Riemannian Geometry [2. rev. ed.] 9783110905120, 9783110145939

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de Gruyter Studies in Mathematics 1 Editors: Heinz Bauer · Jerry L. Kazdan · Eduard Zehnder

de Gruyter Studies in Mathematics 1 Riemannian Geometry, 2nd rev. ed., Wilhelm P. A. Klingenberg 2 Semimartingales, Michel Metivier 3 Holomorphic Functions of Several Variables, Ludger Kaup and Burchard Kaup 4 Spaces of Measures, Corneliu Constantinescu 5 Knots, Gerhard Burde and Heiner Zieschang 6 Ergodic Theorems, Ulrich Krengel 1 Mathematical Theory of Statistics, Helmut Strasser 8 Transformation Groups, Tammo torn Dieck 9 Gibbs Measures and Phase Transitions, Hans-Otto Georgii 10 Analyticity in Infinite Dimensional Spaces, Michel Herve 11 Elementary Geometry in Hyperbolic Space, Werner Fenchel 12 Transcendental Numbers, Andrei B. Shidlovskii 13 Ordinary Differential Equations, Herbert Amann 14 Dirichlet Forms and Analysis on Wiener Space, Nicolas Bouleau and Francis Hirsch 15 Nevanlinna Theory and Complex Differential Equations, Ilpo Laine 16 Rational Iteration, Norbert Steinmetz 17 Korovkin-type Approximation Theory and its Applications, Francesco Altomare and Michele Campiti 18 Quantum Invariants of Knots and 3-Manifolds, Vladimir G. Turaev 19 Dirichlet Forms and Symmetric Markov Processes, Masatoshi Fukushima, Yoichi Oshima, Masayoshi Takeda 20 Harmonic Analysis of Probability Measures on Hypergroups, Walter R. Bloom and Herbert Heyer

Wilhelm P. A. Klingenberg

Riemannian Geometry Second Revised Edition

w

Walter de Gruyter G Berlin · New York 1995 DE

Author Wilhelm P. A. Klingenberg Mathematisches Institut der Universität Bonn Wegeierstraße 10 D-53115 Bonn For my wife Series Editors Heinz Bauer Mathematisches Institut der Universität Bismarckstraße l D-91054 Erlangen, FRG

Jerry L. Kazdan Department of Mathematics University of Pennsylvania 209 South 33rd Street Philadelphia, PA 19104-6395, USA

Eduard Zehnder ETH-Zentrum/Mathematik Rämistraße 101 CH-8092 Zürich Switzerland

7997 Mathematics Subject Classification: 53-02; 53Axx, 53Bxx, 53Cxx, 49Lxx, 49Rxx, 57Rxx, 58Axx, 58Bxx, 58Cxx, 58Dxx, 58Exx, 58Fxx Keywords: Differential Geometry, manifolds, global analysis, analysis on manifolds, calculus of variations © Printed on acid-free paper which falls within the guidelines of the ANSI to ensure permanence and durability.

Library of Congress Cataloging-in-Publication Data Klingenberg, Wilhelm P. ., 1924Riemannian geometry / Wilhelm Klingenberg. - 2nd rev. ed. p. cm. — (De Gruyter studies in mathematics ; 1) Includes bibliographical references (p. ) and index. ISBN 3-11-014593-6 (acid-free) 1. Geometry, Riemannian. I. Title. II. Series. QA649.K544 1995 516.3'73-dc20 95-2589

Die Deutsche Bibliothek -

CIP-Einheitsaufnähme

Klingenberg, Wilhelm:

Riemannian geometry / Wilhelm P. A. Klingenberg.- 2., rev. ed. Berlin ; New York : de Gruyter, 1995 (De Gruyter studies in mathematics ; 1) ISBN 3-11-014593-6 NE: GT

© Copyright 1995 by Walter de Gruyter & Co., D-10785 Berlin. All rights reserved, including those of translation into foreign languages. No part of this book may be reproduced in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. Printed in Germany. Typesetting and Printing: Fa. Tutte, Salzweg. Binding: Dieter Mikolai, Berlin. Cover design: Rudolf Hübler, Berlin.

Preface to the Second Edition

It was with great satisfaction that I accepted the invitation of the publisher to prepare a Second Edition of my Riemannian Geometry. Since the book first appeared, numerous misprints and several errors came to my attention. At an age where printing with the assistance of a computer has become quite simple, it is somewhat paradoxical that at the same time it became very time consuming and costly to make changes in a published text. I highly appreciate that nevertheless the Walter de Gruyter Company was willing to take into account my suggestions in an oldfashioned way. In particular, I want to thank Dr. M. Karbe, who supported me through all stages. Since my Riemannian Geometry first appeared in 1983, the field has experienced a tremendous growth and extension. In the Second Eddition, I only sporadically can hint at some of the high lights. Fortunately, there exists an excellent survey of the state of affairs at the beginning of the Nineties: It is part 3 of volume 54, dedicated to Riemannian Geometry, in the Proc. Symp. Pure Math., Providence, RI: Amer. Math. Soc. 1993. However, there is one result of very recent origin, which I am quite happy to include into the Second Edition. I call it the Main Theorem for Surfaces of Genus 0 and it states: There always exist on such a surface infinitely many geometrically distinct closed geodesies. It is a truly centennial result since, already in 1905, H. Poincare showed that on a convex surface there exists a simple closed geodesic. His work was continued and extended by G. Birkhoff, L. Lusternik, L. Schnirelmann and many others, until finally, in 1993, the combined efforts of V. Bangert and J. Franks led to a proof of the Main Theorem in full generality. In the same year, N. Kingston published a paper which only uses methods which have been developed in this monograph and avoids the approach that Franks used. It therefore became quite natural for me to present in the final section of chapter 3 a complete proof of the Main Theorem. I feel that this constitutes an important and beautiful finale to my work. Bonn, Januar 1995

Wilhelm P. A. Klingenberg

Preface

The present book is an outcome of my course on Riemannian Geometry. Its origin can be traced back to a series of special lectures which I gave during the summer semester 1961 in Bonn. At that time, D. Gromoll and W. Meyer were among my students and in 1967 we jointly published in the Lecture Notes Series our "Riemannsche Geometrie im Großen". These lectures have met with great interest, because for the first time a concise introduction into Riemannian Geometry was combined with global methods culminating in the so-called Sphere Theorem, which states that the underlying topological manifold of a simply connected Riemannian manifold with suitably restricted positive curvature is a sphere. Over the past twenty years, global Riemannian Geometry has experienced considerable growth in various areas. Here I wish to mention in particular the work of Gromov [1], [2], [3] on manifolds with restrictions on the curvature and the numerous results on the eigenvalues of the Laplace operator. Also - and this is the field in which I have been most active myself - a great number of new results on the existence and on the properties of closed geodesies have been obtained. For an excellent survey of the present state of research cf. Yau [1]. It is only natural that in my course I have chosen topics which are close to my own areas of research. But I have always begun with a full exposition of the classical, local Riemannian Geometry. Thus, chapter 1 is devoted to the foundations. What is unusual here is that from the very beginning I have allowed manifolds to be modelled on separable Hubert spaces. This presents no difficulties when compared with the case of finite dimensional manifolds and it has the advantage of yielding the necessary framework for later applications in chapter 2. Of course, there are some differences between Hubert manifolds and finite dimensional manifolds, which appear for the first time when considering the tensor product. However, for most of the basic results, the step from finite dimensions to Hubert space is no bigger than the step from 2 dimensions to n dimensions. Whenever the restriction to the finite dimensional case brings about some simplification, I have pointed this out clearly. In chapter 1 - and the same is true for the later chapters - the first part of a section is usually more basic than the rest. At least this is the case when the sections are longer than 10 pages or so. While the expert will have no difficulty in constructing a 'basic

Preface

vii

course' from our material, the beginner should keep in mind that it might be wise to switch to the next section when he reaches subjects like general vector bundles, submersions or focal points, to mention a few. He can always return to the previous sections when the need arises. In chapter 2, entitled Curvature and Topology, I restrict myself to finite dimensional manifolds because the local compactness of the manifold is needed. Complete manifolds are studied and there follows a rather complete account of the theory of symmetric spaces from the point of view of Riemannian Geometry which differs from the usual, more algebraic approach. After this, I develop in three sections the basic theory of the manifold of curves. Here, I take advantage of the fact that in chapter 1 Riemannian manifolds modelled on Hubert Spaces were allowed. When it comes to the critical point theory I only develop the LusternikSchnirelmann approach. I do not enter into the much more delicate Morse Theory. The full power of this approach has not been sufficiently recognized. Among other things I show that one can give a completely elementary proof of the fundamental estimate on the injectivity radius of 1/4-pinched manifolds. The chapter continues with a simplified proof of the Alexandrov-Toponogov Comparison Theorem. This is an essential tool in the proof of the Sphere Theorem, given in the next section. I conclude with the basic constructions on non-compact manifolds of positive curvature. With the end of chapter 2 I have covered all the material contained in the "Riemannsche Geometrie im Großen" and indeed much more; e.g., symmetric spaces and the manifold of curves with its various important submanifolds. In chapter 3, entitled Structure of the Geodesic Flow, I deal with a subject which, traditionally, is not presented in a course on Riemannian Geometry. I feel, however, that this field should not be left to specialists in ergodic theory or Hamiltonian systems. Rather, it should be tied more closely to Riemannian Geometry proper. In fact, it is one of the oldest fields of research. Thus, e.g., the geodesic flow on the ellipsoid was even studied by C. G. J. Jacobi and the problem of stability for periodic orbits plays a fundamental role in Poincare's investigations on Celestial Mechanics. I present many of the classical results together with numerous examples. Among them, there are theorems for periodic orbits with elementary proofs employing only the LusternikSchnirelmann theory developed in chapter 2. The last two sections deal with manifolds of non-positive curvature. Here, in particular the case of strictly negative curvature is treated for the first time in a monograph, with elementary proofs of many of the basic results in this important area. After this brief description of the contents, it would certainly take more space to describe what has been omitted. To have an idea of topics not covered in this book see e.g. Chem [2] and de Rham [2]. Most notable is the absence of integration methods. It is clear that on Hubert manifolds differential forms are bound to play a lesser role than on manifolds of finite dimension. But the deeper explanation for this and for most of the other omissions is simply that a book on mathematics, like any other literary work, is necessarily prejudiced by the personal experiences of the author and thus reveals

viii

Preface

strong autobiographical traits. As a matter of fact, I have composed this monograph around my own area of research in Riemannian geometry over the past 25 years, thereby including the work of younger colleagues who I had the privilege of meeting to mutual advantage. I have no other excuse to proffer for the selection of the contents, except that I am convinced that my choice represents a lasting contribution to the field and that future fruitful developments seem most likely. Thus, I hope that my efforts in writing this book over many years will not just be a record of results and methods but will also serve as an impetus towards further research. It only remains to express my gratitude to the people who helped me with the manuscript, by reading whole sections. I wish to mention in particular W. Ballmann, V.Bangert, J.Eschenburg, H.Matthias, A.Thimm, G.Thorbergsson and F.Wolter. Finally, I wish to thank Walter de Gruyter & Co. for accepting my manuscript in their new series 'Studies in Mathematics'. Bonn, 1982

Wilhelm Klingenberg

Table of Contents

Chapter 1: Foundations 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 1.12

Review of Differential Calculus and Topology Differentiable Manifolds Tensor Bundles Immersions and Submersions Vector Fields and Tensor Fields Covariant Derivation The Exponential Mapping Lie Groups Riemannian Manifolds Geodesies and Convex Neighborhoods Isometric Immersions Riemannian Curvature Jacobi Fields

1 8 13 23 31 39 53 59 67 78 86 97 109

Chapter 2: Curvature and Topology 2.1 2.1 2.2 2.3 2.4 2.5 2.5 2.6 2.6 2.7 2.8 2.9

Completeness and Cut Locus Appendix - Orientation Symmetric Spaces The Hubert Manifold of H^curves The Loop Space and the Space of Closed Curves The Second Order Neighborhood of a Critical Point Appendix - The S1- and the Z2-action on AM Index and Curvature Appendix - The Injectivity Radius for 1/4-pinched Manifolds Comparison Theorems for Triangles The Sphere Theorem Non-compact Manifolds of Positive Curvature

124 136 141 158 170 181 196 203 212 215 229 240

Chapter 3: Structure of the Geodesic Flow 3.1 Hamiltonian Systems 3.2 Properties of the Geodesic Flow

256 265

χ

3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10

Table of Contents

Stable and Unstable Motions Geodesies on Surfaces Geodesies on the Ellipsoid Closed Geodesies on Spheres The Theorem of the Three Closed Geodesies Manifolds of Non-Positive Curvature The Geodesic Flow on Manifolds of Negative Curvature The Main Theorem for Surfaces of Genus 0

References Index

279 288 303 324 337 350 363 380 393 403

Chapter 1: Foundations

This chapter contains the basic definitions and results on differentiable manifolds, vector and tensor bundles over such manifolds and Riemannian metrics. The material presented here differs little from that in other well-known text books, except that we consider manifolds modelled on Hubert spaces rather than on finite dimensional spaces. This will be useful in Chapter 2 and presents no conceptual difficulties anyway, as was demonstrated by Lang [1]. Not quite standard in our chapter on the Foundations is the discussion of submersions (see 1.11) and Jacobi fields (see 1.12). This constitutes a first step towards global geometry, which is the subject of the remainder of the book.

1.0 Review of Differential Calculus and Topology In this section we set forth some notation and recall some basic properties of differentiable maps between Banach spaces. For details we refer to Dieudonne [1] and Lang [1]. We shall conclude with some facts on topological spaces. Reference will be made to Bourbaki [1]. 1. We denote by E, E; E f , . . . , F, F; F,.,... real Banach spaces. In fact, most of the time these will actually be separable (complete) Hubert spaces. Subspaces are always assumed to be closed and linear mappings are assumed to be continuous. Note. Subspaces of finite dimension or finite codimension are always closed. We say that a closed (linear) subspace E' of E splits if there exists a closed complement E" such that E is isomorphic to Ε' χ Ε". Note that for a Hubert space E, every subspace E' splits: Take for E" the orthogonal complement of E'. Let F: E -> F be an injective linear mapping whose image is a closed subspace f'.F is called a splitting mapping if F' splits, i.e., if F ^ F' χ F". More generally, a linear mapping F: E -» F with closed image is called a splitting mapping if the induced injection E/kerF-* F splits. Again, for Hubert spaces, any closed linear mapping splits - closed means that the image is a closed subspace. Let us denote by L ( E ; F) the vector space of linear mappings F: E -> F. L(E; F) becomes a Banach space by taking as norm \F\ of an Fe L(E; F) the greatest lower bound of all numbers k such that

2

Foundations

\F.X\^k\X\,

forallJireE.

If E and F are finite dimensional, one can define a scalar product on L ( E, F) so as to make it into a (finite dimensional) Hubert space, see (1.0.2). More generally, we define a norm on the space L C F i , . . . , F P ;G) of r-linear mappings from Fj χ . . . χ F, into G by taking for | F\ the greatest lower bound of real numbers k satisfying

where (Xlt . . ., Xr) e ^1 χ . . . χ Fr . With this, the canonical mapping

from the space of iterated linear maps into the space of multilinear maps becomes a Banach space isomorphism. Of particular importance are the various tensor spaces associated to a Hubert space E. Let E* denote the dual of E. Then 7 7 E E E L(E*,...,E*, Ε

; R)

is called the space ofr-fold contravariant and s-fold covariant tensors. We also use 77 E to denote any of the (r + s ) \ / r l s \ spaces L(E l 5 ..., E r + s ; (R), where r of the E,· are equal to E* and the remaining s of the E; are equal to E. Since L(E*; IR) = E; L(E; R) = E* we have for rs> 0 77 E = L ( E * , . . . , E * , E__.JE; R) s L(E*. .... E*. E,__JE; E) r 5 r— l 5

andj

aZ.(E*.....E*.E....E;E*) 5— 1

A word to explain the terminology : Take e.g. Xe TQ E = E, a 1-fold contravariant tensor. Let F: E -> E be an automorphism. Choose an (orthonormal) Hubert basis {ei} and its dual {e1}. Then the z'-th coordinate of A" ist given by X1 = (e', Xy where we denote by the canonical pairing Ε* χ Ε -» (R. The z'-th coordinate of FX is given by Xfi = (el,FXy =

for all permutations σ of the set (1, . . ., s}. (ii) The space ASE of s-fold covariant antisymmetric tensors consisting of the Z s °e£°E satisfying for all permutations σ of (1, . . ., s}. We see that if dim E = η < oo, then aim SsE = (n + s-l)l/s\(n-\)l (nl/s\(n-s)l, = < (Q ,

whereas

(or l^s^n for s > η

We conclude this section by indicating some canonical isomorphisms between spaces of linear maps when all vector spaces have finite dimension. Recall that the tensor product E ® F of two vector spaces E and F is characterized by the following properties : (i) There exists a bilinear mapping Φ: Ex F ^ E ® F ; (JT, Y}^

X®Y

such that the image generates E ® F as a vector space.

Foundations

(ii) Given any bilinear map FeL(E, F;G), there exists a unique G e L ( E ® F;G) with F = G o Φ. In particular, if {e,·}, {j£} are bases of E resp. F, then {e; j£} ^s a basis for E ® F. One has the canonical isomorphisms : L ( E ® F ; G ) ^ L ( E , F; G) ^L(E; L(F;

L(E; F*) ^ E * ® F* For instance, Fe L(E; F*) corresponds to £ (f}, F(et)y e1 fj where {e1}, {fj} are the dual bases of the bases {e,·}, {jj} of E and F. Combining these isomorphisms, we get the 1.0.1 Proposition. Let dim E C/' differentiable, if it is differentiable of class Cr for all r. Sometimes we will find it convenient to use the language of categories and functors. Thus we may speak of the category formed by the open subsets of Banach spaces as objects and the differentiable mappings between them as morphisms. This means that with /i : £ / ! < = £ ! - > i / 2 c E 2 ; F 2 : i/ 2 £ / 3 c E 3 , being differentiable the composition F2 = F i: £/!-> I/,

is also differentiable. Moreover, idv: Uc E -> U · F bijective, i.e., E 2 ^ F via E 2 . r/ze« ί/zere e^/iii a local diffeomorphism h-.^x L/ 2 ,0)cz(E 1 x E 2 , 0 ) - > ( E , 0 ) w/r/z C7; an open neighborhood of 0 e E; swcn //zai

f o / z : t/j x U2 -> U2 c E 2 A F W i/ze projection pr2 onto the second factor. Note. If E, F are Banach spaces, one must assume that ker DF(Q) splits. Proof. For the proof of (1 .0.3) one uses the contraction lemma. For details we refer to the literature, cf. Dieudonne [1], Lang [1]. Corollary 1 is deduced from the theorem by extending F: U -» F to a locally invertible mapping Φ: υ χ F 2 c E x F 2 s F -> F t x F 2 ; (w, u 2 ) '-» ^(") + (0, t> 2 ).

Indeed, Ζ)Φ(0, 0) = Z)F(O) + (0, iW| F2). Taking g as the local inverse of Φ we prove our claim. Similarly, for the proof of Corollary 2, we consider Φ: C / c E l X E 2 ^ E l X F ^ E l X E 2 ; (uit u2) ^(uit F(UI, u2)). Then Ζ)Φ(Ο) = (id\ E l f 0) +(£)F(0)| E 1? £>F(0)| E2), i.e., Ζ)Φ(Ο) is bijective. Taking for h the local inverse of Φ we get a mapping satisfying our requirements. 4. A topological space Μ is called metrizable if there exists a metric on Μ which induces the given topology. Μ is called separable if it possesses a countable base for the open sets. For metric

8

Foundations

spaces this is equivalent to saying that there exists a countable dense set of points in M. Let (MJa€ A be an open covering of M. That is, all ΜΛ are open and every p e M is contained in some Ma. An open covering (M ) eB of M will be called refinement of the covering (Mx)aeA if there exists a mapping σ: 5 ->· Λ such that M^ (R is continuous ^ 0 with {φβ > 0} £Α, for the topological manifold Μ equivalent if their union gives a differentiable atlas. (iii) A differentiable structure on a topological manifold is an equivalence class of differentiable atlases. A differentiable manifold-or simply manifold-is a topological manifold, endowed with a differentiable structure. Notes. 1. If the model space E of the topological manifold Μ has dimension η then M is called η-dimensional. In this case the hypothesis metrizable can be replaced by Hausdorff. 2. Let (M, M') be a chart of the manifold M. That is for each p e M', u(p) is an element of [/+}), (u_, S1^ — {/?_}) define a differentiable atlas for Sg. These charts are called the stereographic projections from p+ and p_, respectively. (iii) Any open subset M' of a differentiable manifold M is a differentiable manifold. Indeed, if (ua,,Ma)xeA is a differentiable atlas for M, its restriction (Μα|ΜαηΜ', Μα η ΛΓ)ae/1 to M' gives a differentiable atlas for M'. (iv) As a particular example for (iii) consider for M the space M(n; [R) of all real 2 (n,n)-matrices. Take as differentiable atlas the isomorphism with IR" , where the (n (i — 1) +j) — th coordinate of A e M(n; IR) is given by the element atj in the /-th row andy'-th column. (v) The general linear group GL(n; IR) is the open subset M'of M(n; IR) defined by del Λ φ 0. Thus, GL(n, IR) is a manifold of dimension n2. (vi) If M and M' are manifolds, modelled on E and E' respectively, the product manifold Μ χ Μ', modelled on E x E', is defined by taking as atlas the product atlas The structure preserving mappings, briefly morphisms, between differentiable manifolds are the differentiable mappings: 1.1.4 Definition, (i) A mapping F: M -»· TV from a manifold M into a manifold TV is called differentiable if Fis continuous and if, for some atlas (ux, Afa)ae A of M and some atlas (ve, N ) eB of TV, the mappings v

° F° (ua\(AfanF

1

N )) 1 : u^M^r^F lN ) -> v (N )

are differentiable, for all (a, )eA χ Β. (ii) If in particular TV is a Hubert space F then, using the canonical chart (id, F), F: M -> F being differentiable means that

is differentiable, for all aeA. In the case F = (R, the real line, we also call F: M-» IR a differ en liable func lion.

Differentiable Manifolds

11

(iii) If F: M -> ./V is a homeomorphism such that Fas well as F~1 is differentiable we also call F a diffeomorphism. We show that this definition does not depend on the atlas. 1.1.5 Proposition. Whether or not F: M -> N is differentiable does not depend on the particular choices of the atlases on M and N. Moreover, if

F! : Mi -»· Λ/2 and F2: M2 -> M3 are differentiable, the composition

F2 ο F! : M! -» M3 is also differentiable. Note. This allows us to speak of the category of differentiable manifolds whose morphisms are the differentiable mappings. Proof. Let (ι*α·,Μα.)Λ.εΑ. and (νβ·,Νβ be a second pair of atlases fo r M and N. Then the mapping (with suitably restricted domains of definition) V . ο Fo Μ"1 = (υβ. ο t»' 1 ) o (V

o Fo M" 1 ) o (ua. ο Μ'1)"1

is differentiable. As for the claim about the composition F2 o F1 of Fj and F2 we reduce its proof to the local representation. Here it reads

which is obviously true.

Π

Example. Given a chart («, M') on M, ul: M' -> IR, the ι'-th coordinate function, is a differentiable function on the manifold Μ'. Μ' is a manifold according to (1.1.3, iii). Actually, its differentiable structure is determined by the atlas with the single chart (u, M'). We add a result on differentiable functions on M. 1.1.6 Proposition. The set & M of differentiable functions on a manifold M is an ^-algebra under the natural compositions (f+g}(p)=f(p}+g(p);

(af)(p)

= af(p);

(fg)(p)=f(p)g(p)

with f,g in &M, ae IR. Moreover, if F: M -+ N is a differentiable mapping, we obtain an algebra morphism

F*: &Ή -> &M\ g t-» g o F. Proof. Evident from the definitions.

D

1.1.7 Definition. Let M, jV be manifolds. A local diffeomorphism at p e M is a bijec-

12

Foundations

tive mapping

of some open neighborhood M'(p) of p onto an open neighborhood N'(q) of the image q = F(p) of p such that F and F~l are differentiable. Remark. The concept of a diffeomorphism F: M -> TV between two manifolds clearly defines an equivalence relation. If M and TV have the same underlying topo logical manifold (which we denote for the moment by M0) then it may happen that an atlas on M is not equivalent to an atlas on TV. Nevertheless, there might exist a diffeomorphism F: M -»· TV. Example. M0 = IR, M = (id, IR); TV = («, IR) with u(x) = x3. The two atlases on M0 are not equivalent. But F: M -»· TV; χι-* l/x is a diffeomorphism. In this connection one can ask whether there exist several non-isomorphic differentiable structures on a topological manifold or perhaps none at all. As for the first question, Milnor discovered that the 7-sphere admits non-isomorphic differentiable structures. Actually, there are 28 such structures on S1'. See Milnor [1]. After this it soon became clear that in general a topological manifold, if it admits a differentiable structure, will also admit more than one, provided the dimension is at least 5. But there also exist topological manifolds which admit no differentiable structure at all. The first example was discovered by Kervaire [1], having dimension 10. We conclude by describing a procedure for extending a locally defined mapping to a globally defined one. 1.1.8 Lemma. Let (u, M'} be a chart on Μ such that U = u(M') is a neighborhood of QeE.Letrbe>Q such that Br(V) [R and r l5 r2 with 0 < rt < r2 < r, then there exists a differentiable /: M -» [R such that

/( ° Tu'1. D Besides the tangent bundle τ : TM ->· M of M, we have as another important object the co-tangent bundle τ* : Γ* Μ -* Μ associated to Μ. For each peM let 7J,* Μ be the dual of Tp M. Denote by Γ* M the collection of the T* M, peM. Define τ* : Γ* Μ -> Λ/ by associating to a X*e T*M its Z>ase pom/ pe M. Recall that any isomorphism F: E -> Ε' determines the contragradient (i.e., inverse transposed) isomorphism F~: E* -*· E'* between the duals. Thus, in particular we have from Tpu: TpM -> r„(p) U the mapping Tpu~:T*M-> T*(p) U. 1.2.5 Proposition. Γ*Μ w a differentiable manifold modelled on E x E * = E x E . τ*: Τ* Μ -> M w differentiable. More precisely, every atlas (ua, Ma),eA of M determines an atlas (T*ua, T*Ma) UUx F L ΡΜ'=π~ιΜ'Π -

/"•ι

π Ι Ρ Μ' '

ι/Γ

ς ι

;

,

u k

77

/

"(Ρ)

with the following properties: (a) Pu is bijective and Ppu = Pu\fp: Fp -»· {«(/>)} χ F is a Banach space isomorphism. Here the Banach space structure on {«(/?)} χ F is defined by the canonical identification pr2: (u(p)} χ F -> F. (b) If (Pu, u, M'), (Pu', u', M") are two bundle charts then

18

Foundations

(*)

Pu' o(pu\P(M'^M"))-1:

u(M'nM")x F -» H'(M"nA/0 x F

is a diffeomorphism. We also denote this mapping briefly by /V ο /Ή"1. The object thus defined is called a vector bundle (over A/) vw/A bundle atlas. (ii) Two bundle atlases (Pu,, u,, MJaeA, (Pua>, u,·, ΜΛ)Λ.€Α. of the same π: Ρ -» Μ, associated with two equivalent atlases (ua,Ma)aeA, (ua; M a ')a'ex' are called equivalent if their union is a bundle atlas associated with the union of (i/ e > AfJ e 6 X and (Μβ·,Λ/β,)β.6Χ.. An equivalence class of vector bundles over M with atlases is called a vector bundle over Μ with fibre modelled on F. If π : Ρ -» Μ, then Ρ is called the total space of η and A/ the base space of π. Remark. Some authors find it preferable to introduce the bundle of frames. At least in the case of finite dimensional manifolds, cf. Kobayashi and Nomizu [1], Sulanke und Wintgen [1], Bishop and Crittenden [1]. For our purposes, it seems more appropriate to restrict ourselves to vector bundles. Note. The fibre Fp = π~ l (p) over/? is isomorphic to F, but there is no canonical isomorphism. Rather, for each bundle chart (Pu, u, A/') covering Fp, we obtain the 'isomorphism Pu(p)=pr2oppU:

Fp -> F .

Here we have introduced the notation Pu (p) in analogy to the differential Du (p) for the tangent bundle. Let (Pu, M, A/'), (Pu', u', M") be two bundle charts. Then we have the transition mapping Pu' ο Pu~ 1 introduced under (b) above. Let us define, for p e M' n M", the isomorphism P(u' * u - l } ( u ( p ) } = Pu'(p)°Pu(p)-v

: F ^ F.

Pu' o Pu~ l may then be written as

(«, {„) H-» ((«' ο «- 1) (u), p(u' ο «- 1) («). g . Since the transition mapping is differentiable so is the mapping

Example. The tangent bundle τΜ : TM -> M of a manifold M is a vector bundle over M with fibre modelled on E, the model for M. The atlas (Γκβ, TMa)xeA of ΓΜ, associated with an atlas (ua, Af J a e X of M, cf. (1.2.3), is a bundle atlas; as such it is denoted by (Τχ, «β, Μ α ) αε ^. From (1.2.8) we immediately obtain the 1.2.9 Proposition. Let π: P ^> Μ be a vector bundle over Μ with fibre modelled on F.

Tensor Bundles

19

Let E be the model of M. Then the bundle atlas (Pua, Ma, Ma)aeA determines an atlas (Pu3, PMx)aeA of the total space P making it into a manifold modelled on E x F. Proof. Clear from the definition.

D

Remark. If F is not a Hubert space, then we have used a slight generalization of the concept of a differentiable manifold: Instead of the model space being a Hubert space as in (1.1.1), in our case it is a Banach space. However, this is not the full picture if we observe that in our case the transition mappings, determined by the overlap of two bundle charts (Pu, u, M'), (Pu', u', M"), have the special feature of being linear isomorphisms on the second factor F. More precisely, the transition mapping Pu'oPu~l may be written in the form u' c u~l χ P(w'° w" 1 ): w(M'nM") χ F -> w'(M"nM') χ F where P(u' o u'1): «(M'nM") -» GL(F) is a differentiable mapping into the group GL( F) of linear isomorphisms. In this way, the transition mapping preserves the product structure of the model Ε χ F, i.e., we obtain the commutative diagram «(M'nM")x

Pu' o Pu - 1

M'(M"nM')x F

/"Ί Μ (MOM")

ll'olT1

We complement the definition of vector bundles over manifolds by introducing the corresponding structure preserving mappings or, briefly, morphisms. This yields the category of vector bundles over manifolds. Cf. Lang [1] and (in the case of finite dimensions) Sulanke und Wintgen [1]. 1.2.10 Definition. Let π: Ρ^ Μ; π*: Ρ* ^ Μ*

be vector bundles. Denote by F, F* the model spaces for the fibres of π and π*, respectively. A morphism from π into π* consists of a pair of differentiable maps F: Μ -» Μ*; PF: P -> P* such as to make the diagram PF

π

Μ

-» Ρ*

20

Foundations

commutative. Moreover, the restriction Pp FoiPFto the fibre Fp over p shall be a linear mapping where Fp, FF*p) = π*~ 1 (F(/?)) are endowed with their Banach space structure. Remark. Let (Pu, u, M'), (Pu', u*, M*') be bundle charts for π and π*. Then the above diagram possesses the local representation Pu*c PFc Pu'1 Ux F -·-> 17* χ F* Pri

U

———

»

£/*

Here we have written £7 for u(M' nF~l M*')and £7* for u*(M*'). From /»«(/»): Fp -» F; />pF: Fp -> FFfp); Pu*(F(p)): FF*p) -* F* we form P(u* o F - u - i } ( u ( p } } = Pu*(F(p)}oppFoPu(pYv:

F ^ F*.

Then ueU^P(u*oFoU-1)(u)eL(f;

F*)

is differentiable. Actually, this characterizes the differentiability of the mapping Pu* o PF o Pu'1 in the above diagram. For the case of the tangent bundle we see that the tangential TF: TM ->· TN of a morphism F: Μ -* Ν is a vector bundle morphism, cf. (1.2.7). In (1 .0) 1 we have constructed from Banach spaces F1? . . ., Fr ; G the Banach space L(F l 5 ... F r ; (B). We complement this by observing that this construction yields a canonical inclusion (*)

L( ,..., ; ) : G L ( F 1 ) x . . . x G L ( F , ) x G L ( G ) - > G L ( L ( l F 1 , . . . , Frr

as follows: If X p ,..., F,. p ;G p ) = (briefly) IH p .

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21

By L(A^ . . ., Ar; B) = (briefly) C we denote the union of the IH p , p e M. Let

or briefly y : C^M

be the projection mapping defined by y ~ l ( p ) = IH p . For each chart (u, Μ ') of Μ we define the bundle chart (Cu, u, M') as follows. For each pe M' let Cu(p): IH p -> IH be given by

Xp H* Bu(p) ο Xp

1.2.12 Lemma.

is a vector bundle over M, with the bundle charts defined in (1.2.11). Proof. We only need to show that the transition mappings Cu' ο CM" 1 : u (M' n Μ") χ IH -» u' (Af " η Λ/') χ ΙΗ for the above defined charts are differentiable. Observe that we may write Cu' = Cu~ 1 ~ u' = u' J χ C(u' = u~ !) with 1(M

/

= M-1) , ....^..(M'oM-1)

;£(«' = I T 1 ) ) .

Since the inclusion (*) L( ,..., ; ) before (1.2.11) is differentiable, we see that

is differentiable.

D

1.2.13 Definition. Γ/ze bundle ofr-foldcontravariant ands-foldcovariant tensors over M is defined by

L(r*, . . ., τ*, τ, . . ., τ ; ρ) : L(T* M, . . ., Γ* Μ, ΓΜ, . . ., TM; R Μ) -» Μ r s or briefly τ/: 77Af-> Μ. Here, ρ : ( Κ Λ / Ξ Μ χ ( β ^ Λ / denotes the trivial (R -bundle over Λ/. Remarks. 1 . One also denotes by the same name any of the (r + s) ! l r \ s ! bundles L(cit . . . , τ Γ + ϊ ; ρ ) : L(T,M,..., Tr+sM; KAf) -> M where r of the τ, are equal to τ* and 5 of the τ, are equal to τ. All these bundles are

22

Foundations

canonically isomorphic to each other. Moreover, since L(E*; (R) = E, we have L(E*,..., Ε*, Ε , . . . , Ε ; IR) £L(E*,..., Ε*, Ε , . . . , Ε ; E).

r

s

r —l

s

Therefore we also denote by τ£, r > Ο, the bundle L(τ*,... τ*, τ,... τ ; τ). 2. Since L(E*; R) = Ε and L(E; IR) = E* we see that τ£ = τ and τ? = τ*. We also define τ° to be the trivial bundle ρ: Μ χ [R -> M. 3. The transition functions for τ£, Γ/κ'οίΤ»- 1 : w (Α/" η Μ") χ L (Ε*,..., E;IR)-*· Μ' (Μ" η Λ/') χ L (E *,..., E;1R), determine for each u 77 E with A ^ H + A T / o ('/)(«' ο i T ^ x ... χ 'D(u' °u~l)x D(u' ° t/" 1 )" 1 χ ... χ D(w' o w' 1 )" 1 ) In (1.0) 1 we defined the concept of a general tensor space as a subspace of Tsr E which is invariant under TsrF, FeGL(E). The construction of rrs: Tsr M -> M shows that for each such general tensor space we obtain a subbundle for τ£. In particular, we get from the 5-fold covariant symmetric {antisymmetric} spaces Ss Ε [As E) the corresponding bundles. 1.2.14 Definition. The bundle of s-fold covariant symmetric [antisymmetric] tensors, as: SSM -> Μ {as: ASM -> M},

is the bundle obtained from restricting the bundle τ° to the subspace SsTpM {AsTpM}. 1.2.15 Remark. We have constructed various tensor bundles from the tangent bundle τΜ: TM -» Μ over M. Most of these constructions can also be carried out with a general vector bundle π: Ρ -» Μ over Μ. At the basis of these constructions is Lemma (1.2.12). For details see Bourbaki [2], §7. Here we mention only the bundle

S2(n):S2P^ Μ of symmetric bilinear forms with fibre isomorphic to S2 F, if the fibres of π are isomorphic to F. 1.2.16 Remark. We conclude with an alternative description of the tensor bundle τ,: TsrΜ -> Μ in the case where Μ is modelled on a space E of finite dimension.

Immersions and Submersions

23

From (1.0.1) we then have that

77 E = E ® . . . ® E ® E * ® . . . ® E * . r

s

An FeGL(E) determines

T;F = F® ... ®F®F V ® ... ®F V eGL(r/E) where Fv = *F~ l is the contragradient of F. With this we put 1J M = Tί'nM® ... ® 71 A/® 71* A/® ... ®f 71*Af. *»f0 f F /·

ί r

77 M is defined as the union of the Ts p M, p e M, and τ£ : Tsr Μ -> Μ is given by (TD * (/>) = Ts'p M. Finally, from an atlas (ux, Mx)aeA of M we construct the bundle atlas (Tsrux,ua,Mx)xeA with

given by

1.3 Immersions and Submersions In (1.0) 3 we characterized locally injective (surjective) differentiable mappings by their tangential at a point. Carrying this over to manifolds we obtain local immersions and submersions, cf. (1.3.3). Also of interest are the global versions of this property. In particular, we have the concept of a submanifold which appeared in the very beginning of Differential Geometry when Gauss developed the theory of surfaces in 3-dimensional Euklidian space IR 3 . Submanifolds possess a local characterization, cf. (1.3.5). They occur as counter image of submanifolds for a transversal (regular) differentiable mapping F: Μ -»· Ν, cf. (1.3.6), (1.3.8). Quadratic hypersurfaces and groups of matrices can, in this way, be seen to have a natural structure of a differentiable manifold, cf. (1.3.9). In (1.3.10) we introduce the important concept of an induced bundle. A first example is the normal bundle of an immersion, see (1.3.11). We conclude with a look at the structure of the tangent bundle of the total space of a vector bundle π : Ρ -> Μ over a manifold M. Here we point out that each tangent space Τξ Ρ contains a distinguished subspace, called the vertical space, i. e., the tangent space to the fibre containing ξ. We begin by reformulating the inverse mapping theorem (1.0.3). 1.3.1 Lemma. Let F: M-+ Ν be differentiable. If, for peM, TpF is a linear bijection from TPM onto TqN, q = F(p), then F is a local diffeomorphism. D

24

Foundations

1.3.2 Definition. Let F: M -+ N be differentiable. (\)F is called an immersion at p e M if TpF is injective and closed. It is called an immersion if Tp F is injective and closed for all peM. (ii) Fis called a submersion at p ifTpFis surjective. If Tp F is surjective for all p e M, then F is called a submersion. With this notation the corollaries (1.0.4) and (1.0.5) read: 1.3.3 Theorem. Let F: Μ ^>· Ν be differentiable. Consider p e M andput F(p) = q. If Fisan immersion {submersion} atp, then there exist charts (u, M') around p and (v, N') around q such that ν ο Fo u~ 1 : U = u(M') -+ V = v(N') = UxV2 is given by

u (-> (u, 0) [v°FoU-1: U = u(M') = U1 χ V^ V = v(N') is given by

u — (ui, ν) ι—> ν] . Π

Among the immersions, the following class is particularly important: 1.3.4 Definition, (i) F: Μ -> Ν is called an embedding if it is an immersion and if F: Μ ->· F(M) is a homeomorphism, where F(M) ^ N is endowed with the induced topology. (ii) If a subset Μ of a manifold TV can be given the structure of a differentiable manifold such that the inclusion i : Μ -»· TV is an embedding, then Μ is called a submanifold of N. For submanifolds we have the following characterization. Note that here our notation differs from the one used in (1.3.3), (1.3.4). 1.3.5 Theorem. A subset Ν of a manifold Μ is a submanifold if and only if the following holds: Let Ε be the model ofM. There exists a splitting Ε ·= Ej χ E 2 and, for every ρ e N, there exists a chart (u, M'} of Μ around ρ of the form C/2,(0,0)). Here, Uj is an open neighborhood of Oe Ej,j = 1,2, and u\(M'r\N) is given by Ul χ {0}.

Proof. Let TV be a submanifold, i:N -» Μ the inclusion. According to (1.3.3) there exists around peN a chart (u, M') such that U — u(M') = C/i x U2, MO j ο (Μ Ι Μ' r\ N)~ J : U± -> Ul χ {0} a linear injection. Conversely, if we have around each ρ e TV such a chart, then ι is an immersion. The topology of TV c Μ is generated

Immersions and Submersions

25

by the open subsets of M' n N which are homeomorphic to the open subsets £/! c E j via u\M' r^N. D 1.3.6 Definition. Let F: M -> W be differentiable. Let Λ^ be a submanifold of N. Fis called transversal to Nl at ρ if either q — F(p) φ Ν ι or else TqN is generated by TqNl and TpF(TpM). For a 1-form ω Φ Ο on Ν, F*co*Q if co\TqNl = Q. Notation: If F/KpNi for all p&M, then Fis called transversal to N^. Notation: Example. Λ^ = {q}. Then F/iA^ is equivalent to saying that, for every/? with F(/>) = q,TpF is surjective. In this case