Differentiable Manifolds: A First Course

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Lawrence Conlon Differentiable Manifolds A First Course

Birkhauser Boston Base1 Berlin

Lawrence Conlon Department of Mathematics Washington University St. Louis, MO 631 30-4899 USA

]Library of Congress Cataloging-in-Publiation Data Conlon, Lawrence,1933Differentiable manifolds : a first course / Lawrence Conlon. -- Boston ; Basel ;Berlin : Birkhguser, 1993 (Basler Lehrbticher, a series of advanced textbooks in mathematics; Vol. 5) p. cm. Includes bibliographical references. ISBN 0-8176-3626-9 (hard : acid-free). -- ISBN 3-7643-3626-9 (hard : acid-free) 1. Differentiable manifolds. I. Title. QA614.3.C66 1992 92-31098 CIP 5 16.3'64~20 Printed on acid-free paper O Birkhiuser Boston 1993 Copyright is not claimed for works of U.S. Government employees. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without prior permission of the copyright owner. Permission to photocopy for internal or personal use of specific clients is granted by Birkhiuser Boston for libraries and other users registered with the Copyright Clearance Center (CCC), provided that the base fee of $6.00 per copy, plus $0.20 per page is paid directly to CCC, 21 Congress Street, Salem, MA 01970, U.S.A. Special requests should be addressed directly to B W u s e r Boston, 675 Massachusetts Avenue, Cambridge, MA 02139, U.S.A. ISBN 0-8176-3626-9 ISBN 3-7643-3626-9 Typeset by the Author in AMS-uTEX. Printed and bound by Quinn-Woodbine, Woodbine, NJ. Printed in the U.S.A. 9 8 7 6 5 4 3 2 1

This book is dedicated to my wife Jackie, with much love.

TABLE OF CONTENTS

Preface

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

Acknowledgments

xi

... . . . . . . . . . . . . . . . . . . . . . . . xi11

.

.............

1

. . . . . . . . . . . . . . . .

1

. . . . . . . . . . . . . . . . . . . . . . .

3

Chapter 1 Topological Manifolds 1.1. Locally Euclidean Spaces

1.2. Manifolds

. . . . . . . . . .7

1.3. Quotient Constructions and 2-Manifolds 1.4. Partitions of Unity

. . . . . . . . . . . . . . . . . .

15

. . . . . . . . . . . . . .

18

. . . . . . . . . . . . . . . .

20

1.5. Imbeddings and Immersions 1.6. Manifolds with Boundary

Chapter 2. Local Theory

. . . . . . . . . . . . . . . . . 25

2.1. Differentiability Classes

25

2.2. Tangent Vectors

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

. . . . . . . . . .

34

2.4. Diffeomorphisms and Maps of Constant Rank

38

2.5. Smooth Submanifolds of Euclidean Space

. . . . . . . . . . . . . .

42

2.3. Smooth Maps and Their Differentials

2.6. Constructions of Smooth Functions 2.7. Smooth Vector Fields 2.8. Local Flows

. . . . . . . . . . . 47

. . . . . . . . . . . . . . . . . 50

. . . . . . . . . . . . . . . . . . . . .

55

. . . . . . . . . . .

64

2.9. Critical Points and Critical Values

.

Chapter 3 Global Theory

. . . . . . . . . . . . . . . . 67

3.1. Smooth Manifolds and Mappings

3.2. Diffeomorphic Structures 3.3. The Tangent Bundle

. . . . . . . . . . . . 67

. . . . . . . . . . . . . . .

73

. . . . . . . . . . . . . . . . . . 74

3.4. Cocycles and Geometric Structures

. . . . . . . . . . .

78

CONTENTS

........

3.5. Global Constructions of Smooth Functions

84

3.6. Manifolds with Boundary

87

3.7. Submanifolds

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91

3.8. Homotopy and Isotopy

93

3.9. Degree Theory

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

95

............

.

Chapter 4 Flows and Foliation

. . . . . 4.2. The Lie Bracket . . . 4.3. Commuting Flows 4.4. Foliations . . . . . . . 4.1. Complete Vector Fields

.

Chapter 5 Lie Groups

. . . .

. . . .

. . . .

. . . . . . . . . . . . 101 . . . . . . . . . . . . 107 . . . . . . . . . . . . 110 . . . . . . . . . . . . 116

................

5.1. Basic Definitions and Facts

101

127

. . . . . . . . . . . . . . 127

. . . . . . . . . . . . . 136 . . . . . . . . . . . . . . . . . . . 139

5.2. Lie Subgroups and Subalgebras 5.3. Closed Subgroups

. . . . . . . . . . . . . . . . . 145 . . . . . . . . . . . . . . . . . . . 150

5.4. Homogeneous Spaces 5.5. Principal Bundles

.

Chapter 6 Covectors and 1-Forms

...........

. . . . . . . . . . . . . 6.2. Line Integrals 6.3. The First Cohomology Space . . 6.4. Some Topological Applications .

. . . .

. 159 . 167 . 173 . 180

............

189

6.1. The Space of 1-Forms

.

Chapter 7 Multilinear Algebra

. . . . . . . 7.2. Exterior Algebra . . . . . . 7.3. Symmetric Algebra . . . . . 7.4. Multilinear Bundle Theory . . 7.5. The Module of Sections . . . 7.1. Tensor Algebra

. . . . .

. . . . . . . . .

. . . . . . . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

159

. . . . . . . . . .189 . . . . . . . . . . 199 . . . . . . . . . . 207 . . . . . . . . . . 209 . . . . . . . . . . 212

CONTENTS

ix

........

.

Chapter 8 Integration and Cohomology

221

. . . . . . . . . . . . . . . . 221 . . . . . . . . 228 8.2. Stokes' Theorem and Singular Homology 8.3. The Poincar6 Lemma . . . . . . . . . . . . . . . . . 243 8.4. Exact Sequences . . . . . . . . . . . . . . . . . . . 249 8.5. Mayer-Vietoris Sequences . . . . . . . . . . . . . . . 252 8.6. Computations of Cohomology . . . . . . . . . . . . . 257 8.7. Degree Theory . . . . . . . . . . . . . . . . . . . . 260 8.8. Poincar6 Duality . . . . . . . . . . . . . . . . . . .263 8.9. The de Rham Theorem . . . . . . . . . . . . . . . .268 8.1. The Exterior Derivative

.

Chapter 9 Forms and Foliations

............

277

. . . . . . . . . . . 277 9.2. The Normal Bundle and Transversality . . . . . . . . . 281 9.3. Closed, Nonsingular 1-Forms . . . . . . . . . . . . . . 286 9.1. The Frobenius Theorem Revisited

.

Chapter 10 Riemannian Geometry

...........

. . . . . . . . . . . . 10.2. Riemannian Manifolds . . . . . . . . . . . . . . . . 10.3. Gauss Curvature 10.4. Complete Riemannian Manifolds . . . 10.5. Geodesic Convexity . . . . . . . . 10.1. Connections

10.6. The Cartan Structure Equations 10.7. Riemannian Homogeneous Spaces

. . . . .

. . . . .

293

. . . . . . .294 . . . . . . . 302 . . . . . . . 306 . . . . . . . 314 . . . . . . . 327

. . . . . . . . . . . . 331

. . . . . . . . . . . 344

. . . . . . . . . . . . 349 . Appendix B . Inverse Function Theorem . . . . . . . . . . . 353 Appendix C. Ordinary Differential Equations . . . . . . . . 359 Appendix A Vector Fields on Spheres

C.1. Existence and uniqueness of solutions C.2. A digression concerning Banach spaces

. . . . . . . . . . 360

. . . . . . . . . 362

CONTENTS

C.3. Smooth dependence on initial conditions C.4. The Linear Case

. . . . . . . . . . . . . . . . . . .366

. Appendix E. de Ftharn-cech

Appendix D Sard9sTheorem

E.1. ~ e c cohomology h

. . . . . . . . . . . . . . . . 367

. . . . . . . . . . . . 371 . . . . . . . . . . . . . . . . . . . 371 Theorem

E.2. The de ~ h a m x e c hcomplex Bibliography Index

. . . . . . . . 364

. . . . . . . . . . . . . . 375

. . . . . . . . . . . . . . . . . . . . . . . . .383

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 385

PREFACE

This book is based on the full year Ph.D. qualifying course on differentiable manifolds, global calculus, differential geometry, and related topics, given by the author at Washington University several times over a twenty year period. It is addressed primarily to second year graduate students and well prepared first year students. Presupposed is a good grounding in general topology and modern algebra, especially linear algebra and the analogous theory of modules over a commutative, unitary ring. Although billed as a "first course", the book is not intended to be an overly sketchy introduction. Mastery of this material should prepare the student for advanced topics courses and seminars in differential topology and geometry. There are certain basic themes of which the reader should be aware. The first concerns the role of differentiation as a process of linear approximation of nonlinear problems. The well understood methods of linear algebra are then applied to the resulting linear problem and, where possible, the results are reinterpreted in terms of the original nonlinear problem. The process of solving differential equations ( 2 . e., integration) is the reverse of differentiation. It reassembles an infinite array of linear approximations, resulting from differentiation, into the original nonlinear data. This is the principal tool for the reinterpretation of the linear algebra results referred to above. It is expected that the reader has been exposed to the above processes in the setting of Euclidean spaces, at least in low dimensions. This is what we will refer to as local calculus, characterized by explicit computations in a fixed coordinate system. The concept of a "differentiable manifold" provides the setting for global calculus, characterized (where possible) by coordinate-free procedures. Where (as is often the case) coordinate-free procedures are not feasible, we will be forced to use local coordinates which vary from region to region of the manifold. When theorems are proven in this way, it becomes necessaxy to show independence of the choice of coordinates. The way in which these local reference frames fit together globally can be extremely complicated, giving rise to problems of a topological nature. In the global theory, geometric topology becomes an essential feature.

xii

PREFACE

These themes of linearization, (re)integration, and global versus local will be emphasized repeatedly. Although a certain familiarity with the local theory is presupposed, we will try to reformulate that theory in a more o r g a n i d and conceptual way that will make it easier to treat the global theory. Thus, this book includes a modem treatment of the elements of multivariable calculus. Fundamental to the global theory of differentiable manifolds is the concept of a vector bundle. The easiest nontrivial example is the tangent bundle to the n-sphere which we introduce from a purely topological point of view in Section 1.2. The subtleties involved in this bundle are illustrated in the discussion of the vector field problem on spheres in Section 1.2 and Appendix A. As the global theory is developed, the tangent bundle, the cotangent bundle, various tensor bundles, and the associated (principal) frame bundles will play increasingly important roles, as will the related notions of infinitesimal G-structures and integrable G-structures. For conceptual simplicity, all manifolds, functions, bundles, vector fields, Lie groups, homogeneous spaces, etc., will be smooth of class Coo. It is possible to adapt the treatment to smoothness of class Ck, 1 5 k < 00,but the technical problems that arise are distracting and the usefulness of this level of generality is limited. On the other hand, in much of the literature, the study of Lie groups and homogeneous spaces is carried out in the real analytic (CW)category. In these treatments, it is customary to note that C" groups can be proven to be analytic, hence that no generality is lost. It seems to the author, however, that nothing would be gained by this approach and that the ideal of keeping this book as self contained as possible would be compromised. Finally, the proofk of certain theorems have been relegated to appendices. Two of these, the inverse function theorem and the uniqueness, existence, and smoothness of solutions of systems of ordinary differential equations, are fund* mental and will be used repeatedly. Their proofs, however, are relatively long, add little insight, and can be distracting in a limited set of lectures. The remaining appendices establish results that are appealed to in the book, but are somewhat less central. These are Sard's theorem, the de R.ham-&~h cohomology theorem, and the Radon-Hurwitz-Eckmann construction of independent vector fields on spheres.

ACKNOWLEDGMENTS I would like to thank Robby Gardner and his students at Chapel Hill who "beta tested" eight chapters of a preliminary version of this book in an intensive, onesemester graduate course. Their many suggestions were most helpful in the find revisions. Others whose input was helpful include Gary Jensen, Alberto Candel, and Nicola Arcozzi. Finally, I particularly want to thank Filippo De Mari, whose beautiful class notes, written when he was one of my students in an earlier version of the course, were useful in subsequent revisions and first suggested to me the idea of writing this book.

CHAPTER 1 Topological Manifolds

This chapter pertains to the global theory of manifolds. See also [3, Chapter I] and [38, Chapter 11.

1.1. Locally Euclidean Spaces Classical analysis is carried out in Euclidean space, the operations being defined by local formulas. One might hope, therefore, to extend this classical theory to all topological spaces that are locally Euclidean. While this is not generally possible without further restrictions on the spaces, the locally Euclidean condition is fundamental. 1.1.1. A topological space X is locally Euclidean if, for every DEFINITION x E X, 3 n 2 0 (an integer), an open neighborhood U C X of x, an open subset W & Rn and a homeomorphism cp : U + W .

If we can show that n is uniquely determined by x, we will write n = d ( x ) and call this the local dimension of X at x.

EXAMPLE1.1.2. Any open subset X R n is a locally Euclidean space that is also Hausdorff and 2nd countable. We will see that the local dimension is n at every x E X. EXAMPLE 1.1.3. Let X = R LI {*I, where * is a single point and 11 denotes disjoint union. Topologize this set so that a basis of open subsets V c X consists of the following: a If * 4 V , then V is open as a subset of R. If * E V, then 0 4 V and there is an open neighborhood W c R of 0 such that V = (W \ (0)) U {*I. This space is locally Euclidean and znd countable. It is not Hausdorff since every open neighborhood of * meets every open neighborhood of 0. In this case, the local dimension is everywhere 1, even a t * and at 0.

-

EXAMPLE 1.1.4. In [38, Appendix A], there is described a bizarre space called the long line. It is connected, Hausdorff, and locally Euclidean with d(x) 1,

2

1. TOPOLOGICAL MANIFOLDS

but it is not 2nd countable. In fact, the long line contains an uncountable family of disjoint, open intervals. The following difficult result, known as the Brouwer theorem on invariance of domain, is needed in order to show that local dimension is always well defined on locally Euclidean spaces. The proof will not be given. It is best carried out by the methods of algebraic topology [9, p. 3031, [37, p. 1991, [12, p. 1101. For amore classical proof, see (20, pp. 95-96]. In the theory of smooth manifolds, differential calculus reduces the appropriate analogue of this theorem to elementary linear algebra.

THEOREM 1.1.5 (BROUWER).If U & Rn is open and f ous and one to one, then f (U) is open in Rn.

: U -,Rn

is continu-

COROLLARY 1.1.6. If U C Rn and V & Rm are open subsets such that U is homeomorphic to V, then n = m .

# n, say, m < n . Define i : Wm i(xl,. . . ,xm) = (x1 , .. . ,xm,0,. . . ,O).

PROOF.Assume that m

L,

Rn by

v n-m

This map is continuous and one to one and i(Rm) is not open in Rn and does not even contain a subset that is open in Rn. By assumption, there is a h o m m morphism cp : U --, V, so the composition

is continuous and one to one. Also, U Rn is open, but f (U) = i(V) C i(Rm) cannot be open in Rn. This contradicts Theorem 1.1.5. [7 COROLLARY 1.1.7. If X is locally Euclidean, then the local dimension is a well defined, locally constant function d : X -,Z+.

PROOF. Let x E X and suppose that there are open neighborhoods U and V of x in X, together with open subsets @ E Rn, Rm, and homeomorphisms

Since V n U is open in X , it follows that

are inclusions of open subsets and, similarly, that $(V

n U) is open in Rm. But

is a homeomorphism, so m = n by the previous corollary. All assertions follow. I7 COROLLARY 1.1.8. If X is a connected, locally Euclidean space, then the local dimension d : X -,Z+ is a constant called the dimension of X .

1.2. MANIFOLDS

3

EXERCISES (1) Prove that each connected component of a locally Euclidean space X is an open subset of X. (2) Prove that a connected, locally Euclidean space X is path connected. (3) Give an example of a connected, 2nd countable, Hausdorff space that is not path connected. (4) Let X and Y be connected, locally Euclidean spaces of the same dimension. If f : X + Y is bijective and continuous, prove that f is a homeomorphism. (5) RRcall that a regular space X is one in which any proper, closed subset C c X and point x E X \ C can be separated by disjoint, open neighborhoods of each. Prove that a locally compact (in particular, a locally Euclidean), Hausdorff space must be regular.

1.2. Manifolds Some authors call any locally Euclidean space a manifold. It is more common, however, to require more. DEFINITION 1.2.1. A topological space X is a manifold of dimension n (an n-manifold) if (1) X is locally Euclidean and d(x) G n = dim X; (2) X is Hausdorff; (3) X is 2nd countable. Of the three examples in the previous section, only the open subsets of R n were manifolds. LEMMA1.2.2. If X is a compact, connected, rnetrizable space that is locally Euclidean, then X is an n-manifold, for some n E Z+. PROOF. Indeed, X is Hausdorff because it is metrizable. It is 2nd countable because it is locally Euclidean and compact. Being locally Euclidean and connected, X has constant local dimension. Here are some examples of manifolds.

I llvll = 1) is an n-manifold. EXAMPLE 1.2.3. The n-sphere Sn = {v f One way to see that it is locally Euclidean is by stereographic projection. Let p+,p- E Sn be the north and south poles, respectively. Then the stereographic projections (see Figure 1.1)

are homeomorphisms and {Sn \ { p - ),Sn \ {p+ )) is an open cover of Sn. Since Sn is compact and metrizable, it is an n-manifold.

1. TOPOLOGICAL MANIFOLDS

FIGURE1.1. Stereographic projection from p +

EXAMPLE 1.2.4. If N is an n-manifold and M is an m-manifold, then N x M is an (n m)-manifold. Indeed, if (x, y) E N x M, let U be a neighborhood of x in N homeomorphic to an open subset of Rn and V a neighborhood of y in M homeomorphic to an open subset of Rm. Then, the neighborhood U x V N x M of (x, y) is homeomorphic to an open subset of Rn x Rm = Rn+m. Since M and N are Hausdorff and 2nd countable, so is N x M.

+

EXAMPLE 1.2.5. The n-torus 7

n factors

is an n-dimensiond manifold. Indeed, by Example 1.2.3, S1 is a 1-manifold and Example 1.2.4, applied successively, then implies that Tn is an n-manifold.

EXAMPLE 1.2.6. A vector w E Itn+'is defined to be tangent to Sn at v E Sn if w I v. This conforms to naive geometric intuition and can be seen to conform to the general definition of tangent vectors to differentiable manifolds (cf. Exercise (5) on page 38). In order to keep track of the point of tangency, we will denote this tangent vector by (v, w) E RnS1 x Rn+'. Thus, the set of all tangent vectors to Sn is T(Sn) = {(v, w)E Rn+' x Rnfl

I IIvII

= 1,w

Iv } .

This space is topologized as a subspace of Rn+l x Rn+l. We also define the continuous map p : T(Sn) + Sn by p(v, w) = v . Thus, p assigns to each tangent vector its point of tangency. For each vo E Sn, consider

1.2. MANIFOLDS

5

the set of all vectors tangent to Sn at vo. This is an n-dimensional vector space under the operations

This structure, p : T(Sn) -, Sn,is called the tangent bundle of Sn. The space T(Sn) is the total space of the bundle, the space Sn is the base space of the bundle, and p is the bundle projection. By a common abuse of terminology, the total space is often referred to as the tangent bundle itself. In Exercise (3),you are going to prove that T(Sn) is a 2n-manifold. DEFINITION 1.2.7. If U Snis an open subset, then T(Sn)IU = T(U) is the space p-l(U). Then p : T(U) + U, the restriction of p to T(U) T(Sn), is called the tangent bundle of U. DEFINITION 1.2.8. If U E Snis open, a vector field on U is a continuous map s : U -,T(U) such that p o s = idv.

A thorough understanding of the tangent bundle of Sn eluded topologists for several decades. For instance, it was long unknown what is the maximum number r(n) of vector fields s, : Sn -, T(Sn), 1 5 a 5 r(n), which are everywhere linearly independent. That is, we require that, for each u E Sn, the vectors {sl (v), . . . , s ~ (u)) ( ~ be ) linearly independent in T. (Sn) and that no set of r(n) +1 fields has this property. The problem of computing r(n) was known as the "vector field problem for spheres". DEFINITION 1.2.9. The sphere Sn is parallelizable if r(n) = n. This brings us to a striking example of global versus local properties. If Sn is parallelizable, Exercise (3) will imply that T(Sn) 2 Sn x Rn. For general n, this same exercise will show that the tangent bundle T ( S n ) is locally a Cartesian product of an open set U C Sn with Rn, but it is only globally such a product when Sn is parallelizable. Not every sphere is parallelizable. For instance, it has long been known (and will be proven in Section 8.7) that r (2n) = 0. This means that every vector field on S2"is somewhere zero. In the case of S2,this is sometimes stated facetiously as "you can't comb the hair on a billiard ball". It was also known for some time that S1,S3,and S7are parallelizable. The following was finally proven in the late 1950s [4], [27].

THEOREM 1.2.10 (BOTT AND MILNOR,KERVAIRE).The sphere Sn as p a d lelizable if and only zf n = 0,1,3, or 7. The case n = 0 is the trivial fact that the O-sphere So= {kl) c R admits 0 independent fields. There is an interesting relationship between Theorem 1.2.10 and the problem of defining a bilinear multiplication on R n without divisors of zero. Such a multiplication is a bilinear map

6

1. TOPOLOGICAL MANIFOLDS

written p(u, w) = vw, such that uw

=0 H

u = 0 or w = 0.

THEOREM 1.2.11. There is a multiplication on Wn+' without divisors of zero if and only if n = 0,1,3, or 7. Indeed, R1 = R, R2 = C, and W4 = W (the quaternions). The multiplication on W8 is given by the Cayley numbers, a nonassociative division algebra whose elements are ordered pairs (x, y) of quaternions [40, pp. 10&109]. This proves the "if" in Theorem 1.2.11. Conversely, such a multiplication can be used to construct n everywhere independent vector fields on Sn (Exercise (6)), so the "only if" part of the theorem is given by Theorem 1.2.10. The full solution to the vector field problem for spheres was given by Frank Adams in the early 1960s [I], culminating a long history of research on that problem by several algebraic topologists. We state Adam1 result. Define the function p(n), n 2 1, by requiring that Sn-' admit p(n) - 1 everywhere linearly independent vector fields, but not p(n) such fields (thus, r(n) = p(n 1) - 1). Write each natural number n uniquely as

+

where r, c, d are nonnegative integers and c 5 3. This uses the unique factorization theorem and the division algorithm mod 4.

Remark that p(odd) = 1, since c = d = 0. This gives the classical result that every vector field on an even dimensional sphere is somewhere zero. The easier part of Adams' theorem is that Sn-' does admit at least 2C 8d - 1 independent vector fields. The original proof, using only linear algebra, was given by W o n and Hurwitz and, in 1942, an improved proof was given by Eckmann [8]. A detailed outline of a modern proof will be found in Appendix A. The harder part of the theorem, that there are at most 2C 8d - 1 such fields, is too advanced for this book.

+

+

EXERCISES (1) In Example 1.2.3, write down formulas for the stereographic projections .~rhand prove carefully that they are homeomorphisms. (2) Given vo E Sn, show that there is an open neighborhood U C Sn of uo and vector fields s j : U -,T(U), 1 5 i 5 n, such that

is a basis of the vector space Tu(Sn),V v E U.

(3) Let U c Sn be as in the previous exercise. Using that exercise, construct a continuous bijection cp : U x Rn -, T(U) and prove that p is a homeomorphism. (This is not very deep. You do not, for instance, need Theorem 1.1.5.) Using this, prove that T(Sn) is a 2n-manifold. Prove also that, for each u E U, the formula pv(w) = cp(v, w) defines an isomorphism cpu : Rn + Tv(Sn) of vector spaces.

1.3. QUOTIENT CONSTRUCTIONS

7

(4) Using Theorem 1.2.12, show that p(n) = n if and only if n = 1,2,4, or 8. This gives back Theorem 1.2.10. (5) Without appeal to Theorem 1.2.12, show that p(2) = 2. (6) If Rn+I admits a multiplication without divisors of zero, prove that Sn is pardelizable. 1.3. Quotient Constructions and 2-Manifolds

We continue the project of constructing manifolds. In this section, examples will be constructed using the quotient topology. Before giving a careful definition of quotient spaces, we look a t some intuitive examples of 2-manifolds constructed in this way. Consider the square D = [O, 11 x [0, 11. In Figure 1.2, the arrows indicate that the opposite sides of D are to be glued together so that the vertical edges are glued bottom to top and the horizontal edges from left to right. When the first pair of edges are glued, the result is a cylinder. When the second pair are glued, the cylinder becomes the 2-torus T 2= S1 x S1.

FIGURE 1.2. The 2-torus

In Figure 1.3, the vertical edges of D are identified in the same way, but the top and bottom are glued together in opposite senses. If the top and bottom are identified first, the result is a Mobius strip. It is not very easy, then, to picture the rest of the identification. If the left and right are identified first, the result is a cylinder whose boundary circles are then to be identified with an orientation reversing "flip". The resulting 2-manifold is called the "Klein bottle" K and can only be pictured in R3 if one allows the surface to intersect itself. Figure 1.4 is an attempt at such a picture, the self-intersection occurring along a circle. This circle of intersection corresponds to the horizontal line L and the circle c in Figure 1.3. In order to view K 2without self intersection, it is necessary to situate it in a 4dimensional framework. This is not as psychologically hopeless as it might seem.

1. TOPOLOGICAL MANIFOLDS

FIGURE 1.3. The Klein bottle

FIGURE 1.4. K~ with self-intersection One can, for example, color K~ by shades of grey, varying continuously over the Klein bottle, in such a way that the circle c in Figure 1.3 has no points with the same shade as any point in the line 1. By continuously assigning numbers from 0 to 1 (lightest to darkest) t o these shades, one introduces a fourth "dimension" and the shaded Klein bottle has no self intersections. What was the circle of intersection is now two disjoint shaded circles l and c. One can think of the shaded Klein bottle as a topologically imbedded Klein bottle in R4. In Figure 1.5, we identify each pair of opposite sides of D with a reverse of orientation. The resulting 2-manifold is called the projective plane P and, once again, it cannot be imbedded in R3.The topologist (but not the geometer) can view D as the unit disk D 2 = {v E R2 1 llvll 5 1 ) in such a way that the gluing identifies antipodal pairs of boundary points. That is, if J J v J=J 1, then v and -v are identified. Another way to think of P2 is to start with S2 C IR3 and to identify the antipodal pairs { w ,-w). To see that this also yields P2,first carry out the identification for the antipodal pairs not lying on the equatorial circle z = 0. The resulting disk has the equatorial circle as boundary and the remaining identifications give the previous description of P 2. DEFINITION 1.3.1. Let MI and M2 be 2-manifolds and let Di C Mi be imbedded disks, i = 1,2. Let Mi' = Mi \ int(Di), i = 1,2, and glue these together

1.3. QUOTIENT CONSTRUCTIONS

FIGURE 1.5. The projective plane by a homeomorphism of a M i = aDl to aMi = a D 2 . The resulting 2-manifold Ml # M2 is called the connected sum of M I and M2. When this definition has been put on a rigorous setting, it can be shown that the connected sum is well defined up to homeomorphism. Thus, strictly speaking, the symbols M I , M2, and MI # M2 should denote homeomorphism classes of 2-manifolds. With this understanding, connected sum can be seen to be commutative, Ml#M2 = M2#Ml, associative, (Ml ttM2) #M3 = M l # (M2 # M 3 ) , (hence parentheses can be dropped) and to admit (the homeomorphism class of) S2 as a 2sided identity

In this way, the set of (homeomorphism classes of) surfaces becomes an abelian semigroup. Of particular interest is the subsemigroup of compact, connected surfaces. THEOREM1.3.2. The semigroup of compact, connected 2-manifolds is generated by T~ and P 2 . Indeed, every element other than the identity S 2 can be uniquely written as a connected sum of finitely many copies of T 2 or of finitely many copies of P 2 . This is the classification theorem for compact, connected surfaces. It is a classical result. For a proof of this classification, see Massey [26, Chapter 11. He also shows that

10

1. TOPOLOGICAL MANIFOLDS

Thus, the connected sum operation does not have cancellation. An important tool in the proof of Theorem 1.3.2 is a "triangulation" of a compact surface M. Tkiangulations are useful for many purposes, so we give a brief discussion. Let A c It2 be the convex hull of the points vo = (0, O), vl = (1,0), and v2 = (0,l). That is,

a closed, triangular region in the plane having vertices (vo, vl, v2) and edges eo = {(x, y E A) I x + y = 1) el = ((0, y) E A) e2 = {(x, 0) E A). The triangle A C Kt2 is called the "standard Z-simplex". One decomposes M into pieces Al ,A2, . . . ,A,, together with homeomorphisms

in such a way that any two of the triangles Ai, Aj are either disjoint, have in common just one vertex v i ( V t ) = vj (vk), or have in common just one edge Pi(et) = vj(ek). Such a decomposition is called a triangulation of M. THEOREM1.3.3 ( R A D ~ )Every . compact surface M admits a triangulation. Equivalently, M can be constructed, up to homeomorphism, by taking finitely many copies of the standani 2-simplex A and gluing them together appropriately along edges. This theorem is intuitive but nontrivial [31, pages 58-64]. The first proof was given by T. Rad6 [36]. The standard 2-simplex is "oriented" by a choice of ordering of its three vertices. Two orderings give equivalent orientations if they differ by an even permutation. Thus, up to equivalence, there are two orientations of A. Such an orientation induces a direction along the edges of A, leading to the terminology "clockwise orientation" and "counterclockwise" orientat ion. The standard orientation of A, given by the ordering (vo, vl ,v2), is the counterclockwise orientation. Note that the homeomorphism a : A + A, defined by a(x, y) = (y, x) is orientation reversing. Triangulations give us a way of defining the notion of "orientability" for a compact surface. The idea is that a triangle A, = vi(A) has the standard orientation (9, (vo), vi(vl), 9, ( ~ 2 ) ) .TWOtriangles Ai and A j that have a common edge e are coherently oriented if their standard orientations induce opposite directions along e. A little thought should convince the reader that this is the natural notion if we are to picture the orientations of both A and A j as counterclockwise. If A, and A j have no edge in common, they are also said to be coherently oriented. Given a triangulation of a connected surface M , one can attempt to make all the orientations coherent as follows. Starting with one triangle, say A 1, look at any triangle, say A2, having an edge in common with A1. If they are coherently oriented, well and good. If not, replace 9 2 with pa o a, making them

1.3. QUOTIENT CONSTRUCTIONS

Vl

FIGURE 1.6. A triangulation of S2 coherently oriented. Continuing in this way either orients the triangulation or leads to conflicting orientations on at least one simplex. In the first case, the triangulation is said to be orientable and, in the second, to be nonorientable. If M has more than one component, each must be treated separately and the triangulation of M is orientable if and only if the triangulation of each component is orientable. It can be proven that some triangulation of M is orientable if and only if every triangulation of M is orientable. DEFINITION 1.3.4. We say that M is orientable if some, hence every, triangulation of M is orientable. EXAMPLE1.3.5. The sphere S2 is orientable. Indeed, define the standard h i m p l e x A3 c W3 to be the convex hull of the set of points

That is, A ~ = { ( X , ~ , IZx), y , z ~ O a n d x + y + z 6 1 ) . Any three of the vertices {vo,vl, v2, v3) correspond to a triangular face of A3. These four triangular faces unite to form a surface homeomorphic to S', defining thereby a triangulation of S2 (Figure 1.6). Orienting these faces by (vl ,v2, us), (vO,v2, vl), (213, 212, vo), and (us, vo, vl ), respectively, gives coherent orientations to all the triangles. An interesting invariant of surfaces is the Euler characteristic. If the compact surface M has a triangulation 7, let V denote the number of points of M that are vertices of the triangulation, E the number of arcs in M that are edges of the triangulation, and F the number of triangular faces.

+

THEOREM1.3.6. The number F - E V depends only on M , not on the choice of triangulation 7. This number is called the Euler characteristic of M and is denoted x(M).

1. TOPOLOGICAL MANIFOLDS

12

This theorem has many proofs. One method, useful only for smoothly imbedded surfaces in R3, is to use the Gauss-Bonnet theorem, which asserts that the integral of the Gauss curvature over M is 27rx(M) ([16, p. 1111, [34, pp. 380-3821, [41, pp. 237-2391). EXAMPLE1.3.7. In the triangulation of S2given in Figure 1.6, V = 4, E = 6, and F = 4, so X(S2)= 2. In Exercise (4), you are invited to show directly that an arbitrary triangulation of S2always has F - E V = 2.

+

Our discussion of the topology of 2-manifolds belongs to the "cut and paste" brand of topology. To put such constructions on a rigorous footing, it is necessary to introduce quotient spaces. Let X be a topological space and let be an equivalence relation on X . That is, xNx,vxEX x-y*y-x x~yandy-Z+X-2. This relation partitions X into a collection {X,),Em of disjoint subspaces, the equivalence classes of -.

-

-

DEFINITION 1.3.8. The set {X,),fm of equivalence classes of X is called the quotient space of X modulo

and is denoted by X/-.

The surjection

is the map that assigns to each x f X its equivalence class ~ ( x E) X/-. A subset U & X/- is said to be open if and only if 7r-'(U) is open in X. It is trivial to check that these open sets constitute a topology on the quotient space X/-. This is the quotient topology, characterized as the largest (i. e., the finest) topology on XI- relative to which the canonical projection

is continuous.

-

EXAMPLE 1.3.9. Let X = W x (0, I), topologized as the disjoint union of two copies of R. Define an equivalence relation on X by setting (x,a) (y,P) if and only if either a = ,d and x = y, or a # P and x = y # 0. Thus {(O,O)) and {(0,1)) each constitute a distinct equivalence class, but all other classes are pairs { (x, 0) , (x, 1)) where x # 0. The quotient space X/- is the non-HausdorfT, locally Euclidean space given in Example 1.1.3. As the above example shows, even if X is Hausdorff, the quotient X/- may fail to be Hausdorff. LEMMA1.3.10. If the space X is compact, so is X/-.

PROOF.The continuous image 7r(X) = X/- of a compact space X is compact. LEMMA1.3.11. If the space X is connected, so is X/-o

1.3. QUOTIENT CONSTRUCTIONS

13

PROOF. The continuous image 7r(X) = XI- of a connected space X is connected. [3 DEFINITION 1.3.12. A map f : X -* Y respects an equivalence relation X if x y + f (x) = f (y). In this case, the induced map

-

on

is well defined by f(r(x))= f ( 4 .

-

LEMMA1.3.13. Let X and Y be topological spaces, let be an equivalence relation on X, and let f : X -* Y be a map respecting this equivalence relation. Then f is continuous i f and only i f f is continuous.

PROOF.Consider the commutative diagram

Since f = f o 7r, continuity of f implies continuity of f. For the converse, assume that f is continuous, hence that f -'(U) is open in X whenever U is open in Y. But this implies, via the commutative diagram, that

is an open subset of X. By the definition of the quotient topology, F 1 ( U ) is open in X/w, so f is continuous.

EXAMPLE 1.3.14. Let X = [-I, 11 and let Y

= S'

c @.

Define

-

by f ( t )= eint, a continuous surjection. It is not quite one to one since f (1) = f (-1). On X , define the equivalence relation x y by requiring either that x = y or that {x, y) = {f1). Clearly, f respects this relation. Denote X/- by [-I, I]/{& 1) and remark that is bijective. By Lemma 1.3.13, f, is also continuous. But [- 1,1]/{&1) is compact

by Lemma 1.3.10 and S1 is Hausdod. A one to one, continuous map from a compact space onto a Hausdod space is a homeomorphism, so f gives a canonical homeomorphism [-I, I]/{& 1) E S1. Generally, if A X , one can define the equivalence relation N A by writing x - A y if and only if either x = y or x, y E A. The quotient space X/ .v A is thought of as the result of crushing A to a single point in X and will be denoted by X/A. Some care should be taken in using this notation. If X = G is a topological group and H Z G is a subgroup, then G/H denotes the space of left cosets of H, not the space obtained by collapsing H alone to a point.

14

1. TOPOLOGICAL MANIFOLDS

-

The context should make clear which interpretation is intended. Remark that G / H is also a set of equivalence classes, the relation being g 1 92 if and only if g;'g2 E H. Thus, G / H can be given the quotient topology and the continuous map T : G 4 G / H is ~ ( g=)g H .

DEFINITION 1.3.15. If G is a topological group and H E G is a subgroup, the quotient space G / H is called the (left) coset space of G mod H.

EXERCISES (1) Let Dn = {v E W n 1 1 1 ~ 1 1 5 I), the unit n-disk (also called the closed n-ball) with boundary aDn = Sn-I . Prove that Dn/Sn-l is h o m e morphic to Sn. (We proved the case n = 1 in Example 1.3.14.) (2) View W n as an abelian group under vector addition. This is a topological group. The integer lattice

is a (normal) subgroup of W n . Let T n = Rn/Zn be the coset space (actually, a group). Prove that this space is homeomorphic to

our definition of the n-torus in Example 1.2.5. Use this to show that the surface constructed by Figure 1.2 is, indeed, T2. (3) Define an equivalence relation on S n C Etn+l by writing v w if and only if v = f w . The quotient space P n = S n / - is called projective n-space. (This is one of the ways that we defined the projective plane P2.) The canonical projection .rr : Sn + Pn is just ~ ( v = ) {fv). Define Ui c P n , 1 5 i 5 n 1, by setting

-

+

Prove (a) U, is open in P n . (b) {Ul, - . - , Un+1) covers Pn. (c) There is a homeomorphism cp, : Ui + Rn. (d) Pn is compact, connected, and Hausdofl. This proves that P n is an n-manifold. (4) Give an intuitive, but completely convincing, proof that V - E F = 2 for every triangulation of S 2 . (5) Produce triangulations of the surfaces T2,K2, and P 2 . Use these to prove that T 2is orientable and that K 2 and p2 are not. Use T h e rem 1.3.2 to determine all orientable, compact, connected surfaces and all nonorientable ones. (6) For compact, connected surfaces M1 and M2, prove that

+

1.4. PARTITIONS OF UNITY

15

Use this, Theorem 1.3.2 and Exercise (5) to compute the Euler characteristics of all compact, connected surfaces. 1.4. Partitions of Unity

Partitions of unity play a crucial role in manifold theory. Their existence is a consequence of the fact that manifolds are paracompact. In this section, we establish these facts. For further information about paracompact spaces, the reader is referred to [7, pp. 162-1691. DEFINITION 1.4.1. A family C = {C,),Em of subsets of X is locally finite if each x E X admits an open neighborhood W, such that W, n C, # 0 for only finitely many indices cu E ?2l. The proof of the following useful lemma is left as an exercise (Exercise (1)).

LEMMA1.4.2. X , then

If C = {C,),E~ is a locally finite

UaEa C, is a closed subset of X .

family of closed subsets of

DEFINITION 1.4.3. Let U = {Ua},Era and V = {Vp)pEg be open covers of a space X. We say that V is a refinement of U if there is a function i : 93 -+ % such that Vp U,(p), V P E 93. DEFINITION1.4.4. A HausdodT space X is paracompact if it is regular and if every open cover of X admits a locally finite refinement. Actually, it is redundant to require regularity, but it will shorten some arguments. By Exercise (5) on page 3, locally compact Hausclod7 spaces, hence manifolds, are regular. DEFINITION 1.4.5. Let U = (Ua)aEll be an open cover of a space X . A partition of unity, subordinate to U, is a collection A = {A,),,% of continuous functions A, : X + [O,11 such that (1) supp(A,) c U,, Vcu E 2l (where the support supp(A,) is the closure of the subset of X on which A, # 0); (2) for each x E X , there is a neighborhood W, of x such that A, IW, $ 0 for only finitely many indices cy E iZL; (3) the sum C,,, A,, well defined and continuous by the above, is the constant functxon 1.

Our goal is to prove that open covers of manifolds admit subordinate partitions of unity. This will require a proof that manifolds are paracompact, a property that could fail had we not required manifolds to be second countable. LEMMA1.4.6. If X is pamwmpact and l.4 = {U,)aEQ is an open cover of X , there is a locally finite refinement V = {Vp}pEg, i : 23 + %, such that Vp C ui(p), V P E 93-

PROOF.Indeed, paracompact spaces are regular, so it is easy to find a refinement W = {Wr)xsa and associated mapping j : R + %, such that W,C Uj(,), Vrc E A. Passing to a locally finite refinement of W gives a refinement of U with the desired properties.

1. TOPOLOGICAL MANIFOLDS

Recall that a topological space X is said to be normal if, whenever A, B c X B = 0. are closed, disjoint subsets, there is an open set U 3 A such that

an

LEMMA1.4.7. If X is paracompact, it is normal.

PROOF.Let A, B c X be closed, disjoint subsets. The space X being regular, there is a family {U,),Ea of open subsets of X , covering A and such that U, c X \ B,V a E ?2l. The space X being paracompact, the open cover of X , obtained by adjoining X \ A to {U,),Em, has a locally finite refinement. Thus, we lose no generality in assuming that {Ua)aEa is a locally finite family of open sets. If x E M, an open neighborhood W of x meets U, if and only if W meets U,, so {Va},Ea is also locally finite. Set U = U,, U,, an open neighborhood of A. By Lemma 1.4.2, U = U,,% U, and this set does not meet B. In the construction of partitions of unity, we will use the following well known property of normal spaces [7, pp. 146-1471. We do not give the proof here since, ultimately, our interest is in the Cooversion which we will prove later (Corollary 3.5.5). THEOREM1.4.8 (URYSOHN'S LEMMA).If X is a normal space and A, B c X are closed, disjoint subsets, then there is a continuous function f : X + [0,1) such that f lA 1 and supp(f) C X \ B.

THEOREM 1.4.9. Every locally compact, grid countable Hausdorfl space X is paracompact. PROOF. By Exercise (5) on page 3, X is regular. Next, we observe that there is a countable, increasing nest

of compact subsets of X such that

Indeed, let ( W i ) z l be a countable base of the topology of X such that each -

W i is compact. We set K t = and, assuming inductively that K j has been defined, 1 5 j 5 r , we let l denote the least integer such that

and set

e+i

K,+i

=

UITi.

i=l

Let U = {U,),Ea be an open cover of X . We select a refinement as follows. We can choose finitely many Vi = Uai f U, 1 5 i 5 L1, that cover the compact set K l . Extend this by {u,~ ):!-el+1 to an open cover of K 2 . Since X is Hausdorff,

1.4. PARTITIONS OF UNITY

17

< <

the compact set K 1 is closed, so V , = Uai \ K1 is open, Lz + 1 i L2, and {V.);Z,is an open cover of K z . We have arranged that K1 does not meet V,, i > el. Proceeding inductively, we obtain a refinement V = {Vi)zl of U with the property that K, only meets finitely mimy elements of V, V r 2 1. Given x E X , choose r 2 1 such that x E int(K,), a neighborhood of x which only meets finitely many elements of V. IJ

COROLLARY 1.4.10. Every manifold is paracompact. The existence of partitions of unity can now be established. THEOREM1.4.11. If X is a paracompact space and U = {Ua)aEll is an open cover of X, then there exists a partition of unity subordinate to 24.

PROOF.Choose a locally finite refinement V = {VP)BEg of U, with associated mapping i : B + Q. By Lemma 1.4.6 we can choose a locally finite refinement W = {W,},Er of V, with associated mapping j : R + 93,such that W,c V K E a. For each K E R, use Theorem 1.4.8 to define a continuous function 7, : X [O, 11 such that -+

Given CY E M, let A, = { K E A I a = i ( j ( ~ ) ) } a, possibly empty set of indices. , the union of these sets is closed Since V is locally finite, so is { s ~ p p y , ) , ~ ~, so in X (Lemma 1.4.2), hence is the support of the continuous function

2=

C 7~: X

-+

10,

m).

Of course, if R, = 0,7, = 0. By these choices, it is clear that {?a)rrEm satisfies properties (1) and (2) in Definition 1.4.5. It is also clear that 7 = GaEB 7, < m is continuous and nowhere 0. Therefore, {A, = ?a/7),Emis a partition of unity subordinate to U. El

COROLLARY 1.4.12. Every open cover of a manifold admits a subordinate partition of unity. One use of partitions of unity will be to allow us to define Riemann integration globally on (smooth) manifolds. Another will be to show that every (smooth) manifold has a Riemannian metric. For these and similar applications we will need smooth partitions of unity, a notion that we are not yet ready to define. In the next section, we will use the existence of continuous partitions of unity to prove a topoIogical imbedding theorem for manifolds.

EXERCISES (1) Prove Lemma 1.4.2. (2) Let M be a manifold, U M an open subset, K C U a set that is closed in M, and let f : U -,R be continuous. Prove that the restriction f (K extends to a continuous function f : M + R.

18

1. TOPOLOGICAL MANIFOLDS

(3) Let K C Sn be a closed subset, U > K an open neighborhood of K, v a vector field defined on U. Prove that vl K extends to a vector field on all of sn. 1.5. Imbeddings and Immersions

We will prove that compact manifolds can always be imbedded in Euclidean spaces of suitably large dimensions. DEFINITION 1.5.1. Let N and M be topological manifolds of respective dimensions n 5 m. A topological imbedding of N in M is a continuous map i : N -,M which carries N homeomorphically onto its image i ( N ) . DEFINITION 1.5.2. If N and M are as above, a topological immersion of N into M is a continuous map i : N -* M such that, for each x E N , there is an open neighborhood W of x in N such that i(W : W -* M is a topological imbedding. For example, Figure 1.4 on page 8 depicts an immersion of the Klein bottle K into W3. Every imbedding is also an immersion, but even one to one immersions can fail to be imbeddings. The immersion of W in W2, pictured in Figure 1.7, is one to one, but is not a homeomorphism onto its image.

FIGURE 1.7. The topologist's sine curve

-

DEFINITION 1.5.3. If M is a manifold and X M is a subspace, we say that X is a submanifold if there is a manifold N and an imbedding i : N M such that X = i ( N ) . DEFINITION 1.5.4. The image of a one to one immersion i : N an immersed submanifold of M.

--, M

is called

Some authors use the term "submanifold" to include immersed submanifolds. From the point of view of a topologist, this seems dangerously misleading.

THEOREM 1.5.5. I f M is a compact n-manifold, then there is an integer k > n and an imbedding i : M -+ Ktk.

1.5. IMBEDDINGS AND IMMERSIONS

19

PROOF.Since M is compact, we can find a finite open cover 24 = {Uj};=, of M, together with homeomorphisms 9j : Uj + W j Rn. Let X = {A j } ; = l be a partition of unity subordinate to U.We wiil take k = r ( n 1) and construct an imbedding i : M r Rk. Define z:M-*W~X~.-XR~XRX-.-XIW

+

-r factors

r factors

by i ( ~= ) (Xl(x)(~l(x), , X r ( x ) ~ r ( ~ ) , X l ( .x.. ) , Xr(z)). Here we make the convention that 0 vj (x) = 6 E IRn, even when V j (x) is undefined. Since supp(Xj) c Ujand dom(cpj) = Uj, the expression Xj (x)vj (x) is identically 6 near the set-theoretic boundary of Uj and on all of M \ Uj. This implies that the map i : M + IRk is continuous. Since M is compact and i(M) is Hausdorff, we only need to prove that i is one to one in order to prove that i is a homeomorphism onto its image. Let x, y E M and suppose that i(x) = i(y). Since X is a partition of unity, there is a value of j such that Xj(x) # 0. But the (nr+j)th coordinates of i(x) and i(y) are Xj(x) = Xj(y), so X, Y E s u ~ ~ ( XCj )Uj- Also, Xj(x)vj(x) = Xj(y)vj (y), SO vj (x) = P j (9). Since V j : Uj -* Rn is one to one, it follows that x = y.

+

The imbedding dimension k = r ( n 1) given by this theorem for compact n-manifolds is often much too generous. For example, Exercise (3) on page 14 gives a covering of p2 by r = 3 open sets homeomorphic to R2, so the theorem only guarantees that P2 can be imbedded in R9. In fact, it is possible to imbed P2into It4, as you will show in Exercise (1).

EXERCISES (1) Define f : S2+ R4 by the formula

f (x, y, 2) = ( y t ,xz, xy, x2 + 2y2 + 3z2). Prove that f passes to a well defined, topological imbedding

(It is known that P2 cannot be imbedded in R3.) (2) Let g : S2+ IR3 be defined by Find six points p l , . . . ,p6 E P2such that itself into IR3 is known is a topological immersion. The mapping B of as Steiner's surface. It is simply the imbedding f into R4 followed by projection onto a threedimensional subspace of R4. Prove that 3 does not restrict to an imbedding of any neighborhood of p i , 1 5 i 6. (It

<

1. TOPOLOGICAL MANIFOLDS

is known that p o p : W4 -* It3.)

f

cannot be an immersion for any linear surjection

1.6. Manifolds with Boundary

Manifolds are modeled locally on Euclidean n-space. Something like the closed n-ball Dn = {v E Rn I llvll 5 1) fails to be a manifold because a point on the boundary aRn = Sn-l does not have a neighborhood homeomorphic to an open subset of Rn. It does, however, have a neighborhood homeomorphic to an open subset of Euclidean half-space.

DEFINITION 1.6.1. The Euclidean half space of dimension n is and the boundary of Wn is The interior of Wn is Remark that BWn is canonically identified with Wn-' by suppressing the coordinate x1 = 0 and renumbering the remaining coordinates as y i = xi+', l
DEFINITION 1.6.2. A topological space X is an n-manifold with boundary if (1) for each x E X , there is an open neighborhood U, of x in X , an open subset W, c Wn, and a homeomorphism cp : U, -* W,; (2) X is Hausdorff; (3) X is 2ndcountable. DEFINITION 1.6.3. Let X be a manifold with boundary. We say that x E X is a boundary point if a suitable choice of cp : U, + W,, as above, carries x to a point cp(x) E aWn. The interior points x E X are those such that a suitable homeomorphism cp : U , -* W, carries x to a point cp(x) E int(Wn). The boundary BX is the set of all the boundary points of X and the interior int(X) is the set of all the interior points of X . It is clear from the definition that every point of X is either a boundary point or an interior point. That is, X = dX U int(X). It is not immediate, however, that a point cannot be both an interior point and a boundary point. You will prove this in Exercise (I), together with the fact that BX is an (n - 1)-manifold.

EXAMPLE 1.6.4. The closed, unit n-ball D n is a manifold with boundary BRn = Sn-I. The interior is the open ball Bn = {v E Rn I llvll < 1). TOsee this rigorously, note first that the standard matrix action of O(n) on Rn restricts to a group of homeomorphisms of D onto itself. If el E Sn-l is the column vector with first entry 1 and remaining entries 0, then Ae 1 is the first column of the matrix A f O(n). Since every unit vector v E Sn-l occurs as the first column of some matrix A E O(n), it follows that v E Sn-I is carried by A-' E O(n) to el. Thus, an arbitrary point of Sn-l will be a boundary point of D n if and only

1.6. MANIFOLDS WITH BOUNDARY

21

if el is a boundary point. Let U C D n be the open neighborhood of e l defined by the condition x1 > 0. Then cp : U -* Wn, defined by

carries the unit vectors in U into aWn and the other points of U into int(lHIn). Furthermore, cp has continuous inverse 1C, given by the formula

We have proven that Sn-l C dDn. Since the open ball B n = Dn \ Sn-lis an open subset of Rn, it is clear that Bn int(Dn). By Exercise ( I ) , all assertions follow. EXAMPLE 1.6.5. The n-torus T n is the boundary of the solid torus D 2 x ~ n - 1 , a compact (n + 1)-manifold with boundary (cj. Exercise (2)). EXAMPLE 1.6.6. If Wl and W2 are compact 3-manifolds with nonempty, connected boundaries, let Di c aWi be a closed, imbedded 2-ball (a disk), i = 1,2. By fixing a homeomorphism j : D l -* D2, we set up an equivalence relation ~f on the disjoint union Wl 11 Wz by defining x - f y if and only if either x = y, y = f (x), or x = j (y). The quotient space is denoted W1 Uf W2 and can be proven to be a compact 3-manifold with boundary the connected sum 8Wl # aW2. This is intuitively obvious, and we will not at tempt a rigorous proof. It follows that, if M and N are compact, connected 2-manifolds which bound suitable compact 3-manifolds, then the connected sum M # N is also the boundary of some compact 3-manifold. By Examples 1.6.4 and 1.6.5, S and T 2 are such boundaries, hence the classification of compact surfaces (Theorem 1.3.2) implies that every compact, connected, orientable surface bounds a compact 3manifold. What about the compact, connected, nonorientable surfaces? It is known that the Klein bottle is such a boundary K = aW.Intuitively, we construct W from the solid cylinder D~ x [O,1] by gluing D 2 x (1) to D2 x (0) by an orientation reversing flip about a diameter. Since P2# p2 = K2, it follows that the connected sum of an even number of projective planes is the boundary of a compact 3-manifold. By Theorem 1.3.2, every compact, connected, nonorientable surface is a connected sum of finitely many copies of p2. Thus, the only compact, connected surfaces that might fail to bound a compact 3-manifold are connected sums of an odd number of projective planes. In fact, these do fail to bound. We will not prove this but, in Exercise (5), you will reduce the assertion to the special case that P2 does not bound. Example 1.6.6 suggests an interesting topological problem. Given a compact n-manifold (connected or not), is it the boundary of some compact (n + 1)manifold? A closely related problem is that of classifying compact n-manifolds up to cobonlzsm, an equivalence relation that was first defined by H. Poincar4 [35, Section 51. DEFINITION 1.6.7. Let M and N be compact n-manifolds without boundary. We say that M and N are cobordant, and write M -a N, if there is a compact (n + 1)-manifold W such that a W is homeomorphic to M II N, where 11denotes

1. TOPOLOGICAL MANIFOLDS

22

disjoint union. The cobordism class of M is denoted [MI. The set of cobordism classes of compact n-manifolds is denoted 9,. The fact that cobordism is an equivalence relation is elementary (Exercise (3)) as is the fact that homeomorphic manifolds are cobordant. If we make the standard convention that the empty set is a manifold of every dimension, the cobordism class [@I E 9, is precisely the set of compact n-manifolds that bound. In Exercise (4), you will show that disjoint union well defines an abelian group structure on 9, under which every element is its own inverse. It is a nontrivial fact that this group is finitely generated, hence 9,E Z2 @ Z2 $ - - - @ Z2. It is rather obvious that go= Z2, the nontrivial element being [point]. We are about to see that, up to homeomorphism, the only compact, connected, boundaryless 1-manifold, is S' = aD2,so 91 = 0. In Exercise ( 5 ) , you will show that 9, = Z2, generated by [P2]. We close with a sketch of the classification of compact 1-manifolds, possibly with boundary. It is enough to classify the connected ones. THEOREM1.6.8. If N is a compact, connected 1-manifold, then N is homeomorphic either to S1 or to [0, 11. The proof of this theorem requires the following lemma, the proof of which will be Exercise (6). LEMMA1.6.9. If N is a compact I-manifold, there is a mm'mal open subset U c N which is homeomorphic to an open interval in R. Such a subset is called a maximal open interval in N.

PROOF OF THEOREM 1.6.8. Let U C N be a maximal open interval and fix a homeomorphism f : (0,l) -,U. Here we use the well known fact that any two open intervals in W are homeomorphic. Since N is a compact HausdoriT space, there is a strictly increasing subsequence ( a C (0,l) such that ak f 1 and

k)z,

lim f ( a k ) = x+ E N

k+oo

\

U

exists. It follows easily that, for every monotonic sequence xk f 1 in (0, I), lim f (xk) = x+.

k-oo

SimiIarly, there is a unique x - E N

\

U with

lim f(yk) = x-

k-00

whenever yk 1 0. Therefore, = U U {x+,x-) is the closure of U in N , a compact, connected subset. Define g : [O, 11 -, 0 to be the extension of f by g(0) = x- and g(1) = x+, clearly a continuous map. We consider the cases x + = x- and x+ # x- . If x+ = x-, g induces a one to one, continuous map : [O, 1]/(0,1) -, 0. Since S1 S [O, 1]/(0,1) is compact and N is Hausdorff, this defines an imbedding S1 N of a compact, connected, boundaryless 1-manifold into a connected 1manifold. By an application of Theorem 1.1.5, the image of this imbedding is

-

1.6. MANIFOLDS WITH BOUNDARY

23

an open subset of N. Being compact, this subset is also closed in N and, being nonempty, it is all of N. This proves that N is homeomorphic to S1. If x+ # x-, it follows that g : [O, 11 -,N is one to one, hence is a topological imbedding. If x+ E a N , let W N be an open neighborhood of x+ and find 0 < S < 1 such that g ( l - 6,1] = J C W. We can assume that there is a homeomorphism h : W -* (-m,0] such that h(x+) = 0, so h ( J ) is a connected subset of (-m,O] containing 0 and having more than one point. Using the standard fact that the only connected subsets of R are the intervals, we conclude that h ( J ) is a nondegenerate interval. By an application of Theorem 1.1.5, one sees that h(J) has the form (-e, 01 for suitable E > 0, from which it follows that J is open in the topology of N. By Theorem 1.1.5, g(0,l) is also open in N, so g(O, 11 is open. Similarly, if x- E a N , g[O, 1) is an open subset of N. Thus, if x+ and x- both belong to a N , the image of g is open and, being compact, is also closed in N. By connectivity, this image is all of N, proving that N is homeomorphic to [0, 11. Suppose, therefore, that x+ E int (N) and deduce a contradiction. The same argument will show that x- 4 int(N). Let h : V -, R be a homeomorphism of an open neighborhood V of x+ in N onto an open subset of IR and let J denote the connected component of g(0, 11 n V containing x + . Since x+ is the limit of a sequence in g(0, I), J does not degenerate to a single point, hence h(J) is a half-open interval with h(x+) as its endpoint. hchoosing h, if necessary, assume that h ( J ) = (-1,Ol. For a small value of e > 0, there is an open set W c V such that h(W) = (-1, e ) and J = g(O, 11n W. It follows easily that W U g(O, 1) = W U U is homeomorphic to an open interval in R, contradicting the maximality of U. The following innocuous corollary will be quite important in Section 3.9.

COROLLARY 1.6.10. Every compact, 1-dimensional manijold M has a M equal to a finite set of points with an even number of elements.

EXERCISES (1) Use Theorem 1.1.5 to prove that a X t l int(X) = 0. Also prove that a X is an ( n - 1)-manifold and that int(X) is an n-manifold. (2) If aM = 0 and a N # 0, show that M x N is a manifold with boundary

(3) Prove that cobordism is an equivalence relation on the set of all compact n-manifolds (ignore ZermeleF'rankel set theoretic scruples about the meaning of "set of all compact n-manifolds" ) . Prove that homeomorphic manifolds are cobordant . (4) Show that the operation [MI [N] = [M II N ] is a well defined binary operation on %., Prove that this operation makes Ttn into an abelian group with 0 element the class and -[N] = [N], V [N] E %.,

+

[a]

24

1. TOPOLOGICAL MANIFOLDS

(5) Using Example 1.6.6 and assuming that p2is not a boundary, prove that Y !12 = Z2, the nontrivial element being [P2].If M and N are compact, connected surfaces, prove that [MI [N] = [M# N ] . N is an open interval in the 1-manifold N if it is (6) We say that V an open subset that is homeomorphic to an open interval in W. You are to prove that there is a maximal open interval in N (Lemma 1.6.9), proceeding as foilows. (a) Let Z be the family of open intervals in N , partially ordered by inclusion. Let V = {V,),Em be an infinite, linearly ordered subset of Z. Prove that there is a sequential nest

+

in V such that, for each a E Z, there is an integer k > 0 with V, c V,, . (Hint: N is 2nd countable.) (b) For {V,,)jEl as in part (a), prove that it is possible to choose the homeomorphisms hk of V,, to open intervals in R so that hk+lJVOlk = hk, Q k 2 1. Show that these assemble to define a homeomorphism h of Ur==, V,, onto an open interval in W. (c) Using the above, show that the partially ordered set Z is inductive, hence contains a maximal element (Zorn's lemma).

CHAPTER 2 The Local Theory of Smooth Functions

In this chapter, we treat the fundamentals of differential calculus in open subsets of Euclidean spaces. Everything will be set up so as to extend naturally to global differential calculus on smooth manifolds.

NOTATION.Elements of Rn, when thought of as vectors, will be written as column n-tuples. When thought of as points, they will be written as rows. 2.1. Differentiability Classes

c

Rn be an open subset. Let x = ( x l , . . . ,xn) denote the general Let U (variable) point of U and let p = . . ,pn) be a fixed but arbitrary point of U . Let f : U + W be a function and let Lp : U + W be an affine (i.e., inhomogeneous linear) map

such that LP (P) = f (P).

DEFINITION 2.1.1. If f and Lp are as above and if lim

f (4- LP(4

= 0) llx - ~ l l Then L, is called a derivative of f at p. If f admits a derivative at p, then f is said to be differentiable at p. x+p

We think of a derivative Lp as a linear approximation of f near p. By the definition, the error involved in replacing f (x) by Lp(x) is negligible compared to the distance of x from p, provided that this distance is sufficiently small. It follows from the definition that an affine map is a derivative of itself. The above definition of "derivative" as a linear approximation embodies the real philosophy of differential calculus. As it stands, however, this definition is a bit unsatisfying. The use of the indefinite article ( a derivative) raises the issue

2. LOCAL THEORY

26

of uniqueness, while the relationship of the notion of a derivative to the familiar operation of differentiation is also unclear. The following lemma, whose proof will be Exercise (I), resolves these doubts. LEMMA2.1.2. If Lp(x) = C +

x7=lbizi is a derivative o f f at p, then

1 5 i 5 n. In particular, i f f is diflerentiable at p, these partial derivatives exist and the derivative L p is unique. Having seen that derivatives are given by partial derivatives, we center our attention on these more familiar operators. DEFINITION 2.1.3. The class of continuous functions f : U -r R is denoted CO(U).If r 2 1, the class Cr (U) of functions f : U -* R that are smooth of order r is specified inductively by requiring that af /axa exist and belong to Cr-I (u), 1 5 i 5 n. The functions that are smooth of order r are also called Crsmooth. One has a decreasing nest

and examples show that these inclusions are all proper. DEFINITION 2.1.4. The set of infinitely smooth functions on U is

It is usual simply to call Cw functions "smooth". We will be concerned primarily with such functions. Remark that the coordinates in U are themselves smooth functions xi : U -, R. Thus, q E U has coordinates xt(q), 1 5 i 5 n, and we can write q = ( x ' ( ~ ) ., .. ,xn(q)).

EXERCISES (1) Prove Lemma 2.1.2 (2) If dimU = 1, prove that the derivative Lp exists if and only if f'b) exists. But if dim U = 2, p = (0,O), find a function f : U -+ R such that both partial derivatives exist at every point of U, but such that the derivative L(olo)does not exist. (3) Let D(U) = { f : U -* R I f is differentiable at x, Vx E U). Show that CO(U) 2 D(U) 2 C1(U). Produce examples to show that both of these inclusions are proper. (Hint: First do this for dim U = 1 and then extend to arbitrary dimensions.) (4) Let U C Wn be open, let f E Cr(U), where 1 5 r 5 oo,and let g : R -* R be Crsmooth also. Prove that the composition g o f belongs to Cr(U).

2.2. TANGENT VECTORS

2.2. Tangent Vectors

We continue to let U C Rn be a fixed but arbitrary open set. We fix p E U and describe the tangent space Tp(U)of U at p. In calculus, it is customary to translate a tangent vector a' at p to the origin 0 E R n , thereby identifying a' canonically with an element of Rn. That is, we set Tp(U) = Rn. This will not do for our purposes since we are trying to set up a local calculus that will make sense on manifolds where, generally, there will be no preferred coordinate system. In standard calculus, the vector

defines a directional derivative Dz at p by the formula

Da(f )

=

lim f(p+h;)-ff(p) h

h+O

=

af C" a'-@), axi i=l

where f is an arbitrary smooth function defined on an open neighborhood of p. Applying this operator to the coordinate functions xi gives so the vector a' is uniquely determined by its associated directional derivative. Another way to obtain this directional derivative is to consider a curve

(where E, 6 > 0 ) , written such that each xi(t) is of class at least C1 and

That is, 5 ' (0)

= pa,

S' (0)= a',

for 1 5 i

5 n. By standard calculus, D a ( f ) = lim f (s(h)) - f @)

h In other words, although the directional derivative was defined as differentiation at t = 0 along a straight line curve with constant velocity a', it could just as well have been defined via any C1 curve out of p with initial velocity a'. While the notion of "straight line" will not have meaning on general manifolds, the notion of "C1 curve" will. h-+O

2. LOCAL THEORY

28

DEFINITION 2.2.1. Given p E U, Coo(U,p) will denote the set of smooth, real valued functions f with dom(f) an open subset of U and p E dom(f).

c1curves s : (-S,e) + U (where S on s) such that s(0) = p is denoted S(U, p).

DEFINITION 2.2.2. The set of all 0 and

E

> 0 depend

>

DEFINITION 2.2.3. If sl ,s2 E S(U, p), we say that sl and sn are infinitesimally equivalent at p and we write sl x, s 2 if and only if

for all f E Cm(U,p). It is easy to check that x, is an equivalence relation on the set S(U,p).

DEFINITION 2.2.4. The infinitesimal equivalence class of s in S(U,p) is denoted (s), and is called an infinitesimal curve at p. An infinitesimal curve at p is also called a tangent vector t o U at p and the set T,(U) = S(U,p)/=, of all tangent vectors at p is called the tangent space to U at p. The following is immediate by the definition of infinitesimal equivalence.

LEMMA 2.2.5. For each (s),

E

T,(U), the operator

is well defined by choosing any representative s E (s), and setting

for all f E Cm(U,p). Conversely, (s), is uniquely determined by the operator

D(9)pBy the definition of infinitesimal curve, there is a natural one t o one correspondence between T,(U) and what, in calculus, is usually called the space of vectors at p, but this notion of tangent vector will make sense on manifolds. Since the whole reason for introducing tangent vectors is to produce linear approximations to nonlinear problems, it will be necessary t o exhibit a natural vector space structure on T,(U), not dependent on the coordinates of Rn.The key lemma for this follows.

LEMMA2.2.6. Let (sl), , (s2), E T,(U) and a, b E B. Then there is a unique infinitesimal cume (s), such that the associated derivatives on Cm(U,p) satisfy

2.2. TANGENT VECTORS

29

PROOF.We are allowed to use the coordinates of R n to prove this assertion. The important point is that the assertion itself is coordinate free. It is clear, then, that s(t) = asl(t) bs2(t) - ( a b - l)p, defined by coordinatewise operations for all values of t sufficiently near 0, is a C curve in U with s(0) = p, and that this curve represents the desired (s) E Tp(U). By the definition of infinitesimal equivalence, (s), is completely determined by D P

+

+

,

DEFINITION 2.2.7. Let (sl),, (s2), E T,(U) and a, b E R. Then is defined to be the element (s), given by Lemma 2.2.6.

LEMMA2.2.8. The operation of linear combination, given by Definition 2.2.7, makes T,(U) into an n-dimensional vector space over R. The zero vector is the infinitesimal curve represented by the constant p. If (s), E T,(U), then - (s), = (s-), where s-(t) = s(-t), defined for all suficiently small values oft. The proof of this lemma is left as an exercise. The operator Db)p,described above, does not "see" all of f E C * (U,p) , only the behavior of f in arbitrarily small neighborhoods of p. The proper way to say this is that D(,!p(f) depends only on the ''germ" of f at p. There is some usefulness in formalizing this.

DEFINITION 2.2.9. We say that the elements f ,g E Cw(U,p) are germinally g if there is an open neighborhood W of p in equivalent (at p) and write f U such that W 5 dom(f) n dom(g) and f (W = g(W.

-,

It is clear that germinal equivalence is an equivalence relation on C * (U,p).

DEFINITION 2.2.10. The germinal equivalence class [f], of f E C w (U, p) is called the germ of f at p. The set CM(U,p)/-,

of germs at p is denoted 8,.

DEFINITION 2.2.11. For each s E S(U,p) the operator is defined by De,,

d

[flp = zf

(s(t))lt=o*

REMARK.The discussion so far would have worked equally well if we had fixed an integer k 2 1, replaced Cw(U,p) with the set C' (u,p) of C k functions defined in neighborhoods of p, and taken 6, = e): to be the germs of these functions. This remark is crucial if one wants to formulate the theory of C manifolds. There is a purely algebraic characterization of T,(U) which, though adrnittedly more formalistic, has its charms. This definition of the tangent space is only valid for the C w case (the default). First we define algebraic operations on germs as follows.

2. LOCAL THEORY

Scalar multiplication: t[f], = [tfIp, V t E R, V [flp E 8,. Addition: [fIp [gIp = [flw glW]p, V[f]p, [glp E @pr where W is an open neighborhood of p in dom(f) n dom(g). Multiplication: [f],[g], = [(flW)(glW)]p, v[f]p,[glp E @pr where W is again as above.

+

+

LEMMA2.2.12. The above operations are well defined and make 8, a commutative and associative algebra over R with unity. The elementary proof of Lemma 2.2.12 is left as an exercise. DEFINITION 2.2.13. The evaluation map e, : 8, + JR is defined by e,[fl, = f @). The following is immediate. LEMMA2.2.14. The evaluation map e, : 8, phism of algebras.

+R

is a well defined hornomor-

DEFINITION2.2.15. A derivative operator (or, simply, a derivative) on 6, is an R-linear map D : 8, + R such that

for all a, b E 8,. A derivative on 6, is also called a tangent vector to U at p. The set of all derivatives on 8, is denoted T, or T,(U) and is called the tangent space to U at p. We define algebraic operations on T,(U). scalar multiplication: (tD)( a ) = t(D(a)), V t E R, V D E T,, V a E 6,; addition: (Dl &)(a) = D l (a) D2(a), V Dl, D2 E T,, Va E 8,.

+

+

LEMMA2.2.16. The tangent space Tp(U) is a vector space over JR under the above operations. Again, the proof will be an exercise.

2.2.17. Define Di,, : 8, EXAMPLE

+R

by

This is a well defined, R-linear map. Furthermore, by the Leibnitz rule for partial derivatives,

Thus DilPis a derivative, 1 5 i 5 n.

2.2. TANGENT VECTORS

EXAMPLE 2.2.18. If ( s ) is ~ an infinitesimal curve, then derivative. Indeed,

31

D(3.p

: Bp + IR is a

where

so the assertion follows from the previous example. It is obvious that

defines a linear map from the space of infinitesimal curves into the space of derivatives of Q3p. It is also clear that this linear map is injective. The fact that it is an isomorphism of vector spaces (Corollary 2.2.22) requires proof. LEMMA2.2.19. If c is a constant function on U and D E Tp(U), then

PROOF. Consider first the case c = 1. Then

from which it follows that D[1Ip = 0. For an arbitrary constant c, D[c], = cD[l], = 0, by linearity. El In order to get more information on Tp(U), we need a technical lemma. Let x = (xl,. .. ,xn) and p = (xl(p), . . . ,xn(p)). LEMMA2.2.20. Let f CE Cm(U,p). Then there exist functions g,, . . . ,g, E C""(U,p) and a neighborhood W C dom(f ) n dom(gi ) f~- - - n do+,) of p such that

PROOF.Define

2. LOCAL THEORY

32

This is clearly a smooth function defined at dl points x sufficiently near p. In order to prove ( 2 ) , consider

In order to prove ( I ) , consider

THEOREM 2.2.21. The set of tangent vectors { D l l p l .. .,Dn,,) is a basis of the vector space T,(U).

PROOF.Suppose that n

For the coordinate functions xj, 1 5 j

5 n,

the Kronecker delta. Thus,

for 1 5 j 5 n. This proves that { D l , , , . . . , D,,,) is a linearly independent subset of T,. We must prove that it is also a spanning set. Let D E T,. Set a' = D[xi],, 1 5 i n. Given an arbitrary I f ] , E 6,,write

<

2.2. TANGENT VECTORS

as in Lemma 2.2.20. Then,

Since [f], E 6 , is arbitrary, it follows that D =

x:=,aiDi,,.

0

REMARK.The above proof would not work for deriwtives of the algebra 6 of germs of C ' functions, k < oo. The problem is that gi E c'-', 1 5 i 5 n, so DlpiIp is not even defined. In fact, for 0 < k < m, the space of derivatives of 6 is infinite dimensional [33]. COROLLARY 2.2.22. The two definitions of T,(U)give canonically isomorphic vector spaces. PROOF.Indeed, the linear injection, defined in Example 2.2.18, between the two vector spaces must be an isomorphism since both are n-dimensional.

REMARK.We can identify T,(U) canonically with Rn via

On the other hand, one should be wary since T,(U) should not be thought of as identical with T, ( U ) when p # q. There will be no such canonical identification on manifolds. Let T ( U ) = LIxEUTx( U ) , a disjoint union. There is a one to one correspondence T ( U ) +-, U x IWn given by

We use this to transfer the topology of U x R n to T ( U ) .

2. LOCAL THEORY

34

DEFINITION 2.2.23. The tangent bundle n : T ( U ) -,U is defined by

Via the canonical identification T ( U ) = U x R n ,?r is just the standard p r e jection onto the first factor. For each x E U , T , ( U ) should be thought of as the linear approximation of U at x. This is going to enable us to approximate smooth maps between open subsets of Euclidean spaces by linear maps.

EXERCISES ( I ) Prove Lemma 2.2.8. So as to assure the usefulness of the infinitesimal curve point of view in the context of ckfunctions, you should not reduce this to Lemma 2.2.16. ( 2 ) Prove Lemma 2.2.12. ( 3 ) Prove Lemma 2.2.16. (4) Let 6; c 6, be the kernel of the evaluation map e , and let 6;' c 6; be the vector subspace spanned by the germs of functions gf , where g, f E C m ( U , p ) and g ( p ) = 0 = f (p). Prove that the quotient space 6;/0% is canonically isomorphic to the vector space dual of T , ( U ) . In particular, this quotient space is n4imensional. 2.3. Smooth Maps and Their Differentials Let U Rn and V Rm be open subsets. We consider functions and their coordinate representations

where each

: U +V

ai: U + R.

DEFINITION 2.3.1. We say that O : U + V is a map of class ck(0 5 k 5 m) if a' E c ~ ( u1)5, i 5 rn. If is of class C", it is called a smooth map. The following lemma is clear by the standard chain rule. LEMMA2.3.2. Wherever defined, compositions of smooth maps are smooth. DEFINITION 2.3.3. If @ : U

-,V

is smooth and if p E

U ,let

be defined by

a*, ( 4 , = (aO s)*(,)

7

for arbitrary ( s ) , E T p ( U ) .This is called the differential of 8 at p.

2.3. SMOOTH MAPS

35

LEMMA2.3.4. The diflerential @,p, as defined above, is a well defined linear map. Relative to the bases {Di,p}y=lof Tp(U)and {Dj,o(p)}$lof T q p ) ( V )the , matrix of a, is the Jawbian matrix

The elementary proof of Lemma 2.3.4 will be an exercise. REMARK.The differentials a, computed at all points x E U , assemble to a mapping @, = d@ : T ( U )-,T ( V ) , given by

Here, we have identified T ( U )with U x Rn and T ( V )with V x Rm. COROLLARY 2.3.5. Under the identifications T ( U ) = U x Rn and T ( V ) = V x Rm, d@ = @, : T ( U )+ T ( V ) is a smooth map from an open subset of R2" to an open subset of R2". COROLLARY 2.3.6. If @ : U

+V

is of the form

@(x= ) L ( x )+ Y o , where L : Rn + W m is linear and yo E Rm is fied, then

In this case,

d@=(@,L):UxRn+VxRm.

THEOREM 2.3.7 (THE GENERAL CHAIN RULE).If U & Rn, V C Rml and W & RQare open subsets and i f @ : U -+ V and \E : V + W are smooth, then

PROOF. Indeed,

2. LOCAL THEORY

36

REMARK.In terms of Jacobian matrices, the general chain rule can be written

One can verify this directly by applying the less general chain rule for real-valued functions and the formulas for matrix multiplication. This is the usual proof in multivariable calculus, but the proof via infinitesimal curves is more elegant and more intuitive. The above discussion works equally well in the ckcategory, Q k 2 1. In the Coocategory, the differential can be treated as a transformation on derivatives of the germ algebra. We turn to this point of view. LEMMA2.3.8. If @ : U -+ V is smooth, p E U , and i f D p E T p ( U ) is viewed as a derivative of 8,) then Dq,) : 8 q P )-+ R, defined by D*(P) i f l o c p ) = DP [f O @I, is an element O ~ T @ ( ~ )W(eVdefine ) . d a P ( D p )= a r p ( D p )= Da(pb PROOF.First we prove that D*(,) is linear. If t E R, then

By entirely similar reasoning,

It remains to prove the Leibnitz rule:

LEMMA2.3.9. Under the canonical equivalence of the two definitions of T p ( U ) , the two definitions of the diflerential d@, also agree. The proof of this lemma will be an exercise. The proof of the chain rule, using the second definition of d@,, will also be an exercise. This point of view, while less appealing intuitively, will be quite useful for developing the global theory of smooth manifolds with boundary (Section 3.6). DEFINITION 2.3.10. If U C Rn and V 5 Rm are open, a map @ : U V is a diffeomorphism if it is smooth and bijective and if :V U is also smooth. -+

-+

2.3. SMOOTH MAPS

PROPOSITION 2.3.11. If @ : U

37

V is a difeomorphism of an open subset of Rn onto an open subset of R m and i f p E U,then d@, : T p ( U )+ T+(,)(V) is a linear isomorphism. In particular, n = m. -+

o @ = idU is the restriction to U of i d p , it linear map, PROOF.Since Corollary 2.3.6 shows that d(@-' o a), = idRn. By the general chain rule, it follows that the linear map d Q p is invertible with inverse d(6-')qp1.

We saw earlier that the Brouwer theorem of invariance of domain implies the equality of dimensions, even if @ were only a homeomorphism. That theorem was very deep, while Proposition 2.3.11 is quite elementary. This is an example of the technique of reducing nonlinear problems to linear ones via derivatives. Finally, the precise sense in which the differential d@, is a linear approximation of a near p is given by the following theorem, in which d@ is interpreted as a linear map of Rn -+ Rm,V y E dam(@). THEOREM 2.3.12. If each p E U ,

U C Rn is open and

lim

:

U + R m is smooth, then, for

@(XI - @(Y)- d@v(x- Y) 1 1 5-

(Z,~)+(P,P)

Y ll

= 0.

PROOF.Using the coordinate representation

we see that it is enough to prove the assertion for maps Lemma 2.2.20, let p be a variable point y and write

@ j=

f :U

Thus,

Since (xi - Yi)/llx- yll is bounded, 1 5 i 5 n, it is clear that lim

R(x,y)=O

(x,Y)+(P,P)

and the assertion follows. In Exercises (6) and (7), you will need the following definition.

-P

R. In

38

2. LOCAL

THEORY

DEFINITION 2.3.13. If A C Rn is an arbitrary subset, define Cm(A) to be the set of all functions f : A + W such that f = f l ~ , where f : U + R is a smooth function defined on some open neighborhood U of A.

EXERCISES (1) Prove Lemma 2.3.4. (2) Prove Lemma 2.3.9. (3) Viewing d a p as a transformation on the space of derivatives of the algebra of germs, give a direct proof of the chain rule. (4) If f E Cm(U, p) and Lp is the derivative of f at p (Definition 2.1. I), use Theorem 2.3.12 to express L, in terms of df,. (5) Let p E Sn C Rn+' and define

Tp(Sn)= { ( s ) ~E Tp(Wn+l) I s : ( - E ,

L) + Rn+l

has im(s) c Sn).

Prove that Tp(Sn) is the linear subspace of Wntl = Tp(Wn+l)consisting of all v Ip. This is what we earlier called the tangent space of Sn at p (Example 1.2.6). (6) If A = [0,1]x [O,1] and f E Cm(A), let f : U -+ R be a smooth extension as in Definition 2.3.13. Prove that df(o,o)depends only on f , not on the choice of f. (7) If A = Sn,f E Cm(Sn),p E Sn, and f is as in Definition 2.3.13, show by an example that djp may well depend on the choice of extension f, but prove that dflTP(sn) depends only on f . 2.4. Diffeomorphisms and Maps of Constant Rank

If @ : U -+ V is a diffeomorphism between open subsets of R n , then the Jacobean matrix J@(p) is nonsingular, Vp E U. While the converse is not exactly true, it is true locally. FUNCTION THEOREM). Let @ : U + V be smooth, THEOREM 2.4.1 (INVERSE where U, V C Wn are open subsets, and let p E U. If d@, : T,(U) + TGIP)(V) is a linear isomorphism, then there is an open neighborhood W , of p in U such that @IWpis a dzfleomorphism of Wp onto an open neighborhood @(Wp)of @(p) in V.

This is a remarkable and fundamental result. From a single piece of linear information at one point, it concludes to information in a whole neighborhood of that point. This theorem is often proven in courses in admced calculus. We give a proof in Appendix B which works for C k maps, 1 5 k 5 m, and even works for maps between open subsets of a Banach space. There is a generalization of Theorem 2.4.1, called the "constant rank theorem", which is actually equivalent to the inverse function theorem. We will prove the constant rank theorem using the inverse function theorem.

2.4. CONSTANT RANK

39

DEFINITION 2.4.2. A smooth map @ : U -+ V, between open subsets of Euclidean spaces of possibly different dimensions, has constant rank k if the rank of the linear map d@, : Tz(U) + T+(,) (V) is k at every point of U. Equivalently, the Jacobian matrix J@ has constant rank k on U. EXAMPLE 2.4.3. Consider the composition

where k

< n, k < m, and

m-k

The Jacobian of i 0 .~ris constantly the n x m matrix having I k as its upper left k x k corner and 0's elsewhere. The rank is constantly k. The constant rank theorem asserts that, in a certain precise sense (see the remark on page 40), maps of constant rank k locally "look like" the above example.

THEOREM 2.4.4 (CONSTANT RANK THEOREM). Let U C_ Rn and V C Rm be open and let @ : U + V be smooth. Let p E U and suppose that, i n some neighborhood of p, @ has constant rank k . Then there are open neighborhoods W of p in U and Z 1 @ ( W )of @(p)in V , together with diffeomorphisms

onto open subsets of Rn and Rm, respectively, such that, throughout the neighborhood % of F b ) ,

PROOF.If we take f : Rn + Rn to be the translation taking p I+ 0 and g : Rm + Rm the translation taking @(p)I+ 0, then we can replace @ with 5 = go@of-', V with = g(V), and U with 6 = f (U), obtaining 6 : 8 + so that &(o) = 0. This reduces to the case in which p = 0 E Rn and @(p)= 0 E Rm. Indeed, if - and are diffeomorphisms that work in this case, we take F = ? o f and G = G o g for the original case. In a similar way, one can use linear maps f : Rn + Rn and g : Rm + Rm that simply permute coordinates in each of these spaces so as to assume that the upper left k x k block

v

v

of J@ is nonsingular at p = 0. Without loss of generality, therefore, we make these assumptions.

2. LOCAL THEORY

40

Let x = ( x l , . . . ,xn) and z = (zl,. . . ,zn). Define F : U + Rn by F(x) =

(a1(x), . . . , ak(x), x k f l , . . . ,xn) .

Then F(0) = 0 and

is a matrix that is nonsingular at p = 0. By Theorem 2.4.1, there is a neighborhood W of 0 on which F is a diffeomorphism to an open set F ( W ) C W". On F(W) , we get (for suitable smooth functions pk+l,. .. ,pm)the formula @ o F-' (z) = (zl, . . . ,zk,pk+'(z), . . . ,pm(2)) ,

hence

-

Since J@has rank k in a neighborhood of 0 and JF-' is nonsingular on the neighborhood W = F ( W ) of 0, we can choose W smaller, if necessary, so as to assume that the above matrix has rank k at every point of F. It follows that the lower right block must consist entirely of O's, hence that the functions p j only depend on ( z l , . . . ,zk), k 1 5 j 5 m. Let y = ( y l , . . . , ym) and define

+

This is defined for all yl,. . . ,y k close enough t o 0 and, in particular, on a neighborhood of 0 in R m that contains @(W) . It is clear that

is a nonsingular matrix, hence Theorem 2.4.1 implies that G is a diffeomorphism of a small enough neighborhood Z of 0 in W m onto a neighborhood 2 = G(Z). Taking W smaller, if necessary, we can assume that @(W) Z. From the formulas, it is clear that

and G G: Z + 2 in the constant rank theorem should be thought of as changes of coordinates in these open sets.

REMARK.The diffeomorphisms F : W +

2.4. CONSTANT RANK

For instance, one could write

. . .,xn) z2 = ~ ' ( x ' ,. . . , xn) z1 = F'(x',

viewing (zl, z2,. . . , zn) as new coordinates of the point ( x l , x2,. . . ,xn). The new coordinates depend smoothly on the original ones and, F being a diffeomorphism, the original coordinates depend smoothly on the new ones. Thus, all of calculus, formulated in the coordinates xi, has a completely equivalent formulation in the coordinates zi. The specific formulas change, but the realities they express do not. According to this philosophy, the point of the constant rank theorem is that the most general map of constant rank can be expressed locally using the same formula as the simple Example 2.4.3, provided the coordinates in the domain and the range are suitably changed. There are two important special cases of Theorem 2.4.4, the immersion t h e e rem and the submersion theorem. DEFINITION 2.4.5. Let U C Rn and V C Rm be open subsets. A smooth map @ : U -+ V is a submersion if it has constant rank m on U. It is an immersion if it has constant rank n on U. Remark that, if @ is a submersion, then n 2 m. If it is an immersion, then rn n. If it is both a submersion and an immersion, then n = m and @ is locally a diffeomorphism by Theorem 2.4.1. The next two corollaries are immediate applications of Theorem 2.4.4.

>

COROLLARY 2.4.6 (SUBMERSION THEOREM). Let @ : U + V be a submersion and let p E U. Then there are open neighborhoods W of p in U and Z 2 @(W)of @(p) i n V , together with diffeomorphisms

onto open subsets of Wn and Rm, respectively, such that, throughout the neighborhood % of F ( p ) ,

COROLLARY 2.4.7 (IMMERSION THEOREM). Let @ : U + V be an immersion and let p E U. Then there are open neighborhoods W of p in U and Z @ ( W ) of @ ( p ) in V , together with diffeomorphisms

>

42

2. LOCAL THEORY

onto open subsets of Rn and Rm, respectively, such that, throughout the nezghborhood of F(p),

Thus, submersions look locally like projections onto the first m coordinates and immersions look locally like the canonical imbeddings

COROLLARY 2.4.8 (IMPLICIT FUNCTION THEOREM). Let U C R n be open and let p E U. Let f : U + R be smooth with f (p) = a. If

then, on some open neighborhood W o f p i n U, the set of solutions to the equation f (9) = a is the graph of a smooth function

EXERCISES (1) Use Corollary 2.4.6 to prove Corollary 2.4.8. (2) Deduce Theorem 2.4.1 from Theorem 2.4.4. Since we deduced Theorem 2.4.4 from Theorem 2.4.1, the two theorems are equivalent. (3) Let @ : U + V be smooth, U c Rn and V c Rm open subsets, and suppose there is a smooth function F : V + R such that F o @ E 0 on U. Show that the rank of J @ is everywhere < rn. (4) If @ : U + V is as in the previous exercise and has constant rank r < rn on U, show that, on some open set W @(U)in V, there is a smooth function F : W --+ R such that F o @ -= 0 on U. (5) Use Theorem 2.4.8 to prove that the unit sphere Sn c Rn+' is a t o p e logical submanifold of dimension n.

>

2.5. Smooth Submanifolds of Euclidean Space

--

We already know what is meant by a topological submanifold of R n (Definition 1.5.3). We extend this notion to the smooth category. The model will be Rn, r 5 n, given by the standard imbedding Rr

(xl,.. . ,xr)

-

(xl,.. . ,xr, 0,. . . ,O). n-r

Whenever we view Rr C Rn, we understand Rr to be the image of this imbedding.

DEFINITION 2.5.1. Let U E Rn be open. A topological subspace N 5 U is said to be a smooth submanifold of U of dimension r 5 n if, for each x E N, 3 U, U, an open neighborhood of x, and a diffeomorphism f : U, -+ Q onto Rr. an open subset Q Rn such that f ( N fl U,)= Q f~

2.5. SUBMANIFOLDS

43

The empty set 0 c U is considered, by convention, to be a smooth submanifold of every dimension P' 5 n. LEMMA2.5.2. If N C_ U is a smooth submanifold of dimension r, then N is also a topological submanzfold of dimension r of U.

PROOF.Let x E N and use the notation of Definition 2.5.1. Then N n U, is an open neighborhood of x in the relative topology of N in U ,V x f N. Since f carries N n U, homeomorphically onto the open subset Q n R r of Rr, it

-

follows that N, with the relative topology, is locally Euclidean of dimension r, the inclusion map i : N U being a topological imbedding. As a topological subspace of Euclidean space, N is Hausdorff and second countable. THEOREM 2.5.3. Let U C Rn and V C Rm be open and let :U + V be a smooth map of constant rank k. Let q E V. Then 9-1(q) is a smooth submanifold of U of dimension n - k. PROOF.If 9-l (q) = 0, the aswtion is true by convention. Assume that this set is nonempty and let x be one of its points. Choose U, to be the neighborhood W as in Theorem 2.4.4. Without loss of generality, we can replace W with and 9lW with G o 9 o F-I on E, all as in that theorem. That is, on U, we assume that 9(y1, . . . , y") = (yl,. . . , y k ,o, . . . ,o) . Thus q = (a1,. . . , ak ,O,. . . ,o)

and U, n 9-l (q) is the set of all points in U, of the form

The desired diffeomorphism f will be

EXAMPLE 2.5.4. Your solution to Exercise (5) on page 42, together with Theorem 2.5.3, implies that Sn C Rn+l is a smooth submanifold of dimension n.

EXAMPLE 2.5.5. Let Gl(n) denote the group of nonsingular matrices over R (the general linear group). The special linear group Sl(n) is defined to be the subgroup of Gl(n) consisting of the matrices of determinant 1. We will show that Gl(n) is an open subset of Rn2 and that Sl(n) is a smooth submanifold of Gl(n). Let %Jl(n) denote the set of all red n x n matrices. One can fix an identification %Jl(n) = Rn2. The determinant function det : m ( n ) -, R is a polynomial, hence is smooth. The set R* of nonzero reals is open in R, hence the general linear group Gl(n) = det (w*) is an open subset of Rn2. We claim that det : Gl(n) + R has constant rank 1. To prove this, we need to show that, for arbitrary A E Gl(n), the linear map

2. LOCAL

44

THEORY

has rank 1. For this, we only need to find an infinitesimal curve (s)A such that det,A ( s ) ~= (det O S ) ~ , ( ~#) 0. Define s(t) to be the matrix obtained by multiplying the first row of A by 1 t. Then s(0) = A and the fact that Gl(n) is open in 8"' implies that s(t) E Gl(n) for it 1 small enough. Since det (s(t)) = (1 t) det (A) and det (A) # 0, it is clear that (det 0s) det(A) # 0. By the above remarks and Theorem 2.5.3, Sl(n) = det-'(1) is a smooth submanifold of Gl(n) of dimension n2 - 1.

+

+

DEFINITION2.5.6. If N C U is an r-dimensional, smooth submanifold of the open set U C Rn, and if x E N, a vector v E T,(U) is tangent to N at x if, as an infinitesimal curve, v = (s), has a representative s : (-E, E) + U such that s(t) E N , -E < t < E. The subset Tx(N) T,(U), consisting of all vectors tangent to N at x, is called the tangent space to N at x. LEMMA2.5.7. If N C U is an r-dimensional, smooth submanijold of the open set U C Rn, and if x E N , the tangent space Tx(N) is an r-dimensional vector subspace of T, (U). PROOF.For the "model" case N = Rr C Rn, the assertion is evident. Let all notation be as in Definition 2.5.1. If the smooth path s : (-E, E) -+ U has image in N and s(0) = x, then the diffeomorphism f : U, + Q sends s to a smooth path f o s in Rr through f (x). Thus, the linear isomorphism

carries T, (N) into the vector space TI(,) (Q n Rr) = Rr. But Q n Rr is mapped onto U, n N by f and the same argument shows that the inverse isomorphism d( f ) ),, = carries Tf),( (QfIRr ) into Tx(N). The assertion follows.

-'

-'

(a)-'

EXAMPLE 2.5.8. The subspace Tz(Sl(n)) C im(n) is the space of matrices of trace 0. To prove this, we first show that this tangent space is the kernel of det .I and then that d e t , ~= t r : m ( n ) + R . Both Tz(Sl(n)) and ker(det,Z) have dimension n2 - 1, so equality will follow if we show that the first is a subspace of the second. If v E Tz(Sl(n)) is thought of as an infinitesimal curve, then v = ( s ) where ~ s : (-E, E) + Sl(n), ~ ( 0 = ) I. Thus, det o s 1 is a constant curve, so det,z(v) = (det os), = 0. The linear functionals det ,I and tr on ?lR(n) will be equal if they agree on a basis. The matrices E i j , 1 5 i, j 5 n, having 1 in the (i,j) position and 0's elsewhere, form a basis. As an infinitesimd curve at I, Eij = ( s i j ) Z where sij (t) = I tEij. But

-

+

det(sij(t) = from which it follows that {det osij)' =

{

1=tr(Eij), i = j 0 = tr(Eij), i # j.

2.5. SUBMANIFOLDS

45

U be a smooth r-dimensional Let U be an open subset of Rn and let N submanifold. Exactly as we did for Sn in Section 1.2, we can define the topological subspace T(N) 5 W n x Rn = to be the set of pairs (x, v) , where x E N and u E Tz(N). We define by p(x, u) = x. The fiber P-' (x) = Tz(N) is a vector space, V x E N.

DEFINITION 2.5.9. The structure p : T ( N ) + N is called the tangent bundle of the submanifold N C U. The total space of the bundle is T(N), the base space is N , and p is called the bundle projection. In the case of the model submanifold Rr Rn,T(Rr) = Rr x Rr & Rn x Rn. We will say that this bundle is trivial (cf. Definition 2.5.10). If Y C N is an open subset, then Y is also a smooth submanifold and T(Y) = p-I (Y) is the total space of the tangent bundle p : T(Y) + Y. In the following definition, the term "diffeomorphism" refers to a bijection between subsets of Euclidean spaces which, together with its inverse, is smooth in the sense of Definition 2.3.13.

DEFINITION 2.5.10. The tangent bundle p : T(N) commutative diagram

-P

N is trivial if there is a

T(N) A N x Rr

where p~ is projection onto the factor N and cp is a diffeomorphism with the property that, V y E N, (plT,(N) -+ {y) x R' is a linear isomorphism.

In Exercise (5) you will show that, in all cases, T(N) is "locally trivial" (cf. Exercise (3) on page 6, also).

DEFINITION 2.5.11. If the tangent bundle p : T(N) manifold N is said to be parallelizable.

+

N is trivial, the sub-

DEFINITION 2.5.12. If N E Rn is a smooth submanifold as above, a smooth which is smooth as a map vector field on N is a map X : N + T(N) C and satisfies p o X = idN. from the subset N Rn into

46

2. LOCAL THEORY

REMARK. When we get to the global theory, smooth manifolds will be defined without need of an ambient Euclidean space and their tangent bundles will be defined in an intrinsic way. A key property of these bundles will be local triviality.

EXERCISES (1) If P E m ( n ) , the left multiple map

is given by Lp (Y) = P Y , V Y E !3l(n). Similarly, the right multiple map is given by Rp(Y) = YP, VY E m(n). Clearly, both L p and Rp are smooth. Prove that

are given, via the natural identifications of these tangent spaces with the underlying Euclidean space m ( n ) , by ~ ( L P ) (A) Y = PA ~ ( R P )(A) Y =AP VA E 11IZ(n). (2) Let Lp and Rp be as above. If P E Gl(n), prove that the restrictions of Lp and Rp to Gl(n) are diffeomorphism of this open set onto itself. These diffeomorphisms Lp, Rp : Gl(n) + Gl(n) are called, respectively, the left and right translations by P. (3) The map @ : Gl(n) + Gl(n), defined by @(Y) = Y T ~ is ,smooth since its coordinate functions are quadratic polynomials. Prove the following: (a) Relative to the standard identification TI(Gl(n)) = ?IXf(n), the differential d a Z : Tz(Gl(n))+ Tz(Gl(n)) has the formula

+

(b) The map @ has constant rank n(n 1)/2. (c) Using the above, conclude that the orthogonal group O(n) c Gl(n) (Definition A.l) is a smooth, compact submanifold of dimension n(n - 1)/2. (d) Show that the vector subspace Tz(O(n)) c m ( n ) is the space of skew symmetric matrices. (4) Show that the r~dimensionalsubmanifold N is parallelizable if and only if there are r smooth vector fields on N which, at each point of N , are linearly independent. Using this, show that the submanifolds Sl(n) C Rn2 and O(n) c Rn2 are parallelizable. (5) Let N C Rn be a smooth submanifold of dimension r and let x E N. Prove the following:

2.6. CONSTRUCTIONS O F SMOOTH FUNCTIONS

(a) There is an open neighborhood Y

47

c N of x such that

is trivial. (We say that the tangent bundle of N is locally trivial.) is a smooth submanifold of dim 2r. (b) The subspace T ( N ) C

2.6. Constructions of Smooth Functions The main goal in this section is the proof of the following special case of the Cw Urysohn Lemma.

THEOREM 2.6.1. Let K & R" be compact and let U C Rn be a n open neighborhood of K . Then, there i s a smooth m a p f : Rn + [O, 11 such that f lK = 1 and supp(f) C U. Define h : R + [O, 1) by

The following standard fact can be proven by induction (Exercise ( 1 ) ) . LEMMA2.6.2. T h e finction h is C w , even at t = 0, where the derivatives are

for all integers n

> 1.

(One says that h is Cw-flat at t = 0.)

The graph of h is depicted in Figure 2.1.

FIGURE 2.1. The graph of h The functions h* : R + [O,1 ) are defined by

These are smooth and Coo-flat at t = 0 by exactly the same reasons that h is. The graphs are depicted in Figures 2.2 and 2.3 respectively.

2. LOCAL THEORY

FIGURE 2.2. The graph of h+

FIGURE 2.3. The graph of h , These functions are then combined to produce a C w function k : R

-+

[O, I),

k ( t ) = h-(t - b)h+(t - a ) , where a < b. This function is Coo.It is positive exactly for a < t < b (see Figure 2.4).

FIGURE 2.4. The graph of k LEMMA2.6.3. Let A = ( a l ,b l ) x - .- x (a,, b,) c Rn be a n open, bounded, n-dimensional interval. T h e n there i s a smooth function g : R n -+ [O, 1) such that g > 0 o n A and gl(Rn \ A) = 0.

PROOF.The definition of k gives functions ki,by taking a = ai and b = bi in that definition, 1 5 i n. Then

<

is as desired.

2.6. CONSTRUCTIONS

OF SMOOTH FUNCTIONS

Next we define a smooth function C : R

-+

[O, 11 by

where a and b are the numbers in the definition of k. This function is weakly monotonic increasing, e(t) zz 0 for t 5 a and C(t) I- 1 for t 2 b. The graph is depicted in Figure 2.5.

FIGURE2.5. The graph of

t

PROOFOF THEOREM 2.6.1. Let K and U be as in the statement of the theorem. For each x E K , let Ax be an open, bounded, n-dimensional interval, centered at x and having 71, C U. Apply Lemma 2.6.3 to obtain a smooth function g, : Rn -+ [O, I ) , strictly positive on A, and vanishing identically outside of A,. Since K is compact, it is covered by finitely many A , , . . . ,AXq. The function G = g,, - - - gXqis Cm on Rn,strictly positive on K, and has

+ +

Since K is compact, we can also find min(G1K) = S > 0. In the definition of C : R -+ [0,11, take a = 0 and b = 6. Then f = e o G : R n -+ [O, 11 is smooth, supp(f) c U, and f l K = 1.

EXERCISES (1) Prove Lemma 2.6.2. (2) Let U Rn be open, f : U -+ R smooth, and p E U. Prove that there is a Cm function f : Rn -+ R such that [f], = [f], in 8,. Conclude that, for arbitrary p E Rn,8, can be identified canonically with the set of germs at p of globally defined, smooth, real valued functions on R n. (3) Let C c Rn be a closed subset, U C Rn an open neighborhood of C . Show that there is a smooth, nonnegative function f : R n -+ R such that f lC > 0 and supp(f) C U.

2. LOCAL THEORY

50

2.7. Smooth Vector Fields

Let U C Rn be an open subset. In particular, this is a smooth submanifold and we consider the set X(U) of smooth vector fields on U (Definition 2.5.12). Remark that X(U) is a vector space over R under the pointwise operations. Throughout this section, the preferred way to think of tangent vectors will be as derivatives of the algebra of germs (Definition 2.2.15). Correspondingly, we will be able to view vector fields on U as first order partial differential operators on the algebra of smooth functions. The vector fields Di E X(U), 1 5 i 5 n, assign to each x E U the partial derivative operator Di,, (Example 2.2.17). The canonical identification T (U) = U x Rn is accomplished by the correspondence

From this, it is clear that the general element X E X(U) can be written

where f Z : U -+ R is smooth, 1 5 i 5 n. Remark that Cm(U) can be viewed as an associative algebra over R under the operations of pointwise addition, pointwise multiplication, and the usual multiplication of functions by scalars. This algebra has as unity the constant function 1. From the above observations, it is clear that X(U) is a module over the algebra Coo(U) . In fact, X(U) is a free Cm(U)-module with canonical basis

{Dl, - . ,Dn). We are going to give a deeper algebraic interpretation of X(U). For this, we need some definitions. Let F be an (associative) R-algebra with unity.

DEFINITION 2.7.1. A derivation A of the R-algebra F is a linear map

such that A(fg) = A(f)g be denoted V(F).

+ fA(g), V f , g E F. The set of derivations of F will

LEMMA2.7.2. The set of derivations V(F) is a vector space over R under the linear operations

Va, b E R, VA,, Az E V(F), V f E F. The proof of this is completely elementary and is left to the reader

2.7. SMOOTH VECTOR FIELDS

DEFINITION 2.7.3. If A l , A2 E V ( F ) ,then the Lie bracket

is the operator defined by

V f E F . This is also called the commutator of A 1 and A2.

LEMMA2.7.4. The Lie bracket satisfies the following properties:

v

(1) [A,,A21 E V ( F ) , Al, A2 E W ) ;

( 2 ) the operution , - 1 : V ( F )x V ( F ) + V ( F ) is R- bilinear; ( 3 ) [ A l ,A2]= - [A2, All, V Al ,Az E V ( F ) (anticommutativity); ( 4 ) [A,,[A,,A311 = [ [ A lA,21, A31 + [A,, D l , A311, v A1 A2, A3 E D ( F ) (the Jacobi identity). [

a

The proof is left as an exercise. Thus, we can think of the operation

as a bilinear multiplication making V ( F ) into a kind of R-algebra. This algebra is nonassociative, however, the Jacobi identity replacing the associative law. The algebra is also anticommutative and does not have a unity.

REMARK.One way to remember the Jacobi identity is to notice that, by this identity, the operator [A,- 1 : V ( F ) -4 V ( F ) is a derivation of the (nonassociative) algebra V ( F ) ,V A E V ( F ) . DEFINITION 2.7.5. A nonassociative algebra having the properties listed in Lemma 2.7.4 is called a Lie algebra. LEMMA2.7.6. If

L

E F is the unity and c E R, then A(cL)= 0 , V A E V ( F ) .

The proof is exactly like that of Lemma 2.2.19. Suppose that the algebra F is commutative and define an operation of "scalar" multiplication

F x D ( F ) -+ V ( F ) ,

LEMMA2.7.7. If F is commutative, the scalar product fA is an element of V ( F ) , V f E F , V A E V ( F ) . This makes V ( F ) a module over the algebra F . The elementary proof will be an exercise. LEMMA2.7.8. The space X ( U ) is a Coo(U)-submodule of V ( C o o ( U ) ) .

2. LOCAL

THEORY

PROOF.Indeed, given X E X(U), write it as

where f i E CM(U), 1 5 i 5 n. The Leibnitz rule for each partial derivative Di implies that X(fg) = X(f)g + fX(917 v ~ , ~ E c ~ ( u a) . We are going to prove the reverse inclusion

hence that X(U) is the full Lie algebra and Cm(U)-module V(CM(U))of derivations of the function algebra CM(U). LEMMA2.7.9 (KEY LEMMA).Let A E V(Cm(U)), f E Cm(U), and suppose that V E U is an open set such that f lV = 0. Then A(f)lV -= 0.

PROOF.Let x

E V.

By Theorem 2.6.1, we find cp E Coo(U)such that

Indeed, since {x) is compact, we find .II, E C" (U) such that @(x) = I and supp(.II,)c V. Then p = 1- $ is as desired. Since f lV 0, we see that cpf = f . Thus, A ( f ) = A(Pf = A(cp)f + vA(f 1, hence A ( f )(XI = A(v)(x)f (4+ v(x)A(f )(x). But f (x) = 0 = cp(x), so A( f)(x) = 0. Since x E V is arbitrary, A(f) lV 0.

>

-

COROLLARY 2.7.10. Let A E V(CM(U)), f E CM(U), and x E U. Then A(f)(x) depends only on A and the germ [f], E 8,. PROOF.Let f , g E CM(U) have the same germ [f], = [g],. Choose an open neighborhood W E U of x such that f 1 W = g ) W.By the Key Lemma 2.7.9,

Given w E @,, x E U, there exists f E CM(U)such that w = [f],. This is by Exercise (2) on page 49. This allows us to make the following definition.

DEFINITION 2.7.11. Given A by

V w = If],

E V(Cm(U)) and x E U, Ax : @, -,R is given

Ax (4= A(f >(x), E 8,, where f E CM(U).

By the above discussion, it is clear that Ax is well defined, Vx E U, V A E N C " (41.

2.7. SMOOTH

VECTOR FIELDS

53

PROPOSITION 2.7.12. If A E V(Cm(U)) and x E U, then Ax E T,(U). PROOF. Let [f],, [g], E 6,, where f , g E Cm(U). Then,

It isclear that A x :6, - + R is linear, so Ax E Tx(U). Given A E V(Cm(U)), define i: U -+ T(U) by i ( x ) = Ax E Tx(U). This is a section of the tangent bundle. If this section is smooth, then i E X(U) . Write

remarking that the smoothness of

is equivalent to

PROPOSITION 2.7.13. If A E V(Cm(U)), then

f' E Cm(u),1 5 i 5 n. E X(U).

PROOF. Consider the coordinate functions x i E U, 1 5 i 5 n. Let

Then

We summarize this discussion as follows.

-

THEOREM 2.7.14. Under the correspondence A L, V(Cm(U)) is canonically identified with X(U) as a Cm(U)-module, thus defining a Lie algebra structure on X(U) . From now on, we use either point of view freely, but we denote this Lie algebra and Cm(U)-module only by X(U). EXAMPLE2.7.15. One is often interested in certain Lie subalgebras of X(M). We give here an example on the group manifold Gl(n) to which we will be returning later. Let g[(n) c X(Gl(n)) denote the subspace of left invariant vector fields. That is, X E gI(n) if and only if Lp,(X) = X, V P E Gl(n). We can identify gI(n) with the space m ( n ) of n x n real matrices. Indeed, for each A E m ( n ) , view the right translation map R A : Gl(n) + !Dl(n) as a vector field on Gf(n). The value of this vector field a t Q E Gl(n) is the matrix QA E 1)31(n) = TQ(Gl(n)). Evidently, Lp,(QA) = PQA, the value of the field RA a t P Q = Lp(Q). That is, Lp+(RA)= R A , proving that RA E g[(n). Of course, the value of a left invariant field on Gl(n) at one point, say the identity I, determines the whole field, so

54

2.

LOCAL THEORY

RA is the unique left invariant field whose value at I is A E T r ( G l ( n ) )= !?R(n). It is evident that this one to one correspondence between g((n)and !?R(n)is an isomorphism of vector spaces. In Dt(n),define the bracket to be the usual commutator of matrices [A, B]= A B - BA, Q A, B E %R(n). The proof of the following lemma will be left as an exercise.

LEMMA 2.7.16. The commutator operation makes m(n)into a Lie algebra, and the canonical identification DZ(n) = g[(n)turns the commutator into the Lie Q A, B E !XQ(n),so bracket of vector fields. More precisely, [RA,RBI = g l ( n ) is a Lie subalgebra of X(Gl(n)) isomorphic t o the Lie algebra m(n)of n x n matrices under the commutator bracket.

NOTATIONAL CONVENTIONS. When v ( C M ( U ) )is thought of as the space X ( U ) of smooth vector fields, it is customary to denote its elements by capital letters from the end of the alphabet. The value of a field X E X ( U ) at a point q E U is usually denoted by X , and, if f E C m ( U ) , one writes Xq(f ) for X q [f],.

As noted earlier, a diffeomorphism z : U -+ V onto an open subset V R n can be thought of as a change of coordinates. The new coordinate functions are

Similarly, x = z-l gives the coordinates xi as functions of (zl,. . . ,zn). Recall that such a change of coordinates translates formulas in differential calculus to new formulas relative to the new coordinates. In the present context, vector fields X E X ( U ), viewed as first order differential operators, are carried to vector fields z , ( X ) E X ( V ) . One can think of this as a change of formula for the operator X in terms of the new coordinate system (V, z l , . . . , zn). In order to define z , ( X ) from this point of view, we need some definitions.

DEFINITION 2 . 7 . 1 7 . If z : U -+ V is a smooth map between open subsets of Euclidean spaces, then z* : C m ( V ) -t C M ( U )is defined by setting

for all f E C m ( V ) .

LEMMA2.7.18. If z : U 4 V is a diffeomorphism between open subsets of Rn, then r* : C M ( V )4 C m ( U ) is an isomorphism of algebras. This is completely straightforward to check. DEFINITION 2.7.19. If z : U 4 V is a diffeomorphisrn between open subsets of Rn, then t , : X ( U ) -t X ( V ) is defined by setting

for all X E X ( U ) and all f E C m ( V ) .

FLOWS

2.8. LOCAL

55

The fact that z,(X) E J ( V ) is elementary, as is the following. These are left as an exercise for the reader. LEMMA2.7.20. If z : U -+ V is a difleomorphisrn between open subsets of then z, : X(U) -+ X(V) is an isomorphism of Lie algebras.

Rn,

EXERCISES (1) Prove Lemma 2.7.4. (2) Prove Lemma 2.7.7. (3) Prove Lemma 2.7.16. (4) Prove Lemma 2.7.20 and the preceding statement. (5) Show that, if X E E(U) is viewed as a smooth section of the tangent bundle .rr : T(U) -+ U, then the section Z = z, ( X ) of the tangent bundle .rr : T(V) + V is given by ZC= d ~ , - l ( ~ ) ( X , - l ( ~VC ) ) ,E V. 2.8. Local Flows

We begin this section by showing how a vector field X E X(U) can be viewed as an (autonomous) system of ordinary differential equations (O.D.E.) on U. DEFINITION 2.8.1. Let s : (a, b) to E (a, b) is

-+ U

be smooth. The velocity vector of s at

REMARK.If T = t + t o , a - to < t < b - to, then a ( t ) = S(T) has velocity b(t) = d(r) = d(t to). In particular, i ( t o ) = b(0) = (a),(,o).We will write this infinitesimal curve as (s) ,(to).

+

DEFINITION 2.8.2. The map i : (a, b) the smooth curve s : (a, b) -+ U.

-+

Remark that p o s = s, where p : T(U) remark that s is smooth.

T(U) is called the velocity field of

-+

U is the bundle projection. Also

DEFINITION2.8.3. Let X E J ( U ) and xo E U. An integral curve to X through xo is a smooth curve s : (-6, E) -+ U, defined for suitable 6, E > 0, such that s(0) = xo and i ( t ) = Xs(,), -6 < t < e. Suppose that s is an integral curve to X E J ( U ) through xo E U. Write

and

2. LOCAL THEORY

56

At each t E (-6, E), the Jacobian matrix of s is

The vector

$1,

E Tt(-6, e ) = W 1 is the canonical basis element 1 E W 1 , so

;[ ] d"'( t )

ds.(:I,,

=

.Rn

dt (t)

But

Thus, s is an integral curve to X if and only if

dx' dt

-( t )= f " x l ( t ) ,. . . zn( t ) ) , )

-6 < t < E, 1 < i 5 n. This is an (autonomous) system of O.D.E. with solution s subject to the initial condition s(0) = xo. The existence and uniqueness of integral curves is guaranteed by the following theorem.

THEOREM 2.8.4. Let V G Rr and U Rn be open subsets, let c > 0, let f a E CDO((-c, c) x V x U ) , I _< i 5 n , and consider the system (nonautonomous with parameters b = ( b l , . . . , 6') E V ) of O.D.E.

Let a = (a1,.. . , an) E U. Then there are smooth functions x i ( t ,b), 1 _< i _< n , defined on some nondegenemte interval [-6, E] about 0, which satisfy the system (*) and the initial condition

firtherrnore, zf the functions Z i ( t , b), 1 5 i < n, gzve another solution, defined on [-6,4 and satisfying the same initial condztion, then these solutions agree on [-6, e] n [-8, q. Finally, if we w i t e these solutions as xi = xs'(t,b, a ) (in order to emphasize the dependence on the initial condition), there is a neighborhood W of a in U,a neighborhood B of 6 in V , and a choice of e > 0 such that the solutions xi(t, z , x) are defined and smooth on the open set (-e, E) x B x W IW"+'+~.

2.8. LOCAL FLOWS

57

This is the well known theorem giving the existence, uniqueness, and smooth dependence on initial conditions and parameters of solutions of systems of ordinary differential equations. The proof is given in Appendix C (where the smooth dependence on initial conditions will require some elementary facts from calculus in Banach spaces). DEFINITION2.8.5. The system (*) is autonomous if the functions f do not depend on t , 1 5 a 5 n. The problem of finding integral curves to a vector field is the problem of solving an autonomous system of O.D.E. without parameters. The proof of the existence, uniqueness, and smooth dependence of these curves on the initial conditions, however, involves an inductive step that requires the general formulation of Theorem 2.8.4. From now on, we consider only autonomous systems without parameters. COROLLARY 2.8.6. If X E X(U) and x E U, then there is an integral curve to X through x. Any two such curves agree on their common domain. DEFINITION2.8.7. A local flow @ around xo E U is a smooth map

(written @(t,x ) = a t ( x ) ) , where W is a suitable open neighborhood of x o in U, such that (1) a0: W --+ U is the inclusion W L) U; (2) @t,+t,(x) = @ t , ( @ t 2 ( x ) )whenever both sides of this equation are defined.

If r E U, the flow line through z is the curve a ( t ) = Qt(z), -E

< t < E.

THEOREM 2.8.8. Let X E X(U)andxo E U . Then there is a EocalfEow around xo such that the flow lines are integral curves to X . Two such local flows agree on their common domain.

PROOF. By Theorem 2.8.4, we find an open neighborhood W of xo in U and a number E > 0 such that the integral curve s,(t) through z is defined for - E < t < E and for all r E W. By the smooth dependence on initial conditions, we define a smooth map @ : (-E, E) X W -+ U

Theorem 2.8.4 also assures us that, if 6 is another local flow around xo with flow lines integral to X, then @ and 6 agree on their common domain. We must show that @ is a local flow. Since a. (r) = S , (0) = r, it is clear that Go is the inclusion map W U. Fix to E (-E, E) and r E W and define the curves

-

s(t) = sz(t

+ t o ) = Gt+to(z)

2. LOCAL THEORY

58

and ~ ( t=) S s z ( t o ) ( t ) = Qt(sz(t0))= %(@to (z)). These are defined for small values of t and are both integral to X. They satisfy s(0) = s z ( t o )and ~ ( 0=) s ~ , ( ~ ~ ) (=O s) z ( t o ) ,so s ( t ) = u ( t ) whenever both sides are defined. That is, @ t + t o ( r = ) a t ( G t 0 ( z ) whenever ) both sides are defined. El The locd flow @ associated to X E X(U) as in Theorem 2.8.8 is said to be generated by X. Also, the vector field X is called the infinitesimal generator of the local flow @. DEFINITION2.8.9.

FIGURE 2.6.

EXAMPLE 2.8.10. Let U

Flow lines for X = x l D l

+ x2D2

+

= R2 and X = X ' D ~ x2D2. The integral curves

s ( t ) = ( x l ( t ) ,x2( t ) ) will satisfy

The solution curve with initial condition s(0) = (a l , a 2 ) is

(see Figure 2.6). All of these solutions are defined for -oo < t < cm. Remark that @ t , + r 1, ( a2 , ~ ) =1 ( tl+t2 a e ,a2et1+t2)=~t,(@s(a1,a2)). This curve is stationary for ( a 1 ,a 2 ) = (0,O) and in all other cases follows a radial trajectory out of the origin, but is not parametrized linearly. The "speed" of the trajectory increases proportionally with the distance from the origin.

2.8. LOCAL

FLOWS

FIGURE 2.7. Flow lines for X EXAMPLE 2.8.11. Let U = It2 and X s(t) = (xl (t),x2(t)) will satisfy

=

= x l D l - x2D2

x1Dl - x2D2. The integral curves

The solution curve with initial condition s(0) = (a1, a 2 ) is

(see Figure 2.7). All of these solutions are defined for -w < t

< oo.

Again

This curve is stationary for (a1, a2) = (0,O). Tkajectories of points on the x2-axis (other than the origin) stay on the x2-axis and head toward the origin. Trajectories of points on the xl-axis (other than the origin) stay on that axis and head away from the origin. The remaining trajectories follow hyperbolic paths asymptotic to the coordinate axes. LEMMA2.8.12. Let X E X(U) generate a local flow @ : (-E,E) x W --+ U. Then, for each q E W, there is a neighborhood V of q in W and a number 6 > 0 such that (1) @,(V) W, -6 < t < 6; (2) at : V + W is an imbedding, -6 < t < 6; (3) (@t)*p(Xq)= Xar(p)7 -6 < t < 6. The proof of Lemma 2.8.12 is left as an exercise. So far, we have used vector fields to differentiate functions. Now we will show how a vector field can be used to differentiate another vector field. Let X E X ( U ) , q E U ,and let @ : (-E, e ) x W --+ U be a local flow about q generated

2. LOCAL

60

THEORY

by X. Let Y E X(U). By Lemma 2.8.12, there is a value 6 > 0 such that the vector (@-t),*,(,)(Ya,(,)) E T,(U) is defined, -6 < t < 6. Generally, this vector differs from Y,, although they are equal if Y = X (Lemma 2.8.12 again). The difference quotient

can be thought of as the average rate of change of Y near q along the integral curve to X through q. This lives in the vector space Tq(U) = Rn,so lirnt,0 Z,(t) makes sense. If this limit exists V q E U, we obtain a (conceivably not smooth) vector field @-t*(Y) - y Lx(Y) = lim t -40 t DEFINITION 2.8.13. If Lx (Y) exists on W, it is called the Lie derivative (on W ) of the field Y by the field X.

REMARK. One can also define the Lie derivative of a function by the formula Lx(f) = lim @t*(f)- f t-0 t The use of at instead of is due to the fact that @,' pulls functions back, while at, pushes vector fields forward. The reader should have no trouble seeing that Lx(f) = X(f).

THEOREM 2.8.14. If X, Y

E

X(U), the Lie derivative of Y by X is defined

and smooth throughout U and

The proof requires a couple of lemmas. We will fix q E U and a neighborhood : ( - E , E ) x W -+ U is defined. Write

W of q so that

so limt+o Zq(t) exists if and only if limt,o Ci(t) exists, 1 5 i

< n.

LEMMA2.8.15. The limit A, = limt,o Zq(t) exists if and only iA for each f E Coo (U) , lirnt+0 Zq(t)( f ) exists, in which case this limit is A , ( f ). PROOF. Assume that the limit A, = ~ : = , a 2 D i ,exists. , That is,

a' = lim

t-40

1 i: i

< n.

exists.

ci(t),

Then, for arbitrary f E Cm(U), it is clear that

2.8. LOCAL FLOWS

For the converse, suppose that, for each f E Cm(U), n

lirn Zq(t)(f) = lirn t+O

t+O

af 1cl(t)-(q) axi i=l

exists. Choose f = x j and define a3 = lirn 2, (t)(x3) = lirn cj (t), t-+O t+O 1 5 j 5 n. Thus, limt,o Zq(t) = A, exists and has coordinates a j , 1 5 j 5 n. For arbitrary f E Cm(U) it is also clear that

LEMMA2.8.16. Given f E Coo(U), there is a function g such that

E

Cm((-E, E) x W)

PROOF.Define h(t,x>= f ( @ - t ( 4 ) - f ( 4 , so h E Cm((-E,E) x W) and h(0,x) = 0. To simplify notation, we denote the partial of h with respect to t by h(t, x). Define g E Cm((-e, e) x W) by

Then

giving the first assertion. For the second, consider g(0, x) = lirn g(t, x) t+O

= lirn

f (@-t(x)>- f (5)

-t - f (4 = lirn f (@t(x>> t+O

t-ro

= X d f 1.

t

2. LOCAL THEORY

62

PROOFOF THEOREM 2.8.14. Let f E C" ( U ) and let g be as in the preceding lemma. Let gt : W --, U be given by g t ( x ) = g(t, x ) . Thus, X ( f ) = go = limt+o gt , so limY@,(q)(gt)= Y , ( X ( f ) ) .

t+O

We now compute

lim Z, ( t )( f ) = lim

t+O

t--0

( a - t t qq

) - Yq

t

-Ydf) t Y@t(q) ( f - t9t) - Y , ( f ) = lirn t+O t Y @ t ( q ) (-f )Y , ( f ) = lirn - t+O lim Y@, ( 9 )(gt) t+O t ) Y ( f) ( q ) = lirn Y ( f) ( @ t ( q ) - t+O lim YQ~(,) (gt) t+O t = X , ( Y ( f )) - Y , ( X ( f 1) = [x,YI,(f ). = lirn

Y@t(q)(f O a-t)

t+O

Since f is arbitrary, Lemma 2.8.15 gives the desired conclusion. Let X, Y E X ( U ) and let a, Q denote the respective local flows generated by these fields about some point q E U . We can choose 6, > 0 so that Qt\ES(q)and Q S a t ( q )are defined, -6, < s, t < 6,. As q E U varies, these bounds 6, will also vary. The local flows vary too, but they agree on overlaps by Theorem 2.8.4.

DEFINITION 2.8.17. The local flows of X and Y commute on U if @t!Ps(q)= @,at(q), -6, < s, t < 6,) V q E U . The vector fields themselves commute on U if [X,Y ]_= 0 on U . THEOREM 2.8.18. The vector fields X and Y commute on U zf and only if their local flows commute on U. Suppose that the local flows commute on U . Then, for -6, < t < 6,) < s < 6,) of \E onto another flow line of 3P. By taking the infinitesimal curve point of view, we see immediately that (at),,(Y,) = Y@,(,),-6, < t < S,, V x E U . That is, PROOF.

at carries any flow line {\Es(x) 1 -6,

[ X ,Y ]= lirn t+O

a-t*(Y)- Y Y-Y - lirn =0 t+o t t

-

throughout U . For the converse, we assume that [ X , Y ] 0 on U and deduce that the local flows commute. Let q E U , fix s E (-6,, 6,), and let q' = \E,(q). Define v : (-6,)6,)--, T,, ( U ) by the formula v(t)= (Y@,(,,)).(Were and elsewhere,

2.8. LOCAL FLOWS

63

<

in an attempt to streamline notation, we drop the subscript on differentials f,<.) Then v(t) is a differentiable curve in the vector space T,)(U) = Rn, and

dv = lim (@-t-h)* ( ~ * ~ + ~ ( q l) )a-t*(Yat(,I)) dt

h @-h*(Y@t+h(,l))- Y@t(,')

h-0

=

=

lim

a_+, lim

a-h*(Y@h(@t(ql))

1 - Y@t(q/)

h

h+O

-6, < t < 6,. It follows that v(t) is constant on (-6,, 6,), so (Y*, (,/I) = Y,,, -6, < t < 6,. But q' ranges over a(s) = \E,(q), -6, < s < 6,, an integral curve to Y . Thus, b(s) = Y,(.) and at.(b(s)) = Y@,(,(,)) as s and t range independently

over (-6,, 6,). Therefore, at 0 a is also an integral curve to Y with initial condition at (a(0)) = @t(q). By the uniqueness part of Theorem 2.8.4,

EXERCISES (1) Carry out a discussion similar to that in Examples 2.8.10 and 2.8.11 for the vector field X = x 2D~ - X ' D ~on IE2. In particular, sketch the flow lines and show that they are defined for -oo < t < oo. (2) On U = R, consider the vector field X = et$. Let a E W and compute the integral curve s, to X through a. Be sure to find the largest open interval (77, W) on which S, (t) is defined. (3) Prove Lemma 2.8.12. (4) Given A E !lX(n), view the right translation operation

as a vector field RA E X(Gl(n)). (a) For

show that the local flow generated by R A has the formula

Vt E R, VQ E Gl(2). In particular, we obtain a global flow

2. LOCAL THEORY

64

(b) Note that the formal definition

yields

Compute etB for the matrix

and make an educated guess of a flow on Gl(2) generated by R B . Prove that your guess is correct. 2.9. Critical Points and Critical Values Let U

Rn and V

Rm be open and let @ : U

DEFINITION 2.9.1. A point x E

-+

V be smooth.

U is a regular point of

if

is surjective. Otherwise, x is a critical point. Thus, smooth maps from lower dimensions to higher dimensions have only has only regular points, it is a subcritical points. At the other extreme, if mersion. DEFINITION 2.9.2. A point y E V is a critical value of if @ ~ at least one critical point of a. Otherwise, y is a regular value of a.

l (contains ~ )

You must take care with these terms. If @-'(y) = 0, then this set contains no critical points. That is, a point y E V which is not a value of at all is a regular value of a! THEOREM 2.9.3 (SARD'S THEOREM).If @ : U -+ V is a smooth map, then the set of critical values has Lebesgue measure zero. That is, Lebesgue almost every (a.e.) point of V is a regular value. A proof of this important result is given in Appendix D. EXAMPLE 2.9.4. It is well known that one can construct a continuous surjection s : R -+ R2 (a "space filling curve"). However, if s is smooth, every true value of s is a critical value, so s(R) c EX2 has measure zero. Smooth curves cannot be space filling. More generally, smooth maps from lower to higher dimensions always have images of measure zero. COROLLARY 2.9.5. Let {ai : Ui -+ v}z1, 1 5 N 5 m, be a n at most countable family of smooth maps, each U i Rni being open. Then a.e. y E V is simultaneously a regular value of ai, 1 5 i < N + 1.

2.9. CRITICAL POINTS

65

PROOF. Let Cidenote the set of critical values of cPi, 1 5 i < N + 1. Then C = Ui=1 Ci is a set of Lebesgue measure zero and the complement of C is the set of simultaneous regular values. The fact that a.e. point is a regular value says that the situation in the following theorem is somehow "generic".

THEOREM 2.9.6. If cP : U -+ V is a smooth map and if y E V is a regular value, then @-'(y) is a smooth submanifold of U of dimension n - m. PROOF.If

@-l ( y ) =

0, this is a submanifold

of U of any desired dimension and we are done. We consider the interesting case, therefore, in which the regular value y is actually a value of cP. Let xo E W 1 ( y ) . Since J @ ( x o )has maximum rank m, there is a neighborhood W,, of xo in U such that JcP(z) has rank m, V z E W,, . Let

w = (J wx, ~€9-'(y)

an open neighborhood of cPll(y) in U on which @ has rank m. The assertion follows from Theorem 2.5.3. IJ

Of special note are the critical points of a smooth, real-valued function. S u p pose that f : U + R is such a map, where U R n is open. We sketch some facts that are the beginnings of "Morse theory", a remarkable application of critical point theory to topology. The proofs of the following two lemmas are left as exercises. LEMMA2.9.7. If f : U -+ R is smooth, x E U , and v E T,(U), then, under the canonical identification Tf(,)(W) = R and for all D x E T x ( U ) ,

LEMMA2.9.8. Let p E U be a critical point of f . If X , Y E 3E(U), then X p ( Y ( f ) )= Y p ( X ( f ) ) ,and this number depends only on the tangent vectors X p , Yp,not on their extensions to fields on U . Each element YpE T p ( U )can be extended to a vector field Y E X ( U ) . Indeed, if Yp= C:=,aQD,,,, we can define Y = Cy=la' Di. This remark, together with the lemma, insures that the following definition makes sense. DEFINITION 2.9.9. If p E U is a critical point of f , the Hessian of f at p is the symmetric, bilinear form H p( f ) : T p ( U )x T p ( U )+ R defined by

The critical point is nondegenerate if the symmetric matrix representing the Hessian relative to any choice of basis is nonsingular. The index X of the critical point is the number of negative eigenvalues of this matrix. The following theorem is due to Marston Morse. We guide you through a proof in the exercises.

2. LOCAL THEORY

66

THEOREM 2.9.10 (THE MORSELEMMA).Let p E U be a nondegenerate critical point of index X of the smooth jbnction f : U -t R. Then there is an open neighborhood Up of p in U and a smooth change of coodinates (i.e., diffeomorphism) z : Up -+ W onto a n open neighborhood W of 0 in R n such that, relative to the new coordinates z = (zl, . . . ,zn), (1) P = 0; (2) f has the formula

EXERCISES (1) Prove Lemma 2.9.7. (2) Prove Lemma 2.9.8. (3) Show that, relative to the standard basis {Dl,,, matrix representing the Hessian is the matrix

. . . ,Dn,p) of Tp(U), the

of 2"* partials of f at p. (4) Let p E U be a nondegenerate critical point of f : U 4 R. Show that, in a suitable neighborhood Up C U of p, there are defined smooth, real valued functions functions hij = h j i , 1 5 i,j 5 n, such that n

f (xl,. . . ,x") = f ( p )

+ C ( x 2- xi(p))(xj - xj(p))h,(xl,.

. . ,xn).

i, j

Show also that the matrix [hij(p)] is the matrix of the Hessian Hp(f ) . (5) Using the above, mimic the standard way of diagonalizing a symmetric matrix so as to prove the Morse Lemma.

CHAPTER 3 The Global Theory of Smooth Functions

Our present goal is to extend the theory of smooth functions, developed on open subsets of Rn in Chapter 2, to arbitrary differentiable manifolds. Geometric topology becomes an essential feature. 3.1. Smooth Manifolds and Mappings

Let M be a topological manifold of dimension n. The locally Euclidean p r o p erty allows us to choose local coordinates in any small region of M.

DEFINITION 3.1.1. A coordinate chart on M is a pair (U, cp), where U C M is an open subset and cp : U -+ Rn is a homeomorphism onto an open subset of Rn . One often writes cp(p) = (xl(p),. . . ,xn(p)), viewing this as the coordinate n-tuple of the point p E U. Relative to such a coordinatization, one can do calculus in the region U of M. The problem is that the point p will generally belong to infinitely many different coordinate charts and calculus in one of these coordinatizations about p might not agree with calculus in another. One needs the coordinate systems to be smoothly compatible in the following sense.

DEFINITION 3.1.2. Two coordinate charts, (U, cp) and (V, $) on M are said to be CW-related if either U n V = 0 or

is a diffeomorphism (between open subsets of Rn). We think of cp o $I-'as a smooth change of coordinates on U n V. Thus, on U nV, functions are smooth relative to one coordinate system if and only if they are smooth relative to the other. Indeed, differential calculus carried out in U nV via the coordinates of cp(U n V) is equivalent to the calculus carried out via the coordinates of $(U n V). The explicit formulas will, of course, change from the one coordinate system to the other. Furthermore, piecing together these local calculi produces a global calculus on M. The concept that allows us to make these remarks precise is that of a smooth atlas.

68

3. GLOBAL THEORY

DEFINITION 3.1.3. A Cm atlas on M is a collection A = {(U,, cp,)),,, coordinate charts such that (1) (U,, 9,) is Cm-related to (Up, pp), V a , ,B E a; (2) M = ,U ,, Ua-

of

DEFINITION 3.1.4. Two Cm atlases A and A' on M are equivalent if A U A' is also a Cm atlas on M. It will be seen that global calculus carried out relative to A will be identical to global calculus carried out relative to the equivalent atlas A'.

LEMMA 3.1.5. Equivalence of Cm atlases is an equivalence relation. Each Cm atlas o n M is equivalent to a unique maximal C m atlas o n M. The proof of this lemma is left as an exercise. DEFINITION 3.1.6. A maximal C m atlas A on M is called a smooth structure on M (also called a differentiable structure or a C m structure). The pair (M, A) is called a smooth (or differentiable or C m )n-manifold. By a typical abuse of notation, we usually write M for the smooth manifold, the presence of the differentiable structure A being understood. By Lemma 3.1.5, any Cm atlas (not necessarily maximal) on M completely determines the differentiable structure.

EXAMPLE 3.1.7. The manifold Rn has a canonical smooth structure, namely the set An of all pairs (U, cp) where U C Rn is open and cp : U + Rn is a diffeomorphisrn onto an open set p(U) C Rn. EXAMPLE 3.1.8. If M is a smooth m-manifold and N a smooth n-manifold with respective smooth structures A = {(U,, pa))aEll and B = {(Vp, +p))p,,, then M x N is canonically a smooth (m + n)-manifold. Indeed, is a Cm atlas, determining uniquely a maximal one, called the Cartesian product of the two smooth structures.

EXAMPLE 3.1.9. If W C M is an open subset of a smooth n-manifold, then W is a smooth n-manifold in a natural way. Details are left as an exercise. REMARK. By substituting Ck for Cm in the above discussion, one obtains the notion of a Ck manifold, 1 5 k < oo. Similarly, one defines the notion of a real analytic ( C W manifold. ) The reader should have no trouble in adapting the following discussion to these cases. We Let M be a smooth n-manifold with a smooth atlas A = {(U,, do not require this atlas to be maximal. Set

These local diffeomorphisms in

R n satisfy the cocycle conditions

3.1. SMOOTH MANIFOLDS

69

( 3 ) spa = g i j . It should be noted that properties ( 2 ) and ( 3 ) follow from property (1). DEFINITION 3.1.10. The system is called a structure cocycle for the smooth manifold M . The term "cocycle" comes from algebraic topology and is used because of a certain formal analogy between the cocycle property ( 1 ) and the definition of ~ech cocycles (cf. page 272).

REMARK. It will be useful to see how to "reassemble" M out of the data (Ua = v a ( U a ) ,g a p ) a , p E a . On the disjoint union

define the relation

x - y o 3 a , p ~ % s u c h t h axt E fi,,

y E fib and y = g p , ( x ) .

By properties (I), (2), and(3), this is an equivalence relation, so we form the topological quotient space MI-. We will show that this space is homeomorphic to M and exhibit a natural smooth structure on it. denote the equivalence class of z E %. Define Let [z]E

by setting cp(x) = (cpa(x)l if x E U,. If a E U p also, then gap ( q p ( x ) ) = v a ( x ) , so cp is well defined. It is also continuous. The map from G to M that takes z E6 ,to 9;' ( z )respects the equivalence relation, hence passes to a continuous map

I/J:G/-+M.

It is easy to see that (o and I/J are mutually inverse, so M and G/ware canonically homeomorphic. Each 0, imbeds canonically in E/- as an open subset and id,) on M I - . These charts id, : U, + 0, Rn defines a coordinate chart (6,, are Cm-related via the cocycle so is canonically identified with M as a smooth manifold via the mutually inverse dzfleomorphisms cp and 11, (see Definition 3.1.16).

-

We turn to the smooth maps defined on a manifold.

DEFINITION 3.1.11. A function f : M + R is said to be smooth if, for each x E M , there is a chart (U,cp) E A such that x E U and

is smooth. The set of all smooth, real valued functions on M will be denoted cm(M). The definition only requires us to be able to find some such chart about each point x E M , but the following assures us that all charts will then work.

3. GLOBAL THEORY

70

LEMMA3.1.12. The function f : M

+

IW is smooth zf and only i f

is smooth, V (U,, 9,) f A.

PROOF.Clearly this condition implies that f is smooth. For the converse, suppose that f is smooth and let x E U , where (U,,cp,) E A. By Definition 3.1.11, choose (Up,cpp) E d such that x E Up and

is smooth. Then,

f

OPE' : v a ( U a n U p ) + R

is given by the composition

As a composition of smooth maps, this is smooth. That is,

is smooth on some neighborhood of the point P,(x). But x E U, is arbitrary, so f o is smooth on all of cp,(U,). We think of f o cp;' as a formula for f JU, relative to the coordinate system cp, = ( x k ,. . . ,zz).We generally write (U,, x:, . . . ,x:) or (U,, x,) for (U,, pa). DEFINITION 3.1.13. Let M and N be Cw manifolds with respective smooth structures A and 23. A mapping f : M -4N is said to be smooth if, for each x E M , there are (U,,cp,) E A and (Vp,$Jp) E B such that x E U,, f ( U a ) C Vp, and $Jp

f

9,'

: ~pa(ua) $ J p ( V p ) +

is smooth. LEMMA3.1.14. The map f : M --, N is smooth i f and on13 if, for all choices of (U,, 9,) E A and (Vp,$ J p ) E L3 such that f (U,)C Vp, the map

is smooth. The proof is similar to the previous one and is left to the reader. Again, we think of $ p o f o p;' as a local coordinate formula for f . These two lemmas give an important part of the content of our remark that differential calculus in one coordinate chart is equivalent, in overlaps, to differential calculus in Cw-related neighboring charts. LEMMA3.1.15. I f f : M --, N and g : N -, P are smooth maps between manifolds, then g o f : M 4 P is smooth. This is also elementary and is left to the reader.

3.1. SMOOTH MANIFOLDS ~ E F I N I T ~3.1.16. ~ N

71

If M and N are smooth manifolds, a map f : M + N

is a diffeomorphism if it is smooth and there is a smooth map g : N that f o g = i d N a n d g o f = i d M .

+

M such

EXAMPLE 3.1.17. The maps

on page 69 are mutually inverse diffeomorphisms.

L E M M A3.1.18. If ( M , A ) is a smooth n-manifold, U & M an open subset, and cp : U -+ Rn a difleomorphism of U onto an open subset cp(U) of Rn, then (U,cp) f A. PROOF.By the definition of diffeomorphism, (U,cp) is Cw-related to every (Ua,va)E A, so (U, cp) f A by the maximality of this atlas. If we write (U,cp) as (U,x l , x 2 , . . . ,x n ) , the above discussion allows us to write f ( x l , .. . ,x n ) for f lU, whenever f : M N is smooth. This is logically a bit sloppy, but it is psychologically helpful. If p E M , it makes good sense to talk about germs at p of real valued Coo functions defined on open neighborhoods of p. As before, these form an associative algebra 6 , over R. The evaluation map ep : 6 , + R is defined exactly as before. -+

DEFINITION 3.1.19. A derivative of 6 , is an R-linear map

such that

D(rc) = D ( C ) ~ ~+ ( C~ ,) ( s ) D ( c ) , Vc,C E Q j p . This operator D is also called a tangent vector to M at p and the vector space T p ( M )of all derivatives of 8 , is called the tangent space to M at P-

DEFINITION 3.1.20. If f p f M, the differential

:M

+N

is a smooth map between manifolds and if

is the linear map defined by

(f*p(D))[gl f ( p ) = DL9 for all D E T p ( M )

flP,

all [ g l f( p ) E @ f ( p ) .

L E M M A3.1.21 (GLOBALCHAIN RULE). I f f : M + N and g : N + P are smooth maps between manifolds and x E M , then d ( g o f ) , = dgf),( o df,.

3. GLOBAL THEORY

PROOF. Consider ( ( 9O f )*p(D))[hIg(f(p)) = D [ hO f

O glp

= (A,(Dl[h 91f = (9.f

sinceIh1g(f ( P I )

E

@g(f (PI)

( P ))

( P ) (f*P(D)))[hl,(f (p)).

and D E T p ( M )are arbitrary, the assertion follows.

It is clear that id,, = id : T p ( M )-,T p ( M ,) so the chain rule has the following consequence.

COROLLARY 3.1.22. If f

: M -+

N is a difleomorphisrn, then

is an isomorphism of real vector spaces, V p f M .

COROLLARY 3.1.23. If M is a smooth manifold of dimension n, then T , ( M ) is a real vector space of dimension n, V x E M . PROOF. Let (U,9 ) be a coordinate patch on M with x E U.Then, T z ( U ) = T, ( M ) and, by the previous corollary, is an IW- linear isomorphism. Since v ( U ) E R n is open, we know that

The same kind of argument gives the following.

COROLLARY3.1.24. If f dim N .

:

M

M

N as a difleomorphism, then dim M =

REMARK. We do not have a canonical basis for T z ( M ) since there is no preferred choice of local coordinates about x. Thus, we cannot write T , ( M ) =

Rn. The notion of infinitesimal curve ( s ) makes ~ sense in our context, as does the derivative D f s ) pE T p ( M ) . Viewing tangent vectors as infinitesimal curves is preferable from an intuitive point of view, making the differential of a map "visible" and making the chain rule evident. If one is developing a theory of C manifolds, defining tangent vectors to be infinitesimal curves rather than first order differential operators is essential [33]. The following is proven much as Corollary 2.2.22 and Lemma 2.3.9. Details are left as an exercise.

LEMMA 3.1.25. The correspondence ( s ) * ~ D ( s ) pis a one to one correspondence between the set of infinitesimal curves at p E M and T p ( M ) . Furthermore, i f f : M -+ N is a smooth mapping between manzfolds, the diflerential f a P : T p ( M )-+ T f(,) ( N ) is given, in terms of infinitesimal curves, by

3.2.

DIFFEOMORPHIC STRUCTURES

73

Finally, for smooth maps f : M -, N, we define the notions of regular point, critical point, critical value, and regular value exactly as before.

EXERCISES (1) Prove Lemma 3.1.5. (2) Supply the details in Example 3.1.9 (3) Prove Lemma 3.1.25. (4) Prove that the topological n-manifold P" (see Exercise (3) on page 14) is a smooth n-manifold. (5) Show that the manifold Sn can be assembled from a Cooatlas with just two charts. 3.2. Diffeomorphic Structures

This section is really an extended remark on some very deep theorems, the point of which can now be easily appreciated. Let M be a differentiable manifold with smooth structure A = {(Ua, a. Let @ : M -,M be any homeomorphism. Set

PROPOSITION 3.2.1. The set A* is a Coostructure on M having the same structure cocycle as A. PROOF.Indeed, gap = pa

0

pi1 = (pa 0 @)

0

(pP 0 b)- l ,

and this map carries the set

onto the set (pa 0 @)(@-'(U,)n @-l (Up)) = pa(Ua n Up). Finally, the maximality of the Cm atlas A* follows from that of A and the fact that A*+-I = A . D

a/w,

COROLLARY 3.2.2. The smooth manifold whether definedhrn the Cm structure A or A*, is the same. Consequently, (M,A) and (M, A+) are canonically difleomorphic. We will let Ma denote the smooth manifold (M, A@). DEFINITION 3.2.3. Two Coostructures A and B, defined on the same topological manifold M , are said to be diffeomorphic structures if B = A*, for some homeomorphism @ : M -,M. It is clear that diffeomorphism is an equivalence relation on the set of smooth structures on a topological manifold M. It is natural to ask how many diffe* morphism classes of smooth structures a given topological manifold can support. The following examples are the deep facts, referred to at the beginning of this section, that we cannot prove here. The methods of proof are quite advanced.

3. GLOBAL THEORY

74

EXAMPLE 3.2.4. If n # 4, any two smooth structures on IWn are diffeomorphic 1391.

EXAMPLE 3.2.5. The case of IR4 was cracked by the combined work of a topologist, M. Freedman, and a global analyst, S. Donaldson, showing that IW admits a differentiable structure not diffeomorphic to the usual one. In this structure, it is possible to find a compact set that cannot be surrounded by any smoothly imbedded S3! (For a discussion of this, see [lo, Section I].) Subsequently, various researchers found more "exotic" differentiable structures on IW4 (the first of these was R. Gompf, who found two new structures [ll]). It is now known that the number of distinct diffeomorphism classes of differentiable structures on R 4 is uncountably infinite. There is even a "universal" smooth (IW4, A,) such that every other smooth (IW4,A) smoothly imbeds as an open subset of (IW4,A).

EXAMPLE 3.2.6. Let a ( n ) denote the number of diffeomorphism classes of differentiable structures on Sn. It was long known that a(n) = 1 for n = 1,2,3. The value of a(4) remains a mystery. The following table for 5 5 n 5 18 was computed by Kervaire and Milnor [21].

TABLE3.1. Differentiable structures on spheres EXAMPLE 3.2.7. A topological manifold is said to be smoothable if it can be given a differentiable structure. For n = 1,2,3, it is known that all topological nmanifolds are smoothable. The first dimension in which there exist nonsmoothable manifolds is n = 4 [lo, p. 231.

EXERCISES (1) Let @ : R + R be the homeomorphism @(x) = x3. Show that the identity map, viewed as id : R* + R, is not a diffeomorphism (although it is clearly a homeomorphism). (2) On the other hand, show that, for any homeomorphism @ : M + M of a differentiable manifold M , @ : Ma + M is a diffeomorphism. 3.3. The Tangent Bundle

Let M be a Cm n-manifold with smooth structure {(U,, pa))aEm.Consider the set

a disjoint union with, as yet, no topological structure. For each U,, cr E f # , define

3.3. TANGENT BUNDLE

Then the individual linear maps dp,,,

75

x E U,, unite to define a set map

More precisely, if v, denotes a tangent vector to M at x E U,, and this defines a bijection of T ( U f f onto ) an open subset of R2". If U, nUp # consider d ~ C l :T(~p(UCt T(pa(UQ U p ) ) . By the chain rule, this is

-

0,

a Cw diffeomorphism between open subsets of R2n. We topologize the set T . If

is an open set, then d v i l ( W ) is to be an open subset of T . The following is left as an exercise.

LEMMA3.3.1. T h e above sets form the base of a topology o n T and, in this topology, T is a topological manifold of dimension 2n. Furthermore, the s y s t e m {(T(U,),dp,)),Ezr i s a (not maximal) Cm atlas o n T determining a maximal such atlas A. Finally, i f T ( M ) denotes the diflerentiable manifold ( T , A ) , the m a p T : T ( M ) + M , defined by ~ ( v=) x H v E T,(M), is smooth. DEFINITION 3.3.2. The system T : T ( M ) + M is called the tangent bundle of M . The total space is T ( M ) ,the base space is M , and 7r is called the bundle projection.

REMARK.It is often convenient to replace cp,(U,) x Rn with U, x R n , identifying dp, with the map v, H ( x ,dpaz(v,). This minor abuse of notation will be a major convenience in what follows. For each a f %, we get a commutative diagram

where pl denotes projection onto the first factor and d y , is a diffeomorphism that restricts to be a linear isomorphism T,(M) + { x ) x R n , V x E U,. Thus, T ( M ) is "locally" a Cartesian product of M and R n , the projection T being "locally" the projection of the Cartesian product onto the first factor, and the fiber 7r-'(z) = T,(M) has a canonical vector space structure, V x E M .

DEFINITION 3.3.3. A vector field on M is a smooth map X : M + T ( M ) ( p H X p ) such that ?r o X = idM. The set of all vector fields on M is denoted by X ( M ) .

76

3. GLOBAL THEORY

REMARK.Let X be a vector field on M, (U, zl,. . . ,xn) a coordinate chart on M. By this point in the course, the reader should be able to justify writing

where f : U -,R is smooth, 1 5 i 5 n. Tangent bundles are examples of vector bundles. Vector bundles play a very important role in manifold theory, so we close this section with a brief discussion of the general theory.

DEFINITION 3.3.4. Let M be a smooth m-manifold, E a smooth ( m + n)manifold, and 7r : E 4 M a smooth map. This will be called an n-plane bundle over M (or a vector bundle over M of fiber dimension n) if the following properties hold. (I) For each x E M, Ex = x-I (x) has the structure of a real, n-dimensional vector space. (2) There is an open cover { W j)j J of M , together with commutative diagrams

such that + j is a diffeomorphism, V j E J . (3) For each j E J and x E W j, $ j X = $ j JEzmaps the vector space isomorphically onto the vector space {x) x Rn.

Ex

As with tangent bundles, we call E the total space, M the base space, and x the bundle projection. We also call each W j a trivializing neighborhood for the bundle and { W j)j E J a locally trivializing cover (of M ) for E. An obvious example of an n-plane bundle is given by p l : M x Rn -, M. Here, M itself is a trivializing neighborhood and the bundle is said to be trivial. DEFINITION 3.3.5. Let .rrl : El -, M and ?rz : Ez -, M be n-plane bundles over M . A bundle isomorphism is a commutative diagram

M-M id

such that cp is bijective, smooth, and carries space) onto Ezx,V x E M.

El, isomorphically (as a vector

Note that we did not explicitly require that cp-' be smooth. One needs this to be true, however, in order that bundle isomorphism be an equivalence relation. The following lemma, the proof of which will be an exercise, comes to the rescue.

3.3. TANGENT BUNDLE

77

Remark that it is useful not to be required to check in each instance that cp is a diffeomorphism. LEMMA3.3.6. The map cp in Definition 3.3.5 is necessarily a difleomorphism. DEFINITION 3.3.7. An n-plane bundle is trivial if it is isomorphic to the product bundle pl : M x Rn + M . DEFINITION 3.3.8. A section of the n-plane bundle ?r : E + M is a smooth map s : M + E such that ?r o s = idM. The set of a 1 such sections is denoted by r ( E ) . Thus, X ( M ) = r ( T ( M ) ) .The following is elementary and is left to the reader. LEMMA3.3.9. The set r ( E ) is a Coo(M)-module under the pointwise operations

where f f C o o ( M )and si f r ( E ) , i = 1,2. DEFINITION 3.3.10. The manifold M is parallelizable if there are fields

such that { X I , , X2x,. . . , X n z ) is a basis of T x ( M ) ,V x f M .

PROPOSITION 3.3.11. The manifold M is pamllelizable i f and only if T ( M ) is a trivial bundle. This proposition is a special case of Exercise (4). So far, the only real examples of vector bundles that we have seen are tangent bundles and trivial bundles. The following is the least complicated example of a nontrivial vector bundle. EXAMPLE 3.3.12. We give an example of a 1-plane bundle (a "line" bundle) over the circle, known as the Mobius bundle. On R x R, define the equivalence relation ( s ,t ) N ( S n, (-l)nt), n f Z . Remark that t H (-1)"t is a linear automorphism of R. The projection ( s ,t ) H s passes to a well defined map ?r : (W x Ill)/(-) + W/Z = S 1 . It should be clear, intuitively, that this is a vector bundle over S1of fiber dimension 1, but a rigorous proof of this involves checking many details. This will be Exercise (3).

+

EXERCISES (1) Prove Lemma 3.3.1. ( 2 ) Prove Lemma 3.3.6. ( 3 ) Give a careful proof that the Mobius bundle (Example 3.3.12) is truly a line bundle over S 1 . Show that this bundle is not trivial. (4) Prove that the n-plane bundle .;lr : E 4 M is trivial if and only if there are sections sl ,. . . ,sn E r ( E ) such that {sl (x), . . . ,sn (x)) is a basis of E,, V x E M . In particular, this proves Proposition 3.3.11.

3. GLOBAL THEORY

3.4. Cocycles and Geometric Structures

Let n : E -,M be an n-plane bundle and let {Wj)jEJ be a locally trivializing open cover for El the trivializations being qh : n-I (Wj) -r Wj xEn. If Wi f~ Wj # 0,consider

This composition must have the form

where yji(x) E Gl(n), V x E Wi n Wj. LEMMA3.4.1. The map yji : W,

n Wj

--t

Gl(n) is smooth.

PROOF. Let ek denote the column vector with 0's in all places except the kth, where the entry is 1. Then ak(x) = @ j ~ ; l ( ~ ek) , defines a smooth map 0,

: W; f l Wj

--t

(W; n Wj) x Rn,

1 5 k 5 n. But a k ( x ) = (x,yji(x) ek), and the second entry is just the kth column of y j i ( ~ ) .Since k is arbitrary, 1 5 k 5 n, we see that the n2 entries of y j i ( ~ are ) smooth functions of x. These smooth maps have the "cocycle" property

Vx E Wi n Wj n Wk, Vi, j, k E J. As usual, this property implies the following also, for all appropriate choices of x and indices i, j E J .

Again, this "cocyc~e"terminology comes from cohomology theory (cf. page 272). DEFINITION3.4.2. A Gl(n)-cocycle on M is a family 7 = { Wj ,yji),, jEJ such that {Wj)jEJ is an open cover of M and yji : W iI I Wj -, Gl(n) is a smooth map, V i , j E J, all subject to the cocycle condition (3.1). If the cocycle y arises as above from an n-plane bundle E , it is said to be a structure cocycle for E. Just as a structure cocycle g = {U,, for M gave all the data necessary for reassembling the smooth manifold, up to diffeomorphism, so a structure cocycle y for E gives all the data necessary for reassembling the bundle, up to bundle isomorphism. Indeed, given any Gl(n)-cocycle y = {Wj ,yji),, jEJ , one assembles an n-plane bundle for which it is a structure cocycle. Here is a quick sketch of the procedure. Set

3.4. COCYCLES

79

-

and define on J!?T an equivalence relation by setting (x, u ) (y, w) if (x, u) E Wj x Rn, (3, W)€ Wi x Rn, x = in M, and w = rij(x) - v . The fact that this is an equivalence relation is an obvious consequence of equations (3.1), (3.2), and (3.3). Then the standard - projections Wj x Rn + Wj fit together as maps into M to define a map ir : EY + M that respects the equivalence relation. We obtain

and the reader can check that this is a smooth n-plane bundle. In case the cocycle came from local trivializations $ j : x-I (Wj) -+ W j x Wn of a bundle n : E + M , the diffeomorphisms fit together to define : $ + E, again respecting the equivalence relation, and this defines a bundle isomorphism

&

as the reader again can check.

DEFINITION 3.4.3. Two Gl(n)-cocycles

on the same manifold M are equivalent if they are contained in a common Gl(n)cocycle on M. The equivalence class of y will be denoted [y]. In order for [y] to make sense, one must prove that equivalence of cocycles is an equivalence relation. LEMMA3.4.4. If two Gl(n)-cocycles on the same manifold contain a common Gl(n)-cocycle, then they are contained in a common Gl(n)-cocycle. The proof of Lemma 3.4.4 will be an exercise. COROLLARY 3.4.5. Equivalence of Gl(n)-cocycles is an equivalence relation.

-

-

PROOF. The only problem is transitivity. If y 8 and 8 6, let $ be a cocycle containing both y and 8, cp a cocycle containing both 8 and 6. Then $ and cp both contain 8, so Lemma 3.4.4 guarantees that they are contained in a common cocycle p. Then y C p and 6 p, so y 6.

-

Let Vectn(M) denote the set of isomorphism classes [El of n-plane bundles E on M and let H' (M; Gl(n)) denote the set of equivalence classes of Gl(n)cocycles (notation borrowed from algebraic topology, again because of formal analogies with ~ e c cohomology). h The proof of the following is essentially mechanical (Exercise (2)). THEOREM 3.4.6 (BUNDLE CLASSIFICATION). If y is a Gl(n)-cocycle on M, the isomorphism class [ E y ] E Vectn(M) depends only on the equivalence class [y] E H 1(M; Gl(n)) . This defines a canonical bijective correspondence

80

3. GLOBAL THEORY

By this theorem we identify Vect,(M) with H'(M; Gl(n)). In the case of the tangent bundle T(M), any smooth atlas {(U,, with associated structure cocycle {gap)a,pfmfor M , provides a structure cocycle {Ua , Jgap)a,PE'ZL. There are, of course, structure cocycles for T(M) that are not obtained in this way, but these special cocycles tie together the bundle structure of T(M) and the smooth structure of M in an important way.

DEFINITION 3.4.7. A structure cocycle {U,, Jgap),,pEm for T(M), associated to a smooth atlas on M , will be called a Jacobian cocycle. DEFINITION 3.4.8. If T(M) admits a Jacobian cocycle such that Jgap r I,, V a ,,O E a, then M is said to be integrably parallelizable. It follows from Exercise (3) and Proposition 3.3.11 that an integrably parallelizable manifold is also parallelizable. We will see later in this book that the converse is far from true.

DEFINITION 3.4.9. Let G Gl(n) be a subgroup and let .rr : E -+ M be an n-plane bundle. We say that the structure group of E can be reduced to G if there is a Gl(n)-cocycle {Wj, yji)i,jEJ representing the isomorphism class of E such that im(yji) C_ G, V j , i f J. Such a cocycle will be called a G-ocycle for E. Equivalence of G-cocycles can be defined exactly as for the case G = Gl(n), giving a "cohomology set" H1(M; G). Distinct elements of this set may correspond to distinct bundles or to the same bundle with inequivalent G-reductions.

'

DEFINITION 3.4.10. If E admits a G-cocycle y, then [y] f H (M; G) is called a G-reduction of E. DEFINITION 3.4.11. If G E Gl(n) is a subgroup, an infinitesimal G-structure on the n-manifold M is a G-reduction [y] of T(M). If [y] contains a Jacobian cocycle, then [y] is said to be integrable and will be called simply a G-structure or a geometric structure on M. In the case of the trivial subgroup I = {I,) c Gl(n), M is parallelizable if there is an infinitesimal Istructure [yj on M. The manifold is integrably parallelizable if there is an integrable [y] f H 1(M; I).

EXAMPLE 3.4.12. Let Gl+(n) c Gl(n) be the subgroup of matrices with positive determinant. Two ordered bases ( u l , . . . , u,) and (wl,. . . ,wn) of an ndimensional vector space V are said to have the same orientation if the unique matrix A E Gl(n) such that

belongs to Gl+ (n). This is an equivalence relation having exactly two equivalence classes, called orientations of V. A linear isomorphism L : V -, W between ndimensional vector spaces carries each orientation p of V to an orientation L(p) of W. The standard orientation p, of Rn is the orientation class of the standard ordered basis ( e l , . . . ,en). The other orientation of Rn will be denoted -pn. A linear automorphism L of R n is orientation preserving if L ( p , ) = pn and is

3.4. COCYCLES

81

orientation reversing if L(pn) = -/I,. Clearly, L is orientation preserving if and only if det (L) > 0. Let 7r : E + M be an n-plane bundle and let p, be an orientation of Ex, Vx E M. We will say that p, depends continuously on x if, for each x E M , there is a trivialization $J : EIU -+ U x R n , x E U, such that $ ~ * ~ (=p pn, ~ ) V y E U. If p = { , v , ) , ~ ~depends continuously on x, we say that p is an orientation of E and that E is orientable. Given p, there is always the opposite orientation -p. If M is connected, the continuity of p implies that an orientable bundle E over M admits exactly two orientations. An orientation of T(M) is also called an orientation of M. The manifold is orientable if it admits an orientation. Not every manifold is orientable. For instance, projective space Pn is orientable if and only if n is odd. An orientation of E determines a Gl+(n)-reduction of E. Indeed, cover M with local trivializations (U,, $Ji) as above and remark that the associated cocycle .yij(x) carries the orientation p, of {x) x Rn to itself, Vx E U, f l Uj. That is, .yij (x) E Gl+(n). Conversely, given a Gl+(n)-reduction, the reader should be able to produce an associated orientation of E. Thus, an infinitesimal Gl+(n)structure on M is an orientation of M. Every infinitesimal GI+ (n)structure is integrable. Indeed, fix an orientation p of T(M) and let {U,, Jgap)a,pE(n be any Jacobian cocycle such that each coordinate chart (U,, x,) is connected. The n-tuple of fields

defines a continuous orientation pa of T(U,). Since U, is connected, either pa = p(U, or pa = -p(Ua. In the latter case, replace the coordinate x i with -x,.1 This is an orientation reversing change of coordinates and the new coordinates give the correct orientation p, to T, (M), V x E U,. Carrying this out for each a E M, we produce an atlas on M with associated Jacobian cocycle Gl+(n)valued.

EXAMPLE 3.4.13. Let G = O(n). An infinitesimal O(n)structure is called a Riemannian structure and an n-manifold M with a Riemannian structure is called a Riemannian manifold. This is the starting point of Riemannian geometry, a subject that we will treat in Chapter 10. A Riemannian structure enables one to define lengths of tangent vectors and of curves, angles between vectors tangent at the same point, curves that locally minimize length (geodesics), curvature of the manifold, etc. The proof of the following is relegated to Exercise (4). LEMMA3.4.14. The dijjerentiable manzfold M has a n infinitesimal O(n)structure i f and only i f there is a positive definite inner product (-,.), o n T,(M), Vx E M, which varies smoothly with x in the following sense: given any local coordinate chart (U, x l , . . . ,xn) i n the Cw structure of M ,

3. GLOBAL THEORY

82

is smooth on U ,15 i,j 5 n. This smoothly varying inner product on the fibers of T(M) is called a Riemannian metric on M. Using this metric, one defines the norm

d m ,

by llvll = IIvII. = whenever v E T,(M). Similarly, angles between e l e ments of Tx(M) are defined in the usual way, using the fiberwise inner product. Lengths of smooth curves s : [a,b] -,M are defined by

A bit later, we will be able to show that all manifolds admit Riemannian metrics. For this, the fact that manifolds are 2"* countable is essential. On the contrary, very few manifolds admit integrable Riemannian structures ("flat" Riemannian manifolds). The obstruction to integrability is the Riemann curvature tensor. This will be treated in the Chapter 10, where we will show (Theorem 10.6.1) that curvature 0 means that the geometry is locally Euclidean.

EXAMPLE 3.4.15. One can take G = O(k, n- k) c Gl(n) the group of matrices that leave invariant the quadratic form

A discussion similar to the one above shows that such an infinitesimal structure corresponds to a nondegenerate, indefinite metric in each fiber T, (M), varying smoothly with x. For instance, the Lorentz manifolds in relativity theory are 4-manifolds with an infinitesimal O(3,l)- structure. It is no longer true that all manifolds admit such structures. Integrability places an even more severe restriction on the manifold. EXAMPLE 3.4.16. Consider the subgroup Gl(k,n - k) all matrices of the form

c Gl(n) consisting of

where A f Gl(k), C E Gl(n - k), and B is an arbitrary k x (n - k) matrix. Since the linear action Gl(k, n - k) x Bn + Bn carries the subspace Elk c Cn isomorphically onto itself, a Gl(k, n - k)-reduction y of the n-plane bundle 7r : E 4 M selects a k-dimensional subspace F, C Ex, V x E M. Indeed, the equivalence relation on matches {x) x Bk c Wj x Bn isomorphicdy to {s) x Bk c W ix Bn, whenever x E Wj I I W*,so the subspace {x) x Bk passes to a well-defined k-dimensional subspace F, c (E,), in the quotient.

ET

3.4. COCYCLES

83

DEFINITION 3.4.17. A k-plane subbundle F of an n-plane bundle E over M is a k-plane bundle, together with a commutative diagram

M-M

id

such that i, : F, + E, is a linear monomorphism, Vx E M. A k-plane subbundle of T(M) is also called a k-plane distribution on M.

-

We can view i as an inclusion map F E. The choices {Fz)zEM, defined by a Gl(k, n - k)-reduction, are the fibers of a k-plane subbundle.

LEMMA 3.4.18. A Gl(k,n - k)-reduction of the bundle E defines a k-plane subbundle F of E with fibers F, as above. Conversely, given a k-plane subbundle F c-)E, there is a Gl(k, n - k)-reduction of E that gives back this subbundle. The proof will be Exercise (5). Thus, an infinitesimal Gl(k, n -k)-structure on M is a k-plane distribution on M. If this structure is integrable, it will be called afolaation of M of dimension k. These geometric structures will be studied in the next chapter, together with the integrability condition, the Fkobenius theorem.

EXAMPLE 3.4.19. The complex general linear group Gl(n, @) is the group of nonsingular, n x n matrices with complex entries. If we write the elements of this group as A + -B, where A and B are real matrices, then one can check that

In fact, this realizes the complex general linear group as a subgroup

A 2n-manifold M , together with an infinitesimal Gl(n, C)structure, is known as an almost complex manifold. In this case, one can define a bundle isomorphism

such that J~ = - idT(M) (the reader who has succeeded with Exercises (4) and (5) will be able to check this). One extends the fiberwise scalar multiplication

to a complex scalar multiplication

by the formula (a

+ -b)v

= av

+ bJ(v),

V a, b t R, V v E T, (M), V x E M. Effectively, the tangent bundle becomes a vector bundle over @ rather than E,of complex fiber dimension n, called the complex tangent bundle.

3. GLOBAL THEORY

84

If the almost complex structure is integrable, it can be shown that the manifold admits an atlas (U,, cp,) where cp, : U, + Cn and the local coordinate changes gap are complex analytic. That is, M has the structure of a complex analytic manifold and the integrable almost complex structure will be called a complex analytic structure. One defines a "holomorphic tangent bundle" and shows that it is isomorphic, as a complex vector bundle, to the complex tangent bundle associated as above to the almost complex structure. The details of all this, as well as the integrability condition, are best left to a more advanced course.

EXERCISES (1) Prove Lemma 3.4.4. (2) Prove Theorem 3.4.6. (3) Prove that the following are equivalent for an n-plane bundle

(a) The bundle 7r : E -,M is trivial. (b) There is a Gl(n)-cocycle { W j,yji) j , i E J for the bundle such that yji(x) = In,V X f W j fI Wi, V Z ,j E J. (c) There is a smooth function f : E --+ R n such that

is a linear isomorphism, V z f M. (4) Prove Lemma 3.4.14. (5) Prove Lemma 3.4.18. 3.5. Global Constructions of Smooth Functions

The proof of Theorem 2.6.1 adapts easily to the global case. PROPOSITION 3.5.1. Let M be a n n-manijold, let U E M be an open subset and K c U a compact subset. Then there is a smooth function f : M -,R such that f lK = 1 and supp(f) c U. PROOF. For each p E K, choose a coordinate neighborhood (Up,z p ) about p, UpC U,and an open, n-dimensional interval A p with & c Up, centered at p. Since K is compact, finitely many of the Ap cover K. The proof of Theorem 2.6.1 now goes through unchanged. This is not quite the global C" Urysohn Lemma (Corollary 3.5.5) in which the set K is only assumed to be closed, not compact. Actually, Proposition 3.5.1 suffices for most purposes and we will use one of its standard consequences, the existence of smooth partitions of unity, in order to prove the general version. One useful application of Proposition 3.5.1 is the following.

LEMMA3.5.2. Let M be a smooth manifold and let x map

CW(M) -, @, that cam'es f

H

[f l s is su jectiue.

E

M . Then the natural

3.5. GLOBAL CONSTRUCTIONS

85

PROOF.Indeed, given [g], E 6,,find cp E Cw(M) with supp(cp) c dom(g) and cp E 1 on some compact neighborhood of x in dom(g). Then cpg extends by 0 to a smooth function f on M and [f], = [g],. Another application is to identify X(M) with the Lie algebra V ( C m ( M ) ) of derivations of the function algebra Cw(M). Indeed, each X E X(M) is a derivation X : Cw(M) 4 C w ( M ) in the obvious way. In the local case, M = U Rn,the proof of the reverse inclusion used Theorem 2.6.1 to show how to Iocalize an arbitrary operator D E D(CmF(U))to be a derivative D, E T,(U), and one uses Proposition 3.5.1 in the same way for D E V ( C w (M)).

PROPOSITION 3.5.3. The set X(M) is the Lie algebra of derivations of the algebra Cm(M) . One advantage to this point of view is that the Lie bracket is defined intrinsically on X(M). The alternative is to use the local definitions of bracket in coordinate charts and show that the formulas for the bracket transform correctly under coordinate changes so that the local definitions fit together to give [X,Y] E X(M) globally. THEOREM 3.5.4. If U = {U,),Em is an open cover of M, there is a Cw partition of unity {A,),cm subordinate to U. PROOF. First remark that if V = {Vp)pEs is a refinement of U, a smooth partition of unity subordinate to V induces a smooth partition of unity subordinate to U. Indeed, let i : 23 --, I2L be a map such that Vp & Ui(,), Vp E 23. If {pp}pE9 is a partition of unity subordinate to V, define A, = Cp,i-l(,)PO, V a E a. If i - l ( a ) = 0,we understand that A, = 0. It is clear that {A,),Em is a partition of unity subordinate to U. By passing to a suitable locally finite refinement of U and applying the above paragraph, we lose no generality in assuming that (1) U is locally finite; (2) each U, E U is the domain of a Cw coordinate chart (U,, pa); (3) there are open n-dimensional intervals A, with 71, c cp, (U,) compact and such that {cpil (Aa))aEmcovers M. By Proposition 3.5.1, define f, E C m ( M ) in such a way that fa lcp;'(z,) =1 and supp(f,) c U,. Since the cover {Ua),Ell is locally finite, we can define

remarking that f > 0 on M. We obtain the smooth partition of unity by setting A, = f,/f, V a E a. At this point, we can remove the compactness hypothesis in Proposition 3.5.1. COROLLARY 3.5.5 (THEGLOBAL SMOOTH URYSOHNLEMMA).Let U C M be an open subset of a smooth manifold and K U a subset that is closed in M. Then there is a smooth function f : M + R such that f lK = 1 and supp(f) C

c

3. GLOBAL THEORY

86

PROOF.Let W = M \ K. Then {U, W ) is an open cover of M, and we take a subordinate smooth partition of unity ( A u , Aw). Since

the function f = X u is as required. Recall that, if X c R n is an arbitrary subset, a function f : X -F R k is said to be smooth if it extends to a smooth function f : U -+ Rk, where U is some open neighborhood of X in Rn. This definition continues to work well when X c M, but it is not very convenient when the target space is a smooth manifold which is not an open subset of Euclidean space. The following result, another application of smooth partitions of unity, suggests a slightly different, backwardly compatible definition of smoothness that is more useful.

PROPOSITION 3.5.6. If M is a smooth manifold and X

M , then

is smooth i f and only zj, V x E X , 3 an open neighborhood U , 2 M of x, and a smooth map f , : U, Rk such that f , 1 (U, n X ) = f 1 (U, fI X). -+

PROOF. This property clearly follows from our definition of smoothness. We must recover our definition from this property. Let U = UZExU,. Then there is a smooth partition of unity {A,)xEx on U , subordinate to the open cover {U,),Ex of the manifold U. Since each f , is Rk-valued, A, f , makes sense and can be interpreted as a smooth map of U into R k , Then define

a smooth map of

V y E X, so

x.

U into Rk. Evidently,

f is the required smooth extension of f

to the neighborhood

U of

DEFINITION 3.5.7. A function f : X + Y from a subset X c M of a smooth manifold M into a subset Y N of a smooth manifold N is said to be smooth if, for each x E X, there is an open neighborhood U , M of x and a smooth map f , : U, + N such that f,l(U, n X ) = fl(U, fI X). Such a map is a diffeomorphism of X onto Y if it is bijective and both f and f are smooth.

-'

As an application, we prove and globalize the smooth version of Theorem 1.1.5.

THEOREM 3.5.8 (SMOOTHINVARIANCE OF DOMAIN). Let M and N be Coo manifolds of the same dimension n. If U C M is open, zf X c N , and i f cp : U -+ X is a difleomorphism, then X is open in N .

3.6. MANIFOLDS WITH BOUNDARY

87

PROOF.Let xo E U and cp(xo) E X. Since cp-' : X + U is smooth, there extension t,b : V -+ IWn is an open neighborhood V of cp(xo) in N and a smooth of cpdll(V n X).Since cp : U + X is continuous, V = p - l ( ~n X ) is an open neighborhood of xo in U and y5ocplv = cp-' oqlV = idv. Since cp : U N is smooth in the usual sense, the chain rule gives --t

dcp,, : T,,( M )+ T,(,,)(N)is a linear isomorphism. By the inverse function C U of xo which is carried by theorem, there is an open neighborhood W C cp diffeomorphically onto an open subset cp(W) C N. But cp(xo) is an arbitrary point of X and cp(xo) E cp(W) C X ,so X is an open subset of N. SO

EXERCISES (1) Use the existence of partitions of unity to prove that every smooth mzc nifold M admits a Riemannian structure. Explain why it is impossible to generalize this argument to prove the existence of an infinitesimal O ( k ,n - k)-structure. (2) Prove that a compact n-manifold M cannot be diffeomorphic to any subset of Rn. (3) Let M be a smooth manifold, K c M a closed subset, U > K an open neighborhood of K, v E X(U).Prove that vlK extends to a smooth vector field on all of M . (4) Let U = {U,),Em be an open cover of the smooth manifold M. For each a, E 2L, let cp, : M + R have a constant value c, > 0 on U, and be identically 0 on the complement M \ U,. Set

and construct a smooth function f everywhere on M .

:

M

+

R such that 0 < f < cp

3.6. Manifolds with Boundary

Manifolds with boundary were introduced in Section 1.6 from the purely topological point of view. Here we introduce the smooth version. Since Euclidean half space Wn is a subset of the smooth manifold Rn, Definition 3.5.7 allows us to talk about smooth maps and diffeomorphisms between open subsets of Wn. Thus, if M is a topological manifold with boundary, it makes sense to talk about two Wn*harts on M being Coo-related, so we can define a differentiable structure on M to be a maximal Wn-atlas A of Cm-related charts. As in the topological case, we define

88

3. GLOBAL THEORY

The pair (M, A) is a (smooth) n-manifold with boundary dM. Of course, all smooth n-manifolds without boundary are special cases, as is Wn itself. The notion of smooth maps between manifolds with boundary is defined exactly as in the boundaryless case. The following is an immediate corollary of Theorem 3.5.8.

PROPOSITION 3.6.1. If U C_ Wn \ dWn is open, then U is not difleomorphic to an open subset V 5 Wn such that V dWn # 8.

COROLLARY 3.6.2. If M is a manifold with boundary, then M d M = int(M). PROOF.Indeed, the proposition shows that d M = {x E M A, x E Ua , qa (Ua) E int(Wn))-

COROLLARY 3.6.3. If cp : M

4

(Ua,cpa) E

N is a difleomorphism, then

COROLLARY 3.6.4. A smooth n-manifold M with boundary is also a smooth n-manifold e M = int(M) e d M = 0. COROLLARY 3.6.5. If M is a smooth n-manifold with boundary, then a M is a smooth ( n - 1)-manifold and int(M) is a smooth n-manifold. For x E int(M), the n-dimensional tangent space T,(M) is defined as before. If x E d M , T, (d M) is only (n - l)-dimensional, but we are going to define an n-dimensional tangent space T, (M) having T, (dM) as a subspace. The more intuitively appealing approach is that of infinitesimal curves. Indeed, the notion of smooth curve

makes good sense and we can define the corresponding infinitesimal curve ( s )., This can be thought of as a tangent vector to M at x. In this way, one obtains all the vectors tangent t o the boundary and all the vectors that point "into" M. Unfortunately, the negatives of the inward vectors are not so obtained and the resulting set of tangent vectors is not a vector space. Alternatively, the definition of tangent vectors as derivatives of the algebra of germs of C O0 functions at x, while less intuitive, yields a vector space exactly as before. However, in order to show that this vector space is n-dimensional, we will find it convenient to turn to infinitesimal curves. If x E M, whether or not x E d M , the set Coo (M, x) of smooth, real valued functions defined in neighborhoods of x is defined and there is no problem defining germinal equivalence on this set. Thus, we obtain the IR-algebra @ ,of germs. We define Tx(M) to be the vector space of derivatives D : 6, + R. If x E int(M), this agrees with our usual definition and is n-dimensional. If f : M -t N is a smooth map between manifolds with boundary, the different i d s dfz = f*x : Tx(M) Tf(,)(N) -+

3.6. MANIFOLDS

WITH BOUNDARY

are defined by the usual formula

d f z ( D ) [ g l j ( z= ) D[g O f l z and are linear. The proof of the global chain rule (Lemma 3.1.21) goes through equally well in our present context. LEMMA3.6.6. If f : M M N and g : N 4 P are smooth maps between manifolds with boundary, then g o f is smooth and, for each x E M ,

COROLLARY 3.6.7. I f f : M + N is a diffeomorphism between manifolds with boundary, then d f , : T,(M) M T f ( , ) ( N ) is a linear isomorphism, V x E M . We have reached the point where a little work has to be done. We must show that Tx(Wn) is n-dimensional, even when x E dWn. The above considerations will then extend this property to arbitrary manifolds with boundary. When x E dWn c Rn, we will use the notation @,(Rn) for the algebra of germs of Coo ( R n ,x ) and @, (Wn) for the germ algebra of C" (Wn,x ) .

LEMMA3.6.8. Let x E dWn and let p : @,(Rn) p [ f I x = [ f 1 (Wn r 7dom(f )I,. Then p is a surjection.

+

@,(Wn) be defined b y

PROOF. Let U Wn be an open neighborhood of x. If g : U M R is smooth, there is a neighborhood V of x in R n and a smooth extension ij : V 4 R of g1 (V n U n Wn). Then [ j ] ,E @, ( R n ) and p [ j ] , = [g],. For x E dWn, define

p* : T,(Wn)

4

Tz(Rn)

by setting

~ * ( D ) [ f= l . D(p[fl,). It is elementary that this is linear.

L E M M A3.6.9. p* is bijective. PROOF. We prove that p* is one to one. If p f ( D 1 )= p*(D2),then D l ( p [ f ] , ) = Dz(p[f V [ f 1, E 6,( R n ) . Since p is surjective, it follows that & [g], = D2 [ g ] ,, V [ g ] ,E @,(Wn), so Dl = D2. We prove that p* is onto. Let v E T x ( R n ) = Rn. As an infinitesimal curve, this vector is represented by s ( t ) = x tv. As an operator on germs, v = D (,)=. Either v points into Wn (we intend this to include the case that v is tangent to dWn) or v points out of Wn, in which case -v points into Wn. If v points into Wn, then s ( t ) E Wn, V t 0. Define D : @,(Wn) M R by

lz),

+

>

D [ g ] ,= lim g ( s ( t ) )- ilk4 t+o+

t

It is elementary that D E T,(Wn) and that p*(D) = D(,)= = v. If v points out of Wn, then s ( t ) E Wn, V t 5 0. Define D : @,(Wn) M R by

D [g],= lim g ( s ( t ) )- g(z) t+O-

t

3. GLOBAL THEORY

90

Again, D E Tx(Hn) and p* (D) = u.

COROLLARY 3.6.10. The vector space Tx(Wn) is n-dimensional, Vx E dWn. COROLLARY 3.6.11. Let M be a smooth n-manifold with boundary, x E a M . Then the vector space Tx(M) is n-dimensional. PROOF. Let

(U, cp) be an Wn-coordinate chart about x. Then (p*x : TX(U)

+

T,(X)(cp(U))

is an isomorphism. But Tx(U) = Tx(M) and T,(x)(~(u))= Tp(x)(Wn). This latter is n-dimensional. At this point, one can define the tangent bundle .rr : T(M) 4 M , this being an n-plane bundle over M. The total space T(M) is a 2n-manifold with boundary, W ( M ) = .rr-'(d~). In Exercise (4), you will be asked to check this carefully. Vector fields on a manifold M with boundary are smooth sections of T(M). The smooth Urysohn lemma and its consequences extend to this context. In particular, vector fields are derivations of the function algebra Coo(M) and open covers always admit smooth, subordinate partitions of unity. In general, the classification of manifolds, with or without boundary, is impossible. In dimensions one and two, however, the classification has been achieved and there is considerable effort being expended in an effort to classify compact 3-manifolds. The task becomes impossible in dimensions four and greater, where it can be reduced to the problem of deciding, via a finite algorithm, whether or not given finite presentations of two groups (that admit such presentations) force the groups to be isomorphic. This problem, often called the "word problem", has been shown by logicians to be unsolvable.

EXERCISES Let M be a manifold with nonempty boundary. Show that there is a smooth function f : M + R+ such that d M = f-l(0). Let M and N be smooth manifolds of dimension m and n respectively. If d M # 8 = d N , show how to use the differentiable structures on these two manifolds to give M x N the structure of a smooth (m+n)-manifold with nonempty boundary. (3) If, in the previous exercise, both M and N had nonempty boundary, show that M x N is a topological manifold with boundary. Discuss the problem you encounter in trying to give M x N a natural smooth structure of manifold with boundary. This is, in fact, an example of what is called a smooth manifold with corners. (4) For an n-manifold M with boundary, mimic the construction of the tangent bundle ?r : T(M) --t M, showing that one obtains a smooth 2n-manifold with boundary, the projection .rr being identified locally with the canonical projection

3.7. SUBMANIFOLDS

91

3.7. Submanifolds We give a definition of "submanifold" that applies to manifolds with boundary. DEFINITION 3.7.1. Let M be an m-manifold, possibly with boundary. A subset X c M is a properly imbedded submanifold of dimension n if and only if, V p E X , there is an Wm-coordinate chart (U,cp) about p in M in which (p(UnX) = cp(U) n Wn, where Wn c Wm is the (image of the) standard inclusion. Remark that, in the above definition, (U t l X, cpl (U r l X)) can be viewed as an Wn-coordinate chart on X and that the collection of all such charts makes X a smooth n-manifold with boundary d X = X n dM. Thus, if d M = 0, then dX = 0 also. In this case, we generally drop the qualifier "properly". Note also that X cannot be tangent to a M at any point of dX. EXAMPLE 3.7.2. The image of the standard inclusion W imbedded submanifold.

n

L,

Wm is a properly

We will need the following globalization of the submersion theorem. THEOREM 3.7.3 (GLOBALSUBMERSION THEOREM).Let f : M --, N be a smooth map between manifolds of respective dimensions m and n. Assume that ON = 0 and that m > n. If y E N is a regular value simultaneously for f and for d f = f ldM, then f -'(y) is a properly imbedded submanifold of dimension m - n.

PROOF.Let p E f -'(y), and find a suitable coordinate chart about p in M. There are two cases. Case 1. Suppose p E int(M). Choose a coordinate neighborhood (U, x) about p such that U E int(M). Then y is also a regular value of f (U,so Theorem 2.9.6 implies that f -'(y) n U is a smooth submanifold of U of dimension m - n. Case 2. Suppose p E OM and let (U, x l , x2, . . . ,xm) be an MIm-chart about p in M. Assume that f (U) c W, where (W, yl, . . . , y n , is a coordinate chart about y in which y = 0. Let d f 1 (U n d M ) be denoted by cp(x 2 , . . . ,xm) with component functions cpl, . . . , cpn relative to the coordinates of W. Since p is a regular point for cp, U can be chosen so small that the matrix

has constant rank n on UndM. By a permutation of the coordinates x 2 , . . . ,xm, it can be assumed that the last n x n block

3. GLOBAL THEORY

92

is nonsingular on U n dM. Choosing U even smaller, if necessary, the corresponding n x n block in the matrix

is also nonsingular. We then resort to the trick of recoordinatizing U near p by setting zi = xi, 1 5 i 5 m - n, and zm-n+j = f j , 1 5 j 5 n. The inverse function theorem shows, by the above remarks, that this will define an W n-chart on a small enough neighborhood (again called U) of p. But, relative to these coordinates, f(z',z2, .. . , z m ) = (2m-n+l , . . . ,zm). Then f (y) n U is the set of points with coordinates (z , . . . , z " y 0,. . . ,0). That is,

'

-'

f

EXAMPLE 3.7.4. Let f : Wn+'

nU +R

=

w ~ n U. - ~

be given by

Then 1 E R is a regular value both for f and for d f . The hemisphere f -'(l) is the intersection SnnIHn+land is an n-manifold with boundary f -l(l)ndWn+l = sn-1

The following lemma shows that there are plenty of regular values as in Theorem 3.7.3.

LEMMA3.7.5. If d M = 8 and f : N + M is smooth, then the set of points in M that are simultaneous regular values for f and d f is dense in M. PROOF. Clearly, if p E a N is a regular point for d f , it is also a regular point for f . Thus, y E M is a regular value both of f and df precisely when it is a regular value both of f 1 int(N) and of d f . Use countable coordinate coverings {Ui)iEI of int(N), (%IjEJ of d N , and {Wk)kEK of M . For each k E K , consider the countable family of smooth maps

obtained by restrictions. By Corollary 2.9.5, a.e. y E W kis a common regular value of all these maps. Doing this for each k E K, we complete the proof. For simplicity, the following discussion will be carried out only for the case of manifolds with empty boundary. DEFINITION 3.7.6. A smooth map f : N -+ M of an n-manifold into an m-manifold is an immersion if f has constant rank n.

3.8.

HOMOTOPY AND ISOTOPY

93

DEFINITION 3.7.7. If i : N + M is a one to one immersion, i ( N ) is called an immersed submanifold of M. If i is also a homeomorphism onto i(N), this is called an imbedded submanifold. The reader should be warned that many authors call an immersed submanifold i(N) c M simply a submanifold. This is misleading because the relative topology inherited from M may not be the manifold topology of N.

EXERCISES (1) For manifolds without boundary, we now have two definitions of imbedded submanifold. Prove that these definitions are equivalent. (2) Let i : N 4 M be a one to one immersion, let X be a manifold, and let f : X + M be a smooth map with f ( X ) E i(N). (a) Show by an example that i-l o f : X + N may fail to be continuous. (b) If i-l o f is continuous, prove that it is smooth. 3.8. Homotopy and Isotopy

Let M and N be manifolds and assume that d M = 0. We will study smooth maps f : M + N. The set of all such maps will be denoted Coo(M,N). If M and N are diffeomorphic, the set of all diffeomorphisms from M to N will be denoted Diff(M, N) C Cm(M,N). When N = M, Diff(M,M) is a group and will be denoted Diff (M). Since d M = 0, one easily verifies that M x [O,1] is a manifold with boundary. DEFINITION 3.8.1. Elements fo, fl E Cm(M,N) are said to be (smoothly) homotopic if there is a smooth map H : M x [O,1] + N such that (1) f o b ) = H(x, O), v x E M; (2) fl(x) = H(x,l), Vx E M. We write fo fl. The map H is called a (smooth) homotopy between f o and

flOne should think of a homotopy as a smooth deformation of one smooth map to another through smooth maps. It is a "smooth curve" in C "(M, N) connecting fo to fi. We write ft(x) = H(x, t ) , 0 t 1. Similarly, smooth curves in Diff (M) will be smooth deformations, called isotopies, of one diffeomorphism to another through diffeomorphisms.

< <

DEF~NITION 3.8.2. If fo, fl E Diff(M), a homotopy f t between fo and fl will be called an isotopy of fo to f l if ft E Diff(M), 0 5 t 1. If such an isotopy exists, we say that fo is isotopic to f l and we write fo = fl.

<

The proof of the following lemma will be the content of Exercise (1). LEMMA3.8.3. Smooth homotopy is an equivalence relation on Coo(M,N ) and isotopy is an equivalence relation on Diff (M).

3. GLOBAL

94

THEORY

DEFINITION 3.8.4. A diffeomorphism f E Diff (M) is compactly supported if there is a compact subset K C M such that f ( ( M \ K ) = idM,K. The set of all compactly supported diffeomorphisrns is denoted Diff ,(M). A compactly supported isotopy between f 0, fl E Diff ,(M) is an isotopy such that there is a compact subset C M with ftl(M \ C) = idM,c, 0 5 t 1.

<

The proof of the next two lemmas will be Exercises (3) and (4), respectively. LEMMA3.8.5. The set Diff,(M) is a group under composition. LEMMA3.8.6. Compactly supported isotopy is an equivalence relation on the group Diff ,(M). REMARK.For a compactly supported isotopy, f E Diff ,(M), 0 5 t 5 1. THEOREM 3.8.7 (HOMOGENEITY LEMMA).I f N is connected, boundaryless, and x, y E N, then there is f E Diff,(N) and a wmpactly supported isotopy f t such that f (x) = y and f o = idlv, f l = f . The proof of this theorem uses flows and will be deferred until the next chap ter.

-

COROLLARY 3.8.8. I f g E Coo(M,N), y E N, and N is connected and boundaryless, then g ij such that y is a regular value of ij. PROOF. By Lemma 3.7.5, we choose a regular d u e x E N of g. Let f and H be as in Theorem 3.8.7. Since f (x) = y and f is a diffeomorphism, it follows that y is a regular value of ij = f o g . But f t o g is a homotopy of g = fo o g with ij=flogThe definition of smooth homotopy and isotopy that we have given does not adapt nicely to manifolds with boundary. The problem is that [ O , l ] is itself a manifold with boundary, hence M x [O, 11 will be a manifold with corners when d M # 8. This minor difficulty can be overcome by slightly modifying our definitions. DEFINITION 3.8.9. If fo, f i E Coo(M,N), these maps are (smoothly) homotopic if there is a smooth map H : M x R + N such that (1) H(x,O) = fo(x), v x E M; (2) H(x, 1) = fi(x), Vx f M. As usual, we set

ft (

4 = H(x, t ) and say that f o , f i E Diff (M) are isotopic if there is a homotopy between them such that ft E Diff(M), V t E R. For manifolds without boundary, these two definitions are equivalent and, in any case, the second definition is an equivalence relation (Exercise (2)).

3.9. DEGREE THEORY

-

EXERCISES

(1) If fo f l (respectively, fo FZ fl), prove that there exists a homotopy (respectively, an isotopy) H : M x [0, I] + N such that f t = fo, 0 5 t 5 e, and f t = f l , 1-6 5 t 5 1, for suitably small e > 0 and 6 > 0. Use this to prove that homotopy (respectively, isotopy) is an equivalence relation on Coo(M, N ) (respectively, on Diff (M)). (2) Prove that, under the second definition of homotopy and isotopy these continue to be equivalence relations. If OM = 0, prove that the second definition of homotopy and isotopy agrees with the first. (3) Prove Lemma 3.8.5. (4) Prove Lemma 3.8.6.

3.9. Degree Theory Throughout this section, dim M = dim N > 0 and d M = 8 = dN. The manifold M will be compact and N will be connected. If f E Coo(M,N), choose a regular value y E N of f . By Theorem 3.7.3, f-'(y) is a O-dimensional submanifold, hence a set of isolated points. Being a closed subset of a compact space, it must therefore be a finite set. Let k = If (y)I denote the cardinality of this set, an integer 0.

>

THEOREM 3.9.1 (STACKOF RECORDS THEOREM).Under the above hypotheses, let f-'(y) = {PI,.* . , ~ k h Then, if k > 0, there exists an open neighborhood U of y in N such that

a disjoint union, where U, is an open neighborhood of pi in M that is carried by f dzfleomorphically onto U, 1 5 i 5 k.

PROOF.By the inverse function theorem, choose an open neighborhood Wi of pi in M that is carried by f diffeomorphically onto an open neighborhood f (W;) of y in N , 1 5 i 5 k. Since M is Hausdorff, the neighborhoods Wi can be k assumed to be pairwise disjoint. Since M is compact, so is X = f ( M \ U,=, Wi), and this set does not contain y. Let U = f(Wl) n . - -n f(Wk) \ X , an open neighborhood of y in N. Let Ui = f - ' ( ~ )r l W,, 1 5 i 5 k. It is obvious that f carries each U, diffeomorphically onto U and that Ui C f -'(U). But, if k x E f-'(U)\Ui=,Ui, then f(x) ~X,contradictingthefactt h a t X n U =a.

u;,

COROLLARY 3.9.2. The set R of regular values of f is an open, dense subset of N . The function At : R + Z+, defined by Xt(y) = If-'(y)l, is constant on each connected component of R. Indeed, we already know that R is dense and Theorem 3.9.1 shows that it is open. The theorem also shows that X is locally constant, hence constant on each component. DEFINITION 3.9.3. If y E N is a regular value of f , then deg2(f , y) E the mod 2 residue class of Xf (y) .

Z2is

3. GLOBAL

96

THEORY

LEMMA3.9.4 (HOMOTOPY LEMMA). If f , g E C m ( M ,N ) are smoothly homotopic and if y E N is a regular value for both f and g, then deg2(f,y) = deg2 (g, Y). PROOF. Let H be a smooth homotopy of f to g. We consider two cases. Case 1. The point y is also a regular value of H : M x [ O , l ] + N. By Theorem 3.7.3, H (y) is a properly imbedded 1-manifold in M x [0,I]. This submanifold is compact (as a closed subset of M x (0, I]) and

-'

a ~ - (y) ' = H-'(p)

n ( M x {0) U M

x (1)) = f

-'(y) x (0) u g-'

(y) x {I).

+

It follows, by Corollary 1.6.10, that X (y) A,( y) is an even integer, hence that deg2(f, Y) = deg2(g, Y). Case 2. The point y is not a regular value of H. It is, however, a regular value for both f and g, so Corollary 3.9.2 implies that there is an open neighborhood W of y in N on which both Xf and A, are defined and constant. The set of regular values of H is dense, so we choose such a regular value z E W. By Case 1, we get deg2(f, z) = deg2(g, z) , but Xf (9) = Xf (z) and X,(y) = A, (z), so deg,(f,y) =deg,(g,y). Let z E N . By Corollary 3.8.8, choose f f for which z is a regular value. Then deg2(f ,z) is independent of this choice, so we set N

unambiguously, obtaining a function

By Theorem 3.9.1, this function is locally constant. By the connectivity of N, deg, (f) is constant. D E F ~ N I T I3.9.5. ~ N The element deg2(f ) E Z2 is called the degree (mod 2) of f E Cm(M,N). COROLLARY 3.9.6. Iff,g E Coo(M,N ) are homotopic, then

LEMMA3.9.7. If f : M

+N

is not surjective, then deg2(f ) = 0.

PROOF.Any z E M which is not a value of f is a regular value, so deg (f) is the residue class mod 2 of Xf(z) = 101 = 0. COROLLARY 3.9.8. If N is not compact, deg2(f ) = 0. DEFINITION 3.9.9. A map f E C m ( M ,N ) is essential if it is not homotopic to a constant map. A manifold M is contractible if idM is not essential. COROLLARY 3.9.10. If degz(f )

# 0, then f

is essential.

COROLLARY 3.9.11. If M is compact and connected with empty boundary, then M is not contractible.

3.9. DEGREE THEORY

97

PROOF. Indeed, deg2(idM)= 1, so id^ is essential.

THEOREM 3.9.12 (BOUNDARY THEOREM).Suppose that M = dW for a compact manifold W and let g E Cm(M,N). If g extends to a smooth map

then deg2(g) = 0. PROOF. Let y E N be regular, both for G and g = dG. Then G l l ( y ) is a compact, onedimensional manifold with dG-'(y) = G-' (y) r l dW = g-' (y). As usual, this set has an even number of elements, so deg2(g) = 0. EXAMPLE 3.9.13. Let f : C -, C be smooth and let W c C be a compact region bounded by smooth, closed curves. A basic question is whether or not f has a zero in W, assuming that f has no zeros on d W. Define g : dW + S by

'

a smooth function between compact 1-manifolds, S being connected. If f (z) # 0, V z E W, then g extends smoothly to G : W + S by

'

hence deg2(g) = 0. This will be enough to prove "half" of the fundamental theorem of algebra. 3.9.14. I f f : C THEOREM a zero in C .

+C

is a polynomial of odd degree m, then f has

PROOF. No generality is lost in assuming that f has leading coefficient 1. Write f (z) = zm alzrn-I . . . am,

+

+ +

and define a homotopy by H(z, t) = f t ( t ) = tf (z)

+ (1 - t)zm = zm + t(alzrn-' + - .. + a,).

Then, fO(z)= zm and fl(z) = f(z). Suppose that W, C C is a closed disk, centered at 0 and of radius r > 0. We claim that, for r sufficiently large, f t has no zeros on dWrj 0 5 t 5 1. Indeed,

and the term in the parenthesis converges to 0 as z Thus, for r > 0 sufficiently large, we define

+ oo.

G : aw, x [o, 11+ s1

3. GLOBAL THEORY

98

This is a homotopy between

and

re re'') = ei n 0 , so deg2(Go) = deg2(G1). But G;' (TJ) contains exactly m points, V y E S1, SO deg2(GI) = 1 since m is odd. It follows, by Example 3.9.13, that f has a zero in w,. There is an integer-valued degree for maps f f C (M, N) when M and N are both orientable. This can be used to give a proof of the full fundamental theorem of algebra. We will take this up in Chapter 8 when we study differential forms and de Rham cohomology.

DEFINITION 3.9.15. Let W be a compact manifold, possibly with boundary, and let X & W. A retraction of W to X is a smooth map f : W -,X such that f lX = idx. THEOREM 3.9.16. If W is a compact manifold with dW # 0 connected, then there is no retraction f : W + dW. PROOF.Indeed, deg2(fldW) = 1, since fldW = idaw, and this contradicts the existence of the extension f : W + d W by Theorem 3.9.12.

is smooth, there is a point x E D n such that f (x) = x.

PROOF.Suppose f has no fixed point. Define g : D n + dDn = Sn-l as follows. For each x E Dn, construct the ray R, starting at f (x) and passing through x # f (x). Let g(x) be the unique point R , n S1. If x E dDn, it is clear that g(x) = x, so if g is smooth, we have contradicted Theorem 3.9.16. The smoothness of g is left for the reader to check (Exercise (1)). Another famous theorem that can be proven using mod 2 degree is the JordanBrouwer separation theorem (smooth version). We introduce the key idea, that of "winding number", and then, in the exercises, lead you through a proof of the separation theorem in the plane (the smooth version of the Jordan curve theorem).

DEFINITION 3.9.18. Let f : S1 4 R2 be a smooth map and let p E R 2 \ f (sl). Define jp: s1+ s1

by the formula

3.9. DEGREE THEORY

99

where 11. I( denotes the usual Euclidean norm. Then the (mod 2) winding number of the closed curve f around p is

Remark that the winding number is defined for an arbitrary smooth closed curve f . It is not required that f be an imbedding or even an immersion. In the case that f is a diffeomorphic imbedding (a smooth Jordan curve), you will show in the exercises that the open set IR2 \ f (sl) has exactly two components, distinguished from one mot her by the fact that w (f , p) = 0, for every point p in one component, and w2(f ,p) = 1, for every point p in the other. The component in which w2(f , p) = 0 is unbounded (and called the "outside" of f ( S I)), while the other component is bounded (the "inside"). A key lemma, to be proven in Exercise (3), is the following.

LEMMA3.9.19. Let f be a smooth Jordan curve, let U be a connected component of R2 \ f(S1),and let p,q E U. Then wz(f,p) = w ~ ( f , q ) . DEFINITION 3.9.20. If p E R2, the ray in R2 out of p and having direction given by the unit vector v E S1 will be denoted Rp(v). The second key lemma, to be proven in Exercise (4), is the following.

LEMMA3.9.21. I f f : S1+ R2 is a smooth Jordan curve and p E R2 \ f (S1), then v E S1 is a critical value of f p : S1 + S1 if and only if the ray RP(v) is somewhere tangent to the Jordan curve f . EXERCISES (1) In the proof of Corollary 3.9.17, prove that the map g is smooth. (2) Prove that Theorem 3.9.17 holds when f is only assumed to be continuous. For this, use the Weierstrass approximation theorem to construct, for each E > 0, a Coofunction f, : D n+ D nsuch that 11 f (x)- f,(x)(l 5 e, Vx E Dn.(The Weierstrass theorem asserts that, on a compact set such as Dn c Rn,any continuous function f : D n + Rn can be uniformly a p proximated to within any S > 0 by a polynomial function P : D + Rn,) (3) Prove Lemma 3.9.19. (Hint: The mod 2 degree is a homotopy invariant .) (4) Prove Lemma 3.9.21. (5) Prove the smooth Jordan curve theorem: I f f is a smooth Jordan curve, then R2 \ f(S1) has exactly two components, one of which (called the inside off) is bounded (i.e., has compact closure) and the other of which (the outside o f f ) is unbounded. For every point p in the outside o f f , w2(f,p) = 0, and for every p on the inside, w2(f , p) = 1. Finally, f (S1) is the set-theoretic boundary of each of these components. Proceed as follows. (a) If p E Kt2 \ f (S1), prove that w2(f, p) is the number of points f (S1),for v E S1 any regular value of f p . (Hint: mod 2 in Rp(v) 17 Use Lemma 3.9.21.)

100

3. GLOBAL THEORY

(b) Use (a) to prove that there are points p, q E R2 \ j(S1) such that ~ ~ ( f# ,w~2 ( )f , q ) By Lemma 3.9.19, conclude that R2 \ f (S1) has at least two components. Also remark that the winding number is 0 about points in at least one of these components and is 1 about points in at least one of the other components. (c) Using the fact that f is a smooth imbedding, choose a coordinate chart (U, u, w) about any point j ( z ) in which and f (S1) n U is just a horizontal line segment w r 0. Show that every point of R2 \ f (sl) can be connected by a continuous path in R2 \ f (S1) either to the point (0,l) E U or (0, -1) E U. This proves that R2 \ f (S1)has at most two connected components. (d) Prove that one of these components is bounded, that the other is not, and that the winding number of f is 0 about the points in the unbounded component. (e) Show that each point of f (S1) lies on an arbitrarily short arc that meets both components. Conclude that f (S1) is the common settheoretic boundary of these components.

CHAPTER 4 Flows and Foliations

In this chapter, we investigate the global theory of ordinary differential equations (flows), referred to as O.D.E., and the Frobenius integrability condition for kplane distributions (foliations). Although this latter topic concerns global partial differential equations (P.D.E.) , our approach will be largely qualitative, with very few explicit partial differential equations in evidence. Unless otherwise indicated, all manifolds will have empty boundary. 4.1. Complete Vector Fields The space X(M) of smooth sections of the tangent bundle is a module over the algebra Coo(M) and a vector space over R. Viewed as the space of derivatives of C m ( M ) , X(M) is a Lie algebra over R. By the local theory of O.D.E., for each q E M , a vector field X E X(M) generates a local flow @ : (-e, E)x V 4 U ,where (U, xl, . . . ,xn) is a local coordinate chart about q, V is an open neighborhood of q with compact closure in U, and E > 0 is sufficiently small. Any two local flows generated by X agree wherever both are defined. The notion of a local flow about a point makes sense even when V and U are not coordinate neighborhoods. A system of suitably coherent local flows covering a manifold M will be called a local flow on M. Here is the precise definition. DEFINITION4.1.1. A local flow

on M is a family of smooth maps

written P ( t , x) = @r(x),such that (1) V, C U, C M are open sets and {Va)aEa covers M; (2) aa and @P agree on ((-c,, c,) rl (-fp, ea)) x (V, rl VP), V a,,B E a; (3) : V, + U, is the inclusion map, V cu E IZl; = atq o wherever both sides are defined, V a E 94. (4)

DEFINITION 4.1.2. If @ is a local flow on M and q E M, curves of the form (q), -E, < t < e,, where q E V,, are called flow lines of through q. s t ( t )=

102

4. FLOWS AND

FOLIATIONS

r

By the definition of local ffow, any two flow lines s and s t through q agree on their common domain (-fa, E m ) f l ( - E @ , E ~ ) .Thus, the velocity vector

is well defined, V q E M , and this yields a vector field X E X ( M ) . The definition also implies that X.: = ir ( t ), V t E (-E, , e,). DEFINITION 4.1.3. The vector field X obtained from the local flow @ as above is called the infinitesimal generator of @. The proof of the following lemma will be Exercise (1). LEMMA4.1.4. Every vector field X E X ( M ) the infinitesimal generator of a local flow @ on M . If two local flows @ and \E have the same infinitesimal generator X , then @ U Q is a local flow with the same infinitesimal generator X . Consequently, by partially ordering local flows by inclusion we see that, given a local flow @ on M , there is one and only one maximal local flow on M containing @. COROLLARY 4.1.5. Them is a one to one comspondence between maximal local flows @ on M and vector fields X E X ( M ) given by letting X be the infinitesimal generator of @. Working with local flows can be somewhat uncomfortable. Happily, there are natural situations in which the maximal local flow of a vector field contains an honest (i.e., global) flow. DEFINITION4.1.6. A (global) flow on M is a smooth map @ : R x M written a t ( x ) = @(t, x), such that

--+

M,

(1) = idM; (2) @ t 1 + t 2 = atl 0 ata, v t l ,t 2 E El. If the maximal local flow of a vector field X E X ( M ) contains a global flow, we say that X is a complete vector field. In this case, X is also called the infinitesimal generator of the global flow. EXAMPLE 4.1.7. We define a global ffow on T~ = S1x S1. Fix p E R and, for each t = (eZRia,elnib) E T' and t E R, define

It is clear that this defines a global flow @ p on T ~Of. some interest is the way in which the qualitative behavior of this flow depends on the value of the constant p. If p is rational, there is a least positive integer k such that pk is also an integer. Thus, (e2nia, elnib) = qj;(e2nia , elnib). One concludes rather easily that each flow line @ p ( R x (2)) is an imbedded circle, the flow being periodic of period k (cj. Exercise ( 5 ) ) . By contrast, if p is irrational, each flow line @p(R x (2)) is a one to one immersed copy of R that is everywhere dense in T ~ This . is the two dimensional version of a theorem of Kronecker that we will prove in Chapter 5 (Example 5.3.8). The two

4.1. COMPLETE VECTOR FIELDS

103

dimensional version will also follow from a result to be proven in Section 4.3 (cf. Corollary 4.3.10 and Exercise (2) on page 116). Remark that the infinitesimal generator X of the flow @ p "lifts" to a well defined vector field on R2 relative to the canonical projection

More precisely, the constant vector field

2 E X(R2), defined by

-

satisfies

P*(",Y)X(",V)= XP(",V)? for all (x, y) E IR2. The reader can check this easily. It is noteworthy that the vector field 2 on lit2 is also complete, generating the translation flow

This lifted flow is quite tame, regardless of whether p is rational or irrational. Not every vector field is complete. For example, Exercise (2) on page 63 showed that et $ E X(R) is not complete.

LEMMA4.1.8. If the maximal local flow of X E X ( M ) contains an element of the form @ : (-€,

with

6

> 0, then X

PROOF.Let t

€1 x

M

+M

is complete.

E R. Then one can find k E Z and

t = r + k 4 2 . Given x E M, define

r E ( - ~ / 2 ,c/2) such that

If @t(x)is well defined by this formula, -oo < t < oo, then it will be an integral curve to X . To see this, remark that (x), - $ < r < ~ / 2 ,is integral to X and use the fact that (X) = X (Lemma 2.8.12). We show that at(x) is well defined. For simplicity, let t > 0. Obvious modifications of the argument give the general case. Suppose that

where s,r E ( - ~ / 2 ,~ / 2 )and k, q E Z. It follows that r - s E (-E, E), hence that q - k = 0,1, or - 1. If q - k = 0, then r - s = 0 and we are done. Assume,

4. FLOWS AND FOLIATIONS

104

therefore, that q - k = f1. Without loss of generality, take q - k = 1. Then, r - S = €/2, SO

Thus, Qt(x) is well defined, -00 < t < oo, Vx f M , and is an integral curve to X. We must show that @ : R x M + M is smooth. Let (to,xo) E R x M. For smallenoughq > 0, we fix ko E Z such that t = r+ko.e/2, V t E (to - q , t o + q ) and suitable r E (-~/2,€/2). Then, (to - q, to q) x M is an open neighborhood of (to,xo) in R x M on which

+

This is a smooth function of (r,x) = (t - koc/2, x), hence a smooth function of (t, 2). THEOREM 4.1.9. If X E X ( M ) has compact support, then X is complete. PROOF.Since supp(X) is compact, cover it with open subsets U1,. . . , Ur C_ M such that the local flow of X contains elements

1 5 i 5 r. Let Uo = M

\

supp(X), an open set with X(UoE 0. Define

by a:(+) = I , Vx E UO,Vt E IR. Since {Ui)i=o covers M and the 0' agree on - - E i and overlaps, we have a local flow on M generated by X. Let e = min l
generated by X . By Lemma 4.1.8, X is complete. COROLLARY 4.1.10, If M is compact, every vector field X E X(M) is complete. As an application, we return to the homogeneity lemma (Theorem 3.8.7). Let

denote the compactly supported isotopy class of idM. We prove a local version of Theorem 3.8.7.

4.1.

COMPLETE VECTOR FIELDS

105

LEMMA4.1.11 (LOCALHOMOGENEITY LEMMA). Let U = int(Dn), xo, yo E U. Then there is rp E iff:(^) such that rp(xo) = yo. PROOF. Use the ordinary Euclidean norm from Rn and choose e, r] E (0,l) such that 0 I max{lfxotl, Ilvoll) < 77 < Let f : U + R be smooth such that

Let and define

Since supp(X) C supp(f) is closed in U and bounded away from d D n , it is compact. By Theorem 4.1.9, X is a complete vector field, and we let

be the flow it generates. This flow is stationary outside the compact set supp(X). Let s(t) = xo tv = xo (tc1,tc2,. . . ,tcn), 0 5 t 5 1. Then

so f(s(t))

-

+

+

IIs(t) ll = lltyo + (1 - t)xo I1 I maxillxo ll, Ilvoll) 1, 0

< 77,

< t 5 1. Thus, <

0 5 t 5 1. Therefore, s(t) is integral to X , 0 t 5 1, and it follows that @l(xo) = yo. Then at, 0 5 t 5 1, defines a compactly supported isotopy of a. = idu to = rp, where ip E ~iff:(U)and rp(xo) = yo. The proof of Theorem 3.8.7 is a fairly easy consequence of the local home geneity lemma (Exercise (2)). The idea is to show that any two points in the same connected component of M are isotopic in the following sense. DEF~NITION 4.1.12. Let xo, yo E M . We say that xo is isotopic to yo, and write xo -1 yo, if there is rp E iff:!^) such that ip(xo) = yo.

EXERCISES (1) Prove Lemma 4.1.4. (2) Prove the homogeneity lemma, Theorem 3.8.7, proceeding as follows. (a) Show that iff:(^) is a (normal) subgroup of Diff .(M). (b) Show that -1 is an equivalence relation on M. (c) Prove that the - I equivalence classes are exactly the connected components of M.

4. FLOWS AND FOLIATIONS

106

$]~[b,

41 [i,i ]

(3) Let 9 : [0, 11 + [O, be a smooth map such that q(x) = 0 on (0, 11 and ~ ( x = ) 4 on (smooth Urysohn lemma). Extend this to a smooth map 9 : R + 10, $1 by requiring periodicity: p(x 1) = ~ ( x ) . This extension is clearly smooth. In X(R) define

+

+

Prove that X and Y are complete, but that X Y is not. (4) Let @ : R x M -t M be a flow. A subset C C_ M is said to be @-invariant if @,(x) E C, Vx E C, Vt E R. If x E M and Rx = {@t(x))tEwis the flow line through x, prove that the closure Rx is a @-invariant set. (5) Let @ be a flow on M and let x E M be a point not fixed by the flow, but such that @,(x) = x for some c > 0. Prove that there is a number Q > 0 and a smooth imbedding

such that @t(4

= cp(e2 r i t / c o

1 7

-00

< t < 00.

In this case, the flow line Rx is the imbedded circle c p ( ~ land ) we say that x is a periodic point of the flow of period co. (6) Let 9 be a flow on M. A subset C M is said to be a minimal set of @ if (a)

C # 0;

(b) C is closed in M; (c) C is @-invariant; (d) C contains no proper subset with all of these properties. For example, if x is a periodic point (Exercise (5)), the flow line R x is a minimal set. Prove that, if M is compact, every closed, nonempty, @invariant subset of M contains at least one minimal set. (In particular, by Exercise (4), every flow line approaches at least one minimal set.) Show by an example that M itself may be a minimal set. (7) Let @ be a flow on M. One defines the a-limit set and the w-limit set of a flow line Rx as follows:

a ( x ) = (y E M 1 3 t k 1-00 such that lim @t,(x) = y) k--roo

w(x) = { y E M 1 3 t k f

00

such that lim k+oo

at,(x) = y).

If M is compact, prove that each of these limit sets is a compact, nonempty, @-invariant set. Show by examples that a(x) and w(x) may or may not be equal and may or may not be minimal. (Remark: The a- and w-limit set terminology is standard and seems to have its origin in a biblical quotation (Revelation 1:8).)

4.2. LIE BRACKET

107

4.2. The Lie Bracket

As we have already remarked, X(M) is a Lie algebra over R under the Lie bracket. Vector fields D, when viewed as derivations of the function algebra C" (M), localize to open subsets U c M. This localization is equivalent to the restriction DIU of D as a smooth section of the tangent bundle, so it is elementary to check the following. LEMMA4.2.1. If X, Y E X(M) and if U

M is open, then

In particular, properties of the bracket that were proven with coordinates can be extended to global properties on M. We apply this remark to Lie derivatives. Since every vector field X E X(M) generates a local flow on M, the definition of the Lie derivative Lx(Y) = lim t+o

a-t*(Y) t

-

y

makes sense pointwise on M and defines a new field L x ( Y ) E X(M). The proof that Lx (Y) = [X,Y] (Theorem 2.8.14) that was given in R can be carried out in local coordinate charts hence, by Lemma 4.2.1, globalizes. THEOREM 4.2.2. If X, Y E X(M), the Lie derivative of Y b y X is defined and smooth throughout M and

Similarly, Theorem 2.8.18 globalizes. Here, commutativity of local flows @ = and !I' = {!I'P}pEB means that !I'f o 8: = o !I'f wherever both sides are defined. Commutativity of vector fields X, Y E X ( M) means that [X,Y] = 0 on M. THEOREM 4.2.3. Vector fields X , Y on M commute if and only zf the local flows they generate on M commute. COROLLARY 4.2.4. Complete vector fields X, Y E X(M) commute if and only if the flows that they generate commute. We are going to be interested in Lie subalgebras of X ( M ) . If F C_ T ( M ) is a k-plane subbundle (also called a k-plane distribution on M), r ( F ) C X(M) is a Coo(M)-submodule and a real vector subspace. It is not generally a Lie subalgebra but, when it is, there are important geometric consequences.

DEFINITION 4.2.5. The k-plane distribution F

C_ T(M) is a F'robenius distri-

bution if r ( F ) is a Lie subalgebra of X(M). REMARK.If f , g E C w ( M ) and X, Y E X(M), it is easy to verify the identity Consequently, if F S T (M) is a k-plane distribution and if X 1, X2, . . . ,Xr E r ( F ) span r ( F ) over C m ( M ) , then F wilI be a F'robenius distribution if and only if [X,, Xj] E I'(F), 1 5 i, j 5 r.

108

4. FLOWS AND FOLIATIONS

EXAMPLE 4.2.6. On the group manifold Gl(n), we will define a couple of interesting Frobenius distributions. First recall (Example 2.7.15) that the space of left invariant vector fields ~ [ ( nc) X(Gl(n)) is a finite dimensional Lie subalgebra, canonically identified with the Lie algebra im(n) of n x n real matrices under the commutator bracket. Let s[(n) c g[(n) be the set of matrices of trace 0 and let o(n) c g[(n) be the subset of skew symmetric matrices. These are clearly vector subspaces and the reader should have little difficulty in computing dim s[(n) = n 2

-

1

n(n - 1) 2 Since tr(AB) = tr(BA), we see that tr[A, B] = 0,hence s[(n) is a Lie subalgebra of gI(n). If A, B E o(n), then dim o (n) =

so o(n) C gl(n) is also a Lie subalgebra. If {S1,.. . ,S n 2 - l ) is a basis of sI(n), extend it to a basis of g[(n) by adjoining a suitable vector S n 2 and view { s ~ )as $ a~set of left invariant vector fields on Gl(n). It is clear that, at each point P E Gl(n), these fields give a basis {PSI,.. ., PSn2)of Tp(G1(n)). Let Sp c Tp(Gl(n)) be the (n2- l)-dimensional subspace spanned by { P S ~ ) ~and ; ' remark that this subspace does not depend on the choice of basis of s[(n), Let

This is an (n2- 1)-plane distribution on Gl(n). Indeed, the vector fields {s~):!~ define an explicit trivialization of T(Gl(n)) r Gl(n) x ~ [ ( n relative ) to which S becomes the trivial subbundle Gl(n) x s[(n). (Warning: We are not using the standard trivialization of T(Gl(n)) = Gl(n) x im(n). This would give an imbedding Gl(n) x sI(n) c T(Gl(n)) different from the one we have defined and not very interesting.) Similarly, extending a basis of o(n) to one of gl(n) and viewing these as left invariant vector fields on Gl(n), we obtain a distribution 0 c T(Gl(n)) of fiber dimension n ( n - 1)/2 and independent of the choices. By the remark preceding this example, the fact that d ( n ) and o(n) are closed under the bracket implies that S and 0 are Frobenius distributions on Gl(n). Also, by our construction, if P E Gl(n), then

Recall that the special linear group is the subgroup Sl(n) c Gl(n) consisting of the matrices of determinant 1 (Example 2.5.5). In Example 2.5.8, we showed that TI(Sl(n)) is the subspace of Tr(Gl(n)) = m ( n ) consisting of the matrices of trace 0. For each Q E Sl(n), LQ carries Sl(n) onto itself, so it follows from our construction that TQ(Sl(n)) = SQ. Also, for arbitrary P E Gl(n),

4.2. LIE BRACKET

109

We say that the left cosets P - Sl(n) are integral submanifolds to the distribution S. In exactly the same way, using Exercise (3) on page 46, we see that the left cosets P - O(n) of the orthogonal group are integral submanifolds to the distribution 0 C T(Gl(n)). These are the first examples of foliations in this course. The left cosets of these groups are the leaves of the foliation integral to the respective distributions. A distribution F on M with (one to one immersed) integral submanifolds through each point of M is said to be integrable. We will see that the Frobenius property is precisely the integrability condition (cf. Example 3.4.16). This is the theorem of Clebsch, Deahna, and Frobenius, commonly called the Frobenius theorem. Generally speaking, a smooth map f : M -+ N does not push a vector field X E X(M) forward to a vector field f, (X) E X(N). There are two problems. If f is not surjective, there would be points of N where f ,(X) would not even be defined. If f is not injective, there could be points of N where f , (X) would be multiply defined. Nevertheless, there are situations in which f fails to be bijective, but the following concept makes sense.

DEFINITION 4.2.7. If f : M -+ N is a smooth map between manifolds (possibly with boundary), vector fields X E X(M) and Y E X(N) are said to be f-related if, for each q E M , f,,(X,) = Yf (,). EXAMPLE 4.2.8. It is possible that X E X(M) is not f-related to any Y E X(N). For example, let f : R -+ S1 be the map f (t) = eZTit.Then

There is clearly no vector field on S1 satisfying this. EXAMPLE 4.2.9. It is possible that X E X(M) may be f-related to many vector fields in X(N). For example, let f : S1 -+ (C be the inclusion map. Let U c @ be an open neighborhood of S1 such that U # @. Let X : @ -t [ O , l ] be smooth such that supp(X) C U and XIS1 1. Define the vector field Y, = iz on C. Since z I iz, we obtain X E X(S1) by setting X, = iz, Vz E S1. Clearly, X is f -related to both Y and XY and these are distinct fields on @

-

--

PROPOSITION 4.2.10. Let f : M -+ N be smooth and let X , Y E X(M) - -be f -related to X , Y E X(N), respectively. Then [X,Y] is f -related to [ X ,Y]. Equivalently, Lx (Y) is f -related to La(?). PROOF.For each h E C w ( N ) and for each q E M , Y(h

f )(q) = Yq(h 0 f ) = f*q(Yq)(h)= Yf (*) (h) = ( F ( h ) 0 f)(q).

That is, Y(ho f ) = F(h) 0 f , Vh E C w ( N ) .

4. FLOWS AND FOLIATIONS

110

There is a similar relation between X and

2.Thus,

V h E Ca(N). That is,

This is the assertion of the proposition.

EXERCISES (1) Let f : M -+ N be a one to one immersion. Let Y E X(N). Prove that there exists a field X E X(M) that is f-related to Y if and only if Yf( q ) E f.,(T,(M)), V q E M. In this case, prove that X is unique (we call X the restriction of Y to the immersed submanifold). (2) Let f : S2n-1L) R2n be the inclusion (an imbedding, hence a one to one immersion). Find Y E X(R2n)that restricts, as above, to a nowhere zero vector field X E X(S2"-I). (By Theorem 1.2.12, this is false for the inclusion f : S2n R2n+1. This fact is more elementary than Theorem 1.2.12, however, and will be proven in Chapter 8, Theorem 8.7.5.)

-

4.3. Commuting Flows

An Rn-chart (U, xl, . . . ,xn) about q E M determines n commuting vector fields

The corresponding local flows about q are of the form

Conversely, the following implies that commuting, linearly independent vector fields correspond to a coordinate chart.

THEOREM 4.3.1. Let M be an n-manifold without boundary, let q E M, let

U be an open neighborhood of q, and let X I , . . . ,Xk E X(U). If these vector fields commute and if {x:, . . . ,x;} is linearly independent, then there is a local coodinate chart (W,(P) about q such that

4.3. COMMUTING FLOWS

111

PROOF.Making U smaller, if necessary, we assume that it is a coordinate chart, hence view it as an open subset of Wn. We can do this so that q becomes the origin 0 and so that the vectors

form a basis of To(U). Let

be a local flow about q = 0 generated by X', 1 5 i 5 k. We can choose of the form (-E, E )with ~ E > 0 so small that the formula 8(x1,. . . , x") = a:, is defined, V (zl,. . . ,xn) E

0

.

0

% to be

02, (0,. . . ,o, x k+l , . . . ,xn)

F.This defines a smooth map

Since [X" X j ] = 0 on U, 1 5 i, j 5 k, we have

for each permutation a of {1,2,.. . ,k). Also,

-

For r = (rl, ... , r n ) E W and 1 < i 5 k,

Here, the notation @fi is a common device for indicating omission of the term . In particular, B,O : Wn + Wn is nonsingular, so we can assume (choosing r > 0 smaller, if necessary) that 0 : f? -+ U carries % diffeomorphically onto an open neighborhood W of 0. By the above,

The desired coordinate chart (W,cp) is obtained by setting cp = 8 -' .

112

4. FLOWS AND

FOLIATIONS

In particular, if X is a nonzero vector field defined in a neighborhood of q E M , then there is a coordinate chart (U, x l , . . . ,xn) about q such that

DEFINITION 4.3.2. A k-flow on M is a smooth map 9 : R x M 9(v, x) = 9,(x), such that (1) 90= idM (2) QVfW= 9, 0 Qw, v v , w E Rk.

-+

M , written

It follows that 9-, = \EL', so 9, E Diff(M), V v E IRk. The map Rk -+ Diff (M), defined by v I+ 9,, is a homomorphism of the additive group JRk into the group Diff (M). We think of 9 as a smooth action of the group R on M. Given q E M define 9 4 : IRk 3 M by setting 'Q4(v) = Qv(q). In particular, \kg (0) = q. DEFINITION 4.3.3. Given a k-flow 9, the S-orbit of q E M is 99(IRk). When a choice of 9 is fixed, the Q-orbit is also called the Rk-orbit of q.

REMARK. Points p, q E M are said to be equivalent under the k-flow if and only if there exists v E Rk such that 9,(p) = q. The fact that this is an equivalence relation is a trivial consequence of Definition 4.3.2. The R k-orbits are the equivalence classes. DEFINITION 4.3.4. The k-flow Q is nonsingular if

is one toone, Qv E Rk, Vq E M. LEMMA4.3.5. The k-flow Q is nonsingular if and only if

is one to one, V q E M .

PROOF. For fixed v E Itk, let

T,

: IRk

-+Rk denote translation byv. Then,

Vv E Rk, Vq E M. By the chain rule,

But (T-~),, = idRk and 9:o is one to one.

is bijective, so \k$, is one to one if and only if

4.3. COMMUTING FLOWS

113

PROPOSITION 4.3.6. The set of k-tuples ( X l , . . . ,x') of complete, commuting vector fields on M is in natural, one to one correspondence with the set of k-Bows Q o n M . The fields X I , . . . , xk are pointwise linearly independent i f and only i f Q is nonsingular.

PROOF. Given the k-tuple (X1,. . . , x k ) of complete, commuting fields, let a', . . . ,@ be the corresponding flows. Since these flows commute, the formula defines a k-flow on M. Conversely, given the k-flow Q, let { e 1, . . . ,e k} be the standard basis of IRk and set

af = Q t e ; , 1 5 i 5 k. This defines k commuting flows and their corresponding infinitesimal generators, X I , . . . are complete, commuting vector fields on M. Finally, these fields are linearly independent at q E M if and only if Q:, is one to one. Thus, the previous lemma gives the final assertion.

,xk

THEOREM 4.3.7. Let M be a connected n-manifold. If there exists a nonsinfor some integer gular n-flow on M , then M is difleomorphic to T~ x k = 0 , 1 , ..., n. Modulo one technical point, the proof of Theorem 4.3.7 is quite straightforward. Nonsingularity of the n-flow implies that, for each q E M , the smooth map 9 9 : Rn -+ M has constant rank n. Consequently, each Rn-orbit is an open subset of M. Being equivalence classes, distinct orbits are disjoint, so the connectivity of M implies that there is only one Rn-orbit and Qq is a surjection. Fix q E M. If v , w E Rn, then

It is clear that G = (Qg)-l(q) is an additive subgroup of R n , so Qq passes to a well defined homeomorphism

If G were the subgroup 2Zk x {O} c C kx

we would have a natural structure of smooth n-manifold on Rn/G = T~ x IRn-k and, 9 9 being locally a diffeomorphism, the homeomorphism

would be a diffeomorphism. Therefore, it will be enough to find a linear automorphism A : Rn + Rn such that A(G) = 2Zk x (0). This is the technical point mentioned above.

4. FLOWS AND

114

FOLIATIONS

DEFINITION 4.3.8. An additive subgroup G C Rn is a kclimensional lattice if it is generated by a linearly independent subset { v . . . ,v k ) C Rn . If G is a k-dimensional lattice for some k = 0,1, . . . , n, then G is called a lattice subgroup.

<

For example, Z n c Rn is an nclimensional lattice. More generally, if 0 k 5 n, then Z k x (0) C Rk x R ~ is-a k-dimensional ~ lattice. In fact, if G is as in Definition 4.3.8, a linear automorphism of Rn taking v, to the standard basis vector ei, 1 5 i k, carries G to 2Zk x (0).

<

THEOREM 4.3.9. A nontrivial, additive subgroup G C R n is a lattice subgroup i f and only if there is a neighborhood U C R n of the origin such that G n u = (0). By the above remarks, this theorem will complete the proof of Theorem 4.3.7. Indeed, the fact that Q4 is locally a diffeomorphism implies immediately that there is a neighborhood of 0 E R n meeting G = (QQ)-'(~)only in the point 0. Before proving Theorem 4.3.9, we discuss some other consequences. COROLLARY 4.3.10. If G C R is an additive subgroup that is isomorphic to Z k , some k _> 2, then G is dense i n W .

PROOF.For dimension reasons, G is not a lattice subgroup. Given E > 0, one can find a E G such that la1 < E (Theorem 4.3.9). Thus, { r ~ ) , ~czG partitions R into intervals of length < E, so every t E R lies within E: of a point of G and G is dense in R. COROLLARY 4.3.11. If G c S 1 is a subgroup that is isomorphic to 2Zk, some k _> 1, then G is dense i n S 1 . PROOF. Indeed, the standard projection p : R -+ S 1 is a group surjection and p-I (G) is an additive subgroup of W that is isomorphic to Z k f l . By Corollary 4.3.10, p-l(G) is dense in R and the assertion follows. Either of these corollaries can be used to give a proof of the assertion in Example 4.1.7 that each flow line of the irrational slope, linear flow on T is everywhere dense (Exercise (2)). For the proof of Theorem 4.3.9, we need two lemmas. LEMMA4.3.12. If there exists a neighborhood U as i n Theorem 4.3.9, then every bounded subset of G is finite. PROOF.If B C_ G is bounded, let { g i ) z l C B. Since B is bounded, we can assume, without loss of generality, that this sequence is Cauchy. Let E > 0 be so small that the €-neighborhood of 0 in R n is contained in U and choose r > 0 such that J J g i - g j l J< E, V i , j 2 . r . Then i , j 2 r =+ gi - g j E G n U = {0), so the sequence must have only finitely many distinct terms. LEMMA4.3.13. If U is as in Theorem 4.3.9 and n = 1, then G is infinite cyclic.

4.3.

COMMUTING FLOWS

115

PROOF. Let g E G n (0, oo) be the element closest to 0. If there were no such element, we could produce an infinite, strictly decreasing sequence in G n (0, oo), contradicting Lemma 4.3.12. Let A = {mg I m E Z), an infinite cyclic subgroup of G. We claim that A = G. Otherwise, find f E G \ A and m E Z such that mg < f < (m 1)g. Then 0 < f - rng < g and f - mg E G, contradicting the choice of g. [7

+

fROOF OF THEOREM 4.3.9. If G C Rn is a lattice subgroup generated by the linearly independent vectors vl, . . . , vk, there is a nonsingular linear automorphism L : Rn -+ W n such that L(vi) = ei, the irnathth standard basis vector, 1 5 i 5 k. Since L is also a homeomorphism, we lose no generality in assuming vi = ei, 1 5 i 5 k . In this case, elementary geometry shows that llgll 2 1, Vg E G \ (0). Thus, U = {v E Rn I llvll < 1) has the property that U n G = (0). For the converse, suppose U exists as desired and proceed by induction on n. The case n = 1 is true by Lemma 4.3.13. For the inductive step, assume the truth of the theorem for some n 2 1 and suppose that G c Rn+l and U c Rn+' satisfy the hypotheses of the theorem. Let 0 # g E G and let V C R n f l be the onedimensional vector subspace spanned by g. The major step in our proof will be to show, via the inductive hypothesis, that G/(G n V) is a lattice subgroup of IWn+l/vz Rn. By Lemma 4.3.13, G n V is infinite cyclic, generated by some go-E G n V. Let {7,)g1be a sequence in G/(G fl V) c Rn+'/V such that lim,,, ji = 0. Write f i = fi (G n V), f, E G c Rn+'. Then the distance of f, from the line V approaches 0 as i --t oo. Thus, for some constant c > 0, one can find m i E Z such that 11 fi-migoll < C , V i -> 1. By Lemma 4.3.12, { fi-migo)El c G contains only finitely many distinct elements. That is, Ti = fi (GnV) = (fi - migo) -( G n V) assumes only finitely many distinct values as i --t oo. Since limi+oo f i = 0, we conclude that Ti = 0 for large enough values of i. Therefore, there is a neighborhood U' C Rn+'/V of 0 such that U' n (G/(G n V)) = (0). By the inductive hypothesis, G/(G n V) is a lattice generated by a linearly independent set {gi (G n v))L, C Rn+'/V. Given g E G, write

+

+

+

+

That is, there is ro E Z such that

Thus, {go,91, . . . ,ge) generates G and this set is clearly linearly independent in IWn+l.

REMARK.Following Milnor [30],one defines the rank of an n-manifold M to be the maximum number of everywhere linearly independent, commuting vector fields that the manifold admits. By Proposition 4.3.6, the rank of a compact n-manifold M is the largest integer r 5 n for which there exists a nonsingular

4. FLOWS AND

116

FOLIATIONS

r-flow on M. It is a celebrated theorem of E. Lima 1251 that the rank of S3is 1. Since S3 is parallelizable, it admits a nowhere 0 vector field, hence a nonsingular 1-flow. On the other hand, Theorem 4.3.7 implies that the only compact 3manifold of rank 3 is T3. The hard part of Lima's theorem is to show that S3 does not have rank 2.

EXERCISES (1) Let M be an n-manifold, aM = 0. Prove that M is integrably parallelizable (Definition 3.4.8) if and only if there exist pointwise linearly independent, commuting vector fields X l, . . . , Xn (not necessarily complete). Use this t o prove that a compact n-manifold is integrably parallelizable if and only if each component is diffeomorphic to T n. (2) Use Corollary 4.3.10 or Corollary 4.3.11 to give a proof of the assertion in Example 4.1.7 that a line l c R2 of irrational slope projects one to one onto an everywhere dense curve in T2. 4.4. Foliations

Let F T ( M ) be a k-plane distribution on M . For simplicity, we consider only the case in which dM = 8. DEFINITION 4.4.1. An integral manifold of F through q E M is a one to one immersion i : N -+ M of a k-manifold such that q E i ( N ) and

We generally identify N and i(N). This is similar to the customary identification of a curve s : [a,b] + M with its image. The correct topology on the subset i ( N ) of M is the manifold topology of N , not the relative topology. DEFINITION 4.4.2. A k-plane distribution F on M is integrable if, through each q E M , there passes an integral manifold of F.

EXAMPLE 4.4.3. Take M

= R3 and let

F be the 2-plane distribution spanned by the pointwise linearly independent vector fields

Let

7r : IR3 -+

R2 be the projection ~ ( xy,, z ) = (x, 9). Since

4.4. FOLIATIONS

117

an integral manifold of F through q will be carried by 7r locally diffeomorphically onto an open subset of W2. That is, an integral manifold of F is locally a graph = f (x,9) and

That is, f (x,y ) solves the system

This overdetermined system of P.D.E. implies that

That is, a necessary condition for F to be integrable is that -a g= - ah

ay

ax'

It turns out, as we will see, that this is also a sufficient condition for integrability. This integrability condition can also be written in terms of brackets. Indeed,

so the integrability condition becomes

[X, Y J= 0. By our theory of commuting vector fields, this condition implies that there is a local coordinate chart (U, u, v, W) about q in which

118

4. FLOWS AND

FOLIATIONS

and, in this coordinate system, the integral manifolds are given by the equations w = const. Finally, in this coordinate neighborhood, arbitrary fields Z 1, Zz E I'(FJU) are function linear combinations of X, Y, hence integrability implies that [Z1,Zz] E r(F1U). This is true in suitable coordinate charts about each point of W3, hence r ( F ) C X(R3) is a Lie subalgebra and F is a Frobenius 2-plane distribution on R3. This exemplifies the following theorem, due to Clebsch, Deahna, and Frobenius, but generally credited only to the last of this trio.

THEOREM 4.4.4 (THE FROBENIUS THEOREM). If F & T(M) is a k-plane distribution on M , the following are equivalent. (1) F is integrable. (2) F is a Fkobenius distribution. (3) About each q E M there is a coordinate chart (W, x l , . . . ,xn) such that

The proof of the Frobenius theorem is the primary goal of this section. Before giving a proof, however, we discuss some of the consequences. A coordinate chart as in (3) of Theorem 4.4.4 will be called a h b e n i u s chart. Theorem 4.4.4 allows us to find a Cm atlas {(U,, cp,)),En on M such that the associated family of local trivializations

is contained in the Gl(k, n - k)-reduction of T ( M ) corresponding to F. This means that the associated Jacobian cocycle satisfies

Jg,p : U,

n Up

-t

Gl(k,n - k),

V a , p E a. That is, the infinitesimal Gl(k,n - k)-structure determined by F is integrable in the sense of Definition 3.4.11. Fix a coordinate cover {Vx,x i , . . . , satisfying (3) in Theorem 4.4.4 and such that (1) the index set C is at most countably infinite and {VA)XEL: is a locally finite cover of M; (2) x i ranges over the open interval (-2,2), V X E C, 1 5 i 5 n; (3) if Wx C VA is defined by the inequalities -1 < x i < 1, 1 i 5 n, then {Wx)xEc is an open cover of M . This is possible since M is 2nd countable and paracompact. We are primarily interested in the coordinate neighborhoods WA,with the VA's playing an auxiliary role.

<

DEFINITION 4.4.5. A coordinate cover {WA,x i , . . . ,~ 1with )all of ~the above properties is called a regular cover for the integrable distribution F.

If a = (ak+', . . . ,a n ) where a' E (-1, I), k + 1 5 i _< n, the equations define an integral manifold PA,,to FIWx, called a plaque of F in Wx. Every connected integral manifold to FJWx lies in some plaque. Similarly, there are

~

~

4.4. FOLIATIONS

119

a,..

plaques FA,, of F in V' with PA,. c Remark that, by the definition of is a compact subset of VAand PA,, is a compact regular cover, the closure = [- 1,1Inin the coordinates of VA. subset of FA,.. In fact,

mA wA

LEMMA4.4.6. A plaque PA,, of F in Wx meets at most finitely many plaques PXl,bl,. . . PA,,*, of F i n other charts of the regular cover.

PROOF.Since the cover (Vx)xEr: is locally finite, each point of PA,, has a neighborhood in 6,.that meets only finitely many plaques PA,,bi.Since FAia is compact, the assertion follows. LEMMA4.4.7. Let F be an integrable distribution on a manifold M and let Nl, N2 c M be integral manifolds of F . Then N1 n N2 is a n integral manifold of F . PROOF. If N1 n N2 = 0, there is nothing to prove. Let p E N1 n N2 and let

(W, x l , . . . ,xn) be a Frobenius chart about p. That is, FIW is spanned by the first k coordinate fields and the plaques Pa are the connected integral manifolds to FIW. Let q E N1 belong to the component of p in Nl n W (with its manifold topology). Then, there is a smooth curve s : [O, 11 -+ N1 with s(0) = p, s(1) = q and s ( t ) E Nl n W, 0 5 t 5 1. Then s ( t ) E T,(,)(Nl) = F,(,), 0 5 t 5 1, so

That is,

x j ( s ( t ) )= x ' ( ~ ) = a j ,

+

a constant, k 1 5 j 5 n, 0 5 t 5 1. This proves that the component of p in Nl n W lies in a plaque Pa. Since dim N1 = k , this component is an open subset of the plaque Pa. Similarly, the component of p in N2 n W is an open subset of Pa. Therefore, the component of p in N1 n N2 n W is an open subset of Pa, hence is an integral manifold to F . Since p E Nl n N2 is arbitrary, Nl n N2 is a (possibly disconnected) integral manifold to F. DEFINITION 4.4.8. If F is an integrable distribution on M , then x, y E M are said to be F-equivalent, and we write x y, if there exist connected integral manifolds N1, N2,. . . ,N, of F such that x E NI, y E N,, and Ni f~Ni+1 # 0, 15i
PROPOSITION 4.4.9. Let F be an integrable k-plane distribution on M. Then the relation N F is a n equivalence relation on M and the equivalence classes are the maximal connected integral manifolds to F .

120

4. FLOWS AND FOLIATIONS

PROOF. The fact that - F is an equivalence relation is practically immediate. We show that each -F equivalence class L C_ M is (the image of) a one to one immersed submanifold integral to F . Fix p E L. Given q E L, choose connected integral manifolds

such that p E N l , q E N,, and Nin Ni+1 # 0, 1 5 i 5 r - 1. By the definition of -F, NiC L, 1 5 i 5 r . In particular, L is a union of integral manifolds to F and we can topologize L by letting the open subsets N C L be exactly the unions of integral manifolds to F that lie entirely in L. Then L itself is open, the empty subset is open by default, arbitrary unions of open sets are open and Lemma 4.4.7 guarantees that finite intersections of open sets are open. This is a locally Euclidean topology of dimension k. Indeed, the topology in each of the integral manifolds in L is the usual manifold topology (open subsets of integral manifolds are integral manifolds). Since connected integral manifolds are path connected, the definition of F implies that L is path connected in this topology. The fact that the topology is Hausdorff is easy. We prove that L is 2nd countable. Let { (W, ,x: , . . . ,x : ) ) , ~ ~ be a regular cover of M. Each plaque Pa c W , is connected, hence each plaque that meets L lies entirely in L. Thus, L is a union of plaques and the chains N 1 , . . . ,NT in the definition of F-equivalence can be taken so that each Ni is a plaque. It will be enough to prove that the set of plaques in L, defined by the given regular cover, is at most countably infinite. L. Recall (Lemma 4.4.6) that each plaque Fix a plaque Po with p E Po meets only finitely many other plaques. Thus, for a fixed integer r > 0, there can be only finitely many plaque chains Po,PI,. . . , Pr with Pi-, n P, # 0 , 0 < i 5 r. Thus, as r ranges over the positive integers, the number of such plaque chains, starting at the fixed Po,is only countable. Since every plaque in L is reached by a finite plaque chain from Po,there are at most countably many distinct such plaques. We have shown that L is a connected topological k-manifold. But property (3) in Theorem 4.4.4 provides a smooth atlas on L and shows that this manifold is smoothly immersed in M. Being locally integral to F, L is itself a connected integral manifold, obviously maximal with this property by the definition of

-

-F'

DEFINITION 4.4.10. The decomposition of M into F-equivalence classes is called a foliation F of M. Each F-equivalence class L is called a leaf of the foliation 3. If dim M = n and the leaves of F are k-dimensional, the dimension of the foliation is d i m F = k and the codimension is codimF = n - k.

is a nonsingular k-flow, the IRk-orbits are the leaves of a k4imensional foliation F of M. Indeed, the k-flow is generated by a family {X l , . . . , X k ) of everywhere linearly independent, commuting vector fields (Proposition 4.3.6) and these fields

4.4. FOLIATIONS

121

span a k-plane distribution F C T(M). Since the fields commute, F is a F'robenius distribution, hence integrable, and there is a corresponding k4imensional foliation F of M. For each q E M ,

~ is an immersion and G = (Pq)-' (q) is a lattice subgroup of Itk. Thus, I t k / is a smooth k-manifold and !Pq passes to a one to one immersion

This immersed submanifold is evidently a connected integral manifold to F, hence is an open neighborhood of q in the leaf L, of 3 through q. The image of $ is, in fact, the Itk%rbit of q. If it were not all of L,, then this leaf would be the disjoint union of two or more such orbits, each open in L,, contradicting the fact that L, is connected. As we have seen, the foliation F can be quite complicated. As in Example 4.1.7, there is such a onedimensional foliation of T 2 with each leaf everywhere dense in T2. In fact, we will see that this is true for Tn, V n 2 (Example 5.3.8). In the same way, higher dimensional nonsingular k-flows on Tn are readily produced having everywhere dense leaves (cf. Exercise (2)).

>

We turn to the proof of Theorem 4.4.4.

PROPOSITION 4.4.12. If F is an integrable distribution, then F is Fkobenius. PROOF. Let X, Y E I'(F), - - q E M , and let i : N + M be an integral manifold of F through q. Let X, Y E X(N) be the unique restrictions of X, Y to N (Exercise (1) on page 110). Then [ X ,Y] is i-related to [X,Y]. In particular, if q = i(p), - [X,Y], = ip*[X,YIP € ip*(Tp(N))= F,.

--

This proves that (1) + (2) in Theorem 4.4.4. The following gives (3)

+ (1).

PROPOSITION 4.4.13. Let F be a k-plane distribution on M . If each q E M has a coordinate neighborhood (U, X I , .. . , xn) such that

then F is integrable. PROOF.Indeed, if ai = xi(q), k + 1 5 i 5 n, the level set {(xl,. . . , x n ) 1 xi = a', k + 1

< i 5 n)

is a k-dimensional integral manifold of F through q. It remains that we prove (2) =+ (3). This is the hard part.

4. FLOWS AND FOLIATIONS

122

-

LEMMA4.4.14. Let U Rn be an open subset and let F be a k-plane distribe the projection ( x l , . . . , x n ) H ( x l , . . . , xk). bution on U. Let ?r : Rn i Let p E U and suppose that ,?.r : Fp T,(,)(W~) is bijective. Then there is an open neighborhood W o f p in U such that a,, : F, -+ T,(,)(IRk) is bijective, vx E W.

PROOF. Let V be an open neighborhood of p in U such that there are fields X1, . . . ,xkE I'(F1U) which give a basis of F,,Vx E V. Write

Then

Consider the k x k matrix

[

f; ( 4 - - - f; ( 4

A(.)

=

f:(x)

.-.

]

.

XCx)

By assumption, det A(p) # 0, so, for a small enough open neighborhood W of p in V, det A ( x ) # 0, Vx E W. That is, n,, : F, -+ T,(,)(IR~) is bijective, V x E W. LEMMA4.4.15. Let F be a k-plane distribution on M and let q E M . Then there is a coordinate chart (W,x l , . . . ,xn) about q such that the map

given by n ( x l , . . . , x n ) = (xl,. . . , x k ) , has

bijective, Vx E W . PROOF.By Lemma 4.4.14, we must choose the coordinates so that

is bijective. Then we restrict to a smaller neighborhood, if necessary. Let (U, xl, . . . ,xn) be a coordinate chart about q and let X . . . ,xk E I'(FIU) give a basis of F,. Permuting the coordinates suitably, we can assume that

',

is a basis of Tq(M). Then the surjection rr,, : Tq(M) i T,(,)(IKk) annihilates the last n - k of these vectors, hence it carries {x: . . . ,x,*} to a basis of ( 9 )R k

4.4.

FOLIATIONS

123

L E M M A4.4.16. Let F , q E M , and n : W + I t k all be as in Lemma 4.4.15. Given x E W and 1 5 i 5 k , let 2: E F, be the unique vector such that

Then Z1,. . . ,ZkE I'(FI W ) . PROOF. The only problem is to prove that Z iis smooth at each x E W , 1 2 i 5 k. We can assume that there is a trivialization T ( W ) W x Rn relative to which FIW E W x w*. The standard trivialization of T ( R * )is given by the coordinate fields azl

7 ' ' '

axk

'

We express the linear map a*, : T,(W) + T,(,)(w*) relative to these trivializations by a matrix [ A ( x ,) B ( x ) ]where A ( x ) is k x k and B ( x ) is k x ( n- k). Since rr,, carries F, = { x ) x ak bijectively onto { n ( x ) )X P * , we see that A ( x ) E Gl(k) and depends smoothly on x. Thus, A(x)-' is also smooth in x and

has image F,. The zth column of this matrix is depends smoothly on x.

21, 1 5 i 5 k, so this vector

We can now complete the proof of Theorem 4.4.4. PROPOSITION 4.4.17. If F is a h b e n i u s distribution on M , then about each q E M there is a coordinate chart (W,x l , . . . ,x n ) such that

PROOF. Let W be a coordinate neighborhood of q and Z I , . . . ,ZkE r(FI W ) , all as in Lemma 4.4.16. Since Zi is is n-related to the zth coordinate field, 1 5 i 5 k , it follows that .rr,[Zi,Zj] = 0, 1 5 i ,j 5 k. But the Frobenius condition implies that [Zi, Z j ] E I'(F I W ) and n, is one to one, so [Z2,Z j ] = 0 , 1 i,j 5 k. Since these fields are pointwise linearly independent, Theorem 4.3.1 furnishes a coordinate chart (U, y , . . . ,yn) around q E W such that

<

'

EXERCISES ( 1 ) Let M = R3 \ (0). (a) Define

Fv = ( W E T v ( M ) I W I v ) , V v E M . Prove that F = UVEM Fv is an integrable distribution and describe the integral manifolds.

4. FLOWS AND FOLIATIONS

(b) Define Ev = { ( a ,b, c) E Tv(M) ( ( b , c, a) 1v),

Qv E M. Prove that E = UVEM EY is a 2-plane distribution and decide whether or not it is integrable. (2) Construct a nonsingular 2-flow on T 3 having each leaf diffeomorphic to the cylinder S1 x B and everywhere dense in T 3 . (3) Let M be an n-manifold without boundary. If f : M -+ N has constant rank k, prove that, as g ranges over f (M), the connected components of the level sets f -l(y) range over the leaves of a foliation F of M of dimension n - k. (4) Let f : R3 -+ R be the submersion

By Exercise (3), the connected components of the level sets of f are the leaves of a twdimensional foliation F of R3. Show the following. (a) The cylinder x2 y2 = 1 is a leaf Lo of F. (b) The leaves interior to this cylinder Lo are diffeomorphic to B2. (c) The leaves exterior to Lo are diffeomorphic to cylinders. (But they are not geometric cylinders.) (d) The foliation F is invariant under translations in the z-coordinate. Although we have considered foliations only on manifolds M without boundary, one often relaxes this assumption by requiring special behavior for foliations near a M . In the case of foliations of codimension one, it is natural to require that each component of a M be a leaf. Use Exercise (4) to produce a foliation P of LI2 x S1 having the boundary torus S1 x S1 as a leaf and having all other leaves diffeomorphic to R2.This is the famous "Reeb foliation" of the solid torus. Let F be a foliation of M and let i : L M be the one to one immersion of a leaf. Let X be a manifold and f : X -+ M a smooth map such that f (X) C i ( L ) . Then i-' o f : X -i L is smooth. (Hint: cf. Exercise (2) on page 93.) Let .F be a foliation of M . A subset C C M is F-saturated if, for each x E C, the entire leaf Lx through x lies in C. Prove that the closure C of an F-saturated set is an F-saturated set. Let .F be a foliation of M. A subset C C M is said to be a minimal set of F if (a) C # 0; (b) C is closed in M; (c) C is F-saturated; (d) C contains no proper subset with all of these properties. For example, a closed leaf is a minimal set. Prove that, if M is compact, every closed, nonempty, F-saturated subset of M contains at least one minimal set. (In particular, by Exercise (7), every leaf of F closes on at least one minimal set.) Show by an example that M itself may be a minimal set.

+

(5)

(6)

(7)

(8)

-

4.4. FOLIATIONS

(9) Let L

125

c M be a leaf of a foliation F. The limit set of L is defined by lim(L) =

n

(L \ K ) ,

KEK

where K: is the family of compact subsets of the leaf L and the overline denotes closure in M. Prove the following. (a) If M is compact, then lim(L) is a compact, F-saturated set. (b) If M is compact, lim(L) = 0 if and only if L is compact. (c) If L is dense in M (but not equal to M) lim(L) = M. (d) If the leaf L is an imbedded submanifold of M , then Lnlim(L) = 0.

CHAPTER 5 Lie Groups and Lie Algebras

Lie groups and their Lie algebras play a central role in geometry, topology, and analysis. Here we can only give a brief introduction to this fascinating topic. 5.1. Basic Definitions and Facts

A topological group is a topological space together with a group structure on that space such that the group operations are continuous. A Lie group is a differentiable manifold together with a group structure on that manifold such that the group operations are smooth. Lie groups are also topological groups, but not vice versa. Here are the precise definitions.

DEFINITION 5.1.1. A topological group G is a topological space which is also a group such that the operations

are continuous maps.

DEFINITION 5.1.2. A Lie group G is a smooth manifold without boundary which is also a group such that the operations

are smooth maps. For example, every finite dimensional vector space over R or C is a Lie group under vector addition. Here are some more interesting examples.

EXAMPLE 5.1.3. The group Gl(n) is evidently a Lie group, the operations being given by rational functions of the coordinates. The subgroups Sl(n) and O(n) are smoothly imbedded submanifolds of Gl(n), hence smoothness of the group operations on Gl(n) implies smoothness of their restrictions to Sl(n) and O(n). It can be shown that Gl(n) and O(n) each have two connected components,

128

5. LIE GROUPS

distinguished by the sign of the determinant. The component of the identity I in each of these groups is itself a Lie group (Exercise (I)), denoted by Gl+ (n) and SO(n) respectively. The group SO(n) is called the special orthogonal group. Recall that the orthogonal group O(n), hence also the special orthogonal group, is compact (Exercise (3) on page 46).

EXAMPLE 5.1.4. The group Gl(k, n - k) consists of matrices

where A E Gl(k), C E Gl(n - k), and B is an arbitrary k x (n - k) matrix. Thus, Gl(k,n - k) is a manifold diffeomorphic to Gl(k) x Gl(n - k) x R ~ ( " - ~This ). is an open subset of R k a + ( n - k ) 2 + k ( n and the group operat ions are rational functions in the coordinates, so this is a Lie group.

EXAMPLE 5.1.5. The group Gl(n, C) of nonsingular, n x n matrices over the complex field C is a Lie group, called the complex general linear group. Remark that Gl(1, C) = C* is the multiplicative group of nonzero complex numbers. The unit circle Sf c @* is a subgroup and a smoothly imbedded submanifold, hence is also a Lie group. EXAMPLE 5.1.6. If G and H are Lie groups, then G x H is a Lie group under the usual Cartesian group operations and the smooth product structure. In particular, Tn = S1 x - .x S1 is a Lie group. EXAMPLE 5.1.7. Define ip : Gl(n,C) + Gl(n, @) by q(A) = A ~ A Here . the overline indicates complex conjugation in each entry of the matrix. This map has constant rank n 2 and q ( I ) = I. We define U(n) = i p - l ( ~ ) , a smoothly imbedded, nonempty submanifold of Gl(n, C) of dimension 2n2 - n2 = n2. It is easy to check that U(n) is also a subgroup, hence it is a Lie group, called the unitary group. For the same reasons that O(n) is compact, the Lie group U(n) is T compact. Since A E U(n) if and only if A l l = 2 , it follows that I det(A)I = 1. Indeed, det : U(n) + S1 is a group homomorphism and a submersion. One defines the special unitary group SU(n) = det-'(1) C U(n), a compact Lie group of dimension n 2 - 1. The reader is invited to check in detail these various assertions in Exercise (2). EXAMPLE 5.1.8. Let W denote the division algebra of quaternions. The nonzero quaternions W* form a multiplicative group and a manifold diffeomorphic to JR4 \ { O ) . It is clear that the group operations are smooth, so W* is a Lie group. The 3-sphere S3 c W*consists of the unit length quaternions, hence it is closed under multiplication and passing to inverses. This gives a Lie group structure on s3. Usually, the identity element of a topological group or Lie group will be denoted by e. For matrix groups, however, the customary symbol for the identity is I . Because the Lie groups S1 and S3are subgroups of the division algebras C and W respectively, the identity elements in these groups are denoted by 1, the unity of the respective algebras.

5.1. BASICS

129

DEFINITION 5.1.9. Let G be a topological group (respectively, a Lie group), a E G. Left translation by a is the continuous (respectively, smooth) map La : G --+ G defined by La(x) = ax, Vx E G. Remark that La-I = L i l , so La E Homeo(G). Similarly, if G is a Lie group, La E Diff(G). Also, the inversion map L : G + G is continuous and equal to its own inverse, so L E Homeo(G) and, when G is a Lie group, L E Diff(G). The following discussion illustrates some of the striking ways in which algebra and topology interact in these structures. LEMMA5.1.10. If G is a topological group and U G is a n open neighborhood of e E G, then there is a n open neighborhood V U of e with the property that v E V w v-l E V. Such a neighborhood V of e is said to be symmetric.

c

PROOF.Indeed, L(U)is also an open neighborhood of e E G, so V = U n L(U) is as desired. LEMMA5.1.11. If Z, W

cG

and if W is open in G, then the set

is open in G.

PROOF.Indeed, Z W = UzEZLz(W) is a union of open sets.

PROPOSITION 5.1.12. Let G be a connected topological group and let U c G be a n arbitrary open neighborhood of the identity e E G. T h e n U generates the group G. PROOF. By Lemma 5.1.10, we lose no generality in assuming that U is a symmetric neighborhood of e. Using Lemma 5.1.11, we define open sets

by induction on n. Since e E U, these form an increasing nest

of open neighborhoods of the identity. By the symmetry of U and the formula for the inverse of a product, each Un is symmetric. Thus, we obtain an open, symmetric neighborhood 00

of the identity. Clearly, U" is closed under group multiplication; hence, being a symmetric neighborhood of the identity, Urn is a subgroup of G. But the left cosets {aUrn)aEGform a cover of G by disjoint, open sets and the connectivity of G implies that there is only one coset. That is, U" = G. We focus our attention on Lie groups G and their associated Lie algebras. It will be seen that the introduction of Lie algebras produces further remarkable interactions of algebra, topology and calculus.

5 . LIE GROUPS

130

DEFINITION 5.1.13. A vector field X E X(G) is left invariant if, for each a E G, La,(X) = X . The set of left invariant vector fields on G is denoted by L(G). The following is quite easy, but very important.

PROPOSITION 5.1.14. T h e subset L(G) c X(G) is a Lie subalgebra. PROOF. Indeed, the bracket of La-related fields is La-related to the bracket of these fields. It follows immediately that the bracket of left invariant fields is a left invariant field. DEFINITION 5.1.15. If G is a Lie group, its Lie algebra is L(G). EXAMPLE5.1.16. We saw in Example 2.7.15 that the Lie algebra L(Gl(n)) = gl(n) consists of the fields RA, A E ?YJl(n), and that this defines a canonical isomorphism of Lie algebras L(Gl(n)) = m(n). Similarly, L(Sl(n)) = sI(n) is the Lie algebra of n x n matrices of trace 0 and L(O(n)) = o(n) is the Lie algebra of skew symmetric matrices (cf. Example 4.2.6). PROPOSITION 5.1.17. T h e evaluation map a n isomorphism of vector spaces.

E : L(G) + T,(G),

E(X) = Xe, is

PROOF. This is clearly a linear map. The fact that it is injective follows immediately from Xa = La, (X,), V a E G. We prove that E is surjective. Let v E Te(G) and define This defines X : G + T(G) carrying each a E G to X a E T,(G) and X, = v. We must prove that X is smooth, hence a vector field, and that it is left invariant. For f E Co0(G), form the function X ( f ) : G + IW by setting

If X (f ) is smooth, V f

E

Cm(G), it will follow that X is smooth. Note that

The function g : G x G + IW, defined by is smooth. Let (U, x l , . . . ,xn) be an arbitrary coordinate neighborhood in G. About e E G, choose coordinates (V, y l , . . . ,yn) relative to which e = (0,. . . ,0). Then (x, e) E U x V has coordinates (x', . . . , x n ,0,. . . ,O). Write

so that

5.1. BASICS

131

a smooth function in the arbitrary coordinate neighborhood (U,x l , . , . , x n ) . This proves that X E X(G). Finally, we prove that X is left invariant. Indeed, if a , b E G ,

(La*(X))b= (La)*a-lb(Xa-lb) = (La)*a-lb((La-lb)*e(Xe))

= (La O La-lb)*e(Xe)

= (Lb)*e ( X e ) = Xb.

Since b is arbitrary, La, ( X ) = X . Since a is arbitrary, X E L(G). COROLLARY 5.1.18. If G is a Lie group, then dim L(G) = dim G. COROLLARY 5.1.19. If G is a Lie group, there is a canonical trivialization of the tangent bundle n : T ( G )+ G . In particular, every Lie group is parallelizable. PROOF. Define cp : G x L(G) + T ( G )by cp(a,X) = X a . Then the restriction pa = cpl({a) x L(G))is an isomorphism (Pa : { a ) x

L(G)

+

Ta(G)

of vector spaces, V a E G. If we show that cp is smooth, it will be a bundle isomorphism. Fix a basis X I , . . . ,Xn of L(G). Coordinatize L(G) via this basis, thereby defining an isomorphism G x L(G) G x Rn. Relative to this coordinatization, cp becomes

Since the fields Xi are smooth, so is

@,

hence cp.

In particular, S 3 is parallelizable (cf. Theorem 1.2.10). This sphere is not, however, integrably parallelizable (cf. Exercise ( 1 ) on page 116). DEFINITION5.1.20. A 1-parameter subgroup of a Lie group G is a smooth , t l , t z E R. map s : R + G such that s(0) = e and s(tl t 2 )= s ( t l ) s ( t z ) V

+

PROPOSITION 5.1.21. If G is a Lie group and X E L(G), there is a unique 1-parameter subgroup s x : R + G such that sx (0) = X e . Furthermore, X is a complete vector field, the flow that it generates being given b y

PROOF.We first prove that X is complete. Indeed, if is a local flow about e E G generated by XIW and if a E G, then the formula

defines a local flow

aa : (-€,

E) X

La(U) + La(W)

5 . LIE GROUPS

132

about a having infinitesimal generator La, (XI W) = XI La(W). These fit tc+ gether to give @ : (we, E) x G - 3 G and the field X is complete by Lemma 4.1.8. Let ax designate the global flow generated by X. Remark that we have also established the identity

If a is any 1-parameter subgroup with initial velocity &(O) = X,, then the identity u(t 7) = u ( t ) a ( r ) , for fixed but arbitrary t , T f R, implies that

+

In particular, take

T

= 0 and conclude that

That is, a is the unique integral curve to X through e . By the previous paragraph, we must define

( e ) and prove that this is a 1-parameter subgroup of G. Indeed, by sx (t) = sx(0) = e and

where the second-twlast equality is by the identity (5.1). Evidently, s x (0) = X,. Finally, another application of the identity (5.1) yields

:P (a) = La o for arbitrary values of t

E

o La-I (a) = a s x (t),

R and a E G.

EXAMPLE 5.1.22. Let A E m(n)= L(Gl(n)). The series

converges absolutely. Set s(t) = exp(tA). Clearly, s(0) = I and, by bat2) = s(tl)s(ta). In particular, sic properties of the exponential series, s(t exp(tA) exp(-tA) = I, so the matrix exp(tA) is invertible, V t E R. By these remarks, s(t) is a 1-parameter subgroup of Gl(n). Finally, s(0) = A and, by Proposition 5.1.21, exp(tA) is the unique 1-parameter subgroup of Gl(n) with

+

5.1. BASICS

133

initial velocity vector A. This example provides the motivation for the following terminology. DEFINITION 5.1.23. If G is a Lie group, the exponential map exp : L(G)

-+G

is defined by exp(X) = s x (1), VX E L(G). In turn, we obtain a perfect generalization of Example 5.1.22.

LEMMA5.1.24. If G is a Lie group and X E L(G), then s x ( t ) = exp(tX), -oo < t < oo. PROOF.Fix T E B and set u(t) = s x ( r t ) . Then b(t) = TX,(~)and u(0) = e. By Proposition 5.1.21, a = s,x. In particular,

for each T E R. Hereafter, the standard notation for the 1-parameter subgroup associated to X E L(G) will be exp(tX). Via the identification L(G) = Te(G), we can view the exponential map as exp : Te(G) + G, a map between smooth manifolds of the same dimension. The following important property of this map is obtained via the inverse function theorem. The reader is invited to supply the details in Exercise (4). PROPOSIT~ON5.1.25.

The map exp : Te(G) + G is smooth, carrying some neighborhood of 0 in T,(G) di,Seomorphically onto a neighborhood of e in G. 5.1.26. If G and H are Lie groups, a Lie group homomorphism DEFINITION cp : G + H is a smooth map which is also a group homomorphism. If, in addition, cp is a diffeomorphism, it is called a Lie group isomorphism and G and

H are said to be isomorphic Lie groups. Remark that a l-parameter subgroup s : B phism, where R is a Lie group under addition.

+

G is a Lie group homomor-

PROPOSITION 5.1.27. If cp : G + H is a Lie group homomorphism, then there is a unique linear map cp, : L(G) + L(H) such that X is cp-related to cp,(X), VX E L(G). Thus, cp, is a Lie algebra homomorphism. PROOF.The R-linear map cp, will be the composition

134

5 . LIE GROUPS

If X E L(G), Y = cp,(X), and a E G, then

Thus, X is p-related to Y. The left invariant field Y is uniquely determined by Ye which, itself, is uniquely determined by the requirement that X be p-related to Y. REMARK.Using the canonical identifications L(G) = Te(G) and L(H) = Te(H), one obtains a Lie algebra structure on Te(G) and Te(H). Then the Lie algebra homomorphism cp, becomes cp,, : Te(G) + Te(H). DEFINITION5.1.28. Let G be a topological group, I' C G a subgroup. If there is a neighborhood U 2 G of the identity e E G such that U n I' = { e ) , then is called a discrete subgroup of G. By Theorem 4.3.9, a lattice in Rn is exactly a discrete subgroup of the additive Lie group Rn. The elementary proof of the following will be Exercise (5). LEMMA5.1.29. If G is a Lie group and r C G is a discrete subgroup, then the space G / r has a canonical smooth manifold structure relative to which the quotient projection p : G -, G / r is a local difleornorphism. If I? i s also a normal subgroup, then G / r is a Lie group and p : G + G/I' is a Lie group homomorphism.

PROPOSITION 5.1.30. Let G and H be Lie groups with H connected. Let cp : G -t H be a Lie group homomorphism such that cp,, : Te(G) + Te(H) is an isomorphism of vector spaces. Then there is a discrete normal subgroup I? C G such that H is isomorphic as a Lie group to G / r .

PROOF.By the inverse function theorem, cp carries some open neighborhood U C G of e E G diffeomorphically onto an open neighborhood cp(U) C H of e E H . Since cp respects left translations in these groups, it follows that cp(G) is an open neighborhood of the identity in H. Since p(G) is a subgroup of H, Proposition 5.1.12 implies that cp(G) = H. Finally, since cp is one to one on U G, the normal subgroup I' = ker(cp) is a discrete subgroup of G and cp induces a Lie group isomorphism @ : G / r + H.

REMARK. In most of the literature, Lie groups are defined to be real analytic. That is, G is a manifold with a CW (real analytic) atlas and the group operations are real analytic. In fact, no generality is lost by this more restrictive definition. Smooth Lie groups always support an analytic group structure, and something even stronger is true. Hilbert's fifth problem was to show that if G is only assumed to be a topological manifold with continuous group operations, then it is, in fact, a real analytic Lie group. This was finally proven by the combined

5.1. BASICS

135

work of A. Gleason, D. Montgomery, and L. Zippin. The details are too deep to be discussed here.

EXERCISES (1) Prove that the component of the identity in a Lie group is itself a Lie

group (2) Check the various assertions in Example 5.1.7, showing that U(n) and SU(n) are compact Lie groups of respective dimensions n 2 and n2 - 1. (3) Identify L(U(n)) and L(SU(n)) as Lie algebras of complex matrices. (4) Prove Proposition 5.1.25. (5) Prove Lemma 5.1.29. ( 6 ) If G is a Lie group and X, Y E Te(G) , show that the curve a(t) = exp(tX) exp(tY) has initial velocity vector &(O) = X + Y. (7) A Lie algebra a is said to be abelian if [A,B] = 0, V A, B E a. Prove the following. (a) A connected Lie group G is abelian if and only if its Lie algebra L(G) is abelian. (b) If the Lie group G is connected and abelian, the map exp : Te(G) -+ G is a surjective Lie group homomorphism, where the vector space Te(G) is viewed as a Lie group under vector addition. (c) Every connected, n-dimensional, abelim Lie group is Lie isomorphic to T~ x Rn-IE for some k = 0,1,. . . ,n. (8) For each z E S3 C W = IR4, define A, : It4 -+ It4 by A,(w) = zwz-l (quaternion operations). Prove the following, using st mdard facts about the skew field W and the norm lzJ= 6on W. (a) A, is a nonsingular, norm-preserving linear transformation. That is, as a matrix, A, E O(4). In fact, show that A, E SO(3) under canonical inclusions SO(3) c O(3) c O(4) of Lie subgroups. (b) The map A : S3 S0(3), defined by A(z) = A,, is a homomorphism of Lie groups. (c) The kernel of the homomorphism A is the normal subgroup

-

hence (for dimension reasons) the associated homomorphism

is an isomorphism of Lie algebras. Conclude that SO(3) and S3/Z2 are canonically isomorphic Lie groups. (d) Let sl and s 2 be one parameter subgroups of S3 such that the initial velocity vector Si(0) E TI (S3) C R4 has Euclidean norm 1, i = 1,2. Using the previous step, show that there is an element z E S3 such that zsl(t)z-' = s2(t), Vt E R. (e) Using the above, prove that, up to parametrization, the l-pararneter subgroups of S3 are exactly the great circles through 1 E S3.

5. LIE GROUPS

5.2. Lie Subgroups and Subalgebras

We fix a choice of the Lie group G and discuss its Lie subgroups.

-

DEFINITION5.2.1. A subset H C G is a Lie subgroup if H has a Lie group structure relative to which the inclusion map i : H G is a one to one immersion and a group homomorphism. In particular, the inclusion i of a Lie subgroup is a Lie group homomorphism. We emphasize that the topology of H as a Lie group may not coincide with its relative topology in G.

EXAMPLE 5.2.2. A nontrivial 1-parameter subgroup s : R -+ G is an immersion, generally not one to one. We will see that, if n 2 2, uncountably many of the 1-parameter subgroups s : R + Tn are one to one immersions, each having image dense in Tn (Example 5.3.8). By our definition, s(R) with its manifold topology and additive group structure is a Lie subgroup with i = s, but the relative topology of this subgroup in Tn is wildly different from its manifold topology.

-

LEMMA5.2.3. Let i : H G be a Lie subgroup. Then i, : L(H) imbeds L(H) as a Lie subalgebra of L(G). Indeed, i,, : T,(H) sition 5.1.27.

-t

4

L(G)

Te(G) is one to one, so the lemma follows from Prop*

THEOREM 5.2.4. The correspondence between Lie subgroups i : H c, G and their Lie subalgebras i, : L(H) -+ L(G) induces a one to one correspondence between the set of connected Lie subgroups ofG and the set of Lie subalgebras of L(G). The proof of Theorem 5.2.4 is the main goal of this section. In light of the preceding lemma, what we have to prove is that, given a Lie subalgebra Ij C_ L(G), there is a unique connected Lie subgroup i : H G such that Ij = i,(L(H)). The principal tool for this will be the Fkobenius theorem. The evaluation map e : L(G) + Te(G) carries Ij one to one onto a vector subspace E, Te(G). For each a E G, define

-

Then E = UaEG Ea is a k-plane distribution on G, where k = dim Ij. Indeed,

LEMMA5.2.5. The subset E C T(G) is an integrable k-plane distribution on G.

PROOF.Let XI,. . . , X I , be a basis of the vector space b. This is a set of everywhere linearly independent, left invariant fields on G, proving that E is a k-plane distribution. Remark that Ij r(E)spans T ( E ) as a Coo(G)-module and [Xj, Xe] E Ij, so E is an integrable distribution by Theorem 4.4.4.

5. LIE GROUPS

138

PROOF.The multiplication map p~

Since ~ H ( xH H)

:H

xH

+ G is given by the composition

i(H), the map

is smooth by Exercise (6) on page 124. This is the group multiplication in H . Similarly, the group inversion L : H + H is smooth. Finally, i *,(Te(H)) = E,, so i,(L(H)) = lj. The following lemma completes the proof of Theorem 5.2.4. LEMMA5.2.10. T h e Lie subgroup i : H that i, (L(H)) = b.

L,

G i s the only one with the property

PROOF. The Lie subgroup H is a leaf of the foliation Fl determined by the Lie subalgebra b. Suppose that it : H' L, G is also a connected Lie subgroup such that i:(L(H1)) = lj. Then H' must be a connected integral manifold through e to the distribution E determined by 5. The maximal such integral manifold is H , and therefore H' C H . Indeed, H' is an open Lie subgroup of the connected Lie group H , so H' = H by Proposition 5.1.12.

EXERCISES (1) Let i : H

L)

G be a Lie subgroup of G. Let e x p :~Te ( H ) + H exp~ : Te(G) + G

be the respective exponential maps. Prove that the diagram

H

-

G i

is commutative. (2) Using the above exercise and the fact that exp : m ( n , IF) -+ Gl(n, IF) is ordinary matrix exponentiation, IF = R or C, give a new proof that L(O(n)) is the algebra of skew symmetric matrices over R and determine the Lie subalgebra L(U(n)) c Dl(n, C). (3) Let cp : H + G be a smooth homomorphism between Lie groups. Prove that p ( H ) is a Lie subgroup of G.

5.3. CLOSED SUBGROUPS

-

5.3. Closed Subgroups While a Lie subgroup i : H G is generally only an immersed submanifold, we have seen a number of examples, such as Sl(n) c Gl(n), O(n) c Gl(n), and U(n) c Gl(n, C ) , in which H is a properly imbedded submanifold. In this case, the Lie subgroup has the relative topology from G, making it easier to work with. Generally, properly imbedded submanifolds are not closed subsets, but this is true for Lie subgroups.

PROPOSITION 5.3.1. If the Lie subgroup H manifold, then H is closed as a subset of G.

C

G is a properly imbedded sub-

PROOF.We use the foliation 'H from the previous section. The components of H are leaves of 'H. Find a neighborhood U of e in which the foliation 'HIU becomes a foliation by plaques Pa. Since H is properly imbedded, this neighborhood can be chosen so that H n U = Po,a single plaque. We can assume that U c G, a compact subset, where HI6 is also a foliation by plaques Fa. Let {hn)F=l c H be a sequence that converges in G to an element g. We must prove that g E H. Let W = Lg(U), a neighborhood of g in which 'HIW is . can show also a foliation by plaques Lg(Pa). Similarly, let ti; = ~ ~ ( 6 If) we that all but finitely many h, lie in a common plaque of 'Hi@, then this plaque must be the one containing g and it must lie in the integral manifold H, so g E H. Since limn,, h;' h,+l = e, there is an integer r 2 1 such that h i 1hn+l E U, Vn 2 r. That is, hzlhn+l E U n H = Po. Also, since e E Po,it follows that both h, E Lhn(Po)and h,+l E Lh,(PO). For r sufficiently large, Lh,(Po) c where is a plaque - of 'HI@. That is, when n 2 r , h, and h,+l lie in a-common plaque P of W. Similarly, h,+l and hn+z lie in a common plaque P'. Since h,+l E n P', it follows that P = F . Proceeding in this way, we see that contains h,, V m 2 r .

F,

-

P

P

This proposition has a surprisingly strong converse.

THEOREM 5.3.2 (CLOSEDSUBGROUP THEOREM). If G is a Lie group and H G is a closed subset that is also an abstract subgroup, then H is a properly imbedded Lie subgroup. Our proof will be modeled, in certain important ways, on the proof given in [15, pp. 105-1061. The problem with that proof is that it is based on a lemma 115, Lemma 1.8, p. 961 which assumes that Lie groups are real analytic groups. The fact that C" groups are, in fact, real analytic, will not be used in our proof. For a somewhat different presentation, also carried out in the smooth category, the reader can consult [44,pp. 110-1 121. Fix the assumptions in Theorem 5.3.2. Define

This contains 0 E L(G) and is closed under scalar multiplication. It is not evident that b is a vector subspace, let alone a Lie subalgebra. It is also unclear

140

5 . LIE GROUPS

that b # 0 if H is not discrete. In fact, b will turn out to be the Lie algebra of an open Lie subgroup of H. Let V C L(G) be the vector space spanned by b. In the following proof, it will be convenient to use the notation a x = La, ( X ) and X a = R,, (X) (where Ra denotes right translation by a ) , a E G, X E X(G). Thinking of vectors as infinitesimal curves makes this notation particularly natural. LEMMA5.3.3. The vector space V is a Lie subalgebra of L(G). PROOF.Since the Lie bracket is bilinear and V is spanned by b, it will be enough to prove that [X,Y] E V, VX, Y E b. By Theorem 4.2.2 and Proposition 5.1.21, Y exp(-tX) - Y [X,Y] = lim t+O t Since Y is a left invariant field,

and, for a fixed t, this is a left invariant field whose corresponding 1-parameter group is O(T)= exp(tX) exp(rY) exp(-tX). Since X, Y E b, a ( r ) is a product of elements of the subgroup H, hence a ( r ) E H, V. Thus, -m < T < m. It follows that, for each value of t, Y exp(-tX) E b [X,Y] E V, as desired. Let Ho c G be the connected Lie subgroup with L(Ho) = V. LEMMA5.3.4. There is an open neighborhood U of e in Ho (in the manifold topology of H o ) which is contained in H . PROOF.Let {Yl, . . . , Y,)

c b be a basis of V.

The map cp : V

-+

Ho, defined

by

satisfies cp,o(Y,)= Y,, 1 _< i 5 q, so the inverse function theorem implies that p carries some neighborhood Uo 5 V of 0 diffeomorphically onto a neighborhood U Ho of e . But exp(tiY,) E H, 1 i 5 q, and H is a subgroup, so U C H.

<

COROLLARY 5.3.5. The Lie group Ho is a subgroup of H. PROOF.By Proposition 5.1.12, Ho is generated by U 2 H, so Ho Remark that, at this point, we know that b = V, hence of Ho. LEMMA5.3.6. The subgroup Ho the relative topology of H .

H.

is the Lie algebra

H , with its manifold topology, is open in

5.3. CLOSED SUBGROUPS

141

PROOF.(Compare [15, p. 1061.) It will be enough to prove that some open neighborhood U of e, in the manifold topology of H o , is a neighborhood of e in the relative topology of H. The problem is that, for each such U, there might be a sequence { x k ) ~ = ,c H \ U such that xk + e in the topology of G. Assuming that this is so, we deduce a contradiction. Find a direct sum decomposition L(G) = Ij @ W and remark that, by the inverse function theorem, the map cp : b $ W -t G, defined by

carries some neighborhood N of 0 in L(G) diffeomorphically onto a neighborhood of e in G. Choose U = exp(tj n N). Thus, for k sufficiently large, we can write

where v k E b n N and wk E W n N. Since xk @ U, it is clear that wk # 0, for all large values of k. Select a bounded neighborhood W o c W of 0 and positive integers nk such that, for k sufficiently large, nkwk E Wo, but (nk 1)wk # Wo. Since Wo is bounded, we can assume that 72.kt.uk + w E W. Since wk -+ 0 and (nk + l)wk @ Wo, we must have w # 0. For arbitrary t E R, we will show that exp(tw) E H. That is, w E b, hence 0 # w E b n W, the desired contradiction. Write tnk = s k tk, where sk E Z and Itkl < 1. Thus, tkwk + 0 and

+

+

= = =

lim exp(skwk)

k+oo

lim e x p ( ~ ~ ) ' ~

k-oo

lim ( ~ X ~ ( - U ~ ) X ~ ) ' " .

k+oo

Since H is closed in G, it follows that exp(tw) E H .

PROOF O F THEOREM 5.3.2. Let : Ho --, H be the inclusion map. We have proven that i carries Ho, with its manifold topology, homeomorphically onto an open subset of H in the relative topology. In particular, the manifold topology of Ho coincides with its relative topology, so Ho is a properly imbedded Lie subgroup of G. By Proposition 5.3.1, Ho is closed in G, so Ho = i(Ho) is a connected, open-losed subset of H. Thus, Ho coincides with the component of the identity in H. The other components La(Ho), a E H, of H are also properly imbedded submanifolds of G, so H is a Lie subgroup. Since H has the relative topology, each of its components is relatively open in H, so H is a properly imbedded Lie subgroup. COROLLARY 5.3.7. Let I' 2 G be an abstract subgroup. Then the closure G is a properly imbedded, Lie subgroup of G.

in

142

5 . LIE GROUPS

EXAMPLE 5.3.8. Let v = (a1,. . . , a n ) E Rn be a point such that, when R is viewed as a vector space over the rational number field Q, the subset {a1,. . . , a n ) C R is linearly independent. Let p : Rn + Tn be the standard projection and let l c Rn be the line through v and 0. A classical theorem of Kronecker asserts that this line projects one to one to a 1-parameter subgroup p(l) c Tn that is everywhere dense in Tn (for the case n = 2, cf. Example 4.1.7 and Exercise (2) on page 116). It is now fairly easy to prove Kronecker's theorem. Indeed, one proves (Exercise (2)) that, if vl, . . . , vk E Zn and

for suitable coefficients xi E R, then k = n. By Corollary 5.3.7, the closure & Tn is a compact, connected, abelian Lie subgroup, hence a toral subgroup of dimension r 5 n (Exercise (7) on page 135). It follows that v E l c V where V c Rn is a subspace and 7 = p(V). In particular, V is spanned by V n Z n , so Exercise (2) implies that dim V = n and 2 = Tn.

e

Following Helgason [15, pp. 107-1081, we deduce the following classical result as another corollary of Theorem 5.3.2. For a proof that does not depend on that theorem, see [44, p. 1091.

THEOREM 5.3.9. Ifcp : G + H is a continuous group homomorphism between Lie groups, then cp is smooth. PROOF.The product G x H is a Lie group and the projections

are smooth group homomorphisms. Let I' C G x H be the graph of cp. That is, I' = {(x,cp(x)) I x f G), clearly a closed subgroup of G x H. Thus, I? is a properly imbedded Lie subgroup. Also, rG(r = -$ : I? -, G is a smooth group homomorphism and is bijective. If it can be shown that $-I : G -t I? is smooth, will be smooth and the assertion will be proven. then cp = 7 r o~ By the inverse function theorem, it will be enough to show that $ *, is bijective, is a smooth homomorphism, it is enough V y E I.-' Since is a Lie group and to prove this at y = ( e ,e). Remark that the exponential maps for the groups G x H , G, and H are related by

+

This, together with the proof of Theorem 5.3.2, implies that L ( r ) = {(X, Y) E L(G) x L(H) I (expGtX,expH tY) E I?, V t E It) Since +.(,,,)(X, Y) = X, we must show that, for each X E L(G), there is a unique Y E L(H) such that (X, Y) E L ( r ) .

5.3. CLOSED SUBGROUPS

143

We first show uniqueness of Y. If (X, Y) E L ( r ) and (X, 2) E L ( r ) , then the difference is (0, Y - 2 ) E L ( r ) , implying that (e, exp, t(Y - 2 ) )E r, Vt E R. Thus, exp,t(Y - 2 ) = cp(e) = e, Vt E R, and Y - 2 = 0. Choose open neighborhoods Uo L(G) and Vo L(H) of the origin and Ue G and Ve H of the identity such that (1) expG : Uo + Ue is a diffeomorphism onto; (2) expH : Vo --+ Ve is a diffeomorphism onto; (3) ~ ( u eC) Ve; (4) exPGxH carries (Uo x Vo) n L ( r ) diffeomorphically onto (U, x Ve) n r .

c

c

c

c

Let X E L(G) and choose an integer r > 0 such that ( l / r ) X E Uo. Thus, cp(expG(l/r)X) E Ve and there is a unique Y,. E Vo such that expH Y, = cp(expG(l/r)X). There is also a unique 2, E (Uo x Vo) n L ( r ) such that Since expG, is one to one on Uo x Vo, this implies that

Take Y =rYT, obtaining ( X , Y ) =rZ, E L(r). COROLLAFW 5.3.10. Let cp : G + H be a continuous homomorphism of Lie groups and let K = ker(cp). Then K is a properly imbedded, normal Lie subgroup of G7 G / K is canonically a Lie group, and the induced map : G / K + H is a one to one immersion of this Lie group as a Lie subgroup of H.

PROOF.Indeed, K is a normal subgroup by standard group theory and K = cp-'(e) is a closed subset of G, so Theorem 5.3.2 guarantees that K is a properly imbedded Lie subgroup of G. By Theorem 5.3.9, cp is a smooth h e momorphism, so Exercise (3) on page 138 guarantees that q ( G ) is a Lie subgroup of H. Obviously, p is an isomorphism of the group G/K onto q(G) , so p can be used to transfer the Lie structure of cp(G) back to G/K.

EXERCISES (1) Let G be a topological group whose underlying space is a topological manifold. Prove that there is at most one differentiable structure on the topological manifold G making G into a Lie group. (The positive solution to Hilbert's fifth problem guarantees that a topological g r o u p manifold does have a smooth (in fact, analytic) structure making it into a Lie group.) (2) Prove the assertion in Example 5.3.8 that the vector v E R n , with rationally independent coefficients, cannot be expressed as a real linear combination of fewer than n elements of the integer lattice Zn. (3) Let v = ( a 1 ,. . . ,a n ) E R" be a point such that 11, a', . . . ,a n ) is linearly independent over Q. Prove that the subgroup A c T n , generated by a = p(v), is everywhere dense. (Hint: Every point in the coset qv Z n has rationally independent coefficients, V q E Z. Prove this and use it

+

5 . LIE GROUPS

144

to show that every 1-parameter subgroup of Tn meeting a nontrivial element of A is dense in Tn.) (4) Show that every finite dimensional Lie algebra contains a nontrivial abelian subalgebra which is not itself contained properly in another such subalgebra. Similarly, show that every compact Lie group G contains a maximal subgroup that is Lie isomorphic to T ~some , k > 1. This is called a maximal t o m of G. (a) When G is compact, prove that the correspondence between Lie subalgebras and connected Lie subgroups sets up a one to one correspondence between the maximal abelian subalgebras of L(G) and the maximal tori in G. (b) Let G be compact and let K E G be a maximal abelian subgroup. Although K need not be connected, you are to use Exercise (3) to prove that there is an element x E K such that the group { x ~ ) , ~ ~ is everywhere dense in K. (c) Let G be compact and connected, K C_ G a maximal abelian subgroup. You are to prove that K is connected, hence that the maximal abelian subgroups of G are exactly the maximal tori. You may use the nontrivial fact that each element of G lies on a t least one 1-parameter subgroup (Corollary 10.7.12 and T h e e rem 10.7.15). (5) Let G be an n-dimensional Lie group and g = L(G) its Lie algebra. Let Gl(g) denote the group of nonsingular linear transformations of the vector space g and let Aut(g) c Gl(g) be the subgroup of Lie algebra automorphisms of 8, Prove the following. (a) Gl(g) has a canonical Lie group structure under which it is (non canonically) isomorphic to Gl(n). Also, for use in Exercise (6), show that L(G1(g)) is canonically the space End(g) of linear endomorphisms of the vector space g, the bracket in End(g) being the commutator product of endomorphisms. (b) Aut(g) is a closed subgroup of Gl(g), hence a properly imbedded Lie subgroup. (c) Assume that G is connected and let C C G denote the center of G, clearly a closed subgroup. Each element a E G determines an inner automorphism of G, denoted by Ad(a) and defined by

Prove that { A ~ ( U ) ) is, ~canonically ~ a Lie subgroup of Aut (g), isomorphic as a group to G/C. This subgroup is denoted by Ad(G) and called the adjoint group of G. (6) Let G be a Lie group and again denote its Lie algebra by g. A derivation D : g + g is a linear transformation such that D [ x , y] = [DX,Y ]

VX, Y

+ [X,D Y ] ,

E g. Let D(g) be the space of derivations of g.

(a) Prove that, under the commutator product, D(g) is naturally identified as a Lie subalgebra of L(Gl(g)) (cf. Exercise (5), part (a)).

5.4. HOMOGENEOUS SPACES

(b) For each X E 0, define ad(X) : g + g by ad(X)Y = [X,Y], V Y E g and prove that ad(X) E D (0). (c) Prove that ad : g + L(Gl(g)) is a homomorphism of Lie algebras. Thus, ad(g) L(Gl(g)) is a Lie subalgebra. (d) Assume that G is connected and prove that the connected Lie subgroup of Gl(g) corresponding to the Lie subalgebra ad(g) is exactly the adjoint group Ad(G). (Hint: Prove that Ad(exp(tX)) = exp(t ad(X)), VX E 0.) 5.4. Homogeneous Spaces Lie groups arise in many natural ways as transformation groups of manifolds. When the group action is transitive, the manifold is called a homogeneous space and one has considerable control over its structure. DEFINITION 5.4.1. Let M be a smooth manifold and G a Lie group. A smooth map p:GxM+M, written p(g, x) = gx, is said to be an action of G (from the left) on M, and G is called a Lie transformation group on M , if (1) g i ( g 2 ~= ) ( 9 1 9 2 ) ~Vgl, ~ 92 E G and Vx E M; (2) ex = x, Vx E M.

REMARK.One can also define a right action

by making the obvious changes in the above definition. DEFINITION 5.4.2. An orbit of the action

is a set of points of the form {gxo I g E G), where xo E M . The action is transitive if M itself is an orbit, in which case M is said to be a homogeneous space of G.

REMARK. It is elementary that the orbits of a group action are equivalence classes, two points x , y E M being equivalent under the action if 3 9 E G such that gx = y. EXAMPLE5.4.3. The orthogonal group O(n) acts on R n in the usual way, leaving invariant the unit sphere Sn-l. Note that, if e l E Sn is the column vector with first entry 1 and remaining entries 0,then Ae 1 is the first column of A E O(n). Every unit vector appears as the first column of suitable orthogonal matrices, so the action 0 ( n ) x sn-'+ sn-'

5 . LIE GROUPS

146

is transitive and Sn-' is a homogeneous space of O ( n ) . In a completely similar way, there is a transitive action

where

s ~c Cn ~ is the - unit ~ sphere in the standard Hermitian metric.

DEFINITION 5.4.4. Let M be a homogeneous space of G and let xo E M. The isotropy group of xo is the set G,, = {g E G I gxo = xo). LEMMA5.4.5. The isotropy group G,, subgroup of G.

as above is a properly imbedded Lie

PROOF.It is obvious that Gxois an abstract subgroup of G. If {gn)F==,c G,, converges to g E G, then, by the continuity of the group action, gxo = lim gnxo = lim xo = xo. n --roo

ndoo

Thus, G,, is a closed subset of G. By Theorem 5.3.2, G,, is a properly imbedded Lie subgroup. EXAMPLE5.4.6. If el E Sn-l is as in Example 5.4.3, the isotropy group

where A E O(n - 1). Similarly, for e 1 E S2n-1,U(n),l is the set

where A E U(n - 1). We are going to show how to put a smooth structure on the quotient space G/G,, and prove that this manifold is diffeomorphic to the homogeneous space M. Under the identification M = G/G,, , the G-action on M becomes the act ion G x G/Gx, GIG,,, g(hG,,) = (gh)G,,. In what follows, we consider an arbitrary properly imbedded Lie subgroup H C_ G, put the quotient topology on G/H, and construct a natural smooth structure on this space. Throughout this discussion, we set tj = L ( H ) . Decompose L(G) = m @ b, where m is any fixed choice of complementary subspace. Let $:m@b--+G be the map $(A, B)= exp(A) exp(B) and choose a neighborhood V of 0 in b and a neighborhood W of 0 in m such that .11, sends W x V diffeomorphically onto a neighborhood U of e in G. Choose a compact neighborhood C c W of 0 with the property that -C = C and exp(C) exp(C) c U. We can assume that coordinates x l , . . . , xk in tj define V by the inequalities - 1 < xi < 1, 1 5 i 5 k. Similarly, coordinates y l , . . . , y4 for m define C by -+

148

5. LIE GROUPS

LEMMA5.4.10. With the above Cooatlas, G / H is a smooth manifold, the projection n : G + G / H is a submersion, and the action

defined b y p(a, bH) = abH is smooth.

COROLLARY 5.4.11. If G is a Lie group and H G is a closed, normal subgroup, then the group G / H has a smooth structure in which it is a Lie group. We return to the smooth transitive action

Let xo E M and let H = G,, be the isotropy group. Define the map by B(aH) = ax0 . This is induced by the smooth map 8 : G -+ M, #(a) = ax0 , so 19 is continuous. Since aH = bH if and only if a-'b E H , we see that xo = a-'bxo, hence ax0 = bxo, if and only if aH = bH. That is, 9 is well defined, one to one, and continuous. Since the action of G is transitive, 13 is a surjection. The following diagram is commutative:

G x M

-

M

P

Thus, if we prove that 0 is a diffeomorphism, 9 will be a canonical identification of G / H with M as a homogeneous space of G.

PROPOSITION 5.4.12. The map I3 : G / H

+M

is a difleomorphism.

PROOF.Let La denote left translation by a E G on both G / H and M. That is,

La(bH) = abH, L ~ ( x=) a x ,

VbH E G / H , V X EM .

Then

8=Lao80La-I, VaEG. Since La : M + M and L a - I : G / H -+ G / H are diffeomorphisrns, it follows that I3 will be smooth at aH if and only if it is smooth at eH. Furthermore, if smoothness has been established, then

will be an isomorphism of T a H( G / H ) onto T a z O ( Mif) and only if

is an isomorphism. We show smoothness at eH. Consider the commutative diagram

5.4. HOMOGENEOUS SPACES

The map

81 exp(Co) : exp(C0) + M is smooth, and the map

is a diffeomorphism onto the coordinate neighborhood U,,so 13 is smooth in a neighborhood of e H . We show that 13*eH : TeH(G/H) -4 Tz,(M) is an isomorphism. Again, this translates, via the commutative triangle, to showing that 8,, is an isomorphism of Te(exp(Co)) onto T,, (M). Let v E Te(exp(Co)) = m and consider the curve s(t) = exp(tv)xo. Since O,,(v) = s(0), we only need prove that i(0) = 0 implies that v = 0. If a = exp(tov), then L,(s(t)) = s(t to), so L,,(s(O)) = i(t0) and i(0) = 0 implies that ~ ( t = ) 0, Vt E R. That is, exp(tv)xo = xo, Vt E R, implying that v E fi m = (0).

+

EXAMPLE 5.4.13. Thus, as a homogeneous space of O(n), where O(n - 1) is properly imbedded as a Lie subgroup of O(n) as in Exam-

ple 5.4.6. Similarly,

~ 2 n - 1- U(n)/

U(n - 1).

EXERCISES (1) Prove Lemma 5.4.8. (2) Prove Lemma 5.4.10. (3) Let Gn,k denote the set of k-dimensional vector subspaces of R n . (a) Show how to make G n l kinto a compact manifold which is a homogeneous space of O(n). This is called the (real) Grassmann manifold of k-planes in n-space. (b) If xo E Gn,k is the standard Rk Rn, identify the isotropy group O(nL0. (c) Show that projective space P"-' is the Grassmann manifold Gntl and identify the standard two-to-one map Sn-I -+ Pn-' as a map O(n)/ O(n - 1) + O(n)/ O(n),, (Remark: Using Cn instead of Rn,one defines in a similar way the complex Grassmann manifolds Gnjk(@)as homogeneous spaces of U(n). Complex projective space is defined to be P n ( @ ) = GnI1(@).The real and complex Grassmann manifolds play an important role in differential geometry and topology.)

150

5 . LIE GROUPS

(4) An orthonormal k-frame in R n is a k-tuple (vl , . . . ,uk) of ort honormal vectors in Rn. Let Vn,k denote the set of d l orthonormal k-frames in Rn, identify this as a homogeneous space of O(n) (called the Stiefel manifold of k-frames in n s p a c e ) . Remark that VnV1= Sn-' and that this case gives back Example 5.4.3. Using C n and the standard positive definite Hermitian inner product on C n , one obtains the complex Stiefel manifolds Vnta(C) and Vn,1(C) = s2"-' . 5.5. Principal Bundles

A good command of the theory of principal bundles is essential for mastery of modern differential geometry (cf. [22], [23]). In recent years, principal bundles have also become central to key advances in mathematical physics (Yang-Mills theory) which have in turn generated exciting new mat hematics ( e.g., Donaldson's work on differentiable 4-manifolds 161, [lo]). In this book, however, principal bundles will only be needed in Section 10.6, where the principal frame bundle will be used to globalize the Cartan structure equations. Let V be a real vector space of dimension n. An n-frame in V is an ordered basis D = (vl,. . . ,v,). If V = Rn, each vi is a column vector and the frame is a nonsingular matrix. That is, the set of all n-frames in R n is naturally identified with the manifold Gl(n). Generally, we denote the set of all n-frames in V by F ( V ) and topologize this set as a subset of Vn = V x . . . x V. Let cp : V -+ R n be an isomorphism of vector spaces and define a diffeomorphism p : Vn -+m ( n ) by ~ ( v l* ., ,vn) = ( ~ ( v l ). ., - , ~ ( v n ) ) . Clearly p ( F ( V ) ) = Gl(n), hence F(V) c Vn is an open subset, diffeomorphic via p t o Gl(n). DEFINITION5.5.1. The smooth manifold F ( V ) is called the frame manifold of V . One might try t o make F(V) into a Lie group via p, but this is a bad idea since, if cpl and cp2 are two isomorphisms of V to R n , it is not generally true that the diffeomorphism pl o (p2)-l : GI(n) -+ Gl(n) is a group isomorphism. That is, there is no canonical way t o make F ( V ) into a Lie group. There is, however, a natural right action

defined by formal matrix multiplication

This is a smooth, transitive action with trivial isotropy group (such an action is said to be simply transitive). Remark that the right action of Gl(n) on itself, defined by the group multiplication, is also smooth and simply transitive and that p : F(V) -+ Gl(n) respects these actions in the following sense.

5.5. PRINCIPAL BUNDLES

LEMMA5.5.2. If cp : V A E Gl(n), then

-+ Rn

151

is a linear isomorphism and if o E F(V),

p(a A) = p(a)A. One says that p is Gl(n)-equivariant. There is a natural map F(V) x R n -+ V defined by formal matrix multiplicat ion

For each fixed a E F ( V ) , this defines a linear isomorphism every linear isomorphism is of this form. We summarize.

P :

Rn

-+

V, and

LEMMA5.5.3. The map defined by (5.2) sets up a canonical identification of F(V) with the set of a11 linear isomorphisms a : R n -+ V. One can canonically reconstruct the vector space V from the frame manifold F(V) as follows. Define the left action of Gl(n) on the space F ( V ) x R n to be the "diagonal action" (5.3)

A . (P,8 ) = (a . A-', AZ),

where A ranges over Gl(n) and (P, a') ranges over F ( V ) x Rn. It is elementary that this is a smooth, left action of the group Gl(n), so there is an associated equivalence relation on F(V) x Rn with the Gl(n)-rbits as equivalence classes. We denote the quotient set by F(V) x ~ l ( Rn. ~ ) The equivalence class of ( P , a') will be denoted by [o, a']. Remark that

We put a vector space structure on the quotient set F(V) x ~ a frame o E F(V) and defining

[o,Zl]

[P, a'z] = [ O , a ' l C[D,

l ( Rn ~ )by

fixing

+Z2]

a'] = [O, &I.

To see that these definitions do not depend on the choice of frame, let u be another choice, let A E Gl(n) be the unique matrix such that u = a A, and use the relation (5.4) to obtain

-. + [u,

[u,all

821 = [O,

= [ol

AZl] + [o,AZ2 ]

+

+ li2)]

= [~,a'l 521 C[U,

a] = c[o,AZ] = [0, A&]

= [.,&].

The map defined by (5.2) passes to a well defined linear map

5 . LIE GROUPS

152

Fixing a frame u E F ( V ) , we define

by

j ( ~Z ) = [o,Z]. It is clear that .1C, and j are mutual inverses, hence that j does not really depend on the choice of frame b. We summarize these remarks in the following.

L E M M A5.5.4. The vector space operations on F ( V ) x ~ l ( Rn ~ ) are well defined, independently of the choice of the frame o, and the map 1C, is a canonical isomorphism of vector spaces, the inverse j being well defined independently of the choice of frame. These remarks generalize, fiber by fiber, to vector bundles. Let 7r : E -+ M be an n-plane bundle over an m-manifold. Each fiber E x = 7rV'(x) is an ndimensional real vector space, so we form the associated frame manifold F ( E x ) and the disjoint union

We also define p : F ( E ) + M by p(F(E,)) = x , V X E M . If

U

-

U

id

is a local trivialization of E, then cpx : E x inducing a diffeomorphism

-+

Rn is a linear isomorphism, V x

E U,

This determines a commutative diagram

where p is bijective, hence defines a topology and smooth structure on p-'(U). It is straightforward to check that, on overlaps p - l ( U ) f7pp1(V),corresponding to two local trivializations of E , the two topologies and smooth structures coincide and t hat this makes F ( E ) into a smooth manifold of dimension m n 2 . Furthermore, p : F ( E ) + M is a smooth submersion and the inclusion map, i x : F ( E x ) F ( E ) smoothly imbeds the fiber p-'(x) as a proper submanifold,

Vx

E

M.

-

+

5.5. PRINCIPAL BUNDLES

153

We have defined a new kind of bundle over M, called the frame bundle of E. The fibers F ( E x ) are not vector spaces. Instead, they are manifolds diffeomorphic to Gl(n) which admit a canonical right action of Gl(n). This defines a right action F ( E ) x Gl(n) -+ F(E) which is simply transitive on each fiber. By Lemma 5.5.2, the local trivializations turn this action into the action

defined by ( x , B ) . A = (x,BA). This is clearly a smooth action, so

is locally, hence globally, a smooth action. By Lemma 5.5.3, the fiber F(Ex)over x E M of the frame bundle of E is exactly the set of linear isomorphisms of the "standard fiber" R n to the specific fiber Ex. The construction for recovering a vector space V from its frame manifold F ( V ) globalizes in a natural fashion to give a canonical way of recovering the vector bundle E from its associated frame bundle F(E). The diagonal action of Gl(n) given by (5.3) extends by the same formula to a left action

and we let F(E) x ~ l ( , )Rn denote the corresponding quotient space (with the quotient topology). The bundle projection p : F(E) + M passes t o a well defined map : F(E) x ~ 1 ( , )

Rn

+M

and the map

F(E) x Rn

+ E,

again defined as in (5.2), passes to a well defined map

These maps are continuous and the diagram

commutes. It has also been arranged, by the very definitions, that $ restricts t o $, on the vector space jT1(x) = F(Ex)x ~ l ( , )Rn c F(E) X G ~ ( Rn, ~ ) carrying it isomorphically onto the vector space Ex, Vx E M. The inverse jx of $,,

5 . LIE GROUPS

154

although defined by a choice of frame tt E F(E,), is independent of that choice (Lemma 5.5.4), V x E M , and these fit together to give a global inverse

In a local trivialization of El one can choose n linearly independent smooth sections, using these t o define j. It follows that j is locally, hence globally, smooth and is a diffeomorphism. We summarize this discussion in a theorem.

+

-

THEOREM 5.5.5. The stmcture jj : F ( E ) x ~ l ( , )Rn + M is a vector bundle, canonically isomorphic t o x : E M . The association E F ( E ) is a one to one correspondence between the set of vector bundles over M and the set of associated frame bundles. The frame bundle F ( E ) is an example of a principal bundle. We give the general definition. -+

DEFINITION 5.5.6. Let M and P be smooth manifolds, G a Lie group, and p :P M a smooth map. Suppose that there is an open cover (Ua)aEaof M and, V a E 24,a commutative diagram -+

where $, is a diffeomorphism. Suppose also that there is a smooth right action P x G -, P, simply transitive on each fiber p - l ( x ) . Finally, assume that is G-equivariant with respect to this action and the standard right G-action on U, x G, Vcu E %. Then, p : P -+ M, together with this G-action, is called a principal G-bundle over M.

+,

EXAMPLE 5.5.7. Let p : P -, M be a principal G-bundle and let a : M --t P be a smooth map such that p o a = idM. As in the case of vector bundles, a is called a section of the principal bundle. While a vector bundle always admits sections (e.g., the 0 section) this fails for principal bundles. Indeed, if a is a section, define cp:MxG+P by 4 x 7 g ) = a ( x ) g7 obtaining a diffeomorphism such that the diagram M x G (P P

commutes. As in the case of vector bundles, cp is called a trivialization of the principal bundle P. Thus, the existence of a section is equivalent t o triviality of a principal G-bundle.

5 . 5 . PRINCIPAL BUNDLES

155

EXAMPLE5.5.8. Suppose that the n-plane bundle .rr : E + M is given an explicit O(n)-reduction. Equivalently, there is a positive definite inner product (. , .), on Ex which varies smoothly with 2 E M. That is, (sl(x), s2(x)), is a smooth function of x, Vsl, s2 E r(E) (cf. Lemma 3.4.14 for the case E = T ( M ) ) . In fact, this smoothly varying inner product can be thought of as a smooth "field" of inner products (cf. Exercise (3)), often called a Riemannian metric on the bundle E. One can then define O(E) C F ( E ) to be the subset of frames that are orthonormal with respect to this smooth field of inner products and restrict p to a projection p : O(E) -+ M. If U M is an open, locally trivializing neighborhood for E, let ( s l , . . . , s,) be a smooth frame field on U (i.e., a smooth section of F(E1U)). By an application of the Gram-Schmidt process to this frame field, we can assume that it is everywhere orthonormal, hence is a smooth section a of O(E1U). All of O(EIU) can be swept out by applying right actions of O(n) to a and one obtains a local trivialization

by setting O(x,A) = a(x) A, Vx E U, VA E O(n). Thus, O(E) c F(E) can be thought of as a subbundle which is invariant under the right action of O(n) c Gl(n). In fact, O(n) is simply transitive on the fibers and O ( E ) is a principal O(n)-bundle. The standard application of partitions of unity shows that there are infinitely many choices of Riemannian metrics on E and corresponding orthonormal frame bundles. EXAMPLE5.5.9. The n-plane bundle .rr : E + M always admits O(n)reductions, but we know that it may or may not admit a Gl(k, n - k)-reduction. In fact, such a reduction is equivalently a k-plane subbundle Ek C E (Exarnple 3.4.16). Given such a subbundle, let Fk(E) F ( E ) consist of the frames of E whose first k entries form a frame of Ek. As in the previous example, this forms a locally trivial subbundle of F ( E ) which is invariant under the right action of Gl(k,n - k), a simply transitive action on each fiber. This is a principal Gl(k, n - k)-bundle.

EXAMPLE 5.5.10. An O(n, n - k)-reduction of E corresponds to an indefinite inner product , on Ez (cf. Example 3.4.15) which varies smoothly with x E M. Such a reduction may or may not exist but, if it does, one obtains a principal O(k, n - k)-bundle of frames that are "orthonormal" with respect to this indefinite metric. The reader can supply the details. (a

a),

EXAMPLE5.5.11. Let G be a Lie group, H G a closed subgroup. Then the quotient projection p : G -+ G / H is a principal H-bundle over the homogeneous space G/H. Indeed, the local triviality is by Lemma 5.4.8 and the required property of the right H-action is obvious. EXAMPLE5.5.12. The case of a principal bundle with d2screte structure group G is noteworthy. A discrete group is simply an abstract group with the discrete

156

5 . LIE GROUPS

topology (each point is an open set). This is a topological group and, if G is at most countably infinite, it can also be viewed as a Lie group of dimension 0. A principal G-bundle p : P + M is called a reguiar covering space if G is a discrete group. It is evident that dim P = dim M and that p is locally a diffeomorphism. In fact, the local triviality of the bundle guarantees that each point x E M has a neighborhood U C M such that p-l(U) Z U x G is a disjoint union of copies of U ,one for each point of G, each of which is carried by p diffeomorphically onto U (compare Theorem 3.9.1). I t can be seen that the group of homeomorphisms g : P -t P such that p o g = p is exactly the group G, acting (from the right) on the total space P of the bundle (Exercise (5)). This is called the group of covering transformations. For instance, the standard projection p : Rn -+ T n is a regular covering with Zn as the group of covering transformations. The projection p : S n -+ Pn is another example, the covering group being Z2.

EXERCISES (1) Let p : P -, M be a principal G-bundle and let F be a manifold. If p : G x F -+ F is a smooth action, define the "diagonal" action of G on P x F in analogy with (5.3) and denote the quotient space by P x , F. Show that the bundle projection p passes to a well defined map p : P x , F -+ M, and that this is a smooth, locally trivial bundle with fiber F (define carefully what this means). It will be helpful to consider first the case in which M reduces to a single point (i.e., prove that G x , F = F canonically), then the case in which P = M x G is a trivial bundle, and finally the general case. (2) Let p : Gl(n) x R n -,Rn be the smooth action defined by

Given an n-plane bundle E over M , define E * = F ( E ) x,Rn as in Exercise (1) and prove that E* is an n-plane bundle over M with each fiber El canonically isomorphic t o the dual space of E x . Again, it will be help ful to consider first the case F ( V ) x ,Rn = V*, then the case of a product bundle, and finally the general case. (The bundle T * ( M )= (T(M))* is called the "cotangent bundle" of M. Its sections are "covector fields" or differential 1-forms, the topic of Chapter 6.) (3) Define a smooth action p : Gl(n) x '33Z(n) -+ m ( n ) by

Given an n-plane bundle E over M , show that S ( E ) = F ( E ) x '33Z(n) is an n2-plane bundle over M with each fiber S(E), canonically isomorphic to the space of symmetric bilinear forms on E x . (A section of the bundle S(T(M)) is called a symmetric 2-tensor on M . Riemannian metrics and indefinite metrics are examples of such tensors.) (4) Let p : P -+ M be a principal G-bundle, H & G a closed subgroup, and Q C_ M a properly imbedded submanifold such that p(Q) = M . Suppose

5.5. PRINCIPAL BUNDLES

157

that Q is invariant under the right action of H (the restriction to H of the G-action on P ) and that p-I (x) n Q is an H-rbit, Vx E M . (a) Prove that p : Q -+ M is a principal H-bundle. This is called an H-reduction of the principal G-bundle p : P -+ M . (b) If X : G x F F is a smooth action, denote the restriction of this action to H by X also and prove that the inclusion -+

induces a commutative diagram

M

-

M id

in which T is a diffeomorphism. (c) Specialize to the case in which F = G / H and X : G x G / H -+ G/ H is the standard homogeneous action. Prove that the principal Gbundle p : P -+ M admits an H-reduction if and only if the associated bundle p : P xx G / H -+ M admits a global smooth section s : M -+ P xx G / H . (d) If ?r : E -+ M is an n-plane bundle, p : F(E) -+ M its associated frame bundle, and if H Gl(n) is a properly imbedded Lie subgroup, prove that 7r : E 4 M admits an H-reduction if and only if p : F ( E ) -+ M admits an H-reduction. Thus, part (c) gives a new existence criterion for infinitesimal H-structures. (5) Let p : P -+ M be a smooth map. A point x E M is said to be "evenly covered" by p if there is a neighborhood U of x such that p-'(U) falls into a disjoint union of open sets, each carried by p diffeomorphically onto U. If every x E M is evenly covered by p, then p : P M is called a covering space. The group G of diffeomorphisms g : P -+ P such that p o g = p is called the group of covering transformations. If this group permutes the set pP1(x) transitively, V x E M, we say that p : P -+ M is a regular covering space. Prove that this definition is equivalent to the one given in Example 5.5.12. -+

CHAPTER 6 Covectors and 1-Forms

An important analytic tool in our study of manifolds M has been the Lie algebra X(M) of smooth vector fields. In this chapter, we begin the study of the dual o b ject, the space A'(M) of differential 1-forms on M. One would expect this space of "covector fields" to be neither more nor less useful than X(M), but for many purposes it is much more powerful. One reason for this is the "functoriality" of A1(M), as will be explained presently. Another is exterior multzplication, an operation which produces higher order objects, called q-forrns. These q-forms can be integrated over suitable q-dimensional domains and differentiated. A version of the fundamental theorem of calculus, called Stokes' theorem, relates these operations and, in the global setting, leads to a remarkable tool (de Rham cohomology) for analyzing the topology of M . This, in fact, is the beginning of a major mathematical discipline called algebraic topology. Finally, theorems stated in terms of differential forms sometimes provide interesting and useful alternatives to equivalent vector field versions. An example of this will be a differential forms version of the Frobenius theorem. These topics will require the next several chapters to do them justice. Here we deal only with 1-forms. 6.1. The space of 1-forms We introduced the cotangent bundle T * ( M ) in Exercise (2) on page 156 as a bundle associated to the principal frame bundle F(T(M)). Here we give a more elementary construction of T * (M), analogous to our construction of the tangent bundle in Section 3.3. Throughout this discussion, we allow the possibility that a M # 8.

DEFINITION 6.1.1. Let M be a differentiable manifold, x E M. The dual space T,*(M) = Homw(T,(M), R) is called the cotangent space of M at x. Each element a E T,'(M) is called a cotangent vector to M at x. A typical cotangent vector is the differential of a map. Let U M be open, x E U ,and let f E Cm(U). Since Tf(,)(R) = R canonically, we obtain a linear

6. COVECTORS AND 1-FORMS

160

functional dfx : Tx(M) R, so df, E T: (M). It is evident that df, depends only on the germ [f 1, E we obtain an R-linear map +

a,, so

LEMMA 6.1.2. For each X , E Tx(M), dfx(Xx) = X x ( f ) . PROOF.Let (U,x l , . . . ,xn) be a coordinate chart about x. Then

-(a.)I

f

[ af

dfx = Jfx = =(x)

, . . . ' ax1

-

If

then dfX(XX) = Jfx . an

i=l

COROLLARY 6.1.3. Relative to local coordinates x ', . . . , xn about x E M, the covectors dx:, . . . , dx; form a basis of T,'(M). PROOF.Since dimT:(M) = n, it will be enough to show that this set of covectors is linearly independent. By the lemma,

Thus

0 COROLLARY 6.1.4. The linear map d : 8,

-+ T,*(M) is

surjective.

DEFINITION 6.1.5. If cp : M + N is a smooth map between manifolds, if x E M , and if u E T&,) ( N ) , then cp: ( a )E T: ( M ) is defined by

Evidently, this defines a linear map cp; : T;(x) (N)

-+

T '(M).

DEFINITION 6.1.6. The linear map c p z is called the adjoint of cp,,.

6.1. SPACE OF 1-FORMS

161

This is just a special case of a standard construction in linear algebra. If Vl, V2 are vector spaces (of respective dimensions m and n) over a field IF and if L : Vl -+ V2 is linear, then the adjoint is a linear map L * : V2* -+ V;C defined exactly as above. In terms of bases { e l , . . . , e m ) of Vl and ( € 1 , .. . , en) of V2,L is represented by an n x m matrix A over IF. The dual basis {e,t)E1 of V;C is defined by

and {E;);=~is defined similarly. Relative to the dual bases, L * is represented by an m x n matrix A*. The following is standard.

LEMMA6.1.7. I f A and A* are as above, then A* = A ~ . We return to

If x l , . . . , x m are coordinates about x and y l , . . . ,yn coordinates about cp(x), we have bases

and the dual bases

We define the adjoint Jacobian by

the matrix representing cp: relative t o these coordinates. The following three lemmas can be seen directly from the definitions or via adjoint Jacobians.

LEMMA6.1.8. If cp : M -+N is smooth, if x E M , and if cp,, is an isomorphism, then cp: is also an isomorphism and (cp:)-l is the adjoint of (cp,,)-l. LEMMA6.1.9. If M = N and cp = idM, then cpz = idT,'(M) LEMMA6.1.10. If M % N

11,

--+

P are smooth maps and x E M , then

162

6. COVECTORS AND 1LFORMS

Lemma 6.1.10 is a version of the chain rule. Let cp : U -+ V be a diffeomorphism between open subsets of Rn. By Lemmas 6.1.9 and 6.1.10 (applied to (p o cp-' and cp-' o cp),

For each x E U , define cp; = (9;I-l : T W )

+

TG(X)(V)

and the associated n x n matrix

Again, we have chain rules

The definition of the cotangent bundle can now be given and agrees, practically word for word, with that of the tangent bundle. Let

be the maximal smooth atlas on M. As a set,

and we topologize each T * (U,) via the bijection

The structure cocycle for M yields the Gl(n)-cocycle {J'g,p)o,pEa for T*( M ) and the local trivializations are

One can show (Exercise (1)) that the tangent and cotangent bundles are isomorphic, but not canonically. We do not identify these bundles. DEFINITION 6.1.11. The Coo(M)-module of covector fields on M is

Covector fields are also (and more commonly) called 1-forms on M. If w E A1(M), then w : M + T * ( M ) is written x I+ w, f T,*(M), V x E M.

6.1. SPACE OF 1-FORMS

163

If w E A1(M) and (U, xl, . . . , x" ) is a coordinate chart on M , then the restriction of w to U can be written

for unique fi E Cm(Ui), 1 5 i 5 n.

REMARKS. Let 7r : E + M be a k-plane bundle with associated Gl(k)+ocycle {yap}a,PE'd. The inverse adjoint defines m o t her Gl(k)-cocycle { ~ h ~ } , ,and p~~ this, in turn, defines a dual bundle 7r : E* + M whose fibers E: are canonically identified with (E,)* = Homw(Ex,R), V x E M. The double dual of an n-plane bundle 7r : E + M is canonically isomorphic to the original bundle. That is, (E*)* = E. Indeed, it is immediate that (?Lo)' =

Denote by (X(M), Cm(M)) the Cm(M)-module of d l Cm(M)linear maps X(M) + C m ( M ) . If w E A1(M) and X E X(M), we obtain w(X) E C m ( M ) by setting

It is clear that w(fX) = fw(X), V so we can view this 1-form as an element

f

E COO(M),

This defines an injective homomorphism

of Cm(M)-modules. We will now show that this is also a surjection, proving that these Cm(M)-modules are canonically isomorphic. Let a E H O ~ ~ ~ ( ~ ) ( X CbD(M)) ( M ) , and let U C M be an open subset.

LEMMA 6.1.12. If X E X(M) and XIU = 0 , then a(X)IU

-

0.

PROOF.Let x E U and choose f E C m ( M ) , vanishing at x and identically equal to 1 on M \ U. Then f X = X and This shows that a ( X ) ( x ) = f(x)a(X)(x) = 0. Since this is true for arbitrary x E U, it follows that a(X)IU

LEMMA 6.1.13. There is a unique

such that Z(XlU) = a(X)IU, VX E X(M).

-

0. El

6. COVECTORS AND 1-FORMS

164

PROOF.Uniqueness is clear. If Y E X(U), define G(Y) E Coo(U) as follows. For arbitrary y E U, choose f E C m ( M ) such that f = 1 on some open neighborhood V c U of y and f ( ( M\ U) r 0. Then we can interpret f Y as a field defined on all of M (= 0 outside of U) and fYlV = YIV. Define &(Y)(y)= a ( f Y ) ( y ) . If f and are different choices, Lemma 6.1.12 implies that the two definitions of Z(Y) agree at y (and, indeed, on the neighborhood V n of y in U). It is clear (X(U), Cm(U)) and that G(XIU) = a(X)IU, that this defines Z E VXEX(M).

P

P

By this lemma, we can define a l U = C, calling this the restriction of a to U.

COROLLARY 6.1.14. If cr: E H O ~ ~ ~ ~ ~ ~ ( X ( M ) , Cthen "(M a(X ) ) ( x ) depends on X, but not otherwise on X , Vx E M , VX E X(M). PROOF.Let x E M . Choose a neighborhood U of x in M over which T ( M ) is trivial and let Y1, . . . , Yn E X(U) give a basis of of the tangent space at each point of U. Then arbitrary X f X(M) can be written on U as

and

n

4x1(5) = ( ~ l u ) ( X l U 2) ) ( = C fi(x)(alu)(yi)(x). i=l

On the right hand side of this equation, the only dependence on X is in the values fi(x), 1 5 z 5 n. The property of a in the above corollary is called the tensor property.

LEMMA 6.1.15. If r] : X(M) --t Coo(M) is an R-linear map, then 7 has the tensor property if and only if r] E Horncm (X(M), C m ( M ) ) . PROOF.We have proven the "if" part. For the converse, assume that 7 has the tensor property and let f E Cm(M),X E X(M). For each x E M , r](fX)(x) depends only on f (x)X,, hence

by R-linearity. Since x E M is arbitrary, q ( f X ) = f q ( X ) . This equivalence between Cm(M)-linearity and the tensor property will recur in the broader context of Cm(M)-multilinearity in Chapter 7. For x E M and a E Homcm(Ml (X(M), Cm(M)), we can define a, E T: (M) as follows. Given u E T,(M), let X E X(M) be any vector field such that X, = u. Define a,(v) = a(X)(x). By the above, this depends only on v, not on the choice of extension X, so we get a, E T,+(M),Vx E M . Using local coordinates, it is elementary to check that the map x I+ a, defines a smooth section of T*(M). This identifies a as an element of A 1 ( ~ ) completing , the proof of the following.

6.1. SPACE OF 1-FORMS

PROPOSITION 6.1.16. There is a canonical isomorphism,

of C" ( M )-modules.

REMARK.Similarly, X(M) = Horncm (A1(M), Cm(M)). Let cp : M -t N be smooth. If w E A1(N), define cp*(w) : M --+ T * ( M )by

If f E C m ( N ) , define cp*(f) = f lemmas will be Exercise (3).

o

cp E C W ( M ) . The proof of the following two

LEMMA6.1.17. If cp : M -+ N is a smooth map of manifolds and w E A' (N), then cp4(w) E A'(M) and, this defines a linear map

of vector spaces over R. Furthermore, i f f E Coo(M),

are smooth maps of manifolds, then ($ o cp)* = cp* A' (P).

o

$* on both C"(P)

and

REMARK.In the language of category theory, A' is a contravariant functor from the category of differentiable manifolds and smooth maps to the category of real vector spaces and linear maps. We explain this in a little more detail. A category is a class C of objects and of maps between these objects, called rnorphisms, with the property that compositions of morphisms, whenever defined, yields morphisms and, for each object 0 E C, ido E C. Thus, the class V of real vector spaces and linear maps between them is a category, as is the class M of differentiable manifolds and smooth maps. A functor is either a "homomorphism" or an "antihomomorphism" between two categories. In the first case, the functor is said t o be covariant and, in the second case, it is contravariant. More precisely F : C1 -+ C2 is a covariant functor if, whenever cp : 0 -+ 0' is a morphism in C1, then F(cp) : F(0) + F(0') is a morphism in C2 and (1) F(cp 0 $) = F(cp)0 F($), v morphisms cp, Q E C1, V object 0 E C1. (2) F(ido) = The functor F is contravariant if F(cp) : F ( 0 ' ) + F ( 0 ) and the first of the above properties is replaced with

In the case of a covariant functor, it is customary to write F(cp) = cp, and, in the contravariant case, to write F(cp)= cp*. Thus, Lemmas 6.1.17, 6.1.18 and the trivial fact that i d L = i d A ~ ( Mimply ) that A' : M V is a contravariant functor. By contrast, X(M) does not define -+

166

6. COVECTORS AND 1-FORMS

a functor since, generally, cp, : X ( M ) -,X ( N ) is not defined. For many purposes, this makes 1-forms significantly more useful than vector fields. Observe also that Coo : M + V is a contravariant functor.

LEMMA6.1.19. I' f E C m ( M ) , then df : M df,, Vx E M , is a 1-form on M .

-+

T * ( M ) , defined by df (x) =

Indeed, the fact that df is a smooth section of T * ( M ) is easily checked via local coordinates, where the formula is

Alternatively, it is an immediate consequence of Lemma 6.1.2 that

DEFINITION 6.1.20. If f E C o o ( M ) ,then df is called the exterior derivative of f . The R-linear map d : C m ( M ) + A 1 ( M )is called exterior differentiation. L E M M A6.1.21. If f , g E C m ( M ) , then d ( f g ) = f d g + g d f .

PROOF.Indeed, if X E X ( M ) , then

Since X E X ( M ) is arbitrary, the assertion follows.

[7

This lemma is a Leibnitz rule for exterior differentiation.

LEMMA6.1.22. If cp : M

N is smooth, then the diagram

is commutative. That is, d(cp*( f ) ) = cp*(df),V f E C m ( N ) . The property of d in this lemma is called the naturality of the exterior derivative. The proof, as the one for the preceding lemma, works itself and is left to the reader.

6.2. LINE INTEGRALS

EXERCISES (1) Exhibit a bundle isomorphism between T(M) and T*(M). In the case that M is an open subset of Euclidean space, show that there is a canonical choice of isomorphism. What is the difficulty about defining a canonical isomorphism in general? (In classical physics and advanced calculus courses, it is quite common not to distinguish tangent vectors and cotangent vectors. It is precisely because these treatments are carried out in Euclidean domains that this identification is legitimate.) (2) Prove that the dual bundle E*,as described in Exercise (2) on page 156, is canonically isomorphic to the dual bundle E* as constructed in this section. (3) Prove Lemmas 6.1.17 and 6.1.18. (4) Let X E X(M) and let @ denote the local flow generated by X. One defines the Lie derivative

@t*(w)- W , vw E A ~ ( M ) , t+O t taken pointwise on M. Recall from Section 2.8 the analogous definitions of the Lie derivatives L x ( f ) and L x ( Y ) for f E Coo(M) and Y E X(M). Prove the following identities for arbitrary f E C OQ ( M ), w E A (M), and Y E X(M). (a) t x ( d f ) = dLx(f)(b) Lx(fw) = L x ( f )w + f t x ( w ) . (c) L x (w(Y)) = Lx (w)(Y) + (Y)). Lx(w) = lim

6.2. Line Integrals

If w E A1(M) and s : [a,b] We can write s*(w) = f dt.

-+

M is a smooth curve, then s * ( w ) E A1([a, 61).

DEFINITION6.2.1. The line integral of w E A 1 ( ~ along ) a smooth curve s : [a,b]-+M is

f (t) dt. Line integrals are insensitive to orientation preserving changes of parameter and only experience a sign change under an orientation reversing repararnetrization. It is not even necessary to require that the change of parameter be nonsingular or monotonic. LEMMA6.2.2. Let s : [a, b] o = S O u. Then,

4

M and u : [c,4

-+

[a,b] be smooth. Set

(1) 2f u ( c ) = a and u(d) = b, J,w = JUw, Vw E A1(M); (2) if u(c) = b and u ( d ) = a, - Js w = Jm w, V w E A'(M).

6.

168

COVECTORS AND 1-FORMS

4.

PROOF.Let t denote the coordinate of [a,b] and r the coordinate of [c, Then

In case (1))the rule for change of variable in integrals gives

In case (2), the same rule gives

M and s2 : [c,4 -+ M be smooth paths with the same initial point and the same terminal point. That is, s l ( a ) = sz(c) = x and sl (b) = s2(d) = y . If f E CM( M ) , then

LEMMA 6.2.3. Let sl

: [a,b] -+

PROOF.By Lemma 6.2.2, we lose no generality in assuming that [a,b] = [c,dl. Then,

6.2. LINE INTEGRALS

169

This lemma is a 14imensional version of Stokes' Theorem. As the proof makes clear, it is just the fundamental theorem of calculus. DEFINITION 6.2.4. A form w E A 1 ( M )is said to be exact if w = df, for some f E Coo(M). Lemma 6.2.3 says that the line integral S,w of an exact l-form w depends only on the endpoints of the path s, not otherwise on s. In physics, the law of conservation of energy is a special case of this result. Lemma 6.2.3 is a part of Theorem 6.2.9, which will be stated and proven shortly. The notion of a line integral can be extended to allow integration of l-forms along paths s : [a, b] -+ M that are only piecewise smooth. That is, s is continuous and there exists a partition a = t o < t l < - . - < t , = b such that si = sl [ti-l, ti]is smooth, 1 5 i 5 q. We write s = sl sz . sq and define

+ + +

Since it is not assumed that the partition contains only points at which s is not smooth, it is necessary to observe that this definition is independent of the choice of allowable partition. This is elementary and is left to the reader. The proof of the following consequence of Lemma 6.2.3 is also left to the reader. COROLLARY 6.2.5. Let sl : [a,b] -+ M and sz : [c,d] -+ M be piecewise smooth paths with the same initial point x and the same terminal point y. Then,

LEMMA6.2.6. If w E A 1 ( M ) and iJ for every piecewise smooth path s, the integral Js w = 0 , then w = 0. PROOF. Otherwise, there is a point z E M and a tangent vector v E T , ( M ) such that w,(v) > 0. Let s : [ - E , E ] -+ M be smooth such that s(0) = z and i ( 0 ) = v. Choosing E > 0 smaller, if necessary, we can assume that

An elementary computation shows that s * ( w ) = ~ ~ , ( t , ( s ( t )dt, ) so

contradicting the hypothesis. DEFINITION 6.2.7. We say that w E A 1 ( M ) has path independent line integrals if, for every piecewise smooth path s : [a,b] -+ M , Js w depends only on s (a) and s(b) and not otherwise on s. DEFINITION 6.2.8. A piecewise smooth path s : [a, b] -+ M is a loop if s(a) = s(b).

170

6 . COVECTORS A N D 1-FORMS

THEOREM 6.2.9. For w E A1(M), the following are equivalent. (1) w is an exact form. (2) Js w = 0, for all piecewise smooth loops s. (3) w has path independent line integrals. PROOF.We prove that (1) + (2). If w = df is exact and s : [a,b] -, M is a piecewise smooth loop, s(a) = q = s(b), then Corollary 6.2.5 implies that

where q denotes the constant path q(t) = q, a 5 t 5 b We prove that (2) + (3). Let sl and s2 be piecewise smooth curves starting at the same point x and ending at the same point y. Without loss of generality, assume that sl is parametrized on [- 1,0] and s2 on [0, 11. Let u : [0, I] [O,1) be defined by u(t) = 1 - t. Then s 2 o u starts at y and ends at x and -+

is a piecewise smooth loop. By our assumption,

where we have used part (2) of Lemma 6.2.2 to write

We prove that (3) + (1) by using (3) to construct f E Coo(M) such that w = df. Without loss of generality, we assume that M is connected (otherwise, carry out the construction off on each component individually). Fix a basepoint xo E M. Given any point x E M , use connectivity to find a piecewise smooth path s : [a, b] -+ M such that s(a) = xo and s(b) = x. (In fact, the homogeneity lemma, Theorem 3.8.7, implies the existence of a smooth path, but the present claim is more elementary and is left to the reader.) Set

By the assumption of path independence, this is independent of the choice of piecewise smooth path s from xo to x. Remark that f (xo)= 0. We prove first that f : M -+ R is smooth. Let q E M be arbitrary and choose a coordinate chart (U, x l , . . . ,xn) about q in which q is the origin and U = int Dn, where D n is the unit ball in Rn. In these coordinates, we can write

6.2. LINE INTEGRALS

For each x E U, let s, : [O,1] express f J U by the formula

-+

U be defined by s,(t) = tx, 0 5 t

171

5 1, and

Thus, on U,

This is clearly smooth. Since q E M is arbitrary, f E Cm(M). Next, we prove that w = df. Let s : [a,b] -, M be an arbitrary piecewise smooth path. Let c < a and let s o : [c,a] + M be piecewise smooth such that s 0 ( c ) = xo and so(a) = s(a). Then

That is

It follows that the form 5 = w - df satisfies

for all piecewise smooth paths s. By Lemma 6.2.6, 5 = 0, so w = df.

DEFINITION 6.2.10. A 1-form

w E

A'(M) is locally exact if, for each x E M ,

there is an open neighborhood U of x such that wl U E A'(U) is exact.

EXAMPLE 6.2.11. On the manifold M = W2 \ ((0, 0)), define the 1-form

We claim that q is locally exact. Indeed, if q E M is not on the y-axis, a branch of 8 = arctan(y/x) is defined and smooth on a neighborhood of q. A direct computation gives d8 = q. Similarly, if q E M is not on the x-axis, select a branch of 8 = - arctan(x/y) and check that dB = q. Since no point of M is on both axes, this proves that q is locally exact. We claim, however, that q is not exact. Indeed, consider the smooth loop s : [O, 11 -+ M defined by s(t) = (cos 2nt, sin 27rt). Clearly, qs(t) = - sin 2rt

+ cos 27rt dys(t),

and s*(dx) = -2n sin 2 r t dt s*(dy) = 27r cos 2nt dt,

6. COVECTORS A N D 1-FORMS

172 SO

s*(7) = 2n(sin2 27rt

+ cos22xt) dt = 2n dt.

Thus

Remark that the form 7 in the above example cannot be extended to a 1-form on IR2. We are going to see shortly (Corollary 6.2.14) that, on R 2 , every locally exact 1-form is, in fact, exact. The above example reflects a topological feature of IR2 \ { ( O , O)}, the missing point, that distinguishes that space from IR2. DEFINITION 6.2.12. Let so,sl : Ia,b] -+ M be piecewise smooth loops. We say that so is homotopic to sl, and write s o $1, if there is a continuous map N

H : [a, b] x [ O , l ] and apartition a = to < tl

<

-, M

= b such that

<

(1) Hl([ti-17ti] x [ O , l ] ) is smooth, 1 5 i r ; (2) H(t,O) = so(t) and H(t, 1) = sl(t), a 5 t (3) H ( a , r ) = H(b,r),0 T 5 1.

<

< b;

As usual, homotopy is an equivalence relation. The proof of the following important result will be Exercise (3). PROPOSITION 6.2.13. If w E A' (M) is locally exact and zf the loops s 1 and s2 are piecewise smooth and homotopic, then

COROLLARY 6.2.14. Every locally exact 1-form on Rn is exact. PROOF.Let s : [a, b]

-+

Rn be a piecewise smooth loop and define

by H ( ~ , T=) r s ( t ) . Then H is a homotopy of the constant loop 0 to the loop s. If w is locally exact, Proposition 6.2.13 implies that

Since the loop s is arbitrary, Theorem 6.2.9 implies that w is an exact form. Corollary 6.2.14 is a special case of the Poincare' Lemma (Theorem 8.3.7).

EXERCISES (1) Show that every 1-form on R is exact, but exhibit a 1-form on IR2 which is not exact. (2) Find a 1-form on S1 which is locally exact but not exact. (3) Prove Proposition 6.2.13. Proceed as follows.

6.3. FIRST COHOMOLOGY

173

(a) Let U EX2 be open, let R = [a,b] x [c,d] c U, and let w E A'(u) be locally exact. For E > 0, let R, = (a - e, b E ) x (c - E , d c). Show that there is E > 0 such that R, G U and w 1 R, is exact. (b) Let cp : M -+ N be a smooth map between manifolds and let w E A1(N) be locally exact. Prove that cp*(w) E A1(M) is locally exact. (c) Use these two results to prove the proposition. (4) If n 2 2, prove that every locally exact 1-form on S n is exact.

+

+

6.3. The First Cohomology Space In Example 6.2.11, we saw that a locally exact form on a manifold can Eail to be exact and that this seems to be related to the topology of the manifold. This insight is formalized and exploited by the de Rham cohomology H '(M), a vector space associated to the manifold M which measures, in some sense, how much the notions of "locally exact" and "exact" differ on M. DEFINITION 6.3.1. The space of (de Rham) l-cocycles on M is Z1(M) = {w E A'(M) I w is locally exact). The space of (de Rham) l-coboundaries is B'(M) = {w E A1(M) I w is exact). Remark that, if we regard A1 (M) as a vector space over R, then Z1(M) and B1(M) are vector subspaces. They are not Cm(M)submodules. It is also clear that B'(M) G Z1(M). DEF~NITION 6.3.2. The vector space

is called the first (de Rham) cohomology space of the manifold M. If w E Z (M), its cohomology class is [w] = w B1(M) E H1(M).

+

'

Although, whenever dim M > 0, the vector spaces Z (M) and B1(M) are infinite dimensional, it frequently happens that H1(M) is finite dimensional. We will see, for instance, that this is the case whenever M is compact. Cohomology is a contravariant functor from the category of differentiable manifolds (smooth maps are the morphisms) to the category of real vector spaces (and linear maps). Indeed, by Lemma 6.1.22 a smooth map cp : M -+ N induces a linear map cp* : Z1(N) -+ Z1(M) and cp*(B1(N)) C B1(M), so cp* passes to a well defined linear map (of the same name)

It is trivial to check that (cp 0 $)* = $* o cp* and (idM)*= idH1(M). PROPOSITION 6.3.3. Let w, G E z l ( M ) . Then [w] = [G] E H1(M) zf and only if w = SsS as s varies over all piecewise smooth loops in M . These numbers are called the periods ofw and of the cohomology class [w].

Ss

174

6. COVECTORS AND 1-FORMS

PROOF.The locally exact forms w, S E z'(M) have the same periods Jsw = Js3, for every piecewise smooth loop s, if and only if Js(w - 5) = 0 for all such loops. By Theorem 6.2.9, this holds if and only if w - L? E B1(M). Equivalently, [w]=[W].

PROPOSITION 66.4. If fo, f l : M -+ N are smooth and homotopic (fo .- fl), then f; = ff : H'(N) -+ H ~ ( M ) . PROOF.Let [w] E H 1 ( N ) . If s : [a,b] -+ M is a piecewise smooth loop, then si = f, o s : [a, b] -+ N is &o a piecewise smooth loop, i = 0 , l . Let H : M x R + M be a homotopy of f o to fi. Then, the composition [a, b] x [O,1]

sxid

M xR

H 4

N

is a homotopy of so to sl, so

=

d.

w

(Proposition 6.2.13)

Since s is an arbitrary piecewise smooth loop, Proposition 6.3.3 implies that

Since [w] E H' (M) is arbitrary, f f = f; at the cohomology level, as desired. DEFINITION 6.3.5. A smooth map f : M -+ N is a homotopy equivalence if there exists asmooth map g : N + M such that f o g .- idN and g o f --idM. COROLLARY 6.3.6. A hornotopy equivalence f : M isomorphism f * : H'(N) -+ H1(M).

-+

N induces a linear

PROOF.Since f o g idN, it follows by functoriality and Proposition 6.3.4 that g * ~ f * = ( f ~ g ) * = i d & =Hi1(N) d N

Similarly, f * o g* = idH' ( M ) , SO f * and g* are mutually inverse isomorphisms on cohomology.

-

6.3. FIRST COHOMOLOGY

175

-

EXAMPLE 6.3.7. Let f : (0) Dn be the inclusion. Let g : Dn -+ {0} be the only map. These maps are smooth and g o f = id ~ 0 )(hence, g o f id{o)). Consider the map f o g : Dn + Dn having image (0). We claim that this is homotopic to idon. Indeed, let cp : R -+ [0, I] be smooth such that cp(0) = 0 and cp(1) = 1. Define H : Dn x R -,Dn by

Then H(x,O) = 0 - f(g(x)), V X E Dn, and

H(x,1) = x = idDn(x), V X E Dn. This establishes the desired homotopy and completes the proof that f is a hw mot opy equivalence, so

is an isomorphism. That is,

H'(D") = o

or, equivalently, every Iocally exact 1-form on D " is exact. A similar proof shows that Rn is homotopically equivalent to a point, and we recover Corollary 6.2.14.

EXAMPLE 6.3.8. Let

i : s1~f R2 \ {(O,O))

be the inclusion. Let g : R2 \

{(O,O)}

4

s1

be the map defined by

These maps are smooth, and goi=idsl. We claim that hence that i is a homotopy equivalence. Indeed, define

bv the formula

+

This is smooth since, llvll being strictly positive, so is t (1 - t) IIvII, 0 5 t 5 1. Then H ( v , 1) = v, V V E R2 \ {(O,O)}, and

v H(v,O) = - =i(g(v)), V V E EX2 llvll

\

{(0,0)}.

It follows that

H ~ ( R \ { ~( o , ~ ) }= ) ~l(sl).

6. COVECTORS AND 1-FORMS

176

PROPOSITION 6.3.9. There i s a canonical isomorphism H l ( S 1 )= IW. We will prove this via three lemmas. First define the map

by

p(t) = (cos 2rt, sin 2nt). This is the map that induces the standard diffeomorphism R/Z = S 1 . Define

a linear map.

LEMMA6.3.10. The linear m a p a passes t o a well defined linear map

PROOF.Indeed,

0

= p1 [O, 11 is a smooth loop and w E

B

'(s') implies that

by Theorem 6.2.9.

LEMMA6.3.11. T h e linear m a p a : H 1( S 1 ) R is injective. -+

PROOF.Let w E Z 1 ( S 1 )be such that a(w) = 0. We must prove that w E B' (5''). For n E Z, let T, : R -+R be the translation rn(t)= t n. Then, p o Tn = p,

+

SO T;

o p * = P* :

~ l ( s-+ l )A ~ ( R ) .

This and the change of variable formula for the integral gives

In particular, since a(w) = 0, we obtain

Define

fw E

C m ( S 1 )by

If this is well defined, it will be smooth. It will be well defined precisely if

6.3. FIRST COHOMOLOGY

If n = 0, this is obvious. If n > 0,

=

l+n 1 p* ('d)

t

=

P*(w).

A similar computation for the case n < 0 is left to the reader. If we define

f;= P*(fW), we get so the fundamental theorem of calculus and Lemma 6.1.22 give

But p : R 4 S1 is a local diffeomorphism, so w and df, are equal locally, hence globally. That is, w E B'(s') as desired. Recall the locally exact form

of Example 6.2.11 and let

where i is the inclusion map of S1 into B2 ((0,O)). For the loop a = p1[0,1], s = i o a is as in Example 6.2.11, and we showed that a*(ij)= o*(i*(7))= (i o a)*(7)= S* (7)= 27rdt.

LEMMA6.3.12. The linear map a : H1(S1)-+ B is surjective. PROOF. It is enough to show that a is nontrivial. But [ij] E H '(S1) and

The proof of Proposition 6.3.9 is complete.

EXAMPLE 6.3.14. Recall that the Brouwer fixed point theorem for smooth (in fact, continuous) maps f : D2 -+ D2 follows from the nonexistence of a smooth retraction p : D 2 + dD2 = s l . Recall that, for p to be a retraction, it is required that the diagram

6. COVECTORS AND 1-FORMS

commute, where L is the inclusion of the boundary circle. By the functoriality of cohomology, this produces a commutative diagram

That is,

commutes, which is absurd. This example illustrates the general philosophy of algebraic topology: use functors from a topological category to an algebraic category to "paint" an algebraic picture of a topological problem, then solve the algebraic problem and try to determine the implications of this solution on the original topological problem. This algebraic picture, to be useful, must leave out a lot of detail but not, of course, too much. For instance, H 1 cannot see the difference between D n , Rn, and a point, but if it could not distinguish these from S l , it would have been useless for proving the nonexistence of the retraction. Our computation of H1(S1) generalizes to the higher dimensional tori. You will be led through a proof of the following in Exercise (1).

PROPOSITION 6.3.15. There is a canonical isomorphism

EXERCISES (1) Let exp : Rn --, Tn be the homomorphism of abelian Lie groups defined by 27rixn exP(xl,.. . , xn ) = 27rix1 (cf. Exercise (7) on page 135). If a : [a,b] -+ Rn is piecewise smooth such that v', = a ( b ) - a(a) E Zn, then exp oa is a piecewise smooth loop on Tn. Prove Proposition 6.3.15, using this observation, as follows.

6.3. FIRST COHOMOLOGY

179

(a) Prove that every piecewise smooth loop on Tn is of the form exp o a as above and that the homot opy class [expoa] is completely determined by v', . (b) If w E Z1(Tn), define cp, : Zn R as follows. Given v' E Zn, choose a piecewise smooth path a : [a,b] -+ Rn such that v' = v', and set -+

vw(q=/

w-

exp o o

Show that this is well defined and that cp, : Zn uniquely to a linear functional of the same name

-+

R extends

(c) Prove that the assignment w cpw passes to a well defined linear injection cp : H ~ ( T " )-+(R")* = R". (d) Show that there are forms e l , . . . ,On E Z1(Tn) such that exp*(8') = dx',

15 i

< n.

Use this to show that

is also surjective, hence is the canonical isomorphism we seek. (2) If n 2 2, show that T n and Sn are not homotopically equivalent. (3) If the vector space H1(M) has finite dimension k, prove that there are piecewise smooth loops 01, . . . ,a k on M such that the map

defined by

is an isomorphism of vector spaces. (4) We will say that a locally exact form w E Z1(M) is integral if all of its periods are integers. For example, jj/27r E Z1(S1) is integral. By Proposition 6.3.3, w is integral if and only if every w' E [w] is integral, in which case we say that [w] is an integral cohomology class. We denote by H1(M; Z) c H1(M) the subset of integral cohomology classes. (a) Prove that H1(M;Z) is a subgroup of the additive group of the vector space H1(M). We call H1(M; Z) the integral cohomology of M . (b) If f : M N is smooth and w E Z1(N) is integral, prove that f*(w) E Z1(M) is also integral. Using this, show that integral cohomology is a contravariant functor from the category M of smooth manifolds and smooth maps to the category G of abelian groups and group homomorphisms. -+

6. COVECTORS AND

1-FORMS

(c) Referring to Exercise ( I ) , prove that H' (Tn;Z) is canonically the integer lattice cI W= ~ HI (T~).

zn

(d) To each smooth map f : Tn -t Tn, show how to assign canonically an n x n matrix Af of integers, depending only on the homotopy class of f , such that Af,, = ABAf(matrix multiplication). If f is a diffeomorphism of T n onto itself, prove that A is unimodular (i.e., has determinant f1). (e) Prove that every n x n unimodular matrix of integers occurs as the matrix Af assigned to some diffeomorphism f : Tn -+Tn.

6.4. Some Topological Applications To begin with, we will classify the smooth maps

up to homotopy. The first remark is that, since H l(S1) = R,f induces a linear map

f*:R-+R depending only on the homotopy class of f . Thus, f * is just multiplication by a certain constant af E R and af depends only on the homotopy class of f . By part (d) of Exercise (4) on page 180, a f is an integer, being the sole entry in the 1 x 1 integer matrix Af. We are going to give alternative ways to see that af E Z, proving in particular that this integer determines f up to homotopy. This will give a one to one correspondence between Z and the set of homotopy classes of smooth maps of S' to itself. LEMMA6.4.1 (LIFTINGLEMMA).If f : S1

smooth map

j :R

-+

-+

S1 is smooth, there exists a

R so that the dzagram

commutes. F u r t h e n o r e , some integer k .

f

is another such lip o f f if and only if j

=

f + k , for

PROOF.We first prove that, if g : R 4 S1 is a smooth map, then there exists a continuous map i j : R -+ R such that p o i j = g. The map i j will actually be smooth because of the smoothness of g and the fact that p is locally a diffeomorphism. Applying this to g = f o p, we see that j = lj is the required lift. First we define such a lift on an arbitrary compact interval [a,b] c R. Let J+ and J- be the complements in S1 c C of the points +1 and -1, respectively. Then p - ' ( J + ) is the disjoint union of all intervals of the form (k, k I ) , k t Z, while p-l(J-) is the union of all (k - 1/2, k + 1/2), k E Z. These intervals are

+

6.4.

TOPOLOGICAL APPLICATIONS

181

carried diffeomorphically by p onto J+ or J-, respectively. By the continuity of g and the compactness of [a, b], find a partition a = t o < tl < . . . < t, = b such that g[ti-l,ti] lies either in J+ or J-, 1 I i 5 q. Then gl = gl[to,tl] has a continuous lift ijl (depending only on the choice of (to) E p-l (g(to))),after which g;! = [ [t t2] can be lifted uniquely so that iz (tl) = gl (tl). Continuing in this way, we produce the lift of gl [a, b] as desired, depending only on the choice of the value of the lift at to = a. Remark that we could just as well have produced a lift by choosing its value at t, = b. Expressing R as the union of intervals [k,k 11, k E Z, we first choose a lift of g on [0, 11, then on [I,21 so that they agree at 1, then on [-I, 01 so that they agree at 0, etc. If f and j are two lifts of f : S1-9 S1,the fact that p is a homomorphism of the Lie group R onto the Lie group S implies that

+

That is f(t) - f(t) E Z, Vt E W. The continuity of this expression in t then implies that this integer value is a constant k, hence that f = f k.

+

PROPOSITION 6.4.2. Iff : S1 -+ S1 is smooth, then a j = j(1) - f(0) E Z, where j is any iifi off as in Lemma 6.4.1. PROOF.Let

f be a lift of f

as in Lemma 6.4.1. Then

SO

!(I) - f(0) = m E Z. Thus, let i j E zl(S1) be as on page 177 and compute

That is, af = m E Z. DEFINITION 6.4.3. If f : S1 -+ S1 is smooth, the integer af is called the degree of f and denoted deg(f ). In Section 3.9, we defined deg,(f) E Z2for smooth maps between manifolds (without boundary) of the same dimension. We will see that, for smooth maps of the circle to itself, degz(f) is just the residue class modulo 2 of deg ( f ) (Corollary 6.4.7). COROLLARY 6.4.4. Iff : S1 -* s1is smooth, if j is a lift as in Lemma 6.4.1, and if t E IR is arbitrary, then deg(f) = j ( t 1) - f(t).

+

6. COVECTORS AND 1-FORMS

182

PROOF.View p : R

-+

S1 c C as a group homomorphism. Then

so f(t + 1) - j ( t ) E Z, Vt E P.This function of t, being continuous and integervalued, is constant on R, hence equal to f(1) - j(0) = deg(f). 17

DEFINITION 6.4.5. The set of homotopy classes of smooth maps f : M will be denoted n[M, N].

-+

N

By the homotopy invariance of cohomology, we have defined deg : n [ s l , s']+ Z. We describe deg(f) in terms of regular d u e s . Let zo E S' be a regular value of f : S1 + S1. Then f-'(zo) = {zl, . . . ,z,). Here, if f -'(to) = 0, we take r = 0. Recall that deg2(f) = r (mod 2). Choose iiE B such that p(&) = t i , 1 4 i j r . The smooth map f : S' -+ S1 preserves orientation s t zi if ?(Hi) > 0 and reverses orientation at Z i if f'(4) < 0. Let ri = j'(ii)/l f (2,) 1 E {-1,l) and remark that this depends only on f and Z* .

PROPOSITION 6.4.6. With the above conventions, deg(f ) =

xi=,

€i.

PROOF.Choose a E R \ {p-'{zl, . . . ,r,)). Then p - l ( ~ i )n (a, a + 1) is a singleton and we choose this point as our Hi, 1 5 i 5 r. We will also use the fact that f(a 1) = f(a) deg(f ). Let p-'(ro) = {b + kIrez and consider the graph of s = f(t), a < t < a + 1, together with the horizontal lines s = b + k, k E Z. Each time the graph crosses a line s = b + k, the parameter t is equal to one of the Zi and e, records whether the graph crosses this lime while increasing (ci = 1) or decreasing (r; = -1). Figure 6.1 illustrates a case in which r = 7, €1 = €2 = €3 = €4 = €7 = 1, and €5 = r s = -1. The sum of the ti's pertaining to a single line s = b + k is 1, - 1, or 0, the net number of directed crossings. Ciearly, the sum of all these net numbers is

+

+

(In Figure 6.1, the degree is 3.)

COROLLARY 6.4.7. deg,(f)

= deg(f)

(mod 2).

6.4. TOPOLOGICAL APPLICATIONS

FIGURE6.1. Graph of

j

EXAMPLE 6.4.8. For each n E Z, let f, : S1 -+ s1be defined by fn(z) = zn. Here, of course, we view S1 C @. We can choose the lift fn : R -+ R to be fn(t) = nt, so deg(fn) = j n ( 1 ) - fn(0) = n. If z t. S1is a regular value of f n , then

where pl, . . . , ars the distinct nth roots of t . Of course, if n = 0, then f o is constant and f&'(z) = 0. If n > 0, all € i = +1 and, if n < 0, all s i = -1. Thus,

in all cases.

THEOREM 6.4.9. The function deg : r[S1,S1]-+ Z is bijective.

-

PROOF.For each integer n, deg(fn) = n, so deg is surjective. We must show that, if deg(f ) = deg(g) = n, then f g. Define

by the formula I;T(t, T) = T j ( t )

+ (1

-

~)j(t).

184

6. COVECTORS AND 1-FORMS

Since f7(t

+ 1, r ) - H(t, r ) = r n + (1 - r ) n = n, the formula

well defines a smooth map

Evidently, H(z,O) = g ( z ) and H ( z , l ) = f ( z ) , V z E S1,s o f - g . THEOREM 6.4.10. A smooth map f : S1 F : D2 S1 if and only if deg(f) = 0.

s1extends

to a smooth map

-+

PROOF.First suppose that the smooth extension F exists. That is, f = F o i where i : S1L) R2 is the inclusion. Then f * = i* o F* and

implying that f * = 0. Therefore, deg(f) = 0. For the converse, suppose that deg(f) = 0. By Theorem 6.4.9, it follows that f -- fo E 1. By the C" Urysohn trick, choose the homotopy

so that

Then H induces a smooth map of the disk to the circle as follows. Define a smooth surjection cp : S' x [0, 11 4 D~ C C by p(z, t ) = tz. Then cp carries S1 x (O,1] diffeomorphically onto D 2\ (0). Define

d}

This is well defined. It is smooth on D 2\ {O) and, on {w E R 2 ( (w(5 it is constant, so F is smooth. Evidently, F(z) = H(z, 1) = f (z), V z E S ', so F extends f . Recall that the mod 2 degree allowed us to prove the fundamental theorem of algebra for polynomials of odd degree (Theorem 3.9.14). The integer degree makes it possible to carry out essentially the same argument for all positive degrees. OF ALGEBRA). Let f : C -+ @. THEOREM 6.4.11 (FUNDAMENTAL THEOREM be a polynomial of degree n 2 1. Then there is zo E @. such that f ( z o ) = 0.

6.4. TOPOLOGICAL APPLICATIONS

PROOF.We can assume that the leading coefficient is 1 and write

f (z) = zn

+ alzn-' + . - .+ a,-lr + a,.

Suppose this has no root. For each positive real number r, define a smooth function Fr : l I 2 -4 s1 by the formula

This is where we use the hypothesis that f has no roots. Let g, = F,.Isl. Then set Gr(z, t) = (rz)" t(al(rz)"-l . an)

+ +

+

and note that, if r is large enough, this vanishes nowhere on S1 x [0,11. Indeed,

approaches 1 as r of r and define

4

oo, uniformly on

s1x [0,11. Thus, fix a large enough value

H : s1x [ O , l ]

4

S'

by the formula

-

Then H ( z , l ) = g,(z) and H(z,O) = zn, V Z E S1. Thus, g, fn and deg(g,) = n > 0. But g, extends smoothly to F, : D 2-+ S1,a contradiction to Theorem 6.4.10.

LEMMA6.4.12. If f , g : s1 S1 are smooth, thendeg(f og) = deg(f)deg(g). Indeed, functoriality of cohomology implies that a f o g = &fas, SO the lemma is immediate. -+

COROLLARY 6.4.13. If f , g : S1 4 S1 are smooth, then f o g and g o f are homotopic. If f , g : S1 -+ S1 are smooth, define the pointwise product f g : S1 + S1 by the Lie group structure of S'. That is,

-

-

It is elementary that, whenever fo f l and go gl, then fogo fig1 (the homotopy is the pointwise product of the given homotopies). This defines a commutative multiplication on the set n[S1, S1].The constant map 1 determines a class [I]E n[S1, S1]which is an identity for this multiplication. If L : S1-+ S1 is the group inversion map, then each [f ] E n[S l , s']has an inverse [ L O f ] relative to this multiplication, so n [ s l , sl]is canonically an abelian group. N

PROPOSITION 6.4.14. The bijection deg : n[S l , S1]+ Z is an isomorphism of groups.

186

6.

-

COVECTORS AND 1-FORMS

+

-

-

PROOF.It is only necessary to prove that deg(fg) = deg(f) deg(g). But f fn and g f m , where n = deg(f) and m = deg(g). Then f g fnfm = fn+mMore generally, the Lie group structure on S1 makes n[M,S1]into an abelian group and one obtains the following. THEOREM 6.4.15. The map

defined by

x[fl = f *[7?/2nl7 is an isomorphism of groups. Here, the integral cohomology H'(M; Z) is defined as in Exercise (4) on page 180. The fact that [ij/2n] is an integral class implies that f *[ij/2n] is also integral by that same exercise. You will be led through a proof of this theorem in Exercise (2).

EXERCISES (1) Use degree theory to show that the group Diff (S1)has exactly two is* topy classes. (2) Prove Theorem 6.4.15. Proceed as follows. (a) Show that x is a group homomorphism. (b) Let w E Z1(M) be an integral form. You are going to define a S1 such that f: (7?/27r) = w. For this, no smooth map f, : M generality will be lost in assuming that M is connected (why?), so make that assumption and fix a basepoint xo E M . For each x E M , choose any piecewise smooth path s : [a, b] -+ M such that s(a) = xo and s(b) = x, and show that -+

depends only on x (and xo), not on the choice of path s. (c) Prove that f, : M -+ S' is smooth and that its homotopy class is independent of the choice of basepoint xo. (d) Prove that f;(ij/2n) = w . In particular, conclude that x is surjective. (e) Let f : M S1be such that w = f *(ij/2n) is an exact form. You are to prove that f 1, SO note that, again, no generality is lost in assuming that M is connected. In this case, show that f,, as defined in step (b), is actually well defined as a map f, : M -+ R and that there is a constant c such that f = p o (f, c ) . Conclude 1, hence that x is one to one. that f (3) Show that the correspondence f H A f of Exercise (4) on page 180 induces a one to one correspondence between the set of isotopy classes in Diff(Tn) and the set of n x n unimodular integer matrices. -+

-

-

+

6.4. TOPOLOGICAL APPLICATIONS

187

(4) Let p = (a, b) E R2 and, in analogy with Example 6.2.11, define a locally exact 1-form qp on R2 \ {p) having periods exactly the set Z of all integers. If s = sl . . . s, is a piecewise smooth loop on R2 \ {p), one defines the winding number of s about p to be the integer

+ +

If s is smooth, show that the mod 2 residue class of w(s,p) agrees with the notion of winding number in Definition 3.9.18.

CHAPTER 7 Multilinear Algebra and Tensors

Smooth functions, vector fields and I-forms are tensors of fairly simple types. In order to handle higher order tensors, we will need some rather sophisticated multilinear algebra. The reader who is well grounded in the multilinear algebra of R-modules can skip ahead to Section 7.4, referring to the first three sections only as needed.

7.1. Tensor Algebra We will be working in the category M ( R ) of R-modules and R-linear maps, where R is a fixed commutative ring with unity 1. In order to study R-multilinear maps, we build a universal model of multilinear objects called the tensor algebra over R. In the typical applications in this book, R will be either the real field R or the ring Cm(M).

DEF~NITION 7.1.1. An R-module V is free if there is a subset B c V such that every nonzero element v E V can be written uniquely as a finite R-linear combination of eIements of B (terms with coefficient 0 being suppressed). The set B will be called a (free) basis of V.

If R is a field, every R module is free. Another example is the integer lattice 2Zk , a free Z-module. At the other extreme, the abelian group Z 2 , when viewed as a Z-module, is not free. A basis would have to contain 1 E Z2, but 0 E Z2 would then have infinitely many representations a . 1, a E 22. The following example will be very important. E -, M be a trivial n-plane bundle. Then r ( E ) is a free Cm(M)-module on a basis of n elements. Indeed, fix a trivialization E 2 M x Rn, let {el,. . . ,en) be the standard basis of R n , and define si E r ( E ) by the formula si(x) = (x,ei), 1 5 i 5 n. An arbitrary section s(x) = (x, f (x), . . . ,fn (x)) has the unique expression

EXAMPLE 7.1.2. Let

7r :

7. MULTILINEAR ALGEBRA

190

REMARKS.There are strong but limited analogies between vector spaces over a field and free R-modules. Here are some of the facts. (1) If V is free on the basis B , then R-linear maps cp : V W into arbitrary R-modules W correspond one to one to set maps p : B + W, the correspondence being p = cpl B. (2) If V is a free R-module, it can be shown that any two bases of V have the same cardinality, called dimR V. For example, dimz Zk = k. (3) On the other hand, there are important dissimilarities. A submodule W c V of a free R-module can fail to be free. Even when the submodule W is free, it may have no basis that extends to a basis of V. For example, the even integers 22 form a free submodule of the free Z-module Z,but neither of its two bases, (2) or (-21, extends t o a basis of Z. --+

Modules will not be assumed free unless that is explicitly stated. DEFINITION7.1.3. If Vl, V2,V3 are objects in M ( R ) , a map cp : Vl x V2 is R-bilinear if

--+

V3

are R-linear, Vvi E Vi, i = 1,2. REMARK.For fixed choices of Vl ,V2, V3 E M (R), the set of R-bilinear maps cp : Vl x V2 --+ V3 is itself an R-module under the pointwise operations. DEFINITION 7.1.4. A tensor product of R-modules Vl, V2 is an R-module Vl 8 V2, together with an R-bilinear map

with the following "universal property": given any R-module V3 and any Rbilinear map cp : Vl x V2 + V3, there is a unique R-linear map : Vl @ V2 -r V3 such that the diagram Vl

'v2

\ 1" @

v 2

v 3

commutes. Write @(v,w) = v 8 w. Thus, the R-module of R-bilinear maps Vl x V2 + V3 is canonically isomorphic to the R-module HomR(Vl @ V2, h)of R-linear maps Vl @ V2 --+ V3. THEOREM 7.1.5. Given R-modules Vl, V2 u s above, a tensor product Vl @ V2 exists and is unique up to a unique isomorphism. That is, if

7.1. TENSOR ALGEBRA

191

are two such t e n ~ o products, r there is a unique zsomorphism 0 : Vl@V2 -+ Vl6fi of R-modules such that the diagram 43

K

x v2

commutes.

PROOF.First we prove uniqueness. If 43 and are two tensor products, the universal property gives unique R-linear maps O1 and O2 making the following diagrams commute:

Then the diagram

also commutes, as does

By the universal property, we conclude that O2 o O1 = id. Similarly, O1 o O2 = id, so O1 and O2 are mutually inverse R-linear isomorphisms. Since 191 is unique, we are done.

192

7. MULTILINEAR ALGEBRA

The existence proof, though elementary, is a bit more long winded. Let W be the free R-module spanned by the set Vl x V2. The module W is just the set of all formal linear combinations

where a* E R and (vi, wi) E Vl x V2. This is an R-module under the obvious operations and each element 0 # w E W is uniquely expressed as an R-linear combination of finitely many members of the basis Vl x V2. Any linear combination with all coefficients 0 is equal to the 0 E W. Let R C W be the submodule spanned by all elements of the form (av

+ h,W) - a(v, w) - b(u, w)

where a, b E R and u, v, w are in Vl or V2 appropriately. We think of R as the submodule of bilinear relations and set

The cosets of the elements of the basis VL x V2 will be denoted by

and we define @:

v, x v2-+ v18 v2

Bilinearity follows immediately from the definition of R. For example,

Note that, as a special case of bilinearity, (av) @ w = a(v @ w) = v @ (aw) and, in particular, v @ 0 = 0 = 0 8 v. We establish the universal property. Let cp : Vl x V2 -+ V3 be an R-bilinear map. Since Vl x V2 is a free basis of W, there is a unique R-linear map

-

such that ~ ( vw) , = ~ ( uw), , V(v, w) E Vl x V2. Since rp is bilinear, it follows that cp vanishes on the generators of R, hence that plR 0. Consequently, cp passes to a well defined R-linear map

such that the diagram

7.1. TENSOR ALGEBRA

-.

commutes. Since V1@V2is spanned by elements of the form v@w, cp is unique. In a completely parallel way, one can consider R-trilinear maps and prove the existence and uniqueness of a universal R-trilinear map

sending

( v 1 , ~ 2 , ~-211 3)

@v2 @v3

It is a trivial exercise to check that the composition

(Vl x v2) x V3

Bxidv,

(Vl @VZ)x

v32 (&

@VZ)@ &

also has the universal property, as does

COROLLARY 7.1.6. If Vi is an R-module, i = 1,2,3, there are unique Rlinear isomorphisms Vl 8 (V2 @ V3) = (Vl @ V2) @ V3 = Vl @ V2 @ & identihing €3212 €9213, Vvi E V , , i = 1,2,3. vl @ ( ~ €9213) 2 = (vl @ v2) € 3 ~ = 3 More generally, for each integer k 2 2, there is a unique universal, k-linear map (over R)

vl x

X

"'Vk % & @ V 2 @ " ' @ V k

and canonical identifications

An obvious induction shows that all groupings by parentheses are equivalent, so parentheses can be dropped or used selectively as desired.

DEFINITION 7.1.7. An element v E Vl 8 . . @ Vk is decomposable if it can be written as a monomial v = v l @ . . - @ vk, for suitable elements vi E V,, 1 5 i 5 k. Otherwise, v is indecomposable. By the construction of the tensor product in the proof of Theorem 7.1.5, the decomposable elements span.

LEMMA 7.1.8. If V and W are free R-modules with respective bases A and B, then V 8 W is free with basis C = { a @ b 1 a E A, b E B).

194

7. MULTILINEAR ALGEBRA

An arbitrary element v E A @ B can be written as a linear combination of decomposables. A decomposable element v @ w can be expanded, via the multilinearity of tensor product, to a linear combination of elements of C, proving that C spans V @ W. It remains for us to show that, if PROOF.

where ai E A and bj E B, 1 5 i 5 p, 1 5 j 5 q then all c i j = d i l j . Subtracting one expression from the other, we only need to prove that

implies that all cij = 0. The bilinear functionals cp : V x W R correspond one to one to arbitrary functions f : A x B + R. The correspondence is cp * pl(A x B). Thus, the linear functionals $ : V @ W -+R also correspond one to one to these functions f : A X B + R. If ( a ,b) E A X B, let fa,b : A x B -+ R be the function taking the value 1 on (a, b) and the value 0 on every other element of A x B. to The corresponding linear functional will be denoted by $a,b. Applying equation (*), we see that all cij = 0 as desired. -+

By an obvious induction on the number of factors, this lemma generalizes to the following.

COROLLARY 7.1.9. If Vl, . . . , Vk are free R-modules on bases B1, . . . ,Bk, respectively, then Vl @ - - - @ Vk is a free R-module with basis

PROPOSITION 7.1.10. If Ai : V, a unique R-linear map

-,

W iis an R-linear map, 1 I i I k , there is

which, on decomposable elements, has the formula

Since the decomposables span, uniqueness is immediate. For existence, define the multilinear map PROOF.

by

X(v1,. . . ,uk) = Xl(vl) 8 .- - 8Xk(vk).

Then X1 @ . . . @ Xk is defined to be the unique associated linear map.

DEFINITION 7.1.11. The dual V * of an R-module V is Hom (V, R) , the module of R-linear functionals.

7.1. TENSOR ALGEBRA

195

LEMMA7.1.12. If V has a finite free basis (vl,... ,v,), then V * has a finite free basis {v;, . . . ,WE}, called the dual basis and defined by

PROPOSITION 7.1.13. There is a unique R-linear map

which, on decomposable elements, has the formula

If the R-modules V , are all free on finite bases, then L is a canonical isomorphism.

PROOF.Uniqueness is immediate by the fact that decomposables span. For existence, define the multilinear functional

by ~ ( v I , - -,-~ k , v l , .,vk) -= ~i(vi)qz(va). . .~k(vk).

By the universal property, this gives the associated linear functional

and we define L :

:v @"'@v~

(vl@"'@vk)*

If {viTl,... ,vi,mi} is a free basis of ir,, 1 5 i 5 k, let {vll,. . . ,},,:v be the ; @ - - - @ Vi dual basis. Let B and B be the respective bases of Vl@ - . - @ Vk and V given by Corollary 7.1.9. The formula

shows that L carries the basis B* one one onto the basis dual to B, so isomorphism.

L

is an

Let V be an R-module and view R as a module over itself.

LEMMA7.1.14. Scalar multiplication

induces canonical isomorphisms R 63 V = V @ R = V relative to which 1 @ v = v@l=v.

7. MULTILINEAR ALGEBRA

196

Indeed, scalar multiplication is R-bilinear, so there are canonical R-linear maps

These are inverted by the R-linear maps

respectively.

DEFINITION 7.1.15. Let V be an R-module. For each integer n 2 0, the n t h tensor power of V is

REMARK.By Lemma 7.1.14, TO(V)@ Tn(V) = Tn(V) = 7"(V) 8 To(V). When n and m are both positive, the identity I n ( V ) @ Tm(V) = 7 n + m ( V ) is given by the associativity of the tensor product. Set 7 ( V ) = {7n(V))r=o and note that 8 defines an R-bilinear map 7"(V) x 7 m ( V )

s Tn(V)

@ Tm(V) = Tn+m(v).

This makes 7 ( V ) into a graded algebra over R in the following sense.

DEFINITION 7.1.16. A graded (associative) algebra A over R is a sequence n of R-modules, toget her with R-bilinear maps (multiplication) {A 00

(written (a, b) H a b or, sometimes, (a, b) +t that the compositions

An x (Am x AT) are equal, V n, m, r 2 0.

ab) which is associative in the sense

id x .

An x Am+T

DEFINITION 7.1.17. The graded algebra A is connected if A0 = R and A0 x Am 3 Am

c A"

x A'

are equal to scalar multiplication, V m 2 0. Remark that a connected graded algebra has unity 1 E R = A'.

DEFINITION 7.1.18. If V is an R-module, then 7 ( V ) , with multiplication 8, is called the tensor algebra of V.

7.1. TENSOR ALGEBRA

197

It is clear that the tensor algebra T ( V ) is connected. DEFINITION 7.1.19. A homomorphism cp : A -+ B of graded R-algebras is a collection of R-linear maps cpn : An B n , V n 2 0, such that the diagrams -+

commute, V n, m 2 0. The homomorphism cp is an isomorphism if cpn is bijective, VnlO. THEOREM 7.1.20. If A : V --, W is an R-linear map, then there is a unique induced homomorphism T(A) : T(V) -+I ( W ) of graded R-algebras such that T0(A) = idR and I1(A) = A. This homomorphism satisfies

V n 2 2, V v i E V, 1 2 i 5 n. Finally, this induced homomorphism makes I a covariant functor from the category of R-modules and R-linear maps to the categov of graded algebras over R and graded algebra homomorphisms.

PROOF.The formula on decomposable tensors is imposed by the requirement that T(A) be a homomorphism of graded algebras, together with the stipulation that T1(A) = A. Existence and uniqueness of the linear maps Tn(A) are given by A) preserves 63 multiplication is immediate. Proposition 7.1.10. The fact that I( The final assertion amounts to the obvious identities

The following is an elementary consequence of Corollary 7.1.9 and Proposition 7.1.13. THEOREM 7.1.21. If V is a free R-module with basis { e 1, . . . ,e m ) then

is a free basis of T k ( V ) and Tk(V*) = T k ( v ) *. REMARK.In particular, if V is a free R-module with dim R V = m, then

TERMINOLOGY. The established terminology "covariant tensor" and "contravariant tensor" in geometry is inconsistent with "covariant" and "contravariant" as used in category theory. For later reference, here are the geometer's definitions. Let V be a finite dimensional vector space over the field IF.

7. MULTILINEAR ALGEBRA

198

DEFINITION 7.1.22. For each integer r 2 0, Tr(V*), viewed as the space of r-linear maps VT IF, is called the space of covariant tensors on V of degree r and is denoted T,'(V). -+

DEFINITION 7.1.23. For each integer s 2 0, T s ( V ) , viewed as the space of s-linear maps (V*)' -+ IF, is called the space of contravariant tensors on V of degree s and is denoted by 'T;(V). DEFINITION7.1.24. The space of tensors on V of type (r,s) is the tensor product T(V) =G(V)0C(V).

A tensor a

E c ( V ) is said to have covariant degree

r and contravariant degree

s.

Obviously, c ( V ) is the space of (r f s)-linear maps

EXERCISES (1) Let V be an R-module, V* its dual. (a) Exhibit a canonical R-linear map a : V * 8 V + R. (b) If V is free, prove that a is a surjection. If, in addition, V has a basis with one element, prove that a is a bijection. (c) If V and W are R-modules (not necessarily free), exhibit a canonical R-linear map P : V * @ W -,Hom (V, W ). (d) If R is a field, prove that P is injective. Do not assume that V and W are finite dimensional. is surjective if and only if either (e) If R is a field, prove that dimR V < oo or dimR W < oo. (2) Let A be a connected, graded R-algebra. (a) Show that there is a unique homomorphism

= idR and = idA1. of graded algebras such that (b) Define a suitable notion of 2sided ideal I 5 A so that A / I = ( A n / I n ) ~ = ois again a graded R-algebra. (c) If A is generated, as a graded algebra, by A', show that there is a canonical ideal I c 7(A1), with I' = (0) = 11,and a canonical isomorphism

such that yo = idR and y' = idA1.

7.2. EXTERIOR ALGEBRA

199

7.2. Exterior Algebra

Let R be any commutative ring with unity 1 such that f E R. That is, if 2 = 1 + 1 E R, then E R has the property that f . 2 = 1. In the case that R = IF is a field, this means that the characteristic of F is not 2. LEMMA7.2.1. Let V be an R-module, v E V. Then v = -v H v = 0. PROOF. Evidently, v = 0 =+v = -v. For the converse,

Let C be the group of permutations of {1,2, . . . ,k) , a group of order k! . DEFINITION 7.2.2. The sign of a E C k is (-1)O =

1, -1,

a an even permutation, a an odd permutation.

D E F I N ~ T7.2.3. ~ ~ N Let V and W be R-modules. An antisymmetric k-linear map cp : Vk + W is a k-linear map such that

Remark that this definition will be useful only because 1 # -1 in R. As in the definition of tensor product, for each k 2 2, define a universal antisymmetric k-linear map

written A(vl, ... ,Vk) = 211 A " ' A Vk. Here, the subspace 72 of relations is generated by the k-linear relations and all elements (211,

.. 7 ~ k -) ( - l ) u ( ~ o ( l ) ,. - , ~ u ( k ) ) 0, E Ck.

Existence and uniqueness, up to unique isomorphism, are established exactly as for tensor product. One sets AO(V)= R and A1(V) = V. DEFINIT~ON 7.2.4. The R-module Ak(V) is called the kth exterior power of V. The connected graded R-algebra A(V) = {A with multiplication

(~)}r=~

is called the exterior algebra of V.

200

7. MULTILINEAR ALGEBRA

As before, R-linear maps X : V + W induce canonical homomorphisms of graded algebras A(A) : A(V) -+ A(W) such that AO(A)= id^ , A' (A) = A, and

This makes A a covariant functor from the category of R-modules and R-linear maps to the category of graded algebras over R and graded algebra homomorphisms.

DEFINITION 7.2.5. An element w E A k ( v ) that can be expressed in the form vl A 712 A . - * Auk, where vi E V, 1 5 i 5 k, is said to be decomposable. Otherwise, w is indecomposable. It is clear that A k ( v ) is spanned by decomposable elements, but generally there are plenty of indecomposable elements as well. It will be useful to relate A(V) more directly to 'T(V).

DEFINITION 7.2.6. The ideal M(V) c T ( V ) is the 2sided ideal generated by all elements in T ~ ( v )of the form vl 8 v2 vz @ vl, vl, v2 E V.

+

Thus,

ak(v)= span

U

TP(V) @ a 2 ( v ) @ 7 (V)

p+q=k-2

Let cp : Vk -+ W be an antisymmetric k-linear map. As a k-linear map, cp can be interpreted as a linear map which, for clarity, we denote by (p : 7 k(V) -+ W.

LEMMA7.2.7. If cp : Vk

-+

W is antisymmetn'c, then (p(Mk(V)) = {O}.

ak(v).

PROOF. It will be enough to show that @ vanishes on a set spanning Thus, if w E 7p(V), u E Tq(V), p q = k - 2, and v l , v2 E V , we will show that

+

But the antisymmetry of cp implies that

and the assertion follows from linearity.

7.2. EXTERIOR ALGEBRA

COROLLARY 7.2.8. There is a canonical isomorphism A(V> = 7(V)/'21(V) of graded algebras.

PROOF.For k = 0,1, it is clear that Ak(V) = T k ( v ) / 2 l k ( ~ )For . each k

> 2,

consider the k-linear map

where 7r is the quotient projection. Given an arbitrary antisymmetric k-linear map cp : Vk -+ W, we obtain the commutative triangle

(v) = 0. Thus, jinduces q : Tk(v)/21k(v)--+ W making the following and @lak diagram commutative:

That is, the triangle

commutes. Since Tk(v)/ak( V) is spanned by

and commutativity of the diagrams forces

we see that q is the only linear map making the triangle commute. That is, T o 8 : Vk -+Tk(V)/21k(V)has the universal property for antisymmetric, klinear maps, hence is uniquely identified with A : V k Ak(V). Finally, this identification gives --+

so it is an identification of graded algebras.

T]

7. MULTILINEAR ALGEBRA

202

DEFINITION 7.2.9. A graded algebra A is anticommutative if a

and

E

E

AT + crp = (-l)kr@a.

COROLLARY 7.2.10. The graded algebra A ( V ) is antico~mmutative. PROOF.It is enough verify Definition 7.2.9 for decomposable elements of A k ( v ) and A r ( V ) . But that case is an elementary consequence of the case k = r = 1, and this latter case is given by VAW=V@W+%~(V) = -w @ v - -W A v ,

+ g2(v)

VV,WEV.

COROLLARY 7.2.11. If w E A2"+'(v), then w A w = 0. PROOF.Indeed, w

w = (- 1 )(2'+1)(2r+1)

/\ w = -w

A W.

By Lemma 7.2.1, w A w = 0.

COROLLARY 7.2.12. If w

E

A k ( v ) is decomposable, then w A w = 0.

For the remainder of this section, we specialize to the case in which V is a free R-module on a basis ( e l , . . . ,em).

LEMMA7.2.13. If V is as above, then {eil A ei2 A

A eik)15il
is a free basis of nk(v), k 2 2. In particular, dimR A ~ ( v = ,

(3

m!

= k ! ( m - k)!

and, if k > m, nk(v) = (0).

PROOF.The basis {ejl

@

@ e j k ) ~ < ~ l ,,jk<m .,.

' '

of Tk(v) projects to a spanning set

{ej, A

.

A ejk)l<j, ,... ,jk
of A k ( v ) . If some jr = j,, r # q, then ej, A 1 5 i 5 k , choose a E C k such that

jo(l) < ja(2) < ' ' Letting

2 , = j,(,),

- - A ejk = 0. < ja(k).

1 5 r _< k , we get ei,A

Aei, = ( - l ) u e j l A - - - A ej,.

If all ji are distinct,

7.2. EXTERIOR ALGEBRA

Thus, the set

B = {ei, A . - .A eik}lSil
whenever 1 5 j l < j2 < - - < jk j m. By antisymmetry and k-multilinearity, this uniquely determines the desired map. As a linear map, cp i,i,. ..ik : Ak (v)-t R has the property that

This proves that the spanning set B is free and provides, simultaneously, the dual basis B* = {~ili2...ik}l
We emphasize that

and, if {el, . . . ,e m } is a basis of V, then e 1 A . - A em constitutes a basis of Am ( V ) . REMARK.If X : V -+ V is linear, then det ( A ) E R is well defined. Indeed, if A, B are m x rn matrices over R that represent X relative to two choices of basis, then there is an invertible matrix P over R such that A = PBP-l, so det ( A ) = det ( B ) . LEMMA7.2.15. If X : V multiplication by det ( A ) .

-+

V is linear, then Am(X) : A m ( V ) 4 A m ( V ) is

PROOF.Relative to a basis { e l , .. . , e m ) of V , write

Then,

204

7. MULTILINEAR ALGEBRA

Any term with a repeated j index vanishes. If J = ( j1,j2,. . . ,j m ) contains no repetitions, there is a unique permutation a J E Em such that jO,(,) --r

, l
Thus,

LEMMA7.2.16. If R is a field, a set of vectors wl, . . . ,wk linearly independent i f and only i f wl r\ - A wk # 0.

E

V, k 2 2, is

PROOF.If the set is dependent, the existence of inverses in R allows us to assume, without loss of generality, that

Then,

Conversely, if the set is linearly independent, extend it to a basis by suitable choices of wk+1,. . . ,Wm E V. Then, wl A . . . A wk A - A w, is a basis of the one-dimensional space Am(V), hence is not 0. In an obvious sense, one can view A(V) and I ( V ) as graded R-modules by ignoring multiplication. It will be convenient to construct a (graded) R-linear imbedding A : A(V) L, I ( V ) . That is,

We emphasize that this will not be an imbedding of graded algebras. Define an antisymmetric, k-linear map Ak : vk-+ I k ( v )by

By the universal property of the exterior powers, we view this as a linear map

The sequence A = {Ak}o
A : A(V) -+ T(V). These definitions do not require V to be free or finitely generated. The following, however, requires both.

LEMMA7.2.17. If V is a free R-module on a finite basis, then each A k is one to one, hence A : A(V) L, I ( V ) is a canonical graded linear imbedding.

7.2. EXTERIOR ALGEBRA

205

PROOF.Let {el, . . . ,e m ) c V be a basis and consider the basis {eil A . ' A eik}lli1<...
which is part of a free basis. Then, since jl

< - - . < jk and il < . . < ik,

and the assertion follows. Since A is canonical, we will generally suppress it from the notation, writing

THEOREM 7.2.18. If V is a free R-module on a finite basis, there is a canonical isomorphism Ak(v*) = Ak(V)*. PROOF.By Lemma 7.2.17 and Theorem 7.1.21,

so Ak(v*) can be viewed as a space of k-linear maps V k -+R. We prove first that each w E Ak(v*), interpreted as w : vk + R, is antisymmetric. Indeed, let {e;, . . . ,eL) c V* be a basis and suppose that w = e:l A - A eik, 1 5 il < - .- < ik 5 m. Then,

That is, w(v1, . . . , vk) = det lei, (ve)] Thus, if

T

E

zk1

proving that w = e:, A . A e:k is antisymmetric, 1 5 i 1 < - - - < i 5 m. These monomids range over a basis of hk(v*), so every element w of this space is antisymmetric as a k-linear map w : vk + R.

7. MULTILINEAR ALGEBRA

206

Consequently, we can view w E A k ( v * )as defining a linear map

That is, w E Ak(V)* and, if

then It follows that el: A . A e:, +

hence that the map w

I+

= (ei, A

- .A ei,)*,

W carries Ak(V*) isomorphically onto Ak(v)*.

EXERCISES (1) If R is a field and dimR V = n, prove that every element of An-'(V) is decomposable. (Hint: Given 0 # a E hn-' (v), consider the linear map a A : V --+ An(V).) (2) If R is a field, dimR V = a,and 0 # a E A2(V), let r 1 be the unique integer such that the r-fold exterior power a A - . A a # 0 , but the ( r 1)-fold power = 0. This integer is called the rank of a. Show that there exists a basis {vl, . . . ,v,) of V so that

>

+

(Hint: Proceed by induction on r. For the inductive step, show that there is always a basis {wl, . . . ,w,) such that a = wl A wz a' where a' is a linear combination of terms wi A wj with 3 2 i < j 5 n.) (3) Recall the Grassmann manifold Gk(Rm)of k-planes in R m (Exercise (3) on page 149). (a) Using exterior algebra, define a canonical imbedding (of sets)

+

(Hint: Consider the decomposable elements of A (Rm)). (b) Exhibit a natural linear (hence, smooth) group action

(c) Using the above, exhibit a smooth action

A ek) E Gl(Ak(Rm))and show that, (d) Let xo = span{el A e l A relative to the above action, the isotropy group of x is Gl(m),, = Gl(k, m - k). (e) Show that ip (Gk(Rm)) = Gl(m) - xo, hence Gk(Rm) is expressed as the homogeneous space Gl(m)/ Gl(k, m - k).

7.3. SYMMETRIC ALGEBRA

(f) Finally, restrict the above action to the orthogonal group

and prove that this has exactly the same orbits as Gl(m). This gives Gk(Rm) as a homogeneous space of O(m). (4) If V is a free R-module on a finite basis and u E V, define the interior product i, : Ak(v*) -+ Ak-I (v*) as follows. Viewing w E Ak(V*) as an antisymmetric k-linear map

let i,(w) be the antisymmetric (k- 1)- linear map defined by the formula

If w E Ap(V*) and q E Aq(V*), prove that i,(w A q) = i, (w) A q

+ ( - l ) p ~A i, (q).

7.3. Symmetric Algebra

This will be an abbreviated treatment, not because the subject is unimportant, but because the ideas and prook are so analogous to those for exterior algebra. There are, however, notable differences. Again, our initial hypothesis is that V is a module over a commutative ring R with unity. DEFINITION 7.3.1. A k-linear map cp : V k --+ W is sy~nmetricif, for each a E Ck, v(vo(l),... , ~ o ( k ) = ) cp(~l,... , ~ k ) , vvl, ... , U k E V. In the usual way, we build a universal, symmetric, k-linear map

usually written with the dots suppressed:

DEFINITION 7.3.2. The space s k ( v ) is called the kth symmetric power of V, = R and S 1 ( v ) = V. The connected, graded algebra where, as usual, SO(v) S(V) = {sk ( v ) } ~ , ,with multiplication , is called the symmetric algebra of "a"

v.

We define the graded, 2sided ideal B(V)

w @ W -U2 and obtain

c T ( V ) , generated by all

€ 9 ~ 1E I~(V)

7. MULTILINEAR ALGEBRA

208

PROPOSITION 7.3.3. There is a canonical isomorphism

of graded algebras.

Remark that, if a E SP(V) and ,B f SQ (V), then

If X : V + W is R-linear, there is induced a homomorphism

of graded algebras such that

Once again, S is a covariant functor. Specializing to the case in which V is a free R-module on a finite basis, we obtain

LEMMA7.3.4. If { e l , ... , e m ) is a basis of V, then {eilei, ' ' ' e i k ) ~ ~ i l ~ i ~ ~ . . . ~ i ~ < m is a basis of S k ( v ) , k > 2.

If the space V is nontrivial, so is S k ( V ) , Vk 2 0, in strong contradistinction to the fact that h k ( v ) = {0), Vk > m. REMARK.

As usual, when V is free on a finite basis, s*(v*) = S k ( v ) *and this is the space of symmetric k-linear functionals on V (Exercise (1)).

DEFINITION 7.3.5. Let V be a finite dimensional vector space over a field IF. A function f : V -,F is a homogeneous polynomial of degree k on V if, relative to some (hence, every) basis { e l , . . . , e,) of V ,

is a homogeneous polynomial of degree k in the variables x 1 , . . . , x,. The vector space of all homogeneous polynomials of degree k on V will be denoted P k(V). In Exercise (2), you will show that the space of homogeneous polynomials of degree k on V is exactly Sk(v*).

EXERCISES (1) If V is free on a finite basis, find a canonical inclusion S(V)

-

I ( V ) of V k 2. graded R-modules and use this to prove that s*(v*) = Sk(v)*, Show that, if { e l , . . . , e m ) is a basis of V, this identifies eTl e:2 . . . e:k = (eilei2- - e i k ) * .

>

7.4. MULTILINEAR BUNDLE T H E O R Y

(2) For all k 2. 0,establish a canonical isomorphism of vector spaces. For the case k = 2, construct 8-' explicitly. (p2(v)is called the space of quadratic forms on V and the process 8-I of recovering the symmetric bilinear form from its associated quadratic form is called polarization.)

7.4. Multilinear Bundle Theory Just as the linear construction of dualizing a vector space passes to the construct ion of dualizing a vect or bundle, so the multilinear constructions of the previous three sections pass to corresponding constructions on vector bundles. Let ?ri : Ei + M be a ki-plane bundle, 1 5 i 5 m. We want to define a bundle El @ . . . @ Em-+ M with fiber over x E M canonically equal to El, 8.. @Em,.The fiber dimension will be k1k2 - - - km. As a set, the total space of this bundle will be +

and ?r will send El, @ ..-@ Em, to x, Vx E M. In order to put a topology and smooth structure on this set, choose an open cover {U,),En of M by neighborhoods that trivialize Ei, 1 i m. For each cu € a, let the corresponding trivializat ions be

< <

15 i

< m.

For each x E U,, we get a linear isomorphism

hence a commutative diagram

$A

.

$zx.

where the bijection $, is given fiberwise by 8 --@ In the usual way, we use $, to transfer the topology and smooth structure of U, x ~ ~ l ~ 2 "to' El. @ . . . @ E m , = (El O . . . O Em)IUaIn order to show that the two smooth structures on

uXEu,

(El @ . . O Em) ((U,n Up)

~

m

7. MULTILINEAR ALGEBRA

210

agree, we use the cocycles { ~ i p ) , p ~ for a E, associated to the above local trivializations, 1 5 i 5 m. For each x E Ua n Up, define

We identify ~

.

~ 8k. l. 8 ~

k

= m ~klk2"'krn

by lexicographic order on the basis

where {ei , . . . ,e;, } is the standard basis of IRkj , 1 j j j m. It is clear that (x) E Gl(klk2 . . . km), V x E Ua n Up.

LEMMA7.4.1. The map

is smooth and y = {yap)a,pEn is a cocycle.

PROOF.The cocycle property is rather obvious. For smoothness, it is enough to show that the vector-valued function

is smooth on Ua n UB. But

and the real-valued functions a: (x) are smooth. Expanding

we obtain a linear combination of the elements of the basis B with smooth functions of x as coefficients. [7 This structure cocycle provides a well defined topology and differentiable structure for the tensor bundle. If { ~ u T ) u , , E e is defined similarly via local trivializations {w,,06)uEB for Ei, 1 5 i j m, it is an easy exercise to show that ('YaP)a,PEa and {TuT)u,,Ee are cohomologous, so the smooth structure on El 8 - . - @ Em defined by the one cocycle is the same as the one defined by the other. In particular, given an n-plane bundle n : E -t M , we can form the tensor powers n : T*(E) + M, the fiber over x E M being, canonically, T*(E,). If E has an associated cocycle then T*(E) has {Tk(yap)},,pEa as an associated cocycle.

7.4. MULTILINEAR BUNDLE THEORY

211

LEMMA7.4.2. The oth tensor power 1 0 ( E ) is canonically isomorphic to the trivial bundle M x W.

PROOF.Indeed, for each x E M, 1 0 ( E x ) = R and, if x 7°(7ap(x)) = id^.

f U,

n UO, then

The following is proven in the same way.

LEMMA7.4.3. The 1" tensor power I 1 ( E ) is canonically isomorphic to E . We denote by s : I ( E ) --, M the collection {s : I * ( E ) + M ) p-o and interpret this system as a "bundle" of graded R-algebras over M. In complete analogy with these construct ions, we form the exterior powers

of the bundle E over M and the "bundle"

of exterior B-algebras. Again, AO(E)= M x B and A'(E) = E . Finally, one forms the symmetric powers S (E) and the bundle S(E) of symmetric algebras, noting the identities S O ( E ) = M x W and S 1 ( E )= E. By identities proven in the previous sections, we obtain the following.

*

LEMMA7.4.4. If E is a vector bundle over M , then

>

canonically for each integer k 0 . Of particular note are the tensor bundles

of covariant degree r and contravariant degree s. Similarly, one can define bundles of linear homomorphisms. If E and F are vector bundles over M , we construct a vector bundle Hom(E, F) whose fiber over each x f M is the vector space HornR(E, ,F,). Since this vector space is canonically equal to E; 8 F, (Exercise (1) on page 198), we define

Likewise, the bundle of k-linear maps (Ex)* + Fx,V x f M , can be defined as

and its antisymmetric and symmetric cousins are

212

7. MULTILINEAR ALGEBRA

respectively. Another example of such a construction is the "Whitney sum" E $ F, the fiberwise direct sum of two vector bundles (Exercise (2)). PROPOSITION 7.4.5. If El, E2 and F are vector bundles over a manifold M , there is a canonical bundle isomorphism (El @ E2) €3 F = (El €3 F) @ (E2 @ F). The proof will be Exercise (3).

REMARK.There is a general philosophical principal at work here. If one views vector spaces as vector bundles over a point, then all "natural" linear and multilinear constructions for combining vector spaces to get new ones extend to analogous operations, fiber by fiber, on vector bundles. Here "natural" means that the constructions can be carried out without reference to choices of bases. Such constructions are also "canonical". Similarly, natural relations between vector spaces, such as the relation

extend to analogous relations between vector bundles.

EXERCISES (1) Let dim M = n 2 2. Prove that M is orient able if and only if A (T*(M)) admits a nowhere 0 section. (2) Recall the operation of direct sum Vl $ V2 of vector spaces. If El and Ez are vector bundles over M, show how to define a bundle El $ E2 over M with fibers (El @ E2), = El ,$ E2,, Vx E M. (3) If Vl, V2, and W are finite dimensional vector spaces, construct a natural bilinear map

hence a natural linear map

Prove that 8 is a linear isomorphism. Use this to prove Proposition 7.4.5.

7.5. The Module of Sections We are going to view the set of all vector bundles over a fixed manifold M as the objects of a category V M . For this, we need to define the morphisms of the category. Let

be vector bundles (of possibly differing fiber dimensions)

7.5. MODULE OF SECTIONS

213

DEFINIT~ON 7.5.1. A homomorphism of the n-plane bundle E to the m-plane bundle F is a commutative diagram

M-M id

where cp is smooth and, for each x E M, cp, = cplE, : Ex 4 Fz is linear. We denote by HOM(E, F ) the set of all bundle homomorphisms from E to F.

EXAMPLE 7.5.2. The canonical fiberwise inclusions

(Sections 7.2 and 7.3) assemble to give bundle monomorphisms

the smoothness of these maps being easily checked via local trivializations. It is clear that the composition of bundle homomorphisms, whenever defined, is a bundle homomorphism and that idE : E -+ E is a bundle homomorphism, so VM is a category with bundle homomorphisms as its morphisms. To each object E E V M we associate a CM(M)-module r ( E ) , the space of smooth sections of E. If cp : E + F is a bundle homomorphiim, there is induced a CM(M)-linear map cp, : r ( E )+ r ( F ) , defined by

It is obvious that (idE)* = idqE) and that (@ o cp), = @, o cp,, so r is a covariant functor r : + M(cM(M)).

v,

(The squiggly arrow

"+"is

commonly used for functors.)

LEMMA7.5.3. If E, F E V M , then there is

a canonical identzfication

HOM(E, F ) = r(Hom(E, F ) ) = r(E*8 F ) . In particular, the set HOM(E, F) of homomorphisms of the bundle E to the bundle F is naturally a CM(M)-module. The proof is fairly straightforward (Exercise (1)). The functor r transforms the R-multilinear bundle construct ions of Section 7.4 into t he corresponding CbO(M)-mult ilinear module const ructions of Sections 7.1, 7.2 and 7.3. A fairly easy case in point is the following.

214

7. MULTILINEAR ALGEBRA

PROPOSITION 7.5.4. I ' E is a vector bundle over M , then there is a canonical isomorphism r ( E * ) = r(E)* of C" (M)-modules. Indeed, Proposition 6.1.16 was a particular case of this proposition and the proof for the general case remains the same. Consider the tensor product T' (E) @ r (F) of C M(M)-modules. Since these are also vector spaces over R, this notation can be ambiguous. In order to avoid such ambiguites the tensor product of R-modules A and B is often written A €3 R B. For the time being, we will denote this CM(M)-module by r ( E ) @ c m ( M ) r(F) and its decomposable elements by s @ c m ( M ) o. The vector space tensor product will be r ( E ) Brwr ( F ) and its decomposables will be s € 3 a. ~ Since one seldom thinks of the set of sections as a real vector space, we will ultimately drop the subscript Cm(M), but retain Bw for the vector space case. Given s f r ( E ) and a E r(F),one produces a(s,a ) f I'(E €3 F ) by setting We will write a ( s ,a ) = s a,the pointwise tensor product of sections. It is not hard to check that this defines a smooth section of E €3 F and that

is CM(M)-bilinear. Denote also by a the associated Cm(M)-1'inear map

We emphasize that s @cm D and s €3 a = a ( s a ) are conceptually distinct . The following theorem asserts that this conceptual distinction can safely be disregarded. THEOREM 7.5.5. The Coo(M)-linear map a is a canonical isomo~phismof C" ( M )-modules r ( E ) €3cm(~) r ( F ) = r(E€3 F).

COROLLARY 7.5.6. There are canonical isomorphisms of C "(M)-modules

PROOF.Indeed, the first of these identities is an immediate consequence of Theorem 7.5.5. There are canonical inclusions

the second of which comes from the bundle inclusion of Example 7.5.2. The images of these inclusions correspond perfectly under the identification 7 *(I'(E)) = r ( T k ( ~ ) proving ), the second identity. The third has exactly the same proof as the second.

7.5. MODULE OF SECTIONS

215

Combining this corollary with Proposition 7.5.4 gives COROLLARY 7.5.7. Them is a canonical isomorphism T{(F(E)) = r(T,'(E)) of C" (M) -modules. EXAMPLE 7.5.8. There are many other natural identifications now available. For example, (Tk( ( E ) ) ) * = ( ( T k ( ~ ) ) ) * by Corollary 7.5.6 = r(Tk(E)*) by Proposition 7.5.4 = I'(Tk(~*))

by Lemma 7.4.4

= ~*(I'(E*))

by Corollary 7.5.6

= I~((~(E)*)

by Proposition 7.5.4,

and there are similar identities for A and S k . 7.5.9. A Riemannian metric (-,-) on a manifold M is a smooth EXAMPLE section of S2(T*(M)).In local coordinates, the metric has the formula

By the above, the metric can ailso be thought of as an element of

We begin the proof of Theorem 7.5.5. It is surprisingly delicate.

LEMMA7.5.10. If both F and E am trivial bundles, then a is an isomorphism of C" (M)-modules. PROOF. In this case, choose global sections {al,. . . ,an}of E and {rr, . . . , 7,) of F which trivialize these bundles. These are b e bases of the respective C" (M)-modules r ( E ) and r ( F ) (Example 7.1.2), so

is a free basis of r(E)@

~ m (r (~F ))

(Corollary 7.1.9). The set

of pointwise tensor products of sections trivializes the bundle E @ F, hence is a free basis of I'(E @ F). Since a(ai @ C W ( M ) rj) = ai @ T j , for all relevant indices, we see that a is an isomorphism of C "(M)-modules. In light of this lemma, we will reduce the general case of Theorem 7.5.5 to the case in which both bundles are trivial. The key to this is the following. THEOREM 7.5.11. Given a vector bundle E over M, there exists a vector bundle E' over M such that the bundle E $ E' is trivial.

7. MULTILINEAR ALGEBRA

216

For the moment, we accept this. Let El and E2 be vector bundles over M and define bundle homomorphisms

It is clear that p o L = idE1, SO the functoriality of

r implies the following.

LEMMA7.5.12. The composition p, o L, is equal to idr(ElI. In particular,

is injective and p* : ~ ( E@I E2)

-*

~ ( E I )

is surjective.

PROOFOF THEOREM 7.5.5. By Proposition 7.4.5, ( E @E I)@ ( F @ F I ) splits off a direct summand E @ F . Consider the commutative diagram

By Lemma 7.5.10, the top horizontal arrow is an isomorphism of Cw(M)modules. The above lemma guarantees that the leftmost vertical arrow is injective. Since

the rightmost vertical arrow is also injective. It follows that

is injective. Similarly, the diagram

r ( ( E CB E') O ( F $ F')) 4 r ( E @ E')

@CM(M)r ( F CB F I )

commutes and the vertical arrows are surjective, implying that

is surjective. U

7.5. MODULE OF SECTIONS

217

Everything now hinges on Theorem 7.5.11. We prove the case in which M is compact and then quote a theorem from dimension theory that extends this proof to the noncompact case. Suppose that E is an n-plane bundle over a compact manifold M . Compactness of M will be used only to find a finite open cover {Ui)T=l such that E 1 Ui is trivial, 1 5 i 5 r . Let {Xi):=, be a subordinate partition of unity and, for each i = 1,2,. . . , r, let s t , . . . , sq E r ( E I U i ) be everywhere linearly independent (hence a basis at each x E Ui). Let a! E r ( E ) be the extension by 0 of Xis!, 1 5 i 5 r , 1 j 5 n. Then, for each x E M, { o i ( ~ ) } : , ' spans ~ . ~ E x . View r ( E ) as a vector space over R and consider the finite dimensional subspace

<

Then pl : M x V homomorphism

-+

M is a trivial vector bundle. We define a surjective p:MxV+E

of vector bundles by setting p(x, 0 ) = a ( x ) The smoothness of p is elementary, as is the fact that it carries { x ) x V linearly onto the fiber Ex, V x E M. Let E: c { x ) x V be the kernel of this linear surjection and set

The fact that E L is a vector subbundle of M x V is a case of the following result.

LEMMA7.5.13. I f p : F -,E is a surjective homomorphism of vector bundles over M , then E' = ker(p,)

u

xE M

is a subbundle of F .

PROOF.TO prove this, it is enough to produce local trivializations. For this, let x E M and choose vectors vl, . . . ,v, E F, that are carried by p, one to one onto a basis of E x . Extend these to a basis {vl,. . . , v,, v,+l,. . . ,Ztn) of F,. By the local triviality of F, there is a neighborhood U of x in M and sections a , of FIU such that a i ( x ) = vi, 1 < i 5 n, and such that { a i ( y ) ] Z , is a basis of F,, V y E U . Consider the sections si = p o ai of E I U.Taking U smaller, if necessary, and appealing to the continuity of p, we arrange that { ~ ~ ( y ) ) is : =a ~basis of E,, V y E U . Then there are unique expressions

7. MULTILINEAR ALGEBRA

218

where the coefficient functions f ; are d l smooth. Consider the smooth sections

It should be clear that these give a basis of ker(p,), V y E U, hence define a local trivialization of E' over U. Fix a positive definite inner product on V, viewing it as a Riemannian metric on the bundle M x V. For each x E M , let c {x) x V be the subspace orthogonal to We claim that the set

Ex

e.

is a subbundle of M x V. More generally, LEMMA7.5.14. If F E is a vector subbundle and i f there is given a Riemannian metric on E, then the subset E , fiberwise perpendicular to F , is a subbundle. PROOF.Again, local triviality is all that needs to be proven. There are sections 01, . . . ,a,, a,+l, . . . ,anof EIU, trivializing that bundle, where U is a neighborhood of an arbitrary point of M. These can be chosen so that a 1, . . . ,a, are sections of FIU which trivialize that bundle. An application of Gram-Schmidt turns these into fiberwise orthonormal sections sl,. . . ,s,, s,+l,. . . , sn with-the same properties. It follows that sr+l,. . . ,s, are trivializing sections of FIU, proving that is a subbundle of E as desired. The bundle homomorphism p ( E : fiber by fiber, so

+E

is an isomorphism, this being true

MXV=E~$E?E~CBE. This completes the proof of Theorem 7.5.11 in the case that M is a compact manifold. The compactness assumption on M is removed by showing that, whether M is compact or not, there is always a finite trivializing cover {Ui)L=l for E. This will follow from a theorem in dimension theory. THEOREM 7.5.15. Let M be a manifold of dimension r. Then every open cover V of M admits a refinement W = {W,),Em such that, whenever the indices ai E 2l, 1 5 i _< r 2, are all diistinct, then

+

For the proof, see [20, Theorem V.8, page 671, together with Example 111.4 on page 25 of that same reference.

7.5. MODULE OF SECTIONS

219

THEOREM 7.5.16. Let M be a manifold of dimension r and let E be a vector bundle over M . Then there is an open cover {u~);:: such that EIUk is trivial, l
us = {x f M I min A, aES

(x) > max Ap(x)). PE%\S

Let IS1 denote the cardinality of S. The following are elementary: (i) (ii) (iii) (iu)

c

{Us I S 2l is finite) is an open cover of M; if S1 # S2 are finite subsets of IZL with IS1( = IS21, then Us, n Us, = 0; IS1 > r 1 + Us = 0; Us W,, vcu E S.

c

+

For each integer k 2 1, set

By (i), {Uk)P=, is an open cover of M. For each k 2 1, El Uk is trivial ((ii) and (iv)) and, by (iii), Uk = 0 for k > r + 1 . [7 The proof of Theorem 7.5.5 is now complete. DEFINITION 7.5.17. The space of covariant tensors of degree k on M is

The of M.

h e b r a I * ( M ) = { ' T * ( M ) } ~is~called the covariant tensor algebra

REMARK.If I* (M) is viewed as r ( I ( T *(M))), the multiplication is by pointwise tensor product of sections. If it is viewed as 'T(r(T* (M))), the multiplication is just that of the tensor algebra of the Cm(M)-module r(T*(M)). By the proof of Theorem 7.5.5, the two graded algebra structures agree. DEFINITION 7.5.18. The space of contravariant tensors of degree k on M is

and Z ( M ) = {?ck(M))p=ois called the contravariant tensor algebra of M. DEFINITION 7.5.19. The space of (mixed) tensors of type (r, s) on M is

7. MULTILINEAR ALGEBRA

220

DEFINITION 7.5.20.The space of k-forms on M is The exterior algebra A*(M) = r(A(T*(M)))= A(I'(T*(M))) = h(A1(M)) is called the Grassmann algebra of M.

DEFINITION 7.5.21.The space of (covariant) symmetric tensors on M is S k ( M ) = r(Sk (T*(M))) = S* (I'(T* (M))) = S*(A1(M)). The graded algebra S*(M) = I'(S(T*(M))) = S ( r ( T *(M))) = s(A'(M)) is called the symmetric algebra of M.

Remark that the graded algebras I* (M), I,(M), A* (M), and S*( M ) are all connected and that Z (M) = X ( M ) and T1(M) = A1 (M) = S1(M).

EXERCISES (1) Prove Lemma 7.5.3. (2) Let M be an oriented n-manifold, let (U, x 1 , . . . ,x,) and (V, yl, . . . ,y,) be coordinate neighborhoods in M respecting the orientation, and let w E An(M) be such that supp(w) is a compact subset of U n V. Let

be the respective formulas for w in these coordinate systems. Finally, let g(xl,.. . ,x,) = (yl,. . . ,yn) be the formula for the change of coordinates. (a) Show that, on U f l V, f = (hog)det(Jg). (b) Show how to define the integral JM w E R and prove that your definition is independent of choices. (c) Denote the oppositely oriented M by -M and show that

CHAPTER 8 Integration of Forms and de Rham Cohomology

In Chapter 6, we studied the first de Rharn cohomology H ' ( M ) of a manifold. This measures the difference between exactness and local exactness of 1-forms on M and was shown to have interesting topological applications. Here we generalize these ideas, using the full Grassmann algebra A * ( M ) to produce a graded algebra H * ( M ) ,the de Rham cohomology algebra. The proper generalization of "locally exact 1-form" is "closed p-form", defined as a p-form that is annihilated by "exterior differentiation". Exact forms are closed and H P ( M ) measures the extent to which closed p-forms may fail to be exact. By Stokes' theorem, the geometric boundary operator and exterior differentiation of forms are mutually adjoint operations in a certain precise sense. This is a generalization of the fundamental theorem of calculus and a powerful tool for computing cohomology. The reader who would like to pursue this theory further could hardly do better than to consult [5]. 8.1. The Exterior Derivative Rn be an open subset. Since A O ( u )= CDO(U),we have already Let U defined the exterior derivative

(Lemma 6.1.19). For p 2 1, we can define the exterior derivative

d : AP((I)+ Ap+l (u) by the following formula:

It is clear that this operator is R-linear. It is not clear, but will be proven shortly, that the definition is invariant under changes of coordinates.

8. INTEGRATION AND COHOMOLOGY

222

LEMMA8.1.1. If V

U is open, then (dw)lV = d(w)V).

LEMMA8.1.2. The composition

is trivial ( d 2 = 0).

PROOF.By the antisyrnmet ry of exterior multiplication, the above formula for d gives the same answer whether or not the indices are in increasing order or even are distinct. Thus

and this vanishes by the equality of mixed partials and the antisymmetry of exterior multiplication.

REMARK.The equation d2 = 0 is equivalent to the equality of mixed partials which, in turn, is equivalent to [a/axk , ,alaxj] = 0, the commutativity of coordinate fields. The proof of the following is mechanical, hence is left to the reader.

LEMMA8.1.3. If w E AP(U) and q f Aq(U), then d(w A q ) = (dw) A q

+ ( - l ) P ~A dq.

REMARK.In particular, writing f q for f A q when f E A O(U) = C" (U), we get If = dfi A , . A d f p , where fi E AO(U),1 5 i 5 p, then repeated use of the above two lemmas yields d q = 0 and

COROLLARY 8.1.4. If U C Rn and V is smooth, then

d for a11 p

> 0.

Rm are open subsets, and p : U

Y* = p* o d : AP(V)+ AP+'(U),

-+ V

8.1. EXTERIOR DERIVATIVE

223

PROOF. In the following computation, the third equality is by the above remark: d(cp*( f dyZ1A

- - A dyip)) = d(cp*( f )cp* ( d y i l )A . . A cp* (dy'p)) = d(cp*(f d(cp*(yil)) A . . A d(cp*( y a p ) ) ) = d(cp*( f )) A d(cp*( y " ) ) A - - - A d(cp*(y")) = cp* ( d f ) A cp* (dyi') A = cp* (df A dy" A

. . A cp* (dyip) *

- . - A dy")

= cp*(d(f dydl A

A dyip)).

Since every q E AP(V) is a sum of forms of the type used in the above computation, the claim follows. We want to extend the exterior derivative to an R-linear operator

on all manifolds M and for all nonnegative integers p. We take an axiomatic approach, requiring that this operator satisfy the following: ( 1 ) f E A O ( M )and x E X ( M ) + d f ( x ) = x(f). ( 2 ) w E AP(M) and 77 E A Q ( M )+ d(w A 77) = dw A 77 (-1)pw A dq. (3) d2 = 0. ( 4 ) U c M open and w E A p ( M ) + (dw)tU = d(w1U). (5) cp:M-+Nsmooth~cp*od=docp*.

+

REMARK.If one only considers manifolds that are open subsets of Euclidean spaces, then all of the axioms hold for the exterior derivative as already defined. For d : A O ( M )+ A 1 ( M ) ,which has been defined on all manifolds, we have seen the truth of the first axiom (Lemma 6.1.2). DEFINITION 8.1.5. If an W-linear operator d : A P ( M ) -+ Ap+l(M), defined for all smooth manifolds M and all integers p 0, satisfies the above axioms, it is called an exterior derivative.

>

THEOREM 8.1.6. There is a unique exterior derivative. PROOF. First we prove uniqueness. By Axiom (4), it will be enough to show that, whenever U M is a coordinate neighborhood and w E A p ( M ) , the form d(w/U)f A ~ + ' ( u )is uniquely determined. Let cp : U + Rn be a diffeomorphism onto an open subset V and set cpi = cp*(x') = x' o cp, 1 5 i < - n. Also, by functoriality, (cp-')* = (cp*)-.l, which is to say that cp* : A * ( V ) + A * ( U ) is an isomorphism. Thus, since dcp" cp*(dxi) (Axiom ( 5 ) ) , the set

(U)-module AP(U). Thus, is a free basis of the Coo

224

8. INTEGRATION AND COHOMOLOGY

and

where the second equality uses Axioms (2) and ( 3 ) . But dfi,...ipis uniquely specified by Axiom (1). Thus, d(wlU) is unique and, as remarked above, the uniqueness in general of the operator d follows. Remark that all five axioms have been used in this argument. We turn to the proof of existence. Let

be the maximal coordinate atlas for M . The diagrams

make sense and commute, V a, p E a, since everything is written for open subsets of Euclidean space. For w E AP(M), define

If U, n Up# 0, we check that the two definitions agree on U, n Up.Indeed, by the commutative diagram,

Thus, the local definitions of dw piece together to give dw E Ap+'(M). Since the axioms are true for open subsets of Euclidean space, they are true locally on M , hence globally.

>

DEFIN~TION 8.1.7. A form w E AP(M), p 0, such that dw = 0 is called a closed p-form on M . Closed p-forms are also called p-cocycles and the real vector space of all such forms is denoted by Z P ( M ) . We set Z* ( M ) = { Z P ( M ) } g o . DEFINITION 8.1.8. A form w E AP(M), p > 1, is said to be exact if there is a form Q E A p - l ( M ) such that dq = w. Exact p-forms are also called pcoboundaries and the real vector space of all such forms is denoted B p ( M ) . For p = 0, we define BO(M) = { 0 ) c A'(M). Finally, B * ( M )= { B P ( M ) } E o .

8.1. EXTERIOR DERIVATIVE

225

REMARK.A form w E AP(M) is locally exact if every point x E M has an open neighborhood U such that wlU E Bp(U). One version of the Poincark Lemma (Section 8.3) asserts that the set of locally exact p-forms on M is precisely Zp(M). In particular, our earlier definition of Z (M) (Definition 6.3.1) agrees with our present one. In anticipation of the Poincarb lemma, you will be invited to prove a special case of this last assertion in Exercise (1). The formula d2 = 0 is equivalent to the inclusion BP(M) C ZP(M), p 0.

>

>

DEFINITION 8.1.9. For each integer p 0, the pth (de Rham) cohomology space of M is the real vector space HP(M) = ZP(M)/BP(M).

If cp : M -+ N is smooth, the formula cp* o d

=d o

cp* implies that

so cp* induces an R-linear map cp* : HP(N) + HP(M).

As usual, we have functoriality. LEMMA8.1.10. The pth cohomology HP is a contravariant functor from the category of diflerentiable manifolds and smooth maps to the category of real vector spaces and linear maps. If w E ZP(M) and q E Zq(M), then

That is, the exterior product of a closed p-form with a closed q-form is a closed ( p + 9)-form. Thus, Z*(M) is a graded algebra over R under exterior multiplication. Both A* (M) and Z*(M) are anticommutative in the sense of Definition 7.2.9. LEMMA8.1.11. The graded R-algebra Z*(M) is connected i f and only i f M is a connected manifold. PROOF.The space ZO(M)consists of all f E C m ( M ) such that df = 0. That is, ZO(M) is the space of locally constant, real-valued functions on M. Identifying R with the space of const ant functions in C "(M), we have R & zO(M). The product in Z*(M) of a constant function and a form becomes naturally identified with scalar multiplication. But locally constant functions are all constant if and only if M is connected. COROLLARY 8.1.12. The space H O ( M ) is one-dimensional i f and only i f M is connected. I n this case, HO(M) = W canonically. Generally, HO(M)is a direct product of copies of R, one for each component of M. PROOF. Indeed, HO(M) = zO(M)/BO(M) = ZO(M),the space of locally constant functions, and all claims follow easily. LEMMA8.1.13. If dim M = n, then HP(M) = 0 , V p > n.

226

PROOF.

8. INTEGRATION AND COHOMOLOGY

Indeed, Ap(M) = 0 for all integers p greater than the dimension of

M.

LEMMA8.1.14. The graded subspace B*(M) Z*(M) is a 2-sided ideal, hence H * ( M ) = Z*(M)/B*(M) is a graded, untiwmmutative algebra over the field R. PROOF.If w E ZP(M) and Ap-'(M), hence

Q

f BQ(M),q

> 1, then 77 = d a for some a E

Since A w = ( - l ) P q ~ ~ 7 ] ,it follows that B*(M) is a 2sided ideal in Z*(M).

[7

Since smooth maps cp : M + N preserve exterior multiplication and pass to well defined maps in cohomology, we see that

is a homomorphism of graded algebras. We have completed the proof of the following.

THEOREM 8.1.15. The graded cohomology construction defines a contruvariant functor H* from the category of diflerentiable manifolds and smooth maps to the category of unticommutative graded algebras over R and graded algebra homomorphisms. The graded algebra H*(M) is connected if and only i f M is connected. The graded algebra H*(M) is called the (de Rharn) cohomology algebra of M. Whether or not it is connected, H '(M) has a unity, namely the constant function 1 E ZO(M)= H'(M).

DEFINITION 8.1.16. The space of compactly supported p-forms on M is denoted by AE(M). It is clear that the exterior product of two compactly supported forms is compactly supported. Indeed, it is enough that one of them be compactly supported. Thus, each Af(M) is a module over Coo(M) and these assemble into a graded algebra AZ(M) over Cm(M). It is also a graded algebra over R, which is more to our purpose. Furthermore,

so one can define the space

and the vector subspace

8.1. EXTERIOR DERIVATIVE

227

If we use the common convention that A-'(M) = A;l(M) = 0, the above definition of BZ(M) includes the case p = 0. As before, Z,*(M) is a graded subalgebra of A;fi(M) and B,* (M) Z,*(M) is a 2sided ideal.

DEFINITION 8.1.17. The (de Rharn) cohomology algebra with compact s u p ports is H,*(M)= Z,*(M)/B,*(M). Remark that H*(M) = H,*(M) if and only if M is compact. At the other extreme, if M has no compact component, the space z ~ ( M ) of compactly s u p ported, locally constant functions on M is trivial, so Hz(M) = 0. In any event, each element of Z:(M) will vanish on all but finitely many components of M. These observations establish the following.

LEMMA8.1.18. The vector space H:(M)

is isomorphic to a direct sum of copies of R, one for each compact component of M . Note the different conclusions in Lemma 8.1.18 and Corollary 8.1.12. Each element of a direct sum has terms from only finitely many summands, while elements of a direct product are allowed to have terms from infinitely many of the factors.

REMARK.The graded algebra H,*(M) generally does not have a unity unless M is compact. DEFINITION 8.1.19. A smooth map cp : M set C N , the set cp-l(C) is also compact.

+N

is proper if, for each compact

For example, id : M + M is always proper. If M is compact, cp is always proper. In any event, the composition of proper maps is proper, so the class of differentiable manifolds and smooth, proper maps between them is a category. If cp : M + N is proper and if w E A!(N), then cp* (w) E Az(M). As usual, cp* o d = d o cp*, so we get an induced homomorphism of graded algebras

THEOREM 8.1.20. Cohomology with compact supports is a contmvariant functor H,* from the category of diflerentiable manifolds and smooth, proper maps to the category of anticommutative graded algebras over R and graded algebra homomorphisms.

EXERCISES (1) Define a manifold M to be contractible if it is homotopy equivalent to a point. Prove that the following three versions of the Poincar6 Lemma for 1-forms are equivalent, and verify (c) when n = 2. (Here, Z '(M) denotes the space of closed 1-forms, not the space of locally exact ones.) (a) The form w E A1(M) is locally exact if and only if it is closed. (b) If M is contractible, then z'(M) = B'(M). (c) Z I ( I W= ~B ) ~(R~)(2) Prove that HE (R) S R. This is the one-dimensional case of the Poincar6 Lemma for compactly supported cohomology.

228

8. INTEGRATION AND COHOMOLOGY

8.2. Stokes' Theorem and Singular Homology In this section, we define integration of forms and give a detailed treatment of two versions of Stokes' theorem. As an application of the second (combinatorial) version, we define the singular homology of a manifold and relate it to de Rham cohomology, stating the celebrated de Rham theorem. A detailed sketch of the proof of this theorem will appear in Section 8.9. Throughout what follows, we assume that dim M = n and that M is oriented. We also allow d M # 8. THEOREM8.2.1. For each oriented n-manifold M , there is a unique R-linear functional

called the integral and having the following property: i f (U, cp) is an orientationrespecting coordinate chart, iif w E AF(M) has supp(w) c U, and if cp-'*(w) = dxl A - A dxn E A: (p(U)), then

g JM

(the Riemann integral).

= J,,u,

PROOF.First we prove uniqueness. Let {(U,, cp,)},€%

be a smooth Wn-atlas be a smooth partition of unity on M respecting the orientation. Let subordinate to the atlas. If w E A,"(M), then X,w E AF(M) and X,w # 0 for only a finite number of a E a. This is because supp(w) is compact and the partition of unity is locally finite. Thus,

and this sum is actually finite. Then, if

JM

exists, linearity gives

and supp(X,w) = supp(X,) n supp(w) is a compact subset of U, property of J M , each X,w is uniquely given as

IM

.

By the local

where g, dxl A A dxn = cp;'*(wlU,). We give one way to define establishing existence. This will depend on a choice of orientation-respecting coordinate atlas {U,, and of subordinate partition of unity {Xcr},Ea. (One could remove some of this arbitrariness by requiring the atlas to be the maximal one, but the choice of partition of unity still could not be made canonical.) We appeal to the uniqueness already proven

L,

8.2. STOKES' THEOREM

229

to show independence of the choices. If w E A,"(M), only finitely many X,w are not identically 0. Define

where g, dxl A

... A dxn = (P;l*(~[U,).

Then define

a finite sum. Defined in this way,

is an R-linear map. We must check that, if supp(w) c U, where (U, cp) is an arbitrary orientationrespecting coordinate chart (not necessarily in our atlas), and if

then

JM l(u) g*

=

First remark that

(A,

0

cp,l)ga

since supp(w) c U. By Exercise (2) on page 220,

for each a E %. The fact that the charts are compatibly oriented is essential. Thus,

230

8. INTEGRATION AND COHOMOLOGY

REMARK.By Exercise (2) on page 220 and the above proof,

The orientation of M induces an orientation of d M in the following way. Let {U,, x,1 , . . . , x : ) , ~ ~ be an Wn-atlas on M respecting the orientation. By the definition of Wn, x: 5 0, wherever defined, and x i = 0 exactly on U, n dM, V a E !2l. Let !2l' = {a E !2l1 U, n d M # 0 ) and consider the Ren-'-atlas

gzp

of dM. Let gap and denote the respective changes of coordinates for these atlases. At x E U, T) Up n 8 M ,

Here, since x: decreases with xi, the upper left hand entry in this matrix is strictly positive. Since det(Jg,p). > 0, it follows that d e t ( ~ ~ z > ~ 0. ) , Thus, this Itn-l-atlas on d M defines an orientation of dM. DEFINITION 8.2.2. The orientation of dM, produced as above, is said to be induced by the given orientation of M. We always assume that, when M is oriented, d M has the induced orientation. The following fundamental result asserts that exterior differentiation is the "adjoint" of passing to the boundary. For this reason, d is often referred to as the "coboundary operator".

THEOREM 8.2.3 (STOKES'THEOREM), Let M be an oriented n-manifold and let i : a M --, M be the inclusion. Then, i f w E A;-l(M),

where, i f d M = 0 , the right hand side is interpreted as 0. PROOF.First we prove the local case. That is, M = W n and d M = iRn-'. Any w E A:-'(HIn) can be written as

-

where fj has compact support, 1 5 j 5 n, and dxj indicates that this term is omitted. Then

and

8.2. STOKES' THEOREM

231

By the fundament a1 theorem of calculus and the compactness of supp( f ) , for j =2,

..., n,

Therefore,

Now we can prove the global case. Let

denote the respective boundary inclusions and let

be the orientation-respecting atlases on M and d M , respectively, a s chosen above. If {A,),,ca is a smooth partition of unity subordinate to the atlas on M, then {A, o ~ ~ is a smooth ) partition ~ ~ of unity ~ subordinate t to the atlas on d M . Note that pgl o zwn = iMo (&)-', v a E a'. For w E A:-'(M), the fact that w = C , X,w is a finite sum gives finite sums

8. INTEGRATION AND COHOMOLOGY

232

Therefore,

The last equality is by the local version proven above. If dM = 0, then

and this integral vanishes. Otherwise, we get

igw. =L M

THEOREM 8.2.4. Let M be an oriented n-manifold with a M = 0. Then

is a well defined, R-linear sudection.

PROOF.Since A:+l(M) = 0, we have Z:(M) = A:(M). If w = dv E B:(M), then Stokes' theorem and the fact that dM = 0 imply that

Thus, the linear map

r

8.2. STOKES' THEOREM

induces a well defined linear map

To prove surjectivity, we only need prove that this map is nontrivial.

Let

(U,xl,. . . ,xn) be a compatibly oriented chart and let X E Coo ( M ) have compact support contained in U ,with X 2 0 everywhere and X > 0 somewhere. Thus w = X dx' A

. A dxn can be interpreted as an elemcnt of Z,"(M)

and of AY(Rn),

SO

A deeper fact, to be proven later (Theorem 8.6.4), is that, if M is both oriented and connected, then JM is a bijection from H,"(M) to W. In order to integrate p-forms, where p < dim M , it is necessary to define suitable p-dimensional domains of integration. For the case p = 1, we have already studied line integrals, the domain of integration being a (piecewise) smooth curve in M. In general, it is convenient to use singular psimplices (defined below) as domains for integrating p-forms. A singular l-simplex is simply a smooth curve. Recall that a subset A c RP is convex if, for each pair of points v, w E A, the straight line segment joining v and w lies entirely in A. If C C_ R P is an arbitrary subset, the convex hull of C is defined to be the smallest convex set containing C. Since an arbitrary intersection of convex sets is convex, and RP is itself convex, is just the intersection of all convex sets containing C.

DEFINITION 8.2.5. The standard psimplex A p c R P is the convex hull of the set {eo,e l , . . . , e,}, where ei is the ith standard basis vector, 1 < i 5 p, and eo = 0.

Thus, A, = {O}, a single point, and A = [0,1]. The cases p = 2 and p = 3 are pictured in Figures 8.1 and 8.2, respectively.

FIGURE 8.1. The standard 2simplex

A more explicit definition of the standard psimplex is

8. INTEGRATION AND COHOMOLOGY

el FIGURE 8.2. The standard 3-simplex It is sometimes convenient to set A -l = 0.

DEFINITION 8.2.6. A singular p-simplex in a manifold M is a smooth map

s:Ap+M. Thus, each point of M can be thought of as a singular Osimplex and smooth curves, up to parametrization, are singular lsirnplices. One could also define piecewise smooth singular psimplices, but we will not do so.

DEFINITION 8.2.7. For 0 5 i 5 p, the ith f x e of the standard psimplex Ap is the singular ( p - 1)simplex Fi: Ap-l + A, defined by

The oth face of A. is considered to be defined but empty. If s : A, + M is a singular psimplex, the ith face of s is the singular (p - l)-simplex ais = s o F,. It is clear that Fi: ApPl + A, is a topological imbedding and that the image of Fiis exactly the subset ordinarily thought of as the "face" of A, opposite the vertex ei. The ith face dis of a singular psirnplex s is essentially the restriction of s to the ith face of Ap, but parametrized on the standard Ap-1.

DEFINITION 8.2.8. If s : A, 4 M is a singular psimplex and w E AP(M), then s*(w) has the form g dxl A . A d x p and we set

where the right hand side is the Riernann integral. If s : (0) + M is a singular Osimplex and w = f E AO(M),the integral is interpreted to mean

8.2. STOKES' THEOREM

235

There is a combinatorial version of Stokes' theorem, according to which the integral of an exact p-form dq over a singular psimplex s is equal to the integral of q over the "boundary" of s. THEOREM 8.2.9 (COMBINATORIAL STOKES'THEOREM).If s : Ap + M is a singular p-simplex and 77 E Ap-I (M), then

REMARKS.The signs in the combinatorial Stokes' theorem are dictated by comparing the standard orientation of A I with the induced orientation of as a part of the boundary of Ap. We write the formal expression

,-

and express Stokes' formula as

This highlights the analogy with Theorem 8.2.3 and agrees with established usage in algebraic topology. For f E AO(M)and a smooth curve s : [0, I] + M , Theorem 8.2.9 asserts that

which is just Lemma 6.2.3. That lemma was a thinly disguised version of the fundamental theorem of calculus and Theorem 8.2.9 is a somewhat less thinly disguised version of the same fundamental theorem. PROOFOF THEOREM 8.2.9. It is clearly sufficient to prove that

where q is a (p - 1)-form defined on an open neighborhood of A, in RP. We can write

and, by the linearity of the integral, prove the formula for each term of the sum. That is, without loss of generality, we assume that 1 5 j 5 p and

By the local formula for exterior differentiation,

8. INTEGRATION AND COHOMOLOGY

236

and we are reduced to proving the formula

The right hand side of equation (8.1) can be simplified. For this, it will be helpful to let xi denote the coordinates in Rp and zi the coordinates in RP-'. Remark that

and, if i

> 0,

One obtains the formula

and, if i

> 0,

-

F ; ( f d x 1 ~ . . . ~ dAx. 3. .

A ~ X P= )

i f i # j, (f o F j ) d ~ ' A . . - A d ~ P - 'i f i = j .

Substituting these terms in the equation (8.1) and multiplying both sides by (- l)j-I reduces us to proving

which, rewritten in terms of the Riemann integral, becomes

The linear change of coordinates in IRp-', defined by

8.2. STOKES' THEOREM

237

preserves volume (i.e., the Jacobian determinant is f1) and carries morphically onto itself, as the reader will easily check. Also,

diffee

While this coordinate change reverses orientation when j is odd, the Riemann integral is insensitive to orientation, so (8.2) becomes

To avoid notational confusion, replace the dummy variables w with zi:

Let

A; = Fj(AP-1) = {(XI,.. . , b )E Ap the face of Ap opposite the vertex e j , 1 5 j 5 p. Since j the linear diffeomorphism

=

01,

> 0, it

is evident that

Fj : Ap-1 + A;

preserves ( p - l)-dimensional volume, so the three integrals in (8.3) can be computed, respectively, by the multiple integrals

With these substitutions, (8.3) is checked by standard manipulation of iterated integrals and an application of the fundamental theorem of calculus. The proof of Theorem 8.2.9 is complete.

COROLLARY 8.2.10. A f o n w E AP(M) is closed i f and only if every singular ( p + 1)-simplex s in M .

Sgsw = 0, for

238

8. INTEGRATION AND COHOMOLOGY

PROOF. If w is closed, then

For the converse, suppose that dw = rj- # 0. Choose a point x E M such that A rj-, # 0. Choose vectors v l , . . . ,vp+l E T,(M) such that rj-,(vl > 0. These vectors must be linearly independent, so we can find a local coordinate chart (U, x l , . . . ,xn) about x in which vi is the value of the ith coordinate field ti = d/dxi at x, 1 _< i 5 p 1. By making this chart sufficiently small, we can guarantee that rj-(Jl A ... A Jp+l) > 0 on all of U. Let s : Ap + U be any orientation-preserving, smooth imbedding into the coordinate ( p + 1)-plane {(xl, , . . ,x n ) E U I X P + ~= . . . = xn = 0). It follows that

+

Singular simplices are used to detect topological features of a manifold. Recall, for instance, how a piecewise smooth, closed curve s = sl s, in the punctured plane It2 \ {x) can detect the missing point, provided that s has nonzero winding number w(s, x) about the point. The closed curve s is assembled from the singular l-simplices s l , , . . ,sqwhich join together, end to end, to form a "l-cycle". The winding number itself was found by integrating a certain locally exact (hence, closed) 1-form 77, over s (Exercise (4) on page 187). Piecewise smooth closed curves s in IR3 \ {x) do not detect the missing point. Indeed, it can be shown (Exercise (2)) that any such loop s is homotopic in IR3 \ {x) to a constant loop. However, a map s : S2 4 IR3 \ {x) can snag the missing point. One effective way to use this observation is to triangulate S2 (Section 1.3) and form singular 2simplices s l , . . . , s, by restricting s to these triangles. It is only necessary to assume that each si is smooth, so we get a piecewise smooth map s : S2 + It3 \ {x) and write s = s l + + s, by analogy with the case of loops. We call this a "singular 2-cycle". One should test whether or not the singular 2-cycle has snagged the missing point x by integrating a suitable closed 2-form over this cycle:

+

+

The possibilities for singular 2-cycles are richer than for I-cycles. For instance, triangulations of T2 and corresponding piecewise smooth maps s of T~ into M define '%oral" singular 2-cycles s = s l + . . s, in the manifold M and such a cycle might well detect a topological feature that would be missed by a "spherical" cycle. Again, a test of what this cycle detects is made by integrating closed 2-forms over the cycle. These remarks are extended and made precise by defining the singular homology of a manifold, a covariant functor H , from the category of smooth manifolds to the category of graded vector spaces over R. The celebrated de Rham theorem asserts that this functor is dual to de Rham cohomology. We sketch the

+

8.2. STOKES' THEOREM

239

main facts, illustrating the importance of the combinatorial Stokes' theorem for algebraic topology.

DEFINITION 8.2.11. The set of all singular pimplices in M , p 2 0, is denoted by Ap(M). The space Cp(M) of singular pchains on M is the free Rmodule (real vector space) generated by the set Ap(M). By convention, if p Ap(M) = 0 and Cp(M) = 0.

< 0,

Each p-form w E AP(M) can be viewed as a linear functional

as follows. An arbitrary p c h a i n c E Cp(M) can be written uniquely (up to terms with coefficient 0) as a linear combination

where

sj

E Ap(M), 1 5 j 5 m. One then defines the value of w on c to be

The face operators 8,s = s 0 Fi have already been defined, V s E Ap(M), and can be viewed as set maps

ai : Ap(M) -+ Cp-l (M). Since Ap(M) is a basis of Cp(M), these set maps extend uniquely to linear maps

DEFINITION 8.2.12. The boundary operator d : Cp(M) -4Cp-l(M), p 2 0, is the linear map P

For w E AP(M) and c E Cp(M), the combinatorial Stokes' theorem asserts that

That is, the operators d and d are adjoint to one another. The boundary operator is an algebraic analogue of the geometric notion of a boundary. The following crucial property can be viewed as the algebraic analogue of the fact that the boundary of the boundary of a manifold is empty. The proof is a nice exercise in combinatorics, left to the reader as Exercise (1).

LEMMA8.2.13. The composition

is trivial (a2= 0).

8. INTEGRATION AND COHOMOLOGY

240

DEFINIT~ON 8.2.14. The space Z p ( M ) C_ C p ( M )of all p-cycles is the kernel of the boundary operator d : C p ( M ) + Cp-l ( M ) . The space B p ( M ) C p ( M )of all p-boundaries is the image of the boundary operator a : C p + 1 ( M )+ C p ( M ) . An immediate corollary of Lemma 8.2.13 is that B p ( M )

Zp(M).

DEFINITION 8.2.15. The pth singular homology of M is the vector space

If z f Z p ( M ) ,the homology class of z is the coset [ z ] E H p ( M ) represented by the cycle z.

EXAMPLE 8.2.16. Since C - l ( M ) = 0, the boundary operator vanishes identically on C o ( M ) . That is, Z o ( M ) = C o ( M ) is the real vector space with basis the set of points of M. Define c : Z o ( M ) 3 R by

where all E R and all xi f M . If s E A 1 ( M ) , € ( a s ) = ~ ( s ( 1-) s ( 0 ) ) = 0 , so B o ( M ) C_ k e r ( ~ )and c passes t o a well defined linear map Z : H o ( M ) 4 R. We assume that M # 8, so there is a point x E M and Z([x])= 1, proving that 2 is a surjection and [x]# 0, Vx E M. If M is connected, then every two . . + s , and points x, y E M can be joined by a piecewise smooth path s = sl a s = y - x. This implies that [x] = [y], hence that H o ( M ) has basis consisting of a single element [a]. We have proven that the oth singular homology of a nonempty, connected manifold is canonically isomorphic to R.

+

cpt

EXAMPLE 8.2.17. Let M be contractible (cf. Exercise ( 1 ) on page 227). Let : M 4 M be the contraction, a homotopy of cpo = idM to a constant map

91.

Using this contraction, we are going to define linear maps

>

-

-

V p 0, with a remarkable property. Rp+l restricts to an inclusion A p AP+l The standard inclusion RP there is a unique which is just the face map Fp+l. For each point v E point v' E A p and a unique number t E [ O , 1 ] such that v = teP+l ( 1 - t ) v f ,and every point of WP+' of such a form is a point in If s : A p + M is smooth, define a smooth map L p ( s ) : Ap+l 4 M by the formula

+

The fact that this is well defined when t We view s I+ L p ( s ) as a set map

==

1 is due to

cpl

being a constant map.

and take the linear map L p to be the unique linear extension of this set map to all of C p ( M ) .In Exercise (6), you are invited t o check that

8.2. STOKES' THEOREM

24 1

>

provided that p 1. This is the remarkable property promised above. If z E Zp(M) and p 2 1, it follows that

hence that Zp(M) Bp(M). The reverse inclusion also holds, so we have the result that the singular p-cycles and the singular p-boundaries in a contractible space are exactly the same, V p 2 1. That is, Hp(M) = 0 in all degrees p > 0. The above two examples give THEOREM 8.2.18. If M is a contractible n-manzfold, then

In particular, this is true for M = Rn. PROPOSITION 8.2.19. If w E ZP(M) and z E Zp(M), then the real number Jz w depends only on the cohomology class [w]E HP(M) and the homology class [ ~ El HP(M). PROOF.Indeed, [w] is the set of all closed pforrns w + d ~ where , q E Ap-' (M). We have by Stokes' theorem and the fact that z is a cycle, so

Similarly, [z]is the set of all pcycles of the form z Since

+ ac, where c E Cp+l (M).

we obtain

Thus, we can define an R-linear map

8.2.20 (THEDE RHAMTHEOREM).The linear map DR is a canTHEOREM onical isomorphism of vector spaces. This is a deep result. For the case in which M is compact, the proof will be discussed in some detail in Section 8.9. In that case, the vector spaces are finite dimensional (Theorem 8.5.8), so we also get Hp(M) = HP(M)*. The following corollary generalizes Proposition 6.3.3.

242

8. INTEGRATION AND COHOMOLOGY

COROLLARY 8.2.21. Let w,3E ZP(M). Then [w] = [GIzf and only zf Jz w = Jz 3 as z ranges over all singular p-cycles in M . These numbers are culled the periods of w and of the cohomology class [w]. In particular, w is an exact form if and only if all of its periods are 0, which generalizes the equivalence of properties (1) and (2) in Theorem 6.2.9.

EXERCISES (1) Prove Lemma 8.2.13. (2) Let s be a piecewise smooth loop in Rn \ {x), where n 2 3. Show that s is homotopic to a constant loop. Without loss of generality, assume that x = 0 and proceed as follows. (a) Show that s is homotopic to a piecewise smooth loop on S c Wn \ (0). (b) Show that every piecewise smooth loop on Sn-' misses a point y E Sn-l, provided that n 3. (Hint: Sard's theorem.) (c) Show that Sn-' \ {y) is contractible and draw the desired conclusion. Note that, by the de Rham theorem, it follows that H1(Wn \ (0)) = 0 = H1(Rn \ {0)), n 3. (3) Let M be an n-manifold and let z E Zn+l(M). Assuming the de Rharn theorem, prove that there is a chain c E Cn+2(M)such that z = dc. (4) Show that singular homology is a covariant functor, proceeding as follows. (a) If f : M + N is a smooth map between manifolds, exhibit a canonical way to induce a linear map f # : Cp(M) + C,(N), v p 2 0. (b) Prove that the diagram

"-'

>

>

commutes, V p 2 0. Conclude that the linear map f # passes to a linear map f, : H, ( M ) + H, (M) of graded vector spaces. (c) Verify the properties (f o g), = f, o g, and id, = id. (d) Under the de Rham isomorphism of H k ( ~with ) Hk(M)*,show that

are adjoint to each other. (5) Without appealing to the de Rharn theorem, extend the argument in Example 8.2.16 to show that Ho(M) is a direct sum of copies of R, one for each connected component of M. (6) Prove the formula for d o L, asserted in Example 8.2.17.

8.3. POINCARE LEMMA

243

8.3. The Poincard Lemma In the following discussion, R will stand for any nondegenerate, compact interval [a,b] or for R. We consider an arbitrary n-manifold M , not necessarily orientable. If M has nonempty boundary, M x R will always denote M x 113, thereby avoiding manifolds with corners. Homotopies, therefore, will be understood in the sense of Definition 3.8.9 whenever convenient. We will agree to denote the standard projections by

and p: M x R 3 R .

The coordinate of R will be denoted by t and p*(dt) A1 ( M x R) will be denoted by dt (an abuse). An atlas {(W,, x , ) ) , ~ ~on M determines an atlas {(W, x R, (x,, t ) ) ) a E ~ on M x R. Here, if d M = 0 and R = [a,b], we model (n + 1)-manifolds with boundary on W n x [a, b] instead of on HIn+'. subordinate to { Wcr)aE%, determines A smooth partition of unity a smooth partition of unity

on M x R subordinate to {W, x R)aE'ZL.Here, -

L ( x , t) = Xa:(x). Ifw E A&(Mx R), let

u, = ul(W, x R) and write

where we use the conventions

and

Since

we write

u= For each cr E a,choose

0, : M

+

[O,l]

8. INTEGRATION AND COHOMOLOGY

244

with supp(8,) and

c W,

and 8,) supp(A,)

1. Let

8, = n*(8,).

--

-

Then, A d , = A,

In each open set n-I (Wo) = Wp x R, only finitely many indices a E 21 correspond to nonzero terms in the expression for w (n-' (Wp). Also, for q = k or k - 1, 8, dx2 A . - ~ d x = z x* (v), for some 7 E AQ(M), so we have proven the following.

LEMMA8.3.1. Each form w E A k ( M x R) can be expressed as a locally finite sum of k-forms, each being one of the following two types:

We construct an important operator which "integrates out" the dt component of forms on M x R.

LEMMA8.3.2. For each r E R and each integer k 2 0, there is a unique R -linear map S, : A*(M x R) A ~ - ~ (xMR) which is additive over locally finite sums and satisfies

(Here we understand that A-'(M x R) = 0, so ST : A'(M x R) -, 0 is trivial, consistent with the fact that all forms in A O ( Mx R) are of type ( i i ) . )

PROOF. By the existence of a decomposition of w

E

A (M x R) into a locally

finite sum of forms of the types ( i ) and (ii),the stipulated properties of ST force that operator to be unique, provided that it exists. But, if we fix the choice of atlas {W,, x , ) , ~ ~ of , subordinate partition of unity {A,),cat, and of the functions (6a}aE%,we then have an algorithm for producing a locally finite decomposition of w f A k ( x~R) into the desired types of summands. We use (a) and ( b ) to define ST on each of these summands and remark that the result is a locally finite system of ( k - 1)-forms on M x R, hence that their sum is a well defined element ST(w) E A*-'(M x R). It is clear that S T , defined in this way, is R-linear. The crucial fact that it is also additive on Iocally finite sums is left as Exercise (1). Uniqueness shows that the definition of S , is really independent of the choices. For each T E R, let iT : M -t M x R be given by i,(x) = ( x , ~ )One . version of the Poincarb Lemma is that, at the cohomology level, n * and i: are mutually inverse isomorphisms. Indeed, it is clear that i: o n* = (ro iT)* is the identity at the level of forms, so it remains to show that .rr * 02: is the identity on cohomology. The main step is the following.

LEMMA8.3.3. O n A k ( x~R), the operator S, sat$es the identity

The reader is invited to check this formula in Exercise (2). THEOREM 8.3.4 (POINCARE LEMMA,VERSIONI). The map n* : H*(M) -t H*(M x R)

is an isomorphism and its inverse is i:, level, i: is independent of r.

QT

E R. In particular, at the cohomology

PROOF.As remarked above, we only need to prove that, at the cohomology level, n* 0 i: = id. If w E ZP(M), we apply Lemma 8.3.3 to obtain w - r *(2: (w)) = d(S, (w))

+ ST(d(w))

= d(S7(~)).

That is, w and n*(i:(w)) differ by a coboundary, and we are done.

THEOREM 8.3.5 PO IN CAR^ LEMMA,VERSION11). If f o , f l : M homotopic, then f ; = f ; : H * ( N ) + H *(M).

+

N are

PROOF.Let F : M x 113 + N be the homotopy. Then f o = F o io and fi = F 0 il . By functoriality,

fi = 28 o F* f ; = i; o F*.

But 2; = ii by Theorem 8.3.4, SO f i = fi. Here are four more versions of the Poincark Lemma. The first of these is immediate by Theorem 8.3.5, and each implies the next. All of the implications are rat her obvious. THEOREM 8.3.6 PO IN CAR^ LEMMA,VERSION111). I f f : M -t N is a homotopy equivalence, then f * : H *(N) 4 H*(M) is an isomorphism of graded algebras. THEOREM 8.3.7

PO IN CAR^

LEMMA,VERSIONIV). For a contractible man-

I n particular, this holds for M = Rn.

THEOREM 8.3.8 PO IN CAR^ LEMMA,VERSIONV). For k > 0, every closed k-form on a contractible manifold is exact. Since manifolds are locally contractible (each point has a neighborhood diffeomorphic to Rn), the next version follows. THEOREM 8.3.9 PO IN CAR^ LEMMA,VERSIONVI). If k > 0, a k-form on a manifold M is closed i f and only if it is locally exact. Thus, the definition of H1(M) given in Chapter 6 agrees with our current definition. It can be shown that Version VI implies Version I, so all versions are mutually equivalent. We will not prove this.

246

8. INTEGRATION AND COHOMOLOGY

DEFINITION 8.3.10. If d M = 8 and fo, fl : M

N are proper smooth maps, they are said to be properly homotopic if there is a proper smooth map -t

such that F(x,O) = fo(x) and F ( x , 1) = fl(x), Vx E M. The map F is called a proper homotopy between f and f 1. For compactly supported cohomology on manifolds without boundary, define the term "homotopy" using R = [O, 11 and remark that both n * and i: are proper maps. Theorem 8.3.4 continues to hold in this situation and a suitably reworded version of Theorem 8.3.5 also holds (Exercise (3)). One could call this the Poincarh Lemma, but what usually goes by that name for compact cohomology takes a rather different form. This is our next topic. First note that w E A: ( M x R) is a finite linear combination of forms of the types

(i) f (x, t) d t ~ n * ( q )where , f (x, t) is compactly supported, but 7 E Ak-'(M) may not be compactly supported. (ii) f (x, t)rr*(7), where f (x, t ) is compactly supported, but q E A *( M ) may not be compactly supported. One then defines an R-linear map

called "integration along R" , by requiring that

As before, there is a unique R-linear operator n, with these properties. Remark that requiring the operator to be additive over locally finite sums is no longer necessary. It will also be convenient to define A;Q(M) = 0, V q > 0,to agree that d (= 0) is defined on this trivial module, hence to have H - Q ( M ) defined and trivial.

LEMMA8.3.11. With the above definitions, n, antiwmmutes with d. That is,

The proof is a straightforward computation on forms of types (i) and (ii). It is analogous to Exercise (Z), only easier.

COROLLARY 8.3.12. The linear map n, passes to a well defined linear map

8.3. POINCARE LEMMA

247

We want to prove that this map is an isomorphism, so we need a candidate for its inverse. Choose a compactly supported function b : IR -P W such that b(t)dt = 1. Let /iJ = b(t)dt E A:@). Finally, for each k E Z, define

It is practically immediate that

and we draw the following conclusion.

L E M M A8.3.13. The linear map P, passes to a well defined linear map

P* : H,L(M)

-+

H,'+'(M x W),

V k E Z. Clearly,

and we will show that, at the level of compact cohomology, P, identity. Once again, we construct an operator

o

n, is also the

such that doS+Sod=id-@,on,. We define S. Set

and define S on forms of type ( i i ) by

and, on those of type (i),by

Remark that this form is, indeed, compactly supported. This is obvious in the x variable and, for t 1 -00, it is also clear. But, as t f oo, the function in the parentheses ultimately becomes

so the support is bounded in all directions, hence compact.

8. INTEGRATION AND COHOMOLOGY

248

LEMMA8.3.14. The formula

holds on A,k(M x R), V k E Z. The somewhat tedious computations that prove this lemma are relegated to Exercise (4). As in the proof of Theorem 8.3.4, this is all that is needed to establish the following.

THEOREM 8.3.15 (POINCARE LEMMAFOR

COMPACT SUPPORTS).

The map

is a canonical isomorphism with inverse P,, V k E Z. In particular, p, does not depend, at the level of compact cohomology, on the b(t)dt = 1. choice of the compactly supported function b(t) such that

Jrm_

REMARK. The operators S, and S, used to prove the Poincare lemmas for ordinary and compact de Rbam theory, are examples of cochain homotopies in algebraic topology. One says that i d A . ( ~ ~ is R )cochain homotopic to n* o i:, writing n* 0 i: idA*(MXR) . N

Similarly,

id^^

P* 0 n*

(MX

W)

.

Exactly as in the proof of Theorem 8.3.4, cochain homotopic maps induce the same map in cohomology. In the present situation, since one of the maps is the identity, they both induce the identity. In Example 8.2.17, we used a chain homotopy between the identity and a map that is 0 in positive degrees to show that the singular homology of a contractible manifold is trivial.

COROLLARY 8.3.16. For each integer n 2 0, H , ~ ( w "= )

R, k = n 0, otherwise.

PROOF. This is clearly true for n = 0. Inductively, suppose that it is true for a given value of n 0 and appeal to Theorem 8.3.15 to get

>

In Exercise (2) on page 227, you proved this for the case n = k = 1.

COROLLARY 8.3.17. The linear map

is an isomorphism.

8.4.

EXACT SEQUENCES

249

Indeed, since Rn is orientable, we have seen that this is a surjection (The* rem 8.2.4). Since the cohomology space is onedimensional, it is an isomorphism.

EXERCISES (1) Prove that the operator ST in Lemma 8.3.2 is, indeed, additive on locally finite sums. (2) Prove Lemma 8.3.3. (3) Show that Theorem 8.3.4 makes sense and holds for compactly s u p ported cohomology, provided that d M = 8 and R = [a,b]. Using proper maps and proper homotopies, formulate and prove the analogue of Theorem 8.3.5. (4) Prove Lemma 8.3.14. 8.4. Exact Sequences

A basic tool for computing cohomology will be the Mayer-Vietoris sequence (Section 8.5). In order to develop and apply this sequence, we will need some properties of exact sequences. This purely algebraic section may be a review for many readers. At any rate, the proofs are elementary and will be relegated to exercises. We fix a commutative ring R and consider modules A, B , C , etc., over R. All maps cp : A + B will be R-linear. We also consider graded R-modules A *, B*, etc., over R, in which case cp : A* --, B* will denote a homomorphism of graded R-modules. We will generally assume that the grading is indexed by Z rather than just Z+. No generality is lost since A* = { A * } E ~can be replaced by { A k } ~ - , by setting A-P = 0, V p > 0. DEFINITION 8.4.1. A sequence

of module homomorphisms is said to be exact at B if im(a) = ker(P). If a sequence of module homomorphisms is exact at each module (except the first and last), it is called an exact sequence. An exact sequence of the form

is called a short exact sequence. Similarly, the notions of exact sequence and of short exact sequence are defined for graded module homomorphisms. Remark that, in the short exact sequence, i is injective and j is surjective. LEMMA8.4.2 (THEFIVE a

LEMMA).

a

A - B - C - D - E

Let Y

5

8. INTEGRATION AND COHOMOLOGY

250

be a commutative diagram in which the two rows are exact. I f / \ , p, p, and n are isomorphisms, then v is a n isomorphism. The proof is a mechanical diagram chase (Exercise (1)).

DEFINITION8.4.3. A cochain complex (A*,S) is a graded R-module, together with a sequence 6 ~ p ++ . . . -,6 Ap + l6 ~

p +! +

. . ~.

such that S2 = 0. Similarly, a chain complex (C,, a) is a graded R-module and a sequence

a cp a - . ---t

cp-la

-3

CP-2

a

+"'

such that d2 = 0.

As usual, one defines ZP = ker(S) n AP (respectively, Zp = ker(d) n Cp) and = im(S) n AP (respectively, Bp = im(d) n C,). In what follows, we explicitly consider cochain complexes, but everything goes through, with the obvious modifications, for chain complexes. In this book, we are mainly interested in the de Rham cochain complexes (A* (M),d) and (A: (M), d) and in the singular chain complex (C, (M) ,d) , although others will be mentioned on occasion. The condition that S2 = 0 implies that B* C Z* and the cohomology of the cochain complex is defined to be B p

a graded R-module. This can be viewed as a measure of the extent to which the sequence in Definition 8.4.3 fails to be exact. The corresponding construction for a chain complex (C,, d) is called the homology of the complex and denoted by H* (C*,a)-

DEFINITION 8.4.4. A homomorphism cp : (A*, S) 4 (C*,6 ) of cochain complexes is a homomorphism of the graded R-modules such that cp o S = 6 o cp.

-

Evidently, a homomorphism cp : (A*, S) (C*,S) of cochain complexes induces a homomorphism cp* : H*(A*,S) -,H*(C*,6) of graded R-modules.

DEFINITION 8.4.5. A homomorphism X p E Z is a sequence of R-linear maps

-00

:

H*(A*,S)

-+

H*(C*,6 ) of degree

< k < oo. This is sometimes written X : H*(A*,S) -+H*+p(C*,6).

For instance, in the Poincarb Lemma for compactly supported cohomology, we defined a homomorphism

of degree - 1.

8.4. EXACT SEQUENCES

L E M M A8.4.6. Let

be a short exact sequence of homomorphisms of cochain complexes. Then, there is canonically induced a homomorphism

S* : H* ( E * ,S )

4

H*+'

(c*, S)

of degree +1, called the connecting homomorphism. This homomorphism is "natural" in the following sense: i f

is a wmmutative diagram with both rows exact, then H* ( E * ,S )

6'

H*+'

(c*, 6)

also commutes. In the case of a short exact sequence of chain complexes, the connecting homomorphism has degree -1. We show how to find S* [el E H k + l ( C * ,S ) , where [el E H k ( E * ,6). Consider the commutative diagram

and choose e E E~ representing [el. In particular, S(e) = 0. Since j is surjective, choose et E Dk such that j ( e f ) = e. Then j ( 6 ( e f ) ) = S(j(e')) = 6(e) = 0 and exactness of the middle row implies that there is a unique c E C such that i(c) = S(el). Then i ( S ( c ) )= S(i(c)) = S(S(el))= 0. Since i is one to one, it follows that b(c) = 0, so we define S* [el = [c] E H ~ + ' ( C *6). , More diagram chasing proves that [c]is independent of the choices of e E [el and of e t E D k such that j ( e t ) = e. The "naturality" of S * , as defined in the statement of Lemma 8.4.6, is proven by yet another diagram chase (Exercise (2)).

8. INTEGRATION AND COHOMOLOGY

252

-

PROPOSITION 8.4.7. A short exact sequence 0

( C * ,6 ) 5 ( D * ,6 ) -2. ( E * ,6 ) +0

of cochain complexes induces a long exact sequence

i n cohornology. The proof is a diagra,-n chase (Exercise (3)). One sometimes writes the long exact sequence more compactly as an exact triangle:

H* (C*,6 )

i*

A/

H * ( D * ,6 )

H* ( E * ,8 )

EXERCISES (I) Prove the Five Lemma. (2) Prove that the connecting homomorphism 6* of Lemma 8.4.6 is naturd. (3) Prove Lemma 8.4.7. (4) Let R be a field. If

A

~

B ~ C is an exact sequence of vector spaces over R, prove that the dual sequence

- i*

B* j C*, where i* and j* are the respective adjoints, is also exact. Find an example showing that this may fail for modules over a commutative ring.

A*

8.5. Mayer-Vietoris Sequences

Let U1 and Uz be open subsets of the n-manifold M and consider the inclusions

jl: Ul n U2 j2 : Ul n Uz

--

Ul U2

and

LEMMA8.5.1. The above inclusions give rise to a short exact sequence 0 -+ (A*(Ul U U 2 ) ,d ) 5 ( A * ( U l )d A*(Uz),d @ d ) f (A*(Ul ilU2),d )

-+

0

of cochain complexes, where i ( w ) = ( i ;( w ) ,ia ( w ) ) , V w E A* (U1 U U z ) , and j ( w 1 , w ~= ) jT(w1) - j;(w2), Vwt E A*(Ue), = 1,2.

8.5. MAYER-VIETORIS SEQUENCES

253

PROOF. Indeed, a nontrivial form on U1UU2 must be nontrivial on either U1 or U2, SO i is one to one. Since j! 0 2 ; = jzoi;, it is clear that im(i) C ker(j). For the reverse inclusion, let (wl, wz) E ker(j). Then wl J (Ul nU2) = w2I (Ul f7U2),SO these forms fit together smoothly to define a form w on U1 U U2 and (wl, up) = i(w). Finally we must prove that j is surjective. Let w be a form on U1 n U2. Let {XI, X2) be a partition of unity on U1 U U2 subordinate to {U1, Uz) and set wl = X2w, wp = X1w. (Note that, since X2 is supported in U2, X2w extends smoothly by 0 to all of U1. Similarly, Xlw is a form on U2.) Then, j ( w l , -w2) = W l + W 2 = w. 0

THEOREM 8.5.2. There is a long exact sequence

called the Mayer- Vietorzs sequence. Indeed, the cohomology of the cochain complex (A * (Ul) @ A*(Uz),d @ d) is clearly H *(Ul) @ H*(U2),so we apply Proposition 8.4.7. We turn to the Mayer-Vietoris sequence for compactly supported cohomology. Again, Ul and U2 are open subsets of some n-manifold M. Clearly, there are inclusions a e : A: (Ul n U2) --r A:(Ut), e = 1,2, and Pe : A: (Ut) A: (Ul U Uz), e = 1,2. It is evident that these inclusions commute with exterior differentiation, hence induce linear maps a;, fl; in compact cohomology, 4? = 1,2.

-

LEMMA8.5.3. The above inclusions induce a short exact sequence

of cochain complexes, where a ( w ) = (a1(w), - a 2 (w)), V w E A: (Ul n Uz), and /3(wl,wz) = PI(WI) PZ(WZ), vut E A;(u~), e = 1,2.

+

PROOF.Everything is clear except, perhaps, the fact that /3 is a surjection. If w is a compactly supported form on U1 U U2 and {Xi, X2) is a partition of unity on Ul U Uz subordinate to {U1, U2), then Alw has compact support in U1 and X2w has compact support in U2 (note the switch from the proof of Lemma 8.5.1). Then, /3(Xlw,X2w) = w and /3 is surjective. REMARK.We have chosen the signs differently than in Lemma 8.5.1. This is not necessary for our present needs, but will be useful in our treatment of Poincark duality. THEOREM 8.5.4. There is a long exact sequence

called the Mayer- Vietoris sequence for compactly supported cohomology.

8. INTEGRATION AND COHOMOLOGY

254

DEFINITION 8.5.5. Let M be an n-manifold without boundary. An open cover {Ua)aEQ of M is said to be simple if it is locally finite and every nonempty, finite intersection U = U, n U,, n . . +n U, is contractible and has H,*(U) = Hz (Rn). By Theorem 8.2.18 and Theorem 8.3.7, simple covers have the following property. LEMMA8.5.6. If U is a simple cover of M and U is any nonempty, finite intersection of elements of 24, then H*(U) = H*(Rn) H, (U) = H , (Rn). THEOREM 8.5.7. If M is a manifold with d M = 8, then every open cover of M admits a simple refinement. We will postpone the proof of this theorem to Section 10.5, since it requires methods from Riemannian geometry. The idea is to produce a locally finite refinement by geodesically convex open sets and to prove that a geodesically convex open set U has the property in Definition 8.5.5. Since finite, nonempty intersections of geodesically convex sets are geodesically convex, we obtain a simple refinement. Using Theorem 8.5.7 and Mayer-Vietoris sequences, we obtain the following interesting result. THEOREM 8.5.8. If M admits a finite simple cover, then H *(M) and H,*(M) are finite dimensional. I n particular, if M is a compact manifold without boundary, then H*( M ) is finite dimensional. PROOF.Select a finite simple cover {Ui)T=l of M. We proceed by induction on r . If r = 1, then M = U1 has the ordinary and compact cohomology of R n. Thus, H*(Ul) = H*(singleton), hence is finite dimensional. For H,*(Ul), the assertion is given by Corollary 8.3.16. Suppose, then, that it has been shown that H,*(N) and H*(N) are finite dimensional whenever N has a simple cover by r - 1 elements, some r _> 2. Let M have the simple cover {Ui)F=l, let U = Ui and remark that {Ul n Ur, Uz n Ur, . . . , U,.-l n Ur) is a simple cover of U n U,.. We consider the compactly supported case. By the inductive hypothesis, H,*(U) and H,*(U n U,) are both finite dimensional as, of course, is H,*(U,).Since M = U UUr , the Mayer-Vietoris sequence gives an exact sequence

:u::

Hf (U) @ Hf (Ur) Z Hz (M)

d*

-+

H,'" (U n u,.).

By standard linear algebra,

Since p* has finite dimensional domain and d * has finite dimensional range, the assertion for H,*(M) follows. The proof for ordinary cohomology uses the appropriate Mayer-Vietoris sequence in the same way.

8.5. MAYER-VIETORIS SEQUENCES

255

REMARK. Even if the compact manifold M has boundary, it is true that H*(M) is finite dimensional. One way to prove this is to show that int(M) has a finite simple cover and that M and int(M) are homotopically equivalent. There is also a Mayer-Vietoris sequence for singular homology. The proof is similar to those for cohomology except for one technical point, the proof of which is very tedious and would take us too far afield. Since we will need this sequence for the proof of the de Rham theorem, we derive it here, referring the reader to standard references in algebraic topology for the bothersome technicality. The inclusions

and

induce an exact sequence

where j(c) = (jl#(c), -j 2 # (c)) and i(c1, cz) = il#(cl) +i2# (CZ)(and the induced homomorphisms il#, jl#, etc., are as in Exercise (4) on page 242). The exactness is immediate. If (*) were a short exact sequence, the Mayer-Vietoris sequence for singular homology would follow immediately, but it is generally false that i is a surjection. This brings us to the technical point.

DEFINITION 8.5.9.Let U = {Ua)aEQbe an open cover of the manifold M. A singular psimplex s : Ap -+ M is said to be Usmall if, for some a E I#, ( A ) U The set of U-small singular psimplices is denoted by A: (M). The vector subspace of Cp(M) spanned by A:(M) is denoted by C;(M). By the definition of the singular boundary operator, it is immediate that

DEFINITION 8.5.10. The chain complex ( C ~ ( M )a) , is called the complex of U-small chains. The homology of this complex is H ~ ( M ) the , Usmall homology of M. It is clear that the natural inclusion of the space of Usmall chains into the space of all chains is a homomorphism of chain complexes

so there is induced a canonical homomorphism in homology

We arrive at the technical result.

8. INTEGRATION AND COHOMOLOGY

256

PROPOSITION 8.5.11. The homomorphism zy is a canonical isomorphism

Proofs of Proposition 8.5.11 will be found in the standard references in algebraic topology, such as f12, pp. 85-88] and [43, pp. 207-2081. The idea is to subdivide the singular simplices in each cycle z until all simplices in the subdivision are U-small. With appropriate choices of signs, there results a U-small cycle z' with [x'] = [ z ] . Thus, homology can be computed using Usmall chains, for any open cover U of M. In our situation, 24 = {U1, U2) is an open cover of the manifold U1U Uz and we replace the sequence (*) with the short exact sequence

THEOREM 8.5.12. There is a long exact sequence

called the Mayer-Vietoris homology sequence. Using the result of Exercise (4) on page 252, we obtain dual Mayer-Vietoris sequences. THEOREM 8.5.13. The Mayer- Vietoris sequences dualize to exact sequences

j'

- - .-, Hp(Ul)*@ Hp(U2)*

Hp(Ul U U2)*

where 'i is the adjoint of i* (respectively, of i,),

at

Hp-l(Ul n U2)*

...

etc.

EXERCISES (1) If the manifold M is connected, but not necessarily compact, prove that the real vector spaces H * (M) and H , (M) have dimension at most countably infinite. You may use the de Rbam theorem. (2) Prove that

H k ( s n )= for all n 2 1.

R, Ic = 0 , n 0, otherwise,

8.6. COMPUTATIONS OF COHOMOLOGY

257

8.6. C o m p u t a t i o n s of Cohomology

In this section, we compute the top dimensional cohomology of connected manifolds.

LEMMA8.6.1. If the open subsets U1, U2, and Ul n U2 of an n-manifold M all have the same compact cohomology as Rn and are coherently oriented, then

is an isomorphism.

PROOF.Consider the diagram

where A(t) = (t, -t) and S(s, t) = s+t, V s, t E R. Commutativity of the diagram is obvious (coherency of orientations is essential), exactness of the bottom row is obvious, and the top row is exact by Theorem 8.5.4. The map JuInu2 is an isomorphism by the hypothesis that H," (Ul n U2) = R (as in Corollary 8.3.17). Similarly, JU, @ Ju2 is an isomorphism. Since the diagram can be extended harmlessly by a commutative square of 0's on the right, it follows from the Five Lemma that Jul uu2 is an isomorphism.

LEMMA8.6.2. Let {Ua)aEQ be a simple open cover of a connected, oriented n-manifold M without boundary, each U, being oriented coherently with the orientation of M . Let w,, wp E A: ( M ) have respective supports in U, and Up, . Then [w,] = [wp] if and only if J, w, = JM wp. a,

PROOF.If [wo] = [wp], then we know that w, = JM wp. For the converse, assume equality of the integrals. Since M is connected, we can find a sequence of indices a = a o , a l , . . . , a , = ,f3 in 24 such that U,,-, n U,, # 8, 1 5 i 5 r. Choose wai E AZ(M) such that supp(w,,) c U,,, 1 5 i 5 r, and such that Wao = W, War = wp, and

This is clearly possible. By Lemma 8.6.1,

JUai-

UUa,

: HP(U,,-,

U U,,)

-+ R

is

an isomorphism, 1 5 i 5 r, so [ L J , ~ - ~ ] = [w,,] E H,"(Uai-, U U,,). That is, there U Uai) such that woi = wai-, dq. These forms all is a form q E A:-l(Uai-, live in A,*(M), so [w,,] = [ w , ~ - ~ ]E H,"(M), 1 5 i 5 r. In particular, [w,] = [wp] as desired.

+

8. INTEGRATION AND COHOMOLOGY

258

LEMMA8.6.3. If (Ua)acO is a simple cover of the connected, oriented nmanifold M with a M = 0, then, for each a0 E 8, the natural inclusion e : A;(U,,)

-

A! ( M )

induces an isomorphism

Sum,

Indeed, since : H,"(Uao) + W is an isomorphism, [w] E H,"(Ua0) is nontrivial if and only if JM w = J w # 0. Since JM vanishes on B:(M), it uao follows that e*[w] # 0,so e* is injective. We prove surjectivity. Let w E A:(M) and use a partition of unity subordinate to the simple cover, to write PROOF.

where A,, w E A: (U,, ), 1 5 i

< r. Choose wi E A:(U,,)

so that

Let r

and remark that, by the above lemma, [w,] = [X,,w] E HF (M), 1 5 i 5 r. Thus, as classes in H,72(M) ,

=

[w].

That is, viewing [Z] E HP(Uao) and [w] E H,"(M), we have proven that e*[S]= [ w ] , SO e* is surjective.

THEOREM 8.6.4. If M is a connected, oriented n-manifold without boundary, then the linear map

is an isomorphism.

8.6. COMPUTATIONS O F COHOMOLOGY

259

PROOF. Fix a simple cover and let U be an element of that cover. Consider the commutative diagram

Since e* and Ju are isomorphisms, so is JM. COROLLARY 8.6.5. If M is a compact, connected, oriented n-man2fold without boundary, then JM : H n ( M ) + R

is an isomorphism. THEOREM 8.6.6. If M is a connected, nonorientable n-manifold with empty boundary, then H,"(M) = 0. I n particular, if M is also compact, H n ( M ) = 0.

A:-'

PROOF.Let w E AF(M). We must show that w = dB for suitable B E (M). Choose a simple cover {U,),Ea. By a partition of unity argument,

write w as a finite sum of forms, each compactly supported in one or another element of the cover. If each of these is the exterior derivative of a compactly supported form, we are done. Thus, without loss of generality, we assume that supp(w) c U,, . By nonorientability of M , there is a sequence U,, ,U,, , . . . ,U,, of elements of the simple cover and orientations p, of Uai, 0 5 i 5 r, with the following properties: (1) u*,-l nu,, # 0, 15 5 r; (2) pi-1 and pi restrict to the same orientation of Uai-, n Uai, 1 5 i 5 r; pr = -Po. (3) U,, = Ua0 Choose forms wi E A:(U,,), 0 5 i 5 r, such that w = wo and

15 i

5 r . Thus, wi = wi-1

+ d~i-1, -

That is, wr = wo

On the other hand,

implying that

1 €

+d

UU

~ ,q € A~-'(M).

) ,I 5 i 5 r.

260

8. INTEGRATION AND COHOMOLOGY

Combining these equations, we conclude that w = wo = d8 for suitable 8 E A;-'(M). o If M is compact, any simple cover, being locally finite, is finite. If M is noncompact, it may or may not admit finite simple covers, but it always admits infinite ones. The proof of the following is carried out by the same geometric arguments that prove the existence of simple covers in general.

LEMMA8.6.7. If M is noncompact and connected, d M = 8, then there is a countably infinite simple cover {Ui)El such that Ui n Ui+' # 0, 1 5 i < 00. EXERCISES (1) Use Lemma 8.6.7 to prove that, if M is a noncompact, connected, boundaryless n-manifold (orientable or not), then H (M) = 0. (2) If M is an n-manifold with d M compact and nonempty, prove that M and int(M) are homotopically equivalent. If M is connected, conclude that H n ( M ) = 0. (Hint: There is a compactly supported vector field on M, pointing inward and nowhere 0 along d M . This generates a "half flow", parametrized on [0,GO) and stationary outside of a neighborhood of dM.) 8.7. Degree Theory

Let M and N be connected, oriented n-manifolds without boundary and f : M -+ N a proper map. Let y E N be a regular value of f . Then f -'(y) is compact and discrete, hence finite. Set f -l(y) = {yl, . . . ,y,) and let E.j

=

+1 if f,,i preserves orientation, - 1 if f,,, reverses orientation,

DEFINITION 8.7.1. With the above conventions, deg,( f) = called the local degree of f at the regular value y.

xP=,

ri. This is

The canonical isomorphisms JM and allow US to identify H,"(M) and H,"(N) with R unambiguously. This identification is understood in the following discussion.

PROPOSITION 8.7.2. If f : M 4 N is proper, where M and N are connected, oriented n-manifolds without boundary, and if y E N is a regular value, then f * : H," (N) + H," (M) is multiplication by degl (f). PROOF.By Theorem 3.9.1, there is an open, connected neighborhood U C N of y such that

f-'(u)

=

u, U - . . u u,,

a union of disjoint open sets such that y i E Ui and f carries Ui diffeomorphically onto U ,I 5 i 5 q. If [w] E HP(N), we can choose a representative n-form w so that supp(w) is a compact subset of U. Then, wi = f * (w)lUi is compactly

8.7. DEGREE THEORY

supported in Ui,1 5 i 5 q, and f : Ui-+ according as c i = 1 or - 1. Thus,

It follows that f * : H,"(N)

-+

26 1

U preserves or reverses orientation

H,"(M) is multiplication by deg,(f).

REMARK. This has several obvious consequences: (1) f * : H,"(N) -+ HF(M) is multiplication by an integer. (2) deg, ( f ) = deg(f ) (the degree of f ) is independent of y. (3) deg(f ) is a proper homotopy invariant of f . (4) If M is compact, deg(f ) is a homotopy invariant of f . (5) deg(f g ) = deg(f deg(g)THEOREM 8.7.3. Let W be an oriented (n + 1)-manifold with nonempty, connected boundary. Let N be a connected, oriented n-manzfold without boundary and let f : dW -+N be proper. Iff extends to a proper map F : W -+ N, then deg(f) = 0. PROOF.Suppose that f extends to a proper map F : W -+ N. If w E A:(N), then F*(w) E A:(W) and d(F*(w)) = F*(dw) = 0, since dw E A:+'(N) = 0. Thus, by Stokes theorem,

By Proposition 8.7.2, it follows that deg(f) = 0. This theorem partially generalizes Theorem 6.4.10. Let Sn C lRn+' be the unit sphere and let a, : Sn 4 Sn denote the antipodal interchange map

PROOF.Since an is a diffeomorphism, every x E Sn is a regular value and has pre-image a singleton. Thus, the question reduces to whether a n preserves or reverses orientation. The linear extension A : lRn+l -+ Rn+l of a, is represented by the matrix -In+1 with determinant (- 1)n+l. This transformation, therefore, is orientation-preserving if and only if n is odd. The restriction of the transformation to the unit ball Dn+l is orientation-preserving if and only if n is

8. INTEGRATION AND COHOMOLOGY

262

odd. But Sn = L3Dn+l has orientation induced by the orientation of D n , hence AISn = an : Sn -, Sn is orientation-preserving if and only if n is odd. THEOREM 8.7.5. The sphere Sn has a nowhere zero tangent vector field if and only if n is odd. PROOF. If n is odd, you constructed such a vector field in Exercise (2) on page 110. Suppose, therefore, that s : Sn -,Rn+' \ (0) is smooth with s(u) 1.v, V v E Sn. Equivalently, s is a nowhere zero section of T(Sn). Note that v cos 8

+ s(v) sin 9 # 0,

V 8 E R, so we can define a smooth map

by F(v, t) =

v cos t.lr + s(u) sin t.lr Ilv cos t.lr + s(u) sint.lrl1'

Then,

so an

-

F(v, 0) = v F ( u , 1) = -v

} vu,

sn,

id and 1 = deg(crn) = (-l)n+l, implying that n is odd.

COROLLARY 8.7.6. Every smooth flow on S2nhas at least one stationary point. For the sphere S2,Theorem 8.7.5 and its corollary are sometimes stated facetiously as "you can't comb the hair on a billiard ball". (People whose sensibilities .) are offended by hairy billiard balls substitute LLcoconut" THEOREM8.7.7. If f : Sn -, Sn is smooth and deg(f) has a jixed point.

# (-I)"+',

then f

PROOF.In fact, we will prove that, if f has no fixed point, then f N a,, hence deg(f ) = (- 1)n+l. We claim that t(f (v) v) # u, Vv E Sn,0 5 t 1. Otherwise, t # 0 and

+

<

f (v) = (1 - t)u/t. Since 11 f (u)ll = I = IIvII, it follows that t = 1/2 and f (v) = v, contrary to assumption. Therefore, we can define F : Sn x [O,1] 4 Sn by

This is a homotopy between f (t = 1) and a, (t = 0).

EXERCISES (1) If f : Sn -, Sn has I deg(f )1 # 1, prove that f has a fixed point and that there is a point that f carries to its antipode.

8.8. POINCARE DUALITY

263

(2) If f : s~~ S2nis smooth, prove that either f has a fixed point or f sends some point to its antipode. (The reader who is familiar with covering space theory will see that smooth maps g : P2"-+ P2n"lift" to smooth maps j : s~~ S2". It follows from the present exercise that g must have a fixed point in P2".) (3) If M is a compact, connected, orientable, boundaryless manifold of dimension n and f : M -+ Rn+l is a smooth imbedding, use degree theory to prove that Rn+l \ f (M) has exactly two connected components and f (M) is the set-theoretic boundary of each. This is the Jordan-Brouwer separation theorem. (Proceed in analogy with Exercise (5) on page 99. In fact, the mod 2 degree theory is adequate for this.) (4) Let 171,0 2 : S1 -+ R3 be smooth maps with disjoint images. Define the linking number Lk(ol, 02) to be the degree of the map f : S1x S1 -+ SZ defined by -+

-+

Intuitively, it seems reasonable to define a1 and a 2 to be topologically unlinked if there is a compact, orientable 2-manifold N with a N = S1 and a1 extends to a smooth map

(or if the parallel condition holds, in which the roles of a1 and 02 are interchanged). Prove that, if a1 and 0 2 are topologically unlinked, then Lk(al, an) = 0. Give an example showing, at least intuitively, that the requirement that N be orientable is necessary. (Hint: Consider the Mobius strip.) 8.8. Poincar6 Duality

Assume that M is a connected, oriented n-manifold with empty boundary. We study the pairing

defined by

The fact that one of the forms is compactly supported guarantees that the integral is defined. One can view this pairing as an R-linear map

where An-k (M)* is the vector space dual of Anek(M) and

264

8. INTEGRATION A N D COHOMOLOGY

If w = dy, some y E A,k-l(M) and dq = 0, it is clear that

by Stokes' theorem. Similarly, if w is closed and q is exact, the integral is 0. Thus, our pairing passes to

,.

Again, this can be interpreted as a linear map

called the Poincare' duality operator.

THEOREM 8.8.1 PO IN CAR^ DUALITYT H E O R E M Suppose ). that M is a connected, oriented n-manifold, d M = 8, and that M admits a finite simple cover. Then the Poincare' duality operator

is an isomorphism, V k . In particular, this holds for M compact and defines cs canonical isomorphism H ( ( M ) = Hn-k ( M ) *= Hn-k(M). O f course, the last equality depends on the de Rham theorem. Remark that, ) , the isomorin the compact case, we can also say that H k ( M )% H ~ - ~ ( Mbut phism is not canonical. Theorem 8.8.1 will be proven by a series of lemmas.

L E M M A8.8.2. If M has the same ordinary and compact cohomology as R n the Poincare' duality operator

,

is an isomorphism.

PROOF.Indeed, if k # n, both H , ~ ( Mand ) H " - ~ ( M ) *are 0, so the assertion is trivially true. If k = n, then [w] E H P ( M ) = R is uniquely determined by JM w, while c E H O ( M )= R is just the constant function e. Then,

That is, PD[w] : R -+ R is just multiplication by JM w, proving that PD is also an isomorphism in this case. The proof of Theorem 8.8.1 will use the Mayer-Vietoris sequences. Let Ul, U2 C M be open subsets. Consider the diagram

8.8. POINCARE DUALITY

265

where a* is a s in the definition of the Mayer-Vietoris sequence for compact supports and j' is as in the dual of the ordinary Mayer-Vietoris sequence.

LEMMA8.8.3. The above diagram is commutative. PROOF.Let [w] E H:(Ul Remark that

n U2) and ([ql].[712]) E H"-k (U1) @ H " - ~(112).

/

ulnu2 w ~ j h 9 = L ~ a e ( w ) ~ v e ,

e = 1,2. T ~ U S ,

=

lUI

a1(w) A

711 -

Juz

a 2 (w)

A

712.

But

giving the asserted commutativity.

LEMMA8.8.4. The diagram

is commutative. The proof is quite similar to the proof of the previous lemma and is left to the reader (Exercise (1)). Remark that the different sign conventions for the Mayer-Vietoris sequences in ordinary and compactly supported cohomology are needed for these two lemmas.

LEMMA8.8.5. The diagram

266

8 . INTEGRATION AND COHOMOLOGY

commutes up to sign. More precisely, d f o PD = ( - l ) k + lPD od*.

PROOF. We fix [w] E H:(Ul U U 2 ) and [q] E H"-*-'(U~ n U2)and verify that

where { X I ,X2} is a partition of unity on U 1 U U2 subordinate to { U 1 ,Uz). In the diagram

we see that P ( k w ,A 2 4 =W d @ d(Xlw, X2w) = (dX1 A W , -dX1 A W ) a(dX1 A w ) == (dX1 A w , -dX1 A w ) ,

where the second equation uses the fact that dX1 + dX2 = d(X1 + X 2 ) = 0. Therefore, d* [w] = [dX1A w] and equation (8.5) follows. ] ) P D ( [ w ] ) ( d[*q ] ) ,we first compute d* [ q ] . In order to compute d f ( P D [ w ] ) ( [ q= Here we consider

and note that

It follows that d*[q]= [-dAl A q ] . Then

8.8. POINCARE DUALITY

267

which is equation (8.6). Here, the sign is due to permuting the 1-form dX 1 past the k-form w. PROOFOF THEOREM 8.8.1. Let {Ui)L=l be a simple cover of M and proceed by induction on r. By the definition of simple covers and Lemma 8.5.6, the case r = I is given by Lemma 8.8.2. If, for a given r 2 2, the assertion has been proven whenever a manifold has a simple cover with r - I elements, then it holds for the manifold U = U1 U - - - U Ur- 1, for Ur , and for U f I Ur (which has the simple cover (Ul n Ur, . . . , U,-1 n Ur)). We must prove it for M = U U Ur. By the Mayer-Vietoris compact cohomology sequence of Theorem 8.5.4 and the dual cohomology sequence of Theorem 8.5.13, together with Lemmas 8.8.3, 8.8.4, 8.8.5, and the Five Lemma (for which commutativity up to sign is fine),

is an isomorphism.

EXERCISES (1) Prove Lemma 8.8.4. (2) Let M be connected, oriented and n-dimensional with d M = 0. Let N C M be a compact, oriented, k-dimensional submanifold with a N = 0, 0 5 k < n, and denote the inclusion map by i : N L) M. If [w]E H* (M), we will write JN [w] for JN i*[w] . (a) Show that there is a unique compactly supported cohomology C ~ W [vN] E e - k ( ~ such ) that

V [ w ] E Hk(M). For fairly obvious reasons,

[vN] is

called the

Poincare' dual of N. (b) If U C M is any open neighborhood of N , prove that the representative, compactly supported form Q N E [qN] can be chosen so that supp(qN) C U. This is the localization principal for the Poincar6 dual of N. (c) If io,il : N -+ M are two smooth imbeddings, we say that they are isotopic if there is a homotopy it : N --, M between io and il such that it is a smooth imbedding, 0 _< t 5 1. In this case we also say that Ne = ie(N) are isotopic submanifolds of M , E = 0 , l . If No and Nl are isotopic submanifolds of M , prove that [vN,,] = [qN1]. (d) Suppose that Pl and P2are compact, oriented, boundaryless s u b manifolds of M of respective dimensions kl and k2 such that n = kl k2. Show that

+

8. INTEGRATION AND COHOMOLOGY

This is called the algebmic intersection number 1(P1, P2)of PI with P2 and is an integer (but you are probably not prepared to prove that). (e) If Pl and P2 as above are isotopic to submanifolds Pi and Pi, respectively, such that Pi n Pi = 0,prove that (PI, P2) = 0. (f) If P is a compact, orientable n-manifold without boundary, let Ap c P x P be the diagonal, A p = { (x, x) I x E P ) . If P has a nowhere vanishing vector field, prove that L(Ap , Ap) = 0. (3) In part ( f ) of the previous problem, you proved half of the Poincark Hopf theorem: There is a nowhere vanishing vector field on P if and , = 0. Assuming this theorem, prove the following. only i f L ( A ~Ap) (a) The diagonal Ap C P x P can be isotoped completely off of itself if and only if its algebraic self intersection number L(Ap, Ap) vanishes. (In particular, by Theorem 8.7.5, in S2kx S2kthe diagonal cannot be isotoped completely off of itself.) (b) Every compact, orientable, odd dimensional manifold without boundary has a nowhere vanishing vector field. 8.9. The de Rharn Theorem

We will prove the following case of Theorem 8.2.20.

THEOREM 8.9.1 (DE RHAMTHEOREM).If M is a manifold without boundary which has a finite simple cover, then the de Rham map

is a canonical isomorphzsm of gmded vector spaces. The proof follows exactly the pattern of proof of Theorem 8.8.1. LEMMA8.9.2. If the n-manifold M has the same singular homology and cohomology as Rn, then the de Rham homomorphism

is an isomorphism. PROOF.If n = 0, the result is immediate, so we assume n > 0. If k # 0, both H ~ ( M and ) Hk(M)* are 0, so the assertion is trivially true. Finally, H0(M) = ZO(M)= R is the space of constant functions on M and Ho( M ) = R has canonical basis the singleton {[XI), where x E M is fixed but arbitrary. If c E H O ( M ) then , DR(c) ([x]) = c(x) = c, so DR is an isomorphism as claimed. We will use the appropriate Mayer-Vietoris sequences. Let Ul ,U2 C M be open subsets. Consider the diagram

8.9. DE RHAM THEOREM

with d* as in Theorem 8.5.2 and

a'

as in Theorem 8.5.13.

LEMMA8.9.3. The above diagram is commutatioe. PROOF.Let [w] E H k ( U l n U2) and [z] E Hk+1 (U1 U U 2 ) Let U denote the open cover { U 1 ,U 2 ) of Ul U U2. We can choose the representative cycle z E [ z ]to b e u s m a l l . That is, z = zl +z2, where zi E Ck+l(U,), i = 1,2. Note that zl and z2 may not, individually, be cycles. All that is required is that a ( z 1 + z2) = 0, SO

a ( z 1 ) = -a(za)

E

Ck(U1 n U2).

As a singular chain, z2 is a linear combination of singular simplices s 1 , . . . ,s,, where S i : Ak+1 + U2,

15 i 5 9.

We define the suppod of this chain to be

a compact subset of U2. It is easy to choose a smooth partition of unity { A 1 , X2) on Ul U U2, subordinate to U and having the property that X 2 = 1 on 1 z2 1, hence X 1 = O on Iz21 Fkmark that, for 15 i 5 q,'

-

proving that dX2 0 on Iz2 1. Similarly, dX1 Given these choices, we will show that

-

0 on 1 z2 1.

Since [wj and [z]were arbitrary elements of the respective vector spaces, commutativity of the diagram will follow. In the diagram

we see that

j(X2w, -X1w) = w d @ d(X2w, -Xlw) = ( d X 2 A W , dXz A W ) i(dX2 A w ) = (dX2 w , dX2 A w ) ,

270

8. INTEGRATION AND COHOMOLOGY

where the second equation uses the fact that dX1 = -dX2. Therefore, d* [w] = [dX2 A w ] and

since dX2 vanishes identically on 1 z2 1. This is equation (8.7). In the diagram

we see that

where the second equation uses the fact that [ a z l ] . Then,

atl

= -at2. It follows that

& [ z ]=

since azl = -dz2 and dX1 vanishes identically on )z21. This is equation (8.8). The following two lemmas have easier proofs (Exercise (1))because the connecting homomorphisms are not involved.

LEMMA8.9.4. The diagram

is commutative.

8.9. DE RHAM THEOREM

LEMMA 8.9.5. The diagram

is commutative.

PROOFOF THEOREM 8.9.1. Let {Ui)r=l be a simple cover of M and proceed by induction on r . By Lemma 8.5.6, the case r = 1 isgiven by Lemma 8.9.2. If, for a given r 2 2, the assertion has been proven whenever a manifold has a simple cover with r - 1 elements, then it holds for the manifold U = U1 U - - - U U,-l, for U,, and for U n U, (which has the simple cover {Ul n U,, . . . ,U,-l n U,)). We must prove it for M = U U U,. By the Mayer-Vietoris cohomology sequence of Theorem 8.5.2 and the dual homology sequence of Theorem 8.5.13, together with Lemmas 8.9.3, 8.9.4, 8.9.5, and the Five Lemma, DR : H ~ ( M )+ Hk(M)* is an isomorphism. There are many versions of the de Rham theorem. The version we have proven identifies de Rham theory as a graded vector spuce with the dual of singular h e mology. Actually, this latter can be defined directly from a singular cochain complex (the d u d of the singular chain complex) and is called singular coh* mology. There is a natural graded algebra structure in singular cohomology (the multiplication is called "cup product" for some obscure reason) and a stronger version of the de Rham theorem asserts that DR is an isomorphism of graded algebras. Also, the requirement that 6 M = 0 was convenient for our approach, but is quite inessential. It can be proven that singular homology and cohomology, defined using d of the continuous singular simplices instead of only the smooth ones, gives exactly the same result. The approach is to approximate continuous simplices with smooth ones, showing that every (co)homology class has a smooth representative (co)cycle. A very interesting consequence is the following. THEOREM 8.9.6. The de Rham whomology algebra H * ( M ) depends only on the underlying topological manifold M , not on the choice of d8erentiable structure. Another form of the de Rham theorem (cf. Appendix E) asserts that H *(M) is isomorphic to the Cech cohomology algebra H * ( M ) , a cohomology theory fashioned out of the family of open subsets of M. Thus, H* (M) is a purely topological invariant of M , so this de Rham theorem gives another proof of Theorem 8.9.6. We discuss here the equality HP(M) = HP(M) for p = 0 , l . This will also motivate the use of the term "cocycle" in our earlier discussion of differentiable structures (Definition 3.1.10) and in vector bundle theory (Definition 3.4.2), as well as the cohomology notation H1(M; Gl(n)) in Theorem 3.4.6. Let U = {U,),Ea be an open cover of the manifold M. A ~ e c O-cochain h on 2-4 is a function B which, to each U, E 2.4, assigns a real number Baa = 8(Uaa).

272

8. INTEGRATION AND COHOMOLOGY

A Cech l-cochain on U is a function y which assigns a real number

to every ordered pair (U,, , U,,) of elements U, , Ua, E U such that U, n U,, # 0. Similarly, a Cech 2-ochain C assigns a real number C,,, to each ordered triple (Uao, U,, ,Ua2) of elements of U with U,, n U,, n U, # 0. The general pattern is clear, but we will stick with p-cochains for p = 0,1,2. The set of p-cochains is denoted by c ~ ( u ) . As real-valued functions on a set, p-cochains can be added and they can be multipiied by real scalars. This makes C ~ ( Uinto ) a vector space over R. Define ~ e c coboundary h operators

It is a moment's work to check that 62 = 0, so we obtain the space ZP(U) of p-cocycles, the space BP(U) of p-coboundaries, and the pth cohomology space H ~ ( u ) ,for p = 0 , l .

REMARK.A cochain y condition

E

~ ' ( 2 4 )is a cocycle precisely if it satisfies the cocycle

Yav = 'Yap f Ypv, whenever Ua n Up n Uv # 0. The Gl(n)-cocycles in bundle theory had a completely analogous definition, except for the multiplicative notation forced by the multiplicative structure of Gl(n). Indeed, if is a G1+(1)-cocycle on U ,then logo? is a Cech l-cocycle on U. The set of G1(1)-cocycles forms an abelian group under operations inherited from G1(1), but, for n > 1, the Gl(n)-cocycles do not form a group of any kind because of the noncommutativity of Gl(n).

LEMMA8.9.7. If each Ua E U is connected, the space fiO(U) is canonzcally isomorphic to the space of locally constant, real-valued functions on M . In particular, HO (u) = H0 (M). PROOF. Remark that k0(u)= z0(u) and that this is the space of k o c h a i n s 0 such that 8, = Bp whenever U, n Up # 0. Thinking of 8 , as a constant function on U,, V a E 31, we see that these constant functions agree on overlaps of their domains, hence unite to form a coherent locally constant function 8 on M. Conversely, if 8 : M -, R is a locally const ant function, its restriction 8, = 81Ua is constant by the connectivity of U, , V a E 24. Remark that simple covers U satisfy Lemma 8.9.7.

LEMMA8.9.8. If U is a simple cover, there is a canonical linear isomorphism F(u) =H~(M).

8.9. DE RHAM THEOREM

273

REMARK. It is an immediate consequence of Lemmas 8.9.7 and 8.9.8 that &P(u) does not depend on the choice of simple cover, hence this vector space can be denoted by HP(M),p = 0 , l . For the case p = 0, the purely topological condition on U in Lemma 8.9.7 proves that H O ( M )and HO(M) are topological invariants. However, the definition of a simple cover requires a differentiable structure, so we cannot conclude from Lemma 8.9.8 that H (M) and H' (M) are h involves topological invariants of M . The proper definition of ~ e c cohomology passing to an algebraic limit over the directed set of all open covers of M, thus obtaining a true topological invariant. We will show that every open cover has a simple refinement (Section 10.5) and, in Appendix E, use this fact to prove Theorem 8.9.6. We sketch the construction of the isomorphism in Lemma 8.9.8 and leave verification of several details to the exercises. Fix the choice of simple cover u = {Ua),Ea We define a linear map

Given [w] E H1(M), select a representative w E [w]. By simplicity, H1(U,) = 0, so the restriction w, = wlU, of the closed 1-form w is exact, V a E %. Thus, we can choose fa E AO(U,) such that w, = df,, V a E a. On U,, n U,, # 0, d(fao - f a l ) = w - w 0, so fao - fal is locally constant on U,, n U,,. The cover being simple, this set is connected, so fa, - f,, = ,c E R is a constant. This defines a Cech lbcochain c E ~ ' ( 2 - 4 ) .But

on U,, fl U,, n U,, , so c E 2' (U). If [c] E we can set

g1(2.4) depends only on [w] E

H' (M),

cp([wl) = lcl. LEMMA8.9.9. The class [c] defined above is independent of the choice of representative w E [w] and of the choices of f, E Ao(U,) such that df, = w,. Consequently, cp is a well defined linear map. The proof of Lemma 8.9.9 will be Exercise (2). We define a linear map

For this, we will need to fix the choice of a smooth partition of unity subordinate to U. Given [c] E H'(u), choose a representative cocycle c E [el. For each a0 E a, define f,, E A0(u,,) by

Then, on U,, fl U,, # 0,

2 74

8. INTEGRATION AND COHOMOLOGY

It follows that df,, = df,, on U,, n U,, , so these exact forms assemble to give a well defined locally exact 1-form w f Z1(M). If [w] E H' (M) depends only on [c] E H' (u),we can set $([el) = [w].

LEMMA8.9.10. The class [w] defined above is independent of the choice of representative c E [c]. Consequently, $ is a well defined linear map. The proof of Lemma 8.9.10 will be Exercise (3). LEMMA8.9.11. The homomorphisms cp and $ are mutually inverse.

PROOF.Given [c] E H'(u), the definition of $([c]) = [w] produces functions f, E A0(u,)such that wlU, = df,, V a E 2l. Using this choice in the definition of cp([w]) gives back the representative cocycle c. That is, cpo$ = id. For the reverse composition, the definition of cp([w]) = [c] selected the functions f , E A1(U,) such that all df, = wl U, and fa, - fa, = c,,,, . The definition of $ produces different functions

where

The closed form 3 obtained by piecing together the exact forms t o w by 3-w=dh,

dj,

is related

so [Z] = [w], proving that $ o cp = id.

In particular, although q!.J was defined relative to a choice of partition of unity, it inverts cp which did not depend on that choice, so q!.J is, in fact, independent of the choice also. We close this section with some remarks about triangulations and cohomology. Let S = {eo,el, . . . ,en) be the set of vertices of the standard n-simplex A,. The convex hull of any subset C C S of cardinality p + 1 is a p-simplex. It lies in the boundary of A, and will be called a p-face of A,. The natural ordering of the indices of the points ei, E C defines a canonical identification of this p-face with the standard p i m p l e x Ap. More precisely, this natural ordering defines a canonical linear imbedding A, r A, with image the given p-face. If s : A, + M is a singular nsimplex, its p-faces are the singular p-simplices obtained by restricting s to the p-faces of A,. Recall from Section 1.3 the fact that compact surfaces can be triangulated. A corresponding theorem for compact, differentiable n-manifolds also holds. That is, the manifold can be divided up into a union of smoothly imbedded nsimplices A:, . . . A:, any two of which either do not meet at all or meet along exactly one common lower dimensional face. This theorem is intuitively plausible, but rather difficult to prove. If A stands for a choice of triangulation of M , we obtain a chain subcomplex (C) (M), a) c (C, (M), a) by using only those singular simplices that are the

8.9. DE RHAM THEOREM

275

inclusion maps of simplices of the triangulation. (By the simplices of the trimgulation, we mean all of the p-faces of the nsimplices of A, 0 < p n.) This is called the simplicial chain complex associated to the triangulation A. Remark ) a finite dimensional vector subspace of Cp(M), 0 2 p 5 n, and that C ~ ( M is vanishes if p > n. Let

<

be the inclusion map, a homomorphism of chain complexes, and let Hf ( M ) be the homology of the simplicid chain complex. The following theorem is standard in algebraic topology (cf. [37, p. 1911, where it is proven more generally for simplicid complexes.) THEOREM 8.9.12. The inclusion homomorphism i A induces a canonical isomorphism Hf (M) = H,(M). The beauty of this result is that the problem of finding the homology of compact manifolds is reduced to a finite set of computations. Note that the theorem assures independence of the choice of triangulation, so one normally chooses A to have the fewest possible simplices. The triangulation of S2depicted in Figure 1.6 (on page 11) has the fewest simplices of any triangulation of S2 . Triangulations can be used to give a proof, without appeal to Riemannian geometry, of the existence of a simple cover of a compact manifold. We do not pursue this, but remark that it leads to a very simple proof that the ~ e c cohomology h of this simple cover and the simplicid cohomology (Hf (M))* are canonically isomorphic.

EXERCISES (1) Prove Lemmas 8.9.4 and 8.9.5. (2) Prove Lemma 8.9.9. (3) Prove Lemma 8.9.10. (4) Using the minimal triangulation of S2 depicted in Figure 1.6, give a direct computation of the homology of S2. (5) Fix a triangulation A of the compact n-manifold M and let cp denote the number of psimplices of A, 0 5 p n. Note that cp = dim C ~ ( M ) . Let hp = dim H : (M) = dim Hp( M ) (called the pth Betti number of M ) . Define the Euler charactemstics of A and M by

<

respectively. (a) Prove that x(A) = X(M). Thus, this important topological invariant can be computed from a triangulation, but does not depend on the choice of triangulation.

8. INTEGRATION AND COHOMOLOGY

(b) Compute x(S2) and give an intuitive proof, not using part (a), that this number is independent of the choice of triangulation. (c) Prove that x(M) = 0 for compact, odd dimensional manifolds M. (d) In fact, it can be proven that, if M is orientable,

the algebraic self intersection number of Exercise (2), part (d), on page 267. The Poincar&Hopf theorem (Exercise (3) on page 268) then asserts: There is a nowhere vanishing vector field on M zf and only if the Euler characteristic of M vanishes. (In fact, the orient ability condition, required in the earlier statement, can now be dropped.) Assuming this theorem, give a new proof that S admits a nowhere vanishing vector field if and only if n is odd.

CHAPTER 9 Forms and Foliations

In Section 4.4, we proved the vector field version of the Frobenius integrability theorem: a k-plane field E o n a manifold M is integrable if and only i f r(E) C X(M) is a Lie subalgebra. In this chapter, we develop an equivalent version of this theorem, stated in terms of the Grassmann algebra A*(M) of differential forms. Useful consequences of this point of view will be treated. 9.1. The Frobenius Theorem Revisited

Let M be an n-manifold without boundary and let E C T ( M ) be a smooth k-plane distribution on M.

DEFINITION 9.1.1. For each integer p 2 0, the degree p annihilator of E is

where we understand that, for p = 0, IO (E) = 0. The annihilator of E is

It is clear that I ( E ) is a graded true.

C" (M)-submodule of A* (M), but more is

LEMMA9.1.2. T h e annihilator I(E) is a 2-sided graded ideal in A * (M). Indeed, this follows by applying the following lemma fiber-by-fiber.

LEMMA9.1.3. Let V be a finite dimensional vector space and let E C V be a subspace. Then, the annihilator

is a 2-sided graded ideal in A(V*)

9. FORMS AND FOLIATIONS

278

PROOF. Let dim V = n, dim E = k, and let {e . . . , ek, f i , . . . ,f n - k ) be a basis of V with ei E E, 1 5 i 5 k. Let {e;, . . . , e;, f;, . . . , f i - k ) be the dual basis of V*. Then, in particular, { f i,. . . , f i - k ) is a basis of I1(E). Consider

where 1 5 il < - - .< 2, 5 k, 1 5 jl< . . < j,-, 5 n - k, and it is allowed that either p = 0 or p = r. If p # r , it is clear that q vanishes on anything of the form em, A - - . A emr, hence it vanishes on all E AT(E). If p = r , q ranges over a basis of Ar (E*). It follows that the set of all the forms q, with p # r, is a basis of Ir(E),r 2 1, clearly implying that I(E) is a 2sided graded ideal. 0

<

DEFINITION 9.1.4. A graded ideal 3 C A* (M) is a differential graded ideal if d(3) G 3. THEOREM 9.1.5 (THE FROBENIUS THEOREM).The following are equivalent for a k-plane distribution E C T(M): (1) E is integrable; (2) I ( E ) is a dzflerential graded ideal; (3) d(ll(E)) C 12(E). For the proof, we need the following. LEMMA9.1.6. Let w E A1(M) and let X, Y E X(M). Then

PROOF. Define i;s : X(M) x X(M) -,Cm(M)

by the formula

This is clearly R-bilinear and antisymmetric. We claim, in fact, that i;s is Cm(M)-bilinear. Indeed, let f E Cm(M) and compute

By antisymmetry, we also have z(X, f Y) = - z ( f Y,X) = -f g(Y,X ) = f z ( X ,Y). Thus, i~ E P ( M ) . By antisymmetry, iJ E A2(M).

9.1. FROBENIUS REVISITED

279

In order to prove the equality of Z;s and dw, it will be enough to show that . . . ,xn). both forms restrict to the same 2-form on any coordinate chart (U, x ', (The previous paragraph was needed so that cIU would make sense.) Write wlU =

cgi dx'

and remark that no generality is lost in assuming that w is of the form g dxi. By permuting the coordinates, assume that w = g dx I , so

Thus, if k

< j,

Since 1 5 k

< j, we have dxl(a/axj)

= 0, so

Since dw[U = JIU for an arbitrary coordinate neighborhood, dw = J.

PROOFOF THEOREM 9.1.5. We prove that (1) (2). Thus, it is assumed that E is integrable and we must prove that, if w E IQ(E),then dw f (E). By the integrability condition, it will be enough to prove this in a coordinate chart (U, xi, - - - xn) such that {a/axl, . . . , a/axk} spans r ( E IU). Then {dxk+l,.. . , dxn) spans I1(EIU). In these coordinates, we write IQ+'

Then, if 1 5 i l

< - - < i, I k,

280

9.

FORMS AND FOLIATIONS

so every nonzero term in the expression for w l U contains at least one dxZj E I 1 ( ~ I U ) The . same will then hold for dwlU and, I(EIU) being an ideal, we see that dulU E Iq+'(EIU). Covering M with such charts, we conclude that dw E I Q + ~ ( E ) . The implication (2) + ( 3 ) is trivial. We prove that (3) + (1). Thus, we are given that d(I1( E ) ) C ( E ) . Let X , Y E r ( E ) be given and choose an arbitrary element w E I1( E ) . Then, since dw E I ~ ( E ) ,

~0

= -w([X,

~0

Y]).

Since w E I 1 ( E ) is arbitrary, it follows that [X,Y] E r ( E ) . Since X, Y E r ( E ) are arbitrary, it follows that r ( E ) E X ( M ) is a Lie subalgebra. By the vector field version of the F'robenius theorem, E is integrable. We note that Lemma 9.1.6, which played a key role in the above proof, is a special case of the following general formula which you are asked to prove in Exercise (1).

THEOREM 9.1.7. If w E A 4 ( M ) and X I , - - - , Xq+l E X ( M ) , then

REMARK.If q = 0, the formula in Theorem 9.1.7 is understood to reduce to

V f E A O ( ~ For ) . q 2 1, the formula is noteworthy in that it gives a completely coordinate free definition of the exterior derivative. It is useful in other ways, one of which is given in the following example.

9.1.8. The formula in Theorem 9.1.7 is closely related to one that EXAMPLE occurs in a purely algebraic context. Let C be a finite dimensional Lie algebra over R and let A(C*) be the exterior algebra of the dual vector space C * . Thus, Aq(C*) is the dual space of AQ(C)or, equivalently, the space of antisymmetric q-linear functionals on C. One defines a coboundary operator

by the formula

where w E Aq(CC)is viewed as a multilinear functional and vi E C is arbitrary, 1 5 i 5 q+ 1. The antisymmetry of Sw is part of Exercise ( 3 ) ,so 6w E AQ+l(C*).

9.2. TRANSVERSALITY

28 1

In Exercise (3), you are also asked to prove that S2 = 0, SO (A(C*),5) is a cochain complex. The cohomology of this complex is denoted by H * (C) and is called the cohomology of the Lie algebra 2. This is of considerable interest in algebra, but we want to remark on its use in studying the de Rham cohomology of Lie groups. If G is a Lie group and we apply the above construction to its Lie algebra C = L(G), we obtain a finite dimensional cochain complex (A(L(G)*), 6). Remark that the subspace of left invariant q-forms in AQ(G)is canonically isomorphic to AQ(Te(G)*)= AQ(L(G)*).This can also be identified as the subspace of all w E AQ(G)such that w (XI, .. . ,X , ) is a constant function, for each choice of left invariant vector fields X I , . . . ,X, E X(G). The formula for dw in Theorem 9.1.7, when evaluated on L(G) c X(G), reduces to Sw, Vw E AQ(L(G)*).It follows that there is an injective homomorphism

of cochain complexes. There is induced a canonical homomorphism

of graded algebras and a surprising theorem, which we will not prove, asserts that, if G is both compact and connected, then L is an isomorphism (cf. [13, Chapter IV]). Since L(G) is completely determined, as a vector space, by T, (G) and, as a Lie algebra, by the Lie derivatives at e of left invariant vector fields, it follows that the cohomology of the manifold G is entirely determined by infinitesimal data at e. This is a remarkable case of recovering global data from linear approximations at a single point.

EXERCISES (1) Prove Theorem 9.1.7. (2) Let F be a foliation of codimension q and integral to the distribution E. Prove that the exterior product of any q 1 elements of I ( E ) vanishes identically. (3) For w E AQ(C*),prove that the coboundary operator 6,defined in Example 9.1.8, does produce an element Sw E AQ+l(C*).Prove also that 62 = 0. (4) Using the theorem cited in the remark above, compute the de Rham cohomology algebra of the torus Tn. (5) Prove that, if G is a Lie group, the set of connected Lie subgroups K C G corresponds one to one to the set of graded ideals 3 C A(L(G)*) such that S(3) C 3.

+

9.2. The Normal Bundle and Transversality

Let I1C A1(M) be a Cm(M)-submodule which is closed under locally finite sums. For example, if E C T(M) is a k-plane distribution on M, we might take I1= I 1 ( E ) .For each x E M , set

9. FORMS AND FOLIATIONS

282

LEMMA9.2.1. For each x E M , Qx is a vector subspace of T,*(M). This is rather obvious. Indeed, if we view I as a vector space over R, we see that w H wX defines an R-linear map I' -+ T,+(M) with image Q,. Let qx = dim Qx .

DEFINITION 9.2.2. The Coo(M)submoduleI' C A1(M) has constant rank q i f q , = q , V x ~M. LEMMA9.2.3. Let M be a n n-manifold and let I' E A1(M) be a C m ( M ) submodule that is closed under locally finite sums. Then the following are equivalent:

( 1 ) I' has constant rank q; ( 2 ) I' = I'(Q), for some q-plane subbundle Q T*(M); ( 3 ) I' = I I (E), for some (n - q ) -plane subbundle E C T(M). The proof of this lemma will be relegated to Exercise (1).

DEFINITION 9.2.4. If E is a p-plane distribution on M, set q = n-p, the codimension of E. Then the q-plane subbundle Q & T'(M) such that I 1(E) = r ( Q ) is called the normal bundle of E. If E is integrable and 7 is the corresponding foliation, then Q is called the normal bundle of 7 and q is called the codimension of F (codim3). This terminology comes from the following observation.

LEMMA9.2.5. Let E be a p-plane distribution o n M and mannian metric o n M . Let

E,I = {V V x E M . Then

EL =

E

fi a choice of Rie-

Tx(M) u IEx},

U E:

T(M)

ZE M

is a vector subbundle isomorphic to the normal bundle Q. PROOF.Denote the Riemannian metric on Tx(M) by (., isomorphism cp : T(M) + T*(M) of bundles by

and define an

It is clear that cp carries E~ onto Q. Our way of defining the normal bundle as a subbundle of the cotangent bundle is intrinsic. Defining it in T(M) via a Riemannian metric gives a subbundle that depends on the choice of the metric.

DEFINITION 9.2.6. Let M be a manifold with a foliation 3that is integral to the distribution E . A smooth map f : N + M is transverse to 3 if, V x E N, ftx(Tx(N))U Ef spans Tf (,) (M). In this case we write f rh 7 .

9.2. TRANSVERSALITY

283

Remark that no assumption is made about the relative dimensions of N and M other than what is implicit in the definition: dim N must be large enough that its sum with the dimension of the leaves of F is at least as large as dim M . Remark also that a submersion f : N + M is automatically transverse to every foliation of M .

LEMMA9.2.7. Let F be a foliation of M with normal bundle Q . A smooth map f : N + M is transverse to F i f and only if f: : T;(,) ( M ) + T,*( N ) is one to one on Q f ( = )V, x E N . PROOF. Assume that f h 3. If a E Q f ( z )and f z ( a ) = 0, then, for every v E Tx(N), a ( f * x ( v ) )= f3C*(a)(v)= 0' But a also vanishes on E f (), hence, by transversality, it vanishes on all of T f ( M ) . That is, a = 0. For the converse, suppose that f $ : Qf(,) + T,'(N) is one to one. Let v E T f(,)(M). We are to prove that v is the sum of an element of E f),( and an element of f,,(T,(N)). Choose a basis { a l , . . . , a q ) C Qf(,) and let a, = a,(v), 1 _< i 5 q. Since { f z ( a l ) ,. . . ,f$(a,)) is linearly independent, there is a vector w E T z ( N )such that a* = f,' ( a i ) ( w ) ,1 i 5 q. That is,

<

from which it follows that f,,(w)

-

v E Ef(,)

THEOREM 9.2.8. Let 3 be a foliation of M , codimF = q, and let f : N + M be smooth and transverse to F . Then there is a canonically defined foliation f -'(F) of N of codimension q such that f carries each leaf of f (F)into a leaf of F .

PROOF. Let 3 be integral to the distribution E and have normal bundle Q. Then f * ( I 1 ( E ) )C_ A 1 ( N ) is a vector subspace and we let I' A ' ( N ) be the Cm(N)submodule, closed under locally finite sums, that is generated by this subspace. An arbitrary element r] E I' can be written locally as

'

where qi = f *(u*) and ui E I ' ( E ) = I'(Q), 1 5 i 5 l . By Lemma 9.2.7, I will have constant rank q, so we can write

as in Lemma 9.2.3. It is clear that f * ( I ( E ) ) q E I'(@ locally as above, we see that, locally,

I ( @ . Furthermore, writing

9. FORMS AND FOLIATIONS

284

since f * (hi) E f * (I(E))by the integrability of E (Theorem 9.1.5). But this implies that E is integrable, again by Theorem 9.1.5. Define f -'(3) to be the foliation integral to E . The normd bundle is 6,so codim f -'(F) = q. It remains to be shown that each leaf of f -l(.F) is carried by f into a leaf of 3 . Since the leaves of 3 (respectively, of f - l ( 3 ) ) are maximal connected integral manifolds to E (respectively, to it will be enough to show that f*.(Ex) G Ef(,), V x E N. But, if a E Qt(,) and v E gx,then

E),

h

since f j( a ) E V V E Ex.

Qx.Since a E Q

(x)

is arbitrary, this proves that f ,,(v) E Ef(,),

EXAMPLE 9.2.9. Let f : N + M be a submersion, dim M = q, dim N = n. Then, as y ranges over M , the connected components of the submanifolds f -'(y) range over the leaves of a foliation of N of codimension q. Indeed, the unique 0-plane distribution on M is trivially integrable, the leaves of the corresponding foliation 3 being the points of M. This foliation is of codimension q, so there is a pull-back foliation f -'(3) on N of codimension q. Each leaf L of f -'(3) is a connected submanifold of N of dimension n - q and is carried by f into a point y of M. By the constant rank theorem, f -l(y) is also a submanifold of dimension n - q, so L must be a connected component of f EXAMPLE9.2.10. Let El, E2 C T ( M ) be integrable subbundles with corresponding foliations Fi of codimension qi, i = 1,2. Let Ei have fiber dimension pi = n - q i , i = 1,2, where n = dim M. If, for each leaf L of F1,the inclusion map L : L M (a one to one immersion) is transverse to 3 2 , we will say that F1 is transverse to 3 2 and write 3 1 rh 32. This simply means that, for each x E M , El, U EZxspans T,(M), so the relation is symmetric (31rh F2 F2rh F1). In this case, if x E M and L~ : L, M is the inclusion of the leaf through x of 31, then L ; ~ ( F ~ ) is a foliation of L, of codimension 92. Thus, the leaves of ~ ~ ~ ( have 3 2 tangent ) spaces contained in the restriction of E2 to L,(L,). These tangent spaces also lie in El and their dimension is p l - 92 = pl p2 - n, the fiber dimension of the bundle El n E2. Since each point of M lies in a leaf of El, it follows that El n E2 is integrable and that the leaf through x E M of the corresponding foliation is the the leaf through x of ~ ; ' ( 3 ~ ) .These leaves are just the connected components of the intersections (when nonempty) of leaves of F1with leaves of 3 2 . We can denote this foliation of M by F1n 32. It is of codimension n - (pl p2 - n) = ql q2.

-

-

*

+

+

+

DEFINITION 9.2.11. A p-plane distribution E on M is transversely orientable if its normal bundle Q is orientable. A foliation 3 of M of dimension p is transversely orient able if it is integral to a transversely orient able p-plane distribution. If 3 is transversely orientable and M is orientable, then each leaf of 3 is an orientable manifold. Indeed, T(M) E' E @ Q and it follows easily that E is an orientable vector bundle. Thus, the tangent bundle to a leaf L, being the restriction of E to L , is orientable. REMARK.

9.2. TRANSVERSALITY

285

PROPOSITION 9.2.12. If 3 is a transversely orientable foliation of M and f : N + M is transverse to F,then f 1 1 ( 3 ) i s transversely orientable.

a

-'

PROOF. Let Q be the normal bundle of 3and the normal bundle of f (3). These are q-plane subbundles of the cot angent bundles of M and N, respectively. Since Q is orientable, there is a nowhere vanishing section w of A4(Q). By Lemma 9.2.7, f * (w) is a nowhere vanishing section of hq proving that is orient able. 0

(a),

8

EXERCISES (1) Prove Lemma 9.2.3. (2) Let E C T ( M ) be a p-plane distribution on the n-manifold M, and let Q T*(M) be its normal bundle, a q-plane bundle where q = n - p. Assume that E is transversely orientable, hence that there is a nowhere zero q-form w E r(AQ(Q)). (a) For each x E M, prove that Ex is the set of all vectors v E Tx(M) such that W , ( V A V ~ A . . . A V , - ~ ) = 0, for all choices of v l , . . . ,vq-1 E T,(M). We call Ex the nullspace of w, and we also say that E is defined by the partial differential equation (P.D.E.) w = 0. If E is integrable, the leaves of the foliation F integral to E are said to be the maximal solutions to the P.D.E. w = 0. In this case, w is said to be integrable. (b) Prove that w is integrable if and only if there is a form q E A (M) such that dw = q A w. (c) Let the foliations Fl and 3 2 in Example 9.2.10 be transversely orientable and let the P.D.E. wi = 0 define the bundle Ei, i = 1,2. Show that the P.D.E. wl A w2 = 0 defines El n E2 and verify the integrability condition

c

'

as a direct consequence of the integrability of w 1 and w2. (3) Let 3be a transversely orientable foliation of codimension q with normal bundle Q and tangent distribution E. Let w E I'(AQ(Q)) be nowhere zero. Let q E A1(M) be such that dcJ = q A w. (a) Prove that dq E 1 2 ( ~ ) . (b) Prove that q A (d71)QE A2Q+l(M) is a closed form. (c) Show that [q A (dq)q] E H2q+'(M) does not depend on the allowable choices of w and of q. (Hint: First hold w fixed and prove independence of the choice of q. Then prove independence of the choice of w.) This class is denoted by gv(3) and called the Godbillon- Vey class of F. (d) If f : N + M is transverse to 3,prove that

This is called the naturulity of the Godbillon-Vey class.

9. FORMS AND FOLIATIONS

286

The Godbillon-Vey class was discovered in the early 1970s, leading to a formidable body of research into the algebraic topology of foliations. 9.3. Closed, Nonsingular 1-Forms

The topology of foliations is a fascinating and subtle topic. In this book we can only scratch the surface, but, with the tools developed so far, there are some interesting questions about foliations of codimension one that are accessible. One of these st arts with the question: which compact, connected, boundaryless n-manifolds M admit closed, nowhere zero I-forms w ? A nowhere zero form is also said to be nonsingular. Throughout this section, we fix the hypothesis that the n-manifold M is compact and connected with aM = 0.

LEMMA9.3.1. In order that M , as above, admit a closed, nonsingular form w E A1 (M), it is necessary that H1(M) # 0. Indeed, such a form determines a nontrivial element [w] E H1(M).

PROOF.If w = df, some f

Cm(M), the compactness of M implies the existence of a critical point x E M of f . For instance, a point where the maximum is attained will do. But df, = 0 at any critical point x, contradicting the assumption that w = df is nonsingular. E

It is known, however, that this condition is not sufficient. The following observation relates the question to foliations.

LEMMA9.3.2. Let w define

E

A1(M) be closed and nowhere 0. At each x E M,

E, = {V E Tz(M)

1 W, (v) = 0).

Then E = UzEM E, is an integrable distribution on M.

PROOF.Indeed, let I' A1(M) be the Cm(M)submodule generated by u. That is, I1is the set of all f w , f E Cm(M), hence is of constant rank 1. Then E is clearly the (n - 1)-plane distribution such that I = I1(E). Since w is closed, we have d(fw) = df A w E 1 2 ( so ~) E ,is integrable by Theorem 9.1.5. We let Fw denote the foliation of M corresponding to the closed, nonsingular 1-form w . The codimension of Fw is 1.

EXAMPLE 9.3.3. Since S1 is parallelizable, it admits a nowhere 0 form 13 E A1(S1). By default, dI3 = 0. The corresponding foliation is the zerdimensional foliation having each point of S1 as a leaf. Admittedly, this is not an interesting example, but it leads to a class of interesting examples, namely the manifolds that fiber over S1. That is, we consider a compact, connected, boundaryless n-manifold M, together with a smooth map x : M + S that is locally trivial in a sense quite similar to the local triviality of vector bundles and of principal bundles: there is a compact (possibly not connected) (n- 1)-manifold F without

9.3. CLOSED, NONSINGULAR 1-FORMS

287

boundary such that each point z E S1 has a neighborhood U and a commutative diagram a-'(U) UxF

u

-

u

id

such that cp is a diffeomorphism. We say that n : M + S1 is a fibration (or a fiber bundle with fiber F ) . In particular, a : M + S1 is a submersion and the (n - 1)-manifolds n-'(t) are all diffeomorphic to F. The connected components of these fibers are the leaves of a (codimension one) foliation, as was observed in Example 9.2.9. Also, w = ~ ' ( 8 )is a closed, nonsingular 1-form on M and the foliation by the components of the fibers is exactly FW.

EXAMPLE 9.3.4. Let T3 = S1 x S1 x S1. There are three obvious fibrations T3 + S1 given by (tl,a2, z3) = z i , 1 5 i 5 3. These are trivial cases of

Ti :

Example 9.3.3, but there are more interesting examples of closed, nonsingular forms w E A1(T3). Indeed, let w, = n:(8), 1 5 i 5 3. These are pointwise linearly independent over W,so every nontrivial linear combination w = aiwi is closed and nonsingular. If, when we view R as a vector space over the rationals 0, the set {al, az, as) is linearly independent, then the corresponding foliation FU of T3 has each leaf diffeomorphic to R2 and dense in T3. Similarly, if we require that two of these numbers, say {a 1, a2), be linearly independent over Q, but not all three, each leaf of FW is diffeomorphic to R x S1 and is dense in T3. Finally, if {al) is linearly independent over Q, but {al, a2) and {al, a3) are not, each leaf of FW is diffeomorphic to T~ and these leaves are the fibers of a fibration of T~ over S1. Remark that any triple {al, an, a3) can be uniformly well approximated by triples of this last type. Thus, there is a sense in which the linear foliations of T3 by dense planes or by dense cylinders can be uniformly well approximated by fibrations over Sl. These assertions are left as an exercise.

xLl

DEFINITION 9.3.5. Let 3 be a foliation of M of codimension one. Let the flow : R x M + M be smooth, nonsingular (i. e., no stationary points), with flow lines everywhere transverse to 3 . If, for each leaf L of 3 and each t E R, a t ( L ) is also a leaf of 3,we say that is a transverse, invariant flow for 3 .

LEMMA9.3.6. The flow @ : R x M + M (not necessarily nonsingular and not necessarily transverse to 3) caries leaves of 3 to leaves of 3 if and only if the infinitesimal generator X E X(M) of satisfies [X,r(E)] I'(E), where E c T(M) is the integrable distribution of tangent spaces to 3. PROOF.If carries leaves to leaves, then kt* (Ex) Vt E R. Thus, if Y E F(E), [X, Y] = lim t+O

c

E*-,(,), Vx E M,

a-t*(Y) - Y E r(E). t

For the converse, suppose that [X,r(E)]c r ( E ) and remark that it will be sufficient to show that, in any F'robenius chart (U, x I , . . . ,xn) for 3,at carries

9. FORMS AND FOLIATIONS

288

plaques to plaques for small enough values oft. Here, we assume that the plaques are the level sets xn = const. In these coordinates, we write

The condition that

implies that f n = fn(xn) is independent of x l , . . . ,xn-l. Thus, the local system of O.D.E. for is

Consequently, given the initial condition a = (a1,. . . , a n ) E U, the nth coordinate xn (a, t) of the flow line at (a) depends only on a n and t. That is, the plaque xn = a n is carried into the plaque xn = xn(an,t). THEOREM9.3.7. The foliation 3 admits a transverse, invariant flow if and on19 zf 3= FU, for some closed, nonsingular I-form w E A' (M).

PROOF.Assume that @ is a transverse, invariant flow for F. Let X

E X(M)

be the infinitesimal generator of the flow and define a nonsingular 1-form w by requiring

where E is the distribution of tangent spaces to 3. Clearly, w, spans the normal fiber Q x , Vx E M . We must prove that dw = 0. For this, let x E M and choose a basis {vl, . . . , vn-1) C Ex. We can extend vi to a field Y , E I'(E), 1 5 i 5 n - 1. Since r ( E ) c X(M) is a Lie subalgebra, 1 5 i, j

5 n - 1. Also, since w(X)

1,

&(X, Y , ) = X(w(Y,)) - Y,(w(X)) - w[X, Y,] = -w[X, Y,], 1 5 i 5 n - 1. By the invariance property and Lemma 9.3.6, [X,Yi] E I'(E), and it follows that h ( X , Y , ) = -w[X, Y,] = 0.

In particular, (dw), vanishes on all pairs from the basis {v 1, . . . , vn-1, X,) of T,(M), hence (dw), = 0. Since x E M is arbitrary, dw = 0. Conversely, suppose that 3 = FU,where dw = 0. We must produce the transverse, invariant flow. Cover M by F'robenius charts { U, , xk , . . . ,x ~ } ~ = , such that the 3-plaques in U, are the level sets xR, = const. Then,

9.3. CLOSED, NONSINGULAR 1-FORMS

where f a

# 0 on U,. Set

x,

1

a

fa

ax: '

= --

a vector field transverse to the plaques and satisfying

Let

be a partition of unity subordinate to the Frobenius atlas and set

This vector field satisfies w(X) 1 and, in particular, the flow @ that it generates is everywhere transverse to 3. Furthermore, if Y E r ( E ) , 0 = &(X, Y) = X(w(Y)) - Y(w(X)) - w[X, Y] = -w[X, Y],

implying that [X,Y] E r ( E ) . Since Y E r ( E ) is arbitrary, Lemma 9.3.6 completes the proof that @ is a transverse, invariant flow for F W . Fix the hypothesis that w, FW, and @ : R x M + M are all as above. Let C denote the 1-dimensional foliation of M by the flow lines of @.

LEMMA9.3.8. For arbitrary leaves L and L' of FU,there are values t E R such that Qt(L) n L' # 0, i n which case at carries L di~eomorphicallyonto L'. PROOF. It is clear that Qt(L) n L' # 0 if and only if at carries L diffeomorphicdly onto L'. Thus, we obtain an equivalence relation on the set of leaves by setting L -, L' if and only if there is such a value of t. Since the flow is leaf preserving and transverse to 3, an easy application of the inverse function theorem proves that @ : R x L + M is a locd diffeomorphism, hence it has as image an open subset of M. This image is the union of the leaves equivalent to L, hence, by the connectivity of M, all leaves are equivalent. If L is a leaf of FW , denote by P(L, w) the set {t E

1 @ t(L) = L)-

LEMMA9.3.9. If L and L' are leaves, then P(L, w) = P(L1,w) and this set, call it P ( w ) , is an additive subgroup of R. PROOF. Let t E P(L, w). Let

T

ER

be such that @,(L) = L'. Then

Thus, P(L, w) P(Lt,w) and the reverse inclusion is proven in the same way. This set P(w) carries every leaf to itself. By the properties of a flow, it is clear that P(w) is closed under addition and multiplication by -1, hence is an additive subgroup of R.

9. FORMS AND

290

FOLIATIONS

LEMMA9.3.10. Let {U,,x: ,... ,x:),"=, be a Fkobenius atlas for 3". Let a : [a,b] + M be a piecewise smooth loop. Then a is homotopic to a piecewise smooth loop

T

with the following property: there is a partition

such that, for 1 5 i 5 p, ~ l [ t ~t i-] c ~ ,Uai and this segment either lies in a plaque o f F or in a plaque of C. The proof of Lemma 9.3.10 will be Exercise (2). COROLLARY 9.3.11. The group P(w) is exactly the set of periods of the closed 1-form w . PROOF.If a E P(w), the segment s 1 ( t )= at (xO),where t ranges from 0 to a , has both endpoints in the leaf L of 3through x 0. Let sz be a piecewise smooth path in L from the endpoint of sl to the initial point of s l . Then s = sl sz is a piecewise smooth loop in M and it is clear that J, w = a. That is, a is a period of w. For the reverse inclusion, choose a piecewise smooth loop a and deform it to r as in Lemma 9.3.10. Let r k = TI [pk,vk], 0 5 k 5 r , (taken in increasing order) be the segments of r that lie in C-plaques and let r1[vk,~ k + ~ 0] 5 , k 5 r , lie in 3-plaques, with = 7(pO).Then

+

and a is a period of w. All periods can be obtained in this way. Let Lk denote the leaf through r ( p k ) , 0 5 k 5 r+ 1 (hence Lr+l = Lo). Then @,,(Lk) = Lk+,, O~k~r,and@,(Lo)=Lo,provingthata~P(w).

LEMMA9.3.12. The subgroup P(w) c R is either infinite cyclic or everywhere dense in R. PROOF. Indeed, if P(w) # 0, this follows from Corollary 4.3.10. But P(w) = 0 implies that [w] = 0 (Theorem 6.2.9), contradicting Lemma 9.3.1. THEOREM 9.3.13. If the period group P ( w ) of a closed, nonsingular 1-form w is infinite cyclic, then the leaves of FWare the fibers of a suitable fibration p : M + S 1 . If P(w) is not inJinite cyclic, each leaf of FWis dense in M . PROOF.If P(w) is not infinite cyclic, then Lemma 9.3.12 implies that a dense set of real numbers t has the property that Qt(L) = L, for an arbitrary leaf L of FW.Since @ : R x L + M is onto, it follows easily that L is dense in M. Suppose that P(w) is infinite cyclic and let a E (0, oo) be the smallest positive period. Then, replacing w by w/a, we lose no generality in assuming that P ( w ) = Z. Fix xo E M and consider all piecewise smooth paths a : [a,b] + M with a ( a ) = xo. If a(b) = x, define

In order to see that this is well defined, let T be another such path from xo to x . Let -a denote the path from x to xo obtained by reversing the parametrization

9.3. CLOSED. N O N S I N G U L A R 1-FORMS

of a and consider the loop C = r it follows that ,2ri

J, w

291

+ (-a). Since the period Jcw = k is an integer,

= e2ni ( k + J

b

u)

- e 2 ~Joi ( v )

Furthermore, since M is assumed to be connected, p(x) is defined for every x E M. The reader can verify that p : M + S 1 is smooth. We claim that p(x) is constant as x ranges over a given leaf L of FU.Indeed, given arbitrary points x, y E L, let a be a path from xo to y and let r be a path in L from y to x. But JT w = 0, since w vanishes on vectors in E, so

This shows that pl L is constant. By the definition of P ( w ) , it is clear that p : M + S1 sets up a one to one correspondence between the leaves of 3, and the points of S1. Finally, we prove local triviality. Indeed, given z E S l , let U c S1 be the open arc {ze2"" 1 -1/2 < t < 1/21. Let L, = p-'(2). The map

defined by q(t,w) = a t ( w ) , is a diffeomorphism and the inverse diffeomorphism cp = 1CI-l makes the diagram

commute. Thus, the possibilities illustrated in Example 9.3.4 are the typical ones. The last statement in that example, that the linear foliations of T 3 with dense leaves can be arbitrarily well approximated by fibrations over S is also typical.

',

THEOREM 9.3.14 (TISCHLER). If w is a closed, nonsingular I-form o n M , then there is a sequence { w i ) Z 1 of closed, nonsingular 1-forms with P(wi) E Z, V i > 1 such that limi,, w, = w uniformly o n M.

COROLLARY 9.3.15. There is a closed, nonsingular form w only i f M fibers over S 1 .

f

A 1 ( M ) i f and

This corollary is an answer to the opening question of this section. The proof of Theorem 9.3.14, which will be found in [42], is not difficult, but it uses some facts from algebraic topology that we will not prove in this book.

EXERCISES (1) Prove the assertions in Example 9.3.4. (2) Prove Lemma 9.3.10.

292

9. FORMS AND FOLIATIONS

(3) Let 3 be a foliation of M of codimension 1 and let

be an atlas of Frobenius charts. Thus, on U , dinates has the form

n Up,the change of coor-

(a) If there is a closed, nonsingular 1-form w such that 3 = FW, prove that the Frobenius atlas can be chosen in such a way that the second equation above always takes the form where cap E R is a constant, defined whenever U, n Up # 0. (b) Conversely, if the Frobenius atlas can be chosen as in part (a), prove that there is a closed, nowhere vanishing 1-form w such that 3 = FW. (c) In this situation, show that cap = c(U,, Up) defines a ~ e c cocycle h on the open cover U. (d) It is a fact that, in the above situation, the Frobenius cover U can be chosen to be simple. Assuming this, prove that the de Rham isomorphism H' ( M ) = H' (u), as defined in Section 8.9, identifies [c]E I;I1(u) with [w]E H ' ( M ) .

CHAPTER 10 Riemannian Geometry

Properly speaking, geometry is the study of manifolds that are equipped with some additional structure that permits measurements. For example, nowhere in the definition of a piecewise smooth curve is there anything that would enable us to measure the length of the curve. Likewise, on a compact, oriented n-manifold, we can integrate n-forms, but which of these integrals should be interpreted as the volume of the manifold? And given intersecting curves, how could we measure the angle they make at an intersection point? The additional structure that is needed is a metric tensor, Riemannian metrics and, to a lesser extent, pseudwrtiemannian metrics, being the main examples. Such a tensor makes it easy to define the quantities mentioned above and provides much more. For instance, the metric tensor gives rise to the "LeviCivita connection", which can be thought of as a way of parallel transporting vectors along curves. One is led to study special curves which are "straight" in the sense that the velocity field is parallel along them. These are "geodesics", the analogues in Riemannian geometry of straight lines in Euclidean geometry. These geodesics are locally length minimizing, but this may fail in the global sense. For instance, if two points on a sphere are not antipodal, then, in the standard metric on the sphere, the great circIe through these points is a geodesic. It falls into two imbedded arcs joining the points, one of which is the shortest curve joining them, but the other clearly is not. Parallel transport along curves holds some surprises for Euclidean "flatlanders". For instance, consider the geodesic triangle on S2in Figure 10.1. Imagine that you are a twdimensional native on S2. Starting at point A, you walk down the first leg of the triangle, holding the initial tangent vector straight ahead so as to keep it (in your world) always parallel to its original position. Upon arriving at point B, you start moving sideways along the equatorial geodesic, determinedly keeping the vector pointing in a direction always parallel to its earlier positions. Finally, at C, you start up the last leg of your journey, walking backwards and again holding the vector in front of you in a constant parallel direction. Upon arriving at A, you find that, despite your best efforts not to change the direction of the vector, it ends up pointing in a different direction at A than it started with! Although, at the beginning of this experiment, you

10. RIEMANNIAN GEOMETRY

294

may have been convinced that your world was a Euclidean plane, you now have evidence of intrinsic "curvature7', something you (as a two-dimensional creature) probably cannot imagine, but can nevertheless conceive. If you are Gauss, you may even be able to figure out how to compute the curvature of your world via experiments such as the above.

FIGURE 10.1. A parallel field along a loop on ,S2 10.1. Connections Let Write

U

Rn be open. Given X, Y E X ( U ) , define D x ( Y ) E X(U) as follows.

and define

We can view D as an R-bilinear map

D : X ( U ) x X ( U ) -t X ( U ) .

It has the following properties:

10.1. CONNECTIONS

This is an example of a connection.

DEFINITION 10.1.1. Let M be a smooth manifold. A connection on M is an R-bilinear map V : X(M) x X(M) + X(M), written V(X, Y) = V x Y or Vx(Y), with the following properties: (1) V j x Y = f V x Y , V f E Cm(M), VX, Y E X(M); (2) V x ( f Y ) = X ( f ) Y + f V x Y , V f E Cm(M), V X , Y E X(M).

The connection D, defined above on open subsets U Euclidean connection.

R", is called the

REMARK.By property (I), V is a tensor in the first argument. Thus, if v E Tx(M) and Y E X(M), VvY E Tx(M) is defined. By property (2), however, V is not a tensor in the second argument. Property (2), together with the Cm Urysohn lemma, allows us to prove in standard fashion that V ,Y E T,(M) only depends on the values of Y in an arbitrarily small neighborhood of x. Thus, connections can be restricted to open subsets of M. We describe an important class of examples, the Levi-Civita connections for submanifolds of Euclidean space. Let M C R m be a smoothly imbedded nmanifold. If x E M, then T,(Rm) = Rm canonically. Also,

where vx(M) = {v f TZ(Rrn) I v 1Tx(M)), perpendicularity being defined by the Euclidean metric in R m . Then

is an (rn - n)-plane bundle over M and

is a canonical direct sum bundle decomposition. The summand v(M) is called the normal bundle of M in IRm. The canonical projection

is a surjective homomorphism of bundles. Let X , Y E X(M). Given x E M, there is a neighborhood U of x in R m and extensions X, Y of X 1 (U n M) and Y I (Un M), respectively, to fields on U. Then, (D-p),depends only on 2, = X, and on Y. Represent X, = (s), as x. an infinitesimal curve, where s : (-E, E) + M is smooth and s(0) = x. Then,

--

-

10. RIEMANNIAN

296

GEOMETRY

and this depends only on YI ( U nM ) , not on the choice of extension D x Y is a well defined element of I? (T(Rm) M ).

?. Therefore,

DEFINITION 10.1.2. If X, Y E X ( M ) , then the operator

defined by

vxy = p ( D x Y ) is called the Levi-Civita connection on M

c Rm.

The following is totally elementary, as the reader can check. LEMMA10.1.3. The Levi-Ciwita connection is a connection on M . If V is a connection on M and (U, x l , . . . ,xn) is a coordinate chart, set Ji = d / d x i and write

10.1.4. The functions rFj E C m ( U ) are called the Christofeel DEFINITION symbols of V in the given local coordinates.

DEFINITION 10.1.5. Let V be any connection on a manifold M . The torsion of V is the R-bilinear map

defined by the formula

T ( X ,Y )= V x Y - V y X - [X,Y]. If T = 0, then V is said to be torsion free or symmetric. Proofs of the following three lemmas are left as exercises. LEMMA10.1.6. The torsion T of a connection V on M is CM(M)-bilinear. That is, T E q 2 ( and ~ )T ( v ,w) E T x ( M )is defined, V v ,w E T x ( M ) ,V x E M . Torsion is a tensor. LEMMA10.1.7. The connection V is torsion free if and only if, in every local coordinate chart (U,x l , . . . ,x n ) , the Christogel symbols have the symmetry

This is the reason that torsion free connections are also said to be symmetric. LEMMA10.1.8. If M C Rm is a smoothly imbedded submanifold, the LeviCivita connection is symmetric.

10.1. CONNECTIONS

297

We use connections to define a way of differentiating vector fields along a curve. Indeed, if X E X(M) and s : [a,b] -+ M is a smooth curve, then, at each point s(t), one can compute

But we will also be interested in differentiating vector fields along s that are only defined along s. In fact, it is often natural to consider fields X,(t) along s that are also parametrized by the parameter t, allowing Xs(t,)# Xs(t,l even if s(tl) = s(t2),tl # t2. For instance, Xs(t)= 3(t), the velocity field, may exhibit such behavior. In these cases, it is not immediately clear how to use a connection to produce the desired derivative.

DEFINITION 10.1.9. Let s : [a, b] -, M be smooth. A vector field along s is a smooth map v : [a, b] + T(M) such that the diagram

commutes. The set of vector fields along s is denoted by X(s). We have already seen two examples, namely, the restriction (Yls)(t) = Y,(t) to s of a vector field Y E X(M) and the velocity field i(t). Via pointwise operations, it is evident that X(s) is a real vector space and, indeed, a CM[a, b]-module.

DEFINITION 10.1.10. Let V be a connection on M. An associated covariant derivative is an operator

defined for every smooth curve s on M , and having the following properties: (1) V/dt is R-linear; (2) (V/dt)(f v) = (df /dt)v (3) if Y E X(M), then

+ f Vv/dt, V f E Coo[a,b], V v E X(s);

REMARK. By property (3), V/dt is associated only to the one connection V. THEOREM 10.1.11. To each connection V on M, there is associated a unique covariant derivative V/dt . PROOF.We prove uniqueness first. For this, it is enough to work in an arbitrary coordinate chart (U, x l , . . . ,xn). Let r: be the Christoffel symbols

10. RIEMANNIAN GEOMETRY

298

for V, set Write

ti = d/dxi,

consider a smooth curve s : [a,b]

-t

U, and let v E X ( s ) .

Then any associated covariant derivative must satisfy

evaluated dong s(t). This is an explicit locd formula in terms of the connection, proving uniqueness. We turn to existence. In any coordinate chart (U, x l , . . . ,xn), use the above formula to define V/dt for curves lying in the chart. The reader can easily check that the three properties in the definition of covariant derivative are satisfied, where M = U. Thus, in U ,the connection V(U has an associated covariant derivative and, by the preceding paragraph, this covariant derivative is unique. Consequently, on overlaps U n V of charts, the two sets of Christoffel symbols must define the same covariant derivative for V1U nV. (Classical geometers and physicists defined connections and covariant derivatives by Christoffel symbols and they checked this invariance via explicit change of coordinate formulas.) Thus, along any smooth curve s : [a,b] + M, these local definitions of V/dt can be pieced together to give a global definition. In particular, for the Euclidean connection the expected formula:

D in R m , all

23

= 0 and we get

For M c Rm, the covariant derivative associated to the Levi-Civita connection V on M is dt

JLP(8),

10.1. CONNECTIONS

299

the orthogonal projection into T(M) of the usual Euclidean covariant derivative. CONVENTION. From now on, we adopt the "summation convention" of Einstein. According to this convention, the summation symbol is omitted and it is understood that any expression is summed over all repeating indices. For example,

It is necessary that the indices repeat for terms in a product, not just in a sum, and it is customary that the repeated index occur once as a superscript and once as a subscript (a custom we will usually honor). Thus,

-

the index k being repeated only in terms separated by

+.

DEFINITION 10.1.12. Let M be a manifold with a connection V. Let v E X(s) for a smooth path s : [a, b] -+ M. If Vvldt 0 on s, then v is said to be parallel along s (relative to the given connection). THEOREM 10.1.13. Let V be a connection on M, s : [a,b] + M a smooth path, c E [a,b], and vo E T,(,) (M). Then there is a unique parallel field v E X(s) such that v(c) = vo. This field is called the parallel transport of vo along s. PROOF.In local coordinates, write S(t) = ~ j ( t ) Cs(t) j ~ ( t= ) vi(t)ti s ( t l vo = aiti s(c)

+

Here, as promised, we are using the summation convention. The condition that v be parallel dong s becomes the following equation

or, equivalently, the system of linear O.D.E. dvk dt

i

- = -v u

j

k rji, 15 k 5 n,

with initial conditions vk (c) = a k , 1 j k j n.

By the existence and uniqueness of solutions of O.D.E., there is E > 0 such that the solutions vk(t) exist and are unique for c - e < t < c + E . In fact, these equations being linear in the v 's, it is standard in O.D.E. theory (Appendix C, Theorem C.4.1) that there is no restriction on 6 , so the unique solutions v k(t) are defined on all of [a,b], 1 5 k 5 n.

300

10. RIEMANNIAN GEOMETRY

EXAMPLE 10.1.14. If s : [a,b] -+ M is only piecewise smooth and vo E Ts(,) ( M ) , we can parallel transport vo along the first smooth segment sl [a,tl], then take the vdue v(t as an initial condition, parallel transporting that vector along the second segment, etc. The result is a piecewise smooth vector field along s , called the parallel transport of vo. An example was given in Figure 10.1, where we intuited the way a flatlander on S2 would try to parallel translate a vector along a piecewise smooth loop. It should be clear that this parallel transport is exactly the one defined by the Levi-Civita connection V on S2 c W3. Indeed, on the first leg of the triangle, u is the unit tangent field to the great circle and Dv/dt is perpendicular to S2. Thus, Vv/dt = p(Dv/dt) = 0. On the second leg of the journey, v is parallel, even from the Euclidean point of view, and Vv/dt = p(Dv/dt) = p(0) = 0. On the third leg, v is again the unit tangent field to the great circle. Note that the final vector in this field is rotated from its initial direction by exactly the angle of the geodesic triangle at A. By varying this piecewise smooth loop, we can produce any rotation we want. This is an example of holonomy, a concept we treat next. DEFINITION 10.1.15. Let s : [a, b] -+ M be a piecewise smooth loop based at s o = s(a) = s(b). Then the holonomy of V around s is the map hs : Txo( M ) --+ Tx,, ( M ) , defined by setting h s (vo) = v(b),

where v E X(s) is the parallel transport of uo E T,,(M). Here, X(s) denotes the space of continuous, piecewise smooth fields along s. Since the parallel transport v of uo along s is the solution of a linear system (10.1) of O.D.E., it follows that, if w is also the parallel transport along s of a vector wo E Tx,(M), then u w E X(s) is the parallel transport of uo wo. Similarly, if a E IW, av E X(s) is the parallel transport of avo. This proves the following.

+

+

LEMMA 10.1.16. The holonomy

of V around the piecewise smooth loop s is a linear transformation. DEFINITION 10.1.17. If s : [a, b] + M is piecewise smooth, a weak reparametrization of s is a curve s o r , where r is a piecewise smooth map r : [c,d] + [a,b] which carries {c, d) onto { a ,b). If s(c) = a and s(d) = b, the reparametrization is said to be orientation preserving. If s(c) = b and s(d) = a, it is said to be orientation reversing. LEMMA10.1.18. Let s : [a,b] + M be a piecewise smooth loop at xo and let 5 = s o r : [c,d] + M be a weak reparametrization. If the reparametrization is orientation preserving, then h,- = h, and, if it is orientation reversing, h,- = h r l .

10.1. CONNECTIONS

30 1

PROOF.Without loss of generality, assume that s and r are smooth. Set

hence

and the linear system

is obtained from the system (10.1) by multiplying through by dr/dr. Since the system (10.1) is assumed to be satisfied, so is the system (10.2). Thus, if r(c) = a and r(d) = b, h,-(vo) = G(d) = ~ ( b = ) h, (vo). If r(c) = b and r(d) = b, then we take the initial condition to be C(c) = v(b) = h , ( v ~ ) and h,-(hs(vo))= h,-(v(b)) = C(d) = v(a) = vo.

Let R(M, xo) denote the set of all piecewise smooth loops in M based at xo, loops being identified if they are reparametrizations of each other. If s 1, s2 E a ( M , xo), then sl s g E a ( M , s o ) and

+

Also, h-, = h;' , so h, is a nonsingular linear transformation of Txo(M). These considerations give the following. LEMMA10.1.19. The set {hs)3,cnc~,xo, is a subgroup Hxo(M) C Gl(Txo(M)), called the holonomy group (at xo) of the connection V. REMARK.If M is connected and xo, XI E M , then the groups Hz, (M) and 7ix, (M) are isomorphic, but generally not canonically isomorphic. Indeed, fix a piecewise smooth path a : [O, 11 + M with a(0) = xo and a(1) = X I . Then, (-a) E R(M, xo) and that the given any s E R(M, X I ) , note that a s h,+,+(-,) defines a group homomorphism from Hz, (M) correspondence h, to Hz, (M). By replacing a with -a, we get the inverse group homomorphism ax,(M) -+ Hz, (M), proving that these are group isomorphisms. Generally, the isomorphism depends on the choice of a, so it is not canonical. Finally, we should point out that connections are ubiquitous.

+ +

THEOREM10.1.20. Every manifold M has a connection.

302

10. RIEMANNIAN GEOMETRY

Let {U,,xi,. clidean connection on U, be a smooth partition of write X, = XIU, and Y,

. . ,x:),~%

PROOF.

be an atlas on M and let D a be the Eurelative to the coordinates x: , . . . ,x: . Let {A,),Ea unity subordinate to the atlas. Given X, Y E X(M), = Y I U,. Then, define

where VxY =

C XaDSaYa

€

X(M).

a€%

It is entirely straightforward to verify that V is a connection.

EXERCISES (1) Prove Lemma 10.1.6. (2) Prove Lemma 10.1.7. (3) Prove Lemma 10.1.8. (4) The standard imbedding of S1as the unit circle in It2defines a standard imbedding of Tn = S1x - x S1in IR2". Prove that the associated LeviCivita connection on T n has holonomy B,(Tn)= {id), V x E Tn. (Hint: Find a coordinate atlas relative to which the Christoffel symbols vanish.) (5) We say that V is a fiat connection if (as in Exercise (4)) its holonomy group X,(M) is trivial, Vx E M. (The reason for this terminology will become apparent when we study the relationship between curvature and holonomy.) Prove that a manifold M has a flat connection if and only if M is parallelizable. (6) By Exercise (5), the spheres S3and S7support flat connections. Prove that, for n 2 2, the Levi-Civita connection relative to the standard imbedding Sn c Rn+l is not flat.

10.2. Riemannian Manifolds

A Riemannian manifold is really a pair (M, consisting of the manifold M and a choice of Riemannian metric on T(M). From now on, such a choice of metric is fixed and we will speak of ''the Riemannian manifold M". ( a ,

a ) )

DEFINITION 10.2.1. If v E T,(M), some x E M, then the length of v is the nonnegative number

I I ~ I I= Jv),. When no ambiguity is likely, we will often dispense with the subscript x on (v, w),, where v, w E Tz(M).

DEFINITION 10.2.2. If v , w E T,(M), some x E M , then the angle between v and w is the unique 9 E (0,?r] such that

10.2. RIEMANNIAN MANIFOLDS

DEFINITION 10.2.3. If s : [a,b]

If s = sl

+ . + s,

+M

303

is smooth, then the length of s is

is piecewise smooth, each si being smooth, then the length is

LEMMA10.2.4. If u : [ c , d

+

[a,b] is a weak repammetrization, then Is] =

1s 0 uI. The proof should be familiar from advanced calculus. Notice that it does not matter whether u preserves orientation or reverses it. If the Riemannian manifold M is oriented, we also get a canonical volume form fl E An(M) (where n = dim M). Consider a local trivialization of T(M). That is, we are given an open set U M, together with a smooth frame (XI,. . . , X,) of vector fields defined on U and determining the correct orientation at each point of U. Relative to this trivialization, we express the Riemannian metric by

That is, relative to the given basis, the Riemannian metric, as a bilinear form, is represented by the smooth, matrix-valued function [hij], where hij = ( X i ,Xj) . This is a symmetric matrix. Since the metric is positive definite,

Let {w', . . . , wn)

c A'(U) be the dual basis: wi(xj) = 6;.

DEFINITION 10.2.5. The Riemann volume element on U, relative to the given local trividization of T(M), is

In particular, if (XI, . . . ,X,) is an orthonormal frame, the volume element becomes w l A . A wn . This agrees with intuition. The following theorem shows that the volume element is independent of the choice of local trivializations.

THEOREM 10.2.6. If M is a n oriented, Riemannian n-manifold, there is a globally defined form 52 E An(M) which, relative to any orientation respecting local triwialization of T(M), coincides with the Riemann volume element.

PROOF.Indeed, let (U, X i , . . . ,X,) and (V, Z1,. . . , Zn) be two such trivializations with UnV # 0. Let the respective dual bases of 1-forms be {w ', . . . , w n ) and {ql,. . . ,qn). Let y(x) = [yij(x)] be the Gl(n)-valued function on U n V such that (zl,...7zn)y = (XI YXn) on U n V. Since the frames are coherently oriented, det[yij] > 0. Let

10. RIEMANNIAN GEOMETRY

304

Then, [hij] = [ ? k i l l

[fkt]

[ytj] and it follows that

Also, det(y')ql A where y' = (y

T

)-I.

- A qn = w1 A

h wn,

Putting this information together, we obtain

Thus, the locally defined volume forms fit together coherently to define R as desired.

DEFINITION 10.2.7. Let (U, x l , . . . , xn) be a coordinate chart with coordinate fields ti= d/axi. Then the functions

are called the metric coefficients. Thus, in correctly oriented local coordinate charts (U, x element is given in terms of the metric coefficients by

',. . . ,xn), the volume

where g = det[gij]

DEFINITION 10.2.8. Let M be an oriented Riemannian n-manifold and let U M be a relatively compact, open subset. Then the volume of U is

REMARK. On the a-algebra of Borel sets of M , this Riemannian volume generates a measure, finite on the relatively compact Borel sets. Even if M is not orientable, R is defined locally up to sign and, for small, connected, open sets U,

This also leads to a Borel measure, finite on relatively compact Borel sets.

DEFINITION 10.2.9. A connection V on the Riemannian manifold M is a Riemannian connection if, for all X, Y,Z E X(M) ,

DEFINITION 10.2.10. A connection V on the Riemannian manifold M is a Levi-Civita connection if it is symmetric and Riemannian.

10+2. RIEMANNIAN MANIFOLDS

305

EXAMPLE10.2.11. Let M Rm be a smoothly imbedded n-manifold. The standard inner product (-, -) on R m , viewed as a Riemannian metric on the tangent bundle T(Rm), restricts to a Riemannian metric (-, -) on T ( M ) C_ T(Rm)IM. By Lemma 10.1.8, the connection V on M , constructed in the previous section and called there the Levi-Civita connection, is symmetric. The Euclidean connection D on R m clearly satisfies X (Y, Z) = (DxY Z)

+ (Y, D x Z) , V X , Y, Z E X(Rm).

If X , Y, Z E X(M), then DxY = VxY everywhere on M. Consequently,

+ W, where W E r(v(M)), hence W IZ

and, similarly, (Y,DxZ) = (Y,VxZ), . Thus, V is Riemannian, hence is a Levi-Civita connection in our new sense as well. In Exercise (I), you will be led through a proof of the following fundamental result. THEOREM10.2.12. A Riemannian manifold M has a unique Levi-Czvita con-

nection. In any local coordinate chart, the matrix [gij] of metric coefficients is nonsingular, so we can define

isk']= [gijl-l. The coefficients g k e are rational functions of the metric coefficients g y . By definition, they satisfy sirsk' = sf. DEFINITION10.2.13. A property of the Riemannian manifold M is intrinsic if it depends only on the metric. Otherwise, the property is extrinsic. It is geometric if it does not depend on choices of local coordinates. For example, the functions g g and sk' are intrinsic, but not geometric. By Exercise (2), the Christoffel symbols I?& for the Levi-Civita connection are also intrinsic, but not geometric. The Levi-Civita connection itself is both intrinsic and geometric. In particular, the Levi-Civita connection for M C R m is intrinsic, even though our initial definition of it used the normal bundle v(M), a structure that can be proven to be extrinsic. Our definition of "intrinsic" may seem a bit too informal by current standards. For a more formal definition, one needs the notion of an isometry. DEFINITION10.2.14. An isometry cp : M I + M2 between two Riemannian manifolds, with respective metrics (., .) i , i = 1,2, is a diffeomorphism such that

for arbitrary x E M I and v, w E Tx(Ml).

306

10. RIEMANNIAN GEOMETRY

Now we see that a property of Riemannian manifolds should be called intrinsic if and only if it is preserved by all isometries.

EXERCISES (1) Given a Riemannian manifold M , proceed as follows to show that it has a unique Levi-Civita connection. ( a ) Uniqueness. Show that a Levi-Civita connection V must satisfy the identity 2 (VxY, 2) = X (Y, 2 )

+ Y (X, 2 ) - Z (X, Y)

+ I[X,YI, 2) + ([Z,XI, Y )+ ([Z,yl, X )

7

VX, Y, Z E X(M). (b) Ezistence. Use the identity in (a) to define

and prove that V is a Levi-Civita connection. (2) Let I'h denote the Christoffel symbols of the Levi-Civita connection and find a formula for r&that involves only the gk"s and first derivatives of the gij's. (3) If f : M + N is a diffeomorphism and V is a connection on N, show how to define the pull-back connection f *V on M . If f is an isometry between Riemannian manifolds and V is the Levi-Civita connection on N, prove that f * V is the Levi-Civita connection on M. This is the formal proof that the Levi-Civita connection is an intrinsic property of a Riemannian manifold. (4) Let v, w E X(s) . Show that the covariant derivative defined by the LeviCivita connection (indeed, by any Riemannian connection) satisfies

(It follows from this exercise that fields parallel along a curve s, relative to a Levi-Civita connection, make a constant angle with each other and have constant lengths along s.)

10.3. Gauss Curvature Throughout this section, unless otherwise stated, we assume that dim M = 2, that a M = 0, and that we are given a fixed imbedding M L, IR3 with normal bundle v(M). As usual, we use the metric induced on M by the Euclidean metric of IR3 and the associated Levi-Civita connection V. The Euclidean connection will be denoted by D. M of x and a smooth Given x E M, we find a connected neighborhood U section 6 E r ( v ( M ) IU)such that Il6ll E 1. Remark that 6 is determined up to sign. Remark also that one can interpret n' as a smooth map

10.3. GAUSS CURVATURE

DEFINITION 10.3.1. The map n' : U

+ S2is

called the Gauss map.

FIGURE 10.2. The flat plane

FIGURE 10.3. The flat cylinder Intuitively, the area of 6(U) & S2 seems to have something t o do with the curvature of M in U. Thus, if U is an open subset of a 2-plane in It3, Z(U) degenerates to a single point, hence has area 0 (Figure 10.2). We say that the plane has (Gauss) curvature 0. Similarly, if U lies in a right circular cylinder, n'(U) will lie in a great circle in S2 (Figure 10.3). Again the area of d(U) is 0 and we say that the cylinder has (Gauss) curvature 0. The point here is that the cylinder can be "unrolled" to a portion of a plane without any metric then n' = fid and the area of n'(U) is distortions. On the other hand, if U C s2, the same as the area of U, this being a positive number. We say that the sphere has positive (Gauss) curvature. It is not possible to flatten out any portion of S2 to be planar without distorting the metric properties. For similar reasons, every convex surface has nonnegative curvature everywhere (Figure 10.4). By introducing a notion of "signed area7', one obtains cases of negative curvature, a saddle shaped surface being the typical example (Figure 10.5 and Exercise (3)). In order to put these ideas into precise form, we introduce the Weingarten map.

10. RIEMANNIAN GEOMETRY

t

FIGURE 10.4. Nonnegative curvature

FIGURE10.5. Negative curvature If v E Ty(M),some y E

U ,then Dun' E IR3 makes sense. The equation

0 = v (n',n') = (D,n',n')

+ (n', DUG)= 2 (Dun',5)

proves that DUG1n', hence Dun' E Ty(M). DEFINITION 10.3.2. The linear map L : T,(M)

-,Ty(M), defined

by

L ( v ) = Dun', is called the Weingarten map. This map is well defined up to sign. As soon as the sign of the Gauss map n' has been fixed, the sign of the Weingarten map is determined also. Remark that T, ( M ) = T6(,)(S2)in IR3, since these 2-planes have the common normal vector n'(y). Thus, we are allowed to view the Weingarten map as

10.3. GAUSS CURVATURE

LEMMA10.3.3. The Weingarten map L : T,(M) tial at y of the Gauss map.

309

+

T5(,)(s2) is the difleren-

PROOF.Represent an arbitrary vector in T , (M) as s(0) for a suitable arc s : ( - E , E ) 4 M. Then

LEMMA10.3.4. The Weingarten map is self adjoint. That is,

PROOF.Let v,w E T,(M) and, by making the open set U & M smaller, if necessary, extend these vectors to fields X, Y E E(U).Then (L(X)l,)'

= ( D ~ z l Y, = X (5,Y) -

(n', D x Y )

= - (n', DxY)

+

= - (5,[X7Y] D y X ) = - (G, DyX)

Thus, relative to any choice of orthonormal basis in Ty(M), the matrix of L is symmetric. By the diagonalization theorem for symmetric matrices, there is an orthonormal basis { e l ,e2) C T,(M) relative to which the matrix for L is

That is, ei is an eigenvector for L with eigenvalue number these so that 61 L ~ 2 .

~

i i ,=

1,2. We agree to

LEMMA10.3.5. As v ranges over the unit circle in T,(M), the quadratic form (L(v), v) takes minimum value ~1 and maximum value ~ 2 . Indeed, if

,

is the matrix of a quadratic form Q on EX2, it is standard that the extreme values of Q (v) on S1 are the eigenvalues of the matrix (the method of Lagrange multipliers). Remark that, for the eigenvectors e , , tci = (Vein',e i ) measures the rate at which the normal vector G is turning in the direction e i. This motivates the following definition.

10. RIEMANNIAN GEOMETRY

310

DEFINITION 10.3.6. The numbers 61 and 1c2 are called the principal curvai tures of M at y. The product K I K ~= det L is called the Gauss curvature of M at y and is denoted by K (or ~ ( y ) ) . REMARK.There is another important kind of curvature, the m e a n curvature h of M, which we will not treat in any detail. It is defined by

Unlike the Gauss curvature, this quantity depends, up to sign, on the choice of the unit normal field v. It can be shown that surfaces of mean curvature h z 0 are exactly the ones that, in a certain precise sense, locally minimize surface area. Such surfaces are called minimal surfaces. They arise, for instance, when one considers the possible shapes of soap films spanning wire loops of various configurations. Such a soap film will be modeled by a 2-manifold S with boundary and M = S \ dS will be a minimal surface. Until recently, soap films were the main examples, but the work of Hoffman and Meeks ([17], [18], [19], et 61.)) inspired and illuminated by some spectacular computer graphics, has revealed an astounding array of complete, unbounded minimal surfaces. Let R' denote the Riemann volume form on S2 and let fl denote the volume form on M. Define the "signed area" of the Gauss map on U to be

and, a s usual, let the area of U be

As U shrinks down on { y ) , one can try to form the "Radon-Nikodym derivative" of the Gauss mar, Z:

This number, if it exists, is characterized as the unique number such that, for each E > 0, there is 6 > 0 for which

U

3 y and

IA(U)I < S +

< E,

The proof of the following will be an exercise. PROPOSITION 10.3.7. For each y E M , the above Radon-Nikodym derivative exists and In the above discussion, the normal field n' played a central role. The curvature of M was seen as a measure of how much this field "spreads" infinitesimally at a point. Thus, curvature appears to be an extrinsic property of the surface. But Gauss proved a remarkable theorem (he c d e d it his "Theorema Egregium") that showed the Gauss curvature to be intrinsic. A two-dimensional inhabitant

10.3. GAUSS CURVATURE

311

of the surface can take measurements leading to the computation of curvature. We turn to this theorem. - - Let X, Y, Z E X(U) and extend these to fields X , Y, Z E X(I?), where 6 & IR3 is an open set such that 6n M = U.

LEMMA10.3.8. For fields chosen as above, along U. PROOF.Along U, Lemma 10.3.4,

D,-Y

depends only on X and Y. As in the proof of

( D x K A) = - (L(X), Y) , so the component of Dx Y perpendicular to M is - (L (X), Y) A. By the definition of the Levi-Civita connection, VxY is the component of DxY tangent to M. The Euclidean connection D satisfies a simple commutator relation:

This is because D,- and DT operate on a vector field 2 by applying 2 and B respectively to the individual components of 2. It turns out that, on M , curvature is an obstruction to this commutator relation for V. DEFINITION 10.3.9. The curvature operator

is defined, for arbitrary X, Y E X(M), by

The fact that this operator is related to curvature is far from obvious. It is the content of the Theorema Egregium. By Lemma 10.3.8, at every point of U we have

Similarly, at every point of U, -

D ~ ( D ~=Z-vY(vXz) ) + (L(x),

Z) L(Y)

+ Y (L(X), Z) 6 + (L(Y),V x Z ) n'.

Findy, at every point of U,

- ~ ~= - V~[ X ,, Y +~~~(L([X, ~ Y]),z z)6.

312

10. RIEMANNIAN GEOMETRY

The sum of left sides of these equations being 0, the tangential parts and normal parts of the right sides separately sum to 0. The first of these zero sums gives the Gauss equation:

Summing the coefficients of n' also gives 0. Applying the fact that V is a Riemannian connection and using the fact that Z varies freely over all tangent fields, the reader can obtain the Codazzi-Mainardi equation (10.4)

L([X,Y]) = VxL(Y) - VyL(X).

One immediate consequence of equation (10.3) is the following. LEMMA10.3.10. The expression R(X, Y)Z has value at y E U depending only on the vectors X,, Y,, 2,. Thus, this expression is a tensor, called the Riemann curvature tensor R. Indeed, the right hand side of equation (10.3) is clearly a tensor in all three vector fields. That is, R E T 3 ( M ) . Note also that, since V is an intrinsic and geometric property of the surface, so is the Riemann curvature tensor.

THEOREM 10.3.11 (THEOREMA EGREGIUM). Let y E U and let {e 1, e2) be an orthonormal basis of T,(M). Then

In particular, the Gauss curvature of a surface in IR3 is intrinsic and geometric. PROOF.By equation (10.3), @(el, e2I.2, el) = (L(ea),ea) (L(ei), el) - (L(el),ea) (L(ea),el) . Since the basis is orthonormal, it is true (and easily checked) that the corresponding matrix representation of L is the 2 x 2 matrix with (i,j) th entry (L(ej),et). Thus, (R(el, ez)ez, el) = det L = ~ ( y ) .

Since the Levi-Civita connection is preserved by isometries (Exercise (3) on page 306), we obtain the more formal version of the statement that n is an intrinsic property. COROLLARY 10.3.12. If f : M1 4 M2 is an isometry between two surfaces in IK3 and if I C ~is the Gauss curvature of M i , i = 1,2, then K.~(f (x)) = r;l (x), V X E M1. Theorem 10.3.11 suggests a definition of curvature for a general connection on an n-manifold M .

10.3. GAUSS CURVATURE

313

DEFINITION 10.3.13. Let M be an n-manifold and let V be a connection on M. The curvature tensor R of V is given, for each choice of X, Y, Z E X(M), by

If M is a Riemannian manifold and V is the Levi-Civita connection, then R is called the Riemann tensor R. The fact that this is a tensor in all three arguments will be Exercise (2). Consequently, for each x E M and u, v, w E Tz(M), R(u, v)w E Tz(M) is well defined. We will return to the study of this tensor later. It turns out that, in Riemannian geometry, the Riemann tensor is exactly the obstruction to the geometry being locally Euclidean. In the non-Ftiemannian geometry of spacet ime, there is an analogue of the Levi-Civita connection and Einstein represents gravity by the curvature tensor of this connection.

EXERCISES (1) Prove Proposition 10.3.7. (2) Verify that the curvature tensor R(X, Y)Z of a connection V is Cm(M)trilinear in the fields X , Y, Z. (3) Let M C R3 be the graph of the equation r = x 2 - y2. Prove that K(X,y, z ) < 0, V (5,y , r ) E M. Can you explain this negative sign intuitively? (4) Let M C R3 be a compact 2-manifold. You are to prove that it is not possible that K 5 0 on all of M. Proceed as follows. (a) By compactness, choose a point vo E M at which the function X : M + R, defined by X(v) = IJvl12,assumes its maximum. Prove that 0 # vo IT,, (M). (b) If s : (-E, E) + M is smooth with s(0) = vo and i(0) # 0, prove that (s(O),n'(vo)) is strictly negative. (c) Using the above, prove that the principal curvatures ~1 and ~2 are both nonzero and have the same sign at vo, hence ~ ( v >~0.) (5) One calls a point v E M at which K I = ~2 an umbilic point. Let U C M be the set of points that are not umbilic. (a) Prove that U is open in M and that and ~2 are smooth functions on U. (b) Prove that each v E U has a neighborhood V C_ U on which there is a smooth, orthonormal frame field (XI, X2) such that L(Xi) = KiXi, i = l,2. (c) For V C U and Xl , X2 E X(V) as in part (b), define f l , f 2 E CM(V)by

10. RIEMANNIAN GEOMETRY

Prove that

and that [Xl,X2] = f l X l - f2X2. (d) Using the formulas in part (c), show that, if v E U is a critical point for both I C ~and 6 2 , then, at v,

(6) Let M C lR3 be a compact, connected 2-manifold with constant curvature IC = a. By Exercise (4), a > 0. You are to prove that M is a 2sphere, centered at some point wo E R3 and of radius l/&. Proceed as follows. (a) Prove that v E M is a point at which I C is ~ maximum if and only if it is a point at which I C ~is minimum. (b) Use part (d) of Exercise (5) to show that I C ~can be maximum only at an umbilic point. (c) Prove that every point of M is an umbilic and that IC 1 & = I C ~ . (Hint: This is the maximum value of K ~ . ) (d) Deduce the form of the Gauss map, drawing the desired conclusion. Be sure to make clear how you use the hypothesis that M is connected.

-

10.4. Complete Riemannian Manifolds

This section presents the Hopf-Know theorem and related matters. The author first learned this materid horn Milnor's beautiful exposition [29, pp. 55-64], and its influence will be evident in what follows. The goal here is to use the Riemannian metric to obtain a topologicd metric on M and to relate the topological notion of "completeness" to the problem of extending geodesics indefinitely. In the process, one also discovers, without the use of variational calculus, that geodesics locally minimize arc length. In Euclidean geometry, straight lines play a central role. They can be characterized as the unique smooth curves whose tangent fields are parallel. Here, parallelism under the Euclidean connection is clearly identical with the absolute parallelism in Euclidean space. Taking our cue from this observation, we define the notion of a geodesic for a general connection.

DEFINITION 10.4.1. Let M be an n-manifold with a connection V. A smooth curve s : [a, b] -+M is a geodesic for V if i(t) is parallel along s(t), a 5 t 5 b. We are interested in the case in which V is the Levi-Civita connection of a Riemannian manifold, so we make that assumption from here on. We emphasize that we are considering general Riemannian manifolds, not just surfaces in R3.

10.4. COMPLETE RIEMANNIAN MANIFOLDS

DEFINITION 10.4.2. A smooth curve o : [a,b] Ilb-(t)ll = C, a 5 t 5 b, for some constant c 2 0.

-, M

315

is evenly parametrized if

By Exercise (4) on page 306, the following is immediate.

LEMMA10.4.3. A geodesic s on a Riemannian manifold M is necessarily evenly parametrized. In local coordinates, the definition of a geodesic translates into a system of nonlinear, second order, ordinary differential equations. Indeed, write

and write down the parallelism condition for d ( t ) :

Equivalently, this is the second order system

By setting ui = xi, 1 5 i 5 n, we get an equivalent, nonlinear, first order system u. k

+

i

j

k rij = 0,

ie=ue,

l S k 5 n l<.t
Given initial conditions

there is a unique solution

defined on some interval

-E

< t < E . The initial condition can be written s(0) = (a1,. . , , a n )

B(0) = (b',.

. . ,bn).

The existence and uniqueness theorem for solutions of O.D.E., together with the smooth dependence of the solutions on initial conditions, gives the following.

316

10. RIEMANNIAN GEOMETRY

THEOREM 10.4.4. Let xo E M . Then, there is an open neighborhood U of xo in M and numbers € 1 , € 2 > 0 such that, for every x E U and every v , E T x ( M ) with 11v, 11 < € 1 , there is a unique geodesic

with a,, ( 0 ) = x bv,( 0 ) = v,. Furthermore, in local coordinates, x = ( x l ,. . . , x n ) , v, = viei x , and ov,( t ) = cp(x1 , . . . , xn, V 1 , . . . ,vn, t )

defines a smooth function of 2n + 1 variables. The domain of the function cp is W x ( - E ~ , E

~ ) ,where

an open neighborhood of Ox,. REMARK.It will be convenient to reparametrize geodesics so that their domain of definition always contains the closed interval [- 1,1]. The following trick will be found in [29,p. 571. The system (10.5) is homogeneous of degree 2. This implies that, if o ( t ) is a geodesic, so is o C ( t )= a(&), for a fixed constant c, and irc(t)= w ( t ) . Let 0 < c < E ~ E ~ If/ ~11v.1t < E and It[ < 2, then

so the curve

~ v ( t=) ozvlE2 ( ~ a t / 2 ) ,-2

< t < 2,

defines a geodesic

yv : (-2,2)

+M

such that

yv ( 0 ) = x i;(o) = v. If

BX(4 = {vx E T A M ) 1 llvxll then y,(t) varies smoothly with ( v ,t ) E

< €1,

(JB x ( 4 x (-2,217 zEU

an open neighborhood of (O,,,

0) in T ( M ) x R.

DEFINITION 10.4.5. For x E U ,v E T x ( M ) ,llvll

< E , a11 as above,

10.4. COMPLETE RIEMANNIAN MANIFOLDS

317

By the homogeneity of the system (10.5), we obtain y V ( r ) = y T V ( l ) . The following is an immediate consequence.

LEMMA10.4.6. For x E U ,v E T x ( M ) ,llvll < E , all as above,

CONVENTIONS. In what follows, we routinely view M C T ( M ) by the imbedding of M as the Osection of T ( M ) . Thus, x € M is identified with O x E T x ( M ) . Since the tangent space T v ( V )at any point v of a finite dimensional vector space V is identified canonically with V itself, we will also identify T o , ( T x ( M ) )with Tx ( M I . THEOREM 10.4.7. If

E,

> 0 is suficiently small, then

called the exponential map at x, is a diffeomorphism onto an open neighborhood of x in M .

PROOF.Write q = exp, : B x ( c ) + M . Clearly, ~ ( 0 , )= x. We compute V*0, : To, (Bx

(€1) = Tx ( M ) -,Tx ( M I ,

For 0 # v E To, (B,(e)) = T x ( M ) ,set s ( t ) = tv, -e/llvll < t < ~/11v11,a smooth curve on B X ( e )with s(0) = O x , s(0) = v. Then 7 o s defines a curve on M , ~(s(0= ) ) ~ ( 0 , )= x and ~ ( s ( t = ) )expx(tv)= y,(t). Thus, V*oz (S( 0 ) )= V*oz (v) = q, ( 0 ) = v .

That is, q*o, is the identity under the identification To=(Bx( E ) ) = Tx( M ). By the inverse function theorem, 7 will be a diffeomorphism for e = e x > 0 sufficiently small. Remark that, if the manifold is compact, the number e = ex to be independent of x.

> 0 can be chosen

10.4.8. The Riemannian manifold M is geodesically complete if DEFINITION expx(v) is defined for all x E M and for all v E T x ( M ) . Equivalently, every geodesic segment extends (uniquely) to a geodesic y ( t ) , -m < t < m.

DEFINITION 10.4.9. A path a : [a,b] + M is regular if it is a smooth immersion. If there is a partition a = to < tl < . - -< t , = b such that ~ l [ t ~ - ~ is , t ~ ] regular, 1 5 t 5 r , then a is piecewise regular. Nondegenerate geodesics a are regular. Indeed, if & ( t o )= 0, for some t o , then the fact that a is evenly parametrized implies that & = 0 and a degenerates to a constant path.

DEFINITION 10.4.10. Let M be a connected Riemannian manifold. Then the Riemann distance function p : M x M + [O, m) is defined by

where a ranges over all piecewise regular paths from x to y in M .

10.

3 18

RIEMANNIAN GEOMETRY

Fixing the hypothesis that M is a connected, Riemannian n-manifold with d M = 0,we are going to prove the following. PROPOSITION 10.4.11. The Riemann distance function is a topological metric o n M and the metric space topology is the same as the manifold topology. THEOREM 10.4.12 (HOPF-RINOWI). If M is geodesically complete, x,y E M, then there is a geodesic y from x to y such that 1 y 1 = p(x, y ) . I n particular, exp, : Tx( M ) -, M is surjective. THEOREM 10.4.13 (HOPF-RINOWXI). The manifold M is geodesically complete i f and only if it is a complete metric space in the metric p. COROLLARY 10.4.14. If M is compact, then M is geodesically complete in any Riemann metric. Because of these results, it is standard to use the term "complete Riemannian manifold" when either geodesic completeness or metric completeness is intended. A number of preliminary considerations are necessary for the proofs of Proposition 10.4.11 and the Hopf-Rinow theorems. To begin with, remark that the distance function p has the following properties: (1) p(x, x) = 0, Vx E M; (2) p(x, Y ) = P(Y,x), v x , Y E M ; (3) p(x1 Y ) I P(X,4 + p(a, Y ) , v x , Y , r E M. Thus, to prove that p is a topological metric, it is only necessary to prove that

Let xo E M C T ( M ) , choose an open neighborhood U M of xo, and let E. > 0. Then W = {v, ET,(M) I x E U and IlvxI[ < E) is an open neighborhood of xo = Ox, in T(M). If U and E. are chosen small enough, G(v) = (*(v) 9 ~xP,(,) (v)) is defined and smooth as a function G : W + M x M. We can take U to be a coordinate neighborhood, with coordinates x l , . . . ,xn. We can also assume that there is a smooth, orthonormal frame field Z1, . . . , Zn defined on U. We obtain, thereby, coordinates

on

(U) such that n

W = {(x ,. . . ,xn, y , . . . ,yn) I (xl, . . . ,rn)E U, 1

1

C(y')2 < e2) i= 1

= U x B(E).

LEMMA10.4.15. Given xo E M , the neighborhood W of xo in T(M) can be chosen, as above, so small that G : W 4 M x M carries W diffeomorphically onto a n open neighborhood of (xo,xo) in M x M .

10.4. COMPLETE RIEMANNIAN MANIFOLDS

319

PROOF.Indeed, let tirepresent the basic coordinate fields for x h n d cj those for yj, 1 5 i,j 5 n, and observe that

Thus, G.o,o is bijective and the assertion follows by the inverse function t h e rem. We fix the choice of W as in this lemma. Let V be an open neighborhood of xo in M such that VxVcG(W)andVCU. Given (x,y) E V x V, let v, E W be the unique element such that (x,y) = G(vX)= (x, exp,(v,)). That is, exp,(tv,), 0 5 t 5 1, is the unique geodesic of length < E going from x to y and parametrized on [0, 11. The point exp,(tv,) depends smoothly on (v,, t) E W x [O, 11 and G is a diffeomorphism on W, so exp, (tv,) also depends smoothly on (x, y, t). Less formally, we say that the unique geodesic of length < E joining x, y E V depends smoothly on its endpoints. Remark that, even though x, y E V, there is no reason why the geodesic exp, (tv,) should stay in V for all values of t E [O, I]. It is a fact, as we will see later, that V can be chosen so that the geodesics of length < E. joining any points x, y E V do remain in V. This is J. H. C. Whitehead's theorem on the existence of geodesically convex neighborhoods which will give us the existence theorem for simple covers that we used in the discussion of de Rham cohomology. Finally, note that this discussion also shows that, for each x E V E U ,exp, maps the open €-ball B,(E.)C T,(M) diffeomorphically onto an open neighborhood of x in M that contains V. IR2 be open and consider smooth maps s : U 4 M. In complete Let U analogy with the case of curves, we obtain the space X(s) of all vector fields dong s, these being all smooth maps v : U 4 T(M) such that the diagram

commutes.

EXAMPLE 10.4.16. Let (r, t) be the coordinates of U . Then we define d s l d r and a s / % E X(s) by

10. RIEMANNIAN GEOMETRY

320

Just as for curves, given v E X(s), we have partid covariant derivatives

-

obtained by taking covariant derivatives along the respective curve families t constant r =: constant.

LEMMA10.4.17. For s : U

+M

as above,

The proof of this lemma will be Exercise (4). Let x E V and E > 0 be as before. For 0 < a around x of radius a to be

< E, define the spherical shell

clearly a smooth hypersurface (submanifold of codimension one).

LEMMA10.4.18. The geodesics out of x meet the spherical shell S , orthogonally, 0 < a

< E.

PROOF. Let v(t) be a smooth curve in T , ( M ) , parametrized on B,with 11v(t)ll = 1. For Irl < E , define a smooth function

For fixed to E R, f (r, to) describes the geodesic out of x with initial velocity v(to). For fixed ro E (-E, E), f (ro,t) describes a smooth curve on the shell S,,. Since v(t) was arbitrary, it will be enough to prove that the curves f ( T O , t) and f (r,to) meet orthogonally at the point f (ro,t o ) Since (TO, to) is to be arbitrary, what we really need to show is that

Since the r-curves (t

-

constant) are geodesics, we have

Also, by Lemma 10.4.17,

v af v af ar a t a t a~

-- = --

10.4. COMPLETE RIEMANNIAN MANIFOLDS

-

The last equality is due to the fact that 118f /drll is the length of the velocity field along the geodesic r H exp,(rv(t)) and this length is the constant Ilv(t)ll 1. It follows that (df/dr, d f / d t ) is constant in r . But, a t r = 0, d f / & = 0, so

Let x E V and set V, = exp,

(B,(E)). Then V

a : [a, b]

-+

V,

\

V,. Let

{x}

be a piecewise regular curve. Since exp, : B, ( E ) -,V, is a diffeomorphism, there is a unique piecewise regular curve 6 in Bx(c) \ {0} , parametrized on [a,b] and such that a = exp, 08. We write

where

Thus, a ( t ) = exp,(r(t)v(t)) with r(t) and v(t) subject to these conditions. LEMMA10.4.19. For a as above,

V equality holds,

then r ( t ) is strictly monotone and piecewise regular and v(t) is constant. Consequently, any shortest path joining two concentric shells around x is a piecewise regular reparametrization of a radial geodesic segment.

PROOF.Let f (r, t) = exp, (rv(t)). Thus, a ( t ) = f (r(t), t), so

Clearly, Ildf/dr() EE 1 and, by Lemma 10.4.18, df/dr 1a f / a . Thus,

10. RIEMANNIAN

322

If equality holds, then d f /at

GEOMETRY

= 0, so v ( t )z 0 and v ( t ) is constant.

Therefore,

If equality holds, not only is v ( t ) = v constant, but d r l d t cannot change sign and, since r ( t ) v = 8 ( t )is piecewise regular, drldt can never be 0. That is, r ( t ) is strictly monotonic and piecewise regular.

THEOREM 10.4.20. For V and

> 0 as above, let y

[O, 11 -+ M be the unique geodesic of length < E joining two points x,y E V . Let a : [O, 11 + M be an arbitrary piecewise regular path joining the same two points. Then 1 y 1 5 101, where equality holds if and only i f a is obtained from y by a piecewise regular reparametriration. E

:

PROOF.Set y = exp,(rv), where 0 < r < E and jlvll = 1. Then, if 0 < 6 < r < e, a contains a segment joining the spherical shell Sa to Sr and lying between these shells, hence lying in V,. By Lemma 10.4.19, this segment has length at least r - 6, so 101 2 r - 6. Letting 6 J, 0, we conclude that la1 2 r = 171. If (a( = r , then the segment from each S6 to Sr must be a reparametrization of (the same) radial geodesic. Since a is piecewise regular, the reparametrization r ( t ) is continuous and (t) has only jump discontinuities, occurring only finitely often as 6 1 0, so a itself is a piecewise regular reparametrization of y. Conversely, if a is a piecewise regular reparametrization of y, then 1 ol = I y 1.

+

COROLLARY 10.4.21. Let a : [a,b] -,M be piecewise regular ( p . ~ . )and have minimal length for any p.r. path from a ( a ) to a ( b ) . Then a is obtained from a geodesic by p.r. reparametrization. V a is regular, it is a regular reparametriration of a geodesic. If IIbII is constant, a is a geodesic.

PROOF.Consider any segment of a lying in an open set V as above and having length < e. By the above, this segment must be a p.r. reparametrization of a geodesic. Since every interior point of a lies in the interior of such a segment and o is made up of finitely many such segments, a must be a par.reparametrization y ( r ( t ) ) of a geodesic y . If a is regular, then

and this implies that r ( t ) is a regular change of parameter. Since Ilj.(r(t))11 is constant, d r l d t will be constant if Illr(t)11 is. In this case, r ( t ) = d +e for suitable constants c # 0 and e , so a ( t )= y (ct + e ) is a geodesic. The next two results complete the proof of Proposition 10.4.11.

10.4. COMPLETE RIEMANNIAN MANIFOLDS

323

PROPOSITION 10.4.22. If x, y E M and the Riemann distance p(x, y) = 0, then x = y. PROOF.Suppose x # y. Choose V and E > 0 as usual, but such that x E V, y 4 V. For a suitable choice of q E (0, E), exp,(B,(q)) C V. Then, every p.r. path a from x to y must meet the spherical v h e l l centered at x, so la/ 2 q and P(X,Y)2 r l > 0. Thus, the Riemann distance function is a topological metric on M. PROPOSITION 10.4.23. The topology induced on M by the metric p coincides with the manzfold topology. PROOF.Let x E M and choose E > 0 so small that exp, : B,(E) 4 M is a difieomorphism onto an open neighborhood U,(E) of x in the manifold topology. The set of all such U,(E) is a base for the manifold topology of M. But

Indeed, if y E U, (E) , then p(x, y) < E. Furthermore, if z E M \ U, (E), every p.r. a from x to z must meet every m h e l l , 0 < Q < E, centered at x, so J a J2 E and, consequently, p(x, z) 2 E. This proves the assertion (10.6), showing that {U,(E) I x E M, E > 0) is also a base for the topology induced by the metric P. We note also the following useful fact. 10.4.24. If C C M is compact, there &sts 6 > 0 such that any PROPOSITION two points x, y E C with p(x, y) < 6 are joined by a unique geodesic (parametrized on [0, I]) of length < 6. This geodesic is the shortest p.r. path from x to y and depends smoothly on its endpoints. In particular, if M is compact, 6 can be chosen uniformly for all of M. PROOF.Cover C by open sets V, with corresponding E, as in the above discussion. Select a finite subcover {V,, )r=l. Let 6 > 0 be a Lebesgue number for this cover (i. e., if p(x, y) < S and x, y E C, then x and y lie in a common E,, . All assertions now follow from V,,). We can also demand that S 5 minl 0 be chosen as usual, with x E V, and let Sg C V be a spherical shell around x, 6 < r sufficiently small. Since Sa is compact, there is xo E S6 such that ~ ( ~ Y) 0 = 9 P(~,Y,.

2;

Write xo = exp,(6vo), lbo11 = 1. The geodesic y(t) = exp,(tvo) is defined for all real values of t because M is geodesically complete (see Figure 10.6).

10. RIEMANNIAN GEOMETRY

FIGURE 10.6. The Hopf-Rinow setup We will prove that, contrary to the possibility allowed in Figure 10.6,

thereby proving Theorem 10.4.12. Our procedure will be to prove, for each t E [6,r], the proposition

In particular, F, asserts that p(y(r), y) = 0, giving y = y (r). (a) We prove F6. Every p.r. curve from x to y must cross S6, hence

=6

+ inf

zES6

p(z7 y)

Therefore, ~ ( ~ ( 6Y)1 = , ~ ( ~Y)0 = 1 r - 6-

(b) Assuming the truth of Ft,, some to E [6,r), we will prove the existence of a maximal half-open interval [to,q) such that q 5 r and Ft is true for all t E [to,rl). Indeed, for 6' > 0 sufficiently small (as usual), let S6, be the spherical shell of radius 6' around ?(to). Let xb E S6t be a point with p(xb, y) minimum. Then,

10.4. COMPLETE RIEMANNIAN MANIFOLDS

as before,

SO

p(xb,y) = r - to - S' = r - (to +St). We will show that xb = y (to + 6'). Indeed,

But the path consisting of the segment of y from x to ?(to), followed by a minimal geodesic from ?(to) to xb, has length to S', hence is a piecewise geodesic, parametrized by arc length, joining x to xb and of minimal length. By Corollary 10.4.21, this path is a geodesic. It coincides with y on [0,to] and to > 0, hence it coincides with y on [0,to + 6'1. That is, xb = y (to 6') , as claimed. We have proven that

+

+

which is the assertion Ft,+61. Since 6' > 0 was arbitrarily small, there is a halfopen interval [to,q'), 77' < r , on which Ft holds. The union of all such intervals produces the maximal one [t0, 7). (c) Since F6 holds, let [ 6 , ~be ) the maximal half-open interval on which Ft holds. But the truth of Ft on ( 6 , ~ implies ) F,, by continuity. Thus, if 17 < r , we could apply (b) to obtain a contradiction to the maximality of [6,7). Consequently, v = r and F, holds.

PROOF O F THEOREM 10.4.13. We first assume that M is geodesically complete and we prove that M , as a metric space under p, is complete. For this, M is a pbounded subset, the it will be enough to prove that, whenever B closure B is compact. Choose any x E B and consider the continuous map

c

exp, : T,(M)

+ M,

defined because M is geodesically complete. Since B is bounded, there is a number r > s u p y c p(x, ~ 9). If D C T,(M) is the closed ball of radius r, then B exp, (D) and exp, (D) is compact. Thus, B exp, (D) is compact. Next, assuming that M is complete in the metric p, we prove that M is geodesically complete. Let x E M and let v E T,(M) have unit norm. Let (a, b) denote the maximal open interval about 0 in R such that exp,(tv) is defined, V t E (a, b). We must show that a = -w and b = w. If b < m, choose { t k ) g l c (a, b) such that tk b strictly. This is a Cauchy te - tk, whenever sequence. Set xk = exp,(tkv) and remark that p(xe7xk) I k < 4?. Indeed, the segment of exp, (t), tk 5 t 5 te, is a geodesic of length te - tk

c

c

326

10. RIEMANNIAN GEOMETRY

joining these two points. Therefore, {x k)r=, is Cauchy in the metric p and, by the completeness of this metric, xk -+ y E M. Define y : [0,b] + M by

By the previous paragraph, y is continuous on [0, b]. It is smooth on [0,b). If y is also smooth at b, it is a geodesic and can be extended as a geodesic to [0,b 7)) some q > 0. This would contradict the maximality of (a, b), proving that b = w. In order to prove that y is smooth at b, choose a neighborhood V C M of y and a number E > 0 such that exp,(w) is defined, V z E V, Vw E T,(M) with 11 w 11 < 6. Fbr a Cauchy sequence {xk = exp, (tkv))p==l, chosen as above, xk E V and b - tk < c, for all sufficiently large values of k. Let k be large enough and set vk = j.(tk) E T,, (M). Since llvk11 = 1, exp (tvk) is defined for c,'. 0 5 t 5 b-tk < E. But, for 0 5 t < b-tk, this curve coincides with y ( t k + t ) . By continuity, y(b) = expXi( ( b - tk)vk), completing the proof that y is smooth at b. A completely parallel argument shows that a = -m.

+

EXERCISES (1) Let M C iR3 be a smoothly imbedded surface with the relativized metric and Levi-Civita connection. Let a : [a,b] -, M be evenly parametrized and suppose that im(a) C_ P n M , where P is a 2-plane in IW such that v o ( t ) ( M )c P, a t 5 b. Prove that a is a geodesic. (2) If M is a surface of revolution (as defined in freshman calculus), identify a natural, infinite family of geodesics. Discuss whether or when the "circles of latitude" are geodesics. (3) Show that every geodesic on the unit sphere s2 C IR3 must lie along a great circle. (4) Prove Lemma 10.4.17. (5) Let M be a complete Riemannian manifold, F a foliation of M, and let L be a leaf of F. The Riemannian metric (-, -) on M induces a Riemannian metric (-, -)L on L via the one to one immersion i : L M. Let p~ denote the corresponding topological metric on L. Generally, this is not the restriction of the metric p of M. Let { x k ) ~ c , L be pL-Cauchy. (a) Prove that { x k ) E 1 is p-Cauchy, hence that xk -+ x E M . yl, . . . ,yn) be a F'robenius neighborhood of x. Prove that (b) Let (U, all but finitely many of the points xk lie on the same F-plaque in U as x. (c) Using the above, conclude that, as a Riemannian manifold in the induced metric (-, - ) L , L is complete. In particular, a leaf L in a compact, foliated manifold (M, F) is complete in any metric (., -) that arises, as above, by restricting to L an arbitrary Riemannian metric (., -) on M. The leaf L need not, itself, be compact.

<

-

10.5. GEODESIC CONVEXITY

327

The geodesics on the Riemannian manifold (L, (-, -) L ) are not, generally, geodesics in (M, (-, -)). 10.5. Geodesic Convexity We will prove a theorem of J. H. C. Whitehead which, in particular, will guarantee the existence of simple refinements of open covers. This result was anticipated and used in our treatment of de Etharn cohomology (Chapter 8). Throughout this section, M is a Riemannian n-manifold with empty boundary. M is star shaped with respect to a point DEFINITION 10.5.1. A subset X so E X if each x E X can be joined to xo by a unique shortest geodesic in M and if this geodesic always lies in X. DEFINITION 10.5.2. A subset X with respect to each of its points.

M is geodesically convex if it is star shaped

Equivalently, X is geodesically convex if any two of its points are joined by a unique shortest geodesic in M and this geodesic lies in X. The following is immediate. LEMMA10.5.3. An arbitrary intersection of geodesically convex sets is gwdesically convex. THEOREM 10.5.4 (WHITEHEAD). Let W M be open, x E W . Then there is a geodesically convex, open neighborhood U C W of x. Before proving this result, we show how it implies the existence of simple refinements. LEMMA10.5.5. A set X C_ M, star shaped with respect to xo E X, is wntractable.

PROOF.Indeed, each x E X determines uniquely v, E Txo(M) and t, > 0 such that llvrll = 1 and x = expxo(txvx).Then, F : X x [O,1] --t X , defined by

is the desired contraction. It seems that open, star shaped sets U M are always diffeomorphic to Rn, but this is extremely difficult to prove. The problem is that the set theoretic boundary dU may be very badly behaved. For instance, the "radius function" r : s:,-I

+

[o, 001,

even if it only takes finite vdues, may not be continuous, let alone smooth. This function is defined on the sphere of unit vectors in Txo(M) and assigns to the supremum r (v) of the numbers T > 0 such that exp,, (tv) E U , v E S:,-l 0 < t 5 T . Keep the possible bad behavior of r in mind while formulating a proof of the the following (Exercise (1)).

10. RIEMANNIAN GEOMETRY

328

LEMMA10.5.6. Let U M be open and star shaped with respect to xo E U and let C c U be compact. Then there is an open set V C U, also star shaped with respect to xo, such that is a compact subset of M and C c V c c U.

v

PROPOSITION 10.5.7. If U C M is an open set, star shaped with respect to xo, then Hz (U) = HE (Rn). PROOF.For a suitable value po > 0, the open ball Bxo(po)c T,,(M), centered at 0 with radius po, is carried by exp,, diffeomorphically onto an open set Up, c U with compact closure in U. Since Bxo(po)is diffeomorphic to Rn, so U induces is Up,. In particular, Hz (Up,) = H,*(Rn). The inclusion i : Upo homomorphisms

-

so it will be enough to prove that the second of these is an isomorphism. Let C c U be compact. Using Lemma 10.5.6, find open sets V c W c U, star shaped with respect t o xo and such that C W C W C U, and W are U C c V. Also, fix 0 < a < b < po and the corresponding compact, and open balls Ua c Ub c Up,. By the smooth Urysohn lemma, find f : U -r [O, 11 such that

v

upo

v

Let Z E X(U \ (xo)) be nowhere 0, tangent to the radial geodesics out of xo, and everywhere pointing toward xo. Let Ft : U --, U denote the flow, defined for all time t, generated by the compactly supported vector field f Z E X(U). Then, since C \ Upo is contained in the interior of the support of f 2, as is agpo, = F-,. Since F-, is there is a value T > 0 such that F,(C) C Up,. Set a compactly supported diffeomorphism of U onto itself, isotopic through such diffeomorphisms Ft to Fo = idv, it follows that q* : H,*(U) 4 H,*(U) is the identity. Let w E Z,P(U) and let C = supp(w). By the previous paragraph, we obtain : U --, U such that +* (w) E Z,P(Up, ) and +* [w] = [w] E H,P(U). It follows that [w] E im(i,), hence that i, carries Hz(Upo) onto Hz(U). Suppose that w E Z,P(Upo)and that i. [w] = 0. Choose a > 0 as above such that supp(w) C U,. Viewing w = i. (w) in Z,P(U), we find 7 E A!-' (u) such that w = dq. Let C = supp(q) and obtain : U -, U, as above, so that

+

+

+

but dqo = d$* (7) = +*(dq) = +*(w) = W. That is, [w] = 0 in H,'(Upo), proving that i, is one to one.

10.5. GEODESIC CONVEXITY

329

COROLLARY 10.5.8. Every open cover of M admits a simple refinement. PROOF. By Theorem 10.5.4, each open cover admits a refinement by open, geodesically convex sets. With a little care, one chooses this refinement to be locally finite (Exercise (2)). By Lemma 10.5.3, any finite intersection of elements of this refinement is also an open, geodesically convex set, hence star shaped. By Lemma 10.5.5 and Proposition 10.5.7, this refinement is simple. We turn to the proof of Theorem 10.5.4. Let x E W, as in the statement of the theorem. As in Section 10.4, choose a neighborhood V of x in W and a number e > 0 such that any two points y, z E V can be joined by a unique geodesic a , in M of length < E . As usual, a,,, is parametrized on [ O , l ] and depends smoothly on (y, z) E V x V. Choose 6 > 0 such that the open ball B,(6) c T,(M) of radius 6 is carried diffeomorphically by exp, onto a neighborhood U, C V of x. Let (vl, . . . , u,) be an orthonormal frame for T,(M) and coordinatize this vector space by (xl, . . . ,xn) * xiui. Under the diffeomorphism exp;l : U, -+ B,(6), these become coordinates on U, (called a normal coordinate system on U,). The corresponding coordinate fields are ti E X(U,), 1 5 i 5 n. If y E U, has coordinates (bl,. . . ,b,), then

If 0 < 6, < 6, if Ss, C U, is the spherical shell of radius 6,, centered at x, if Y = (bl,. . . , bn) E S6,, and if v = E Ty(Ss,), then

The key lemma for the proof of Theorem 10.5.4 is the following. LEMMA10.5.9. If 6,

E

(0,6) is small enough, then evey geodesic

such that y(0) = y E S g , and T(0) E Ty(S6,), has the property that p(x, y(t)) 6, for all suficiently small values of It1 # 0.

>

PROOF.As above, denote the normal coordinates of y by (b 1, . . . ,b,). Let 6, be so small that, for C:=, b: 5 6,, the symmetric matrix

is so close to [26ke]as to be positive definite. Let be a geodesic in M, tangent to S6,at y(0) = y = (bl,. . . ,b,), and let T(0) = a'&. Define n

F(t) = p(x, y(t))2 - 61 =

~ ' ( t -) ~ 61.

10. FUEMANNIAN GEOMETRY

330

For small values of It 1, this is a smooth function and

Since ~ ( t is) a geodesic, it satisfies

giving = 2((aa)2- aka'(birh(bl,

. . . , b,)))

the value of a positive definite quadratic form on the vector +(O) # 0. That is, Ft'(0) > 0. Plugging this data into the 2nd order Taylor expansion of F(t) about t = 0, we see that t2 F(t) = Ftf(0) O(t3) > 0,

+

for small enough values of It 1 # 0.

PROOFOF THEOREM 10.5.4. Choose N, = exp,(B,(G,)), where 6, is chosen by the above lemma. Let R N , x N, be the subset of all (y, E) such that ay,, lies entirely in N,. By the smooth dependence of this geodesic on its endpoints, R is an open subset. It is also clear that R # 8. If we prove that R is also a closed subset, then, by the connectivity of Nz x N,, R = Nz x N, and Nz is geodesically convex. Let {(gk,z k ) ) E l c R with limk+oc(yk,zk) = (yo, zO) in N, x N,. If (yo,zO)&I R, then ayo ,, meets dNz = Ss,. If a,,,, is tangent to the spherical shell at some point of intersection, an application of the lemma shows that a,,,, contains points in U,\ Nx. But smooth dependence on (yo, ro) implies that this remains true for all values of (y, r) sufficiently near (yo, .to), hence for (yk,rk), k sufficiently large. This contradicts the fact that (yk,zk) E R. But if an intersection point of a,, ,, with the shell is not a point of tangency, it is clear that ox,, contains points in 11, \ leading to the same contradiction. Thus, (xo, yo) E R, proving that R is closed in N , .

,

x,

EXERCISES (1) Prove Lemma 10.5.6. (2) Check the assertion in the proof of Corollary 10.5.8 that the refinement by open, geodesically convex sets can be chosen to be locally finite.

10.6. CARTAN STRUCTURE EQUATIONS

331

(3) Let x E U M where U is open in M and star shaped with respect to x. Let r : S,"-l -, [O,m] be the radius function for U. That is, 27,"-l c Tz(M) is the unit sphere and

U = {exp,(tv) I v E s:-' and 0 < t < r(v)}. If r is finite-valued of class Cm, prove that U is diffeomorphic to Rn. (4) Let x E U C_ M and r : 9,"-' + [0, m ] be as in the preceding exercise, but do not assume that r is smooth or even continuous. Prove that r is lower semicontinuous. That is, r ( a ,m ] is open in SF', V a E W. M and r : S,"-' -+ [0,m ] be as in the preceding exer(5) Let x E U cises. Construct an example in which r is finite-valued everywhere and discontinuous on a dense subset of S,"-'.

-'

10.6. The Cartan Structure Equations

We return to the torsion and curvature tensors that were introduced earlier for a connection V. The key to understanding the geometric significance of these tensors is a pair of equations, written in terms of differential forms, called the equations of structure. One application will be a proof that the Riemann tensor is exactly the obstruction to the integrability of the infinitesimal O(n)structure defined by the Riemannian metric. More, precisely, we will prove

THEOREM 10.6.1. Let M be a Riemannian manifold, V the Levi-Ciwita connection, and R the Riemann tensor. The following are equivalent. (1) R = 0. (2) There is a smooth atlas i n which the metric coeficients are everywhere gij &j. (3) There is a smooth atlas in which the Christoflel symbols are everywhere k G 0. r.. 13 (4) Each x E M has a neighborhood U, such that the holonomy of V around each loop a E S2(Uz,x) is the identity transformation. ( 5 ) The Riemannian metric, as an infinitesimal O(n)-structure, is integrable. A Riemannian manifold in which one, hence all, of these holds is said to be flat. In what follows, V is a general connection on the n-manifold M, n 2 2. To begin with, we will work in an open, trivializing neighborhood U C M for T(M) and fm the trivialization by a choice of a smooth frame (XI, . . . , X,) on U. Define Li' E A1(U) by Zi(Xj) = S;, 1 5 i,j 5 n. Then, each X E X(U) can be written x =Zi(x)x,. Define forms

6; E A1(U), 1 5 i,j 5 n, by

The fact that these are forms, i.e., that 6j(f X) = f r ) j ( ~ ) V, f E Cm(U), follows from the fact that VxXj is a tensor in X .

10. RIEMANNIAN GEOMETRY

332

In order to express the torsion and curvature tensors of V in terms of this locd trividization of T ( M ) ,we introduce forms Ti, fij E A2(u) by the formulas

The fact that these are antisymmetric tensors (i.e., 2-forms) follows from the same properties of T and R. THEOREM 10.6.2 (CARTANSTRUCTURE EQUATIONS). The above f o m s satisfy the identities

PROOF. The proof is a computation. We carry this out for equation (10.7). The computation of equation (10.8) is more of the same and will be left as an exercise. For arbitrary X, Y E X(U),

Fi(x, Y)Xi = VXY - vyx - [X, Y] = vX(Z'(Y)X~) -

vY(Z'(X)X~) - ~ ' ( [ x Y])Xj ,

= (x(S~(Y))- Y(S'(X))

- ijj([x, Y ] ) ) x ~

+ s ( y ) v X x j - iZj(x)vYXj &j(x,y ) x j + s ~ ( Y ) B ~ ( x ) x ,- G ~ ( X ) ~ ; ( Y ) X , = (&;(x, Y) + ~ ( Y ) % ( x ) - ij-'(x)6;(y))xi. =

But we claim that

Indeed, the standard inclusion A2(u)

-

G2(U) takes

(Lemma 7.2.17). Thus, the coefficients of Xion each side of the above being equal, 1 5 i 5 n, we obtain

Since X and Y are arbitrary, equation (10.7) follows. 0

10.6. CARTAN STRUCTURE EQUATIONS

333

Using matrix notation, we can write the equations of structure more compactly. Set

The n-tuples 5 and ? can be thought of as Wn-valued forms. The matrices 6 and fi can be thought of as L(Gl(n))-valued forms. The tildes on these symbols are reminders that these forms are dependent on the choice of local trivialization. We will remove the tildes by lifting these local forms to the frame bundle of T(M), where they fit together coherently to define global forms.

DEFINITION 10.6.3. The Wn-valued forms i Z and 7 are called the trivializing coframe field and the torsion form, respectively. The L(Gl(n))-valued forms 8 and

6 are called the connection form and the curvature form, respectively.

The structure equations are

where we multiply matrices of forms by the usual rules of matrix multiplication, but use exterior multiplication for products of entries.

EXAMPLE 10.6.4. When the connection is associated to an infinitesimal Gstructure on M, its connection and curvature forms will be L(G)-valued. The Levi-Civita connections of Riemannian and "pseuddtiemannian" manifolds are important examples. Recall that the group O(k, n - k) c GI(n) consists of the matrices that leave invariant the quadratic form

+ + ( x ~ -) (~x ~ + ' ) ~. .- ( x " ) ~ .

Q ~ ( X '.,. . ,xn) = ( x ~ ) - ~- -

,

Recall from Example 3.4.15, that an infinitesimal O(k, n - k)structure on M is a smooth, nondegenerate, symmetric tensor (., .) E S2(M) of constant index n - k ( 2 . e., n - k is the maximal dimension of a subspace of Tz(M) on which this tensor is negative definite, Vx E M). Such a tensor is called an indefinite metric on M and we say that M, with this metric, is a pseudeRiemannian manifold. A particular example is a Lorentz manifold, a four-dimensional manifold with an indefinite metric of index 1. These are models of space-time in general relativity. Of course, Riemannian metrics are also a special case with k = n. For an indefinite metric, there is a unique associated Levi-Civita connection V (i.e.,

334

10. RIEMANNIAN GEOMETRY

the torsion is 0 and X (Y, 2 ) = (VxY, 2 ) + (Y, VxZ)). The proof of this is by exactly the same argument as you used in Exercise (1) on page 306. Given such a metric, choose the locally trivializing frame (XI, . . . ,X,) to be "orthonormal" in the sense that 0, i#j 1, i=j k. This can be done by the Gram-Schmidt process. In Exercise (3), you are asked to prove that the forms and fi, computed relative to this frame, take values in the Lie algebra L(O(k, n - k)). In particular, for the Levi-Civita connection of a Riemannian metric, 6 and fi are L(O(n))-valued. The statement of Threm 10.6.1 will be true, with the obvious modification of parts (2) and (5), for Levi-Civita connections of indefinite metrics (see Exercise (5)), giving several characterizations of flat, pseuddtiemannian manifolds. The Lorentz manifold for special relativity is flat. The equations of structure depend on the choice of local trivializations or "framings" of T(M). In order to remove this dependence on local choices and thereby globalize the structure equations, we will lift them to the principal frame bundle. For the case in which V is a general connection, let p : P + M denote the frame bundle p : F(M) 4 M. If V is the Levi-Civita connection of a Riemannian manifold, we let P = O(M), the bundle of orthonormal frames. In the first case, P is a principal Gl(n)-bundle and, in the second, it is a principal O(n)bundle, namely, the O(n)-reduction of F ( M ) determined by the metric. Indeed, with little extra effort, we can include the case in which V is the Levi-Civita connection of an indefinite metric of index n - k. In this case, the associated reduction P = Ok(M) c F ( M ) is a principal O(k, n - k)-bundle. In order to discuss these cases simultaneously, we w ill denote the respective structure groups, GI(n), O(n), and O(k,n - k), all by G. We will lift equations (10.9) and (10.10), as promised, by finding R "-valued forms w and T on P and L(G)-valued forms 8 and fl on P satisfying

Furthermore, the choice of frame ( X I , . . . ,Xn) on U is a choice of smooth section a :U -,PIU and we will prove that

Thus, the local equations of structure are the a pull-backs of equations (10.11) and (10.12).

10.6. CARTAN STRUCTURE EQUATIONS

Let

C E P, x = p(C), and set P,

= p-'(x).

335

The vertical space at

C E P will

be = Tc(Pz) c Tc(P).

DEFINITION 10.6.5. The vertical subbundle V C T ( P ) is V = elements E E I'(V) are called the vertical fields on P.

UcEPVC. The

Each E E L(G) can be viewed as a vertical field on P. Indeed, E is an n x n matrix and etE is the one parameter subgroup of G generated by E. Using the right action P x G -+ P , we obtain, for each C E P, a curve C - e t E which, when t = 0, is at C. This curve lies in the fiber of P through C, so the corresponding infinitesimal curve is C - E = (C - etE),=, E Q. As varies over P, this defines a vertical field on P. The mapping E H F is a canonical linear injection L(G) -+ I'(V). From now on, we denote F by E, identifying L(G) as a vector subspace of I'(V). M is an open, triviaiizing neighborhood for P, the trivializations If U P(Ur U x G are in one to one correspondence with the sections a E I'(PJU). Indeed, a(x) - B e, (x,B). Fix the choice of a. Since B E G C Gl(n) is a nonsingular n x n matrix and, as remarked above, each E E L(G) is an n x n matrix, the value of the vertical field E at ( = (x, B) E P(Uis C . E = (x, B E ) , where B E is the matrix product. This is because E, as a left invariant vector field on G, has value at B E G given by BE. Thus, this way of viewing a matrix E E L(G) as a vertical field on P is quite analogous to the way that E is viewed as a left invariant field on G. The right action P x G -+ P induces a linear action

via the differential. If X E X(P) and B E G, it will be natural to denote this action by X H X . B. Consider also the automorphism Ad(B) : G -, G, called the adjoint act ion and defined by Ad (B)(A) = B - AB ( cf. Exercise (5) on page 144). The differential Ad(B) : X(G) -,X(G) restricts to an automorphism of L(G) where it will also be called Ad(B) and written Ad(B)(E) = B-'EB. The following is practically immediate.

'

.

-

LEMMA10.6.6. Under the inclusion L(G) I'(V), Ad(B)(E) = E B , VB E G, V E E L(G). I n particular, L(G) c r ( V ) is invariant under the right action of G.

REMARK.Any basis of L(G) c I'(V) gives a trivialization of V. If desired, it is possible to specify a canonical choice of this basis. In the case that G = Gl(n), this will be {Ej}15*,j<,, where E; is the n x n matrix having (i,j)th entry 1 and a l l remaining entries 0. The Lie algebra of O(k, n - k) consists of all matrices

where A is a skew symmetric, k x k matrix and C is (n - k) x (n - k) and skew symmetric. The reader is invited to check this (Exercise (2)). For 1 5 i < j _< k

10. RIEMANNIAN GEOMETRY

336

and k + 1 j i < j 5 n, set A: = E; - E:. For all other i < j, set A: = E; Then, {A;)l
+ E:.

We are going to show that the connection V defines a direct sum decomposit ion T ( P ) = VfB H,

and a canonical choice of basis {Ei)l
<

commutes. We think of s b as a "horizontd" lift and say that the initial velocity db(0) E Tc0(P)is horizontal. If we can show that the set of all vectors in Tco(P) that can be obtained in this way is a vector subspace of dimension n, this will be our candidate for HCo.Similarly, i b ( t ) E H,,(,), justifying the term "horizontal lift". A section a E r(Pl U),where U is a suitably small neighborhood of x 0, gives another way to lift s. If a = (XI,. . . ,Xn) and u(xo) = 6, set

Then, ~ ' ( 0 )= 6 and the diagram

commutes. Since sb(t)and sd(t) lie in the fiber P,(,),there is a unique element A(t) E G such that

Clearly, A : [O, c)

+G

is smooth and A(0) = I, the identity matrix.

10.6. CARTAN STRUCTURE EQUATIONS

337

-

-

Finally, we use a = ( X I , .. . ,X,) to define the forms G, 7,8, and 0 on U . LEMMA10.6.7 (KEY LEMMA).A(O) = B ( i ( 0 ) )E L ( G ) . Before proving Lemma 10.6.7, we deduce some important consequences. 10.6.8. ~f u E r ( P l u ) with a ( x o ) = b E P,,, if s : 10, E ) -+ u COROLLARY is smooth with s(0) = so, and if sb : [0,e ) + PlU is the horizontal Zifl with sb(0)= b,then i b ( 0 )= -6 . B ( i ( 0 ) ) a*,, ( ~ ( 0 ) ) .

+

PROOF.Indeed, differentiate sd(t) = sb( t ) ~ (at t )0, obtaining

But sd(0) = a,,, ( ~ ( 0 ) ) . We define h g : T x , ( M ) + T<,(P)

by h g ( i ( 0 ) )= i b ( 0 ) ,

this being linear by Corollary 10.6.8. Furthermore,

since

-6.9(i(0)), being vertical, is annihilated by p . 6 ,

and

In particular, h g is injective. DEFINITION10.6.9. The horizontal space a t vector space Heo = i m ( h g ) . The horizontal bundle is H= He.

Co

E

P is the n-dimensional

U

6EP

LEMMA10.6.10. The horizontal bundle H is a bundle and T(P)= V $ H , each summand being an invariant subbundle under the right action of G . The proof of Lemma 10.6.10 will be Exercise (4). PROOFOF THE KEY LEMMA. We take U to be a coordinate neighborhood and write s ( t )= ( x l ( t ), . . . , xn ( t ) .) Write

where a i ( 0 ) = 6:. The smooth frame o = ( X I , .. . ,X,) can also be written as a matrix valued function. Writing X k s ( t ) = ufL(t)&s(l) as a column vector, we get

10. RIEMANNIAN GEOMETRY

338

Similarly, write

s b ( t )= ( ~ , s(o b , . . ,x; .(t,) = ;I[. ( t ) ]. In this notation, equation (10.13) becomes

Using the covariant derivative to express the fact that s is a parallel frame along s gives

(10.15)

tj; ( t ) =

-vy ( t )x P ( t ) r p a( s ( t ) ) .

Differentiate equation (10.14) and get

z;;(t)a;( t )+ v; ( t ) b i( t )=

(10.16)

14.;

(t).

In equation (10.16), set t = 0, note that

and get

Use equation (10.15) at t = 0 to conclude that

zii ( 0 )+ u i ( 0 ) k p( 0 ) ~ ; ~= ui.( 0 ) ~(:0 ) .

(10.17)

(20)

By the definition of the local connection form

6, we obtain

6: ( ~ ( O ) ) U ; ( O ) ~ =~ 6; ( ~ ( o ) ) x , = Vi(0)Xj

= Vi(0) (u;.Sa)

' q (0)ta + 21," ( 0 ) V i,,~(to) ~ ~ = (zif(o)+ ~ ~ ( o ) i P ( o ) r $ G , ( ~ .~ ) ) =

z0

lo

Setting the coefficients of tkzo equal and applying equation (10.17),we obtain

In terms of matrix products, this says

Since the matrix L(G).

[U;(O)]

is nonsingular, we conclude that A(O) = B ( i ( 0 ) ) E

10.6. CARTAN STRUCTURE EQUATIONS

339

DEFINITION 10.6.11. The tautological horizontal frame (E;, . . . ,Er), at C E P, is the unique n-frame of vectors in HCsuch that Equivalently, in terms of a local section a = ( X I , . . . ,X,) of P,

As ( varies over P, the reader can check that E: varies smoothly. That is, Ej E F(H) and the tautological frame field (E , . . . ,En) is a canonical trivialization of H . This, together with the remark on page 335, gives the following.

'

LEMMA 10.6.12. Given the connection V , the manifold P is canonically parallelizable. Indeed, for the case P = F ( M ) , the canonical basis of X(P) is {~'}:=l

'J {E;}l
and, for the case P = Ok(M), it is

{E'}:=l U {A~}lji<j
DEFINITION 10.6.13. The canonical coframe field for V on P is the Rn-valued 1-form w such that

Remark that, for each C E P, the vertical space is canonically Vc = L(G). Indeed, the evaluation map r ( V ) 4 Tc(P) carries L(G) c r ( V ) isomorphically onto TC(P). This identification is understood in the following definition.

DEFINITION 10.6.14. The connection form of V on P is the L(G)-valued 1form 8 such that

ejH = 0; eCIVS= id, REMARK. If we write

vc E P.

340

10. RIEMANNIAN GEOMETRY

then, in the case that G = Gl(n), the set {wi}y=l

u {@:}l5;,jsn

of 1-forms is dual to the canonical basis {~'};=l u { ~ ; } l ~ , , j ~ n * For G = O(k, n - k), the dual to the canonical framing of P is {wi}:=l

U {e:}~
LEMMA10.6.15. For B E G, let RB : P + P denote the right action of B. Then R k ( 8 ) = Ad(B) o 8 . That is, if C E P and v E TC(P),then

REMARK.One can use this to generalize the notion of a connection to principal G-bundles p : P 4 M, where G is a general Lie group. A connection on P will be an L(G)-valued 1-form 8 on P such that (1) if C E P and L(G) = Vc c TC(P) is the vertical space at C, then 8, = id : Vc + L(G); (2) R; (8)= Ad(b) 0 8, v b E G. By the first of these conditions, H = ker(8) is an n-plane subbundle of T(P) complementary to V. That is, T ( P )= V @ H, and one calls H the "horizontal distribution". By the second condition, the horizontal distribution is invariant under the right action P x G + P . Piecewise smooth curves s : [a, b] + M have "horizontal lifts" s b : [a, b] + P, uniquely determined by the initial point s (a) = C E Ps(,) and the requirement that db(t) E H s , ( t ) , a t 5 b. This assertion is proven by the basic theorem of O.D.E. The horizontal lift is interpreted as "parallel translation" of C along s. Furthermore, if F is some manifold and G x F 4 F is a smooth left act ion, there is an associated bundle .rr : P x F 4 M, with fibers diffeomorphic to F, and the connection on P defines a notion of parallel translation (or horizontal lifting) in this bundle along curves s : [a,b] -+ M. Using the notation [C, x] E P x G F for the equivalence class of (C, a) E P x F, we define the parallel translate of [C, x] E ?r (s(a)) to be i(t) = [sb( t ) ,XI, where sb(a)= 5. We return to the frame bundle P = F ( M ) or O k ( M ) and the connection V. The following definitions are dictated by equations (10.11) and (10.12).

<

-'

DEFINITION 10.6.16. The torsion form of V on P is the Rn-valued 2-form on P given by r=dw+eAw.

T

DEFINITION 10.6.17. The curvature form of V on P is the L(G)-valued 2form 52 on P given by R=de+@A@.

10.6. CARTAN STRUCTURE EQUATIONS

341

LEMMA10.6.18. Let o = ( X I , .. . ,X,) E r ( P l U ) and let S, 6, ?, associated forms o n U as in equations (10.9) and (10.10). Then,

6

be the

PROOF.If we verify the first two equations, the remaining two follow from the equations of structure. We use the notation in the first remark on page 340. For the first equation, we use wlV = 0. Thus, for arbitrary x E U and IIjSn,

That is, a*(wi)= ZIi, 1 5 i 5 n, so o*(w) = ZI. For the second equation, we use B: I H G 0 and consider the case of a general connection ( P = F ( M ) ) . Thus, for arbitrary x E U and 1 5 j 5 n,

For the case P = O r ( M ) ,replace ~?3;,(~) in the above with A;,(,) over the indices 1 5 r < q 5 n, again obtaining

Since x E U and 1 5 j 5 n are arbitrary, this gives 8. 0

and sum only

o*(e:)= @,

so 0*(9) =

LEMMA10.6.19. The following are equivalent for the connection V :

10. RIEMANNIAN GEOMETRY

(3) V is symmetric. (1) (2) is immediate. We prove (2) w (3). Let Co E P, set xo = p ( b ) E M , and let U M be an open neighborhood of xo such that P p is trivial. We can choose the section u E I'(PlU) so that u(xo) = Co and so that a,xo(Txo(M)) = HCo. For this choice, we have PROOF.

reo I (Heo x Heo)

-- 0 w 0 = 05, (reo)= Fx0 @

the torsion tensor Tzo = 0.

Since xo E M is arbitrary, the assertion follows. In order to prove (3) + (I), let Co, xo, and U be as above, and assume that V is symmetric. Given any direct sum decomposition Teo(P) = vco @ one can choose a E I'(P(U) such that o.,, (Txo(M)) = gc0. Then, the fact that o:,(rs0) = 7. = 0 implies that

&,,

Given u E VC, and w E Heo, choose He,, as above, and u E He, such that v u, w E Heo. This is clearly possible. Then

+

-

0 = T o (v + 21, w)=

(v, w)

+ reo(21, w) = re, (v, w).

This proves that ~eol(50

H
By antisyrnmetry, we also have

In order to prove that

TC,

= 0, it remains to prove that

Given u, w E Vco, let ul, u2 E HC, be linearly independent. Then there is a complementary space Heo, as above, such that v u l , w u2 E Thus,

+

We have proven that

+

&,.

rcO= 0 for arbitrary CO E P.

The following is proven by exactly the same argument.

LEMMA10.6.20. The following are equivalent for the connection V: (1) 0 -= 0; (2) 0I(H @ H) -= 0; (3) The curvature tensor R = 0. We can now deduce the following result, which contains Theorem 10.6.1 implicitly (Exercise (5)).

THEOREM 10.6.21. The following properties of the connection V are equivalent:

10.6. CARTAN STRUCTURE EQUATIONS

The n-plane distribution H on P is integrable; R _= 0; R = 0; for each x E M , there is an open neighborhood U of x on which the holonomy group of V is Hz(U) = {id). Furthermore, if V is symmetric, these properties are equivalent to (1) (2) (3) (4)

PROOF.We prove (1) u (2) Thus,

(3). The distribution H is exactly ker(8).

-

Therefore, integrability of H is equivalent to RI(H $ H ) = 0, which is equivalent to 0 = 0 and to R 0 by Lemma 10.6.20. We prove that (1) + (4). Thus, assume H to be integrable and choose x E M , C E p-l (x), and let L be the leaf through C. Since p , ~carries T<(L) = HC isomorphically onto T, (M), there is an open neighborhood W _C L of C carried diffeomorphically by p onto an open neighborhood U 5 M of x. Let s : [a, b] -+ U be a piecewise smooth loop at x. Let h, : T,(M) + T,(M) be the holonomy transformation around s. The horizontal lift s b : [a, b] + P with initial point sb(a) = (' = (vl , . . . ,v,) has terminal point sb(b) = (h, (vl ) , . . , h, (vn)). The curve s b must lie in W, hence it must be be the loop in W carried by p back to s. That is, sb(b) = sb(a) = (' and h,(v,) = v*, 1 5 i 5 n. Since the vi7sform a basis of Tz(M), h, = Since s E R(M, x) was arbitrary, the holonomy group H,(U) of V in U is trivial. Conversely, supposing that (4) holds, we deduce (1). Let (N, x ,. . . ,xn) be a coordinate chart in M and set W = p-I (N). It will be enough to show that HI W is integrable. Let Zi E I'(HlW) be the unique horizontal field that is p-related to ti, 1 5 i 5 n. These fields span HI W and their brackets [Zi,Zj] are p-related to [ti,(j] 0. This implies that [Z', Zj] E I'(VJW), 1 5 i,j 5 n, but we will show the stronger fact that [Zi,Zj] = 0, 1 5 i,j 5 n, so HIW is integrable. Let qi be the local flow in N generated by t i , 1 5 i 5 n. These flows lift to local flows 9' on p-' (N) with infinitesimal generator Zi E I'(H IW), 1 5 i 5 n. Let x E N and let U N be an open neighborhood of x so small that Hz(U) = {id). Let st be the flow line of 9' in N from x to y = 9fl(x). Similarly, s i is the flow line of Qj in N from y to z = 9:,(y), -s: is the flow line of (@')-I in N b m z to w = 9 y t l , and -s?, is similarly defined from w to sf = a<,,(w). For sufficiently small values of t 1 and tz , the p.r. curve

.

'

stays in U. Since these local flows commute, x' = x and s must be a loop at x. Since the holonomy group H,(U) is trivial, the horizontal lift s b to any

10. RIEMANNIAN GEOMETRY

344

< E p-l(x)

is a loop at

C, and this implies that

<

for all small values of t l and t2. Since x E N and E p-I(x) are arbitrary, it follows that [Zi,Zj] 0, 1 5 i,j 5 n. We turn to property (5). If [ E k EEL] , z 0, 1 5 k,e 5 n, it is clear that H is integrable. To complete the proof of the theorem, we assume that V is symmetric and prove that the integrability of H implies that [ E k EEe] , G 0. Indeed,

This implies that [ E k Eel , E I'(V), hence, when H is integrable, that this bracket vanishes identically, 1 5 k, C 5 n. DEFINITION 10.6.22. If one, hence all of the properties in Theorem 10.6.21 hold, the connection V is said to be flat.

EXERCISES (1) Verify equation (10.8). (2) Check that the Lie algebra L(O(k, n - k)) is correctly characterized in the remark on page 335. (3) Check that each pseud*Ftiemannian manifold (Example 10.6.4) has a unique Levi-Civita connection V and prove that the connection form T ) and the curvature form 6 are L(O(k, n - k))-valued. (4) Prove Lemma 10.6.10. (5) We stated Theorem 10.6.1 for the Levi-Civita connection of a Riemannian metric, but you are to modify the statement to allow indefinite metrics and prove this modified theorem. 10.7. Riemannian Homogeneous Spaces

This will be a very quick, introductory look at a large topic. We assume that (M, .)) is a connected, Riemannian manifold. The group of all isometries cp : M + M will be denoted by I (M). The action of I ( M ) on M preserves all intrinsic properties. In particular, it preserves the Ftiemann distance function, hence is a group of metric space isometries. We note, without proof, the following classical result [32]. (a,

THEOREM 10.7.1 (MYERSAND STEENROD). If M is a Riemannian manifold, the group I ( M ) , with the compact-open topology, is isomorphic, as a topological group, to a Lie group such that the natural action,

10.7. RIEMANNIAN HOMOGENEOUS SPACES

345

defined by (cp, x) H cp(x), is smooth.

DEFINITION 10.7.2. A Riemannian manifold M is homogeneous if the action I ( M ) x M -,M is transitive. The fact that I ( M ) is a Lie group implies that a homogeneous Riemannian manifold is of the form M = I(M)/K, where K is a closed subgroup, hence a properly imbedded Lie subgroup of I ( M ) . We will not use this fact.

PROPOSITION 10.7.3. If M is a homogeneous Riemannian manifold, then it is a complete Riemannian manifold. PROOF. At any x E M , let B,(E) be the diffeomorphic image, under exp,, of the open €-ball in T,(M). For every y E M , choose cp! E I ( M ) such that (p,Y(z)= y. Then By(€) = &!(B,(E)) is the diffeomorphic image, under expy, of the open &-ball in Ty(M). The point is that this E is uniform for all points of M . If s : (a, b) + M is a geodesic with maximal parameter interval, we must show that b = m . The same proof will give a = -m. We may assume that s is parametrized by arclength. If b < oo, find c E (a, b) such that b - c < E. Then B,(,) (E)is as above, so there is a geodesic segment a of length E out of s(c), having initial velocity s(c). Then o must agree with s on [c, b) and la1 = E > b - c, so s extends to the interval (a, c + E) where c + E > b, contradicting maximality. DEF~NITION 10.7.4. A symmetric space M is a Riemannian manifold with the property that, for each x E M, there is $J= E I ( M ) such $,(x) = x and ($z) *z = - id'I'= ( M ) . Remark that $z reverses every geodesic s through x. That is, if s(0) = x, then qz(s(t) = s(-t). As obvious examples, Rn with the Euclidean metric and Sn with its usual metric are both symmetric spaces. There are many other examples and the theory of symmetric spaces is quite highly developed (cf. [15]).

PROPOSITION 10.7.5. If M is a symmetric space, then M is a complete Riemannian manifold. PROOF. Let s : (a, b) + M be a maximal geodesic. If b < m , choose c E (a,b) closer to b than to a. By reparametrizing, if necessary, we can assume that c = 0, hence that b < -a. Then, for -b < t < -a, o(t) = $ s ( o ) (s(-t)) is a geodesic that coincides with s on (4, b). These fit together to form a geodesic parametrized on (a, -a), contradicting the maximdity of b. Thus, b = oo and, similarly, a = -00. 13 COROLLARY 10.7.6. If M is a connected symmetric space, then M is a homogeneous Riemannian manifold. PROOF.Indeed, if x, y E M , completeness and connectedness allow us to find a (shortest) geodesic s joining them (Theorem 10.4.12). Parametrize this geodesic on [-I, 11, s(-1) = x and s(1) = y. Then $,(0) E I ( M ) reverses this geodesic, hence carries x to y. C]

10. RIEMANNIAN GEOMETRY

346

DEFINITION 10.7.7. Let (., .) be a Riemannian metric on a Lie group G. This metric is left invariant if every left translation in G is an isometry. It is right invariant if all right translations are isometries. If the metric is both right and left invariant it is said to be bi-invariant. LEMMA 10.7.8. Relative to a left invariant (respectively, right invariant) Riemannian metric, a Lie group is a homogeneous Riemannian manifold. This is clear, as is the existence of such metrics.

LEMMA 10.7.9. If

(a,

a)

is a right invariant metric on the Lie group G, if

Y, Z E X(G), and if X E L(G), then

PROOF.Since X is left invariant, it generates the flow

at (x) = x . exp(tX), where exp(tX) is the one-pararneter group of X. Thus, Y . exp(-tX) t

([X,Y], 2 ) = lim t+O

-Y

= lim

A ((Y - exp(-tX), 2) - (Y, 2))

= lim

- ((Y, Z - exp(tX)) - (Y, 2 ) ) t

t-0

t+o

t

1

= - (Y,

[x, a).

COROLLARY 10.7.10. If V is the Levi-Ciwita connection for a bi-invariant metric on the Lie group G, then V x X = 0, VX E L(G). PROOF.Indeed, by Exercise (1) on page 306, if X , Y E L(G), we get (VxX, Y) = ([Y,XI, X ) = - ([X,Y],X ) = (Y, [X,XI)

= 0.

COROLLARY 10.7.11. The geodesics on G, relative to a bi-invariant metric, are the cosets (left or right) of the one-parameter subgroups.

PROOF.Indeed, due to the bi-invariance, we only need to show that the geodesics out of e are exactly the one-pararneter subgroups. But s(t) = exp(tX) is integral to the left invariant field X and V x X = 0, so the one-parameter groups are geodesics. But these groups are in one to one correspondence with their initial velocity vectors X,, as are the geodesics, so every geodesic out of e must be of this form.

COROLLARY 10.7.12. If the connected Lie group G has a bi-invariant metric, then every element of G can be reached from e by a one-parameter subgroup. Indeed, this follows from the geodesic completeness of G by Theorem 10.4.12.

10.7. RIEMANNIAN HOMOGENEOUS SPACES

34 7

EXAMPLE 10.7.13. This last corollary gives a necessary condition for a Lie group to admit a bi-invariant metric. Consider the connected Lie group Sl(2). If

then det(A) = 1 and

These give hence Let

so tr(A) = -3. If there is E E L(Sl(2)) such that A = exp(E) = e E , set B = eEi2 and get A = B2.This gives

a clear contradiction. Thus, this element A E Sl(2) cannot be reached from I by a one-parameter Lie group. The group Sl(2) cannot have a bi-invariant metric. PROPOSITION 10.7.14. If G h a a bi-invariant metric, it is a symmetric space relative t o that metric. PROOF.By homogeneity, it will only be necessary to produce $, for x = e. For this, we take $,(y) = y-l, Vy E G. To see that this is an isometry, let X , Y E L(G), remark that, , )T (X) and $, (Y) are right invariant fields, hence the constant (X, Y }and the constant ($, (X), $,, (Y)) are both equal to ( Y e=( e K ) THEOREM 10.7.15. Eve y compact Lie group has a bi-invariant metric. PROOF.Choose a left invariant Riemannian metric (-, on G. The corresponding Riemann volume form fl is left invariant also, so the Bore1 measure p that it defines is left invariant. The corresponding integral satisfies a)

10. RIEMANNIAN GEOMETRY

348

By multiplying the metric by a suitable positive constant, we normalize this integral: P

J G 1 d p = 1. We define an inner product on L(G) by

It is trivial that this is a positive definite bilinear form. As an inner product on L(G), it determines a left invariant Riemannian metric on G. But we claim that this is also a right invariant metric. Indeed, if a E G and X, Y E L(G)

=

( R(X), R ( Y ) d

= (X, Y)'

) (left invariance of p)

.

Thus, the compact Lie groups provide an interesting set of examples of symmetric spaces.

EXERCISES (I) Let G be an n-dimensional, connected Lie group with a given bi-invariant Riemannian metric. Prove that G is flat if and only if G is abelian. (By Exercise (7) on page 135, G = T~ x R"-~.) (2) Let G be a compact, connected Lie group, a : G + G a Lie group automorphism such that a2 = id, and let K = {x E G 1 a(x) = x). (a) Prove that K is a compact Lie subgroup of G. (b) Find a bi-invariant Riemannian metric g on G relative to which a is an isometry. (c) Show that g passes to a Riemannian metric on the homogeneous space G/K, making that manifold a symmetric space. (d) Show that the map cp : G + G defined by cp(x) = xa(x-l) passes to a smooth imbedding j5 : G/K G. (e) Write L(G) = L(K) @ 9X, where 9X IL(K) relative to the metric g. Prove that exp(9X) = cp(G/K). (f) Prove that cp carries each geodesic in G / K onto a geodesic in G, exactly doubling arclength and preserving angles. (g) Conclude that the properly imbedded submanifold M = exp(9X) of G, under the metric glT(M), is itself a symmetric space and a totally geodesic submanifold of G (i-e., the geodesics of M are also geodesics in G).

-

APPENDIX A Vector Fields on Spheres

We sketch a proof of the easier half of Theorem 1.2.12. Throughout this ap0 5 c 5 3. We will pendix, we set ~ ( n = ) 2' 8d, where n = (2r sketch the proof that Sn-l admits a ( n ) - 1 independent fields. Every n x n matrix A can be viewed as a linear map A : R n + Rn. The set of all such matrices with det(A) # 0 forms the general linear group Gl(n) of nonsingular linear transformations of R n. We denote the transpose of the matrix A by AT. We denote the n x n identity matrix by Inor, if the context makes it clear what the dimension is, simply by I.

+

+

DEFINITION A.1. The orthogonal group 0 ( n ) consists of those matrices A E Gl(n) such that AT = A-' . The elements of O(n) are called orthogonal matrices. It is clear that O(n) is a subgroup of Gl(n). The essential property of orthogonal matrices is that they preserve the Euclidean inner product in R n , hence they are the linear transformations that respect Euclidean geometry. Indeed, define the inner product by viewing Rn as the space of n x 1 matrices and setting

Then, if A E O(n),

That orthogonal matrices are the only ones with this property follows by varying v and w independently over the standard basis of R ". THEOREM A.2 (RADON,HURWITZ,ECKMANN). For each n 2 1, there are E O(n) such that a ( n ) - 1 matrices Al, Az, . . . , (1) A: = -1, 1 5 i 5 a(n) - 1; (2) AiAj = -AjAi, i # j. COROLLARY A.3. For each n 2 1, Sn-l admits a ( n ) - 1 everywhere independent vector fields.

A. VECTOR FIELDS ON SPHERES

350

First remark that AT = -A,. fact that Ai E O(n). Define s; : Sn-' + Rn x Rn by PROOF.

This follows from A: = -I and the

s,(v) = (v, Aiv). We claim that si is a vector field on Sn-'. Indeed,

so Aiv I v. Since I I A ~ V I= I

Jw

=I

=

this vector field is nowhere zero. Finally, if i

I ~ I I= 1,

# j,

implying that Aiv I Ajv. Since v E s"-' is arbitrary, we have everywhere independent vector fields s 1, s2, . . . ,s,(,)- 1. In order to sketch a slick proof of Theorem A.2, we introduce the Clifford algebras.

DEFINITION A.4. For each integer k 2 0, the Clifford algebra Ck is the a s w ciative algebra over R that is generated by a unity 1 and elements

subject only to the relations

where i # j and 1 L i , j 5 k. It is evident that Co = R, viewed as a l-dimensional algebra over itself. The algebra C1 is generated by 1 and el, subject only to the relation e: = -1. But this is just the way that the complex numbers are described when viewed as a two-dimensional algebra over R. Thus, we set e l = a, obtaining C1 = C. The quaternions W are a four-dimensional algebra over R, generated by 1,i, j, k with relations i2 = j2 = k2 = -1, i j = -ji = k, jk = -kj = 2, and ki = -ik = j. If we set el = i, en = j, and elez = k, we easily check that W = C2. All of the Clifford algebras have been explicitly computed by Atiyah, Bott, and Shapiro [2]. In order to describe this, we use the notation R(m) to denote the algebra of rn x m matrices with entries from an algebra R. The multiplication in R(m) is defined by matrix multiplication, using the multiplication in the algebra R to make sense out of the usual matrix multiplication formula. We must also define the direct sum R1 $ R2 of algebras R1 and R2. This is the Cartesian product R1 x R2 with all algebra operations defined coordinatewise. For 0 5 k 7, the Clifford algebras are given in Table A.2. The numbers ak+l in the table will be explained shortly. All other Clifford algebras can now be computed, using the following [2].

<

A. VECTOR FIELDS ON SPHERES

TABLEA.2. The first eight Clifford algebras

LEMMAA.5. Ck+s= Ck(16) The mysterious numbers in the third column of the table are similarly defined for all indices by setting a(k+l)+g = 16ak+l. For the reader who is familiar with tensor products of algebras, the proofs in [2] of the above facts are quite accessible and elegant. In order to explain the numbers ak+l, we need the following notion. DEFINITIONA.6. Let R be an associative algebra over R with unity 1. A representation of R on R* is a bilinear map

(variously written as p(c, v) = p,(v) = c . v) such that (1) 1 - v = v , v v ~ R Q ; (2) ( c & ) + v = c + ( d ~ v ) , v v ~ ~ ~ , v c , d E R .

The point is that p represents each c E R as a q x q real matrix p c (a linear transformation on RQ) in such a way that multiplication in R is turned into matrix multiplication and the unity 1 E R becomes the identity matrix I,. PROPOSITION A.7. For each k 2 0, Ck has a representation on W a k + l .

PROOF. For 0 < k 5 2, this is easily seen from Table A.2. Multiplication of reals gives the representation of Co = R on IR = Ral. Complex multiplication C x C + C can be interpreted as a representation of C1 = C on W2 = Was. Similarly, quaternionic multiplication is interpreted as a representation of C2 = Won W4 = Ra3. Remark that, if an algebra R has a representation on Rq, then R(m) has an obvious representation on RQmand R $ R has an obvious representation on Rq. In the first case, write the elements of Rqm as m x 1 matrices with entries from R4 and let the elements of R(m) act by matrix multiplication. In the second case, let p(c,d) = pc, Vc, d f R. For the remaining entries in Table A.2, these remarks make clear what the desired represent ations are. Inductively, assume that Ck has a representation on Rak+l, set C k + g = Ck(16) and Ra(k+l)+8 = ~ l ~ ~ k and + l , use the matrix representation.

A. VECTOR FIELDS ON SPHERES

352

LEMMAA.8. For each natural number n, the largest power of 2 that divides n i s a,(,).

PROOF. For 0 5 c 5 3, a direct check of Table A.2 shows that azc = 2". Thus, c+4d . a,(,) = azc+8d = (16Ida2== 24 d 2 c -2 Thus, we can write

-

2r+l factors

and we can apply Proposition A.7 to each factor to obtain the following.

COROLLARY A.9. T h e Clifford algebra C,(n)-l has a representation o n W n . Thus, the elements e l , . . . ,e,(,)-l determine n x n matrices A1,. . . , satisfying the relations in Theorem A.2. The following completes our sketch of the proof of that theorem.

LEMMAA.lO. By a suitable change of basis in Rn, the matrices Ai become orthogonal, 1 5 i 5 o(n) - 1. PROOF.

Remark that the set

is a finite group under Clifford multiplication. A new positive definite inner product on Rn is defined by

As g ranges over G,so does g h for fixed but arbitrary h E G, implying that (h . v, h - w)' = (v, w)', V v, w E Rn. Recoordinatizing by a basis that is orthonormal in this new inner product makes the matrices A, simultaneously orthogonal. El

APPENDIX B The Inverse Function Theorem

The following simple lemma will be used in the proof of Theorem 2.4.1 and in that of Theorem 2.8.4. LEMMAB.l (CONTRACTION MAPPING LEMMA).Let X be a complete metric space with metric p and let T : X + X be a mapping. If there is a constant c E ( 0 , l ) such that

then T has a unique fixed point xo E X . Furthermore, Tn" V n > 1, and

( x )&

Tn(X),

nT ~ ( x ) 00

= ixo).

n=l

PROOF.Let x E X and remark that 1

+ c + c2 + - .. = C < oo, so

k

p(Tn(x),Tn+*(x))5 cn

p(T2-' ( x ) ,~ ' ( x )5)cnCp(x,T ( x ) ) , i=l

implying that {Tn(x)):=, is a Cauchy sequence. Our hypothesis also implies that T is continuous. By completeness, set

xo

=

lim T n ( x ) .

n-00

Then,

T ( x o )= T ( Ern T n ( x ) ) n+oo

=

lim T ( T n( x ) ) (by continuity)

n+m

= lim T ~ + ' ( x ) n+oo

= 50.

Since T fixes so and strictly reduces distances, the fact that xo is the only fixed point is clear, as are the remaining assertions of the lemma.

B. INVERSE FUNCTION THEOREM

354

DEFINITION B.2. A mapping T : X + X, as above, is called a contraction mapping. Another useful tool in the proof of the inverse function theorem is the mean value theorem from multivariable calculus. Let U and V be open subsets of Rn, @ : U -+ V a map of class C k for some 1 5 k 5 oo. If the line segment { t q (1 - t)p)05t is contained in U, then the mean value theorem asserts that

+

@(PI

-

@(PI

=

(1'

+

J@(ta ( 1 - t ) p ) dt

)-

( q - p).

-

Turning to the proof of Theorem 2.4.1, we let @ : U V be a C k map between open subsets of Rn and assume that J@(p)is nonsingular, for some p E U . We must find an open neighborhood W , of p in U which @ carries Ckdiffeomorphically onto an open neighborhood of @ ( p ) . By appropriate changes of coordinates, we lose no generality in assuming that p = 0 = @(p)and that J@(O)= I n . Thus, the associated mapping Q : U -+ Rn, defined by

satisfies Q(0) = 0 and JQ(0) = 0. Finally, for each positive real number q, let B, denote the closed ball in Rn of radius q and centered at 0. Since J @ ( x )is continuous in x, we can choose q > 0 so small that J @ ( x )is nonsingular, Vx E B,. We fix this condition and, in fact, the following.

LEMMA B.3. There is a value of q > 0 so small that, if x l , x2 E B,, then

'

PROOF.Since Q is of class C and JQ ( 0 ) = 0, an application of equation (B.1) to the mapping Q implies that, for q > 0 sufficiently small, Since @ ( x )= x - Q ( x ) ,it follows that

COROLLARY B.4. F o r each y E BVl2,there is a unique x E B,, such that @ ( x )= y.

+

PROOF.Define Ty on B, by T, ( 2 ) = y Q ( z ) . By the first inequality in Lemma B.3, it is clear that T, (B,) E B,. By this same inequality, so Ty is a contraction mapping on the complete metric space B,. Let x E B,, be the unique fixed point and remark that

is satisfied if and only if @ ( x )= y.

B. INVERSE FUNCTION THEOREM

Let Z = int(BVl2) and W = 8-'(2). V and U ,respectively.

355

These are open neighborhoods of 0 in

COROLLARY B.5. The mapping @ : W + Z as a homeomorphism. PROOF.We have shown that @ maps W oneone onto Z, so it remains to be proven that @-I is continuous. But the equations @(xl) = yl and @(xz)= y2 and the second inequality in Lemma B.3 imply that

proving the assertion. LEMMAB.6. The map @-'is differentiable at each point of Z and

PROOF.Let b = @(a)E Z, a E W. By differentiability at a, we can write

where lim Z(x, a) = 0.

2-+a

Since J@(a)is nonsingular, we can write x - a = J@(u)-'

- (@(x) - @(a))- Ilx - all J@(a)-' . Z(x, a).

Writing x = @-l(y), we obtain

where

But y + b if and only if x

+a

and we have the inequality

by Lemma B.3. That is, lim

67y, b) = O

y+b

and

is differentiable at b E Z with

Since b f Z is arbitrary, all assertions follow. COROLLARY B.7. The map Q-' is of class ckon Z.

B. INVERSE FUNCTION THEOREM

356

PROOF.Since @ is of class c k , the entries in the matrix ( J @ ) - ~are functions of class Ck-I and Corollary B.5, together with Lemma B.6, implies that J ( @-') is continuous. That is, @-I is of class C1. If k = 1, we are done. Otherwise, feeding this new fact back into Lemma B.6 implies that is of class C2. Continuing in this way (forever, if k = oo), we complete the proof. We have proven the Ck version of Theorem 2.4.1, 1 5 k 5 oo. REMARK.It is not hard to adapt the above proof to work for mappings

where U C E is open and E, F are Banach spaces over R. One says that F is differentiable at p E U if there exists a bounded linear transformation

(the Jacobian of F at p) such that lim F(x) - F(P) - J F b ) . ( x - P) = 0. llx - PII

x-P

As usual, one shows that such a linear transformation is unique and that its existence implies the continuity of F at p. If this condition holds for all p E U, we obtain a map JF : U -,C(E, F), where C(E, F ) denotes the Banach space of bounded linear transformations from E to F. If the map JF is continuous, we say that F is of class C1 on U. As usual, one obtains the chain rule

for C1 functions, as well as the fact that a bounded linear transformation is its own Jacobian. Inductively, one defines F to be of class C k on U ,k 2 1, if JF is defined and of class Ck--' on U. If F is of class Ck on U, Vk 1, then F is of class CO" on U. If F is a Ck mapping of U, one-one onto an open subset V E F, k 2 1, and if F-I : V + U is also of class Ck,then F is said to be a c k diffeomorphism of U onto V. In our proof of the inverse function theorem, we chose 7 > 0 SO small that J@(x) is invertible, V x E B,. The usual determinant argument for this is unavailable in infinite dimensions, but it remains elementary that the subset of elements in L(E, F ) with bounded inverses is open (cf. 124, pp. 71-72]). Finally, the mean value theorem (equation (B.l)) is completely elementary for general Banach spaces (cf. [24, p. 107]), so the proof that we have given for the finite dimensional case of the inverse function theorem goes through unchanged.

>

THEOREM B.8. Let F and E be Banach spaces, U C E an open subset, and let F : U + F be of class Ck on U, 1 5 k 5 oo. If p E U and JF(p) is an isomorphism of Banach spaces, then there is an open neighborhood W of p in U that is carried ckdifleomorphically by F onto an open neighborhood F(W) of F(p) in F. The Jacobian of F-l at F(x) is the inverse of J F ( x ) , V x E W.

B. INVERSE FUNCTION THEOREM

357

Here, of course, by an "isomorphism of Banach spaces" we mean a bounded linear transformation with bounded inverse. The final statement of Theorem B.8 is just the equation J(F-l) = (JF)-I o F-I (Lemma B.6). There is a corresponding version of the implicit function theorem (Corollary 2.4.8) for Banach spaces. This will give a remarkably elegant way of proving the smooth dependence on initial conditions in the fundamental theorem of O.D.E. In order to state this implicit function theorem, we will need some notation. Let E, F, and H be Banach spaces, U C E and V E F open subsets, and let F : U x V + H be of class Ck on U x V, I 5 k oo. Denoting the variables in U and V by x and y, respectively, let (p, q) E U x V and form the "partial Jacobian" J, F[p, q) E L(F, H) (respectively, JzF(p, q) E L(E, H)) by holding x (respectively, y) fixed and treating F as a function of the remaining variable.

<

ck

THEOREM B.9. Let F : U x V + H be of class as above, let (p, q) E U x V, F(p,q) = c E H, and assume that J,F(p,q) : F + H is a n isomorphism of Banach spaces. T h e n there exists a n open neighborhood W of p in U and a unique ck m a p cp : W + V such that cp@) = q and, o n W, PROOF.Let G:UxV+ExH be defined by the formula

a

ckmap with Jacobian

This is an isomorphism of the Banach space E x F onto E x H, so the inverse function theorem provides an open neighborhood of (p, q) in U x V which is carried by G diffeomorphically onto an open neighborhood of ( p , c ) in E x H . Since the Banach space E x F has the Cartesian product topology, this neighborhood of (p, q) can be taken to be of the form W x W', where W is an open neighborhood of p and W' an open neighborhood of q. On G(W x W'), the inverse transformation has a formula G-'(x, z ) = (x, H(x, 2)) for a unique Ck map H : G(W x W') formula cp(x) = H (x, c). Indeed,

+ W'.

Since G is a diffeomorphism, cp is unique.

Thus, the desired map p has the

APPENDIX C Ordinary Differential Equations

We prove Theorem 2.8.4. Remark that the general system (time dependent with parameters z = (zl,. . . ,zr) E V IIQ')

on an open subset U C Rn, can be viewed as an autonomous system without parameters on the open subset (-6, E) x V x U C RnSrS1, by adjoining the equations

Consequently, we formulate the proof for the autonomous case without parameters on an open subset U Rn:

We will assume that 1 5 k 5 oo and that f a E C k ( u ) , 1 5 i 5 n, and prove that the solution defines a local flow of class Ck. This will involve an induction on k in which the remark in the previous paragraph becomes crucial. No generality will be lost by taking U = Wn and assuming that the vector field X = ( f l , . . ., f n, is compactly supported. Indeed, we are proving a local theorem near xo E U,so X can be damped off to 0 outside of a relatively compact region W C W C U, containing a given closed ball B,(xo), then extended by 0 to all of Rn. In this way, we will be considering a complete vector field X on W n and will find a uniform parameter interval (-c, c) on which the solution curves are defined for all choices of initial condition x E Wn. Restricting to x E B,(xo)

360

C. ORDINARY DIFFERENTIAL EQUATIONS

and taking c > 0 smaller, if necessary, we see that integral curves to X, starting in B,(xo) and parametrized on (-c, c), must stay in the region W where X has not been altered.

C .I. Existence and uniqueness of solutions Since we will not be thinking of the vector field X as a differential operator, we will write X(x) for X,. A curve s : (-6, E) -,Rn is integral to X if and only if s(t) = s(0)

+

1 t

X(s(u)) du, -6

ir .

This formula suggests a mapping of a certain complete metric space into itself which will turn out to be a contraction mapping (Definition B.2). Let K be a Lipschitz constant for X. This exists because X is C1 and compactly supported. Let 0 < c < 1 / K . In what follows, E(Rn) will denote the Banach space of all continuous paths

with the sup norm.

LEMMAC.1 .I. For each a E R n , the transformation

Ta (s)(t) = a

+ 1 0 X(s(u)) du,

-c 5 t 5 c,

is a contraction mapping.

PROOF.If sl, s 2 E E(Rn), then

That is,

IITa(s1) - Ta(~2)ll5 ~ K l l ~ ~l 2 1 1 and 0 < cK

< 1, so Ta is a contraction mapping.

By the contraction mapping lemma, it follows that there is a unique curve s E E(Rn) with T,(s) = s. As remarked above, this says, equivalently, that there is a unique solution s(t) = (xl(t), . . . ,xn(t)) to (*), parametrized on [-c, c] and having initial condition s(0) = a.

LEMMAC.1.2. The solution curve s : [-c, c] -+ R n is of class ck+'.

C.1. EXISTENCE AND UNIQUENESS OF SOLUTIONS

PROOF. Indeed, the equation

together with the fact that X is C k , implies that, if s is of class Cj, some 0I j k, then s is actually of class Cj+'. But s E E ( R n ) is of class CO,by definition, hence induction on j gives the assertion.

<

REMARK.If the vector field X is only required to be Lipschitz, the above argument goes through to provide a unique C solution parametrized on [-c, c]. We have not quite proven the uniqueness of solutions as formulated in T h e rem 2.8.4. Suppose that sl and $2 are two solutions, parametrized on respective closed, nondegenerate intervals J1 and J2 about 0, and both satisfying the initial condition s l ( 0 ) = sz(0) = a. Let J Jr fl J2 be the largest closed, nondegenerate subinterval about 0 on which sl = s2. By what has just been proven, J is not empty. If J # J1 n J2,then one endpoint ii of J lies in both J1 and J2 and 5 = s ~ ( T= ) s 2 ( r ) . Then a i ( t ) = S ~ ( T t ) is a solution of (*) with ai(0) = ii, i = 1,2, so it follows from what has just been proven that s 1 and s2 agree on a larger subinterval than J after all. That is, J = J1 n J2 and we have proven the existence and uniqueness part of Theorem 2.8.4. Define @ : [ - C , C ] x Rn + Rn

+

<

such that, for each x E Rn and -c I t c, @ ( t ,x) describes the unique integral curve to X with @(O,x) = x. Denote this integral curve by s x and define

LEMMAC.1.3. The map cp admits a Lipschitz constant 23. That is,

In particular, cp is continuous. PROOF. Let x, y E W and remark that

Using notation established above, set E = cK E (0,l) and write

Since T,'(s,)

+ s,

in E ( R n ) as q + m and

the assertion is established.

C . ORDINARY DIFFERENTIAL EQUATIONS

362

COROLLARY C.1.4. T h e m a p

is a local C0 flow. We emphasize that, because of our simplifying assumption that X is compactly supported, the parameter interval (-c, c) is uniform for all initial values x E Rn. The proof of Lemma 4.1.8 is applicable, therefore, and gives COROLLARY C.1.5. The compactly supported vectorfield X generates a unique COflow @ : R x R n+Rn. The hardest part of the proof of Theorem 2.8.4 is to show that this flow is of class Ck. It turns out that, if we can prove it to be C1, a rather ingenious recursive argument yields an inductive proof that Q is of class C k. In order to prove that Q is C1, we will verify the hypotheses of the implicit function theorem (Theorem B.9) for the map

defined by F(x, s) = x - s

+

This notation is understood to define a continuous curve in Rn by the formula

One has F(x, s) = 0 if and only if s = s, is the integral curve to X with initial value s,(O) = x. The implicit function theorem will guarantee that the map cp(x) = s,, satisfying F(x, cp(x)) = 0, is of class C1 (in fact, of class c k ) . It is not obvious that this implies C1 smoothness for the flow @ itself, but this will be the case. Before giving the details, we make a small digression.

C.2. A digression concerning Banach spaces Let G : F + H be a Ck map of Banach spaces, 1 5 k 5 m, and let E(F) and E ( H ) denote the associated Banach spaces of continuous paths, parametrized on a fixed, bounded, nondegenerate interval [-c, c]. We define a map

by the formula

We are going to prove that F is also of class Ck. First, remark that

l*

J G o s t E(L(F,H))

C.2. A DIGRESSION

363

can be interpreted as an element of C ( E( F ) ,E ( H ) ) , V s E E ( F ) . Indeed, if a E E ( F ) , then

so

So*J G o s can be viewed as a transformation sending a E E ( F ) to

lo( J G

o s) .o E E(H).

This is evidently a linear transformation of E ( F ) into E ( H ) and it is bounded because J G ( s ( u ) )is uniformly bounded, -c 4 u 5 c.

LEMMAC.2.1. The map F is of class at least C1and, as s ranges over E ( F ) ,

PROOF.Let so E E ( F ) . Then lim

s-so

F ( s ) - F ( s o ) - S,'(JG 0 so) . ( s 11s - sol1

since the integrand converges to 0 uniformly on [-c, c]. Thus, J G o so satisfies the definition of J F ( s o ) ,V s o E E ( F ) . To see that J F is continuous, write

and appeal to the continuity of JG. Suppose, now, that it has been proven that F is of class C r , some 1 5 r < k , and that J r F ( s ) = J r G o s , V s E E ( F ) . The lemma gives the case r = 1 and it also provides the inductive step: J r F is of class C1 and

S,'

COROLLARY C.2.2. I f G : F of class Ck and

+H

is of class C k , 1 5 k 4 00, then F is also

364

C. ORDINARY DIFFERENTIAL EQUATIONS

C.3. Smooth dependence on initial conditions We return to the map F of Rn x E(Rn) into E(Rn) given by the formula F(x,s) = x - s +

1' + 1'

Xos.

From the previous section, this has the same smoothness class ckas the vector field X and JSF(x,r) = - i d E ( p )

J X o a.

0

Let 11 J X 11 = M . For the following lemma, we take c so that c < 1/M.

> 0 smaller, if necessary,

LEMMAC.3.1. F o r each (x, s) f Rn x E(Rn), the bounded linear operator JsF ( x , s) has a bounded inverse. PROOF. Indeed, for arbitrary a E E ( R n ) ,

It follows that the operator L =

Jof

J X o s has norm 11 Lll 5 C M < 1, hence that

converges. That is, R E C(E(Rn),E(Rn)) and

as asserted. Thus, the hypothesis of Theorem B.9 is verified.

COROLLARY C.3.2. The map cp : R n + E(Rn), deJined by p(x) = s,, is smooth of class Ck. In fact, we only need to know that cp is of class C1.

COROLLARY (2.3.3. The global flow 9 o n Rn is smooth of class at least C1. PROOF.By Corollary C.3.2, cp is smooth of class at least

where lim 6(h) = 0 in E(Rn).

h-0

That is, lim 6(h)(t) = 0 uniformly on [-c, c].

h+O

Thus, from

c', so

C.3. SMOOTH DEPENDENCE

we deduce that the partial Jacobian

exists and, for each h E Rn, is continuous in (t, x). By successively substituting h = ei, 1 5 i 5 n, we conclude that all entries of the matrix J,@(t, x) are continuous functions of (t, x) E [-c, c] x Rn. Since the flow lines are integral to X, we also see that

at

x, = X(@(t,x))

is continuous in (t, x). Thus, all entries of J@(t,x) are continuous on [-c, c] x R n and O is of class C 1there. The parameter interval [-c, c] of this local C 1flow being uniform for all initial values x E R n , we obtain the unique global C 1flow

generated by X .

REMARK.In order to prove Lemma C.3.1, we had to allow c to be chosen small enough. Our final conclusion, however, was that the global flow 9 is C l. We are going to prove inductively that, for 1 5 q 5 k, @ is of class Cq (the case q = 1 is Corollary C.3.3). At each step of the induction, c may have to be chosen smaller. Nonetheless, at each step the conclusion is about the global flow, so the induction works even for the case k = oo. Let 2 5 j 5 k and assume that it has been shown that the flow @(t,x) is of class Cj-l.We will show that d@(t,x)/dt and Jz@(t,x) are both of class Cj-l in (t, x) , concluding that 9(t, x) is of class Cj.

LEMMAC.3.4. The expression d@(t,x)/dt is of class Cj-' i n (t, x).

PROOF.Indeed, a q t , x)/at = x p ( t , x)) and X is of class ck, @ of class Cj-l,and j 5 k.

LEMMA(3.3.5. The expression J x 9 ( t ,x) is of class Cj-l in (t,x). PROOF.Since we know that @ is of class at least C l , we can differentiate

under the integral sign with respect to the variables x, obtaining

Then,

a

, = JX(@(t,x))Jx (@(t,x)), at Jx ( @ ( t2)) and this can be interpreted as a time dependent system of O.D.E. with parameters x. The unknown functions are the entries of the matrix Jx(O(t,x)), so the system is linear. The coefficients are entries of the matrix JX(O(t,x)), hence are

366

C. ORDINARY DIFFERENTIAL EQUATIONS

of class Cj-' in (t, x). Thus, this system is of class Cj-'. As remarked at the beginning of this appendix, time dependent systems with parameters are equivalent to autonomous systems without parameters, so the inductive hypothesis guarantees that the entries of Jz (O(t, 2)) are smooth of class Cj- .

'

The proof of Theorem 2.8.4 is complete. REMARK.It would have been possible to formulate and prove the O.D.E. t h e orem entirely in the context of Banach spaces (see, e-g., Lang [24, pp. 132-1451, who credits the idea of using the implicit function theorem to Pugh and Robbin), but we have avoided significant technical details by resisting the temptation to do so. On the other hand, judicious use of the implicit function theorem in infinite dimensions does seem to simplify the proof of the finite dimensional theorem. C.4. The Linear Case

In the proof of Theorem 10.1.13, we appealed to the theorem that a linear system of 0.D.E. has solutions that are defined on the largest parameter interval (b, c) on which the system itself is defined. Here is the formal theorem. THEOREM C.4.1. Let A : (b, c) -t 9Jt(n) be smooth of class system s(t) = A(t) .s(t), b < t < c, with initial condition s ( t o ) = a has solution s(t) defined for b < t

c'.

Then the

< c.

PROOF.Let (b', c') E (b, c) be the maximal open interval about t o on which the solution s(t) is defined. If c' < c, we deduce a contradiction as follows. Fix a basis {wl, . . . ,w,) of Rn and, for 1 5 i 5 n, let ai(t) be the unique solution of

defined on some interval (c' - E,c' + E) C (b,c). Choose t, E ( d - E, c'), so close to c' that {al(t,), . . . ,an(t,)) is also a basis of Rn. This is possible by the continuity of the solutions aim Thus, there are constants cl, . . . ,cn E R such that

By the linearity of the system, it is clear that

is also a solution and that a ( t ,) = s(t, ). By uniqueness of solutions, a and s agree on (c' - E , c'), so a extends the solution s to the larger interval (b', c' + E). This contradicts the maximality of (b', c'), proving that c' = c. Similarly, b' = b.

APPENDIX D Sard's Theorem

In this appendix, we prove Theorem 2.9.3 (following Milnor [28]). Accordingly, let : U + V be a smooth map, where U C_ R n and V C_ R m are open. The critical set C E U consists of those points x for which rank Ja, < m and we must prove that @(C)5 V has Lebesgue measure zero. As usual, write a = (a1,. .. , a m ) .

DEFINITION D.1. For each integer hi 2 1, Ck C C is the set of points x E U such that all mixed partials of Qi of order 5 k vanish at z, I 5 i 5 m. It is clear that we obtain a nest

of closed subsets of U.The basic estimates behind Sard's theorem are contained in the following proof.

PROPOSITION D.2. For k 2 1 suficiently large, the set @(Ck)has Lebesgue measure zero. PROOF.Let Q C U be a compact cube. Since Ck n U is covered by countably many such cubes, it will be enough to find a value of k, depending only on m and n (not on Q) such that @ ( C k n Q) has Lebesgue measure zero. Let S denote the edgelength of Q. Let p range over Ckn Q. The kth order Taylor series, expanded about p, takes the form @(P + v) - Q(P) = Rb, v), where the remainder term satisfies a uniform estimate for all p E Ck n Q and all v E Rn such that p + v E Q. Subdivide Q into r n subcubes of edgelength S/r and let Q' be one of these subcubes containing a point p E Ck. Every point of Q' has the form p v, where

+

368

D. SARD'S

THEOREM

By the above estimate on the remainder term, we see that @ ( Q f lies ) in a cube, centered at @(p) and having edgelength e/rk+', with e = 2 ~ ( b f i ) ~ + lThus, . @(Ckn Q) is covered by a union of at most r n cubes with total measure

, For large enough k, the exponent of r is negative and lim,

V ( r ) = 0.

Sard's theorem is trivial when n = 0, so we make the inductive assumption that the theorem has been proven for the case U C Rn-l, some n 1, and deduce the case U C Rn.

>

LEMMAD.3. Let p E C \ C1. Then there are coordinate neighborhoods o f p in U and of @(p) in V relative to which the formula for @ becomes

PROOF.By the definition of C1, p $ Cl implies that some first order partial of some coordinate function of @ fails to vanish at y. Suitably permuting the coordinates in Rn and Rm, we lose no generality in assuming that

Thus, by an application of the inverse function theorem, the map

is a diffeomorphism of some open neighborhood W of p onto an open subset @(W) 5 Rn. This defines a coordinate system with the required property.

PROPOSITION D.4. The set @(C\ C1) has Lebesgue measure zero. PROOF.It will be enough to show that, for each p E C \ C1, there is a neighborhood Wp of p in U such that @(Cn Wp) has measure zero. We assume that m > 2 since, when m = 1, C = C1 and the assertion is vacuously true. By Lemma D.3, we can assume that there is a neighborhood Wp in which each point ( t ,x 2 , .. . ,xn) is carried by @ into the hyperplane { t ) x Rm-l. For fixed t , the restriction of @ to Wp n ( { t ) x Rn-l) can be viewed as a map Bt of that set into the hyperplane { t ) x Ktm-'. Since the coordinate t is preserved, the critical set of at is C n Wp n ( { t) x Rn-l ) and the inductive hypothesis implies that this is carried by at onto a set of (m - 1)4imensional measure zero. Integrate with respect to the t coordinate, concluding by F'ubini's theorem that @ ( Cn W,) has mdimensional measure zero.

PROPOSITION D.5.For each integer k 2 1, the set @(Ck\Ck+l)has Lebesgue measure zero.

D. SARD'S THEOREM

369

PROOF. Again, it will be enough to show that, for each p E Ck \ Ck+l,there is a neighborhood W pof p in U such that Q(Ck n W p )has measure zero. Let cp : U + R be a kth order partial of a coordinate function Q r such that some first order partial of cp fails to vanish at p. Without loss of generality, we assume that Of course, cp vanishes identically on Ck. The inverse function theorem again gives a change of coordinates defined on some neighborhood W p of p, thereby coordinatizing Ck n Wp as a subset of the hyperplane {0) x Rn-l. Every point in this set is a critical point of the restriction Qo of @ t o U n ((0) x Ikn-I), so the inductive hypothesis implies that Q(Ck n W p )= QO(Ckn W p )has Lebesgue measure zero. COROLLARY D.6. For each integer k

2 1, the set @(C\ Ck) has Lebesgue

measure zero. PROOF.Indeed, C \ Ck = (C \ Cl) u (Cl \ C2)u - - - u (Ck-l

\

Ck).

By this corollary and Proposition D.2, the proof of the inductive step is complete.

APPENDIX E The de

ham-cech Cohomology Theorem

In this appendix, our goal is to prove a version of the de Rham theorem that shows the de Rham cohomology algebra H * ( M ) to be a purely topological invariant of the manifold M (Theorem 8.9.6). For this, we must first define the ~ e c cohomology h algebra of an arbitrary topological space.

E. 1. Cech cohomology This section is largely definitions and statements of basic properties. Few proofs will be given because they are mostly routine (and tedious) computations. The first step is to define the cohomology of an open cover U = {Ua)aEll of the space X. DEFINITION E. 1.1. If p 2 0 is an integer, a psimplex of (p+ 1)-tuple (U,, ,U, , . . . , Uap) of elements of U such that U,,

U is an ordered n . - - n Uap # I.

DEFINITION E.1.2. If p >_ 0 is an integer, an R-valued p-cochain on U is a function cp which, to each psimplex (Uao, U,, , . . . ,Uap), assigns an element

The set of all R-valued p-cochains on U is denoted by @(U; R). Evidently, the operations of "simplexwise addition" of cochains and "simplexwise scalar multiplication" make &(u;R) into an R-module. More precisely,

and

-

(aip)ooa~..-a,- a ( ~ a o o l . . . a ~ ) r V a E R a n d V c p , $ ' ~@(u;R). DEFINITION E.1.3. The coboundary operator 6 : CP(U;R) the R-linear map defined by the formula

-+

CP+

(U; R) is

372

E. DE R H A M - ~ E C HTHEOREM

Thus, we obtain a sequence

The following is a routine computation.

LEMMAE.1.4. The sequence of coboundary operators satisfies b2 = 0 .

DEFINITION E. 1.5. The module of ~ e c pcocycles h is

and the module of p-coboundaries is

BP(U; R) = im(6) n Cp(Lf;R). " P (U; R) c - i p ( u ; R). As usual, B

DEFINITION E.1.6. The

pth

~ e c hcohomology of 2.4, with coefficients in the

ring R, is HP(U;R) = BP(U;R ) / z ~ ( u ;R). We have defined H*(u; R) as a graded R-module. It is made into a graded algebra by the "cup product".

DEFINITION E.1.7. If ip E @(u; R) and $ E @(u; R), the cup product CP+Q(U;R) of these cochains is defined by

ip$

E

This makes c*(u;R) into a graded algebra. Another straightforward computat ion proves the following.

LEMMAE.1.8. If p E @(u; R) and $ E CQ(U;R), then

This is formally the same as the formula for the exterior derivative of the wedge product of a p-form and a q-form. As in that case, we get the following consequence.

LEMMAE.1.9. The graded module z*(u;R) of Cech cocycles is a graded subalgebra of c*(u; R) and B*(u; R) c z*(u; R) is a 2-sided ideal. Consequently, cup product is well defined on H*(u; R), making that graded R-module into a graded algebra over R. Finally, we will define the ~ e c cohomology h algebra of the space X to be the "direct iimit" H * ( xR); = lim H*(u; R) +

over finer and finer open covers. We make this precise. Let {V,*),Ea be a family of graded R-algebras, indexed by a directed set U. That is, there is a partial ordering a 5 ,O on U such that, whenever a,,O E U, 3y E U with a 5 y and ,O 5 y. Assume also that, whenever a 5 ,O, there is given a homomorphism p! : V,' -+ V; of graded algebras and that, whenever

E.1. CECH COHOMOLOGY

a j p j 7, then 9; o 9f = 92. We say that {V,', pf},,pEa of graded R-algebras. On the disjoint union

define the equivalence relation

-

373

is a directed system

v* = LI v;, a€%

generated by

where v E V,* and a j p. Then V * / - has a natural graded R-algebra structure. Indeed, scalar multiplication a[v] = [av] is clearly well defined. As for addition, if [v], [w] E V * /- are represented by v E V," and w E Vp' , find y E U, cr 5 y, p 5 7,and set [vl + Iwl = [9',(~) + 9;cw)l. It is trivial to check that this is well defined and that these operations make V* /N into a graded R-module. Similarly, the algebra multiplication passes to a well defined multiplication making V * /- into a graded R-algebra.

DEFINITION E.l.lO. In the above situation, we set lim V; = V*/4

and call this graded R-algebra the direct limit of the directed system of graded R-algebras.

EXAMPLE E.l.ll. For a differentiable manifold M, let U, denote the set of open neighborhoods U of x E M. This is a directed set under the partial order U5 V# U V. The graded algebras {A*(U))uEu, form a directed system under the restriction homomorphisms

>

where w E A*(U) and U 2 V. Then

-

A: = lim A*(U) is just the graded algebra of germs at x E M of smooth forms. Let D(X) denote the set of open covers of X. This is partially ordered by: U 5 V a V is a refinement of U. Since any two open covers of X have a common refinement, this makes D(X) a directed set. If U = {Ua),Ea, V = {Vp)pEB,and U 5 V , then there is a choice function i : 93 + U such that Vp C Ui(p), Q p E 93. This induces a homomorphism ir : c*(u; R) 3 c*(v; R) of graded algebras, where -

id(9)pobl...pP- Vi(po)i(~I)-.i(P,) The following is trivial.

374

E. DE RHAM-CECH THEOREM

LEMMAE.1.12. The homomorphism i d : c*(u; R) 4 c*(v; R), as above, satisfies i d 06 = 6 o i n . We cannot use i d as a homomorphism P(: : t"(u; R) -4 C* (V; R) for a directed system of algebras. The problem is that ill depends on arbitrary choices, so we could never guarantee that oP (: =. P ;( But the above lemma implies that ill induces i* : H*(u; R) + H*(V; R) and it turns out that, at the level of cohomology, the arbitrariness disappears.

(PF-

DEFINITION E.1.13. If U 5 V in D(X), if a, j

:B+Q

are two choice func-

tions, as above, and if p f Z, define

by the formula

As usual, if p - 1 < 0,we understand that CP-'(2.4; R) = 0 and S = 0. The following is checked by a routine (if somewhat tedious) computation, left to the reader.

LEMMAE.1.14. If i,, j, and S are as above, then

Consequently, i* = j* : H*(u; R) 4 H*(v; R). By this lemma, whenever U 3 V in D(X), we define a homomorphism

of graded algebras that is independent of the (allowed) choice of i : B + Q.

LEMMAE.1.15. If U jV jW in D(X), then (PYo P(;

=

(PU.

PROOF.Indeed, write U = {Ua)aEa, V = (Vp)pEs, and W = {W,),Ee and let i : B 4 U, j : C 4 B be suitable choice functions. Then i o j : C + 2l is an allowed choice function relative to the refinement U 5 W. But

Thus, we get a directed system { H* (u; R), ( over R.

P ~ } ~ , of ~ graded ~ ~ ( algebras ~ )

DEFINITION E.1.16. The ~ e c hcohomology of the space X with coefficients in R is the direct limit

-

H*(x; R) = lim H*(u; R), taken over the directed set D(X).

E.2. DE RHAM-CECH COMPLEX

37s

Let f : X + Y be a continuous map between spaces. Given UID(Y), define f (U) E D(X) by the usual pull-back construction. If we are given a psimplex (f-'(U,,), . . . , fd1(u,,)) of f-'(U), then it is clear that (U,,, . . .,U,,) is a p simplex of U. Consequently, each ~ e c hcochain 0 E CP(U;R ) has a natural pull-back f d(0) E CP( f -'(u); R ) . This defines a homomorphism

of graded algebras. LEMMAE.1.17. Iff : X + Y , as above, then f f l o 6 = 6 o ffl and there is canonically defined an induced homomorphism

of gmded algebras over R. This makes &h cohomology into a contravariant functor on the category of topological spaces and continuous maps. DEFINITION E.1.18. If I# is a directed set, a cofinal subset C E ?2iis a directed subset with the property that, whenever a E I#, 3 y E C such that a 5 y. Finally, the proof of the following lemma is a straightforward application of definitions. LEMMAE.1.19. Let {V,', cpg},,pEabe a directed system of graded R-algebras. If C 2 I# is a cofinal subset, then there is a canonical isomorphism

-

lim V,' = lim V; 4

of graded R-algebras, where the first limit is taken over a11 a E I# and the second is taken over a11 y E C. By Corollary 10.5.8, the family of simple covers of a differentiable manifold is a cofinal subset of D (M). COROLLARY E.1.20. The cech cohomology H*(M; R) of a diflerentiable manifold can be computed by tahng the limit only over the directed set of simple covers.

E.2. The de

ham-tech complex

The proof we will give of the de Rham theorem is essentially that of Andre Weil [45].The main step is to prove the following. THEOREM E.2.1. If U is a simple cover of M , there is a canonical zsomorphism : H*(u;R)

+ H*(M)

of graded algebras and, if U 5 V, where both are simple covers, then the diagram

E. DE RHAM-CECH THEOREM

is commutative. The equivalence of de Rham theory and &ch theory follows easily. Indeed, the Cech cohomology can be computed by passing to the limit over the simple covers only (Corollary E.1.20), so the isomorphisms $u induce a well defined homomorphism $ : H*(M;R) 4 H*(M). The fact that each $u is an isomorphism implies the same for I). THEOREM E.2.2 (DE RHAM).There is a canonical isomorphism

of graded R-algebras. In order to prove Theorem E.2.1, we will build an enormous cochain complex of graded algebras that includes both (A*(M), d) and (c* (u;B),6) as subcomplexes. If U is simple, we will prove that the inclusions of these subcomplexes induce isomorphisms in cohomology. Fix an open cover U = {Ua)aEll of M. For the following definitions, it is not necessary that U be simple. DEFINITION E.2.3. A ~ e c p-cochain h on U with values in Aq is a function cp which, to each psimplex (U,, , U,, , . . . , Uap) of U assigns

The set of all such cochains will be denoted EP*q(U)= CP(U;Aq). Although the coefficient ring AQ(U,, n U,, n . . . n U,, ) changes with each simplex, one can still add cochains simplexwise and multiply them by real scalars. These operations make EPjq(U) = CP(U; Aq) into a real vector space. There is also a bigruded multiplication

In defining this and other operations, we make the notation less bewildering to the eye by abusing it (the notation, that is). Whenever respective forms have been defined on respective open sets with common, nonempty intersection, addition of such forms and exterior products of such forms are understood to be defined on their common domain. For instance, if (U,, , U,, , U,,) is a 2simplex of U and wffiaj E AQ(U,, n U,,), then

Similarly, if $'a0,,

E Aq(U,, f U, l , ) and ,$ ,,,

E As (U,,

n u,,), then

E.2. DE RHAM-CECH COMPLEX

377

With these conventions understood, we define the bigraded multiplication as follows. If cp E Ep*Q(U) and @ C, Erys(U),then p@ E Ep+r*Q+s(U) k defined on a (P+ r )-simplex (Uao, . . . ,Uap, . . . ,Uap+,) by ~

o

f

f

p

+

qr r (-1)

A ,@ ,

..,

+ ,,

E Aq+'(Uff,n

.. - n u,,,,).

Often we suppress explicit reference to the simplex on which this formula is being evaluated and write cp$ = ( - l ) " p A @ .

We say that E**( U ) = { E P I ~ ( U ) ) Eis~ a= bigraded ~ algebra under this multih plication. Note how this operation combines the cup product from ~ e c theory with the exterior multiplication from de Rham theory. The strange sign ( - l ) q r will be needed in the proof of Lemma E.2.7.

DEFINITION E.2.4. The de Rham operator d

:

Ep79 ( U ) + Ep*q+l( U ) is d,

fined by setting V cp E CP(U;Aq) and for every psimplex ( U f f ,U,, , , . . . , UaP)of U .

DEFINITION E.2.5. The ~ e c operator h 6 : EP3Q(U)-+ E P + ' * ~ ( Uis)defined by setting

Vcp E CP(U;Aq) and for every p-simplex (U,, , U,,

,. . . ,U,,)

of U .

The following are evident:

a2=o b2=0 aos=-boa Remark that the sign ( - l ) p in the definition of d is responsible for the anticommutativity of d and 6.

FIGURE E. 1. The de

ham-~ech complex

E. DE RHAM-CECH

378

THEOREM

DEFINITION E.2.6. For each integer m 2 0, Em(U) = @P+q=m EP,q(U) and the operator D : Em(U) 4 E ~ + ' ( U ) is D = d + 6. It is a good idea to picture E**(U) laid out as a first quadrant array in the (p, q)-plane, having Ep*Q(U)at the point (p, q) of the integer lattice as in Figure E.l, with the de Rharn operators d as vertical arrows and the ~ e c h operators 6 as horizontal arrows. This array is called the de Rharn- ~ e c complex. h The total degree of an element cp E EPyq(U) is p q and Em(U) is spanned by the elements of total degree m. One can view Em(U) in this diagram as lying along the diagonal p q = m. If cp E EplQ(U),where p q = rn,then

+

+

+

LEMMAE.2.7. The pair (E*(U),D) is a cochain complex in which E*(U) is a graded algebra over R and

where cp E Em(U).

PROOF.Indeed, E*(U)= {Em(U))z=ois a graded vector space and it is clear that the bigraded multiplication in E ** (U) induces a graded algebra structure on E*(U). Because of the anticommutativity of d and 6,

Finally, it is enough to verify the Leibnitz formula for cp E EP*Q(U)and $ E ErtS(U),p q = m. Suppress reference to the ( p r)simplex (U,, , . . . , U,,,,) and compute

+

+

That is, Similarly, suppressing reference to the ( p + r we compute

+ 1)simplex (U,,, . . . , Uap+r+l),

That is,

By adding equation (E.l) and equation (E.2), we obtain the desired Leibnitz rule for D.

E.2. DE R H A M - ~ E C H COMPLEX

There are canonical inclusions

(A*( M I ,d ) -i, ( E *( U ) ,D )

(c* (u;R ) ) L ( E *( U ) ,D ) of subcomplexes. Indeed, if w E A Q ( M ) ,i ( w ) E CO(U;A Q )= E'VQ(U)assigns to each O-simplex (U,,) the element i(w),, = w(U,,. Likewise, if cp E R), j ( 9 ) E CP(U;A') assigns to each p-simplex (U,, , . . . ,U,, ) the &form on the open set U,, n . . . n U,, which is the constant function cp,o...,p E R. It is clear from these definitions that

@(u;

whenever w E A Q ( M )and 7 E A S ( M ) ,and that

whenever cp E CP(U;R) and @ E

c'(u;R).

LEMMAE.2.8. There are canonical homomorphisms i* : H* ( M ) + H* ( E *( U ) ,D ) and

j* : H*(u; R ) 4 H*(E*(U)D , ) of graded algebras. We augment the rows of the diagram in Figure E.l by i. That is, the new rows are 6 E O ' ~ ( U ) f E'.Q(u)+ . - . A (M )

Similarly, we augment the columns by j:

LEMMAE.2.9. The augmented diagram has exact rows.

PROOF.If w

E

A Q ( M ) ,we evaluate 6(i(w)) on the lsimplex (U,, ,U,, ),

getting

(S(i(w)))aoal= wlUolo n Ual - wlUa0 n ua, = 0. I f ip E c'(u;Aq) and 6 ( 9 ) = 0, then cp,, E Aq(UaO)and cp,, E AQ(U,,) must agree on U,, n U,, , i f this intersection is nonempty. Hence, the forms c p , E AQ(U,) piece together smoothly to give a form w E AQ( M ) such that i ( w ) = cp. This proves exactness at E O ~ Q ( U ) . We prove exactness at EP?Q(U) = CP(U;AQ)when p _> 1. Let {A,),Ea be a + Ep-l>Q(U) as smooth partition of unity subordinate to U . Define A : EP*Q(U) follows. Given p E &(u; A Q )= Ep*q(U),define A ( 9 ) E CP-'(U; AQ)to be the element whose value on the (p - 1)simplex (U,, , . . . ,U,,-, ) is

E. DE RHAM-CECH THEOREM

380

where each term of this locally finite sum is interpreted, in the obvious way, as a q-form on U,, n - . . nU a p - l .If 6 ( p ) = 0, the reader can check that cp = 6 ( A ( p ) ) , proving exactness at EP?Q(U).

LEMMAE.2.10. If the cover U is simple, the augmented diagram has exact columns.

PROOF.If C E CP(U;B ) , then, on an arbitrary p-simplex (U,, j(C),, ...,, E Ao (Uao n . . - n U a p )is constant, so

, . . . ,U a p ) ,

If (o E @(u; A O ( M ) )and a ( p ) = 0, then d((o,o...ap)= 0, for each psimplex (UaO7 . . . ,UaP).The fact that U,, n. .n UaPis connected implies that E AO(Uaon . . . n Ua,) is constant. Thus, we can define C E ~ ~ ( 2 .W) 4 ; by +

<,o...ap

= the constant

pao..,ap

and j ( c ) = p. This proves exactness at EpgO. If q 2 1, exactness at EPvQ(U)follows from the Poincarb lemma and the fact that U,, n . - n Uap, if not empty, has the de Rham cohomology of Bn.

LEMMAE.2.11. If the cover U is simple, the homomorphisms i * and j* are surjective.

C E Em(U) be a D-cocycle.

[c]

We show that E H * ( E * ( U )D ,) is in the image both of i* and j * . If m = 0, then = 0 = 6c and the assertions follow from Lemmas E.2.9 and E.2.10, respectively. Assume, therefore, that m 2 1. The cocycle lies along the diagonal p q = m, so we write

PROOF.Let

c

a<

+

where Cp E Eptm-p(U), 0 5 p 5 rn. The equation D ( c ) = 0 implies that d(co) = 0, so Lemma E.2.10 allows us to find to E E O ~ ~ - ' ( Uwith ) a(to)= Then C - D(Co) has 0 as its component in EO.m(U)and is D-cohomologous to 5. Suppose, inductively, that Ck E Em-I ( U ) has been found so that C - D(Ck) has 0 as its component in Epjm-p(U), 0 5 p 5 k. Since this is still a D-cocycle, exactness of the column p = k + 1 allows us to find B E E ~ + ~ *(U)~ such - ~ - ~ that 5 - D(&) - D(0) has 0 as its component in Epvm-p(U), 0 5 p 5 k + 1. We take &+' = Ck + 0. By finite induction, there is Cm E Em-'(U) such that C - D(&) is concentrated in Em?O(U). Thus, without loss of generality, we assume that C E Em?O(U)= C m ( ~ ; A 0 )The . fact that this is a D-ocycle implies that a(<)= 0 = 6(<). Thus, there is c E Cm(u; R) such that j(c) = C and j ( 6 ( c ) ) = 6(C) = 0. Since j is one-one, 6 ( c ) = 0, so [c] E H ~ (Bu) and ; j* [cl = ICl. An entirely parallel argument, using exactness of the rows (Lemma E.2.9), shows that we can assume that is concentrated in E Oym ( U ) and that there is w E A m ( M ) such that dw = 0 and i*[w]= [
co.

<

E.2. DE RHAM-CECH

COMPLEX

38 1

LEMMAE.2.12. If the cover U is simple, the homomorphisms i * and j* are injective. PROOF. Suppose that w E A m ( M ) has dw = 0 and i*[w]= 0. If m = 0, then w is a locally constant function and i ( w ) vanishes on every O-simplex (U,,). That is, w 0, so [w]= 0 E H O ( M ) . If m 1, then i ( w ) E EOlm(U)is of the form i ( w ) = D(<), E Em-'(U). Write

>

<

ep

<

where E E p ~ ~ - ' - p( U ) ,0 5 p m- 1. Since the component of i ( w ) in E "vO(U) is 0, 6 ( t m - i ) = 0 and Lemma E.2.9 implies that Jm-l = 6 ( 0 ) , 0 E E " - ~ ~ ~ ( U ) . Thus, 6' = - D(B) has component 0 in E ~ - ' Y O ( U ) and D((') = i ( w ) . Again using Lemma E.2.9 and finite induction, we see that no generality is lost in assuming that ( is concentrated in ~ ~ (24) and v that ~ 6 ( J- ) = 0.~ By one more appeal to Lemma E.2.9, we find a unique r] E A m - ' ( M ) such that i ( q ) = <. Then, i(dr])= d(i(r]))= i ( w ) and, i being injective, w = dr]. That is, [w] = 0 as desired. A completely parallel argument, using Lemma E.2.10, proves that j * is injective.

<

If the cover U is simple, we define an isomorphism of graded R-algebras by Lemmas E.2.11 and E.2.12. The following completes the proof of Theorem E.2.1. LEMMAE.2.13. If U 5 V are simple covers of M , then the diagram

is commutative.

pz

PROOF.Indeed, if U = {Ua),Eaand V = {Vp)4EB,recall that is induced by a choice function Q : 23 + B such that V , G Uq,), V/3 E '33. The same choice function Q defines a homomorphism

and the diagram

E. DE RHAM-CECH THEOREM

commutes. 0

REMARK. In fact, the de ham-Cech isomorphism : H * ( M ;R)-+ H*(M) is jknctorial. That is, whenever f : M + N is a smooth map between manifolds, $J

the diagram

&*(N;R)--L &*(M;R)

is commutative. On smooth manifolds, Cech theory and de Rham theory are equivalent as functors. Verification of this functoriality is straightforward.

BIBLIOGRAPHY [l] F. Adams, Vector fields on spheres, Ann. Math. 75 (1962), 603-632. [2] M. F. Atiyah, R. Bott, and A. Shapiro, Clifford modules, Topology 3 (1964), 3-38. [3] W. Boothby, An Introduction to Differentiable Manifolds and Differential Geometry, Academic Press, New York, New York, 1975. (41 R. Bott and J. Milnor, On the pamllelizability of the spheres, Bull. Amer. Math. Soc. 64 (1958), 87-89. [5] R. Bott and L. Tu, Differential Forms in Algebraic Topology, Springer-Verlag, New York, New York, 1982. [6] S. K. Donaldson, An application of gauge theory to the topology of 4-manifolds, J. Diff. Geo. 18 (1983), 26S316. [7] J. Dugundji, Topology, Allyn and Bacon, Boston, Massachusetts, 1966. [8] 3. Eckmann, Gruppentheoretischer Beweis des Satzes won Hurwitz-Radon uber die Komposition quadratischer Fownen, Comm. Math. Helv. 15 (1942), 358-366. [9] S. Eilenberg and N. Steenrod, Foundations of Algebmic Topology, Princeton University Press, Princeton, New Jersey, 1952. [lo] D.Freed and K. Uhlenbach, Instantons and Four-Manijolds, Springer-Verlag, New York, New York, 1984. [ll] R. Gompf, Three ezotic W4's and other anomalies, J. Diff. Geo. 18 (1983), 317-328. [12] M. Greenberg and H. Harper, Lectures on Algebraic Topology, T h e Benjarnin/Cummings Publishing Co., Reading, Massachusetts, 1981. [13] W . Greub, S. Halperin and R. Vanstone, Connections,Curvature and Cohomology, Volume 11, Academic Press, New York, New York, 1972. [14] V. Guillemin and A. Pollack, Differential Topology, Prentice-Hall, Englewood Cliffs, New Jersey, 1974. [15] S. Helgason, Differential Geometry and Symmetric Spaces, Academic Press, New York, New York, 1962. 1161 N. J. Hicks, Notes on Differential Geometry, D. Van Nostrand, New York, New York, 1965. [17] D. Hoffman and W. H. Meeks 111, Minimal surfaces based on the catenoid, Amer. Math. Monthly, 97 (1990), 702-730. [18] , A complete embedded minimal surface with genus one, three ends and finite total curvature, J. Diff. Geo. 21 (1985), 109-127. [19] , Embedded minimal surfaces offinite topology, Ann. Math. 131 (19901, 1-34. [20] W. Hurewicz and H. Wallman, Dimension Theory, Princeton Univ. Press, Princeton, New Jersey, 1948. [21] M. Kervaire and J . Milnor, Groups of homotopy spheres, I, Ann. Math. 77 (1963), 505537.

384

BIBLIOGRAPHY

1221 S. Kobayashi and K. Nomizu, The Foundations of Differential Geometry, I, Wiley (Interscience), New York, New York, 1963. (231 , The Foundations of Differential Geometry, 11, Wiley (Interscience), New York, New York, 1969. [24] S. Lang, Real Analysis, Addison-Wesley, Reading, Massachusetts, 1983. [25] E. L. Lima, Commuting vector fields on S3,Ann. Math. 81 (1965), 7Ck81. [26] W. S. Massey, Algebraic Topology: An Introduction, Harcourt, Brace, and World, Inc., New York, New York, 1967. (271 J. Milnor, Some consequences of a theorem of Bott, Ann. Math. 68 (1958), 444-449. [28] , Topology from the Differentiable Viewpoint, T h e University Press of Virginia, Charlottesville, Virginia, 1965. [29] , Morse Theory, (Notes by M. Spivak and R. Wells), Annals of Math. Studies, No. 51, Princeton University Press, Princeton, New Jersey, 1963. [30] , Problem List in the Seattle Topology Conference, 1963. [31] E. E. Moise, Geometric Topology in Dimensions 2 and 3, Springer-Verlag, New York, New York, 1977. [32] S. B. Myers and N. E. Steenrod, The group of isometries of a Riemannian manifold, Ann. of Math. 40 (1939), 4 0 M 1 6 . [33] W. F. Newns and A. G. Walker, Tangent planes to a differentiable manifold, J . London Math. Soc. 31 (1956), 40G407. [34] B. O'Neill, Elementary Differential Geometry, Academic Press, New York, New York, 1966. [35] H. Poincark, Analysis situs, Journal de 1 ' ~ c o l ePolytechnique 1 (1895), 1-121. [36] T. Rad6, ~ b e den r Begriff der Riemannsche Fliiche, Acta Univ. Szeged 2 (1924-26), 101-121. [37] E. Spanier, Algebraic Topology, McGraw-Hill, Inc., New York, New York, 1966. [38] M. Spivak, Differential Geometry, Volume I, Second Edition, Publish or Perish, Inc., Houston, Texas, 1979. [39] J . Stallings, The piecewise linear structure of Euclidean space, Proc. Camb. Phil. Soc. 58 (1962), 481-488. [40] N. Steenrod, The Topology of Fiber Bundles, Princeton University Press, Princeton, New Jersey, 1951. [41] J. J. Stoker, Differential Geometry, Wiley (Interscience), New York, New York. 1969. [42] D. Tischler, On fibering certain foliated manifolds over sl, Topology 9 (1970), 153-154. [43] J. W. Vick, Homology Theory, Academic Press, New York, New York. 1973. [44] F. Warner, Foundations of Differentiable Manifolds and Lie Groups, Springer-Verlag, New York, New York, 1983. [45j A. Weil, Sur les the'ortmes de de Rham, Comm. Math. Helv. 26 (1952), 11S145.

INDEX l-coboundaries 173 l-cocycles 173 l-cycle 238 l-form 159 ff. 2-manifolds 7 ff. 2-torus 7 2-cycle 238 2nd countable 82 3sphere as a Lie group 128 4-manifolds 74, 150

A, 68 A*(M) 220 A' ( M ) 159, 162 A ~ ( M 220 ) Ad(a) 144 Aut(g) 144 ad 145 a.e. (almost every) 64 Abelian Lie algebra 135 Lie group 135 group 189 Adams 6 Adjoint 160, 230, 239, 242 Jacobian 161 group 144 of a linear map 161 Algebra of germs 30 Algebraic intersection number 268 Algebraic operations on germs 29-30 Algebraic topology 159, 178, 235, 239 Almost complex manifold 83 Analytic group structure 143 Angle between vectors 81, 293, 302 Anticommutative graded algebra 226 Antisymmetric multilinear map 199 ff. Associativity of tensor product 193 Autonomous system 55, 57, 359, 366

Banach spaces 356 Base space 5, 45, 75, 76 Basis free 189

of cotangent space (T,*(M))160 of tangent space ( T p ( U ) )32 of tensors 197 of (v), 202 of S k ( V ) , 208 Betti number 275 Bi-invariant metric 346 Bigraded algebra 377 Bilinear 190 Bore1 measure 304 Bott 5 Boundary 20 ff. of a l-manifold 23 operator 239 Boundary theorem 97 Bracket of f-related fields 109 Brouwer fixed point theorem 98, 177 invariance of domain theorem 2 Bundle classification 79 isomorphism 76 of exterior algebras 211 of graded algebras 211 of symmetric algebras 211 projection 5, 45, 75, 76 of linear homomorphisms 21 1 @* 128 CP (U; Aq) 376 CP(U) 272 ( C ? ( M ) , 8) 255 C)(M) 274 Cp(M) 239 C?(M) 255 C k (Clifford algebra) 350 C r ( U ) 26

cm

function 26 atlas 68 manifold 68 partition of unity 85 structure 68 CbO(M) 69 C m ( M , N) 93 C W ,P) 28 C m ( U ) 26 Cm-related charts 67 IHInxharts 87 Canonical coframe field 339

386 Canonical smooth structure on Rn 68 Cartan structure equations 331 ff. Cartesian product of smooth structures 68 Category 165, 197, 212 of R-modules 189, 200 of graded algebras 200 of vector bundles 212 ~ech coboundary operator 272, 371 cochains 271, 371 cochains with values in Aq 376 cohomology 271, 371 cohomology algebra 271, 372 Center of a Lie group 144 Chain complex 250 Chain homotopy 248 Chain rule 34, 89 for inverse adjoints 162 Change of parameter 167 Christoffel symbols 296 are intrinsic 306 Classification of manifolds of dimension one 22 ff. impossible in dimensions 2 4 90 Clebsch 109, 118 Clifford Algebra 350 Closed form 221, 224, 237, 245 nonsingular 286 ff. Closed subgroup theorem 139 Closure of a flow line 106 Cobordant 22 Cobordism 21 Coboundary 224, 272 Coboundary operator 230, 272, 371 Cochain complex 250 Cochain homotopy 248 Cocycle 78 ff., 210, 224, 271, 372 conditions 68-69, 272 property 78 Codazzi-Mainardi equation 312 Codimension of a foliation 120 Cofinal subset 375 Coherently oriented 10 Cohomology 173 ff., 221 ff. as a contravariant functor 226 class 173 of a contractible manifold 245 of a cochain complex 250 of spheres 256 Combinatorial Stokes' theorem 235 Commutator product 144 Commuting flows 62, 107, 110 ff. local flows 107 vector fields 62 as coordinate fields 110

INDEX Compact support 104 Compactly supported diffeomorphism 94 supported form 226 supported isotopy 94, 105 Complete Riemannian manifold 314 ff., 345 vector field 101 ff. Complex Grassmann manifold 149 Stiefel manifold 150 analytic manifold 84 analytic structure 84 general linear group 83, 128 projective space 149 tangent bundle 83 Component of the identity 128, 135 Composition of bundle homomorphisms 213 of smooth maps 70 Connected graded algebra 196, 225 sum 9 tensor algebra 197 Connecting homomorphism 251 Connection 294 ff. form 339 existence 301 Levi-Civita 293, 295, 296, 304-306, 333 Conservation of energy 169 Constant rank theorem 38 Continuous choice of orientation 81 homomorphism of Lie groups 142 singular simplices 271 Contractible manifold 96, 227, 240 Contraction mapping 354 Contraction mapping lemma 353 Contravariant degree 198 functor 165, 173, 227 tensor 197 tensor algebra 219 Convex hull 10, 233 set 233 Convexity (see Geodesic) Coordinate chart 67, 110 Coordinates 26 Coset space 14 Cotangent bundle 156, 159, 162, space 159 vector 159 Covariant

INDEX degree 198 ) 297 derivative uniqueness and existence 297 functor 165, 200, 242 symmetric tensors 220 tensor 197 tensor algebra 219 Covector field 159, 162 Critical point 64 Critical value 64 Cup product 271, 372 Curvature 81, 294 form 333, 340 operator 311 tensor 312, 313, 331 Cut and paste 12

(g

Ap(M) 239 A!(M) 255 BM 87 8 s 235 ais 234 ax 20 D ( C w ( M ) ) 85 Diff(M) 93, 112 Diff(S1) 186 Diff(Tn) 186 Diff,(M) 94 Diff(M, N) 93 DR (the de Rham map) 241, 268 ff. 29 dega 95 ff. 181 deg 181, 261 ff. dimR 190

Deahna 109, 118 Decomposable element 200 Decomposable tensor 193 Degree (mod 2) of a map 96 ff. of a map 261 ff. o f f : 5'' + S1 181 of a homomorphism 250 theory 95 ff., 260 ff. De Rham cochain complex 250 cohomology algebra 226 cohomology 98, 159, 225 ff. functoriality 225 H'(M) 173 ff. is a topological invariant 271 with compact supports 227 map 241, 268 ff. theorem 228, 238, 241, 264, 268 ff., 371, 376

De

ham-~ech

cohomology theorem 371 ff. complex 375 ff. isomorphism is functorial 382 Derivation of an R-algebra 50 of a Lie algebra 144 of CM( M ) 107 Derivative 25, 30, 71 directional 27 of the algebra of germs 30, 71, 88 of 6; 33 operator 30 Determinant 43, 203 Diagonal action 151 Diffeomorphic structures 73 ff. Diffeomorphism 36 between subsets of manifolds 86 between manifolds 71 between manifolds with boundary 87 Differentiability class 25 ff. Differentiable function 25 ff. manifold 67 ff. structure 67 ff. for manifolds with boundary 87 Differential 34 as a linear approximation 37 equations 55 ff., 359 ff. form 98, 159 ff., 221 ff. of a linear map 35 of a map of manifolds 71, 89 two definitions of 36 Dimension local 1 of a foliation 120 of a locally Euclidean space 2 of a topological manifold 3 of the tangent space a t the boundary 90 theory 217, 218 Direct limit of algebras 372, 373 Direct sum of bundles 212 Directed set 373 Directed system of graded algebras 373 Directional derivative (see Derivative) Discrete subgroup 134 Distribution k-plane 83, 107, 116 horizontal 340 integrable 83, 109, 116 ff. F'robenius 107, 108, 118 Donaldson 74, 150 Dual bundle 156, 163 Mayer-Vietoris sequences 256

INDEX of a free R-module 195 of an R-module 194 of the dual 163

E* (dual bundle) 156, 163 ( E * ) * (double dual bundle) 163 exp(tX) 133 exp, ( v ) 316 e, 30 Einstein summation convention 299 Equations of structure 331 ff. Equivalence classes 12 of Gxocycles 80 of Gl(n)-cocycles 79 relation 12 Equivalent points under a k-flow 112 Essential map 96 Euclidean connection 295 half space 20, 87 Euler characteristic 11, 275 Evaluation map 30, 71, 130 Even permutation 199 Evenly parametrized curve 315 Exact form 169, 170, 221, 224 Exact sequence 249 ff. Exactness of dual sequence 252 Existence and uniqueness of exterior derivative 223 of solutions of O.D.E. 56, 360 ff. of tensor product 190 of the integral 228 Existence of infinitesimal H s t r u c t u r e s 157 of Riemannian metrics 87 of minimal sets 106 Exponential map 133, 138, 317 of a matrix 132 series 132 Exterior algebra (Grassmann algebra) 199 ff. of a free R-module 202 of M 220 derivative 221 ff. axioms for 223 of a function 166 differentiation 166, 221 ff. multiplication 159, 199 ff. power 199, 211 Extrinsic property 305 F-saturated set 124

F-equivalence 119 F ( M ) (frame bundle) 334 F ( E ) (frame bundle) 153 ff. F(E) x ~ ~ ( IWn n ) 153 F(V) (frame manifold) 150 F(V) XGI(,) IWn 151 f -related bracket 109 fields 109 f * V 306 Face of a singular simplex 234 of the standard v i m p l e x 234 operator 239 First cohomology (see Cohomology) Five Lemma 249 Flat connection 302, 344 Riemannian manifold 82, 331 Flow line 101, 106 Flow global 102 ff. local 55 ff., 101 Foliation 83, 101, 109, 116 ff., 277 ff. defined by a map of constant rank 124 integral to a distribution 109 Frame bundle 153, 159, 334 ff. manifold 150 ff. Framing 334 Free basis 189 R-module 189, 192 Freedman 74 Frobenius chart 118 distribution 107, 108, 118 theorem 83, 109, 118 ff., 136, 159, 277

ff. Fubini's theorem 368 Functor 165 notation -.+ 213 Functoriality 159 Fundamental theorem of algebra 97, 184 of calculus 221, 235 r ( E ) 77, 213 r ( E ) @ r ( F ) 214 6, 29, 71 6, 160 gL(n) 108, 130 G-cocycle 80 G-reduction 80 Gstructure 80

INDEX

Gauss curvature 306 ff. infinitesimal spread of normal field 310 equation 312 map 307 Gauss-Bonnet theorem 12 General chain rule 35 General linear group 43, 349 General relativity 333 Geodesic 81, 293, 314 convexity 327 ff. existence and uniqueness 316 smooth dependence on endpoints 319 Geodesically complete 317 Geodesically convex 327 Geometric property 305 Geometric structure 78, 80 Geometric topology 67 Germinal equivalence 29 Germs of Ck functions 29, 33 of Cw functions 29, 71 Gleason 135 Global 5 calculus 67 chain rule 71 flow (see Flow) smooth Urysohn lemma 84, 85 submersion theorem 91 Graded algebra 196 ff. Graded ideal 198 Grassmann algebra of M 220 Grassmann manifold 149 as a homogeneous space 206 Group (see Lie group) of Lie algebra automorphisms 144 of permutations 199 h!z(M) 301 W (quaternions) 350 W* 128 Wn-atlas 87 Wn-charts 87

H1(M) 173 H p ( M ) 225 H r ( M ) 227 Hp(M) 240 H ~ ( M ; G )80 H1(M; Gl(n)) 79 H1(M;Z) 179 H,'~(M) 275 H ~ ( M 255 ) R*(M) 271 HP(U) 272 H*(u; R) 372 H*(x; R) 372 H * ( M ) 221, 226 H,'(M) 227 Ha ( M ) 240 HOM(E, F ) 213 Hom(E, F ) 211 Homcm (M) (%MI, C W 0 )163 Halfspace 20 Hessian 65 Hilbert's fifth problem 134, 143 Hoffman and Meeks 310 Holomorphic tangent bundle 84 Holonomy 300 group 301 Homogeneity lemma 94, 104 local 105 Homogeneous polynomial 208 Riemannian manifold 345 space 145 ff., 155 (see also Riemannian homogeneous space) Homology class 240 of chain complex 250 of simplicia1 chain complex 275 Homomorphism of (co)chain complex 250 of Lie groups 133, 134, 142 of graded algebras 197, 226 of vector bundles 213 Homotopic loops 172 maps 94, 174 Homotopy 93 ff. equivalence 174 invariance of degs 96 invariance of cohomology (see Poincark Lemma) Hopf-Rinow Theorem 314 versions I and I1 318 Horizontal bundle 336, 337 Horizontal lift 336, 337, 340 Horizontal space 337

390

Identity component of a Lie group 128, 135 Imbedded submanifold 18, 93 Immersed submanifold 18, 93 Immersion 18, 41, 92 Immersion theorem 41 Implicit function theorem 42 for Banach spaces 357 Indecomposable 200 tensor 193 Indefinite metric 82, 333 nondegenerate 82 Index of a critical point 65 Induced homomorphism in cohomology 173, 225-227, 250, 375 in homology 242 of graded algebras 226, 227 Induced map of a quotient space 13 Induced orientation of the boundary 230 Infinitesimal G s t r u c t u r e 80 for G = Gl(k, n - k) 83, 118 for G = Gl(n, C) 83 for G = Gl+(n) 81 for G = O(n) 81 for G = O(k,n - k) 82 Infinitesimal curve 28, 72, 88 as a derivative 31 Infinitesimal equivalence 28 Infinitesimal generator 58, 102, 113 Integer lattice 189 Integrability condition 109, 117, 118 Integrability of O(n)structure 331 Integrable distribution (see Distribution) Integrable structure 80 almost complex structure 84 Gl(k, n - k)-structure 83, 118 Riemannian structures 82 Integrably parallelizable 80, 116 Integral 1-form 179 Integral cohomology 179 Integral curve 55 Integral manifold 109, 116, 118, 119 maximal 119, 137 Integral 220, 228 ff. of a p f o r m over a singular psimplex 234 Interior of a manifold 20, 97 Interior product 207 Intrinsic properties 305-306, 312, 344 Invariance of domain 2, 37 Invariance of line integrals 167 Inverse function theorem 38, 353 ff. Involutive distribution (see Distribution)

INDEX Involutivity 107, 109 Inward vectors at a boundary point 88 Isometry 305 Isometry group 344 Isomorphism of Lie groups 134 of bundles 76 of graded algebras 197 of tangent and cotangent bundles 167 Isotopic submanifolds 267 Isotopy 93 between points 105 class 186 compactIy supported 94, 104 Isotropy group 146, 206 Jacobian 161 cocycle 80 matrix 35 for Banach spaces 356 Jordan curve theorem 98 Jordan-Brouwer separation theorem 98,263 n, nl, K2 310

k-dimensional lattice 114 k-flow 112 k-plane distribution (see Distribution) k-plane subbundle 83 KerMire 5, 74 Klein bottle 7 Kronecker 142 A(E) 211 A(V) 199 Ak((E 211 L(G) 130 L(G)-valued form 333 L(Gl(n)) 130 L(O(n)) 130, 138 L(Sl(n)) 130 L(U(n)) 138 Lattice subgroup 14, 114 Leaves of a foliation 109, 120 Lebesgue measure 64, 367-369 Left coset 109 Left invariant metric 346 vector field 108, 130 Left multiple map 46 Left translation 46, 129 Leibnitz rule 30 for exterior differentiation 166 Length minimizing curves 81

INDEX Length of a curve 82, 293, 303 of a vector (norm) 81, 302 Levi-Civita (see connection) Lie algebra 101, 107, 127 homomorphism of 133 of derivations of CM(M)85 of a Lie group 130 of linear endomorphisms 144 Lie bracket 85, 107 Lie derivative and the Lie bracket 107 of a form 167 of a vector field 60 Lie group 127 ff. homomorphism of 133, 134, 136 isomorphism of 134 transformation group 145 Lie subalgebra 107, 108, 136 of X(M) 107 and Lie subgroups 136 ff. Lie subgroup 136 as a leaf 137 Lift 180 Lifting lemma 180 Lima 116 Limit set a-limit set 106 w-limit set 106 of a leaf 125 Line bundle 77 Line integral 167 ff. Linear approximation 25 Linear combination of derivatives 30 of infinitesimal curves 29 Linear endomorphism 144 Linear flow on T~ 102, 116, 142 Linking number 263 Local Cartesian product 5, 75, 147, 152 ff. coordinate formula 70 degree of a map 260 dimension 1 flow (see Flow) homogeneity lemma 105 theory of smooth functions 25 ff, Localization of vector fields 107 Locally constant functions 225 Euclidean space 1 Euclidean geometry 82 exact form 171, 172, 221, 225, 245 finite family of sets 15 sum 244

length minimizing 293, 314, 321 trivial 47 trivializing cover 76 Long exact sequence in (co)homology 252 Long line 1 Lorentz manifold 82, 333

Manifold smooth 67 ff. topological 3 ff. topology 116 with boundary 20 ff., 87 ff. with corners 90 Matrix form of the chain rule 36 Maximal Cm atlas 68 connected integral manifold 119 integral manifold 137 local flow 102 torus 144 Mayer-Vietoris sequence 252 ff. for ordinary cohomology 253 for compactly supported cohomology 253 for singular homology 256 Mean curvature 310 Mean value theorem 354 in Banach spaces 356 Measure zero 64, 367 ff. Metric coefficients 304 Metric tensor 293 Metric topology 318 Milnor 5, 74, 115, 314, 367 Minimal set of a flow 106 of a foliation 124 Minimal surface 310 Mijbius bundle 77 Module free 189 ff. of bilinear maps 190 over a ring 189 ff. over Coo (U)50 of sections 212 ff. Montgomery 135 Morphism 165, 212 Morse lemma 66 Morse theory 65 Multilinear algebra 189 ff. bundle theory 209 ff. Myers-Steenrod theorem 344

392

INDEX

n' (the Gauss map) 307 n-frame 150 n-plane bundle 76

Natural constructions 212 homomorphism 251 Naturality of exterior derivative 166 Non-Hausdorff 1, 12 Nondegenerate (see indefinite metric) Nonsingular closed 1-form 286 ff. flow 287 k-flow 112, 113, 120 Nonzero quaternions 128 Norm 82 Normal bundle 295 lattice 134 space 16 Nowhere vanishing vector field 262, 268 O(M, xo) 301 w-limit set 106 o(n) 108, 130 O.D.E. (see Ordinary differential equations) O(k, n - k) 82 O ( M ) 334 Odd permutation 199 Omission symbol 111 One to one correspondence between vector bundles and frame bundles 154 One t o one immersion 93 Oneparameter group 131 ff., 136, 140,346, 347 Orbit 112, 113, 120, 121, 145, 151, 157, 207 Ordinary differential equations 55 ff., 101, 359 ff. Orientable bundle 81 Orientable manifold 11, 14, 21, 81, 212, 228 Orientation induced on 8 M 230 of a manifold 80, 81 of a triangle 10 Orthogonal group (O(n)) 46, 109, 127, 349 Ort honormal frame 150, 155 frame bundle 155, 334 Overdetermined system of P.D.E. 117

0

@-invariant set 106 r[S1,S1]182 ff. r [ M , N ] 182

Pk(V)208 P D (the PoincarC duality map) 264 P2 8 Pn 14 P n ( C ) 149 pboundary 240 p-cycle 240 Paracompact 15 Parallel field 299 translation 340 transport 293, 299 existence and uniqueness 299 Parallelizability of S3 5, 131 of S7 5 of Lie groups 131 Parallelizable 5, 45, 46, 77, 302 (see also Integrably parallelizable) Partial covariant derivative 320 derivative 30 differential equations (P.D.E.) 101 order 373 Partition of unity 15 ff., 85 smooth 85 Path independent line integrals 169, 170 Periodic point of a flow 106 Periods of a I-form 173 of a p-cycle 242 of a cohomology class 173, 242 Piecewise regular (p.r.) 317 Piecewise smooth paths 169 vector field along a curve 300 Plaque 118 Plaque chain 120 Poincark Duality Theorem 264 PoincarC duality operator 264 Poincark dual of a submanifold 267 PoincarC Lemma 172, 225, 243 ff. for compact supports 248 Versions I through VI 245 Poincar&Hopf theorem 268, 276 Pointwise linearly independent 113 Pointwise tensor product of sections 214 Polarization 209 Positive definite inner product 81 Principal bundle 150 R. of frames 334 ff. Principal curvatures 310 Projective space 14, 149 plane 8, 9 Proper homotopy 246

INDEX Proper map 227 Properly imbedded Lie subgroup 139, 141 Properly imbedded submanifold 91 PseudwRiemannian manifold 333 Pull-back of a connection 306 Quadratic form 209 Quaternions 128, 350 Quotient spaces 7, 12 Quotient topology 7

Radius function 327 Radon-Nikodym derivative 310 Rank 38, 206 Rank of a manifold 115 Real analytic group 134, 139 Reduction of principal bundle 157 of the structure group 80 Reeb foliation 124 Refined cover (refinement) 15 Regular cover 118 path 317 point 64 space 3 value 64, 92, 94, 182 Relative topology 116 Relativity theory 82 Representation of an algebra 351 Retraction 98, 177 Riemann curvature tensor 82, 312 distance function 317 integral 228 integration 17 volume element 303 Riemannian connection 304 geometry 81, 293 homogeneous space 344 ff. manifold 81, 302 ff. complete 318 metric 17, 82, 155, 215 Right action 145 of Gl(n) on F ( V ) 150 Right invariant metric 346 Right multiple map 46 Right translation 46

S ( E ) 211 s ~ ( E ) 211 S"M) 220 S ( V ) 207 s k ( v ) 207 S1 as a Lie group 128

S3 as a Lie group 128 parallelizability of 5, 131 s7(parallelizability of) 5 S, 320 Sl(n) 43, 108, 127 SU(n) 135 S(U, P) 28 Sard's theorem 64, 367 Section of a bundle 77 Self intersection number 268, 276 Short exact sequence 249 Sign of a permutation 199 Signed area 310 Simple cover 254, 275, 375, 380, 381 Simple refinement 327 Simplicia1 chain complex 275 cohomology 275 Simply transitive action 150 Simultaneous regular values 64, 92 Singular boundary operator 239 chain complex 250 homology 228, 238, 240 as a covariant functor 242 of Rn 241 of a contractible manifold 240, 241 p-chain 239 p-simplex 234 Skew symmetric matrices 46, 108, 130, 138 Smooth Jordan curve 99 action of IRk 112 atlas 67 ff. change of coordinates 67 extension 184 function 25 ff. between arbitrary subsets of manifolds 86 on an arbitrary subset of Rn 38 homotopy 93 invariance of domain 86 manifold 67 ff. manifold with boundary 87 ff. mapping between manifolds 70 ff. mapping between open subsets of EUclidean spaces 34 ff. partition of unity 17, 85

394 structure 68 for manifolds with boundary 87 submanifold 42 real valued functions on manifolds 69 Smoothly compatible charts 67 Smoothness class 25, 26 Soap film 310 Space filling curve 64 Space of k-forms 220 of contravariant tensors 198, 219 of covariant tensors 198, 219 of homogeneous polynomials 208 of mixed tensors 219 of quadratic forms 209 of smooth sections 213 Spacetime 333 Special linear group 43, 108, 127 Special orthogonal group 128 Sphere as a homogeneous space of 0 ( n )and U(n) 146 parallelizable 5 Spherical shell 320 Squiggly arrow (-) for functors 213 Stack of records theorem 95 Standard simplex 10, 11, 233 Standard fiber 153 Standard orientation 10 Star shaped 327 Steiner's surface 19 Stereographic projection 3, 4 Stiefel manifold 150 Stokes' theorem 159, 169, 221, 228 ff. Structure cocycle 69, 78 for T*( M ) 162 Subbundle 83 Submanifolds 18 ff., 42 ff., 91 ff. Submersion 41 theorem 41 Submodule of bilinear relations 192 Summation convention 299 Support 15 Symmetric algebra 207 ff., of M 220 bilinear form 209 connection 296 multilinear map 207 neighborhood of the identity 129 power of a bundle 211 power of a free R-module 208 power of an R-module 207 space 345

INDEX T ( V ) 196 T k ( ~210 ) ' T k ( ~219 ) Tk(M) 219 T n ( V ) 196 T ( M ) 219 T~ (the 2-torus) 7 Tn torus 4 as a coset space 14 as a Lie group 128 T a ( M ) (cotangent bundle) 156, 159, T(Sn) 5 TP(U) 28 Tz(Wn) 90 Tangent bundle 34, 45, 74 of sn5 of a Lie group 131 of a manifold with boundary 90 Tangent space 28, 30, 44 to a manifold 71 t o a manifold with boundary 88 t o sphere 4, 38 two definitions of 33 vector 27, 28, 30, 44 a t a boundary point 88 to a manifold 71 Tautological horizontal frame 339 Tensor 189 Tensor algebra 189 ff. Tensor power 196 bundle 210 lSt211 oth 211 Tensor product of bundles 209 of free R-modules 193, 194 of R-linear maps 194 of R-modules 190 Tensor property 164 Theorema Egregium 312, 312 Top dimensional cohomology 257 of nonorientable manifolds 259 Topological group 127 imbedding 18 immersion 18 submanifold 18 Topology of a Lie subgroup 136 Torsion free connection 296 Torsion form 340 of a connection 296 tensor 331 Torus 4, 7, 14, 128

INDEX Total degree 378 Total space 45, 75, 76 Trace 44 Trace 0 112, 134 Transformation group 145 Triangulation 10, 274 Trilinear 193 Trivial bundle 45, 76, 77 Trivializing neighborhood 76

Whitehead (J.H.C.) 327 Whitney sum 212 Winding number 238 (mod 2) 99 Word problem 90

U s m a l l singular complex 255 U s m a l l singular homology 255 U-small singular simplex 255 U(n) 128, 135

Yang-Mills 150

Umbilic point 313 Unimodular matrix 180, 186 Unit circle 128 Unitary group 128 Unity in a graded algebra 196 in cohomology 226 Universal model 189 multilinear map 190 ff. antisymmetric 199 bilinear 190 k-linear 193 trilinear 193 symmetric 207 property 190, 192, 201 Urysohn lemma 16 smooth version 47, 84, 85 for manifolds with boundary 90

Vector bundle 76 Vector field 45, 75 along a curve 297 problem for spheres 5, 349 ff. on manifolds with boundary 90 Velocity 55 Vertical fields 335 space 335 subbundle 335 Volume 293, 304 Weak reparametrization 300 Weingarten map 308, 309 as differential of the Gauss map 309

Zn (integer lattice) 14

zk (free %module) ZP(U) 272 ZP(U, R) 372 Z1(M) 173 ZP(M) 224 z,!?(M) 227 Zp(M) 240

189