaces correspond to the two inequivalent spin representations obtained by restricting the spin representation of Spin9 to some Spin8 ⊂ Spin9 . Consider an oriented 2-dimensional subspace V in the first ℝ8 , which we might view as ℂ ⊂ 𝕆 with its natural orientation. Then the isotropy group of Spin9 at V is SO2 Spin6 . It follows that the orbit through V is a 20-dimensional singular orbit, and as a homogeneous space it is isomorphic to Spin9 /SO2 Spin6 . Consider a 2-dimensional subspace V which has a 1-dimensional intersection with each of the two ℝ8 , which we might view as ℝ2 ≅ ℝ ⊕ ℝ ⊂ 𝕆 ⊕ 𝕆 ≅ 𝕆2 with its natural orientation. Then the isotropy group of Spin9 at V is SO2 G2 . It follows that the orbit through V is a 21-dimensional singular orbit, and as a homogeneous space it is isomorphic to Spin9 /SO2 G2 . The

234 | 6 Real hypersurfaces in complex quadrics principal orbits have dimension 27 and as homogeneous spaces are isomorphic to Spin9 /SO2 SU 3 ≅ Spin9 /U3 . Theorem 6.3.1 ([60]). Let M be a homogeneous real hypersurface in the complex quadric Qn , n ≥ 3. Then M is congruent to one of the following homogeneous real hypersurfaces: π around the totally geodesic Qn−1 ⊂ Qn (or equiv(A) the tube with radius 0 < r < 2√ 2 alently, around the totally geodesic Sn ⊂ Qn ) (as a homogeneous space, any such tube is isomorphic to the Stiefel manifold SOn+1 /SOn−1 ), (B) (only for n = 2m, m ≥ 2) the tube with radius 0 < r < π2 around the totally geodesic ℂP m ⊂ Q2m , (C) (only for n = 4m − 2, m ≥ 2) a tube around the equivariant embedding of the 4mdimensional homogeneous space Spm Sp1 /U1 Spm−1 SO2 in Q4m−2 (or equivalently, of the (8m − 7)-dimensional homogeneous space Spm Sp1 /SO2 Spm−2 Sp1 in Q4m−2 ), (D) (only for n = 14) a tube around the equivariant embedding of the 20-dimensional homogeneous space Spin9 /SO2 Spin6 in Q14 (or equivalently, of the 21-dimensional homogeneous space Spin9 /SO2 G2 in Q14 ). We will now investigate the geometry of the homogeneous real hypersurfaces of types π (A) and (B). We start with the tubes around Qn−1 ⊂ Qn . Let 0 < r < 2√ and Σ = Qn−1 . 2 We can assume that o ∈ Σ. We have dim(Σ) = 2(n − 1) and hence codim(Σ) = 2. The map Qn−1 → Qn ⊂ ℂP n+1 ,

[z1 , . . . , zn+1 ] 󳨃→ [z1 , . . . , zn+1 , 0]

provides an embedding of Qn−1 into Qn as a totally geodesic complex hypersurface. From the construction of A it is clear that To Qn−1 and νo Qn−1 are C-invariant for each conjugation C ∈ Ao . Moreover, since the real codimension of Qn−1 in Qn is 2, there exists for each unit normal vector ζ of Qn−1 at o a real structure C ∈ Ao such that Cζ = ζ . It follows that the normal space νo Qn−1 consists of A-principal singular tangent vectors of Qn and To Qn−1 = (V(C) ⊖ ℝζ ) ⊕ J(V(C) ⊖ ℝζ ). Let ζ ∈ νo Σ be a unit normal vector and C ∈ Ao be a real structure with Cζ = ζ . From Lemma 1.4.2 we see that the Jacobi operator R̄ ζ leaves νo Σ and To Σ invariant. Therefore Σ is a curvature-adapted submanifold of Qn . We can therefore apply Corollary 2.6.2 to calculate the principal curvatures and corresponding principal curvature spaces of the tube Σr with radius r around Σ. Using the notations of Section 2.6, we have Spec(Kζ ) = {0, 2}, Spec(Aζ ) = {0} and Spec(R̄ ⊥ ζ ) = {2} and corresponding eigenspaces E0 = JV(C) ⊖ ℝJζ , E2 = V(C) ⊖ ℝζ ,

6.3 Homogeneous real hypersurfaces | 235

T0 = To Σ,

V2 = ℝJζ .

Let γζ be the geodesic in Qn with γζ (0) and γ̇ζ (0) = ζ . It follows from Corollary 2.6.2 that the principal curvatures of Σr at γζ (r) with respect to −γ̇ζ (r) are λ = 0, μ = −2 tan2 (r) = −√2 tan(√2r) and α = cot2 (r) = √2 cot(√2r). The corresponding principal curvature spaces are the parallel translates of E0 , E2 and V2 along γζ from 0 = γζ (0) to p = γζ (r). Since Σr is homogeneous, these conclusions hold at any point. Thus we have proved the following theorem. Theorem 6.3.2. Let M be the tube with radius 0 < r < n−1

π 2√2 n

around the complex totally

geodesic embedding of the complex quadric Q into Q . Then the unit normal vector field ζ of M is singular and satisfies Cζ = ζ for some real structure C ∈ A (pointwise). In particular, ζ is A-principal. Moreover, M has three distinct constant principal curvatures α = √2 cot(√2r),

λ = 0,

μ = −√2 tan(√2r)

with corresponding principal curvature spaces Tα = ℝJζ ,

Tλ = J(V(C) ⊖ ℝζ ),

Tμ = V(C) ⊖ ℝζ .

The corresponding multiplicities of the principal curvatures are mα = 1,

mλ = n − 1,

mμ = n − 1.

From this we can deduce some geometric information about the tubes around the totally geodesic Qn−1 ⊂ Qn . Proposition 6.3.3. Let M be the tube with radius 0 < r < n−1

geodesic embedding of the complex quadric Q statements hold: (i) The mean curvature of M is equal to

π 2√2 n

around the complex totally

into Q , n ≥ 3. Then the following

tr(A) = √2 cot(√2r) − √2(n − 1) tan(√2r). π There exists exactly one radius r with 0 < r < 2√ so that tr(A) = 0, that is, so that 2 M is a minimal hypersurface. (ii) The tube M is a Hopf hypersurface, namely Aξ = √2 cot(√2r)ξ . (iii) The maximal holomorphic subbundle 𝒞 of TM is invariant under the shape operator of M, that is, A𝒞 ⊆ 𝒞 . (iv) The tube M is curvature-adapted, that is, AK = KA.

236 | 6 Real hypersurfaces in complex quadrics (v) The tube M is a contact hypersurface and Aϕ + ϕA = −√2 tan(√2r)ϕ. (vi) The Reeb vector field ξ is an eigenvector of the normal Jacobi operator K everywhere, namely Kξ = 2ξ . Proof. (i): We can calculate the mean curvature of M using the expression of the principal curvatures in Theorem 6.3.2: tr(A) = √2 cot(√2r) − √2(n − 1) tan(√2r). π This is a strictly decreasing continuous function on the open interval (0, 2√ ) with 2 π limr→0 tr(A) = +∞ and limr→ tr(A) = −∞. It follows from the mean value theorem 2√2

π that there exists exactly one r ∈ (0, 2√ ) for which tr(A) = 0. 2 (ii): Since Tα = ℝξ , we have Aξ = αξ and thus M is a Hopf hypersurface. (iii): The maximal holomorphic subbundle 𝒞 of TM is invariant under the shape operator of M by (ii). (iv): Since Σ = Qn−1 is curvature-adapted in Qn , the tubes around it are curvatureadapted by Corollary 2.6.2. (v): From the description of the principal curvature spaces in Theorem 6.3.2 we know that ϕTλ = Tμ and ϕTμ = Tλ . For X ∈ Γ(Tλ ) we have

ϕAX = λϕX = 0

and AϕX = μϕX = −√2 tan(√2r)ϕX.

For X ∈ Γ(Tμ ) we have ϕAX = μϕX = −√2 tan(√2r)ϕX

and

AϕX = λϕX = 0.

This implies (Aϕ + ϕA)X = −√2 tan(√2r)ϕX for X ∈ Γ(Tλ ⊕ Tμ ) = Γ(𝒞 ). We also have (Aϕ + ϕA)ξ = 0 = −√2 tan(√2r)ϕξ . Altogether this implies Aϕ + ϕA = −√2 tan(√2r)ϕ. From Proposition 3.5.5 we can conclude that M is a contact hypersurface. (vi): From Theorem 6.3.2 we know that the unit normal vector ζ is an A-principal singular tangent vector of Qn everywhere. From Lemma 1.4.2(i) we see that ξ is an eigenvector of K everywhere, with corresponding eigenvalue 2. Next, we investigate the geometry of the tubes around the complex totally geodesic embedding of Σ = ℂP m into Q2m . We have dim(Σ) = 2m and hence codim(Σ) = 2m.

6.3 Homogeneous real hypersurfaces | 237

Since Σ is a complex submanifold, we see that the normal space νo Σ is a complex subspace of To Q2m . The map ℂP m → Q2m ⊂ ℂP 2m+1 ,

[z1 , . . . , zm+1 ] 󳨃→ [z1 , . . . , zm+1 , iz1 , . . . , izm+1 ]

provides an embedding of ℂP m into Q2m as a totally geodesic complex submanifold. We define a complex structure j on ℂ2m+2 by j(z1 , . . . , zm+1 , zm+2 , . . . , z2m+2 ) = (−zm+2 , . . . , −z2m+2 , z1 , . . . , zm+1 ). Note that ij = ji. We can then identify ℂ2m+2 with ℂm+1 ⊕ jℂm+1 and get T[z] ℂP m = {v + jiv : v ∈ ℂm+1 ⊖ [z]} = {v + ijv : v ∈ V(Az̄ )}, where Az̄ is the shape operator of the submanifold Q2m of ℂP 2m+1 with respect to the unit normal vector z.̄ Note that the complex structure i on ℂ2m+2 corresponds to the complex structure J on T[z] Q2m via the obvious identifications. For the normal space ν[z] ℂP m of ℂP m at [z] we have ν[z] ℂP m = Az̄ (T[z] ℂP m ) = {v − ijv : v ∈ V(Az̄ )}. It is easy to see that both the tangent bundle and the normal bundle of ℂP m consist of A-isotropic singular tangent vectors of Q2m . We assume again that o = [z] ∈ Σ. Let ζ ∈ νo Σ be a unit normal vector, which is an A-isotropic singular tangent vector of Q2m . Thus there exists a real structure C ∈ Ao and orthonormal vectors u, w ∈ V(C) such that ζ = √12 (u+Jw). From Lemma 1.4.2 we see that the Jacobi operator R̄ ζ leaves νo Σ and To Σ invariant. Therefore Σ is a curvature-adapted

submanifold of Q2m . We can therefore apply Corollary 2.6.2 to calculate the principal curvatures and corresponding principal curvature spaces of the tube Σr with radius r around Σ. Using the notations of Section 2.6, we have Spec(Kζ ) = {0, 1}, Spec(Aζ ) = {0} and Spec(R̄ ⊥ ζ ) = {1, 4} and corresponding eigenspaces E0 = ℂCζ ,

E1 = To ℂP m ⊖ (ℂζ ⊕ ℂCζ ),

T0 = To Σ,

V1 = νo ℂP m ⊖ (ℂζ ⊕ ℂCζ ),

V4 = ℝJζ .

Let γζ be the geodesic in Q2m with γζ (0) and γ̇ζ (0) = ζ . It follows from Corollary 2.6.2 that the principal curvatures of Σr at γζ (r) with respect to −γ̇ζ (r) are μ = 0, λ = − tan1 (r) = − tan(r), β = cot1 (r) = cot(r) and α = cot2 (r) = 2 cot(2r). The corresponding principal curvature spaces are the parallel translates of E0 , E1 , V1 and V4

238 | 6 Real hypersurfaces in complex quadrics along γζ from 0 = γζ (0) to p = γζ (r). Note that the parallel translate of ℂCζ corresponds to 𝒞 ⊖ 𝒬 at γζ (r), where 𝒬 is the maximal A-invariant subbundle of Σr (see also Lemma 6.1.2). Since Σr is homogeneous, these conclusions hold at any point. Thus we have proved the following theorem. Theorem 6.3.4. Let M be the tube with radius 0 < r < π2 around the complex totally geodesic embedding of the complex projective space ℂP m into Q2m , m ≥ 2. Then the unit normal vector field ζ of M is singular and A-isotropic. Moreover, M has four distinct constant principal curvatures α = 2 cot(2r),

β = cot(r),

λ = − tan(r),

μ=0

with corresponding principal curvature spaces Tα = ℝξ ,

Tβ = νℂP m ⊖ ℂνM,

Tλ = TℂP m ⊖ (𝒞 ⊖ 𝒬),

Tμ = 𝒞 ⊖ 𝒬.

The corresponding multiplicities of the principal curvatures are mα = 1,

mβ = 2m − 2,

mλ = 2m − 2,

mμ = 2.

From this we can deduce some geometric information about the tubes around the totally geodesic ℂP m ⊂ Q2m . Proposition 6.3.5. Let M be the tube with radius 0 < r < π2 around the complex totally geodesic embedding of the complex projective space ℂP m into Q2m , m ≥ 2. Then the following statements hold: (i) The mean curvature of M is equal to tr(A) = (4m − 2) cot(2r). In particular, the tube with radius r = π4 around the complex totally geodesic embedding of the complex projective space ℂP m into Q2m is a minimal hypersurface. (ii) The tube M is a Hopf hypersurface, namely Aξ = 2 cot(2r)ξ . (iii) The maximal holomorphic subbundle 𝒞 of TM and the maximal A-invariant subbundle 𝒬 of TM are invariant under the shape operator of M, that is, A𝒞 ⊆ 𝒞 and A𝒬 ⊆ 𝒬. (iv) The tube M is curvature-adapted, that is, AK = KA. (v) The tube M has isometric Reeb flow, or equivalently, Aϕ = ϕA. (vi) The Reeb vector field ξ is an eigenvector of the normal Jacobi operator K everywhere, namely Kξ = 4ξ .

6.3 Homogeneous real hypersurfaces | 239

(vii) The normal Jacobi operator K and the structure tensor field ϕ commute, that is, Kϕ = ϕK. (viii) The Ricci tensor Ric and the structure tensor field ϕ commute, that is, Ric ϕ = ϕ Ric. (ix) The shape operator A of M is Reeb parallel, that is, ∇ξ A = 0. Proof. (i): We can calculate the mean curvature of M using the expression of the principal curvatures in Theorem 6.3.4: tr(A) = 2 cot(2r) + (2m − 2)(cot(r) − tan(r)) = 2 cot(2r) + (2m − 2)2 cot(2r) = (4m − 2) cot(2r).

For r = π4 this gives tr(A) = 0 and hence a minimal hypersurface. (ii): Since Tα = ℝξ , we have Aξ = αξ and thus M is a Hopf hypersurface. The value for α has been computed in Theorem 6.3.4. (iii): The maximal holomorphic subbundle 𝒞 of TM is invariant under the shape operator of M by (ii). For the maximal A-invariant subbundle 𝒬 of TM we have 𝒬 = Tβ ⊕ Tλ by Theorem 6.3.4, which implies that 𝒬 is invariant under the shape operator of M. (iv): Since Σ = ℂP m is curvature-adapted in Q2m , the tubes around it are curvatureadapted by Corollary 2.6.2. (v): From the description of the principal curvature spaces in Theorem 6.3.4, and using 𝒬 = Tβ ⊕ Tλ and Lemma 6.1.2(iii), we see that Tβ , Tλ and Tμ are ϕ-invariant. This implies Aϕ = ϕA and hence M has isometric Reeb flow by Proposition 3.4.1. (vi): From Theorem 6.3.4 we know that the unit normal vector ζ is a singular A-isotropic tangent vector of Q2m . From Lemma 1.4.2(ii) we see that ξ is an eigenvector of K everywhere with corresponding eigenvalue 4. (vii): From Theorem 6.3.4 we know that the unit normal vector field ζ is a singular A-isotropic tangent vector field of Q2m . From Lemma 1.4.2(ii) we see that the maximal holomorphic subbundle 𝒞 decomposes into 𝒞 = E0 ⊕ E1 with E0 = ℂCζ ,

E1 = TQ2m ⊖ (ℂζ ⊕ ℂCζ ),

with some real structure C, where 0 and 1 are eigenvalues of K and E0 and E1 are the corresponding eigenspaces in 𝒞 . It is easy to see that both E0 and E1 are ϕ-invariant. This shows that Kϕ = ϕK. (viii): This is an immediate consequence of Theorem 6.1.11 and parts (v) and (vii) of this proposition. (ix): We have (∇Y A)ξ = ∇Y (Aξ ) − A∇Y ξ = ∇Y (αξ ) − A∇Y ξ = αϕAY − AϕAY.

240 | 6 Real hypersurfaces in complex quadrics For Y = ξ this gives (∇ξ A)ξ = αϕAξ − AϕAXξ = α2 ϕξ − αAϕξ = 0. The Codazzi equation implies for Y ∈ Γ(𝒞 ) (∇ξ A)Y = αϕAY − AϕAY + ϕY + ρ(ξ )BY − ρ(Y)Bξ + η(Bξ )ϕBY − η(BY)ϕBξ

− η(Bξ )ρ(Y)ξ + η(BY)ρ(ξ )ξ .

Since ζ is A-isotropic, the vectors ζ , ξ , Bξ , ϕBξ are pairwise orthonormal and we have Cζ = CJξ = −JCξ = −JBξ = −ϕBξ . It follows that (∇ξ A)Y = αϕAY − AϕAY + ϕY − ρ(Y)Bξ − η(BY)ϕBξ .

(6.9)

Let ρ ∈ {β, λ}. Then ρ is a solution of the quadratic equation x2 −αx−1 = 0. For Y ∈ Γ(Tρ ) we have ρ(Y) = 0 = η(BY) and AϕY = ρϕY because of (v). Equation (6.9) then becomes (∇ξ A)Y = αρϕY − ρ2 ϕY + ϕY = −(ρ2 − αρ − 1)ϕY = 0. For Y ∈ Γ(Tμ ) = ℝBξ ⊕ ℝϕBξ we have Y = g(Y, Bξ )Bξ + g(Y, ϕBξ )ϕBξ = η(BY)Bξ − η(BϕY)ϕBξ = η(BY)Bξ − ρ(Y)ϕBξ and hence ϕY = η(BY)ϕBξ + ρ(Y)Bξ . We also have ϕAY = AϕY = μϕY = 0 because of (v). Equation (6.9) then becomes (∇ξ A)Y = ϕY − ρ(Y)Bξ − η(BY)ϕBξ = 0. Altogether this shows that ∇ξ A = 0.

6.4 Hopf hypersurfaces In this section we deduce some basic equations for Hopf hypersurfaces in the complex quadric Qn , n ≥ 3. Let M be a Hopf hypersurface in Qn . Then, by Proposition 3.3.2, the Reeb vector field ξ on M is a principal curvature vector of M everywhere. Thus we have Aξ = αξ with α = g(Aξ , ξ ). Equivalently, the maximal holomorphic subbundle 𝒞 of TM is invariant under A, that is, A𝒞 ⊆ 𝒞 .

6.4 Hopf hypersurfaces | 241

For X, Y, Z ∈ X(M) we rewrite the Codazzi equation (6.3) in the following form: g((∇X A)Y − (∇Y A)X, Z)

= η(X)g(ϕY, Z) − η(Y)g(ϕX, Z) − 2η(Z)g(ϕX, Y) + ρ(X)g(BY, Z) − ρ(Y)g(BX, Z)

− η(BX)g(BY, ϕZ) − η(BX)ρ(Y)η(Z)

+ η(BY)g(BX, ϕZ) + η(BY)ρ(X)η(Z).

(6.10)

Inserting Z = ξ into (6.10) and using the assumption Aξ = αξ leads to g((∇X A)Y − (∇Y A)X, ξ ) = −2g(ϕX, Y) + 2ρ(X)η(BY) − 2ρ(Y)η(BX). On the other hand, we have g((∇X A)Y − (∇Y A)X, ξ )

= g((∇X A)ξ , Y) − g((∇Y A)ξ , X)

= dα(X)η(Y) − dα(Y)η(X) + αg((Aϕ + ϕA)X, Y) − 2g(AϕAX, Y). Comparing the previous two equations and inserting X = ξ yields dα(Y) = dα(ξ )η(Y) + 2η(Bξ )ρ(Y).

(6.11)

Reinserting this into the previous equation yields g((∇X A)Y − (∇Y A)X, ξ )

= 2η(Bξ )ρ(X)η(Y) − 2η(Bξ )ρ(Y)η(X) + αg((ϕA + Aϕ)X, Y) − 2g(AϕAX, Y).

Altogether this implies 0 = 2g(AϕAX, Y) − αg((ϕA + Aϕ)X, Y) − 2g(ϕX, Y)

+ 2η(Bξ )η(X)ρ(Y) − 2η(Bξ )ρ(X)η(Y) + 2ρ(X)η(BY) − 2ρ(Y)η(BX)

= g((2AϕA − α(ϕA + Aϕ) − 2ϕ)X, Y)

+ 2ρ(X)η(BY − η(Bξ )Y) − 2ρ(Y)η(BX − η(Bξ )X)

= g((2AϕA − α(ϕA + Aϕ) − 2ϕ)X, Y)

+ 2ρ(X)g(Y, Bξ − η(Bξ )ξ ) − 2ρ(Y)g(X, Bξ − η(Bξ )ξ )

= 2g(AϕAX, Y) − αg((ϕA + Aϕ)X, Y) − 2g(ϕX, Y) + 2g(X, Cζ )g(Y, Bξ ) − 2g(Y, Cζ )g(X, Bξ )

+ 2g(Bξ , ξ )(g(Y, Cζ )η(X) − g(X, Cζ )η(Y)).

242 | 6 Real hypersurfaces in complex quadrics If Cζ = ζ , we have ρ = 0; otherwise we can use Lemma 6.1.2 to calculate ρ(Y) = g(Y, Cζ ) = g(Y, CJξ ) = −g(Y, JCξ ) = −g(Y, JBξ ) = −g(Y, ϕBξ ). Thus we have proved the following lemma. Lemma 6.4.1. Let M be a Hopf hypersurface in Qn , n ≥ 3. Then we have (2AϕA − α(ϕA + Aϕ) − 2ϕ)X = −2ρ(X)(Bξ − η(Bξ )ξ ) − 2g(X, Bξ − η(Bξ )ξ )ϕBξ for all X ∈ X(M). If ζ is A-principal, we can choose a real structure C such that Cζ = ζ . Then we have ρ = 0 and ϕBξ = −ϕξ = 0, and therefore 2AϕA − α(ϕA + Aϕ) = 2ϕ. If ζ is not A-principal, we can choose a real structure C as in Lemma 6.1.2 and get ρ(X)(Bξ − η(Bξ )ξ ) + g(X, Bξ − η(Bξ )ξ )ϕBξ

= −g(X, ϕ(Bξ − η(Bξ )ξ ))(Bξ − η(Bξ )ξ ) + g(X, Bξ − η(Bξ )ξ )ϕ(Bξ − η(Bξ )ξ ) 󵄩 󵄩2 = 󵄩󵄩󵄩Bξ − η(Bξ )ξ 󵄩󵄩󵄩 (g(X, U)ϕU − g(X, ϕU)U) = sin2 (2t)(g(X, U)ϕU − g(X, ϕU)U),

which is equal to 0 on 𝒬 and equal to sin2 (2t)ϕX on 𝒞 ⊖ 𝒬. Altogether we have proved the following lemma. Lemma 6.4.2. Let M be a Hopf hypersurface in Qn , n ≥ 3. Then the tensor field 2AϕA − α(ϕA + Aϕ) leaves 𝒬 and 𝒞 ⊖ 𝒬 invariant. Moreover, we have 2AϕA − α(ϕA + Aϕ) = 2ϕ

on 𝒬

and 2AϕA − α(ϕA + Aϕ) = 2η(Bξ )2 ϕ

on 𝒞 ⊖ 𝒬.

We will now prove that the principal curvature function α of a Hopf hypersurface is constant if the normal vectors are A-isotropic. Assume that ζ is A-isotropic everywhere. Then we have η(Bξ ) = 0 by Corollary 6.1.4, and from (6.11) we get dα(Y) = dα(ξ )η(Y)

6.4 Hopf hypersurfaces | 243

for all Y ∈ X(M). Since gradM α = dα(ξ )ξ , we can compute the Hessian hessM α by hessM α(X, Y) = g(∇X gradM α, Y)

= d(dα(ξ ))(X)η(Y) + dα(ξ )g(ϕAX, Y)

for all X, Y ∈ X(M). As hessM α is a symmetric bilinear form, the previous equation implies dα(ξ )g((Aϕ + ϕA)X, Y) = 0 for all vector fields X, Y ∈ Γ(𝒞 ). Now let us assume that Aϕ + ϕA = 0. For every principal curvature vector X ∈ 𝒞 with AX = λX this implies AϕX = −ϕAX = −λϕX. We assume ‖X‖ = 1. Using Lemma 6.4.1 we get 1 ≤ λ2 + 1 = ρ(X)η(BϕX) − ρ(ϕX)η(BX)

= g(X, Cζ )2 + g(X, Bξ )2 = ‖X𝒞⊖𝒬 ‖2 ≤ 1,

where X𝒞⊖𝒬 denotes the orthogonal projection of X onto 𝒞 ⊖ 𝒬. This implies ‖X𝒞⊖𝒬 ‖2 = 1 for all principal curvature vectors X ∈ 𝒞 with ‖X‖ = 1. This is only possible if 𝒞 = 𝒞 ⊖ 𝒬, or equivalently, if 𝒬 = 0. Since n ≥ 3, this is not possible. Hence we must have Aϕ + ϕA ≠ 0 everywhere, and therefore dα(ξ ) = 0, which implies gradM α = dα(ξ )ξ = 0. Since M is connected this implies that α is constant. Thus we have proved the following. Proposition 6.4.3. Let M be a Hopf hypersurface in Qn , n ≥ 3. If the normal vector field ζ is A-isotropic everywhere, then the principal curvature function α is constant. On the other hand, assume that ζ is A-principal everywhere, say Cζ = ζ with a real structure C. Then we have ρ(Y) = g(CY, ζ ) = g(Y, Cζ ) = g(Y, ζ ) = 0 for all Y ∈ X(M). It follows from (6.11) that dα(Y) = dα(ξ )η(Y) for all Y ∈ X(M). As above, using the symmetry of the Hessian of α, we conclude that dα(ξ )g((Aϕ + ϕA)X, Y) = 0 for all vector fields X, Y ∈ Γ(𝒞 ). Assume that Aϕ+ϕA = 0. For every principal curvature vector X ∈ 𝒞 with AX = λX we have AϕX = −ϕAX = −λϕX. Using Lemma 6.4.1, the

244 | 6 Real hypersurfaces in complex quadrics assumption Aϕ + ϕA = 0 and the fact that ρ = 0 and Bξ = −ξ in this situation, we get −2(λ2 + 1)ϕX = (2AϕA − α(ϕA + Aϕ) − 2ϕ)X = 0. This is a contradiction and it follows that Aϕ + ϕA ≠ 0 and hence dα(ξ ) = 0. This implies gradM α = dα(ξ )ξ = 0. Since M is connected this implies that α is constant. Furthermore, if ζ is A-principal everywhere, then 𝒞 = 𝒬 and hence 2AϕA − α(ϕA + Aϕ) = 2ϕ on 𝒞 . Thus, if X ∈ 𝒞 with AX = λX, we have (2λ − α)AϕX = (2 + αλ)ϕX. If 2λ = α, then we must have 0 = 2 + αλ = 2 + 2λ2 , which is a contradiction. It follows that 2λ ≠ α and therefore AϕX =

2 + αλ ϕX. 2λ − α

We conclude the following proposition. Proposition 6.4.4. Let M be a Hopf hypersurface in Qn , n ≥ 3. If the normal vector field ζ is A-principal everywhere, then the principal curvature function α is constant. Moreover, if X ∈ 𝒞 is a principal curvature vector with corresponding principal curvature λ, then 2+αλ ϕX ∈ 𝒞 is a principal curvature vector with corresponding principal curvature 2λ−α .

6.5 Real hypersurfaces with isometric Reeb flow In Proposition 6.3.5 we proved that the tube with radius 0 < r < π2 around the complex totally geodesic embedding of the complex projective space ℂP m into Q2m has isometric Reeb flow. In this section we will prove that the converse statement holds. Theorem 6.5.1 ([17]). Let M be a connected real hypersurface in the complex quadric Qn , n ≥ 3. Then M has isometric Reeb flow if and only if n is even, say n = 2m, and M is congruent to an open part of the tube with radius 0 < r < π2 around the complex totally geodesic embedding of the complex projective space ℂP m into Q2m . Taking account of Theorem 6.3.1, this has the following interesting consequence. Corollary 6.5.2. Let M be a connected complete real hypersurface in the complex quadric Qn , n ≥ 3. If M has isometric Reeb flow, then M is a homogeneous real hypersurface in Qn .

6.5 Real hypersurfaces with isometric Reeb flow

| 245

It is remarkable that in this situation the existence of a particular one-parameter group of isometries implies transitivity of the isometry group. As another interesting consequence we get the following corollary. Corollary 6.5.3. There are no real hypersurfaces with isometric Reeb flow in the odddimensional complex quadric Q2m+1 , m ≥ 1. To our knowledge, the odd-dimensional complex quadrics are the first examples of homogeneous Kähler manifolds which do not admit a real hypersurface with isometric Reeb flow. We already studied real hypersurfaces with isometric Reeb flow in Kähler manifolds in Section 3.4. In particular, from Proposition 3.4.1 we already know that any such hypersurface is a Hopf hypersurface and satisfies Aϕ = ϕA. Let M be a real hypersurface in Qn , n ≥ 3, with isometric Reeb flow, and let X, Y, Z ∈ X(M). Differentiating the equation 0 = Aϕ − ϕA covariantly leads to 0 = (∇X A)ϕY + A(∇X ϕ)Y − (∇X ϕ)AY − ϕ(∇X A)Y = (∇X A)ϕY + A(η(Y)AX − g(AX, Y)ξ )

− (η(AY)AX − g(AX, AY)ξ ) − ϕ(∇X A)Y

= (∇X A)ϕY + η(Y)A2 X − αg(AX, Y)ξ

− η(AY)AX + g(AX, AY)ξ − ϕ(∇X A)Y.

If we define Ψ(X, Y, Z) = g((∇X A)Y, ϕZ) + g((∇X A)Z, ϕY), the previous equation implies Ψ(X, Y, Z) = αη(Z)g(AX, Y) − η(Z)g(AX, AY)

+ η(AY)g(AX, Z) − η(Y)g(A2 X, Z).

Evaluating Ψ(X, Y, Z) + Ψ(Y, Z, X) − Ψ(Z, X, Y) leads to 2g((∇X A)Y, ϕZ) = Ω(X, Y, Z) − Ω(Y, Z, X) + Ω(Z, X, Y)

+ 2αη(Z)g(AX, Y) − 2η(Z)g(A2 X, Y),

where Ω(X, Y, Z) = g((∇X A)Y − (∇Y A)X, ϕZ).

246 | 6 Real hypersurfaces in complex quadrics The three Ω-terms can be evaluated using the Codazzi equation, which leads to 2g((∇X A)Y, ϕZ) = ρ(X)(g(BY, ϕZ) − g(BZ, ϕY))

+ η(BX)(g(JCY, ϕZ) − g(JCZ, ϕY))

− ρ(Y)(g(BX, ϕZ) + g(BZ, ϕX))

− η(BY)(g(JCX, ϕZ) + g(JCZ, ϕX))

+ ρ(Z)(g(BX, ϕY) + g(BY, ϕX))

+ η(BZ)(g(JCX, ϕY) + g(JCY, ϕX))

+ 2η(Z)g(ϕX, ϕY) − 2η(Y)g(ϕX, ϕZ) + 2αη(Z)g(AX, Y) − 2η(Z)g(A2 X, Y).

Replacing ϕZ by JZ − η(Z)ζ , and similarly for X and Y, one can easily calculate that g(JCY, ϕZ) − g(JCZ, ϕY) = η(Y)η(BZ) − η(Z)η(BY),

g(JCX, ϕZ) + g(JCZ, ϕX) = 2g(BX, Z) − η(X)η(BZ) − η(Z)η(BX),

g(JCX, ϕY) + g(JCY, ϕX) = 2g(BX, Y) − η(X)η(BY) − η(Y)η(BX). Inserting this into the previous equation gives 2g((∇X A)Y, ϕZ) = ρ(X)(g(BY, ϕZ) − g(BZ, ϕY))

− ρ(Y)(g(BX, ϕZ) + g(BZ, ϕX))

+ ρ(Z)(g(BX, ϕY) + g(BY, ϕX))

− 2η(BY)g(BX, Z) + 2η(BZ)g(BX, Y)

+ 2η(Z)g(ϕX, ϕY) − 2η(Y)g(ϕX, ϕZ) + 2αη(Z)g(AX, Y) − 2η(Z)g(A2 X, Y). Since g(BY, ϕZ) − g(BZ, ϕY) = η(Y)ρ(Z) − η(Z)ρ(Y),

g(BX, ϕY) + g(BY, ϕX) = 2g(BX, ϕY) + η(Y)ρ(X) − η(X)ρ(Y), g(BX, ϕZ) + g(BZ, ϕX) = 2g(BX, ϕZ) + η(Z)ρ(X) − η(X)ρ(Z),

we get g((∇X A)Y, ϕZ) = ρ(X)η(Y)ρ(Z) − ρ(X)ρ(Y)η(Z)

− ρ(Y)g(BX, ϕZ) + ρ(Z)g(BX, ϕY) − η(BY)g(BX, Z) + η(BZ)g(BX, Y)

+ η(Z)g(ϕX, ϕY) − η(Y)g(ϕX, ϕZ) + αη(Z)g(AX, Y) − η(Z)g(A2 X, Y).

6.5 Real hypersurfaces with isometric Reeb flow

| 247

Replacing Z by ϕZ and using ϕ2 Z = −Z + η(Z)ξ gives − g((∇X A)Y, Z) + η(Z)g((∇X A)Y, ξ )

= ρ(X)η(Y)ρ(ϕZ) + ρ(Y)g(BX, Z)

− ρ(Y)η(Z)η(BX) + ρ(ϕZ)g(BX, ϕY)

− η(BY)g(BX, ϕZ) + η(BϕZ)g(BX, Y) + η(Y)g(ϕX, Z). Since g((∇X A)Y, ξ ) = dα(X)η(Y) + αg(AϕX, Y) − g(A2 ϕX, Y), this implies g((∇X A)Y, Z) = dα(X)η(Y)η(Z) + η(Z)g((αAϕ − A2 ϕ)X, Y) − ρ(X)η(Y)ρ(ϕZ) − ρ(Y)g(BX, Z)

+ ρ(Y)η(Z)η(BX) − ρ(ϕZ)g(BX, ϕY)

+ η(BY)g(BX, ϕZ) − η(BϕZ)g(BX, Y) − η(Y)g(ϕX, Z).

From this we get an explicit expression for the covariant derivative of the shape operator, namely (∇X A)Y = (dα(X)η(Y) + g((αAϕ − A2 ϕ)X, Y) − η(Bξ )η(Y)ρ(X) + η(BX)ρ(Y) − η(Bξ )g(BX, ϕY))ξ

+ (η(Y)ρ(X) + g(BX, ϕY))Bξ + g(BX, Y)ϕBξ − ρ(Y)BX − η(Y)ϕX − η(BY)ϕBX. Inserting Y = ξ and choosing X ∈ Γ(𝒞 ) then leads to αAϕX − A2 ϕX = −η(Bξ )ρ(X)ξ + ρ(X)Bξ + η(BX)ϕBξ − ϕX − η(Bξ )ϕBX. On the other hand, from Lemma 6.4.1 we get αAϕX − A2 ϕX = −η(Bξ )ρ(X)ξ + ρ(X)Bξ + η(BX)ϕBξ − ϕX. Comparing the previous two equations leads to η(Bξ )ϕBX = 0 for all X ∈ Γ(𝒞 ).

(6.12)

248 | 6 Real hypersurfaces in complex quadrics Let us first assume that η(Bξ ) ≠ 0. Then we have ϕBX = 0 for all X ∈ Γ(𝒞 ), which implies BX = η(BX)ξ for all X ∈ Γ(𝒞 ), and therefore CX = BX + ρ(X)ζ = η(BX)ξ + ρ(X)ζ for all X ∈ Γ(𝒞 ). This implies C 𝒞 ⊆ ℂζ , which gives a contradiction since C is an isomorphism everywhere and the rank of 𝒞 is equal to 2(n − 1) and n ≥ 3. Therefore we must have η(Bξ ) = 0, which means that ζ is A-isotropic. We thus have proved the interesting fact that the normal vector field of a real hypersurface with isometric Reeb flow must be a singular tangent vector of the complex quadric at each point. Proposition 6.5.4. Let M be a real hypersurface in Qn , n ≥ 3, with isometric Reeb flow. Then the normal vector field ζ is A-isotropic everywhere. From Proposition 6.5.4 and Proposition 6.4.3 we conclude that the principal curvature function α is constant. Since the normal vector field ζ is A-isotropic, the vector fields ζ , ξ , Bξ and ϕBξ are pairwise orthonormal. This implies that 𝒞 ⊖ 𝒬 = ℂBξ .

From Lemma 6.4.2 we know that the tensor field 2AϕA − α(ϕA + Aϕ) = 2ϕ(A2 − αA) leaves 𝒬 and 𝒞 ⊖ 𝒬 invariant and ϕ(A2 − αA) = ϕ on 𝒬 and ϕ(A2 − αA) = 0

on 𝒞 ⊖ 𝒬.

Since ϕ is an isomorphism of 𝒬 and of 𝒞 ⊖ 𝒬, this implies A2 − αA = I

on 𝒬

and A2 − αA = 0

on 𝒞 ⊖ 𝒬.

As M is a Hopf hypersurface, we have A𝒞 ⊆ 𝒞 . Let X ∈ 𝒞 be a principal curvature vector of M with corresponding principal curvature λ, that is, AX = λX. We decompose X into X = Y + Z with Y ∈ 𝒬 and Z ∈ 𝒞 ⊖ 𝒬. Then we get (λ2 − αλ)Y + (λ2 − αλ)Z = (λ2 − αλ)X = (A2 − αA)X = Y.

6.5 Real hypersurfaces with isometric Reeb flow

| 249

If λ2 − αλ = 0, we must have Y = 0 and therefore X ∈ 𝒞 ⊖ 𝒬. If λ2 − αλ ≠ 0, we must have Z = 0 and therefore X ∈ 𝒬. Altogether this implies the following. Proposition 6.5.5. Let M be a real hypersurface in Qn , n ≥ 3, with isometric Reeb flow. Then the distributions 𝒬 and 𝒞 ⊖ 𝒬 = ℂBξ are invariant under the shape operator A of M. Assume that AX = λX with X ∈ 𝒞 . From Lemma 6.4.2 we get λ2 − αλ − 1 = 0 if X ∈ 𝒬 and λ2 − αλ = 0 if X ∈ 𝒞 ⊖ 𝒬. Recall that α is constant. We put α = 2 cot(2r) with 0 < r
α + √α2 + 4 or β < α − √α2 + 4. This finishes the proof of Theorem 6.8.1.

Next, we investigate the case when the unit normal vector field ζ is A-isotropic. Theorem 6.8.2 ([103]). Let M be a Hopf hypersurface with harmonic curvature in the complex quadric Qn , n ≥ 3, with A-isotropic unit normal vector field ζ . Assume that 𝒬 (or equivalently, 𝒞 ⊖𝒬) is invariant under the shape operator and Aϕ = ϕA holds on 𝒞 ⊖𝒬. Then M is congruent to an open part of the tube with radius 0 < r < π2 around the complex totally geodesic embedding of the complex projective space ℂP m into Q2m , where n = 2m, or M has at most six distinct constant principal curvatures α, γ = 0, λ1 , μ1 , λ2 , μ2 whose corresponding principal curvature spaces satisfy Tα = ℝξ , Tγ = 𝒞 ⊖ 𝒬, ϕTλ1 = Tμ1 , ϕTλ2 = Tμ2 and dim(Tλ1 ) + dim(Tλ2 ) = n − 2,

dim(Tμ1 ) + dim(Tμ2 ) = n − 2.

The four principal curvatures λ1 , μ1 , λ2 , μ2 are solutions of the quadratic equation 2x2 − 2βx + 2 + αβ = 0 with β=

α2 + 2 ± √(α2 + 2)2 + 4αh . α

Proof. Since ζ is A-isotropic, the vector fields ζ , ξ , Bξ , ϕBξ are pairwise orthonormal and 𝒞 ⊖ 𝒬 = ℝBξ ⊕ ℝϕBξ . From Proposition 6.4.3 we know that the Hopf principal curvature function α is constant.

6.8 The Ricci tensor of Hopf hypersurfaces | 271

According to (6.24) we have Ric(Y) = (2n − 1)Y + hAY − A2 Y − 3η(Y)ξ + η(BY)Bξ − η(BϕY)ϕBξ for all Y ∈ X(M). Taking the covariant derivative leads to (∇X Ric)Y = dh(X)AY + h(∇X A)Y − (∇X A2 )Y − 3((∇X η)Y)ξ − 3η(Y)∇X ξ

+ (∇X η)(BY)Bξ + η((∇X B)Y)Bξ + η(BY)(∇X B)ξ + η(BY)B∇X ξ

− (∇X η)(BϕY)ϕBξ − η((∇X B)ϕY)ϕBξ − η(B(∇X ϕ)Y)ϕBξ

− η(BϕY)(∇X ϕ)Bξ − η(BϕY)ϕ(∇X B)ξ − η(BϕY)ϕB∇X ξ . We have ∇X ξ = ϕAX,

(∇X η)Y = ∇X η(Y) − η(∇X Y) = g(∇X ξ , Y) = g(ϕAX, Y),

(∇X η)BY = g(ϕAX, BY),

(∇X B)Y = ρ(Y)AX + q(X)ϕBY − g(AX, Y)ϕBξ − q(X)ρ(Y)ξ ,

η((∇X B)Y) = αρ(Y)η(X) − q(X)ρ(Y), (∇X B)ξ = q(X)ϕBξ − η(X)ϕBξ ,

(∇X η)BϕY = g(ϕAX, BϕY),

(∇X B)ϕY = ρ(ϕY)AX + q(X)ϕBϕY − g(AX, ϕY)ϕBξ − q(X)ρ(ϕY)ξ ,

η((∇X B)ϕY) = αρ(ϕY)η(X) − q(X)ρ(ϕY), (∇X ϕ)Y = η(Y)AX − g(AX, Y)ξ ,

B(∇X ϕ)Y = η(Y)BAX − g(AX, Y)Bξ ,

η(B(∇X ϕ)Y) = η(Y)η(BAX),

(∇X ϕ)Bξ = −g(AX, Bξ )ξ ,

ϕ(∇X B)ξ = η(X)Bξ − q(X)Bξ . Inserting all this into the previous equation leads to (∇X Ric)Y = dh(X)AY + h(∇X A)Y − (∇X A2 )Y − 3g(ϕAX, Y)ξ − 3η(Y)ϕAX + g(ϕAX, BY)Bξ + αρ(Y)η(X)Bξ − q(X)ρ(Y)Bξ

+ η(BY)q(X)ϕBξ − η(BY)η(X)ϕBξ + η(BY)BϕAX

− g(ϕAX, BϕY)ϕBξ − αρ(ϕY)η(X)ϕBξ + q(X)ρ(ϕY)ϕBξ

− η(Y)η(BAX)ϕBξ + η(BϕY)η(BAX)ξ − η(BϕY)η(X)Bξ

+ η(BϕY)q(X)Bξ − η(BϕY)ϕBϕAX.

272 | 6 Real hypersurfaces in complex quadrics Using the fact that ρ(Y) = η(BϕY) and ρ(ϕY) = −η(BY) and rearranging the terms, we get (∇X Ric)Y = dh(X)AY + h(∇X A)Y − (∇X A2 )Y

− 3η(Y)ϕAX + η(BY)BϕAX − η(BϕY)ϕBϕAX

− 3g(ϕAX, Y)ξ + η(BϕY)η(BAX)ξ

+ g(ϕAX, BY)Bξ + αη(X)η(BϕY)Bξ − η(X)η(BϕY)Bξ

− g(ϕAX, BϕY)ϕBξ + αη(X)η(BY)ϕBξ − η(X)η(BY)ϕBξ

− η(BAX)η(Y)ϕBξ . We have

0 = BϕX + ϕBX − η(X)ϕBξ − ρ(X)ξ . This implies ϕBϕAX = −ϕ2 BAX − αη(X)Bξ = BAX − η(BAX)ξ − αη(X)Bξ . Moreover, we have g(ϕAX, BϕY) = −g(ϕBϕAX, Y) = −g(BAX, Y) + η(BAX)η(Y) + αη(X)η(BY). Inserting this into the previous equation for (∇X Ric)Y gives (∇X Ric)Y = dh(X)AY + h(∇X A)Y − (∇X A2 )Y

− 3η(Y)ϕAX + η(BY)BϕAX − η(BϕY)BAX − 3g(ϕAX, Y)ξ + 2η(BϕY)η(BAX)ξ

+ g(BϕAX, Y)Bξ + 2αη(X)η(BϕY)Bξ − η(X)η(BϕY)Bξ

+ g(BAX, Y)ϕBξ − 2η(BAX)η(Y)ϕBξ − η(X)η(BY)ϕBξ . Since M has harmonic curvature, this implies 0 = (∇X Ric)Y − (∇Y Ric)X

= dh(X)AY − dh(Y)AX + h(∇X A)Y − h(∇Y A)X − (∇X A2 )Y + (∇Y A2 )X − 3η(Y)ϕAX + 3η(X)ϕAY + η(BY)BϕAX − η(BX)BϕAY

− η(BϕY)BAX + η(BϕX)BAY

− 3g((Aϕ + ϕA)X, Y)ξ + 2η(BϕY)η(BAX)ξ − 2η(BϕX)η(BAY)ξ

+ g(BϕAX, Y)Bξ + 2αη(X)η(BϕY)Bξ − η(X)η(BϕY)Bξ − g(BϕAY, X)Bξ − 2αη(Y)η(BϕX)Bξ + η(Y)η(BϕX)Bξ

+ g(BAX, Y)ϕBξ − 2η(BAX)η(Y)ϕBξ − η(X)η(BY)ϕBξ

6.8 The Ricci tensor of Hopf hypersurfaces | 273

− g(BAY, X)ϕBξ + 2η(BAY)η(X)ϕBξ + η(Y)η(BX)ϕBξ . Taking inner product with the Reeb vector field ξ gives 0 = αdh(X)η(Y) − αdh(Y)η(X)

+ hη((∇X A)Y) − hη((∇Y A)X) − η((∇X A2 )Y) + η((∇Y A2 )X) + η(BY)η(BϕAX) − η(BX)η(BϕAY)

− η(BϕY)η(BAX) + η(BϕX)η(BAY)

− 3g((Aϕ + ϕA)X, Y) + 2η(BϕY)η(BAX) − 2η(BϕX)η(BAY). Using (6.26) and (6.27) we obtain 0 = αdh(X)η(Y) − αdh(Y)η(X)

+ αhg((Aϕ + ϕA)X, Y) − 2hg(AϕAX, Y)

− α2 g((Aϕ + ϕA)X, Y) + g((A2 ϕA + AϕA2 )X, Y) + η(BY)η(BϕAX) − η(BX)η(BϕAY) − η(BϕY)η(BAX) + η(BϕX)η(BAY)

− 3g((Aϕ + ϕA)X, Y) + 2η(BϕY)η(BAX) − 2η(BϕX)η(BAY)

= (αh − α2 − 3)g((Aϕ + ϕA)X, Y) − 2hg(AϕAX, Y) + g((A2 ϕA + AϕA2 )X, Y) + αdh(X)η(Y) − αdh(Y)η(X)

+ η(BY)η(BϕAX) − η(BX)η(BϕAY)

+ η(BϕY)η(BAX) − η(BϕX)η(BAY). Inserting X = ξ implies αdh(ξ )η(Y) = αdh(Y), and reinserting this and the corresponding equation for X gives 0 = (αh − α2 − 3)g((Aϕ + ϕA)X, Y) − 2hg(AϕAX, Y) + g((A2 ϕA + AϕA2 )X, Y) + η(BY)η(BϕAX) − η(BX)η(BϕAY)

+ η(BϕY)η(BAX) − η(BϕX)η(BAY), or equivalently, 0 = (αh − α2 − 3)(Aϕ + ϕA)X − 2hAϕAX + (A2 ϕA + AϕA2 )X

+ η(BϕAX)Bξ − η(BϕX)ABξ − η(BAX)ϕBξ + η(BX)AϕBξ .

(6.34)

274 | 6 Real hypersurfaces in complex quadrics From Lemma 6.4.2 we know that the tensor field 2AϕA − α(Aϕ + ϕA) leaves 𝒬 and 𝒞 ⊖ 𝒬 invariant. Moreover, we have 2AϕA − α(Aϕ + ϕA) = 2ϕ

on 𝒬

(6.35)

on 𝒞 ⊖ 𝒬.

(6.36)

and 2AϕA − α(Aϕ + ϕA) = 0 This implies α α (AϕA + A2 ϕ) + Aϕ + (ϕA2 + AϕA) + ϕA 2 2 α = (A2 ϕ + ϕA2 ) + αAϕA + (Aϕ + ϕA) 2 α 2 α = (A ϕ + ϕA2 ) + α( (Aϕ + ϕA) + ϕ) + (Aϕ + ϕA) 2 2

A2 ϕA + AϕA2 =

α 2 α2 (A ϕ + ϕA2 ) + (Aϕ + ϕA) + αϕ + (Aϕ + ϕA) 2 2 2 α +2 α 2 (Aϕ + ϕA) + αϕ on 𝒬 = (A ϕ + ϕA2 ) + 2 2 =

and α α (AϕA + A2 ϕ) + (ϕA2 + AϕA) 2 2 α 2 2 = (A ϕ + ϕA ) + αAϕA 2 α 2 α2 = (A ϕ + ϕA2 ) + (Aϕ + ϕA) on 𝒞 ⊖ 𝒬. 2 2

A2 ϕA + AϕA2 =

By assumption, A leaves 𝒬 and 𝒞 ⊖ 𝒬 invariant. For X ∈ 𝒞 ⊖ 𝒬 we get from (6.34) 0 = (αh − α2 − 3)(Aϕ + ϕA)X − 2hAϕAX + (A2 ϕA + AϕA2 )X

+ η(BϕAX)Bξ − η(BϕX)ABξ − η(BAX)ϕBξ + η(BX)AϕBξ

=

α 2 α2 + 6 (A ϕ + ϕA2 ) − ( )(Aϕ + ϕA)X 2 2

+ η(BϕAX)Bξ − η(BϕX)ABξ − η(BAX)ϕBξ + η(BX)AϕBξ . By assumption, 𝒞 ⊖ 𝒬 is ϕ-invariant and the equation Aϕ = ϕA holds on 𝒞 ⊖ 𝒬. Since dimℂ (𝒞 ⊖ 𝒬) = 1, we see that A restricted to 𝒞 ⊖ 𝒬 is a multiple ρ of the identity. It

6.8 The Ricci tensor of Hopf hypersurfaces | 275

follows that η(BϕAX)Bξ − η(BϕX)ABξ − η(BAX)ϕBξ + η(BX)AϕBξ = ρη(BϕX)Bξ − ρη(BϕX)Bξ − ρη(BX)ϕBξ + ρη(BX)ϕBξ =0 and hence 0=

α 2 α2 + 6 (A ϕ + ϕA2 ) − ( )(Aϕ + ϕA)X 2 2

for all X ∈ 𝒞 ⊖ 𝒬. This implies 0 = αρ2 ϕX − (α2 + 6)ρϕX = ρ(αρ − α2 − 6)ϕX. This implies ρ = 0 or αρ − α2 − 6 = 0. On the other hand, from (6.36) we have 0 = 2AϕAX − α(Aϕ + ϕA)X = 2ρ(ρ − α)ϕX, which implies ρ = 0 or ρ = α. Altogether this implies ρ = 0 and hence A|𝒞⊖𝒬 = 0.

(6.37)

For X ∈ 𝒬 we have AX, BX, ϕX ∈ 𝒬, and using (6.35) we obtain from (6.34) 0 = (αh − α2 − 3)(Aϕ + ϕA)X − 2hAϕAX + (A2 ϕA + AϕA2 )X = (αh − α2 − 3)(Aϕ + ϕA)X − αh(Aϕ + ϕA)X − 2hϕX

α 2 α2 + 2 (A ϕ + ϕA2 )X + (Aϕ + ϕA)X + αϕX 2 2 α2 + 4 α )(Aϕ + ϕA)X + (α − 2h)ϕX. = (A2 ϕ + ϕA2 )X − ( 2 2 +

(6.38)

Let X ∈ 𝒬 with AX = λX. From (6.35) we obtain (2λ − α)AϕX = (2 + αλ)ϕX. If 2λ = α, then 0 = 2 + αλ = 2 + 2λ2 , which is a contradiction. Thus 2λ − α ≠ 0 and AϕX =

2 + αλ ϕX. 2λ − α

276 | 6 Real hypersurfaces in complex quadrics We define μ =

2+αλ . 2λ−α

From (6.38) we then obtain α 2 α2 + 4 (μ + λ2 ) − (μ + λ) + (α − 2h) 2 2 α α2 + 4 = (μ + λ)2 − αλμ − (μ + λ) + (α − 2h). 2 2

0=

(6.39)

From (6.35) we obtain λμ =

α (μ + λ) + 1. 2

Inserting this into (6.39) implies 0=

α (λ + μ)2 − (α2 + 2)(λ + μ) − 2h. 2 2

If α = 0, we get λμ = 1 and h = −(λ + μ) = −λ − λ1 = − 1+λ from (6.39). This implies λ that the two principal curvatures λ and μ satisfy the quadratic equation x 2 + hx + 1 = 0. It follows that 2λ = −h + √h2 − 4 and 2μ = −h − √h2 − 4 (or vice versa), both with multiplicity n − 2. Together with (6.37), this implies 1 h = tr(A) = (n − 2)(−h + √h2 − 4 − h − √h2 − 4) = −(n − 2)h, 2 which implies h = 0 and leads to a contradiction. It follows that α ≠ 0. Without loss of generality we can assume α > 0. The solutions of the quadratic equation 0 = x2 −

2(α2 + 2) 4h x− α α

are β=

α2 + 2 ± √(α2 + 2)2 + 4αh . α

Since β=λ+μ=λ+

2 + αλ 2λ2 + 2 = , 2λ − α 2λ − α

we have 2λ2 − 2βλ + 2 + αβ = 0. The same argument gives 2μ2 − 2βμ + 2 + αβ = 0.

6.8 The Ricci tensor of Hopf hypersurfaces | 277

Subtracting both equations gives 0 = λ2 − βλ − μ2 + βμ = (λ − μ)(λ + μ − β). If λ = μ, then λ and μ are solutions of the quadratic equation x2 − αx − 1 = 0. If we put α = 2 cot 2r, then λ = μ ∈ {cot(r), − tan(r)}. As a special case, when we choose cot(r) and − tan(r) with the same multiplicities, we can show with our Jacobi field arguments, as in the proof of Theorem 6.5.1, that M is congruent to an open part of a tube around the complex totally geodesic embedding of the complex projective space ℂP m into Q2m , where n = 2m. If λ + μ = β, then λ and μ are solutions of the quadratic equation 2x2 − 2βx + 2 + αβ = 0. This finishes the proof of Theorem 6.8.2. From the above results and computations we can deduce some results about Hopf hypersurfaces with parallel Ricci tensor. For example, it follows that there are no Hopf hypersurfaces with parallel Ricci tensor and A-principal normal vector field in the complex quadric Qn , n ≥ 3 (see also [101]). This result was improved by Lee and Suh [65]. Theorem 6.8.3 ([65]). There are no Hopf hypersurfaces with parallel Ricci tensor in the complex quadric Qn , n ≥ 3. When the Ricci tensor Ric of M satisfies Ric ϕ + ϕ Ric = 0, then M is said to have ϕ-anti-invariant Ricci tensor. Jeong and Suh [51] generalized this concept by requiring that Ric ϕ + ϕ Ric = 2cϕ for some c ∈ ℝ \ {0}. In this situation M is said to have pseudo-ϕ-anti-invariant Ricci tensor (or pseudoanticommuting Ricci tensor). If the Ricci tensor Ric of M satisfies Ric(X) = aX + bη(X)ξ

for some a, b ∈ ℝ,

278 | 6 Real hypersurfaces in complex quadrics then M is said to be pseudo-Einstein. Let M be a pseudo-Einstein real hypersurface. The Ricci tensor is given by Ric(X) = aX + bη(X)ξ for some a, b ∈ ℝ. Then we have Ric(ϕX) = aϕX and ϕRic(X) = aϕX. This implies Ric(ϕX) + ϕRic(X) = 2aϕX. Thus M has pseudo-ϕ-anti-invariant Ricci tensor. We now discuss a particular example. Let M be the tube with radius 0 < r
3) classified the cohomogeneity one actions on the complex hyperbolic quadric ∗ Qn , n ≥ 3, up to orbit equivalence. This naturally leads to the classification of ho∗ mogeneous real hypersurfaces in the complex hyperbolic quadric Qn . Geometrically, n∗ one can divide the homogeneous real hypersurfaces in Q into two types: ∗ (I): There exists a closed subgroup H of SOo2,n such that the action of H on Qn is ∗ of cohomogeneity one and the orbits of H form a Riemannian foliation on Qn . Every n∗ orbit is a homogeneous real hypersurface in Q . ∗ (II): There exists a closed subgroup H of SOo2,n such that the action of H on Qn is of cohomogeneity one and there exists exactly one singular orbit F. Every orbit of H different from F is a tube with some radius r ∈ ℝ+ around F and a homogeneous real ∗ hypersurface in Qn . The homogeneous real hypersurfaces of type (I) were classified by Berndt and Tamaru in [20] in the more general context of irreducible Riemannian symmetric spaces of non-compact type. We now describe the construction of these homoge∗ neous real hypersurfaces for Qn . Let g be the Lie algebra of G = SOo2,n and k be the Lie algebra of the isotropy group ∗ K = SO2 SOn of G at o ∈ Qn . Let B be the Killing form of g and denote by p the orthogonal complement of k in g with respect to B. The resulting decomposition g = k ⊕ p is a Cartan decomposition of g. The Cartan involution θ ∈ Aut(g) on g is given by θ(X) = I2,n XI2,n

−I2 0n,2

with I2,n = (

02,n ), In

where I2 and In are the identity (2 × 2)-matrix and (n × n)-matrix, respectively. Then ⟨X, Y⟩ = −B(X, θY) is a positive definite Ad(K)-invariant inner product on g. Its restric∗ tion to p induces a Riemannian metric g̃ on Qn = SOo2,n /SO2 SOn , which is also known ∗ as the Killing metric on Qn . Note that g̃ = 4ng, where g is the Riemannian metric on ∗ Qn that we introduced in Section 1.4.3. Hence the minimum of the sectional curvature ∗ of (Qn , g)̃ is − n1 .

288 | 7 Real hypersurfaces in complex hyperbolic quadrics The Lie algebra k decomposes orthogonally into k = so2 ⊕ son . The first factor so2 is the 1-dimensional center of k. The adjoint action of 0 1 (0 Z=( .. . (0

−1 0 0 .. . 0

0 0 0 .. . 0

0 0 0) ) ∈ so2 ⊂ k = so2 ⊕ son .. . 0)

⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ .. . ⋅⋅⋅

(7.5)

on p induces the Kähler structure J on Qn . We identify the tangent space To Qn of Qn at o with p in the usual way. Let a be a maximal Abelian subspace of p and a∗ be the dual vector space of a. For each α ∈ a∗ we define ∗

∗

∗

gα = {X ∈ g : ad(H)X = α(H)X for all H ∈ a}. If α ≠ 0 and gα ≠ {0}, then α is a restricted root and gα is a restricted root space. Let Σ ⊂ a∗ be the set of restricted roots. For each α ∈ Σ we define Hα ∈ a by α(H) = ⟨Hα , H⟩ for all H ∈ a. The root spaces provide a restricted root space decomposition g = g0 ⊕ (⨁ gα ) α∈Σ

of g, where g0 = k0 ⊕ a and k0 ≅ son−2 is the centralizer of a in k. The corresponding restricted root system is of type B2 . We choose a set Λ = {α1 , α2 } of simple roots of Σ such that α1 is the longer root of the two simple roots and denote by Σ+ the resulting set of positive restricted roots. If we write, as usual, α1 = ϵ1 −ϵ2 and α2 = ϵ2 , the positive restricted roots are α1 = ϵ1 − ϵ2 ,

α2 = ϵ2 ,

α1 + α2 = ϵ1 ,

α1 + 2α2 = ϵ1 + ϵ2 .

The multiplicities of the two long roots α1 and α1 + 2α2 are equal to 1, and the multiplicities of the two short roots α2 and α1 + α2 are equal to n − 2. We denote by C̄ + (Λ) the closed positive Weyl chamber in a that is determined by Λ. Note that C̄ + (Λ) is the closed cone in a bounded by the half-lines spanned by Hα1 +α2 and Hα1 +2α2 . We define a nilpotent subalgebra n of g by n = ⨁ gλ . λ∈Σ+

Then g = k ⊕ a ⊕ n is an Iwasawa decomposition of g, which induces a corresponding Iwasawa decomposition G = KAN of G. The subalgebra s = a ⊕ n of g is solvable, and the corresponding connected closed subgroup S = AN of G with Lie algebra s is ∗ solvable and simply connected and acts simply transitively on Qn . In this way we can

7.3 Homogeneous real hypersurfaces | 289

identify Qn with the solvable Lie group S = AN equipped with a suitable left-invariant Riemannian metric. Let ℓ be a linear line in a. Then the orthogonal complement ∗

sℓ = (a ⊕ n) ⊖ ℓ = (a ⊖ ℓ) ⊕ n of ℓ in a ⊕ n is a subalgebra of a ⊕ n of codimension 1. Let Sℓ be the connected Lie ∗ subgroup of AN with Lie algebra sℓ . Then the orbits of the action of Sℓ on Qn form a ∗ homogeneous foliation Fℓ on Qn of codimension 1. It was proved in [20] that any two leaves of the foliation Fℓ are isometrically congruent to each other. We denote by Mℓ the leaf containing the point o, that is, Mℓ = Sℓ ⋅ o. Let i ∈ {1, 2}. For each unit vector ζ ∈ gαi the subspace sζ = a ⊕ (n ⊖ ℝζ ) is a subalgebra of a ⊕ n. Moreover, if ζ , η are two unit vectors in gαi , then there exists an isometry k in the centralizer of a in K o such that Ad(k)(sζ ) = sη (see [20], Lemma 4.1). This implies that for each i ∈ {1, 2} we obtain a congruence class of homogeneous ∗ foliations of codimension 1 on Qn . More precisely, let αi ∈ Λ, ζ ∈ gαi be a unit vector and Sζ be the connected Lie subgroup of AN with Lie algebra sζ . Then the orbits of the ∗ ∗ action of Sζ on Qn form a homogeneous foliation Fζ of codimension 1 on Qn . If η ∈ gαi is another unit vector, then the induced foliation Fη is isometrically congruent to Fζ under an isometry in the centralizer of a in K o . We denote by Fi a representative of this ∗ congruence class of homogeneous foliations of codimension 1 on Qn , that is, Fi = Fζ for some unit vector ζ ∈ gαi . Analogously, we denote by Si the corresponding solvable subgroup of AN. The geometry of the leaves of the foliation Fi was investigated in [20]. We denote by Mi the orbit of Si containing o. Then Mi is a minimal homogeneous ∗ real hypersurface in Qn . Any other two leaves of Fi are isometrically congruent if and only if they have the same distance to the minimal leaf Mi . The following result is a consequence of the main theorem in [20]. Theorem 7.3.1 ([20]). Let M be a homogeneous real hypersurface of type (I) in Qn , n ≥ 3. Then M is isometrically congruent to Mℓ for some ℓ ∈ a or to M1 or M2 or one of its equidistant hypersurfaces. ∗

We now turn our attention to the homogeneous real hypersurfaces in Qn of type (II), which are precisely the homogeneous real hypersurfaces with a non-empty focal set. Any such homogeneous real hypersurface can be realized as a tube with radius r ∈ ℝ+ around its focal set. The homogeneous real hypersurfaces with a totally geodesic focal set were classified in [21]. ∗

Theorem 7.3.2 ([21]). Let M be a homogeneous real hypersurface in Qn , n ≥ 3, with a totally geodesic focal set. Then M is isometrically congruent to one of the following homogeneous real hypersurfaces: (i) the tube with radius r ∈ ℝ+ around the complex totally geodesic embedding of the ∗ ∗ complex hyperbolic quadric Qn−1 into Qn ; ∗

290 | 7 Real hypersurfaces in complex hyperbolic quadrics (ii) the tube with radius r ∈ ℝ+ around the totally real totally geodesic embedding of ∗ the real hyperbolic space ℝH n into Qn ; (iii) (only if k = 2m is even) the tube with radius r ∈ ℝ+ around the complex totally ∗ geodesic embedding of the complex hyperbolic space ℂH m into Q2m . The structure of homogeneous real hypersurfaces with a non-totally geodesic focal set was investigated in [23]. There it was shown that any such homogeneous real hypersurface can be obtained through one of two possible construction methods, namely the canonical extension method or the nilpotent construction method. We describe the first method now in more detail. The Lie algebra g = so2,n has two maximal parabolic subalgebras. For a more explicit description, we first define the reductive subalgebras l1 = g−α1 ⊕ g0 ⊕ gα1 ≅ ℝ ⊕ sl2 (ℝ) ⊕ son−2 , l2 = g−α2 ⊕ g0 ⊕ gα2 ≅ ℝ ⊕ so1,n−1 of g. Note that we have g0 = a ⊕ k0 ≅ ℝ2 ⊕ son−2 . We then define the nilpotent subalgebras n1 = n ⊖ gα1

and

n2 = n ⊖ gα2

q1 = l1 ⊕ n1

and

q2 = l2 ⊕ n2

of g. Then

are maximal parabolic subalgebras of g. Any parabolic subalgebra of g is conjugate to a subalgebra of q1 or q2 . We denote by L1 , L2 , N1 , N2 , Q1 , Q2 the closed subgroup of G with Lie algebra l1 , l2 , n1 , n2 , q1 , q2 , respectively. The orbit L1 ⋅ o is isometric to ℝ × ℝH 2 and the orbit L2 ⋅ o is isometric to ℝ × ℝH n−1 . Both orbits are totally geodesic submanifolds ∗ of Qn . Let g1 = sl2 (ℝ) and g2 = so1,n−1 , and let G1 ≅ SL2 (ℝ) and G2 ≅ SOo1,n−1 be the corresponding closed subgroup of G with Lie algebra g1 , g2 , respectively. We define B1 = G1 ⋅ o = ℝH 2 and B2 = G2 ⋅ o = ℝH n−1 . The totally geodesic submanifolds B1 and ∗ B2 are boundary components of the maximal Satake compactification of Qn . Now let Hi be a closed subgroup of Gi acting on Bi with cohomogeneity one so that Hi ⋅ o is a homogeneous hypersurface in Bi . Let hi be the Lie algebra of Hi . We then define the subalgebra h̃ i = hi ⊕ ni

7.4 Horospheres | 291

of qi and denote by H̃ i the corresponding closed subgroup of Qi . Then the orbit M = ∗ H̃ i ⋅ o is a homogeneous real hypersurface in Qn . This construction method is called

the canonical extension method. In our context it says that from every homogeneous hypersurface in ℝH 2 or in ℝH n−1 one can construct a homogeneous real hypersurface ∗ in Qn by canonical extension. The nilpotent construction method is more involved and we do not explain it explicitly here since it was proved by Berndt and Domínguez-Vázquez [10] that the nilpotent construction method does not provide any new homogeneous real hypersurfaces ∗ in Qn . Theorem 7.3.3 ([10, 23]). Every homogeneous real hypersurface in Qn with a nontotally geodesic focal set is obtained by canonical extension of a homogeneous hypersurface in the boundary component B1 = ℝH 2 or in the boundary component B2 = ℝH n−1 . ∗

The homogeneous hypersurfaces in real hyperbolic spaces are all classified, ∗ which leads to a complete classification of homogeneous real hypersurfaces in Qn with non-totally geodesic focal sets that are obtained by the canonical extension method. For the classification of homogeneous hypersurfaces in real hyperbolic spaces we refer to Theorem 13.5.2 in [9]. In the following two sections we will discuss the geometry of some of these homogeneous real hypersurfaces.

7.4 Horospheres The family Mℓ of homogeneous real hypersurfaces that we constructed in Section 7.3 ∗ contains the family of horospheres in Qn . We will now discuss more thoroughly the geometry of these horospheres. Let Hℓ ∈ a be a unit vector, that is, ⟨Hℓ , Hℓ ⟩ = 1, such that ℓ = ℝHℓ . We recall from [20] that the shape operator AHℓ of Mℓ with respect to Hℓ is the adjoint transformation AHℓ = ad(Hℓ ) restricted to sℓ . It follows that the subspace a ⊖ ℓ is a principal curvature space of (Mℓ , g)̃ with corresponding principal curvature 0, and for each λ ∈ Σ+ the root space gλ is a principal curvature space of (Mℓ , g)̃ with corresponding principal curvature λ(Hℓ ). This implies that for the (constant) mean curvature tr(Aℓ ) of each leaf of (Mℓ , g)̃ we have tr(Aℓ ) = ∑ (dim gλ )λ(Hℓ ). λ∈Σ+

We now work this out in more detail.

292 | 7 Real hypersurfaces in complex hyperbolic quadrics We denote by Mk1 ,k2 (ℝ) the real vector space of all (k1 ×k2 )-matrices with real coefficients and by 0k1 ,k2 the (k1 × k2 )-matrix with all coefficients equal to 0. For a = (a1 , a2 ) ∈ ℝ2 we put a1 0

Δ2,2 (a) = (

0 ). a2

Then we have g = {(

A C∗

C ) : A ∈ so2 , B ∈ son , C ∈ M2,n (ℝ)} , B

k = {(

A 0n,2

02,n ) : A ∈ so2 , B ∈ son } , B

p = {(

02,2 C∗

C ) : C ∈ M2,n (ℝ)} , 0n,n

02,2 { { a = {(Δ2,2 (a) { { 0n−2,2

Δ2,2 (a) 02,2 0n−2,2

02,n−2 } } 02,n−2 ) : a ∈ ℝ2 } . } 0n−2,n−2 }

The two vectors 02,2 e1 = (Δ2,2 (1, 0) 0n−2,2

Δ2,2 (1, 0) 02,2 0n−2,2

02,n−2 02,n−2 ) , 0n−2,n−2

02,2 e2 = (Δ2,2 (0, 1) 0n−2,2

Δ2,2 (0, 1) 02,2 0n−2,2

02,n−2 02,n−2 ) 0n−2,n−2

form a basis for a. We denote by ϵ1 , ϵ2 ∈ a∗ the dual vectors of e1 , e2 . Then the root system Σ, the positive roots Σ+ and the simple roots Λ = {α1 , α2 } are given by Σ = {±ϵ1 ± ϵ2 , ±ϵ1 , ±ϵ2 }, Σ+ = {ϵ1 + ϵ2 , ϵ1 − ϵ2 , ϵ1 , ϵ2 }, α1 = ϵ1 − ϵ2 , α2 = ϵ2 . For each λ ∈ Σ we define the corresponding restricted root space pλ in p by pλ = (gλ ⊕ g−λ ) ∩ p.

7.4 Horospheres | 293

Then we have p0 = a and 0 { { { { 0 { { { { { ( 0 { { {( ( pϵ1 = {( 0 { ( v { { ( 1 { { { .. { { { . { { v {( n−2

0 { { { { 0 { { { { ( { 0 { { {( (0 pϵ2 = {( {(0 { ( { { { { .. { { { . { { {(0

pϵ1 −ϵ2

pϵ1 +ϵ2

0 { { { { 0 { { { { (0 { { {( { ( = {( x (0 { { ( { { { { .. { { { . { { {(0

0 { { { { 0 { { { { (0 { { { {( ( = {(−x (0 { { ( { { { { .. { { { . { { 0 {(

0 0 0 0 0 .. . 0

0 0 0 0 0 .. . 0

0 0 0 0 0 .. . 0

v1 0 0 0 0 .. . 0

⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ .. . ⋅⋅⋅

0 0 0 0 v1 .. . vn−2

0 0 0 0 0 .. . 0

0 0 0 0 0 .. . 0

0 v1 0 0 0 .. . 0

⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ .. . ⋅⋅⋅

0 0 x 0 0 .. . 0

0 x 0 0 0 .. . 0

0 0 x 0 0 .. . 0

0 x 0 0 0 .. . 0

x 0 0 0 0 .. . 0 −x 0 0 0 0 .. . 0

0 0 0 0 0 .. . 0

0 } } } } vn−2 } } } } } 0 ) } } ) } n−2 0 ) ) : v1 , . . . , vn−2 ∈ ℝ} ≅ ℝ , } ) } 0 ) } } } } .. } } } . } } 0 ) }

0 } } } } 0 } } } } } 0) } } ) } ) 0) : x ∈ ℝ ≅ ℝ, } } } 0) } ) } } } .. } } } . } } 0) }

⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ .. . ⋅⋅⋅ 0 0 0 0 0 .. . 0

vn−2 } } } } 0 } } } } } 0 ) } } ) } n−2 0 ) ) : v1 , . . . , vn−2 ∈ ℝ} ≅ ℝ , } ) } 0 ) } } } } .. } } } . } } 0 ) }

⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ ⋅⋅⋅ .. . ⋅⋅⋅

0 } } } } 0 } } } } } 0) } } ) } ) 0) : x ∈ ℝ ≅ ℝ. } } ) } 0) } } } } .. } } } . } } 0) }

For t ∈ [0, π4 ] we define Ht = cos(t)e1 + sin(t)e2 ∈ a and denote by Mt the horosphere which coincides with the orbit SHt ⋅ o. Every horosphere in SOo2,n /SO2 SOn is isometrically congruent to Mt for some t ∈ [0, π4 ], and two horospheres Mt1 and Mt2 are isometrically congruent if and only if t1 = t2 . The principal curvatures of (Mt , g)̃ with respect to Ht are 0 and λ(Ht ) (λ ∈ Σ+ ), and a ⊖ ℝHt

294 | 7 Real hypersurfaces in complex hyperbolic quadrics and pλ are the corresponding principal curvature spaces. We now take into account our renormalization g̃ = 4ng and compute the principal curvatures and corresponding eigenspaces and multiplicities of the horospheres Mt = (Mt , g), which we list in Table 7.1. Table 7.1: Principal curvatures of the horospheres Mt . Principal curvature

Eigenspace

Multiplicity

0 √2(cos(t) − sin(t)) √2(cos(t) + sin(t)) √2 cos(t) √2 sin(t)

a ⊖ ℝHt pϵ1 −ϵ2 pϵ1 +ϵ2 pϵ1 pϵ2

1 1 1 n−2 n−2

Thus the number of distinct principal curvatures is 5 unless t ∈ {0, arctan( 21 ), π4 }. In these three cases we get Table 7.2. Table 7.2: Principal curvatures of the horospheres M0 , Marctan( 1 ) and M π . 2

t

Principal curvature

Eigenspace

Multiplicity

0

0 √2

ℝe2 ⊕ pϵ2 pϵ1 ⊕ pϵ1 −ϵ2 ⊕ pϵ1 +ϵ2

n−1 n

0

ℝ(e1 − 2e2 )

1

arctan( 21 )

π 4

√2 √5 2√2 √5 3√2 √5

0 1 2

pϵ2 ⊕ pϵ1 −ϵ2

n−1

pϵ1

n−2

pϵ1 +ϵ2

1

ℝ(e1 − e2 ) ⊕ pϵ1 −ϵ2 pϵ1 ⊕ pϵ2

2 2n − 4

pϵ1 +ϵ2

4

1

We now investigate the maximal complex subbundle 𝒞t of TMt . We recall that the com∗ plex structure J on p ≅ To Qn is given by JX = ad(Z)X for all X ∈ p, where Z is as in (7.5). In particular, we get JHt = iHt ∈ pϵ1 +ϵ2 ⊕ pϵ1 −ϵ2 . The maximal complex subbundle 𝒞t of Mt is invariant under the shape operator of Mt if and only if JHt is a principal curvature vector. Using the above tables and root space descriptions it is easy to see that JHt is a principal curvature vector of Mt if and only if t ∈ {0, π4 }. These two values for t correspond exactly to the boundary of the closed

7.4 Horospheres | 295

positive Weyl chamber C̄ + (Λ) and therefore to the two types of singular geodesics in ∗ Qn . Thus we conclude the following theorem. Theorem 7.4.1. Let M be a horosphere in Qn , n ≥ 3. The following statements are equivalent: (i) the center of M is a singular point at infinity; (ii) the normal vector field ζ of M is A-principal or A-isotropic; (iii) M is a Hopf hypersurface. ∗

We now study the geometry of the horospheres with singular points at infinity in more detail. Note that t = 0 corresponds to an A-principal point at infinity (that is, ζ is A-principal) and t = π4 corresponds to an A-isotropic point at infinity (that is, ζ is A-isotropic). From the above computations we obtain Theorem 7.4.2. Let M be a horosphere in Qn , n ≥ 3, with A-isotropic center at infinity. Then M has three distinct constant principal curvatures ∗

α = 2,

β = 1,

μ=0

with corresponding principal curvature spaces Tα = ℝξ ,

Tβ = 𝒬,

Tμ = 𝒞 ⊖ 𝒬.

The corresponding multiplicities of the principal curvatures are mα = 1,

mβ = 2n − 4,

mμ = 2.

From this we can deduce some geometric information about the horosphere in ∗ Qn , n ≥ 3, with A-isotropic center at infinity. Proposition 7.4.3. Let M be a horosphere in Qn , n ≥ 3, with A-isotropic center at infinity. Then the following statements hold: (i) The mean curvature of M is equal to tr(A) = 2n − 2. (ii) The horosphere M is a Hopf hypersurface, namely Aξ = 2ξ . (iii) The maximal holomorphic subbundle 𝒞 of TM is invariant under the shape operator A of M, that is, A𝒞 ⊆ 𝒞 . (iv) The horosphere M is curvature-adapted, that is, AK = KA. (v) The horosphere M has isometric Reeb flow, or equivalently, Aϕ = ϕA. (vi) The Reeb vector field ξ is an eigenvector of the normal Jacobi operator K everywhere, namely Kξ = −4ξ . (vii) The normal Jacobi operator K and the structure tensor field ϕ commute, that is, Kϕ = ϕK. (viii) The Ricci tensor Ric and the structure tensor field ϕ commute, that is, Ric ϕ = ϕ Ric. ∗

296 | 7 Real hypersurfaces in complex hyperbolic quadrics Proof. (i): We can calculate the mean curvature of M using the principal curvatures and their multiplicities in Theorem 7.4.2. (ii): Since Tα = ℝξ , we have Aξ = αξ and thus M is a Hopf hypersurface. (iii): The maximal holomorphic subbundle 𝒞 of TM is invariant under the shape operator of M by (ii). (iv): It follows from Lemma 1.4.4(ii) and Theorem 7.4.2 that A and K have the same eigenspaces, and hence AK = KA. (v): From the description of the principal curvature spaces in Theorem 7.4.2 we see that Tλ and Tμ are ϕ-invariant. This implies Aϕ = ϕA and hence M has isometric Reeb flow by Proposition 3.4.1. (vi): By assumption, ζ is A-isotropic. From Lemma 1.4.4(ii) we see that ξ = −Jζ is an eigenvector of K everywhere with corresponding eigenvalue −4. (vii): By assumption, ζ is A-isotropic. From Lemma 1.4.4(ii) we see that the maximal holomorphic subbundle 𝒞 decomposes into 𝒞 = E0 ⊕ E−1 , where 0 and −1 are eigenvalues of K and E0 = 𝒞 ⊖ 𝒬 and E−1 = 𝒬 are the corresponding eigenspaces. It is easy to see that both E0 and E−1 are ϕ-invariant. This shows that Kϕ = ϕK. (viii): This is an immediate consequence of Theorem 7.1.11 and parts (v) and (vii) of this proposition. For the horosphere with A-principal singular center at infinity we obtain (see also [58]) the following theorem. Theorem 7.4.4. Let M be a horosphere in Qn , n ≥ 3, with A-principal center at infinity. Then there exists a real structure C with Cζ = ζ . Then M has two distinct constant principal curvatures ∗

α = √2,

λ = 0,

with corresponding principal curvature spaces Tα = ℝξ ⊕ (V(C) ⊖ ℝζ ),

Tλ = J(V(C) ⊖ ℝζ ).

The corresponding multiplicities of the principal curvatures are mα = n,

mλ = n − 1.

From this we can deduce some geometric information about the horosphere in ∗ Qn with A-principal singular center at infinity (see also [58]). Proposition 7.4.5. Let M be a horosphere in Qn , n ≥ 3, with A-principal singular center at infinity. Then the following statements hold: (i) The mean curvature of M is equal to tr(A) = √2n. (ii) The horosphere M is a Hopf hypersurface, namely Aξ = √2ξ . ∗

7.5 Tubes around totally geodesic submanifolds | 297

(iii) The maximal holomorphic subbundle 𝒞 of TM is invariant under the shape operator of M, that is, A𝒞 ⊆ 𝒞 . (iv) The horosphere M is curvature-adapted, that is, AK = KA. (v) The horosphere M is a contact hypersurface and Aϕ + ϕA = √2ϕ. (vi) The Reeb vector field ξ is an eigenvector of the normal Jacobi operator K everywhere, namely Kξ = −2ξ . Proof. (i): We can calculate the mean curvature of M using the principal curvatures and their multiplicities in Theorem 7.4.4. (ii): Since ℝξ ⊂ Tα , we have Aξ = αξ and thus M is a Hopf hypersurface. (iii): The maximal holomorphic subbundle 𝒞 of TM is invariant under the shape operator of M by (ii). (iv): It follows from Lemma 1.4.4(i) and Theorem 7.4.4 that A and K have the same eigenspaces, and hence AK = KA. (v): From the description of the principal curvature spaces in Theorem 7.4.4 we compute the following. For X ∈ Γ(Tα ⊖ 𝒞 ⊥ ) we have ϕAX = √2ϕX

and AϕX = λϕX = 0.

For X ∈ Γ(Tλ ) we have ϕAX = λϕX = 0

and AϕX = √2ϕX.

This implies (Aϕ + ϕA)X = √2ϕX for X ∈ Γ((Tα ⊖ 𝒞 ⊥ ) ⊕ Tλ ) = Γ(𝒞 ). We also have (Aϕ + ϕA)ξ = 0 = 2ϕξ . Altogether this implies Aϕ + ϕA = √2ϕ. From Proposition 3.5.5 we can conclude that M is a contact hypersurface. (vi): From Lemma 1.4.4(i) we see that ξ is an eigenvector of K everywhere, with corresponding eigenvalue −2.

7.5 Tubes around totally geodesic submanifolds We will now investigate the geometry of the tubes around totally geodesic submanifolds that we encountered in Theorem 7.2.1. ∗ ∗ ∗ We start with the tubes around Qn−1 ⊂ Qn . Let r ∈ ℝ+ and Σ = Qn−1 . We can assume that o ∈ Σ. We have dim(Σ) = 2n − 2 and hence codim(Σ) = 2. As in the dual

298 | 7 Real hypersurfaces in complex hyperbolic quadrics compact situation, both To Σ and νo Σ are C-invariant for each real structure C ∈ Ao . ∗ Moreover, since the real codimension of Σ in Qn is 2, there exists for each unit normal vector ζ of Σ at o a real structure C ∈ Ao such that Cζ = ζ . It follows that the normal ∗ space νo Σ consists of A-principal singular tangent vectors of Qn and To Σ = (V(C) ⊖ ℝζ ) ⊕ J(V(C) ⊖ ℝζ ). Let ζ ∈ νo Σ be a unit normal vector and C ∈ Ao be a real structure such that Cζ = ζ . From Lemma 1.4.4 we see that the Jacobi operator R̄ ζ leaves νo Σ and To Σ invari∗ ant. Therefore Σ is a curvature-adapted submanifold of Qn . We can therefore apply Corollary 2.6.2 to calculate the principal curvatures and corresponding principal curvature spaces of the tube Σr with radius r around Σ. Using the notations of Section 2.6, we have Spec(Kζ ) = {0, −2}, Spec(Aζ ) = {0} and Spec(R̄ ⊥ ζ ) = {−2} and corresponding eigenspaces E0 = JV(C) ⊖ ℝJζ ,

E−2 = V(C) ⊖ ℝζ , T0 = To Σ,

V−2 = ℝJζ . Let γζ be the geodesic in Qn with γζ (0) and γ̇ζ (0) = ζ . It follows from Corollary 2.6.2 that the principal curvatures of Σr at γζ (r) with respect to −γ̇ζ (r) are λ = 0, μ = 2 tan−2 (r) = √2 tanh(√2r) and α = cot−2 (r) = √2 coth(√2r). The corresponding principal curvature spaces are the parallel translates of E0 , E−2 and V−2 along γζ from 0 = γζ (0) to p = γζ (r). Since Σr is homogeneous, these conclusions hold at any point of Σr . Thus we have proved the following theorem. ∗

Theorem 7.5.1. Let M be the tube with radius r ∈ ℝ+ around the complex totally ∗ ∗ geodesic embedding of the complex hyperbolic quadric Qn−1 into Qn . Then the unit normal vector field ζ of M is singular and satisfies Cζ = ζ for some real structure C ∈ A (pointwise). In particular, ζ is A-principal. Moreover, M has three distinct constant principal curvatures α = √2 coth(√2r),

λ = 0,

μ = √2 tanh(√2r)

with corresponding principal curvature spaces Tα = ℝJζ ,

Tλ = J(V(C) ⊖ ℝζ ),

Tμ = V(C) ⊖ ℝζ .

7.5 Tubes around totally geodesic submanifolds | 299

The corresponding multiplicities of the principal curvatures are mα = 1,

mλ = n − 1,

mμ = n − 1.

From this we can deduce some geometric information about the tubes around the ∗ ∗ totally geodesic Qn−1 ⊂ Qn . Proposition 7.5.2. Let M be the tube with radius r ∈ ℝ+ around the complex totally ∗ ∗ geodesic embedding of the complex hyperbolic quadric Qn−1 into Qn , n ≥ 3. Then the following statements hold: (i) The mean curvature of M is equal to tr(A) = √2 coth(√2r) + √2(n − 1) tanh(√2r). (ii) The tube M is a Hopf hypersurface, namely Aξ = √2 coth(√2r)ξ . (iii) The maximal holomorphic subbundle 𝒞 of TM is invariant under the shape operator of M, that is, A𝒞 ⊆ 𝒞 . (iv) The tube M is curvature-adapted, that is, AK = KA. (v) The tube M is a contact hypersurface and Aϕ + ϕA = √2 tanh(√2r)ϕ. (vi) The Reeb vector field ξ is an eigenvector of the normal Jacobi operator K everywhere, namely Kξ = −2ξ . Proof. (i): We can calculate the mean curvature of M using the expression of the principal curvatures in Theorem 7.5.1: tr(A) = √2 coth(√2r) + √2(n − 1) tanh(√2r). (ii): Since Tα = ℝξ , we have Aξ = αξ and thus M is a Hopf hypersurface. (iii): The maximal holomorphic subbundle 𝒞 of TM is invariant under the shape operator of M by (ii). ∗ ∗ (iv): Since Σ = Qn−1 is curvature-adapted in Qn , the tubes around it are curvature-adapted by Corollary 2.6.2. (v): From the description of the principal curvature spaces in Theorem 7.5.1 we know that ϕTλ = Tμ and ϕTμ = Tλ . For X ∈ Γ(Tλ ) we have ϕAX = λϕX = 0

and AϕX = μϕX = √2 tanh(√2r)ϕX.

For X ∈ Γ(Tμ ) we have ϕAX = μϕX = √2 tanh(√2r)ϕX

and

AϕX = λϕX = 0.

This implies (Aϕ + ϕA)X = √2 tanh(√2r)ϕX

300 | 7 Real hypersurfaces in complex hyperbolic quadrics for X ∈ Γ(Tλ ⊕ Tμ ) = Γ(𝒞 ). We also have (Aϕ + ϕA)ξ = 0 = √2 tanh(√2r)ϕξ . Altogether this implies Aϕ + ϕA = √2 tanh(√2r)ϕ. From Proposition 3.5.5 we can conclude that M is a contact hypersurface. (vi): From Theorem 7.5.1 we know that the unit normal vector ζ is an A-principal ∗ singular tangent vector of Qn everywhere. From Lemma 1.4.4(i) we see that ξ is an eigenvector of K everywhere, with corresponding eigenvalue −2. Next, we investigate the tubes around the totally real totally geodesic embedding ∗ of the real hyperbolic space ℝH n into Qn . Let r ∈ ℝ+ and Σ = ℝH n . We can assume that o ∈ Σ. We have dim(Σ) = n and codim(Σ) = n. As in the dual compact situation, ∗ both To Σ and νo Σ consist of A-principal tangent vectors of Qn . For each unit normal vector ζ of Σ at o there exists a real structure C ∈ Ao such that Cζ = ζ , To Σ = ℝJζ ⊕ J(V(C) ⊖ ℝζ ), νo Σ = ℝζ ⊕ (V(C) ⊖ ℝζ ).

Let ζ ∈ νo Σ be a unit normal vector and C ∈ Ao a real structure such that Cζ = ζ . From Lemma 1.4.4 we see that the Jacobi operator R̄ ζ leaves νo Σ and To Σ invariant. ∗ Therefore Σ is a curvature-adapted submanifold of Qn . We can therefore apply Corollary 2.6.2 to calculate the principal curvatures and corresponding principal curvature spaces of the tube Σr with radius r around Σ. Using the notations of Section 2.6, we have Spec(Kζ ) = {0, −2}, Spec(Aζ ) = {0} and Spec(R̄ ⊥ ζ ) = {−2} and corresponding eigenspaces E0 = J(V(C) ⊖ ℝζ ),

E−2 = ℝJζ , T0 = To Σ,

V−2 = V(C) ⊖ ℝζ . Let γζ be the geodesic in Qn with γζ (0) and γ̇ζ (0) = ζ . It follows from Corollary 2.6.2 that the principal curvatures of Σr at γζ (r) with respect to −γ̇ζ (r) are λ = 0, α = 2 tan−2 (r) = √2 tanh(√2r) and μ = cot−2 (r) = √2 coth(√2r). The corresponding principal curvature spaces are the parallel translates of E0 , E−2 and V−2 along γζ from 0 = γζ (0) to p = γζ (r). Since Σr is homogeneous, these conclusions hold at any point of Σr . Thus we have proved the following theorem. ∗

Theorem 7.5.3. Let M be the tube with radius r ∈ ℝ+ around the totally real totally ∗ geodesic embedding of the real hyperbolic space ℝH n into Qn . Then the unit normal

7.5 Tubes around totally geodesic submanifolds | 301

vector field ζ of M is singular and satisfies Cζ = ζ for some real structure C ∈ A (pointwise). In particular, ζ is A-principal. Moreover, M has three distinct constant principal curvatures α = √2 tanh(√2r),

λ = 0,

μ = √2 coth(√2r)

with corresponding principal curvature spaces Tα = ℝJζ ,

Tλ = J(V(C) ⊖ ℝζ ),

Tμ = V(C) ⊖ ℝζ .

The corresponding multiplicities of the principal curvatures are mα = 1,

mλ = n − 1,

mμ = n − 1.

From this we can deduce some geometric information about the tubes around the ∗ totally geodesic ℝH n ⊂ Qn . Proposition 7.5.4. Let M be the tube with radius r ∈ ℝ+ around the totally real totally ∗ geodesic embedding of the real hyperbolic space ℝH n into Qn , n ≥ 3. Then the following statements hold: (i) The mean curvature of M is equal to tr(A) = √2 tanh(√2r) + √2(n − 1) coth(√2r). (ii) The tube M is a Hopf hypersurface, namely Aξ = √2 tanh(√2r)ξ . (iii) The maximal holomorphic subbundle 𝒞 of TM is invariant under the shape operator of M, that is, A𝒞 ⊆ 𝒞 . (iv) The tube M is curvature-adapted, that is, AK = KA. (v) The tube M is a contact hypersurface and Aϕ + ϕA = √2 coth(√2r)ϕ. (vi) The Reeb vector field ξ is an eigenvector of the normal Jacobi operator K everywhere, namely Kξ = −2ξ . Proof. (i): We can calculate the mean curvature of M using the expression of the principal curvatures in Theorem 7.5.3: tr(A) = √2 tanh(√2r) + √2(n − 1) coth(√2r). (ii): Since Tα = ℝξ , we have Aξ = αξ and thus M is a Hopf hypersurface. (iii): The maximal holomorphic subbundle 𝒞 of TM is invariant under the shape operator of M by (ii). ∗ (iv): Since Σ = ℝH n is curvature-adapted in Qn , the tubes around it are curvatureadapted by Corollary 2.6.2.

302 | 7 Real hypersurfaces in complex hyperbolic quadrics (v): From the description of the principal curvature spaces in Theorem 7.5.3 we know that ϕTλ = Tμ and ϕTμ = Tλ . For X ∈ Γ(Tλ ) we have ϕAX = λϕX = 0

and AϕX = μϕX = √2 coth(√2r)ϕX.

For X ∈ Γ(Tμ ) we have ϕAX = μϕX = √2 coth(√2r)ϕX

and

AϕX = λϕX = 0.

This implies (Aϕ + ϕA)X = √2 coth(√2r)ϕX for X ∈ Γ(Tλ ⊕ Tμ ) = Γ(𝒞 ). We also have (Aϕ + ϕA)ξ = 0 = √2 coth(√2r)ϕξ . Altogether this implies Aϕ + ϕA = √2 coth(√2r)ϕ. From Proposition 3.5.5 we can conclude that M is a contact hypersurface. (vi): From Theorem 7.5.3 we know that the unit normal vector ζ is an A-principal ∗ singular tangent vector of Qn everywhere. From Lemma 1.4.4(i) we see that ξ is an eigenvector of K everywhere, with corresponding eigenvalue −2. Finally, we investigate the geometry of the tubes around the complex totally ∗ geodesic embedding of the complex hyperbolic space Σ = ℂH m into Q2m . We have dim(Σ) = 2m = codim(Σ). By applying duality, we can use some geometric information that we proved in the dual compact setting. This tells us in particular that both the tangent bundle and the normal bundle of ℂH m consist of A-isotropic singular tangent ∗ vectors of Q2m . We assume that o ∈ Σ. Let ζ ∈ νo Σ be a unit normal vector, which is an A-isotropic ∗ singular tangent vector of Q2m . There exists a real structure C ∈ Ao and orthonormal vectors u, w ∈ V(C) such that ζ = √12 (u + Jw). Using again duality and the results from the dual compact setting, we see that the Jacobi operator R̄ ζ leaves νo Σ and To Σ invari-

ant. Therefore Σ is a curvature-adapted submanifold of Q2m . We can therefore apply Corollary 2.6.2 to calculate the principal curvatures and corresponding principal curvature spaces of the tube Σr with radius r around Σ. Using the notations of Section 2.6, we have Spec(Kζ ) = {0, −1}, Spec(Aζ ) = {0} and Spec(R̄ ⊥ ζ ) = {−1, −4} and corresponding eigenspaces ∗

E0 = ℂCζ ,

E−1 = To ℂH m ⊖ (ℂζ ⊕ ℂCζ ), T0 = To Σ,

7.5 Tubes around totally geodesic submanifolds | 303

V−1 = νo ℂH m ⊖ (ℂζ ⊕ ℂCζ ),

V−4 = ℝJζ .

Let γζ be the geodesic in Q2m with γζ (0) and γ̇ζ (0) = ζ . It follows from Corollary 2.6.2 that the principal curvatures of Σr at γζ (r) with respect to −γ̇ζ (r) are μ = 0, λ = tan−1 (r) = tanh(r), β = cot−1 (r) = coth(r) and α = cot−2 (r) = 2 coth(2r). The corresponding principal curvature spaces are the parallel translates of E0 , E−1 , V−1 and V−4 along γζ from 0 = γζ (0) to p = γζ (r). Note that the parallel translate of ℂCζ corresponds to 𝒞 ⊖ 𝒬 at γζ (r), where 𝒬 is the maximal A-invariant subbundle of Σr (see also Lemma 7.1.2). Since Σr is homogeneous, these conclusions hold at any point. Thus we have proved the following theorem. ∗

Theorem 7.5.5. Let M be the tube with radius r ∈ ℝ+ around the complex totally ∗ geodesic embedding of the complex hyperbolic space ℂH m into Q2m , m ≥ 2. Then the unit normal vector field ζ of M is singular and A-isotropic. Moreover, M has four distinct constant principal curvatures α = 2 coth(2r),

β = coth(r),

λ = tanh(r),

μ=0

with corresponding principal curvature spaces Tα = ℝξ ,

Tβ = νℂH m ⊖ ℂνM,

Tλ = TℂH m ⊖ (𝒞 ⊖ 𝒬),

Tμ = 𝒞 ⊖ 𝒬.

The corresponding multiplicities of the principal curvatures are mα = 1,

mβ = 2m − 2,

mλ = 2m − 2,

mμ = 2.

From this we can deduce some geometric information about the tubes around the ∗ totally geodesic ℂH m ⊂ Q2m . Proposition 7.5.6. Let M be the tube with radius r ∈ ℝ+ around the complex totally ∗ geodesic embedding of the complex hyperbolic space ℂH m into Q2m , m ≥ 2. Then the following statements hold: (i) The mean curvature of M is equal to tr(A) = (4m − 2) coth(2r). (ii)

The tube M is a Hopf hypersurface, namely Aξ = 2 coth(2r)ξ .

304 | 7 Real hypersurfaces in complex hyperbolic quadrics (iii) The maximal holomorphic subbundle 𝒞 of TM and the maximal A-invariant subbundle 𝒬 of TM are invariant under the shape operator of M, that is, A𝒞 ⊆ 𝒞 and A𝒬 ⊆ 𝒬. (iv) The tube M is curvature-adapted, that is, AK = KA. (v) The tube M has isometric Reeb flow, or equivalently, Aϕ = ϕA. (vi) The Reeb vector field ξ is an eigenvector of the normal Jacobi operator K everywhere, namely Kξ = −4ξ . (vii) The normal Jacobi operator K and the structure tensor field ϕ commute, that is, Kϕ = ϕK. (viii) The Ricci tensor Ric and the structure tensor field ϕ commute, that is, Ric ϕ = ϕ Ric. Proof. (i): We can calculate the mean curvature of M using the expression of the principal curvatures in Theorem 7.5.5: tr(A) = 2 coth(2r) + (2m − 2)(coth(r) + tanh(r)) = 2 coth(2r) + (2m − 2)2 coth(2r)

= (4m − 2) coth(2r).

(ii): Since Tα = ℝξ , we have Aξ = αξ and thus M is a Hopf hypersurface. The value for α has been computed in Theorem 7.5.5. (iii): The maximal holomorphic subbundle 𝒞 of TM is invariant under the shape operator of M by (ii). For the maximal A-invariant subbundle 𝒬 of TM we have 𝒬 = Tβ ⊕ Tλ by Theorem 7.5.5, which implies that 𝒬 is invariant under the shape operator of M. ∗ (iv): Since Σ = ℂH m is curvature-adapted in Q2m , the tubes around it are curvature-adapted by Corollary 2.6.2. (v): From the description of the principal curvature spaces in Theorem 7.5.5, and using 𝒬 = Tβ ⊕ Tλ and Lemma 7.1.2(iii), we see that Tβ , Tλ and Tμ are ϕ-invariant. This implies Aϕ = ϕA and hence M has isometric Reeb flow by Proposition 3.4.1. (vi): From Theorem 7.5.5 we know that the unit normal vector ζ is a singular ∗ A-isotropic tangent vector of Q2m . From Lemma 1.4.4(ii) we see that ξ is an eigenvector of K everywhere with corresponding eigenvalue −4. (vii): From Theorem 7.5.5 we know that the unit normal vector field ζ is a singular ∗ A-isotropic tangent vector field of Q2m . From Lemma 1.4.4(ii) we see that the maximal holomorphic subbundle 𝒞 decomposes into 𝒞 = E0 ⊕ E−1 with E0 = ℂCζ ,

E−1 = TQ2m ⊖ (ℂζ ⊕ ℂCζ ), ∗

with some real structure C, where 0 and −1 are eigenvalues of K and E0 and E−1 are the corresponding eigenspaces in 𝒞 . It is easy to see that both E0 and E−1 are ϕ-invariant. This shows that Kϕ = ϕK.

7.6 Hopf hypersurfaces | 305

(viii): This is an immediate consequence of Theorem 7.1.11 and parts (v) and (vii) of this proposition.

7.6 Hopf hypersurfaces In this section we deduce some basic equations for Hopf hypersurfaces in the complex ∗ ∗ hyperbolic quadric Qn , n ≥ 3. Let M be a Hopf hypersurface in Qn . Then, by Proposition 3.3.2, the Reeb vector field ξ on M is a principal curvature vector of M everywhere. Thus we have Aξ = αξ with α = g(Aξ , ξ ). Equivalently, the maximal holomorphic subbundle 𝒞 of TM is invariant under A, that is, A𝒞 ⊆ 𝒞 . For X, Y, Z ∈ X(M) we rewrite the Codazzi equation (7.2) in the following form: g((∇X A)Y − (∇Y A)X, Z)

= η(Y)g(ϕX, Z) − η(X)g(ϕY, Z) + 2η(Z)g(ϕX, Y) − ρ(X)g(BY, Z) + ρ(Y)g(BX, Z)

+ η(BX)g(BY, ϕZ) + η(BX)ρ(Y)η(Z)

− η(BY)g(BX, ϕZ) − η(BY)ρ(X)η(Z).

(7.6)

Inserting Z = ξ into (7.6) and using the assumption Aξ = αξ leads to g((∇X A)Y − (∇Y A)X, ξ ) = 2g(ϕX, Y) − 2ρ(X)η(BY) + 2ρ(Y)η(BX). On the other hand, we have g((∇X A)Y − (∇Y A)X, ξ )

= g((∇X A)ξ , Y) − g((∇Y A)ξ , X)

= dα(X)η(Y) − dα(Y)η(X) + αg((Aϕ + ϕA)X, Y) − 2g(AϕAX, Y). Comparing the previous two equations and inserting X = ξ yields dα(Y) = dα(ξ )η(Y) − 2η(Bξ )ρ(Y). Reinserting this into the previous equation yields g((∇X A)Y − (∇Y A)X, ξ )

= 2η(Bξ )ρ(Y)η(X) − 2η(Bξ )ρ(X)η(Y) + αg((ϕA + Aϕ)X, Y) − 2g(AϕAX, Y).

Altogether this implies 0 = 2g(AϕAX, Y) − αg((ϕA + Aϕ)X, Y) + 2g(ϕX, Y)

− 2η(Bξ )η(X)ρ(Y) + 2η(Bξ )ρ(X)η(Y) − 2ρ(X)η(BY) + 2ρ(Y)η(BX)

(7.7)

306 | 7 Real hypersurfaces in complex hyperbolic quadrics = g((2AϕA − α(ϕA + Aϕ) + 2ϕ)X, Y)

− 2ρ(X)η(BY − η(Bξ )Y) + 2ρ(Y)η(BX − η(Bξ )X)

= g((2AϕA − α(ϕA + Aϕ) + 2ϕ)X, Y)

− 2ρ(X)g(Y, Bξ − η(Bξ )ξ ) + 2ρ(Y)g(X, Bξ − η(Bξ )ξ )

= 2g(AϕAX, Y) − αg((ϕA + Aϕ)X, Y) + 2g(ϕX, Y) − 2g(X, Cζ )g(Y, Bξ ) + 2g(Y, Cζ )g(X, Bξ )

− 2g(Bξ , ξ )(g(Y, Cζ )η(X) − g(X, Cζ )η(Y)). If Cζ = ζ , we have ρ = 0; otherwise we can use Lemma 7.1.2 to calculate ρ(Y) = g(Y, Cζ ) = g(Y, CJξ ) = −g(Y, JCξ ) = −g(Y, JBξ ) = −g(Y, ϕBξ ). Thus we have proved the following. Lemma 7.6.1. Let M be a Hopf hypersurface in Qn , n ≥ 3. Then we have ∗

(2AϕA − α(ϕA + Aϕ) + 2ϕ)X = 2ρ(X)(Bξ − η(Bξ )ξ ) + 2g(X, Bξ − η(Bξ )ξ )ϕBξ for all X ∈ X(M). If ζ is A-principal, we can choose a real structure C such that Cζ = ζ . Then we have ρ = 0 and ϕBξ = −ϕξ = 0, and therefore 2AϕA − α(ϕA + Aϕ) = −2ϕ. If ζ is not A-principal, we can choose a real structure C as in Lemma 7.1.2 and get ρ(X)(Bξ − η(Bξ )ξ ) + g(X, Bξ − η(Bξ )ξ )ϕBξ

= −g(X, ϕ(Bξ − η(Bξ )ξ ))(Bξ − η(Bξ )ξ ) + g(X, Bξ − η(Bξ )ξ )ϕ(Bξ − η(Bξ )ξ ) 󵄩 󵄩2 = 󵄩󵄩󵄩Bξ − η(Bξ )ξ 󵄩󵄩󵄩 (g(X, U)ϕU − g(X, ϕU)U) = sin2 (2t)(g(X, U)ϕU − g(X, ϕU)U),

which is equal to 0 on 𝒬 and equal to sin2 (2t)ϕX on 𝒞 ⊖ 𝒬. Altogether we have proved the following. Lemma 7.6.2. Let M be a Hopf hypersurface in Qn , n ≥ 3. Then the tensor field ∗

2AϕA − α(ϕA + Aϕ) leaves 𝒬 and 𝒞 ⊖ 𝒬 invariant. Moreover, we have 2AϕA − α(ϕA + Aϕ) = −2ϕ

on 𝒬

7.6 Hopf hypersurfaces | 307

and 2AϕA − α(ϕA + Aϕ) = −2η(Bξ )2 ϕ

on 𝒞 ⊖ 𝒬.

Assume that ζ is A-isotropic everywhere. Then we have η(Bξ ) = 0 by Corollary 7.1.4, and from (7.7) we get dα(Y) = dα(ξ )η(Y) for all Y ∈ X(M). Since gradM α = dα(ξ )ξ , we can compute the Hessian hessM α by hessM α(X, Y) = g(∇X gradM α, Y) = d(dα(ξ ))(X)η(Y) + dα(ξ )g(ϕAX, Y) for all X, Y ∈ X(M). As hessM α is a symmetric bilinear form, the previous equation implies dα(ξ )g((Aϕ + ϕA)X, Y) = 0 for all vector fields X, Y ∈ Γ(𝒞 ). Now let us assume that Aϕ + ϕA = 0. For every principal curvature vector X ∈ 𝒞 with AX = λX this implies AϕX = −ϕAX = −λϕX. Using Lemma 7.6.1 we get (1 − λ2 )X = g(X, Bξ )Bξ + g(X, ϕBξ )ϕBξ = X𝒞⊖𝒬 , where X𝒞⊖𝒬 denotes the orthogonal projection of X onto 𝒞 ⊖ 𝒬. The components of this equation in 𝒬 and in 𝒞 ⊖ 𝒬 then give (1 − λ2 )X𝒬 = 0

and λX𝒞⊖𝒬 = 0.

It follows that either X𝒬 = 0 or X𝒞⊖𝒬 = 0. If X𝒬 = 0, then λ = 0, and if X𝒞⊖𝒬 = 0, then λ2 = 1. On the other hand, if Aϕ + ϕA ≠ 0 everywhere, we get dα(ξ ) = 0, which implies gradM α = dα(ξ )ξ = 0. Since M is connected this implies that α is constant. Thus we have proved the following proposition. Proposition 7.6.3. Let M be a Hopf hypersurface in Qn , n ≥ 3. If the normal vector field ζ is A-isotropic everywhere, then at least one of the following two statements holds: (i) We have Aϕ + ϕA ≠ 0 and the principal curvature function α is constant. (ii) We have Aϕ + ϕA = 0 and both 𝒞 ⊖ 𝒬 and 𝒬 are invariant under the shape operator of M. Moreover, for X ∈ 𝒞 ⊖ 𝒬 we have AX = 0 and for all principal curvature vectors X ∈ 𝒬 we have AX = ±X and AϕX = ∓ϕX. ∗

308 | 7 Real hypersurfaces in complex hyperbolic quadrics We now assume that ζ is A-principal everywhere, say Cζ = ζ with some real structure C. Then we have ρ(Y) = g(CY, ζ ) = g(Y, Cζ ) = g(Y, ζ ) = 0 for all Y ∈ X(M). It follows from (7.7) that dα(Y) = dα(ξ )η(Y) for all Y ∈ X(M). As above, using the symmetry of the Hessian of α, we conclude that dα(ξ )g((Aϕ + ϕA)X, Y) = 0 for all vector fields X, Y ∈ Γ(𝒞 ). Assume that Aϕ+ϕA = 0. For every principal curvature vector X ∈ 𝒞 with AX = λX we have AϕX = −ϕAX = −λϕX. Using Lemma 7.6.1, the assumption Aϕ + ϕA = 0 and the fact that ρ = 0 and Bξ = −ξ in this situation, we get (1 − λ2 )ϕX = 0. It follows that λ2 = 1, and if AX = ±X, then AϕX = ∓X. Otherwise, if Aϕ + ϕA ≠ 0, then dα(ξ ) = 0. This implies gradM α = dα(ξ )ξ = 0. Since M is connected this implies that α is constant. Furthermore, if ζ is A-principal everywhere, then 𝒞 = 𝒬 and hence 2AϕA − α(ϕA + Aϕ) = −2ϕ

on 𝒞 .

Thus, if X ∈ 𝒞 with AX = λX, we have (2λ − α)AϕX = (αλ − 2)ϕX. If 2λ = α, then we must have 0 = αλ − 2 = 2(λ2 − 1), and hence λ = ±1. Otherwise we get AϕX =

αλ − 2 ϕX. 2λ − α

We conclude the following. Proposition 7.6.4. Let M be a Hopf hypersurface in Qn , n ≥ 3. If the normal vector field ζ is A-principal everywhere, then one of the following statements holds: (i) We have Aϕ + ϕA ≠ 0 and the principal curvature function α is constant. Moreover, if X ∈ 𝒞 with AX = λX, then one of the following statements holds: (a) If 2λ − α ≠ 0, then ∗

AϕX =

αλ − 2 ϕX. 2λ − α

7.7 Real hypersurfaces with isometric Reeb flow

| 309

(b) If 2λ − α = 0, then λ = 1 or λ = −1. (ii) We have Aϕ + ϕA = 0. Moreover, for all principal curvature vectors X ∈ 𝒞 we have AX = ±X and AϕX = ∓ϕX.

7.7 Real hypersurfaces with isometric Reeb flow In Proposition 7.4.3 we proved that a horosphere in Qn with A-isotropic center at infinity has isometric Reeb flow. We also proved in Proposition 7.5.4 that the tube with radius r ∈ ℝ+ around the complex totally geodesic embedding of the complex hyper∗ bolic space ℂH m into Q2m has isometric Reeb flow. In this section we will prove that the converse statement holds. ∗

Theorem 7.7.1 ([106]). Let M be a connected real hypersurface in the complex hyper∗ bolic quadric Qn , n ≥ 3. Then M has isometric Reeb flow if and only if M is an open part of one of the following real hypersurfaces: ∗ (i) a horosphere in Qn with A-isotropic center at infinity; (ii) (only if n = 2m is even) a tube with radius r ∈ ℝ∗ around the complex totally ∗ geodesic embedding of the complex hyperbolic space ℂH m into Q2m . As we saw in Section 7.3, the hypersurfaces in (i) and (ii) are homogeneous real hypersurfaces in the complex hyperbolic quadric. We therefore get the following interesting consequence of Theorem 7.7.1. Corollary 7.7.2. Let M be a connected complete real hypersurface in the complex hyper∗ bolic quadric Qn , n ≥ 3. If M has isometric Reeb flow, then M is a homogeneous real ∗ hypersurface in Qn . As in the dual compact case, it is remarkable that in this situation the existence of a particular one-parameter group of isometries implies transitivity of the isometry group. We already studied real hypersurfaces with isometric Reeb flow in Kähler manifolds in Section 3.4. In particular, from Proposition 3.4.1 we already know that any such hypersurface is a Hopf hypersurface and satisfies Aϕ = ϕA. ∗ Let M be a real hypersurface in Qn , n ≥ 3, with isometric Reeb flow, and let X, Y, Z ∈ X(M). Differentiating the equation 0 = Aϕ − ϕA covariantly leads to 0 = (∇X A)ϕY + A(∇X ϕ)Y − (∇X ϕ)AY − ϕ(∇X A)Y = (∇X A)ϕY + A(η(Y)AX − g(AX, Y)ξ )

− (η(AY)AX − g(AX, AY)ξ ) − ϕ(∇X A)Y

= (∇X A)ϕY + η(Y)A2 X − αg(AX, Y)ξ

− η(AY)AX + g(AX, AY)ξ − ϕ(∇X A)Y.

310 | 7 Real hypersurfaces in complex hyperbolic quadrics If we define Ψ(X, Y, Z) = g((∇X A)Y, ϕZ) + g((∇X A)Z, ϕY), the previous equation implies Ψ(X, Y, Z) = αη(Z)g(AX, Y) − η(Z)g(AX, AY)

+ η(AY)g(AX, Z) − η(Y)g(A2 X, Z).

Evaluating Ψ(X, Y, Z) + Ψ(Y, Z, X) − Ψ(Z, X, Y) leads to 2g((∇X A)Y, ϕZ) = Ω(X, Y, Z) − Ω(Y, Z, X) + Ω(Z, X, Y)

+ 2αη(Z)g(AX, Y) − 2η(Z)g(A2 X, Y),

where Ω(X, Y, Z) = g((∇X A)Y − (∇Y A)X, ϕZ). The three Ω-terms can be evaluated using the Codazzi equation, which leads to 2g((∇X A)Y, ϕZ) = −ρ(X)(g(BY, ϕZ) − g(BZ, ϕY))

− η(BX)(g(JCY, ϕZ) − g(JCZ, ϕY)) + ρ(Y)(g(BX, ϕZ) + g(BZ, ϕX))

+ η(BY)(g(JCX, ϕZ) + g(JCZ, ϕX))

− ρ(Z)(g(BX, ϕY) + g(BY, ϕX))

− η(BZ)(g(JCX, ϕY) + g(JCY, ϕX))

− 2η(Z)g(ϕX, ϕY) − 2η(Y)g(ϕX, ϕZ) + 2αη(Z)g(AX, Y) − 2η(Z)g(A2 X, Y).

Replacing ϕZ by JZ − η(Z)ζ , and similarly for X and Y, one can easily calculate that g(JCY, ϕZ) − g(JCZ, ϕY) = η(Y)η(BZ) − η(Z)η(BY),

g(JCX, ϕZ) + g(JCZ, ϕX) = 2g(BX, Z) − η(X)η(BZ) − η(Z)η(BX),

g(JCX, ϕY) + g(JCY, ϕX) = 2g(BX, Y) − η(X)η(BY) − η(Y)η(BX). Inserting this into the previous equation gives 2g((∇X A)Y, ϕZ) = −ρ(X)(g(BY, ϕZ) − g(BZ, ϕY))

7.7 Real hypersurfaces with isometric Reeb flow

+ ρ(Y)(g(BX, ϕZ) + g(BZ, ϕX))

− ρ(Z)(g(BX, ϕY) + g(BY, ϕX))

+ 2η(BY)g(BX, Z) + 2η(BZ)g(BX, Y)

− 2η(Z)g(ϕX, ϕY) − 2η(Y)g(ϕX, ϕZ) + 2αη(Z)g(AX, Y) − 2η(Z)g(A2 X, Y).

Since g(BY, ϕZ) − g(BZ, ϕY) = η(Y)ρ(Z) − η(Z)ρ(Y),

g(BX, ϕY) + g(BY, ϕX) = 2g(BX, ϕY) + η(Y)ρ(X) − η(X)ρ(Y), g(BX, ϕZ) + g(BZ, ϕX) = 2g(BX, ϕZ) + η(Z)ρ(X) − η(X)ρ(Z),

we get g((∇X A)Y, ϕZ) = −ρ(X)η(Y)ρ(Z) + ρ(X)ρ(Y)η(Z)

+ ρ(Y)g(BX, ϕZ) − ρ(Z)g(BX, ϕY) + η(BY)g(BX, Z) − η(BZ)g(BX, Y)

− η(Z)g(ϕX, ϕY) + η(Y)g(ϕX, ϕZ) + αη(Z)g(AX, Y) − η(Z)g(A2 X, Y). Replacing Z by ϕZ and using ϕ2 Z = −Z + η(Z)ξ gives − g((∇X A)Y, Z) + η(Z)g((∇X A)Y, ξ )

= −ρ(X)η(Y)ρ(ϕZ) − ρ(Y)g(BX, Z)

+ ρ(Y)η(Z)η(BX) − ρ(ϕZ)g(BX, ϕY)

+ η(BY)g(BX, ϕZ) − η(BϕZ)g(BX, Y) − η(Y)g(ϕX, Z). Since g((∇X A)Y, ξ ) = dα(X)η(Y) + αg(AϕX, Y) − g(A2 ϕX, Y), this implies g((∇X A)Y, Z) = dα(X)η(Y)η(Z) + η(Z)g((αAϕ − A2 ϕ)X, Y) + ρ(X)η(Y)ρ(ϕZ) + ρ(Y)g(BX, Z)

− ρ(Y)η(Z)η(BX) + ρ(ϕZ)g(BX, ϕY)

− η(BY)g(BX, ϕZ) + η(BϕZ)g(BX, Y) + η(Y)g(ϕX, Z).

| 311

312 | 7 Real hypersurfaces in complex hyperbolic quadrics From this we get an explicit expression for the covariant derivative of the shape operator, namely (∇X A)Y = (dα(X)η(Y) + g((αAϕ − A2 ϕ)X, Y) + η(Bξ )η(Y)ρ(X) − η(BX)ρ(Y) + η(Bξ )g(BX, ϕY))ξ − (η(Y)ρ(X) + g(BX, ϕY))Bξ − g(BX, Y)ϕBξ + ρ(Y)BX + η(Y)ϕX + η(BY)ϕBX.

(7.8)

Inserting Y = ξ and choosing X ∈ Γ(𝒞 ) then leads to αAϕX − A2 ϕX = η(Bξ )ρ(X)ξ − ρ(X)Bξ − η(BX)ϕBξ + ϕX + η(Bξ )ϕBX. On the other hand, from Lemma 7.6.1 we get αAϕX − A2 ϕX = η(Bξ )ρ(X)ξ − ρ(X)Bξ − η(BX)ϕBξ + ϕX. Comparing the previous two equations leads to η(Bξ )ϕBX = 0 for all X ∈ Γ(𝒞 ). Let us first assume that η(Bξ ) ≠ 0. Then we have ϕBX = 0 for all X ∈ Γ(𝒞 ), which implies BX = η(BX)ξ for all X ∈ Γ(𝒞 ), and therefore CX = BX + ρ(X)ζ = η(BX)ξ + ρ(X)ζ for all X ∈ Γ(𝒞 ). This implies C 𝒞 ⊆ ℂζ , which gives a contradiction since C is an isomorphism everywhere and the rank of 𝒞 is equal to 2(n − 1) and n ≥ 3. Therefore we must have η(Bξ ) = 0, which means that ζ is A-isotropic. If 0 = Aϕ + ϕA = 2ϕA, then A restricted to 𝒞 vanishes. It follows that case (ii) in Proposition 7.6.3 cannot occur if M has isometric Reeb flow. From Proposition 7.6.3 we therefore conclude that the principal curvature function α is constant. We thus have proved the interesting fact that the normal vector field of a real hypersurface with isometric Reeb flow must be a singular tangent vector of the complex hyperbolic quadric at each point and the Hopf principal curvature function must be constant. Proposition 7.7.3. Let M be a real hypersurface in Qn , n ≥ 3, with isometric Reeb flow. Then the normal vector field ζ is A-isotropic everywhere and the principal curvature function α is constant. ∗

7.7 Real hypersurfaces with isometric Reeb flow

| 313

Since the normal vector field ζ is A-isotropic, the vector fields ζ , ξ , Bξ and ϕBξ are pairwise orthonormal. This implies that 𝒞 ⊖ 𝒬 = ℂBξ .

From Lemma 7.6.2 we know that the tensor field 2AϕA − α(ϕA + Aϕ) = 2ϕ(A2 − αA) leaves 𝒬 and 𝒞 ⊖ 𝒬 invariant and ϕ(A2 − αA) = −ϕ

on 𝒬

and ϕ(A2 − αA) = 0

on 𝒞 ⊖ 𝒬.

Since ϕ is an isomorphism of 𝒬 and of 𝒞 ⊖ 𝒬, this implies A2 − αA = −I

on 𝒬

and A2 − αA = 0

on 𝒞 ⊖ 𝒬.

As M is a Hopf hypersurface, we have A𝒞 ⊆ 𝒞 . Let X ∈ 𝒞 be a principal curvature vector of M with corresponding principal curvature λ, that is, AX = λX. We decompose X into X = Y + Z with Y ∈ 𝒬 and Z ∈ 𝒞 ⊖ 𝒬. Then we get (λ2 − αλ)Y + (λ2 − αλ)Z = (λ2 − αλ)X = (A2 − αA)X = −Y. If λ2 − αλ = 0, we must have Y = 0 and therefore X ∈ 𝒞 ⊖ 𝒬. If λ2 − αλ ≠ 0, we must have Z = 0 and therefore X ∈ 𝒬. Altogether this implies the following. Proposition 7.7.4. Let M be a real hypersurface in Qn , n ≥ 3, with isometric Reeb flow. Then the distributions 𝒬 and 𝒞 ⊖ 𝒬 = ℂBξ are invariant under the shape operator A of M. ∗

Assume that AX = λX with X ∈ 𝒞 . From Lemma 7.6.2 we get λ2 − αλ + 1 = 0 if X ∈ 𝒬 and λ(λ − α) = 0 if X ∈ 𝒞 ⊖ 𝒬. Recall that α is constant. We define Tα = ℝξ . Without loss of generality we can assume that α ≥ 0. The equation λ2 − αλ + 1 = 0 has a solution if and only if α ≥ 2. We therefore distinguish two cases. If α = 2, then the quadratic equation 0 = x2 −αx+1 = x 2 −2x+1 = (x−1)2 has exactly one solution, namely β = 1. Thus A restricted to 𝒬 is the identity transformation. We define Tβ = 𝒬. The quadratic equation x(x − α) has two solutions. Moreover, the rank

314 | 7 Real hypersurfaces in complex hyperbolic quadrics of 𝒞 ⊖ 𝒬 is equal to 2 and 𝒞 ⊖ 𝒬 is both A- and ϕ-invariant. It follows that A restricted to 𝒞 ⊖ 𝒬 is equal to μI, where μ ∈ {0, 2}. We define Tμ = 𝒞 ⊖ 𝒬. If α > 2, we choose r ∈ ℝ+ with α = 2 coth(2r). Then the quadratic equation 0 = x2 − αx + 1 = x 2 − 2 coth(2r)x + 1 has two solutions, namely β = coth(r),

λ = tanh(r).

We denote by Tβ and Tλ the principal curvature spaces corresponding to A restricted to 𝒬. The quadratic equation x(x − α) has two solutions. Moreover, the rank of 𝒞 ⊖ 𝒬 is equal to 2 and 𝒞 ⊖ 𝒬 is both A- and ϕ-invariant. It follows that A restricted to 𝒞 ⊖ 𝒬 is equal to μI, where μ ∈ {0, 2 coth(2r)}. We define Tμ = 𝒞 ⊖ 𝒬. Note that in both cases, since Aϕ = ϕA, we have JTρ = ϕTρ = Tρ for ρ ∈ {β, λ, μ}. According to Lemma 7.6.1 we have (αAϕ − A2 ϕ)X = ϕX − ρ(X)Bξ − η(BX)ϕBξ

= ϕX − η(BϕX)Bξ − η(BX)ϕBξ .

Inserting this into the expression (7.8) for the covariant derivative of A and replacing ρ(X) and ρ(Y) by η(BϕX) and η(BϕY), respectively, leads to (∇X A)Y = (g((αAϕ − A2 ϕ)X, Y) − η(BX)η(BϕY))ξ

− (η(Y)η(BϕX) + g(BX, ϕY))Bξ − g(BX, Y)ϕBξ + η(BϕY)BX + η(Y)ϕX + η(BY)ϕBX

= (g(ϕX, Y) − η(BϕX)η(BY))ξ

− (η(BϕX)η(Y) − g(ϕBX, Y))Bξ − g(BX, Y)ϕBξ + η(BϕY)BX + η(Y)ϕX + η(BY)ϕBX.

This implies g(BX, Y) = −g((∇X A)Y, ϕBξ ). Let X, Y ∈ Γ(Tλ ). Then the previous equation implies g(BX, Y) = −(λ − μ)g(∇X Y, ϕBξ )

= (λ − μ)g(∇X (ϕBξ ), Y)

= (λ − μ)g(ϕ∇X (Bξ ), Y).

7.7 Real hypersurfaces with isometric Reeb flow

| 315

Since Bξ = Cξ , the Gauss formula for M in Qn gives ∗

∇X (Bξ ) = ∇X (Cξ ) = ∇̄X (Cξ ), ∗ where ∇̄ is the Riemannian connection of Qn . Since A is parallel and of rank 2, there n∗ exists a 1-form q on Q so that

(∇̄X C)Y = q(X)JCY for all X, Y ∈ X(Qn ). We then get ∗

∇̄X (Cξ ) = C ∇̄X ξ + q(X)JCξ

= C∇X ξ + q(X)JCξ

= CϕAX + q(X)JCξ = λCϕX + q(X)JCξ

for X ∈ Γ(Tλ ). We have JCξ ∈ ℂBξ and thus g(ϕJCξ , Y) = 0 for Y ∈ Γ(Tλ ). Altogether we therefore get g(BX, Y) = (λ − μ)λg(ϕCϕX, Y) = (λ − μ)λg(CX, Y)

= (λ − μ)λg(BX, Y). If α = 2, we have λ = 1 and μ ∈ {0, 2}. Putting Y = BX ≠ 0 implies 1 = (λ − μ)λ = (1 − μ) and hence μ = 0. Assume that g(BX, Y) ≠ 0 and α > 2. Then we have (λ −μ)λ = 1. Recall that μ ∈ {0, α}. If μ = 0 we get λ2 = 1, and if μ = α we get −1 = (λ − α)λ = 1, both of which are contradictions. Therefore we must have g(BX, Y) = 0 for all X, Y ∈ Γ(Tλ ). The same argument can be repeated for Tβ . Since 𝒬 = Tβ ⊕ Tλ and B𝒬 = 𝒬, we conclude that BTβ = Tλ

and BTλ = Tβ .

Since B = C on Tλ and Tβ , we can replace B by C here. As both Tλ and Tβ are com∗ plex, we see that 𝒬 and hence Qn must have even complex dimension if α > 2. We summarize this in the following proposition. Proposition 7.7.5. Let M be a real hypersurface in Qn , n ≥ 3, with isometric Reeb flow. Then the Hopf principal curvature function α is constant and satisfies α2 ≥ 4. We may assume α > 0. (i) If α = 2, then M has three distinct constant principal curvatures α = 2, β = 1 and μ = 0 with corresponding principal curvature spaces Tα = ℝξ , Tβ = 𝒬 and Tμ = 𝒞 ⊖ 𝒬, respectively. (ii) If α > 2, then n is even, say n = 2m. There exists r ∈ ℝ+ with α = 2 coth(2r) and one of the following two possibilities holds: ∗

316 | 7 Real hypersurfaces in complex hyperbolic quadrics (a) The real hypersurface M has four distinct constant principal curvatures α = 2 coth(2r), β = coth(r), λ = tanh(r) and μ = 0. For the corresponding principal curvature spaces we have Tα = ℝξ and Tμ = 𝒞 ⊖ 𝒬, and there exists a real structure C such that CTλ = Tβ and CTβ = Tλ . (b) The real hypersurface M has three distinct constant principal curvatures α = μ = 2 coth(2r), β = coth(r) and λ = tanh(r). For the corresponding principal curvature spaces we have Tα ⊕Tμ = ℝξ ⊕(𝒞 ⊖ 𝒬), and there exists a real structure C such that CTλ = Tβ and CTβ = Tλ . For p ∈ M we denote by cp the geodesic in Qn with cp (0) = p and ċp (0) = ζp and by Φr the smooth map ∗

Φr : M → Qn , ∗

p 󳨃→ cp (r).

Geometrically, Φr is the displacement of M at distance r in direction of the normal vector field ζ . For each p ∈ M the differential dp Φr of Φr at p can be computed using Jacobi fields by means of dp Φr (v) = Zv (r) (see Section 2.5). Here, v ∈ Tp M and Zv is the Jacobi field along cp with initial values Zv (0) = v and Zv′ (0) = −Av. We know from Proposition 7.7.3 that ζ is A-isotropic everywhere. Using the explicit description of the Jacobi operator R̄ ζ for the A-isotropic case in Lemma 1.4.4, we see that Tμ is contained in the 0-eigenspace of R̄ ζ , Tλ and Tβ are contained in the −1-eigenspace of R̄ ζ and Tα is contained in the −4-eigenspace of R̄ ζ . We first consider case (i) in Proposition 7.7.5. Then we have Bv (t), if v ∈ Tμ , { { { Zv (t) = {e−t Bv (t), if v ∈ Tλ , { { −2t {e Bv (t), if v ∈ Tα for all t ∈ ℝ, where Bv is the parallel vector field along cp with initial value Bv (0) = v. ∗ Now consider a geodesic variation in Qn consisting of geodesics cp . The corresponding Jacobi field is a linear combination of the three types of the Jacobi fields Zv listed above, and hence its length remains bounded when t → ∞. This shows that all ∗ geodesics cp in Qn are asymptotic to each other and hence determine an A-isotropic ∗ singular point z ∈ Qn (∞) at infinity. Therefore M is an integral manifold of the dis∗ tribution on Qn given by the orthogonal complements of the tangent vectors of the geodesics in the asymptote class z. This distribution is integrable and the maximal ∗ leaves are the horospheres in Qn whose center at infinity is z. Uniqueness of integral manifolds of integrable distributions finally implies that M is an open part of a ∗ horosphere in Qn whose center is the singular point z at infinity.

7.7 Real hypersurfaces with isometric Reeb flow

| 317

We now consider case (ii) in Proposition 7.7.5. Then we have n = 2m and (cosh(2t) − α2 sinh(2t))Bv (t), { { { Zv (t) = {(cosh(t) − ρ sinh(t))Bv (t), { { {(1 − μt)Bv (t),

if v ∈ ℝξp ,

if v ∈ Tρ (p) and ρ ∈ {β, λ}, if v ∈ Tμ (p),

where Bv denotes the parallel vector field along cp with Bv (0) = v. This shows that the kernel of dΦr is ℝξ ⊕ Tβ and that Φr is of constant rank dim(Tλ ⊕ Tμ ). So, locally, Φr is a submersion onto a submanifold P of Q2m . Moreover, the tangent space of P at Φr (p) is obtained by parallel translation of (Tλ ⊕ Tμ )(p) (see Theorem 2.5.1), which is ∗

a complex subspace of Tp Q2m . Since J is parallel along cp , also TΦr (p) P is a complex ∗

subspace of TΦr (p) Q2m . Thus P is a complex submanifold of Q2m . Moreover, since dimℂ (Tλ ) = m − 1 and dimℂ (Tμ ) = 1, we see that dimℂ (P) = m. The vector ηp = ċp (r) is a unit normal vector of P at Φr (p). The shape operator APηp of P with respect to ηp is given by ∗

∗

APηp Zv (r) = −Zv′ (r) (see Theorem 2.5.1). From this we immediately get that for each ρ ∈ {λ, μ} the parallel translate of Tρ (p) along cp from p to Φr (p) is a principal curvature space of P with respect to ρ. Moreover, the corresponding principal curvature is 0 for ρ = λ. Recall that coth(2r) Bv (r). Since every complex submanμ ∈ {0, α}. If μ = α we have APηp Zv (r) = 2r2coth(2r)−1 ifold of a Kähler manifold is minimal, this leads to a contradiction, and therefore we must have μ = 0. The vectors of the form ηq , q ∈ Φ−1 r ({Φr (p)}), form an open subset of the unit sphere in the normal space of P at Φr (p). Since APηq vanishes for all ηq it now

follows that P is totally geodesic in Q2m . Rigidity of totally geodesic submanifolds now implies that the entire submanifold M is an open part of the tube with radius r ∗ around a connected, complete, totally geodesic, complex submanifold P of Q2m with dimℂ (P) = m. Klein classified in [56] the totally geodesic submanifolds in complex quadrics. By means of duality, we can apply this classification also to the complex hyperbolic quadrics. According to this classification the submanifold P is either a totally ∗ ∗ ∗ ∗ geodesic Qm ⊂ Q2m or a totally geodesic ℂH m ⊂ Q2m . The normal spaces of Qm ∗ are Lie triple systems and the corresponding totally geodesic submanifolds of Q2m are again m-dimensional complex quadrics. Since m ≥ 2, it follows that the normal ∗ ∗ spaces of Qm contain all types of tangent vectors of Q2m . This implies that the nor∗ mal bundle of the tubes around Qm contains regular and singular tangent vectors of 2m ∗ Q . Since the normal bundle of M consists of A-isotropic tangent vectors only we conclude that P is congruent to ℂH m . It follows that M is congruent to an open part of a tube around ℂH m . This concludes the proof of Theorem 7.7.1. ∗

318 | 7 Real hypersurfaces in complex hyperbolic quadrics

7.8 Contact hypersurfaces In this section we classify contact hypersurfaces in the complex hyperbolic quadric ∗ Qn , n ≥ 3. We already investigated contact hypersurfaces in Kähler manifolds in ∗ Section 3.5. We already encountered examples of contact hypersurfaces in Qn : horospheres with A-principal singular center at infinity (Proposition 7.4.5); tubes around ∗ ∗ the totally geodesic Qn−1 ⊂ Qn (Proposition 7.5.2); and tubes around the totally ∗ geodesic ℝH n ⊂ Qn (Proposition 7.5.4). A natural question is whether there are other ∗ contact hypersurfaces in Qn . We will see that there are no other contact hypersurfaces. ∗ Let M be a connected contact hypersurface in Qn , n ≥ 3. From Proposition 3.5.4 and Proposition 3.5.5 we see that there exists a constant f ≠ 0 such that Aϕ + ϕA = 2fϕ. Moreover, from Proposition 3.5.6 we know that M is a Hopf hypersurface and the mean curvature of M is given by tr(A) = α + (2n − 2)f , where α = g(Aξ , ξ ) is the Hopf principal curvature function. We also know that for all X ∈ Γ(𝒞 ) we have AX = λX

󳨐⇒

AϕX = (2f − λ)ϕX.

From Proposition 3.5.9 we have the useful equation ̄ , ζ )ϕX) 2(A2 − 2fA + αf )X = −(R(ξ 𝒞 for all X ∈ Γ(𝒞 ). Using the explicit expression for the Riemannian curvature tensor R̄ ∗ of Qn in Theorem 1.4.3 we compute ̄ , ζ )ϕX = −g(ζ , ϕX)ξ + g(ξ , ϕX)ζ − g(Jζ , ϕX)Jξ + g(Jξ , ϕX)Jζ + 2g(Jξ , ζ )JϕX R(ξ − g(Cζ , ϕX)Cξ + g(Cξ , ϕX)Cζ − g(JCζ , ϕX)JCξ + g(JCξ , ϕX)JCζ

for all X ∈ Γ(𝒞 ), where C is an arbitrary real structure in A. We have ϕX = JX,

g(ζ , ϕX) = 0,

g(ξ , ϕX) = 0,

g(Jζ , ϕX) = −g(ξ , ϕX) = −η(ϕX) = 0,

g(Jξ , ϕX) = g(ζ , ϕX) = 0,

7.8 Contact hypersurfaces | 319

g(Jξ , ζ ) = g(ζ , ζ ) = 1, JϕX = J 2 X = −X,

g(Cζ , ϕX) = g(ζ , CϕX) = ρ(ϕX),

g(Cξ , ϕX) = −g(CJζ , ϕX) = g(JCζ , JX) = g(Cζ , X) = g(ζ , CX) = ρ(X), Cζ = CJξ = −JCξ ,

g(JCζ , ϕX) = g(JCζ , JX) = g(Cζ , X) = g(ζ , CX) = ρ(X),

g(JCξ , ϕX) = g(JCξ , JX) = g(Cξ , X) = −g(CJζ , X) = g(JCζ , X)

= −g(Cζ , JX) = −g(Cζ , ϕX) = −g(ζ , CϕX) = −ρ(ϕX),

JCζ = −J 2 Cξ = Cξ .

Inserting all this into the previous equation gives ̄ , ζ )ϕX = −2X − 2ρ(ϕX)Cξ − 2ρ(X)JCξ R(ξ for all X ∈ Γ(𝒞 ). We have Cξ = Bξ + ρ(ξ )ζ , and therefore the 𝒞 -component of Cξ is equal to Bξ − η(Bξ )ξ . We also have JCξ = J(Bξ + ρ(ξ )ζ ) = JBξ + ρ(ξ )Jζ

= ϕBξ + η(Bξ )ζ − ρ(ξ )ξ . The 𝒞 -component of JCξ therefore is equal to ϕBξ . Altogether we have proved the following. Proposition 7.8.1. Let M be a connected contact hypersurface in Qn , n ≥ 3. Then we have ∗

(A2 − 2fA + αf − 1)X = −ρ(ϕX)η(Bξ )ξ + ρ(ϕX)Bξ + ρ(X)ϕBξ

(7.9)

for all X ∈ Γ(𝒞 ). Proposition 7.8.2. Let M be a connected contact hypersurface in Qn , n ≥ 3. Then the unit normal vector field ζ of M is A-principal everywhere. ∗

Proof. From Theorem 3.5.19 we know that the Reeb vector field ξ is an eigenvector of the normal Jacobi operator K everywhere. From Proposition 7.1.9 we then see that ζ ∗ is a singular tangent vector of Qn everywhere. Thus ζ is A-principal or A-isotropic everywhere.

320 | 7 Real hypersurfaces in complex hyperbolic quadrics Assume that ζ is A-isotropic everywhere. Then ζ , ξ , Bξ , ϕBξ are pairwise orthonormal and we have Cξ = Bξ , ρ(ξ ) = 0 and η(Bξ ) = 0 by Corollary 7.1.4. Thus (7.9) simplifies to (A2 − 2fA + αf − 1)X = ρ(ϕX)Bξ + ρ(X)ϕBξ for all X ∈ Γ(𝒞 ). We also have ρ(Bξ ) = g(CBξ , ζ ) = g(C 2 ξ , ζ ) = g(ξ , ζ ) = 0, and therefore Cζ = CJξ = −JCξ = −JBξ = −ϕBξ . Furthermore, we have ρ(X) = g(CX, ζ ) = g(X, Cζ ) = −g(X, ϕBξ ) and ρ(ϕX) = −g(ϕX, ϕBξ ) = −g(JX, JBξ ) = −g(X, Bξ ). Thus we have (A2 − 2fA + αf − 1)X = −g(X, Bξ )Bξ − g(X, ϕBξ )ϕBξ for all X ∈ 𝒞 . Since 𝒞 ⊖ 𝒬 = ℂBξ , it follows that (A2 − 2fA + αf − 1)X = 0 for all X ∈ 𝒬 and (A2 − 2fA + αf )X = 0 for all X ∈ 𝒞 ⊖ 𝒬. Let X ∈ 𝒞 be a principal curvature vector with corresponding principal curvature λ, so AX = λX. We decompose X into X = Y + Z with Y ∈ 𝒬 and Z ∈ 𝒞 ⊖ 𝒬. Then we get (λ2 − 2fλ + fα)Y + (λ2 − 2fλ + fα)Z = (λ2 − 2fλ + fα)X

= (A2 − 2fA + αf )X

= (A2 − 2fA + αf )Y + (A2 − 2fA + αf )Z = Y.

7.8 Contact hypersurfaces | 321

If Z ≠ 0, this implies λ2 − 2fλ + fα = 0 and hence Y = 0. It follows that 𝒬 and 𝒞 ⊖ 𝒬 are ∗ invariant under the shape operator A of M. Recall that there exists a 1-form q on Qn ∗ n such that ∇̄X C = q(X)JC for all X ∈ X(Q ). By differentiating the equation 0 = g(Cζ , Jζ ) with respect to X ∈ X(M) we get 0 = g(∇̄X (Cζ ), Jζ ) + g(Cζ , ∇̄X (Jζ )) = g((∇̄X C)ζ ), Jζ ) + g(C ∇̄X ζ , Jζ ) + g(Cζ , J ∇̄X ζ ) = q(X)g(JCζ , Jζ ) − g(CAX, Jζ ) − g(Cζ , JAX) = q(X)g(Cζ , ζ ) − g(AX, CJζ ) + g(JCζ , AX) = −2g(AX, CJζ )

= −2g(X, ACJζ ), which implies ABξ = ACξ = −ACJζ = 0. Thus Bξ is a principal curvature vector of M with corresponding principal curvature 0. By differentiating the equation g(Cζ , ζ ) = 0 we get 0 = g(∇̄X (Cζ ), ζ ) + g(Cζ , ∇̄X ζ ) = g((∇̄X C)ζ ), ζ ) + g(C ∇̄X ζ , ζ ) + g(Cζ , ∇̄X ζ ) = q(X)g(JCζ , ζ ) − g(CAX, ζ ) − g(Cζ , AX) = −q(X)g(Cζ , Jζ ) − 2g(AX, Cζ )

= −q(X)g(ϕBξ , ξ ) − 2g(X, ACζ )

= −2g(X, ACζ ), which implies

AϕBξ = −ACζ = 0. Thus ϕBξ is a principal curvature vector of M with corresponding principal curvature 0. On the other hand, we proved above that Bξ is a principal curvature vector of M with corresponding principal curvature 0. According to Proposition 3.5.6, this implies that ϕBξ is a principal curvature vector of M with corresponding principal curvature 2f . Altogether this implies f = 0, which is a contradiction to Proposition 3.5.5. It follows that ζ cannot be A-isotropic. Hence ζ is A-principal. We now investigate the case when ζ is A-principal. If ζ is A-principal, that is, if Cζ = ζ , we have Bξ = −ξ and ρ(X) = g(CX, ζ ) = g(X, Cζ ) = g(X, ζ ) = 0 for all X ∈ X(M). Equation (7.9) therefore simplifies to (A2 − 2fA + αf − 1)X = 0

322 | 7 Real hypersurfaces in complex hyperbolic quadrics for all X ∈ Γ(𝒞 ). Therefore, if X ∈ 𝒞 with AX = λX, we have λ2 − 2fλ + fα − 1 = 0.

(7.10)

Recall that f is constant by Proposition 3.5.4 and α is constant by Theorem 3.5.19. It follows that there are at most two distinct constant principal curvatures λ and μ = 2f −λ ∗ on 𝒞 . We again use the fact that there exists a 1-form q on Qn such that ∇̄X C = q(X)JC n∗ for all X ∈ X(Q ). By differentiating the equation 0 = g(Cζ , Jζ ) with respect to X ∈ X(M) we get 0 = g(∇̄X (Cζ ), Jζ ) + g(Cζ , ∇̄X (Jζ )) = g((∇̄X C)ζ ), Jζ ) + g(C ∇̄X ζ , Jζ ) + g(Cζ , J ∇̄X ζ ) = q(X)g(JCζ , Jζ ) − g(CAX, Jζ ) − g(Cζ , JAX)

= q(X)g(Jζ , ζ ) + g(AX, Jζ ) + g(Jζ , AX)

= q(X) − 2g(AX, ξ ) = q(X) − 2g(X, Aξ ) = q(X) − 2αg(X, ξ ) = q(X) − 2αη(X). It follows that q(X) = 2αη(X)

for all X ∈ X(M)

and hence ∇̄X C = 0

for all X ∈ Γ(𝒞 ).

From Cζ = ζ we get CJζ = −JCζ = −Jζ . Differentiating this equation with respect to X ∈ Γ(𝒞 ) implies CAX = AX

for all X ∈ Γ(𝒞 ).

Thus, for all X ∈ 𝒞 with AX = λX resp. AX = μX we get λCX = λX and μCX = μX. If both λ and μ are non-zero, this implies CX = X for all X ∈ 𝒞 and hence tr(C) = 2(n−1), which contradicts the fact that C is a real structure and hence tr(C) = 0. We thus may assume that λ = 0. If the corresponding principal curvature space Tλ is J-invariant, this implies f = 0, which is a contradiction. We thus must have 0 ≠ μ = 2f and JTλ = Tμ . Thus we have shown that there are exactly two distinct constant principal curvatures λ = 0 and μ = 2f on 𝒞 . Moreover, we have JTλ = Tμ for the corresponding principal curvature spaces Tλ and Tμ . From (7.10) we finally conclude αf = 1. Thus we have proved the following. Proposition 7.8.3. Let M be a contact hypersurface in the complex hyperbolic quadric ∗ Qn , n ≥ 3, and assume that the normal vector field ζ is A-principal. Then M has three

7.8 Contact hypersurfaces | 323

distinct constant principal curvatures 1 α= , f

λ = 0,

μ = 2f

with corresponding principal curvature spaces Tα = ℝξ ,

Tλ = JV(C) ∩ 𝒞 ,

Tμ = V(C) ∩ 𝒞

satisfying JTλ = Tμ . Here, V(C) = {X ∈ TQn : CX = X}. ∗

We now state the main result of this section, which was proved by the authors in [18] under the additional assumption that the contact hypersurface has constant mean curvature. This assumption now turns out to be redundant by Theorem 3.5.19. Theorem 7.8.4. Let M be a connected real hypersurface in the complex hyperbolic ∗ ∗ quadric Qn , n ≥ 3. Then M is a contact hypersurface in Qn if and only if M is congruent to an open part of one of the following real hypersurfaces: (i) the tube with radius r ∈ ℝ+ around the complex totally geodesic embedding of the ∗ ∗ complex hyperbolic quadric Qn−1 into Qn ; (ii) the tube with radius r ∈ ℝ+ around the totally real totally geodesic embedding of ∗ the real hyperbolic space ℝH n into Qn ; n∗ (iii) the horosphere in Q with A-principal center at infinity. Proof. We proved in Propositions 7.4.5, 7.5.2 and 7.5.4 that the hypersurfaces listed in ∗ the previous theorem are contact hypersurfaces in Qn . ∗ Conversely, assume that M is a contact hypersurface in Qn . Proposition 7.8.2 tells us that the unit normal vector field ζ of M is A-principal everywhere. We choose a real structure C so that Cζ = ζ . Since M is a contact hypersurface, there exists a non-zero constant f ∈ ℝ so that Aϕ + ϕA = 2fϕ (see Section 3.5). Without loss of generality, we can assume that ζ is chosen so that f > 0. We distinguish the three cases f ∈ (0, √2), f = √2 and f ∈ (√2, ∞). In the first case we can write f = √2 tanh(√2r) with some r ∈ ℝ+ , and in the third case we can write f = √2 coth(√2r) with some r ∈ ℝ+ . From Proposition 7.8.3 we see that M has distinct constant principal curvatures α = √2 coth(√2r),

λ = 0,

μ = √2 tanh(√2r),

or α = μ = √2,

λ = 0,

324 | 7 Real hypersurfaces in complex hyperbolic quadrics or α = √2 tanh(√2r),

λ = 0,

μ = √2 coth(√2r),

with corresponding principal curvature spaces Tα = ℝξ ,

Tλ = JV(C) ∩ 𝒞 ,

Tμ = V(C) ∩ 𝒞

satisfying JTλ = Tμ . ∗ For p ∈ M we denote by cp the geodesic in Qn with cp (0) = p and ċp (0) = ζp and by Φr the smooth map

Φr : M → Qn , ∗

p 󳨃→ cp (r).

Geometrically, Φr is the displacement of M at distance r in direction of the normal vector field ζ . For each p ∈ M the differential dp Φr of Φr at p can be computed using Jacobi fields by means of dp Φr (v) = Zv (r) (see Section 2.5). Here, v ∈ Tp M and Zv is the Jacobi field along cp with initial values Zv (0) = v and Zv′ (0) = −Av. Using the explicit description of the Jacobi operator R̄ ζ for the A-principal case in Lemma 1.4.4, we see that Tλ is contained in the 0-eigenspace of R̄ ζ and Tα and Tμ are contained in the (−2)-eigenspace of R̄ ζ . For the Jacobi fields Zv along cp we thus get the expressions (cosh(√2t) −

Zv (t) = {

Bv (t),

ρ √2

sinh(√2t))Bv (t), if v ∈ Tρ and ρ ∈ {α, μ}, if v ∈ Tλ ,

where Bv denotes the parallel vector field along cp with Bv (0) = v. If α = √2 coth(√2r), then the kernel of dΦr is Tα = ℝξ and Φr is of constant rank ∗ 2(n − 1). So, locally, Φr is a submersion onto a submanifold P of Qn . Moreover, the tangent space of P at Φr (p) is obtained by parallel translation of (Tλ ⊕ Tμ )(p) = 𝒞 (p) ∗ (see Theorem 2.5.1), which is a complex subspace of Tp Qn . Since J is parallel along n∗ cp , also TΦr (p) P is a complex subspace of TΦr (p) Q . Thus P is a complex submanifold ∗ of Qn with dimℂ (P) = n − 1. The vector ηp = ċp (r) is a unit normal vector of P at Φr (p). The shape operator APηp of P with respect to ηp is given by APηp Zv (r) = −Zv′ (r)

7.8 Contact hypersurfaces | 325

(see Theorem 2.5.1). From this we immediately get that for each ρ ∈ {λ, μ} the parallel translate of Tρ (p) along cp from p to Φr (p) is a principal curvature space of P with respect to ρ. Moreover, the corresponding principal curvature is 0 in both cases. The vectors of the form ηq , q ∈ Φ−1 r ({Φr (p)}), form an open subset of the unit sphere in the normal space of P at Φr (p). Since APηq vanishes for all ηq it now follows that P is totally

geodesic in Qn . Rigidity of totally geodesic submanifolds now implies that the entire submanifold M is an open part of the tube with radius r around a connected, complete, ∗ totally geodesic, complex submanifold P of Qn with dimℂ (P) = n − 1. Klein classified in [56] the totally geodesic submanifolds in complex quadrics. By means of duality, we can apply this classification also to the complex hyperbolic quadrics. According to ∗ ∗ this classification, the submanifold P is a totally geodesic Qn−1 ⊂ Qn . It follows that ∗ M is congruent to an open part of a tube with radius r ∈ ℝ+ around Qn−1 . If α = √2 tanh(√2r), then the kernel of dΦr is Tμ = V(C) ∩ 𝒞 and Φr is of con∗ stant rank n. So, locally, Φr is a submersion onto a submanifold P of Qn . Moreover, the tangent space of P at Φr (p) is obtained by parallel translation of (Tα ⊕ Tλ )(p) = ∗ (ℝξ ⊕ (JV(C) ∩ 𝒞 ))(p) (see Theorem 2.5.1), which is a totally real subspace of Tp Qn . n∗ Since J is parallel along cp , also TΦr (p) P is a totally real subspace of TΦr (p) Q . Thus P ∗ is a totally real submanifold of Qn with dimℝ (P) = n. The vector ηp = ċp (r) is a unit normal vector of P at Φr (p). The shape operator APηp of P with respect to ηp is given by ∗

APηp Zv (r) = −Zv′ (r) (see Theorem 2.5.1). From this we immediately get that for each ρ ∈ {α, λ} the parallel translate of Tρ (p) along cp from p to Φr (p) is a principal curvature space of P with respect to ρ. Moreover, the corresponding principal curvature is 0 in both cases. The vectors of the form ηq , q ∈ Φ−1 r ({Φr (p)}), form an open subset of the unit sphere in the normal space of P at Φr (p). Since APηq vanishes for all ηq it now follows that P is totally

geodesic in Qn . Rigidity of totally geodesic submanifolds now implies that the entire submanifold M is an open part of the tube with radius r around a connected, complete, ∗ totally geodesic, totally real submanifold P of Qn with dimℝ (P) = n. Klein classified in [56] the totally geodesic submanifolds in complex quadrics. By means of duality, we can apply this classification also to the complex hyperbolic quadrics. According to this classification the submanifold P is an open part of the totally geodesic embedding ∗ of the real hyperbolic space ℝH n into Qn , or of the totally geodesic embedding of the ∗ a n−a Riemannian product ℝH × ℝH of real hyperbolic spaces ℝH a and ℝH n−a into Qn , where 0 < a < n. However, the tangent vectors to ℝH a are either in V(C) or in JV(C) and the tangent vectors to ℝH n−a are then in the other eigenspace JV(C) or in V(C), ∗ respectively. It follows that P is an open part of a totally geodesic ℝH n in Qn . Rigidity of totally geodesic submanifolds finally implies that M is an open part of the tube with ∗ radius r around a totally geodesic ℝH n in Qn . ∗

326 | 7 Real hypersurfaces in complex hyperbolic quadrics If α = √2, we consider a geodesic variation in Qn consisting of geodesics cp . The corresponding Jacobi field is a linear combination of the three types of the Jacobi fields Zv listed above, and hence its length remains bounded when t → ∞. This shows that ∗ all geodesics cp in Qn are asymptotic to each other and hence determine a singular ∗ point z ∈ Qn (∞) at infinity of A-principal type. Therefore M is an integral manifold ∗ of the distribution on Qn given by the orthogonal complements of the tangent vectors of the geodesics in the asymptote class z. This distribution is integrable and the ∗ maximal leaves are the horospheres in Qn whose center at infinity is z. Uniqueness of integral manifolds of integrable distributions finally implies that M is an open part ∗ of a horosphere in Qn whose center is the A-principal singular point z at infinity. This concludes the proof of Theorem 7.8.4. ∗

7.9 Further results In this section we summarize a few further results. The interested reader can find more details by consulting the relevant references. The shape operator In Section 6.7 we saw that there are no real hypersurfaces in the complex quadric with parallel shape operator or, more generally, whose shape operator is a Codazzi tensor. With entirely analogous arguments we can show that the corresponding statements hold in complex hyperbolic quadrics. Theorem 7.9.1. There are no real hypersurfaces in the complex hyperbolic quadric Qn , n ≥ 3, whose shape operator is a Codazzi tensor. In particular, there are no real hyper∗ surfaces with parallel shape operator in the complex hyperbolic quadric Qn , n ≥ 3. ∗

This result motivated Suh and Hwang [111] to consider the weaker condition of Reeb parallel shape operator ∇ξ A = 0, that is, the shape operator A is parallel with respect to the Reeb vector field ξ . They established the following partial classification. Theorem 7.9.2 ([111]). Let M be a connected Hopf hypersurface with Reeb parallel ∗ shape operator in the complex hyperbolic quadric Qn , n ≥ 3. If the Hopf principal curvature function α is non-zero everywhere, then M is an open part of: ∗ (i) a tube around a totally geodesic ℂH m ⊂ Q2m , where n = 2m, ∗ ∗ (ii) a tube around a totally geodesic Qn−1 ⊂ Qn , ∗ (iii) a tube around a totally geodesic ℝH n ⊂ Qn or (iv) a horosphere with a singular center at infinity,

7.9 Further results | 327

or otherwise (v) M has two distinct constant principal curvatures α and λ = n and n − 1, respectively.

α2 −2 α

with multiplicities

The normal Jacobi operator We computed the normal Jacobi operator K of a real hypersurface M in the complex ∗ hyperbolic quadric Qn in Theorem 7.1.8. Explicitly, we have KX = −X + η(Bξ )BX + ρ(ξ )BϕX − ρ(ϕX)Bξ + η(Bξ )η(X)Bξ

− ρ(ξ )η(X)ϕBξ − ρ(X)ϕBξ − 3η(X)ξ + ρ(X)ρ(ξ )ξ + ρ(ξ )2 η(X)ξ

for all X ∈ X(M). Suh proved in [107] the non-existence of Hopf hypersurfaces with parallel normal Jacobi operator in complex hyperbolic quadrics. Theorem 7.9.3 ([107]). There are no Hopf hypersurfaces with parallel normal Jacobi op∗ erator in the complex hyperbolic quadric Qn , n ≥ 3. It remains an open problem whether or not the assumption in Theorem 7.9.3 that the real hypersurface is Hopf is really necessary. The structure Jacobi operator We computed the structure Jacobi operator Rξ of a real hypersurface M in the complex ∗ hyperbolic quadric Qn in Theorem 7.1.10. Explicitly, we have Rξ X = −X + η(X)ξ + ρ(ξ )ϕBX − ρ(X)ϕBξ − η(Bξ )BX + η(BX)Bξ + η(Aξ )AX − η(AX)Aξ for all X ∈ X(M). Suh, Pérez and Woo proved in [114] the non-existence of Hopf hypersurfaces with parallel structure Jacobi operator in complex hyperbolic quadrics. Theorem 7.9.4 ([114]). There are no Hopf hypersurfaces with parallel structure Jacobi ∗ operator in the complex hyperbolic quadric Qn , n ≥ 3. It remains an open problem whether or not the assumption in Theorem 7.9.4 that the real hypersurface is Hopf is really necessary. Lee and Suh [64] then considered the weaker condition of Reeb parallel structure Jacobi operator, that is, ∇ξ Rξ = 0, and obtained the following classification result.

328 | 7 Real hypersurfaces in complex hyperbolic quadrics Theorem 7.9.5 ([64]). Let M be a connected Hopf hypersurface with Reeb parallel struc∗ ture Jacobi operator in the complex hyperbolic quadric Qn , n ≥ 3. Then M is either an open part of a horosphere with A-isotropic center at infinity or an open part of a tube ∗ around the totally geodesic ℂH m ⊂ Q2m , where n = 2m. Suh and Hwang [113] considered the condition that the structure Jacobi operator is a Codazzi tensor, that is, (∇X Rξ )Y = (∇Y Rξ )X for all X, Y ∈ X(M). They obtained the following classification result. Theorem 7.9.6 ([113]). Let M be a connected Hopf hypersurface in the complex hyper∗ bolic quadric Qn , n ≥ 3. If the structure Jacobi operator is a Codazzi tensor and the Hopf principal curvature function is constant along the Reeb flow lines, then M is either an open part of a horosphere with A-isotropic center at infinity or an open part of a tube ∗ around the totally geodesic ℂH m ⊂ Q2m , where n = 2m. The Ricci tensor ∗ Let M be a real hypersurface in Qn and Ric the Ricci tensor of M. We already computed the Ricci tensor in Theorem 7.1.11. More precisely, we have Ric(X) = −(2n − 1)X + hAX − A2 X

− η(Bξ )BX − ρ(ξ )BϕX + ρ(ϕX)Bξ − η(Bξ )η(X)Bξ

+ ρ(ξ )η(X)ϕBξ + ρ(X)ϕBξ + 3η(X)ξ − ρ(X)ρ(ξ )ξ + ρ(ξ )2 η(X)ξ for all X ∈ X(M). Kim and Suh determined in [53] necessary conditions for the principal curva∗ tures of Hopf hypersurfaces with parallel Ricci tensor in Qn . They also showed that such hypersurfaces must have singular normal vectors. It remains an open problem whether or not real hypersurfaces with parallel Ricci tensor exist in complex hyperbolic quadrics. ∗ Let M be a Hopf hypersurface in Qn . When the Ricci tensor Ric and the structure tensor ϕ satisfy Ric ∘ ϕ + ϕ ∘ Ric = 2cϕ,

c ∈ ℝ \ {0},

then the Ricci tensor of M is said to be pseudo-ϕ-anti-invariant. If c = 0 in the previous equation, then the Ricci tensor of M is said to be ϕ-anti-invariant. Recall from Lemma 3.3.11 that this property can be motivated by Ricci solitons.

7.9 Further results | 329

Suh classified in [108] the Hopf hypersurfaces with pseudo-ϕ-anti-invariant Ricci tensor in complex hyperbolic quadrics. Theorem 7.9.7 ([108]). Let M be a Hopf hypersurface with pseudo-ϕ-anti-invariant ∗ Ricci tensor in the complex hyperbolic quadric Qn , n ≥ 3. Then M is an open part of: m 2m ∗ (i) a tube around a totally geodesic ℂH ⊂ Q , where n = 2m, ∗ ∗ (ii) a tube around a totally geodesic Qn−1 ⊂ Qn , ∗ (iii) a tube around a totally geodesic ℝH n ⊂ Qn or (iv) a horosphere with a singular center at infinity. Suh and Hwang [112] investigated real hypersurfaces with ϕ-invariant Ricci tensor in complex hyperbolic quadrics. We saw in Proposition 7.4.3 that horospheres with A-isotropic center at infinity have this property. In Proposition 7.5.6 we saw that tubes ∗ around the totally geodesic ℂH m in Q2m have this property as well. Suh and Hwang ∗ proved that among Hopf hypersurfaces there are no other real hypersurfaces in Qn satisfying Ric ϕ = ϕ Ric with the possible exception of two types of hypersurfaces with singular normal vectors whose principal curvatures satisfy certain relations. It remains an open problem whether or not such real hypersurfaces exist. Suh investigated in [110] Ricci soliton and pseudo-Einstein real hypersurfaces in complex hyperbolic quadrics. The two main results are the following. Theorem 7.9.8 ([110]). There are no Hopf–Ricci solitons in the complex hyperbolic ∗ quadric Qn , n ≥ 3. Theorem 7.9.9 ([110]). There are no pseudo-Einstein Hopf hypersurfaces in the complex ∗ hyperbolic quadric Qn , n ≥ 3. In [109], Suh investigated real hypersurfaces with harmonic curvature in complex hyperbolic quadrics. Recall that harmonic curvature means that the Ricci tensor is a Codazzi tensor, that is, (∇X Ric)Y = (∇Y Ric)X holds for all X, Y ∈ X(M). Suh first proved that, under the additional assumption that the mean curvature is constant, the unit normal vector field ζ is either A-isotropic or A-principal. A thorough investigation of the two possible cases leads to the following theorem. Theorem 7.9.10 ([109]). Let M be a Hopf hypersurface with harmonic curvature and ∗ constant mean curvature in the complex hyperbolic quadric Qn , n ≥ 4. If the unit normal vector field ζ is A-principal, then M is locally congruent to a horosphere with A-principal center at infinity or M has at most five distinct constant principal curvatures α, λ1 , μ1 , λ2 , μ2 with corresponding principal curvature spaces Tα = ℝξ ,

ϕTλ1 = Tμ1 ,

ϕTλ2 = Tμ2

330 | 7 Real hypersurfaces in complex hyperbolic quadrics satisfying dim(Tλ1 ) + dim(Tλ2 ) = n − 1,

dim(Tμ1 ) + dim(Tμ2 ) = n − 1.

The principal curvatures λ1 , μ1 , λ2 , μ2 are solutions of the quadratic equation 2x2 − 2βi x + αβi − 2 = 0 with β1 =

α2 − 1 + √(α2 − 1)2 − 4αh , α

β2 =

α2 − 1 − √(α2 − 1)2 − 4αh , α

and the Hopf principal curvature function α is constant and satisfies 0 < α2 < 4. Here, h = tr(A) denotes the constant mean curvature of M. Theorem 7.9.11 ([109]). Let M be a Hopf hypersurface with harmonic curvature and ∗ constant mean curvature in the complex hyperbolic quadric Qn , n ≥ 4. If the unit normal vector field ζ is A-isotropic, then M is locally congruent to a horosphere with A-isotropic center at infinity or M has at most six distinct constant principal curvatures α, γ = 0, λ1 , μ1 , λ2 , μ2 with corresponding principal curvature spaces Tα = ℝξ ,

Tγ = ℂBξ ,

ϕTλ1 = Tμ1 ,

ϕTλ2 = Tμ2

satisfying dim(Tλ1 ) + dim(Tλ2 ) = n − 2,

dim(Tμ1 ) + dim(Tμ2 ) = n − 2.

The principal curvatures λ1 , μ1 , λ2 , μ2 are solutions of the quadratic equation 2x2 − 2βi x + αβi − 2 = 0 with β1 =

α2 − 1 + √(α2 − 1)2 − 8αh , 2α

β2 =

α2 − 1 − √(α2 − 1)2 − 8αh , 2α

and the Hopf principal curvature function α is constant. Here, h = tr(A) denotes the constant mean curvature of M.

8 Real hypersurfaces in Hermitian symmetric spaces In the previous four chapters we investigated some classes of real hypersurfaces in some particular Hermitian symmetric spaces. In this chapter we want to generalize some of these aspects to arbitrary Hermitian symmetric spaces. For this purpose we will first discuss aspects of the general structure theory of Hermitian symmetric spaces. We will then discuss real hypersurfaces with isometric Reeb flow and contact hypersurfaces in irreducible Hermitian symmetric spaces of compact type.

8.1 Structure theory of compact Hermitian symmetric spaces In this section we present algebraic models of irreducible Hermitian symmetric spaces of compact type using structure theory of semisimple real and complex Lie algebras. A good reference for the algebraic concepts used here is the book [88] by Samelson. Let M be an irreducible Hermitian symmetric space of compact type and G = I o (M) be the identity component of the isometry group of M. Let o ∈ M and denote by K the isotropy group of G at o. Then M can be realized as the homogeneous space M = G/K in the usual way. We denote by g and k the Lie algebras of G and K, respectively. Then g is a simple real Lie algebra. Let g = k ⊕ p be the corresponding Cartan decomposition. Let Z be the center of K and z its Lie algebra. Then dim(Z) = 1 and K is the centralizer of Z. Therefore K has maximal rank in G. There exists a unique element zo in the Lie algebra z of Z such that the complex structure J on To M ≅ p is given by J = ad(zo ). We will give a more explicit description of zo further below. Let h be a Cartan subalgebra of k. Since K has maximal rank in G, h is a Cartan subalgebra of g. Thus the complexification hℂ of h is a Cartan subalgebra of the complexification gℂ of g. Let gℂ = hℂ ⊕ (⨁ gα ) α∈Δ

be the root space decomposition of gℂ with respect to hℂ . We fix a set {α1 , . . . , αr } of simple roots of Δ and denote by Δ+ the resulting subset of Δ consisting of all positive roots. Since [hℂ , kℂ ] ⊂ kℂ and [hℂ , pℂ ] ⊂ pℂ , we either have gα ⊂ kℂ or gα ⊂ pℂ for each α ∈ Δ. If gα ⊂ kℂ , then the root α is compact, and if gα ⊂ pℂ , then the root α is noncompact. A root α ∈ Δ is compact if and only if α(zℂ ) = {0}. Denote by ΔK the set of compact roots and by ΔM the set of non-compact roots. Then we have kℂ = hℂ ⊕ (⨁ gα ), α∈ΔK

https://doi.org/10.1515/9783110689839-008

pℂ = ⨁ gα . α∈ΔM

332 | 8 Real hypersurfaces in Hermitian symmetric spaces We can be more explicit about this. Let H 1 , . . . , H r ∈ hℂ be the dual basis of α1 , . . . , αr defined by αi (H j ) = δij . There exists an integer k ∈ {1, . . . , r} such that ΔK is generated by the simple roots α1 , . . . , αk−1 , αk+1 , . . . , αr and the multiplicity of αk in the highest root δ ∈ Δ+ is equal to one. The complex structure J = ad(zo ) on p ≅ To M is given by zo = iH k and we have ΔK = {α ∈ Δ : α(H k ) = 0},

ΔM = {α ∈ Δ : α(H k ) = ±1},

and thus kℂ = hℂ ⊕ ( ⨁ gα ), α∈Δ α(H k )=0

pℂ =

⨁ gα .

α∈Δ α(H k )=±1

Conversely, if there is a simple root whose coefficient in the highest root is equal to one, we can construct a Hermitian symmetric space of compact type from it by applying the Borel–de Siebenthal construction method [25]. For each α ∈ Δ there exists a unique vector hα ∈ [gα , g−α ] ⊂ hℂ , the so-called coroot corresponding to α, such that α(hα ) = 2. Then we have ih = span of {hα : α ∈ Δ} and α(ih) = ℝ for all α ∈ Δ. The Killing form B of gℂ is non-degenerate on hℂ and positive definite on ih. Define Hα ∈ hℂ , the so-called root vector of α, by α(H) = B(Hα , H) for all H ∈ hℂ , and define an inner product (⋅, ⋅) on the dual space (ih)∗ by linear extension 2 of (αν , αμ ) = B(Hαν , Hαμ ). Then we have hα = (α,α) Hα for all α ∈ Δ. For α, β ∈ Δ with β ≠ ±α the α-string containing β is the set of roots β − pα,β α, . . . , β − α, β, β + α, . . . , β + qα,β α ∈ Δ with pα,β , qα,β ∈ ℤ and pα,β , qα,β ≥ 0 so that β − (pα,β + 1)α ∉ Δ and β + (qα,β + 1)α ∉ Δ. The α-string containing β contains at most four roots. The Cartan integer cβ,α of α, β ∈ Δ is defined by cβ,α = β(hα ) = B(hα , Hβ ) =

(β, α) 2 B(Hβ , Hα ) = 2 ∈ {0, ±1, ±2, ±3}. (α, α) (α, α)

The Cartan integer cβ,α is related to the α-string containing β by cβ,α = pα,β − qα,β . For each non-zero eα ∈ gα there exists e−α ∈ g−α such that B(eα , e−α ) = such vectors we have [eα , e−α ] = hα ,

[hα , eα ] = 2eα ,

[hα , e−α ] = −2e−α .

2 . (α,α)

For

8.1 Structure theory of compact Hermitian symmetric spaces | 333

Since all root spaces are 1-dimensional, there exists for all α, β ∈ Δ with α + β ∈ Δ numbers Nα,β ∈ ℂ such that [eα , eβ ] = Nα,β eα+β . We put Nα,β = 0 if α + β ∉ Δ. It is possible to choose the vectors eα in such a way that Nα,β = −N−α,−β holds. Lemma 8.1.1. The integers Nα,β have the following properties: (i) Nα,β = −Nβ,α = −N−α,−β = Nβ,−α−β = N−α−β,α for all α, β ∈ Δ; Nα,β (γ,γ)

N

N

β,γ γ,α = (α,α) = (β,β) for all pairwise linearly independent α, β, γ ∈ Δ satisfying α + β + γ = 0; 2 (iii) Nα,β = 21 (pα,β + 1)qα,β (α, α) for all α, β ∈ Δ with β ≠ ±α;

(ii)

(iv) Nδ,−ϵ Nγ,ζ −γ + N−ϵ,γ Nδ,ζ −δ = Nγ,δ N−ϵ,−ζ (η,η) for all γ, δ, ϵ, ζ ∈ Δ+ with γ + δ = ϵ + ζ and δ ≤ ζ ≤ ϵ ≤ γ. (ζ ,ζ )

From (iii) we deduce pα,β + 1 = pα,β = 0. We put hν = hαν .

1 q (α, α). 2 α,β

In particular, qα,β =

2 (α,α)

= B(eα , e−α ) if

Lemma 8.1.2. The vectors hν , eα (ν ∈ {1, . . . , r}, α ∈ Δ) form a basis of gℂ , a so-called Chevalley basis, with the following properties: (i) [hν , hμ ] = 0 for all ν, μ ∈ {1, . . . , r}; (ii) [hν , eα ] = α(hν )eα = cα,αν eα for all ν ∈ {1, . . . , r} and α ∈ Δ; (iii) for all α ∈ Δ there exists c1 , . . . , cr ∈ ℤ such that [eα , e−α ] = c1 h1 + ⋅ ⋅ ⋅ + cr hr ; (iv) for all α, β ∈ Δ with α + β ≠ 0 we have [eα , eβ ] = Nα,β eα+β , where (a) Nα,β = 0 if α + β ∉ Δ; (b) Nα,β = ±(pα,β + 1) if α + β ∈ Δ (see [88] about the sign ambiguity). For each α ∈ Δ we now define uα = eα − e−α ,

vα = i(eα + e−α ).

Then the compact real form g of gℂ is given by g = h ⊕ (⨁ (ℝuα ⊕ ℝvα )). α∈Δ+

The Cartan decomposition g = k ⊕ p then obviously is given by k = h ⊕ (⨁ (ℝuα ⊕ ℝvα )) , α∈Δ+K

p = ⨁ (ℝuα ⊕ ℝvα ), α∈Δ+M

334 | 8 Real hypersurfaces in Hermitian symmetric spaces where Δ+K = ΔK ∩ Δ+ and Δ+M = ΔM ∩ Δ+ . The complex structure J = ad(iH k ) acts on p ≅ To M by Juα = vα ,

Jvα = −uα

(α ∈ Δ+M ).

By defining ℂuα = ℝuα ⊕ ℝJuα = ℝuα ⊕ ℝvα for α ∈ Δ+M , we can write p = ⨁ ℂuα . α∈Δ+M

Lemma 8.1.3. The vectors uα and vα have the following properties: (i) [h, uα ] = −iα(h)vα for all h ∈ h and α ∈ Δ; (ii) [h, vα ] = iα(h)uα for all h ∈ h and α ∈ Δ; (iii) [uα , vα ] = 2ihα for all α ∈ Δ; (iv) [uα , uβ ] = Nα,β uβ+α − N−α,β uβ−α for all α, β ∈ Δ with β ≠ ±α; (v) [vα , vβ ] = −Nα,β uβ+α − N−α,β uβ−α for all α, β ∈ Δ with β ≠ ±α; (vi) [uα , vβ ] = Nα,β vβ+α − N−α,β vβ−α for all α, β ∈ Δ with β ≠ ±α; (vii) B(uα , uβ ) = B(uα , vβ ) = B(vα , vβ ) = 0 for all α, β ∈ Δ with β ≠ ±α; (viii) B(uα , uα ) = B(vα , vα ) = −κ for some κ ∈ ℝ+ and all α ∈ Δ. 1 uα , α ∈ Δ+M , provide a complex orthonorFrom (vii) and (viii) we see that the vectors √κ mal basis of the tangent space To M ≅ p. For all α, β ∈ Δ+M we have α + β ∉ Δ, since (α + β)(H k ) = 2 and the coefficient of αk in the highest root δ is 1. This implies the following.

Lemma 8.1.4. The vectors uα and vα have the following properties: (i) [uα , uβ ] = −N−α,β uβ−α for all α, β ∈ Δ+M ; (ii) [vα , vβ ] = −N−α,β uβ−α for all α, β ∈ Δ+M ; (iii) [uα , vβ ] = −N−α,β vβ−α for all α, β ∈ Δ+M . Note that, since α + β ∉ Δ, the (−α)-string containing β starts with β and therefore N−α,β = ±1. Recall that the Riemannian curvature tensor R of M satisfies R(x, y)z = −[[x, y], z]

(x, y, z ∈ To M ≅ p).

8.1 Structure theory of compact Hermitian symmetric spaces | 335

For x ∈ p we define the Jacobi operator Rx on To M by Rx y = R(y, x)x. Let α, β ∈ Δ+M with α ≠ β. We have Ruα uα = 0, since R is skew-symmetric in the first two variables. Next, Ruα vα = −[[vα , uα ], uα ] = 2i[hα , uα ] = 2α(hα )vα = 4vα . Next, if β − α ∈ Δ, we get Ruα uβ = −[[uβ , uα ], uα ] = [[uα , uβ ], uα ] = −N−α,β [uβ−α , uα ] = N−α,β [uα , uβ−α ] = N−α,β (Nα,β−α uβ − N−α,β−α uβ−2α )

and Ruα vβ = −[[vβ , uα ], uα ] = [[uα , vβ ], uα ] = −N−α,β [vβ−α , uα ] = N−α,β [uα , vβ−α ] = N−α,β (Nα,β−α vβ − N−α,β−α vβ−2α ).

Otherwise, if β − α ∉ Δ, we have Ruα uβ = 0 and Ruα vβ = 0. We now list the root systems and corresponding Hermitian symmetric spaces. We also include the extended Dynkin diagrams for {α1 , . . . , αr , −δ}. (Ar ) V = {v ∈ ℝr+1 : ⟨v, e1 + ⋅ ⋅ ⋅ + er+1 ⟩ = 0}, Δ = {eν − eμ : ν ≠ μ};

α1 = e1 − e2 ,

...,

r ≥ 1;

Δ = {eν − eμ : ν < μ}; +

αr = er − er+1 ;

δ = α1 + ⋅ ⋅ ⋅ + αr = e1 − er+1 ; ×

∘

α1

∘

α2

∘

αr−1

∘

αr

For each k ∈ {1, . . . , r}, the coefficient of αk in δ is equal to 1 and the corresponding Hermitian symmetric space is the complex Grassmann manifold Gk (ℂr+1 ) = SU r+1 /S(Uk Ur+1−k ). Since αk and αr+1−k lead to isometric Grassmann manifolds Gk (ℂr+1 ) ≅ Gr+1−k (ℂr+1 ), we will always assume 2k ≤ r + 1. Then Δ+M = {αν + ⋅ ⋅ ⋅ + αμ : 1 ≤ ν ≤ k ≤ μ ≤ r} = {eν − eμ+1 : 1 ≤ ν ≤ k ≤ μ ≤ r}.

336 | 8 Real hypersurfaces in Hermitian symmetric spaces (Br ) V = ℝr ,

r ≥ 2;

Δ = {±eν ± eμ : ν < μ} ∪ {±eν };

α1 = e1 − e2 ,

Δ+ = {eν ± eμ : ν < μ} ∪ {eν };

αr−1 = er−1 − er ,

...,

αr = er ;

δ = α1 + 2α2 + ⋅ ⋅ ⋅ + 2αr = e1 + e2 ; ×

∘

∘

α1

∘

α2

? ∘

∘

αr−2

αr−1

αr

The coefficient of α1 in δ is equal to 1 and the corresponding Hermitian symmetric space is the real Grassmann manifold G2+ (ℝ2r+1 ) = SO2r+1 /SO2r−1 SO2 . Then Δ+M = {α1 + ⋅ ⋅ ⋅ + αμ : 1 ≤ μ ≤ r} ∪ {α1 + ⋅ ⋅ ⋅ + αμ + 2αμ+1 + ⋅ ⋅ ⋅ + 2αr : 1 ≤ μ < r} = {e1 ± eμ+1 : 1 ≤ μ < r} ∪ {e1 }. (Cr ) V = ℝr ,

r ≥ 3;

Δ = {±eν ± eμ : ν < μ} ∪ {±2eν };

α1 = e1 − e2 ,

...,

αr−1 = er−1 − er ,

δ = 2α1 + ⋅ ⋅ ⋅ + 2αr−1 + αr = 2e1 ; ×

? ∘

∘

α1

Δ+ = {eν ± eμ : ν < μ} ∪ {2eν }; αr = 2er ; ∘

α2

αr−2

∘ ?

αr−1

∘

αr

The coefficient of αr in δ is equal to 1 and the corresponding Hermitian symmetric space is Spr /Ur . Then Δ+M = {αν + ⋅ ⋅ ⋅ + αμ−1 + 2αμ + ⋅ ⋅ ⋅ + 2αr−1 + αr : 1 ≤ ν < μ ≤ r} ∪ {2αν + ⋅ ⋅ ⋅ + 2αr−1 + αr : 1 ≤ ν ≤ r}

= {eν + eμ : 1 ≤ ν < μ ≤ r} ∪ {2eν : 1 ≤ ν ≤ r}. (Dr ) V = ℝr ,

r ≥ 4;

Δ = {±eν ± eμ : ν < μ};

α1 = e1 − e2 ,

...,

Δ+ = {eν ± eμ : ν < μ};

αr−1 = er−1 − er ,

αr = er−1 + er ;

8.1 Structure theory of compact Hermitian symmetric spaces | 337

δ = α1 + 2α2 + ⋅ ⋅ ⋅ + 2αr−2 + αr−1 + αr = e1 + e2 ; × ∘

αr−1

∘

α1

∘

α2

∘

αr−2

∘

αr

The coefficient of α1 in δ is equal to 1 and the corresponding Hermitian symmetric space is the real Grassmann manifold G2+ (ℝ2r ) = SO2r /SO2r−2 SO2 . Then Δ+M = {α1 + ⋅ ⋅ ⋅ + αμ : 1 ≤ μ ≤ r} ∪ {α1 + ⋅ ⋅ ⋅ + αr−2 + αr } ∪ {α1 + ⋅ ⋅ ⋅ + αμ−1 + 2αμ + ⋅ ⋅ ⋅ + 2αr−2 + αr−1 + αr : 2 ≤ μ ≤ r − 2} = {e1 ± eμ : 2 ≤ μ ≤ r}. The coefficient of αr in δ is equal to 1 and the corresponding Hermitian symmetric space is SO2r /Ur . (The Dynkin diagram symmetry implies that αr−1 leads to an isometric copy of SO2r /Ur , so we omit this case.) Then Δ+M = {αν + ⋅ ⋅ ⋅ + αr−2 + αr : 1 ≤ ν ≤ r − 2} ∪ {αr }

∪ {αν + ⋅ ⋅ ⋅ + αμ−1 + 2αμ + ⋅ ⋅ ⋅ + 2αr−2 + αr−1 + αr : 1 ≤ ν < μ ≤ r − 1}

= {eν + eμ : 1 ≤ ν < μ ≤ r}. (E6 ) V = {v ∈ ℝ8 : ⟨v, e6 − e7 ⟩ = ⟨v, e7 + e8 ⟩ = 0}; 8 1 8 Δ = {±eν ± eμ : ν < μ ≤ 5} ∪ { ∑ (−1)n(ν) eν ∈ V : ∑ n(ν) even}; 2 ν=1 ν=1 5 1 5 Δ+ = {eν ± eμ : ν > μ} ∪ { ( ∑ (−1)n(ν) eν − e6 − e7 + e8 ) : ∑ n(ν) even}; 2 ν=1 ν=1

1 α1 = (e1 − e2 − e3 − e4 − e5 − e6 − e7 + e8 ), 2 α2 = e1 + e2 , α3 = e2 − e1 , α4 = e3 − e2 , α5 = e4 − e3 , α6 = e5 − e4 ; 1 δ = α1 + 2α2 + 2α3 + 3α4 + 2α5 + α6 = (e1 + e2 + e3 + e4 + e5 − e6 − e7 + e8 ); 2

338 | 8 Real hypersurfaces in Hermitian symmetric spaces

× ∘ α2

∘

∘

α1

∘

α3

∘

α4

∘

α5

α6

The coefficient of α6 in δ is equal to 1 and the corresponding Hermitian symmetric space is E6 /Spin10 U1 . (The Dynkin diagram symmetry implies that α1 leads to an isometric copy of E6 /Spin10 U1 , so we omit this case.) The set Δ+M consists of the following 16 positive roots: 0

0

0 0

0

1

0

0

0 0

1

1

0

0

0 1

1

1

0

1

0 1

1

1

0

0

1 1

1

1

0

1

1 1

1

1

0

1

1 2

1

1

0

1

1 2

2

1

1

1

0 1

1

1

1

1

1 1

1

1

1

1

1 2

1

1

1

2

1 2

1

1

1

1

1 2

2

1

1

2

1 2

2

1

1

2

1 3

2

1

1

2

2 3

2

1

Equivalently, Δ+M consists of Δ+M = {e5 ± eμ : 5 > μ} 4 1 4 ∪ { ( ∑ (−1)n(ν) eν + e5 − e6 − e7 + e8 ) : ∑ n(ν) even}. 2 ν=1 ν=1

(E7 ) V = {v ∈ ℝ8 : ⟨v, e7 + e8 ⟩ = 0};

Δ = {±eν ± eμ : ν < μ ≤ 6} ∪ {±(e7 − e8 )} 8 1 8 ∪ { ∑ (−1)n(ν) eν ∈ V : ∑ n(ν) even}; 2 ν=1 ν=1

Δ+ = {eν ± eμ : ν > μ} ∪ {e8 − e7 } 6 1 6 ∪ { ( ∑ (−1)n(ν) eν − e7 + e8 ) : ∑ n(ν) odd}; 2 ν=1 ν=1

1 α1 = (e1 − e2 − e3 − e4 − e5 − e6 − e7 + e8 ), 2

α2 = e1 + e2 ,

8.1 Structure theory of compact Hermitian symmetric spaces | 339

α3 = e2 − e1 ,

α4 = e3 − e2 ,

α5 = e4 − e3 ,

α6 = e5 − e4 ,

α7 = e6 − e5 ;

δ = 2α1 + 2α2 + 3α3 + 4α4 + 3α5 + 2α6 + α7 = e8 − e7 ; ∘ α2

×

∘

∘

α1

∘

α3

∘

α4

∘

α5

∘

α6

α7

The coefficient of α7 in δ is equal to 1 and the corresponding Hermitian symmetric space is E7 /E6 U1 . The set Δ+M consists of the following 27 positive roots: 0

0 0

0

0 1

1

1 1

1

1 2

1

1 2

1

1 2

2

1 3

1

2

1 3

3

2

1

1

2

2 4

3

2

1

0 0 0 0 1 1 1

0 1 1 2 1 2 2

0 1 1 2 1 2 1

1 1 1 1 1 1 1

0

0 0

1

0 1

1

1 2

1

0 1

1

1 2

2

1 2

2

1 3

1

2

2 3

2

2

1

1

2

2 3

1

3

2 4

3

2

1

2

3

2 4

0 0 0 1 1 1 1

0 1 1 1 2 2 2

1 1 1 1 1 1 2

1 1 1 1 1 1 1

0 0 0 1 1 1 1

0

0 0

1

1

1

0

1 1

1

1

1

1

1 2

2

1

1

1

1 1

1

1

1

2

1 2

1

1

1

2

1 2

2

2

1

2

2 3

2

1

1

3

2

1

3

2

1

Equivalently, Δ+M consists of Δ+M = {e6 ± eμ : 6 > μ} ∪ {e8 − e7 } 5 1 5 ∪ { ( ∑ (−1)n(ν) eν + e6 − e7 + e8 ) : ∑ n(ν) odd}. 2 ν=1 ν=1

340 | 8 Real hypersurfaces in Hermitian symmetric spaces

8.2 Real hypersurfaces with isometric Reeb flow In this section we investigate real hypersurfaces with isometric Reeb flow in irreducible Hermitian symmetric spaces of compact type. We are going to use some of the results in Section 3.4, where we investigated real hypersurfaces with isometric Reeb flow in Kähler manifolds and, more specifically, in Hermitian symmetric spaces. Our aim is to prove the following result. Theorem 8.2.1 ([19]). Let M be a connected orientable real hypersurface in an irreducible Hermitian symmetric space M̄ of compact type. If the Reeb flow on M is an isometric flow and the Hopf principal curvature function α is constant, then M is congruent to an open part of a tube with radius 0 < t < π/√2 around the totally geodesic submanifold Σ in M,̄ where (i) M̄ = ℂP r = SU r+1 /S(U1 Ur ) and Σ = ℂP k , 0 ≤ k ≤ r − 1; (ii) M̄ = Gk (ℂr+1 ) = SU r+1 /S(Uk Ur+1−k ) and Σ = Gk (ℂr ), 2 ≤ k ≤ r+1 ; 2 + 2r r−1 ̄ (iii) M = G2 (ℝ ) = SO2r /SO2r−2 SO2 and Σ = ℂP , 3 ≤ r; (iv) M̄ = SO2r /Ur and Σ = SO2r−2 /Ur−1 , 5 ≤ r. Conversely, the Reeb flow on any of these hypersurfaces is an isometric flow. Note that in [19], Proposition 2.11, we claim that α is constant for real hypersurfaces with isometric Reeb flow in irreducible Hermitian symmetric spaces. However, there is an error in the proof and it remains an open problem whether or not α is constant in this situation. The underlying Riemannian metric on M̄ in Theorem 8.2.1 is the one that is induced naturally by the Killing form of the Lie algebra of the isometry group. As an immediate consequence of Theorem 8.2.1 we obtain the following non-existence result. Corollary 8.2.2. There are no real hypersurfaces with isometric Reeb flow and constant Hopf principal curvature function in the Hermitian symmetric spaces G2+ (ℝ2r+1 ) (2 ≤ r), Spr /Ur (3 ≤ r), E6 /Spin10 U1 and E7 /E6 U1 . In Section 8.1 we explained how all irreducible Hermitian symmetric spaces of compact type are constructed from the root systems of type (Ar ), (Br ), (Cr ), (Dr ), (E6 ) and (E7 ). Corollary 8.2.2 can be rephrased in terms of root systems as follows. Corollary 8.2.3. An irreducible Hermitian symmetric space of compact type admits a real hypersurface with isometric Reeb flow and constant Hopf principal curvature function if and only if the underlying root system is of type (Ar ) or (Dr ). We emphasize that this is an observation based on the classification result. We have no direct explanation of this fact and it would be interesting to find a direct argument.

8.2 Real hypersurfaces with isometric Reeb flow

| 341

All the real hypersurfaces in Theorem 8.2.1 are orbits of cohomogeneity one actions and therefore homogeneous. This gives as another consequence the following corollary. Corollary 8.2.4. Any connected orientable real hypersurface M with isometric Reeb flow and constant Hopf principal curvature function in an irreducible Hermitian symmetric space of compact type is locally homogeneous. In particular, if M is complete, then M is homogeneous. It is remarkable that the existence of this one-parameter group of isometries implies (local) homogeneity and therefore has such a strong influence on the geometry of M. We first outline our approach. Using methods from Riemannian geometry and the recently developed theory about the index of symmetric spaces, we first show that M has constant principal curvatures, provided that the Hopf principal curvature function is constant, and that the geometry of M is in some sense nicely adapted to the geometry ̄ We then reformulate this in algebraic terms using algebraic of the ambient space M. models for the Hermitian symmetric spaces of compact type that are based on Borel– de Siebenthal theory. Using Lie algebraic methods we show that the normal vectors of M correspond to the highest root in the algebraic model. The geometric interpretation of this is that the geodesics in M̄ that intersect M perpendicularly are closed geodesics in M̄ of shortest length. We then apply Jacobi field theory to prove that M has two totally geodesic complex focal sets. Rigidity of totally geodesic submanifolds tells us that M must lie on a tube around each of its two focal sets. The problem we are now facing is that totally geodesic submanifolds in Hermitian symmetric spaces are not classified (except for rank 1 and 2), so we need further arguments to determine the structure of the focal sets. This is finally achieved by investigating representations of certain compact Lie algebras. We now provide the details. For this we are going to use results and notations introduced in Sections 3.4 and 8.1. Let M be a connected orientable real hypersurface in an irreducible Hermitian symmetric space M̄ of compact type. We assume that M has isometric Reeb flow and ̄ o ∈ M and that the Hopf principal curvature function α is constant. Let G = I o (M), K = Go the isotropy group of G at o. We denote by g and k the Lie algebras of G and K, respectively. Let g = k ⊕ p be the Cartan decomposition of g. Recall that p = ⨁ ℂuα . α∈Δ+M

By construction, the highest root δ is in Δ+M , so uδ ∈ p.

342 | 8 Real hypersurfaces in Hermitian symmetric spaces ̄ and δ ∈ Ω so that ‖uα ‖ = Lemma 8.2.5. There exists a subset Ω of Δ+M with |Ω| = rk(M) ‖uβ ‖ for all α, β ∈ Ω and a = ⨁ ℝuα α∈Ω

is a maximal Abelian subspace of p. Proof. We construct Ω explicitly using the root systems that we introduced in Section 8.1: (Ar ) for αk : Ω = {e1 − er+1 , . . . , ek − er+2−k }; (Br ) for α1 : Ω = {e1 + e2 , e1 − e2 }; (Cr ) for αr : Ω = {2e1 , . . . , 2er }; (Dr ) for α1 : Ω = {e1 + e2 , e1 − e2 }; for αr : Ω = {e1 + e2 , . . . , er−1 + er } if r is even and Ω = {e1 + e2 , . . . , er−2 + er−1 } if r is odd; (E6 ) for α6 : Ω = { 21 (e1 + e2 + e3 + e4 + e5 − e6 − e7 + e8 ), e5 − e4 }; (E7 ) for α7 : Ω = {e8 − e7 , e6 − e5 , e6 + e5 }. It is straightforward to verify the properties stated in the lemma. Since every tangent vector of M̄ at o lies in a maximal Abelian subspace of p and all maximal Abelian subspaces of p are congruent to each other, we can assume that Jξo ∈ a. Thus we can write Jξo = ∑ aα uα α∈Ω

with some aα ∈ ℝ. Lemma 8.2.6. If x = ∑α∈Ω cα uα ∈ a is perpendicular to Jξo ∈ a, then aα = 0 or cα = 0 for each α ∈ Ω. Proof. Note that x ∈ a∩ 𝒞o . Since both Jξo and x are in a, we have R̄ Jξo x = 0. Since x ∈ 𝒞o , we also have R̄ Jξo Jx = 0 by Proposition 3.4.6. This implies R̄ ξo x = 0, as the curvature tensor in a Kähler manifold is J-invariant. We have ξo = −J(Jξo ) = − ∑ aα Juα = − ∑ aα vα α∈Ω

α∈Ω

and therefore 0 = R̄ ξo x = −[[x, ξo ], ξo ] = − ∑ cα aβ aγ [[uα , vβ ], vγ ]. α,β,γ∈Ω

8.2 Real hypersurfaces with isometric Reeb flow

| 343

We work out the right-hand side of this equation in more detail. Firstly, we have 0

[uα , vβ ] = {

if α ≠ β,

2ihα

if α = β.

Inserting this into the previous equation yields 0 = ∑ cβ aβ aγ [ihβ , vγ ]. β,γ∈Ω

Next, we have [ihβ , vγ ] = iγ(ihβ )uγ = −γ(hβ )uγ = −2

0 (γ, β) uγ = { (β, β) −2uγ

if β ≠ γ,

if β = γ.

Inserting this into the previous equation implies 0 = ∑β∈Ω cβ a2β uβ . Since the vectors uβ , β ∈ Ω, are linearly independent, we get cβ = 0 or aβ = 0 for each β ∈ Ω. Lemma 8.2.7. The vector Jξo is proportional to uα for some α ∈ Ω. Proof. Let Ω0 = {α ∈ Ω : aα ≠ 0} and define a subspace a0 of a by a0 = ⨁ ℝuα . α∈Ω0

Let d = dim(a0 ) and assume that d ≥ 2. Then we can find a vector 0 ≠ x ∈ a0 that is perpendicular to Jξo . Since x ∈ a, we can write x = ∑α∈Ω cα uα . Moreover, since x ∈ a0 , we have cα = 0 for all α ∉ Ω0 . As x ≠ 0, we must have cα ≠ 0 for some α ∈ Ω0 . According to Lemma 8.2.6 this implies aα = 0, which contradicts α ∈ Ω0 . It follows that d = 1, which means that Jξo is proportional to uα for some α ∈ Ω. The Weyl group of Δ acts transitively on roots of the same length. By a suitable transformation in the Weyl group we can arrange that α ∈ Ω (α as in Lemma 8.2.7) becomes the highest root δ. We can thus assume, without loss of generality, that Jξo = 1 u . √2 δ We now define Δ+M (0) = {α ∈ Δ+M : δ − α ∉ Δ+ ∪ {0}} , Δ+M (1) = {α ∈ Δ+M : δ − α ∈ Δ+ } ,

which gives the disjoint union Δ+M = Δ+M (0) ∪ Δ+M (1) ∪ {δ}. Explicitly, we have

344 | 8 Real hypersurfaces in Hermitian symmetric spaces (Ar ) (Br ) (Cr ) (Dr ) (E6 ) (E7 )

Δ+M (0) = 0 if k = 1; Δ+M (0) = {eν − eμ+1 : 2 ≤ ν ≤ k ≤ μ ≤ r − 1} if k ≥ 2; Δ+M (0) = {e1 − e2 }; Δ+M (0) = {eν + eμ : 2 ≤ ν < μ ≤ r} ∪ {2eν : 2 ≤ ν ≤ r}; Δ+M (0) = {e1 − e2 } if M̄ = G2+ (ℝ2r ); Δ+M (0) = {eν + eμ : 3 ≤ ν < μ ≤ r} if M̄ = SO2r /Ur ; Δ+M (0) = {e5 − e4 , e5 − e3 , e5 − e2 , e5 − e1 , 21 (−e1 − e2 − e3 − e4 + e5 − e6 − e7 + e8 )}; Δ+M (0) = {e6 ± ej : 6 > j}.

The set Δ+M (1) can be worked out explicitly from Δ+M and Δ+M (0) by using Δ+M (1) = Δ+M \ (Δ+M (0) ∪ {δ}). These sets arise naturally as eigenspaces of the Jacobi operator R̄ Jξo . Lemma 8.2.8. The Jacobi operator R̄ Jξo is given by 0, if x ∈ ℝuδ ⊕ (⨁α∈Δ+ (0) ℂuα ), { { M {1 ̄ RJξo x = { 2 x, if x ∈ ⨁α∈Δ+ (1) ℂuα , M { { {2x, if x ∈ ℝvδ . Proof. Recall that the Riemannian curvature tensor R̄ on M̄ satisfies ̄ y)z = −[[x, y], z] R(x,

(x, y, z ∈ To M̄ ≅ p).

We have R̄ uδ uδ = 0, since R̄ is skew-symmetric in the first two variables. Next, R̄ uδ vδ = −[[vδ , uδ ], uδ ] = 2i[hδ , uδ ] = 2δ(hδ )vδ = 4vδ . If α ∈ Δ+M (0), we have R̄ uδ uα = 0 and R̄ uδ vα = 0. Finally, if α ∈ Δ+M (1), we get R̄ uδ uα = −[[uα , uδ ], uδ ] = [[uδ , uα ], uδ ] = −N−δ,α [uα−δ , uδ ] = N−δ,α [uδ , uα−δ ] = N−δ,α Nδ,α−δ uα .

The coefficient p−δ,α in the (−δ)-string containing α must be zero, which implies N−δ,α = ±1. Similarly, the coefficient pδ,α−δ in the δ-string containing α − δ must be zero, which implies Nδ,α−δ = ±1. Since M̄ has non-negative sectional curvature, we get g(R̄ uδ uα , uα ) ≥ 0, which implies N−δ,α Nδ,α−δ = 1. It follows that R̄ uδ uα = uα . Analogously, we can prove R̄ uδ vα = −[[vα , uδ ], uδ ] = [[uδ , vα ], uδ ] = −N−δ,α [vα−δ , uδ ] = N−δ,α [uδ , vα−δ ] = N−δ,α Nδ,α−δ vα = vα .

Taking into account that Jξo =

1 u , √2 δ

we get the desired expression for R̄ Jξo .

8.2 Real hypersurfaces with isometric Reeb flow

| 345

We now give a geometric interpretation of the sets Δ+M (0) and Δ+M (1). For this we decompose To M̄ = p = ⨁α∈Δ+ ℂuα into M

p = p(0) ⊕ p(1) ⊕ ℂuδ with p(0) = ⨁ ℂuα , α∈Δ+M (0)

p(1) = ⨁ ℂuα . α∈Δ+M (1)

Let k(0) = [p(0), p(0)] and g(0) = k(0) ⊕ p(0). Explicitly, we have: (Ar ) k(0) = {0} if k = 1; k(0) = ℝihαk ⊕ suk−1 ⊕ sur−k ≅ u1 ⊕ suk−1 ⊕ sur−k if k ≥ 2, where suk−1 is generated by the simple roots α2 , . . . , αk−1 and sur−k is generated by the simple roots αk+1 , . . . , αr−1 ; (Br ) k(0) = ℝihα1 ≅ u1 ; (Cr ) k(0) = ℝihαr ⊕ sur−1 ≅ ur−1 , where sur−1 is generated by the simple roots α2 , . . . , αr−1 ; (Dr ) k(0) = ℝihα1 ≅ u1 if M̄ = G2+ (ℝ2r ); k(0) = ℝihαr ⊕ sur−2 ≅ ur−2 if M̄ = SO2r /Ur , where sur−2 is generated by the simple roots α3 , . . . , αr−1 ; (E6 ) k(0) = ℝihα6 ⊕ su5 ≅ u5 , where su5 is generated by the simple roots α1 , α3 , α4 , α5 ; (E7 ) k(0) = ℝihα7 ⊕ so10 ≅ so2 ⊕ so10 , where so10 is generated by the simple roots α2 , α3 , α4 , α5 , α6 ; and (Ar ) (Br ) (Cr ) (Dr ) (E6 ) (E7 )

g(0) = {0} if k = 1 and g(0) ≅ sur−1 if k ≥ 2; g(0) ≅ su2 ; g(0) ≅ spr−1 ; g(0) ≅ su2 if M̄ = G2+ (ℝ2r ); g(0) ≅ so2r−4 if M̄ = SO2r /Ur ; g(0) ≅ su6 ; g(0) ≅ so12 .

The Cartan decomposition of the semisimple Lie algebra g(0) is g(0) = k(0) ⊕ p(0). Let G(0) be the connected closed subgroup of G with Lie algebra g(0). Since g(0) = (g(0) ∩ k) ⊕ (g(0) ∩ p), the orbit Σ(0) = G(0) ⋅ o = G(0)/K(0) of G(0) containing o is a ̄ Explicitly, we have: totally geodesic submanifold of M. r−1 (Ar ) Σ(0) ≅ Gk−1 (ℂ ); (Br ) Σ(0) ≅ ℂP 1 ; (Cr ) Σ(0) ≅ Spr−1 /Ur−1 ; (Dr ) Σ(0) ≅ ℂP 1 if M̄ = G2+ (ℝ2r ); Σ(0) ≅ SO2r−4 /Ur−2 if M̄ = SO2r /Ur ; (E6 ) Σ(0) ≅ ℂP 5 ; (E7 ) Σ(0) ≅ G2+ (ℝ12 ).

346 | 8 Real hypersurfaces in Hermitian symmetric spaces In particular, p(0) is a Lie triple system in p. Obviously, ℂuδ is a Lie triple system in p and the corresponding totally geodesic submanifold of M̄ is ℂP 1 . The subspace ℂuδ ⊕ p(0) is also a Lie triple system in p and the corresponding totally geodesic submanifold of M̄ is ℂP 1 ×Σ(0). Geometrically, ℂP 1 ×Σ(0) is a meridian in M̄ (see [36]). The complementary subspace p(1) is also a Lie triple system in p and the corresponding totally geodesic submanifold Σ(1) of M̄ is a polar of M̄ (see [36]). Explicitly, we have: (Ar ) Σ(1) ≅ ℂP k−1 × ℂP r−k ; (Br ) Σ(1) ≅ G2+ (ℝ2r−1 ); (Cr ) Σ(1) ≅ ℂP r−1 ; (Dr ) Σ(1) ≅ G2+ (ℝ2r−2 ) if M̄ = G2+ (ℝ2r ); Σ(1) ≅ G2 (ℂr ) if M̄ = SO2r /Ur ; (E6 ) Σ(1) ≅ SO10 /U5 ; (E7 ) Σ(1) ≅ E6 /Spin10 U1 . The sets Δ+M (0) and Δ+M (1) are therefore intimately related to a particular polar/merid̄ ian configuration in the Hermitian symmetric space M. We now use Jacobi field theory to investigate the structure of the focal sets of M (see Section 2.5 for details about the methodology). From Proposition 3.4.12 we know that M is curvature-adapted and thus we can diagonalize the shape operator A and the normal Jacobi operator R̄ Jξo simultaneously. In particular, the maximal complex subbundle 𝒞 of TM is invariant under the shape operator A of M. Note that 𝒞o = p(0) ⊕ p(1). Let X ∈ p(0) with AX = λX. Then we also have AϕX = λϕX, since Aϕ = ϕA by Proposition 3.4.1. From Corollary 3.4.9 we then get λ(λ − α) = 0. Next, let X ∈ p(1) with SX = λX. From Corollary 3.4.9 we see that 2λ2 − 2αλ − 1 = 0. Since α is constant by assumption, it follows that M is a real hypersurface in M̄ with constant principal curvatures. We write α = √2 cot(√2t) with some 0 < t
2. We thus conclude the following. i(M) Theorem 8.2.12. There exist no real hypersurfaces with isometric Reeb flow and constant Hopf principal curvature function α in the Hermitian symmetric spaces Spr /Ur (r ≥ 3), E6 /Spin10 U1 and E7 /E6 U1 . Case 2. The representation of k(0) on p(1) is reducible The representation of k(0) on p(1) is reducible precisely if the root system and corresponding simple root is ((Ar ), αk ) (4 ≤ 2k ≤ r + 1) or ((Dr ), αr ) (r ≥ 4). Note that the index of G2 (ℂ4 ) ≅ G2+ (ℝ6 ) and of SO8 /U4 ≅ G2+ (ℝ8 ) is equal to 2, so we exclude these low-dimensional cases here. As above, we know that [k(0), (Tλ )o ] ⊂ (Tλ )o and [k(0), (Tμ )o ] ⊂ (Tμ )o . However, the representation of k(0) on p(1) has exactly two irreducible components in both cases, which must coincide with (Tλ )o and (Tμ )o . This allows us to work out P and Q explicitly. We start with ((Ar ), αk ) (k ≥ 2). In this case we have Δ+M (0) = {eν − eμ+1 : 2 ≤ ν ≤ k ≤ μ ≤ r − 1} and Δ+M (1) = {eν − er+1 : 2 ≤ ν ≤ k} ∪ {e1 − eμ+1 : k ≤ μ ≤ r − 1}. These two subsets of Δ+M (1) induce the reducible decomposition p(1) = ⨁ ℂuα ≅ ℂk−1 ⊕ ℂr−k . α∈Δ+M (1)

The roots Δ+M (0) ∪ {eν − er+1 : 2 ≤ ν ≤ k} = {eν − eμ+1 : 2 ≤ ν ≤ k ≤ μ ≤ r} and Δ+M (0) ∪ {e1 − eμ+1 : k ≤ μ ≤ r − 1} = {eν − eμ+1 : 1 ≤ ν ≤ k ≤ μ ≤ r − 1} define Lie triple systems in p for which the corresponding totally geodesic submanifolds are Gk−1 (ℂr ) and Gk (ℂr ), respectively. Since totally geodesic submanifolds are uniquely determined by their Lie triple systems, we see that P and Q coincide with Gk−1 (ℂr ) and Gk (ℂr ) (in no particular order). Thus we proved the following theorem. Theorem 8.2.13. Let M be a real hypersurface with isometric Reeb flow and constant Hopf principal curvature function α in the complex Grassmann manifold Gk (ℂr+1 ), 4 ≤ 2k ≤ r + 1, (k, r) ≠ (2, 3). Then M is congruent to an open part of a tube around the totally geodesic submanifold Gk (ℂr ) in Gk (ℂr+1 ).

8.2 Real hypersurfaces with isometric Reeb flow

| 351

Note that for k = 2 the assumption for the Hopf principal curvature function α to be constant is redundant by Proposition 4.6.3. Next, we consider ((Dr ), αr ). In this case we have Δ+M (0) = {eν + eμ : 3 ≤ ν < μ ≤ r} and Δ+M (1) = {e1 + eμ : 2 < μ ≤ r} ∪ {e2 + eμ : 2 < μ ≤ r}. These two subsets of Δ+M (1) induce the reducible decomposition p(1) = ⨁ ℂuα ≅ ℂr−2 ⊕ ℂr−2 . α∈Δ+M (1)

The roots Δ+M (0) ∪ {e1 + eμ : 2 < μ ≤ r}

and

Δ+M (0) ∪ {e2 + eμ : 2 < μ ≤ r}

define Lie triple systems in p for which the corresponding totally geodesic submanifolds are isometric copies of SO2r−2 /Ur−1 . We conclude that P and Q both coincide with a totally geodesic SO2r−2 /Ur−1 . Thus we proved the following theorem. Theorem 8.2.14. Let M be a real hypersurface with isometric Reeb flow and constant principal curvature function α in the Hermitian symmetric space SO2r /Ur and r ≥ 5. Then M is congruent to an open part of a tube around the totally geodesic submanifold SO2r−2 /Ur−1 in SO2r /Ur . From Theorems 8.2.11–8.2.14 we obtain the first part of Theorem 8.2.1. It remains to prove that any real hypersurface listed in Theorem 8.2.1 has isometric Reeb flow. We first describe To Σ as a Lie triple system ⨁α∈Δf ℂuα = f ⊂ p, using the root systems at the end of Section 8.1. For ℂP k ⊂ ℂP r the root system is (Ar ) and Δf = {e1 − eμ+1 : 1 ≤ μ ≤ k}. For Gk (ℂr ) ⊂ Gk (ℂr+1 ) the root system is (Ar ) and Δf = {eν − eμ+1 : 1 ≤ ν ≤ k ≤ μ ≤ r − 1}. For ℂP r−1 ⊂ G2+ (ℝ2r ) the root system is (Dr ) and Δf = {e1 − eμ+1 : 1 ≤ μ ≤ r − 1}.

352 | 8 Real hypersurfaces in Hermitian symmetric spaces For SO2r−2 /Ur−1 ⊂ SO2r /Ur the root system is (Dr ) and Δf = {eν + eμ : 2 ≤ ν < μ ≤ r}. In all cases uδ is perpendicular to f and thus uδ /|uδ | is a unit normal vector of Σ at o. ̇ Consider the geodesic γ in M̄ with γ(0) = o and γ(0) = uδ /|uδ |. The point γ(t) is on the ̄ tube Mt of radius t around Σ. The Jacobi operator Ruδ /|uδ | is given by 0, if x ∈ ℝuδ ⊕ (⨁α∈Δ+ (0) ℂuα ), { { M {1 ̄ Ruδ /|uδ | x = { 2 x, if x ∈ ⨁α∈Δ+ (1) ℂuα , M { { 2x, if x ∈ ℝv . δ { We decompose To Σ into To Σ = To0 Σ ⊕ To1 Σ with To0 Σ = To Σ ∩ p(0) and To1 Σ = To Σ ∩ p(1). Since Δ+M (0) ⊂ Δf , the normal space νo Σ decomposes into νo Σ = νo1 Σ ⊕ ℂuδ with νo1 Σ = νo Σ ∩ p(1). Denote by γ ⊥ the parallel subbundle of T M̄ along γ defined by ⊥ ̇ γγ(t) = Tγ(t) M̄ ⊖ ℝγ(t).

Moreover, define the γ ⊥ -valued tensor field R̄ ⊥ γ along γ by ̄ ̇ γ(t). ̇ R̄ ⊥ γ(t) X = R(X, γ(t)) Now consider the End(γ ⊥ )-valued differential equation Y ′′ + R̄ ⊥ γ ∘ Y = 0. Let D be the unique solution of this differential equation with initial values I 0

D(0) = (

0 ), 0

0 0

D′ (0) = (

0 ), I

where the decomposition of the matrices is with respect to ̇ γo⊥ = To Σ ⊕ (νo Σ ⊖ ℝγ(0)) and I denotes the identity transformation on the corresponding space. Then the shape ̇ is given by operator A(t) of Mt with respect to −γ(t) A(t) = D′ (t) ∘ D−1 (t) (see Section 2.6). If we now decompose γo⊥ into γo⊥ = To0 F ⊕ To1 F ⊕ νo1 F ⊕ ℝvδ ,

8.3 Contact hypersurfaces | 353

we get by explicit computation that 0 0 A(t) = ( 0 0

− √12

0 tan( √12 t) 0

0

1 √2

0 0

0 0

0

√2 cot(√2t)

cot( √12 t)

0

)

with respect to that decomposition. The principal curvatures of Mt therefore are 0,

−

1 1 tan( t), √2 √2

1 1 cot( t), √2 √2

√2 cot(√2t)

with multiplicities (i) 0, 2k, 2(r − k − 1), 1 for ℂP k ⊂ ℂP r ; (ii) 2(k − 1)(r − k), 2(r − k), 2(k − 1), 1 for Gk (ℂr ) ⊂ Gk (ℂr+1 ); (iii) 2, 2(r − 2), 2(r − 2), 1 for ℂP r−1 ⊂ G2+ (ℝ2r ); (iv) (r − 3)(r − 2), 2(r − 2), 2(r − 2), 1 for SO2r−2 /Ur−1 ⊂ SO2r /Ur . Note that in case (i) the number of distinct principal curvatures is 2 (for k = 0) or 3 (for k > 0). In the other three cases there are four distinct principal curvatures. The corresponding principal curvature spaces are obtained by parallel translation of the corresponding subspaces in γo⊥ along γ from o to γ(t). By construction, To0 Σ, To1 Σ, νo1 Σ are J-invariant subspaces and span 𝒞o . Since J is parallel, it follows that A(t)ϕ = ϕA(t). From Proposition 3.4.1 we finally conclude that the Reeb flow on Mt is an isometric flow. This finishes the proof of Theorem 8.2.1.

8.3 Contact hypersurfaces In this section we investigate contact hypersurfaces in Hermitian symmetric spaces of compact type. We have the following conjecture. Conjecture 8.3.1. Let M be a connected real hypersurface in an irreducible Hermitian ̄ ≥ 3. Then the following statements symmetric space M̄ of compact type and dimℂ (M) are equivalent: ̄ (i) M is a contact hypersurface in M; (ii) M is congruent to an open part of a tube around the real form Σ of M,̄ where: (a) Σ = ℝP k and M̄ = ℂP k (k ≥ 3); (b) Σ = Sk and M̄ = G2+ (ℝk+2 ) (k ≥ 3); (c) Σ = ℍP k and M̄ = G2 (ℂ2k+2 ) (k ≥ 2); (d) Σ = 𝕆P 2 and M̄ = E6 /Spin10 U1 . The focal sets of these real forms are homogeneous complex hypersurfaces and thus we have the following equivalent formulation of our conjecture.

354 | 8 Real hypersurfaces in Hermitian symmetric spaces Conjecture 8.3.2. Let M be a connected real hypersurface in an irreducible Hermitian ̄ > 2. Then the following statements symmetric space M̄ of compact type and dimℂ (M) are equivalent: ̄ (i) M is a contact hypersurface in M; (ii) M is congruent to an open part of a tube around the homogeneous complex hypersurface F in M,̄ where: (a) M̄ = ℂP k , k ≥ 3, and F = Qk−1 = SOk+1 /SOk−1 SO2 ⊂ SU k+1 /S(Uk U1 ) = ℂP k ; (b) M̄ = G2+ (ℝk+2 ), k ≥ 3, and F = G2+ (ℝk+1 ) ⊂ G2+ (ℝk+2 ); (c) M̄ = G2 (ℂ2k+2 ), k ≥ 2, and F = Spk+1 /Spk−1 U2 ⊂ SU 2k+2 /S(U2k U2 ) = G2 (ℂ2k+2 ); (d) M̄ = E6 /Spin10 U1 and F = F4 /Spin7 U1 ⊂ E6 /Spin10 U1 . This seems to be related to the classification of homogeneous hypersurfaces in Kähler C-spaces [62]. We shall prove here that any tube as in Conjecture 8.3.1, or Conjecture 8.3.2, is a contact hypersurface. Proofs are known for ℂP k (Theorem 3.6.6), G2+ (ℝk+2 ) ≅ Qk (Theorem 6.6.4) and G2 (ℂ2k+2 ) (Theorem 4.7.3). Here we give a unifying proof that also works for the other cases. Consider a compact Riemannian symmetric space Σ of rank 1. Then Σ is isometric to a sphere Sk or to a projective space ℝP k , ℂP k , ℍP k or 𝕆P 2 (k ≥ 2). Let G be the Lie group of conformal transformations of Sk or the Lie group of projective transformations of ℝP k , ℂP k , ℍP k or 𝕆P 2 . The Lie algebra g of G is so1,k+1 , { { { { { {slk+1 (ℝ), { { { g = {slk+1 (ℂ), { { { ∗ { { {slk+1 (ℍ) = su2k+2 , { { −26 {e6 ,

if Σ = Sk ,

if Σ = ℝP k , if Σ = ℂP k ,

if Σ = ℍP k , if Σ = 𝕆P 2 .

Since G acts transitively on Σ, we can write Σ as a homogeneous space Σ = G/U. The Lie algebra u of U is a parabolic subalgebra of g.

8.3 Contact hypersurfaces | 355

Let ḡ and ū be the complexifications of g and u, respectively. Thus sok+2 (ℂ), { { { { { { slk+1 (ℂ), { { { ḡ = {slk+1 (ℂ) ⊕ slk+1 (ℂ), { { { { sl2k+2 (ℂ), { { { { {e6 (ℂ),

if Σ = Sk ,

if Σ = ℝP k , if Σ = ℂP k ,

if Σ = ℍP k , if Σ = 𝕆P 2 .

Let Ḡ be the connected complex Lie group with trivial center and Lie algebra g.̄ Morē Ū over, define Ū = U ∩G.̄ Then ū is the Lie algebra of U.̄ The homogeneous space M̄ = G/ ̄ is a compact Hermitian symmetric space and G is the identity component of the group ̄ of holomorphic transformations of M.

Let g = k ⊕ p be a Cartan decomposition of g. Then k is a maximal compact subalgebra of g. Explicitly, we have sok+1 , { { { { { { so { k+1 , { { k = {suk+1 , { { { { spk+1 , { { { { {f4 ,

if Σ = Sk ,

if Σ = ℝP k , if Σ = ℂP k ,

if Σ = ℍP k , if Σ = 𝕆P 2 .

Then gu = k ⊕ ip is a compact real form of g.̄ Explicitly, we have sok+2 , { { { { { { {suk+1 , { { gu = {suk+1 ⊕ suk+1 , { { { { su2k+2 , { { { { {e 6 ,

if Σ = Sk ,

if Σ = ℝP k , if Σ = ℂP k ,

if Σ = ℍP k , if Σ = 𝕆P 2 .

Let τ be the complex conjugation of ḡ with respect to gu . Then τ(g) = g and τ restricted to g (which we also denote by τ) is the Cartan involution of g with Cartan decomposition g = k ⊕ p. There exists a grading g = g−1 ⊕ g0 ⊕ g1 of g so that τ(ga ) = g−a , a ∈ {−1, 0, 1}. Then we have u = g0 ⊕ g1 . There exists a unique element Z ∈ g0 such that ga = {X ∈ g : ad(Z)X = aX}.

356 | 8 Real hypersurfaces in Hermitian symmetric spaces Since [τ(Z), τ(X)] = τ[Z, X] = aτ(X) for X ∈ ga and τ(ga ) = g−a , we have τ(Z) = −Z and hence Z ∈ p. Let K and Gu be the connected subgroups of Ḡ with Lie algebras k and gu , respectively. We define Ko = {k ∈ K : Ad(k)Z = Z} and Ku = {k ∈ Gu : Ad(k)Z = Z} and denote by ko and ku the corresponding Lie algebras. Explicitly, we have sok , if Σ = Sk , { { { { { { sok , if Σ = ℝP k , { { { ko = {uk , if Σ = ℂP k , { { { { spk ⊕ sp1 , if Σ = ℍP k , { { { { if Σ = 𝕆P 2 {so9 , and sok ⊕ so2 , { { { { { { uk , { { { ku = {uk ⊕ uk , { { { { su2k ⊕ su2 ⊕ u1 , { { { { {so10 ⊕ u1 ,

if Σ = Sk ,

if Σ = ℝP k , if Σ = ℂP k ,

if Σ = ℍP k , if Σ = 𝕆P 2 .

Then we can write Σ = K/Ko

and

M̄ = Gu /Ku .

Explicitly, we have Qk = SOk+2 /SOk SO2 , if Σ = Sk , { { { { { { ℂP k = SU k+1 /S(Uk U1 ), if Σ = ℝP k , { { { k M̄ = {ℂP × ℂP k = (SU k+1 × SU k+1 )/(S(Uk U1 ) × S(Uk U1 )), if Σ = ℂP k , { { { { G2 (ℂ2k+2 ) = SU 2k+2 /S(U2k U2 ), if Σ = ℍP k , { { { { if Σ = 𝕆P 2 . {E6 /Spin10 U1 , We define an involutive automorphism Θ ∈ Aut(G)̄ by Θ(g) = Exp(πiZ)gExp(πiZ)−1 . Then we have Θ(K) = K and Θ(Gu ) = Gu . Next, define KΘ = {k ∈ K : Θ(k) = k} and

(Gu )Θ = {g ∈ Gu : Θ(g) = g}.

8.3 Contact hypersurfaces | 357

Then we have KΘo ⊂ K ⊂ KΘ

and Ku = (Gu )Θ ,

where KΘo is the identity component of KΘ . It follows that (K, Ko ) and (Gu , Ku ) are compact symmetric pairs. Next, we define m = {X ∈ k : θ(X) = −X} and mu = {X ∈ gu : θ(X) = −X}, where we denote by θ ∈ Aut(g)̄ the differential of Θ ∈ Aut(G)̄ at the identity e ∈ G.̄ This gives direct sum decompositions k = ko ⊕ m

and gu = ku ⊕ mu

and we can identify To Σ and To M̄ with m and mu , respectively. The element H0 = −iZ is in the center of ku and J = ad(H0 )|mu is the complex structure of M̄ at the base point o. Let B be the Killing form of g.̄ We define an Ad(Gu )-invariant inner product on gu by ⟨X, Y⟩ = −B(X, Y). This inner product induces a K-invariant Riemannian metric on Σ and a Gu -invariant ̄ The canonical embedding K ⊂ Gu induces a canonical emRiemannian metric on M. ̄ ̄ bedding Σ ⊂ M. This embedding is totally geodesic and realizes Σ as a real form of M. The action of G on M̄ is of cohomogeneity one and Σ is a singular orbit of this action. Since M̄ is compact and simply connected, there is exactly one other singular orbit F of this action. Explicitly, we have Qk−1 = SOk+1 /SOk−1 SO2 , { { { { { { Qk−1 = SOk+1 /SOk−1 SO2 , { { { F = {SU k+1 /S(Uk−1 U1 U1 ), { { { { Spk+1 /Spk−1 U2 , { { { { {F4 /Spin7 U1 ,

if Σ = Sk ,

if Σ = ℝP k , if Σ = ℂP k ,

if Σ = ℍP k , if Σ = 𝕆P 2 .

̄ Geometrically, F is the focal set The singular orbit F is a complex hypersurface in M. ̄ and vice versa. of Σ in M, ̄ The exponential map is defined Let expM̄ : T M̄ → M̄ be the exponential map of M. ̄ ̄ by expM̄ (v) = γv (1), where v ∈ Tp M and γv : ℝ → M is the geodesic in M̄ with γv (0) = p and γ̇v (0) = v. The restriction of expM̄ to the normal bundle νΣ of Σ defines the so-called

358 | 8 Real hypersurfaces in Hermitian symmetric spaces normal exponential map expνΣ : νΣ → M̄ of Σ. For r ∈ ℝ+ we define Σr = {expνΣ (rz) ∈ M̄ : z ∈ νΣ, ⟨z, z⟩ = 1}. Then either Σr is a tube around Σ (for generic r) or it coincides with F or Σ (if r is an integer multiple of the distance between Σ and F). We now investigate the geometry of Σr for generic r, that is, when Σr is a hyper̄ For this we calculate its shape operator. Since Σr is an orbit of a cohomosurface in M. geneity one action, it suffices to calculate the shape operator at one point. We choose the base point o = eKo ∈ Σ. Then, by the above construction, To Σ ≅ m, νo Σ = JTo Σ ≅ Jm and To M̄ ≅ mu = m ⊕ Jm. Let z ∈ Jm be a unit normal vector of Σ at o. We want to calculate the shape operator of Σr at γz (r) ∈ Σr with respect to the unit normal vector ̇ ∈ νγz (r) Σr . For this we use the Jacobi field methodology that is explained in detail −γ(r) ̄ Here, R̄ is in Section 2.6. The only piece of data we need is the Jacobi operator R̄ z of M. ̄ v 󳨃→ R(v, ̄ z)z. the Riemannian curvature tensor of M̄ and R̄ z : To M̄ → To M, First of all we note that, since m is a Lie triple system, also Jm is a Lie triple system. The corresponding totally geodesic submanifold Σ′ of M̄ is congruent to Σ and To Σ′ = νo Σ. We see that R̄ z restricted to Jm is the curvature tensor of compact symmetric space of rank 1, which is well known. By a suitable rescaling of the Riemannian metric on M̄ we can write

0, if v ∈ m0 , { { { R̄ z v = {v, if v ∈ m1 , { { {4v, if v ∈ m4 , where Jm = m0 ⊕m1 ⊕m4 and m0 = ℝz. The space m1 is the 1-eigenspace of R̄ z restricted to Jm. If Σ′ ∈ {Sk , ℝP k }, then m4 = {0}. If Σ′ ∈ {ℂP k , ℍP k , 𝕆P 2 }, then 4 is an eigenvalue of R̄ z and m4 is the corresponding eigenspace of R̄ z restricted to Jm. In this situation, if 𝔽P 1 is the 𝔽-line in 𝔽P k generated by z, then m4 is the orthogonal complement of ℝz in T0 𝔽P 1 . Thus dim(m4 ) = 1 if Σ′ = ℂP k , dim(m4 ) = 3 if Σ′ = ℍP k and dim(m4 ) = 7 if Σ′ = 𝕆P 2 . Next, m0 ⊕ m4 is a Lie triple system and the corresponding totally geodesic submanifolds are S1 ⊂ Sk , S1 ≅ ℝP 1 ⊂ ℝP k , S2 ≅ ℂP 1 ⊂ ℂP k , S4 ≅ ℍP 1 ⊂ ℍP k and S8 ≅ 𝕆P 1 ⊂ 𝕆P 2 . Then also (m0 ⊕ m4 ) ⊕ J(m0 ⊕ m4 ) is a Lie triple system and the corresponding totally geodesic submanifold is Q1 ≅ S 2 , if M̄ = Qk , { { { { 1 1 { { Q ≅ ℂP , if M̄ = ℂP k , { { { 2 Q̄ = {Q ≅ ℂP 1 × ℂP 1 , if M̄ = ℂP k × ℂP k , { { { { Q4 , if M̄ = G2 (ℂ2k+2 ), { { { { 8 if M̄ = E6 /Spin10 U1 . {Q ,

8.3 Contact hypersurfaces | 359

The curvature tensor of a complex quadric is known explicitly and we obtain 0, if v ∈ Jm4 , { { { ̄ Rz v = {v, if v ∈ Jm0 and M̄ = Qk , { { k ̄ {4v, if v ∈ Jm0 and M ≠ Q . For the remaining space Jm1 we obtain 0, if v ∈ Jm1 and M̄ = Qk , R̄ z v = { v, if v ∈ Jm1 and M̄ ≠ Qk . With this information we can now apply Jacobi field theory as described in Section 2.6 and obtain the following. Case 1: The real form S k of Qk The tube with radius r around Sk in Qk has three distinct constant principal curvatures. Principal curvature

Eigenspace

Multiplicity

tan(r) 0 − cot(r)

Jm0 Jm1 m1

1 k−1 k−1

The eigenspace is the parallel translate along γz of the space listed in the table. We have Aϕ + ϕA = − cot(r)ϕ and hence the tube with radius r around Sk in Qk is a contact hypersurface by Proposition 3.5.5. Case 2: The real form ℝP k of ℂP k The tube with radius r around ℝP k in ℂP k has three distinct constant principal curvatures. Principal curvature

Eigenspace

Multiplicity

2 tan(2r) tan(r) − cot(r)

Jm0 Jm1 m1

1 k−1 k−1

We have Aϕ + ϕA = −2 cot(2r)ϕ and hence the tube with radius r around ℝP k in ℂP k is a contact hypersurface by Proposition 3.5.5.

360 | 8 Real hypersurfaces in Hermitian symmetric spaces Case 3: The real form ℂP k of ℂP k × ℂP k The tube with radius r around ℂP k in ℂP k × ℂP k has five distinct constant principal curvatures. Principal curvature

Eigenspace

Multiplicity

2 tan(2r) tan(r) 0 − cot(r) −2 cot(2r)

Jm0 Jm1 Jm4 m1 m4

1 2k − 2 1 2k − 2 1

We have Aϕ + ϕA = −2 cot(2r)ϕ and hence the tube with radius r around ℂP k in ℂP k × ℂP k is a contact hypersurface by Proposition 3.5.5. Case 4: The real form ℍP k of G2 (ℂ2k+2 ) The tube with radius r around ℍP k in G2 (ℂ2k+2 ) has five distinct constant principal curvatures. Principal curvature

Eigenspace

Multiplicity

2 tan(2r) tan(r) 0 − cot(r) −2 cot(2r)

Jm0 Jm1 Jm4 m1 m4

1 4k − 2 3 4k − 2 3

We have Aϕ + ϕA = −2 cot(2r)ϕ and hence the tube with radius r around ℍP k in G2 (ℂ2k+2 ) is a contact hypersurface by Proposition 3.5.5. Case 5: The real form 𝕆P 2 of E6 /Spin10 U1 The tube with radius r around 𝕆P 2 in E6 /Spin10 U1 has five distinct constant principal curvatures. Principal curvature

Eigenspace

Multiplicity

2 tan(2r) tan(r) 0 − cot(r) −2 cot(2r)

Jm0 Jm1 Jm4 m1 m4

1 8 7 8 7

8.3 Contact hypersurfaces | 361

We have Aϕ + ϕA = −2 cot(2r)ϕ and hence the tube with radius r around 𝕆P 2 in E6 /Spin10 U1 is a contact hypersurface by Proposition 3.5.5. This finishes the proof for “(ii) ⇒ (i)” in Conjecture 8.3.1 resp. Conjecture 8.3.2. The construction suggests that contact geometry for hypersurfaces in Hermitian symmetric spaces is somehow related to the conformal geometry of spheres and the projective geometry of projective spaces over the four normed real division algebras.

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Index ϕ-anti-invariant 142, 328 A-isotropic tangent vector – of complex hyperbolic quadric 30 – of complex quadric 26 A-principal tangent vector – of complex hyperbolic quadric 30 – of complex quadric 26 almost complex manifold 10 almost complex structure 10 – integrable 10 almost contact metric structure 56 almost Hermitian structure 10 anti-de Sitter space 14 anti-holomorphic submanifold 53 anti-invariant submanifold 53 basic M-Jacobi field 44, 47 basic structure equations for real hypersurfaces – in complex hyperbolic quadrics 283 – in complex hyperbolic two-plane Grassmannians 154 – in complex quadrics 225 – in complex two-plane Grassmannians 92 Bianchi identity – first 6 – second 6 bisectional curvature – holomorphic 12 canonical basis 15 canonical local basis 15 canonical Riemannian metric – on a complex vector space 13 – on a real vector space 3 Cartan decomposition 18 Cartan Theorem on existence of totally geodesic submanifolds 39 Codazzi equation 37 Codazzi equation for real hypersurfaces – in complex hyperbolic quadrics 284 – in complex hyperbolic two-plane Grassmannians 155 – in complex quadrics 226 – in complex two-plane Grassmannians 93 codimension 37 cohomogeneity of isometric action 49 https://doi.org/10.1515/9783110689839-010

complete vector field 9 complex 2-plane Grassmannian 31 complex Euclidean space 13 complex hyperbolic 2-plane Grassmannian 34 complex hyperbolic quadric 27 complex hyperbolic space 14 complex projective space 14 complex quadric 24 complex space form 15 complex structure – almost 10 – induced 10 complex submanifold 53 complex subspace 10 congruent submanifolds 38 connection – Levi-Civita 4 – linear 4 – Riemannian 4 contact distribution 72 contact form 72 contact hypersurface 72 contact hypersurfaces – in complex hyperbolic quadrics 318 – in complex hyperbolic spaces 87 – in complex hyperbolic two-plane Grassmannians 212 – in complex projective plane 74 – in complex projective spaces 87 – in complex quadrics 252 – in complex two-plane Grassmannians 129 – in Hermitian symmetric spaces 76, 353 – in Kähler manifolds 71 contact manifold 71 covariant derivative 4 – normal 38 curvature – harmonic 7 – scalar 7 – sectional 8 curvature tensor – normal 38 – Riemannian 6 curvature-adapted submanifold 40 decomposition – Cartan 18

370 | Index

duality between Riemannian symmetric spaces 21 Einstein hypersurfaces – in complex hyperbolic two-plane Grassmannians 217 – in complex quadrics 279 – in complex two-plane Grassmannians 139 Einstein manifold 7 equation – Codazzi 37 – Gauss 37 – Ricci 38 equidistant hypersurface 42 – shape operator of 43, 44 expanding Ricci soliton 7 exponential map 6 flow of vector field 9 focal manifold – shape operator of 43 focal point – multiplicity of 42 formula – Gauss 37 – Weingarten 37 Fubini–Study metric – on complex hyperbolic space 14 – on complex projective space 14 fundamental 2-form 57 fundamental equations of submanifold geometry 37 Gauss equation 37 Gauss equation for real hypersurfaces – in complex hyperbolic quadrics 284 – in complex hyperbolic two-plane Grassmannians 154 – in complex quadrics 226 – in complex two-plane Grassmannians 93 Gauss formula 37 geodesic 5 – maximal 5 geodesic symmetry 16 – local 16 Grassmann manifold – complex 22 – quaternionic 23 – real 21

harmonic curvature 7 harmonic Weyl tensor 7 Hermitian manifold 11 Hermitian metric 10 Hermitian structure 11 – almost 10 Hermitian symmetric space 23 holomorphic bisectional curvature 12 holomorphic sectional curvature 11 holomorphic subbundle – maximal 56 homogeneous hypersurface 48 homogeneous hypersurfaces – in Euclidean spaces 49 – in real hyperbolic spaces 50 – in spheres 50 homogeneous real hypersurfaces – in complex hyperbolic quadrics 287 – in complex hyperbolic spaces 85 – in complex hyperbolic two-plane Grassmannians 158 – in complex projective spaces 84 – in complex quadrics 232 – in complex two-plane Grassmannians 99 homogeneous submanifold 48 Hopf hypersurface 59 Hopf hypersurfaces – in complex hyperbolic quadrics 305 – in complex hyperbolic two-plane Grassmannians 179 – in complex quadrics 240 – in complex space forms 60 – in complex two-plane Grassmannians 105 – in Kähler manifolds 59 Hopf map – onto complex hyperbolic space 14 – onto complex projective space 14 Hopf principal curvature function 61, 106 – constancy of 61 – differential of 61 Hopf–Rinow Theorem 5 horospheres – in complex hyperbolic quadrics 291 – in complex hyperbolic two-plane Grassmannians 163 hypersurface – contact 72 – equidistant 42 – homogeneous 48

Index | 371

– isoparametric 49 integrable almost complex structure 10 invariant submanifold 53 isometric Reeb flow – in complex hyperbolic quadrics 309 – in complex hyperbolic spaces 86 – in complex hyperbolic two-plane Grassmannians 204 – in complex projective spaces 86 – in complex quadrics 244 – in complex two-plane Grassmannians 122 – in Hermitian symmetric spaces 69, 340 – in Kähler manifolds 65 isometry 3 isometry group 3 isoparametric hypersurface 49 Jacobi field 8 – basic 20 – basic M- 44, 47 – M- 41 Jacobi identity 2 Jacobi operator – along geodesic 9 – normal 40 – structure 148 – with respect to tangent vector 9 – with respect to vector field 9 Kähler form 11 Kähler manifold 11 – quaternionic 15 Kähler metric 11 Kähler structure 11 – quaternionic 15 Killing vector field 9 Koszul formula 4 Levi-Civita connection 4 linear connection 4 – metric 4 – torsion-free 4 local geodesic symmetry 16 locally symmetric space – Riemannian 16 M-Jacobi field 41 manifold – almost complex 10

– contact 71 – Einstein 7 – Hermitian 11 – Kähler 11 – quaternionic Kähler 15 – Riemannian 3 maximal geodesic 5 maximal holomorphic subbundle 56 metric – Hermitian 10 – Kähler 11 – Riemannian 2 metric linear connection 4 multiplicity of focal point 42 normal covariant derivative 38 normal curvature tensor 38 normal Jacobi operator 40 normal Jacobi operator of real hypersurfaces – in complex hyperbolic quadrics 284, 327 – in complex hyperbolic two-plane Grassmannians 155 – in complex quadrics 226 – in complex two-plane Grassmannians 94, 143 parallel transport 5 parallel vector field 5 potential field 7 principal curvature spaces 38 principal curvatures 38 product Riemannian metric 3 pseudo-ϕ-anti-invariant Ricci tensor 142, 277, 328 pseudo-Einstein real hypersurfaces – in complex hyperbolic quadrics 329 – in complex hyperbolic two-plane Grassmannians 215 – in complex quadrics 279 – in complex two-plane Grassmannians 143 pullback of tensor field 2 quaternionic Kähler manifold 15 quaternionic Kähler structure 15 quaternionic subspace 16 rank of symmetric space 20 real hypersurfaces with ϕ-invariant Ricci tensor – in complex two-plane Grassmannians 139 real hypersurfaces with harmonic curvature – in complex hyperbolic quadrics 329

372 | Index

– in complex quadrics 266 – in complex two-plane Grassmannians 141 real hypersurfaces with parallel Ricci tensor – in complex hyperbolic quadrics 328 – in complex quadrics 277 – in complex two-plane Grassmannians 139 real hypersurfaces with Reeb parallel shape operator – in complex hyperbolic quadrics 326 – in complex quadrics 259 real space form 8 real structure on complex vector space 26 Reeb flow 56 Reeb parallel shape operator 262, 326 Reeb vector field 56 regular tangent vector 21 representation – s- 20 Ricci equation 38 Ricci soliton 7 – expanding 7 – shrinking 7 – steady 7 Ricci soliton real hypersurfaces – in complex hyperbolic quadrics 329 – in complex two-plane Grassmannians 142 Ricci tensor 6 – of complex hyperbolic quadric 30 – of complex hyperbolic two-plane Grassmannian 35 – of complex quadric 26 – of complex two-plane Grassmannian 32 Ricci tensor of real hypersurfaces – in complex hyperbolic quadrics 285 – in complex hyperbolic two-plane Grassmannians 157 – in complex quadrics 228 – in complex two-plane Grassmannians 96 Riemannian connection 4 Riemannian curvature tensor 6 – of complex hyperbolic quadric 30 – of complex hyperbolic two-plane Grassmannian 35 – of complex quadric 26 – of complex space form 12 – of complex two-plane Grassmannian 32 Riemannian locally symmetric space 16 Riemannian manifold 3

Riemannian metric 2 – product 3 Riemannian symmetric pair 18 Riemannian symmetric space 16 – rank of 20 – semisimple 20 Riemannian symmetric space of compact type 20 Riemannian symmetric space of non-compact type 20 rigidity of totally geodesic submanifold 39 s-representation 20 scalar curvature 7 scalar curvature of real hypersurfaces – in complex hyperbolic quadrics 286 – in complex hyperbolic two-plane Grassmannians 158 – in complex quadrics 229 – in complex two-plane Grassmannians 97 second fundamental form 38 – covariant derivative of 38 sectional curvature 8 – holomorphic 11 semisimple Riemannian symmetric space 20 shape operator 38 shape operator of equidistant hypersurface 43, 44 shape operator of focal manifold 43 shape operator of tube 46, 47 shrinking Ricci soliton 7 singular tangent vector 21 – of complex hyperbolic 2-plane Grassmannian 35 – of complex hyperbolic quadric 30 – of complex quadric 26 – of complex two-plane Grassmannian 33 space form 8 – complex 15 – real 8 steady Ricci soliton 7 structure – almost complex 10 – almost Hermitian 10 – Hermitian 11 – Kähler 11 structure Jacobi operator 148 structure Jacobi operator of real hypersurfaces – in complex hyperbolic quadrics 285, 327

Index | 373

– in complex hyperbolic two-plane Grassmannians 156 – in complex quadrics 228 – in complex two-plane Grassmannians 95, 148 structure tensor field 56 submanifold – anti-holomorphic 53 – anti-invariant 53 – complex 53 – curvature-adapted 40 – homogeneous 48 – invariant 53 – totally geodesic 39 – totally real 53 submanifolds – congruent 38 subspace – complex 10 – quaternionic 16 – totally complex 16 – totally real 10, 16 symmetric pair – Riemannian 18 symmetric space – Hermitian 23 – Riemannian 16 – Riemannian locally 16 – semisimple 20 symmetry 16 – geodesic 16 tangent vector – regular 21 – singular 21 tensor field – pullback of 2 Theorem – Cartan (on existence of totally geodesic submanifolds) 39

– Hopf–Rinow 5 – Newlander–Nirenberg 11 torsion-free linear connection 4 totally complex subspace 16 totally geodesic submanifold 39 – rigidity of 39 totally geodesic submanifolds – of complex hyperbolic quadrics 286 – of complex hyperbolic two-plane Grassmannians 158 – of complex quadrics 229 – of complex two-plane Grassmannians 97 totally real submanifold 53 totally real subspace 10, 16 transport – parallel 5 tube – shape operator of 46, 47 tubes around totally geodesic submanifolds – of complex hyperbolic quadrics 297 – of complex hyperbolic two-plane Grassmannians 171 – of complex quadrics 234 – of complex two-plane Grassmannians 100 vector field 1 – complete 9 – flow of 9 – Jacobi 8 – Killing 9 – parallel 5 – Reeb 56 Weingarten formula 37 Weyl tensor – harmonic 7

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