Geometry of Manifolds

Geometry

of Manif0Ids

PURE A N D APPLIED MATHEMATICS A Series of Monographs and Textbooks

Edited by

PAULA. SMITHand SAMUEL EILENBERG Columbia University, N e w York

I: ARNOLDSOMMERFELD. Partial Differential Equations in Physics. 1949 (Lectures on Theoretical Physics, Volume VI) 11: REINHOLD BAER.Linear Algebra and Projective Geometry. 1952 111: HERBERT BUSEMANN AND PAULJ. KELLY.Projective Geometry and Projective Metrics. 1953 AND M. SCHIFFER. Kernel Functions and IV: STEFANBERGMAN Elliptic Differential Equations in Mathematical Physics. 1953 V: RALPHPHILIPBOAS,JR. Entire Functions. 1954 VI: HERBERT BUSEMANN. The Geometry of Geodesics. 1955 VII: CLAUDE CHEVALLEY. Fundamental Concepts of Algebra. 1956 Hu. Homotopy Theory. 1959 VIII: SZE-TSEN IX: A. OSTROWSKI. Solution of Equations and Systems of Equations. 1960 X: J. DIEUDONNB. Foundations of Modern Analysis. 1960 XI: S. I. GOLDBERG. Curvature and Homology. 1962 HELGASON. Differential Geometry and Symmetric XII: SIGURDUR Spaces. 1962 XIII: T. H. HILDEBRANDT. Introduction to the Theory of Integration. 1963 XIV: SHREERAM ABHYANKAR. Local Analytic Geometry. In preparation. XV: RICHARDL. BISHOPAND RICHARDJ. CRITTENDEN. Geometry of Manifolds. 1964 XVI: STEVEN GAAL.Point Set Topology. In preparation.

Geometry

of Manifolds

Richard L. Bishop

Richard I . Crittenden

Department of Mathematics University of Illinois Urbana, Illinois

Department of Mathematics Northwestern University Evanston, Illinois

1964

ACADEMIC

PRESS

NEW YORK AND LONDON

COPYRIGHT 0 1964,

BY

ACADEMIC PRESSINC.

ALL RIGHTS RESERVED.

NO PART OF THIS BOOK MAY BE REPRODUCED I N ANY FORM, BY PHOTOSTAT, MICROFILM, OR ANY OTHER MEANS, WITHOUT WRITTEN PERMISSION FROM THE PUBLISHERS.

ACADEMIC PRESS INC. 111 Fifth Avenue, New York, New York 10003 United Kingdom Edition published by ACADEMIC PRESS INC. (LONDON) LTD. Berkeley Square House, London W.l LIBRARY OF CONGRESS CATALOG CARDNUMBER:64-20317

PRINTED I N THE UNITED STATES OF AMERICA

PREFACE Our purpose in writing this book is to put material which we found stimulating and interesting as graduate students into book form. I t is intended for individual study and for use as a text for graduate level courses such as the one from which this material stems, given by Professor W. Ambrose at M I T in 1958-1959. Previously the material had been organized in roughly the same form by him and Professor I. M . Singer, and they in turn drew upon the work of Ehresmann, Chern, and E. Cartan. Our contributions have been primarily to fill out the material with details, asides and problems, and to alter notation slightly. We believe that this subject matter, besides being an interesting area for specialization, lends itself especially to a synthesis of several branches of mathematics, and thus should be studied by a wide spectrum of graduate students so as to break away from narrow specialization and see how their own fields are related and applied in other fields. We feel that at least a part of this subject should be of interest not only to those working in geometry but also to those in analysis, topology, algebra, and even probability and astronomy. In order that this book be meaningful, the reader’s background should include real variable theory, linear algebra, and point set topology. T o get an idea of the scope of this book we refer to the table of contents and the introductory paragraphs to the chapters. We have not included the study of integration theory, for example, the de Rham’s theorems and the Gauss-Bonnet theorem, because we did not wish to get involved in the theory of topological invariants. However, the background for these topics is thoroughly treated, and Morse theory is carried to the point where topology takes over from analysis. T he theorems, lemmas, propositions, and problems are numbered consecutively within each chapter. Our use of these numbers in cross references should be transparent. Th u s in the text of Chapter 6, “theorem 7” refers to the seventh theorem in Chapter 6, while “problem 5.4” refers to the fourth problem in Chapter 5. Definitions are generally distinguished only by italics. T h e word “section” is usually omitted in this usage ; that is, an unmodified number reference is to the corresponding section. In this case, the chapter number is always given. V

vi

PREFACE

T h e problems range from trivial to very difficult, from essential to the text to clearly tangential. T h e subjects of holonomy groups and complex manifolds are developed exclusively in problems. Some problems almost certainly will require recourse to the reference given, namely, problems 1.11, 2.7, 2.13, 2.14, and 8.15. A brief appendix is provided with a statement of the theorem on existence and uniqueness of solutions of ordinary differential equations most appropriate for our needs. T h e reader is referred to [33]and [50] for their extensive bibliographies as well as to their very fine treatment of much of the subject matter of the present text. Italic numbers in brackets are, of course, references to entries in the bibliography. R.L.B. R. J.C.

April 1964

C O NT E NT S Preface

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

1

.

2 7 13 21 22

v

CHAPTER 1

Manifolds 1.1

1.2 1.3 1.4 1.5 1.6

Introductory Material and Notation Definition of a Manifold . . Tangent Space . . . . Vector Fields . . . . Submanifolds . . . . Distributions and Integrability .

.

. .

.

.

.

.

. .

.

.

.

. .

.

. .

.

. .

.

. .

.

. .

.

.

.

.

.

. .

. .

. .

. .

. .

. .

. . .

. . .

. . .

. . .

. . .

. . . . . .

25 26 28 29 30 34

. . . .

. . . .

. . . .

. . . .

. . . .

. . . .

38 41 45 49

. . .

. . .

. . .

. . .

. . .

. . .

53 53 54

.

.

.

.

.

. .

.

.

.

. .

.

.

.

CHAPTER 2

Lie Groups 2.1 2.2 2.3 2.4 2.5 2.6

Lie Groups . . . . . . Lie Algebras . . . . . . Lie Group-Lie Algebra Correspondence Homomorphisms . . . . . Exponential Map . . . . . Representations . . . . .

.

.

.

.

.

.

. .

.

CHAPTER 3

Fibre Bundles 3.1 3.2 3.3 3.4

Transformation Groups . . . Principal Bundles . . . . Associated Bundles . . . . Reduction of the Structural Group

. . . .

. . . .

CHAPTER 4

Differential Forms 4.1 Introduction . . . . . 4.2 Classical Notion of Differential Form 4.3 Grassmann Algebras . . . . vii

. . .

. . .

...

Vlll

4.4 4.5 4.6 4.7 4.8 4.9 4.10

CONTENTS

Existence of Grassmann Algebras . Differential Forms . . . . Exterior Derivative . . . . Action of Maps . . . . . Frobenius’ Theorem . . . . Vector-Valued Forms and Operations Forms on Complex Manifolds . .

.

.

.

.

.

.

.

.

57

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

68 70 71 72

. .

. .

.

. .

.

.

. .

.

.

. .

.

.

. .

.

.

. .

.

.

. 62 . 64

.

.

.

.

.

CHAPTER 5

Connexions 5.1

5.2 5.3 5.4 5.5 5.6

Definitions and First Properties . . . . . . . Parallel Translation . . . . . . . . . . Curvature Form and the Structural Equation . . . . . Existence of Connexions and Connexions in Associated Bundles Structural Equations for Horizontal Forms . . . . . Holonomy . . . . . . . . . . .

. . .

. . .

. .

. .

74 77 80 83 84 87

.

.

CHAPTER 6

Affine Connexions 6.1 6.2 6.3 6.4

Definitions . . . . . . . . T h e Structural Equations of an Affine Connexion T h e Exponential Maps . . . . . Covariant Differentiation and Classical Forms

.

. .

. . . .

. . . .

. . . .

. . . .

. . . .

89 99 108 111

. .

. .

. .

. .

. .

122 127 129 132

CHAPTER 7

Riemannian Manifolds 7.1 7.2 7.3 7.4

Definitions and First Properties T h e Bundle of Frames . . Riemannian Connexions . . Examples and Problems . .

. .

. .

. .

. .

. .

. .

. .

. .

. .

. .

. .

. .

. .

CHAPTER 8

Geodesics and Complete Riemannian Manifolds 8.1 Geodesics . . . . . 8.2 Complete Riemannian Manifolds 8.3 Continuous Curves . . .

.

.

.

.

.

.

.

.

. 145

.

.

.

.

.

.

.

.

. 158

.

.

.

.

.

.

.

.

.

152

ix

CONTENTS

CHAPTER 9

Riemannian Curvature 9.1 9.2 9.3 9.4 9.5

Riemannian Curvature . . . . Computation of the Riemannian Curvature Continuity of the Riemannian Curvature Rectangles and Jacobi Fields . . . Theorems Involving Curvature . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . .

.

161 165 166 172 178

CHAPTER 10

Immersions and the Second Fundamental Form 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.9

. . . . . . . Definitions T h e Connexions . . . . . . Curvature . . . . . . . T h e Second Fundamental Form . . . Curvature and the Second Fundamental Form . . . . . T h e Local Gauss Map Hessians of Normal Coordinates of N . . A Formulation of the Immersion Problem . Hypersurfaces . . . . . . .

. .

.

. .

.

. . .

. . .

. .

. .

.

.

. .

. . . . .

. .

. . . . . . .

. . . . . . .

.

.

187 189 190 192 195 . 197 . 199 . 207

. .

.

.

.

. 185

.

. . . .

CHAPTER 11

Second Variation of Arc Length 1 1 . 1 First and Second Variation of Arc Length 11.2 T h e Index Form . . . . . 11.3 Focal Points and Conjugate Points . I I .4 T h e Infinitesimal Deformations . . 1 I .5 T h e Morse Index Theorem . . . 11.6 T h e Minimum Locus . . . . 1 I .7 Closed Geodesics . . . . . 11.8 Convex Neighborhoods . . . . 1 I .9 Rauch’s Comparison Theorem . . 11.10 Curvature and Volume . . . .

.

.

.

.

. .

. .

. .

. .

. .

. .

. .

. .

. .

. . .

. . .

. . .

. . .

. . .

.

.

.

.

.

.

.

APPENDIX: Theorems on Differential Equations

.

.

.

. . .

.

. . .

. . . .

. 213 220

. 224 . 226

. . .

233 237 241 . 246 . 250 . 253

. . . .

258

BIBLIOGRAPHY .

.

.

.

.

.

.

.

.

. 260

.

.

.

.

.

.

.

.

.

. 265

SUBJECT INDEX

This Page Intentionally Left Blank

Geometry

of

Manifolds

This Page Intentionally Left Blank

CHAPTER 1

M an i fo Ids In this chapter the basic tools of manifold theory are introduced and the main theorems are stated without proof. Lie derivatives are discussed via local one-parameter groups of transformations, and various interpretations of the bracket of vector fields are given. Frobenius' theorem on the integrability of p-plane distributions is given in outline form [4, 24, 25, 33, 50, 78, 831. 1.l Introductory Material and Notation

+

If is a map of M into N andJ! , I a map of P into T , then I,!J o will denote their composition, that is, J!,I o is followed by 4. Here M , N , P, T are any sets and we understand that the domain of I,!J o is #-l(P) n M (in particular, $J c may have an empty domain). T h e same sort of convention, namely, that the domain is the largest meaningful set, will be used in the formation of sums, products, and other combinations of maps. If U c M we use +ILI for the restriction of to u.

+ +

+

+

+

FIG. 1.

1

2

1. MANIFOLDS

T h e d-dimensional Euclidean space will be denoted by Rd,provided with the usual coordinate functions, {ui}, that is, if t = ( t l , ...,td) E Rd, then ui(t) = t i . In the case d = 1 we write R1 = R and u1 = u. If U is open in Re, then a map 4 : U -+ Rd is said to be of class C" (written 4 E C") if the real-valued functions ui o 4, i = 1, ..., d, have all kth order continuous partial derivatives for every non-negative k. C" maps are not necessarily analytic, as is shown by the example: f(x) = exp(-l/x2) if x # 0, f(0) = 0. In fact, there exist nontrivial C" real-valued functions on Rd which vanish outside a given compact set. (See [85], pp. 25 and 26 for the construction of C" Urysohn functions.) Problem 1.

Define exp(-I/.)

if x > O if x 0.

<

Let {rm}be an ordering of the rational numbers and

22 3 . W

).(g

=

- rJ.

n=l

Show that g is C" but nowhere analytic.

1.2 Definition of a Manifold

If X is a Hausdorff topological space, a d-dimensional coordinate system in X is a homeomorphism of an open set in X onto an open set in Rd. X is called a d-dimensional topological manifold if X is covered by domains of d-dimensional coordinate systems. T h e domain of a coordinate system 4 is called the coordinate neighborhood and if x is in the coordinate neighborhood of 4, 4 is said to be a coordinate system at x. If 4 is a coordinate system, we often write ( x l , ..., xd) for the functions (ul o 4, ..., ud o 4). Either 4 or (xl, ..., xd) will be referred to as a coordinate system. Let 4, $ be d-dimensional coordinate systems on X . Then 4, $ are C"-related if 4 o $-l and $ o 4-l are of class C". (Fig. 2.) Consider the following properties of a set of coordinate systems %? on a topological manifold X :

1.2. Definition of a Manifold

3

(1) X is covered by the domains of coordinate systems in %?. (2) Every two coordinate systems in %? are C" related. (3) %? is maximal with respect to (1) and (2). A C" manifold (or just manifold) is a pair ( X , %), where X is a topological manifold, and %? is a set of coordinate systems satisfying (1),

R

FIG. 2.

(2), and (3). %? is said to be a C" structure on X . (We shall usually

omit W in the future, and just write X for a manifold.) A basis for the C" structure W is a subset Vo of %? satisfying (1) and (2). Given a set goof coordinate systems on a set X satisfying (1) and (2), requiring that they be homeomorphisms defines a topology on X so that X becomes a topological manifold. Then there exists a unique C" structure W on X with basis eoand it is obtained by adjoining all C" related coordinate systems. (We have ignored the assumption that X be Hausdorff.) If M , N are manifolds, a map $ : M -+ N is of class C" ($ E C") if for every two coordinate systems on M , 8 on N , the function 0 o $ o 4-l is of class C". For $ : M + N to be of class C" it is sufficient that for every m E M , there are coordinate systems at m, and 8 at $(m) such that 8 o $ o +-I E C". If M is paracompact, since partitions of unity which are subordinate to a given covering can be constructed from rational combinations

+

+

4

1. MANIFOLDS

of Urysohn functions, we can get C" partitions of unity. Because C" partitions of unity are an indispensible device for much of the analysis on manifolds, and because Riemannian manifolds, our ultimate object of study, are metrizable, hence paracompact, we shall assume henceforth that manifolds are paracompact. It then follows from point-set topology that manifolds are separable and so satisfy the second axiom of countability if connected. Examples

(1) Euclidean space. T h e ordinary C" structure on Rd is obtained by taking as basis V o = {identity}.

( 2 ) Open submanifolds. Let M be a manifold with C" structure V , let U be an open subset of M , and let V,, = {#JE V j domain of # J C U}. Then U is a manifold with V,, as its C" structure. U is called an open submanifold of M . ( 3 ) General linear group. Gl(d, R ) = {nonsingular d x d matrices with real entries} is an open submanifold of Rd2,since GZ(d,R ) = Rd2- det-l(O).

(4) Ordinary sphere. Let Sd = {x E Rd+lI C $(x) = l}, and define #J : S d - ((0,..., 0, 1)) -+ Rd, # : S d - ((0,..., 0, - 1)) -+ Rd by stereographic projection from (0, ..., 0, l), (0, ..., 0, - l ) , respec-

I

FIG. 3

Rd+'

1.2. Definition of a Manifold

5

tively. Then Vo = {+, +} is a basis for a C" structure on Sd. [Stereographic projection: +(.) is the point where the straight line from (0, ..., 0, 1) through x intersects ud+,-l(0) = Rd.] (5) Real projective space. Let Pa be real projective d-space, that is, the collection of straight lines through the origin in Rd+'. T h e natural covering map : Sd + P d , which takes x into the line through x,induces a C" structure on Pa,that is, there is a unique C" structure on Pa such that is a C" map with local C" inverses. (See problem 4 below.)

+

+

(6) Low dimensions. Essentially the only 1-dimensional connected paracompact manifolds are R1 and S1. If the restriction of paracompactness is omitted, then this is no longer true, as an example of a long line shows-this is obtained by connecting together an uncountable well-ordered collection of half-closed intervals [39, pp. 55 and 561. A 2-dimensional manifold is called a surface. T h e objects traditionally called "surfaces in 3-space" can be made into manifolds in a standard way. T h e compact surfaces have been classified as spheres or projective planes with various numbers of handles attached [80].

(7) Product manifolds. Let M , N be C" manifolds with structures 93 and dimensions d, e, respectively. Let p , : M x N -+ M , p , : M x N + N be the projections. Then Vo = {(+ o p , , I) o p z ) = **., uao + o P , , ~ 1 0O P , ( ~ 1 0 ueo $ 0 ~ 2 )I + E V , I ) E ~ } is a basis for a Cm-structure on M x N. T h e same structure could also be obtained by using only a basis for V, 9 in their places. More specific examples are: %,

+

Cylinder = R x S1 2-dimensional (ordinary) torus d-dimensional torus = S1 x

9

S1 x S1= T 2 x S1 ( d factors) = T d .

=

Note also that Rd+ecan be canonically identified with Rd x Re.

(8) Non-Hausdorfl manifold. T h e following example shows that the Hausdorff property of a manifold does not follow from the existence of a C" structure. T h e underlying point set consists of the interval (0, 3) with topology described in terms of neighborhoods as follows. T h e neighborhood of a point in (0, 1) u (1, 2) u (2, 3) would be as in the topology induced from the reals.

6

1. MANIFOLDS

A neighborhood of i

=

1 or 2 is, for 0 < E < 1,

(i - E , i] u (2, 2 + €).

----.,.1 2

O

-2

->

3

1

We leave as a problem to show that this has a C" structure. Problem 2. Show that if M is compact then a basis for a C" structure on M must contain more than one coordinate system.

Problem 3.

Show that a C" map is necessarily continuous.

+

A covering map : M e N is a continuous map such that for every n E N there is a neighborhood U of n such that +-l( U ) is the disjoint union of neighborhoods of points of +-l(n) such that is a homeomorphism on each such neighborhood [41,pp. 89-97; 801. is said to evenly cover U and U is said to be a distinguished neighborhood of +. When M and N are C" manifolds, then M is said to be a C" covering of N if is a C" map and if the local inverses of are C" maps.

+

+

+

+

FIG.4.

1.3. Tangent Space

7

+

Problem 4. Prove that if : M + N is a covering map and N has a C" structure, then there is a unique C" structure on M such that M is a C" covering of N.

+

Problem 5. Prove that if : M - t N is a covering map and M has a C" structure such that for every Ui, U j open sets in M on which o 4 is q5 is a homeomorphism and +( Ui) = +( U j ) we have (4 a C" map on U i ,then N has a unique C" structure such that M is a C" covering of N . Problem 6. Prove that if N is a connected C" manifold, then there exists an essentially unique simply connected C" covering of N . (Part of the problem is to show that manifolds satisfy a condition sufficient for the existence of simply connected topological coverings.) Problem 7. A d-dimensional complex manifold is a topological space locally homeomorphic with complex space Cd, and these homeomorphisms are complex analytically related. Make this precise and show that a complex manifold is an even-dimensional (real) manifold. Examples of complex manifolds are Cd itself, the Riemann surface of a complex analytic function of one complex variable, complex d-dimensional projective space CPd, and the set of all nonsingular linear transformations of Cd, denoted by Gl(d, C).

1.3 Tangent Space

Let M be a manifold, m E M , and denote by F ( M , m) the set of C" real-valued functions with domain a neighborhood of m. A C" curve in M is a map of a closed interval [a, b] into M which can be extended to a C" map of an open interval. T h e notion of a tangent can arise from the following considerations. Let y be a C" curve in M . Then y gives rise to a linear function y * ( t ) : F ( M , y ( t ) ) -+ R as follows: if f e F ( M , m), m = y(t), then y * ( t ) ( f ) = (fo y)'(t), which may be described as a directional derivative off at m in the direction of y . [Although F(M, m) is not a

FIG.5.

8

1. MANIFOLDS

linear space, linearity of real-valued functions has an obvious meaning.] f ( m ) y*(t)(g). This linear derivation does everything required of “the tangent to y” and subsequently we show that every such linear derivation is associated with a curve (in fact many curves) in this way. If m E M , a tangent to M a t m is a map t : F ( M , m) -+ R such that

+

y * ( t ) is a derivation, that is, y * ( t ) ( f g ) = y * ( t ) ( f ) g ( m )

+

+

( 4 t(af bg) = a t ( f > bt(g) (b) t(f-1 = t ( f )g ( m ) f ( m ) t(g), for a, b E R, f , g E q M , m).

+

T h e tangents at m form a linear space, denoted by M , . If c is a g are constant function, then t ( c) = 0. We recall that fg and f the usual product and sum, but defined only on the intersections of the domains o f f and g. Letting l u be the function defined only on U and there equal to 1, we see from (a) and (b) that t(flu) = t ( f ) and hence t ( f ) = t ( f lu); that is, t( f ) depends only on the local behavior off.

+

Prove that for constant function c and neighborhood U 1”) = 0. T h e problem includes proving t ( c ) = 0. If = (xl, ..., x d ) is a coordinate system the partial derivative a t m with respect to x i , Dzi(m), is the tangent defined by (DZi(m))f = ( a ( f o+-‘)/aui) (+(m)), which is also denoted by Dzif ( m ) . When the coordinates are {ui} on Rd we shall write Di instead of D U i . Problem 8.

of m that t(c

FIG.6.

I t is easily seen that Dzixi(m) = aij (Kronecker delta), and, hence, {D5i(m)} is linearly independent, as can be seen by evaluating a linear combination on each of the functions xi in turn. Problem 9 . Give an example to show that DZi depends on xl, and not just on x i .

..., xd

9

1.3. Tangent Space Tangents are completely characterized by the following:

If (x, , ..., x,) is a coordinate system at m E M , t a tangent at m, then t = C (tx,) D,i(m).

Theorem 1.

For the proof, we assume the: Lemma.? If f € F [ R d a), , a = ( a l , ..., a,), then there are functions g, , ..., g, € F ( R da) , such that f = f ( a ) C (u, - a,) g, in a neighborhood of a [72; 94, p. 2211. Note. From this it follows that g,(a) = Dif(a).

+

+

Proof of Theorem 1. Let f € F ( I M , m), a coordinate system at m. Then by the lemma, there are g, such that if a = +(m), then giE F(Rd,a), f o 4-l = f 3 +-'(a) C (u, - a,) g, in a neighborhood of a, and g,(a) = D i ( f o +-')(a). Hence, f = f ( m ) C (xi-xi(m)) hi in a neighborhood of m, where h,(m) = D r i f ( m ) , hi E F ( M ,m). Therefore,

+

tf = 2 t ( x d U r n ) + 2 =

Corollary. (Proof:

2 t(xi) oz,(m)(f).

+

-

.,(m)) (4

QED

T h e dimension of Mrn is d, the dimension of M.

{Dzi(m))is a basis.)

We have already defined the tangent vector y * ( t ) for parameter value t of a C" curve y in M . We point out that every tangent at m is such a y*(t). For if xl, ..., x, is a coordinate system at m and s = C a,Dz,(m), then s is clearly the tangent to the curve given by: y ( t ) = that point whose coordinates are x,(m) ta, .

+

+

If : M + N is C", we define the dtfferential of 4, d+ : Mm -P N+(nL,, by: if t E M , , f € F ( N ,+(m)), then d + ( t ) ( f )= t ( f o 4). d+ is clearly a linear map. Problem 10. Prove that the following is an alternate definition of d+: for every C" curve y in M and parameter value t, d+(y*(t))= (+oy)*(t). t In the case of Ck manifolds (which we have not defined) the corresponding lemma is not true, since the gi's will not always be Ck.In fact, for Ckmanifolds the space of derivations at m is infinite in dimension, so the tangent space is defined to be the space spanned by {D,i(m)} [64A].

1. MANIFOLDS

10

+

If : M -t N , m E M , (xl , ..., x d ) is a coordinate system at m, (yl , . . . , y e )is a coordinate system at +(m), then the matrix of d+ with respect to the bases { D J m ) } and {Duj+(m))} is the Jacobian (Dzj(yio +)(m)).

lacobian matrices.

FIG. 1.

Chain rule.

If

+ : M - t N , i,/J : N-+ P are C", then d($ o +) = di,/Jo d+.

We can rephrase our previous definition of the tangent y * ( t ) to a C" curve y as: y,(t) = dy(D(t)). (D = D , = d/du.) Later (3.3) we shall give the collection of all tangents to M , T ( M ) ,a C" structure, so that y* will become a C" curve in T ( M ) . Tangent space of a product. If M , N are manifolds, then there is a natural isomorphism T between M x N,,,,) and M , N,, (direct sum). If p : M x N - t M and q : M x N --f N are the ) dp(t) dq(t). projections and t E M x N(,,,), then ~ ( t= Now let : M x N - + P be C". For (m,n) E M x N , define C" maps +,,,IN + P @:M-tP

+

+

+

by +,,(n) = # ~ ~ ( m =)+(m,n). Let s E M,, , t E N , as an element of ( M x N)(,,,,,, omitting T . Theorem 2.

d+(s

+ t)

= d+%(S)

+ d+m(t)*

T h e proof is left as an exercise. (Fig. 8.)

, and view

s

+t

1.3. Tangent Space

11

A dz..eomorphism of M onto N is a one-to-one map 4 ; M - P N such that 4 and 4-l are C". Existence of a diffeomorphism is the natural equivalence relation for manifolds. Difficult results of Milnor and others [44, 52, 53, 58, 841 show that this equivalence relation is not the same as topological equivalence, at least for manifolds of dimension greater than six. For manifolds of dimensions one, two, and three it is known that both equivalence relations are the same, as is true allegedly also in dimensions four through six according to unpublished work of Cerf.

FIG. 8.

It is easy to give examples to show that a C" homeomorphism need not be a diffeomorphism, that is, that the condition that 4-' be C" is independent. For any integer n > 1 the map : R -+ R is such an example, since the inverse does not have a derivative at 0. If we change our viewpoint and consider the C" homeomorphism as the identity map on an underlying topological manifold, we get examples of different C" structures on the same space, although the resulting manifolds may be diffeomorphic under another map. T h e example given then shows that if we take { P - l } as a basis for a C" structure on R we get different structures for different n's. (However, these manifolds are equivalent in the above sense.) Characterization of diffeomorphism. 4 : M -P N is a diffeomorphism if and only if 4 is a C" homeomorphism with range N and for every m E M , y5 a coordinate system at +(m),y5 o is a coordinate system at m [25, p. 751.

+

Inverse function theorem. Let (xl, at m E M , fl , ..., fd E F ( M ,m). Then

..., x d ) be a coordinate system 4 = ( fl , ..., fd) restricts to a

12

1 . MANIFOLDS

coordinate system at m if and only if det(DXjf i ( m ) ) # 0, that is, d# is nonsingular on M , [25, p. 701. Corollary. I f # : M -+ N , # E C", m E M such that d# : M , -+ N4(,, is one-to-one into, then there is a neighborhood of m in M which is mapped homeomorphically (into) under #. Moreover, if (yl , ..., ye) is a coordinate system at #(m), then a coordinate system at m may be chosen from restrictions of y1 o +, . . . , y eo # [25, p. 791.

If # : M -+ N , # E C", m E M such that d+ : M , -+ N4(,, is onto, then the image under # of every neighborhood of m in M is a neighborhood of #(m) in N . Moreover, if (yl , ...,ye) is a coordinate functions x , + ~ ,..., xd defined in a system at #(m), then there are neighborhood of m such that (yl o #, . . . , y eo #, x ~ , . ..., ~ , xd) is a coordinate system at m [25, p. SO]. Corollary.

c"

# : M -+ N , is a diffeomorphism of an open neighborhood of m onto an open neighborhood of #(m) if and only if d# is an isomorphism onto at m [25, p. SO]. Corollary. (Inverse function theorem for manifolds.) If

# E C", m E M , then #

Corollary. If # : M-+ N , 4 E C", d# = 0 everywhere, and M is connected, then # is constant [25, p. SO]. Problem 11.

Measure zero is a sensible notion on manifolds:

S C M has meczsure zero if for every coordinate map #, # ( S )C Rd has

measure zero. Prove Surd's theorem:

If # : N + M is C", then S = { m E M I m = #(n) for some n such that d# : N , -+ M , is not onto} has measure zero [79]. Problem 12. Prove the following generalization of the first corollary above. Let C be a compact subset of M , # : M -+N , # E C", such that # is one-to-one on C and for every m E C, d&, is one-to-one into. Then there is a neighborhood of C which is mapped homeomorphically under # [83,p . VI-451. Differentials of functions. Every element f of F ( M , m ) gives rise, via its differential, to an element of the dual space M,* of M , as follows: we may identify R,(nl)with R [ a D ( f ( m ) )is identified with a ] , and hence df : M m -+ R,(,,, e R. Note that under this identification, if

13

1.4. Vector Fields

t E M , , then
A vectorfield, X , is a function defined on a subset E of a manifold M which assigns at each point m E E an element X ( m ) of M,,, . If X is a vector field, f E F ( M , m),then Xf is the function defined on the intersection of the domains of X and f by: Xf(n) = X(n)(f). A vector field X is of class C" if its domain is open and for every m in the domain of X and ~ E F ( Mm), , Xf E F ( M ,m) also. Frequently we shall consider a C" vector field X to be the map on C" functions given by f -+ X f , since X is entirely determined this way by varying f . If (xl, ..., xd) is a coordinate system, then Dxi is a C" vector field. If X is a vector field with its domain contained in the coordinate system, we may write X = C f i D x , , where the fi are real-valued functions. P r o b l e m 13.

Prove that X

E

C" if and only if the f i

E

C".

Iff is a C" map of M into Re, so f = ( fi , ...,f e ) withfi real valued, and X a vector field on M , we write Xf for ( X f l , ..., Xfe). Similarly ~ for a tangent t. we define t f Re, It is clear that if X E C", then Xf E C". If X , Y are C" vector fields, then we define a C" vector field [ X , Y ] ,called the bracket of X and Y, on the intersection of their domains by [ X , Y ] = X Y - Y X . Multiplication of vector fields here is composition of their action on functions.

If X and Y are C" vector fields prove: (a) [ X , Y ] actually is a vector field. (b) X Y is not a vector field unless one of them is 0. (c) I f f and g are real-valued C" functions, then

Problem 14.

rfK gY1 = f g [ X

YI

+ f(Wy - d Y f ) x-

Coordinate expression for bracket. If X,Yare C" vector fields, ( x l , ..., xd) a coordinate system, X = C f i D , , Y = C g i D x z on the common

1. MANIFOLDS

14 part of the domains, then

[X, YI

=

2(fiD&

- g,D,,f,) DZj.

1.i

T h e bracket operation is bilinear with respect to real coefficients. = 0, or equivalently, [ X , Y] = -[Y, XI.

It is also skew-symmetric, that is, [ X , XI lacobi identity.

If X , Y, 2 are C" vector fields, then

"X,YII 21

+ "Y, 21, XI + "2, XI,YI = 0.

Another way of expressing this is to say that the map Y --t [ X , Y ] , the Lie derivative with respect to X , is a derivation of the algebra of C" vector fields, where multiplication in that algebra is bracket: [X, [Y,41 = " X , YI, 21 [Y, [ X Z l l .

+ If + : M - t

+

Vector fields and maps. N , E C", X , Y vector fields on M, N , respectively, then X , Y are related if for every m in the domain of X , d+(X(m))= Y(+(m)). If has the property that d+ is one-to-one at every point then is called regular.

+

+

+

4 is regular and Y is a C" vector field on N such that for every m E +-l(domain of Y ) , Y(+(m))E d4(MWJ,then there is a unique C" vector field X on M which is related to Y [25, p. 841. Problem 15. If

+

+

+

Note that in general, if : M -+N , E C", X a vector field on M , then d+ Xis not defined. Namely, if m, n E M a r e such that +(m) = +(n) but d+(X(m))# d+(X(n)),then d+ X is not single valued at +(m). Give an example for which d+ X is not defined when R, N = S', and 4 is regular.

Problem 16.

M

=

+

+

Problem 17. Brackets and maps. If : M + N , E C", X,, X, C" vector fields on M,Y,, Y, C" vector fields on N such that X i is related to Yi (i = 1, 2) , then [ X I , X,] is 9 related to [Y, , Y,] [25, p. 851.

+

Integral curves. If X is a C" vector field, then y is the integral curve of X starting at m if y(0) = m and for every t in the domain of y , y*(t) = X ( y ( t ) ) .T h e existence of integral curves and their essential uniqueness are immediate consequences of the corresponding theorems

15

1.4. Vector Fields

for solutions of systems of ordinary differential equations: at any point we merely take a coordinate system and transfer everything to an open set of Rd. By essential uniqueness we mean: if y and T are integral curves starting at m, then their restrictions to the common interval of definition are the same.

FIG. 9.

Problem 18. Let (xl,..., xd) be a coordinate system at m, y the integral curve of X starting at m, fi = xi o y, so that each fi is a real-valued C" function defined on an interval of the real axis containing 0. Show that the equation y * ( t ) = X ( y ( t ) ) is equivalent to an equation involving thefi and their derivatives, so that the problem of finding integral curves is equivalent to solving systems of ordinary differential equations as claimed.

Let X be a C" vector field. We associate with X a local one-parameter group of transformations T which for every m E M and real number t sufficiently close to 0 assigns the point T ( m , t ) = y ( t ) , where y is the integral curve of X starting at m. By theorems in differential equations on the dependence of solutions on initial conditions, for every m there is a positive number c and a neighborhood Uof m such that T is defined and C" on U x (-c, c). Since the real numbers used as the second variable of T a r e parameter values along a curve, they satisfy an additive property: if n E U , t , s, s t E (-c, c), then T(T(n, t ) , s) = T(n,s t). Conversely, if we are given a C" map having domain of the same type as T and satisfying the additive property, then again calling it T , we get a vector field having T as its local one-parameter group as follows: let the injection j,, be defined by j,(t) = (m, t). Then at m the value of the vector field is X ( m ) = ( T oj,,).+.(O).

Local one-parameter group.

+

+

16

1. MANIFOLDS

+

Problem 19. Let X = ulD, u 2 D , . Find explicit equations for T : R2 x R -+ R2. Do the same for Y = -u2D, ulD,.

+

Functions, differentials, vector fields, and other geometric objects on M are acted upon by transformations: functions by composition with the transformation, differentials by composition with the differential of the transformation, and vector fields by the

Lie derivatives.

FIG. 10.

action of the differential of the transformation. It should be noticed that functions and differentials are pulled back from the range to the domain of the transformation whereas vector fields are pushed forward from the domain to the range. Now if we consider the values of one of these geometric objects along y , the integral curve of X starting at m, then by using the transformations T , = T o 'j, , where 'jt(a)= (n, t ) , we can get a curve of values at m, that is either a real-valued function of t , or a curve in M,* or in M,. Since in each case the values are in a vector space we can differentiate. T h e derivative at 0 is then called the Lie derivative of the object with respect to X at m. In the case of a function f, since we are just differentiating the value o f f along y with respect to the parameter of y , we get the tangential derivative with respect to the curve, that is, we get X ( m ) f . In the case of a differential of a function, df, the curve in M,* is given by t 4df(y(t))o ( d T , ) , , where (dT,),, denotes the restriction of d T , to M , . I t is not difficult to show that the Lie derivative in this case is ( d ( X f ) ) , [24, p. 751. In the case of a vector field Y we must pull back the value at y ( t ) to m, which we do via d T - , , so the curve in M , is t-+ dT-,( Y ( y ( t ) ) .

1.4. Vector Fields

17

Since we shall return to this case in the chapter on bundles we state the result as a theorem; the result agrees with the terminology already introduced above under Jacobi identity. Theorem 3.

T h e Lie derivative of Y with respect to X is

[X, yl.

FIG.1 1 .

Proof.

We must show that for f E F ( M , m)

Noticing that d T - , ( Y ( y ( t ) ) ) f = Y ( y ( t ) ( f oT P t ) we , see that we must consider derivatives with respect to Y , and hence we introduce S, the one-parameter group associated with Y . Then if we define G : V -+ R, where V is a neighborhood of (0,O) in R2, by G(t,r )

= f ( T ( S ( y ( t )r,) , -

t))

it is immediate from the definition of S that

18

1. MANIFOLDS

Now letting H ( t , r , s) = f ( T ( S ( y ( t ) ,r ) , s)), it follows from the chain rule that D,D,G(O, 0) = D,D,H(O, 0, 0) - D D H(0, 0,O). Since H ( t , Y , 0) = f ( T ( S ( y ( t ) ,I ) , 0 ) ) = f ( S ( y ( t ) , r)),' It follows that D,H(t, 0,O) = Y f ( y ( t ) ) , and thus D,D,H(O, 0,O) = x ( m ) Y f . Since H(0, r , s) = f ( T ( S ( m , r ) , s)), it follows that D,H(O, r , 0) = X f ( S ( m ,r ) ) , and thus D,D,H(O, 0,O) = Y ( m ) X f . QED Let X , Y be C" vector fields both defined at m E M . We shall define a curve which has [ X , Y ] ( m ) as a limit of its tangents. Let g , be the integral curve of X starting at m. Then for sufficiently small positive c the remainder of the construction works. Let g , be the integral curve of Y starting at gl(c). Let g, be the integral curve of - X starting at g2(c), g , the integral curve of - Y starting at g3(c).Define a curve g by g(cz) = g4(c). Then [ X , Y ] ( m ) = limf+o+g*(t),that is, for every ~ E F ( Mm), , [ X , u l ( m )f = limf-0, g*(t) f. Theorem 4. Geometrical interpretation of bracket.

Define maps h , , h , , h, of a neighborhood of (0,O) in R2 into M by: h,(t, c) = gz(t), h2(t,c ) = g3(t), h3(t,c ) = g4(t). These maps are C" because they can be expressed as compositions of the one-parameter groups of X and Y ;they make explicit the dependence of g , ,g , ,g , on c. Also let h ( t ) = h3(t, t ) , so that g ( t z ) = h(t). Proof.

FIG.12.

First it will be shown that h,(O) = 0, and then according to problem 20, which follows, it will remain to prove that 2[X, Ylf(m)

= (foW O ) .

1.4. Vector Fields

19

T h e following facts are immediate from the definitions:

(f0 h)"(O) = Dl"f =

=

0 h3)(0)

+2DZW

0 h3)(0)

+ DZYf

@

hdO) KO)

=

(O,O>l

Y y m ) - 2 D 1 ( Y f o h,)(O) - 2 D 2 ( Y f 0 h,)(O) D,"(fo h,)(O) 2 D , D d f o h,)(O) D z 2 ( f ch,)(O) YZf(m) 2 X Y f ( m ) - 2D1(Y f 0 h,)(O) - 2D,(Yf 0 h,)(O)

+ + + + + X2f(m) 2 D d X f o M O ) - 2DZ(XfO hl)(O) + Dl"f hl)(O) + 2D,Dl(f hl)(O) + 4Yf hl)(O) -

0

0

+

0

Y2f(m) 2 X Y f ( m ) - 2Y2f(m) - 2 X Y f ( m ) X2f(m) - 2 Y A p ) - 2X2f(m) Y 2 f ( m ) 2 X Y f ( m ) X2f(m) = 2XYf(m) - 2YXf(m). QED

=

+

+

+

+

Problem 20. Let h be a C" curve such that h,(O) = 0, and define g on small positive numbers by g(P) = h(t), and let m = h(0). Show that: (1) T h e map defined on F ( M , m) by f + ( f c h)"(O) is a tangent at m. (2) T h e same tangent is 2 liml+o+g*(t),where the meaning of lim is the same as above. If we call this the 2nd order tangent to h at a point where the 1st order tangent vanishes, generalize (1) so as to define an nth order tangent to h at a point where the 1st order, 2nd order, ..., (n - 1)st order tangents vanish, and prove a result analogous to (2).

20

1. MANIFOLDS

Problem 21. If X is a C" vector field and X ( m ) = 0, show that the integral curve of X starting at m is the constant curve: y ( t ) = m for every t. Hence if y is a C" curve with y*(O) = 0, then y* does not have an extension to a C" vector field unless y is a constant curve. Problem 22. For X as in problem 19 and f = u p show explicitly that the Lie derivative of df with respect to X is d ( X f ) = df.

Use theorem 3 and also theorem 4 to show that [ X , Y ] = 0, where X and Y are as in problem 19. I t is a consequence of the following theorem that if [ X , Y ] = 0 in a neighborhood of m then the broken curves of theorem 4 are actually closed for sufficiently small t, that is, g ( t ) = m for all t near 0. Problem 23.

Let X i , i = 1, ..., e, be C" vector fields defined and linearly independent in a neighborhood of m E M , where M is a , = 0 in that neigh&dimensional manifold, and such that [ X iXj] borhood for every i , j . Then there is a coordinate system (x, , ..., xd) such that Xi coincides with Dz, in the coordinate neighborhood (i e ) and xi(m) = 0, i = 1, ..., d.

Theorem 5.

<

Proof. Choose a coordinate system ( y , , ..., y d ) such that yi(m) = 0, all i, and such that X , ( m ) , ..., X,(m), Dye+l(m),:,.,D,,(m) are a basis of M , . Define $ : U + M , where U is a neighborhood of 0 E Rd by the conditions

t -+ $(O, 0, ..., 0, t , a{+, , ..., ad) is the integral curve of Xistarting e. at $(O, 0, ..., 0, ai+,,..., ad), i T h e map $ can also be expressed as compositions of the oneparameter groups of X , , X,,..., X, and of the inverse of the y-coordinate map, so for this reason or directly from theorems on differential equations, we see that $ is C". From the definition it is immediate that d$(D,(O)) = Xi(m), i e, and d$(Di(0)) = D,,(m), i > e. By the inverse function theorem $-l is defined and C" in a neighborhood of m and thus defines a coordinate system (x, , ..., x d ) such that xi(m) = 0, all i. Because the integral curves of D , always correspond to integral curves of X , under 4, it is clear that DZ1= X , in the coordinate neighborhood. We complete the proof by showing by induction on i

<

<

1.5. Submanifolds

21

that Xix, = Sij , Since the integral curves of Di starting at points of the form (0, ..., 0, a , , ..., a d ) correspond to integral curves of Xi under +, Xixi = a,, at points of the form $(O, ..., 0, a , , ..., ad). But X,Xi = XiX,,so that for K < i, X k ( X i x j )= X,(X,x,) = 0 by the induction assumption. Thus Xp, is constant along the integral curves of X, , k < i. But every point in the coordinate neighborhood can be reached via chains of such integral curves starting at points of the form $(O, ..., 0, a, , ..., ad),so X,x, = Sij everywhere. QED Problem 24. Find the expression for in the above proof as compositions of the one-parameter groups of X,, X , , ..., X , and of the inverse of the y-coordinate map.

+

1.5 Submanifolds

Let M be a C" manifold. A manifold N is a submanifold of M if there is a one-to-one C" map i : N + M such that di is one-to-one at every point. We call i an imbedding and say that N is imbedded in M by i. Frequently we start with a subset of M , describe a C" structure for it, so that it becomes a submanifold under the inclusion map. For example, in this way an open submanifold of M becomes a submanifold of M . T h e d-sphere as a submanifold of Rd+l and Rd, imbedded as Rd x (0) in Rd+e,are examples where the dimension of the submanifold is less than that of the containing manifold. Problem 25. Compute that the inverse of stereographic projection Rd -+ Rd+l has one-to-one differential everywhere, and point out how this shows Sd is a submanifold of R d f 1 . Striped pants example. This example i : N -+M , although continuous, need

illustrates that the map not be a homeomorphism into, that is, the topology on i ( N ) induced as a subset of M need not be the same as the manifold topology of N . Let M be the twodimensional torus, N = R, imbedded as a dense one-parameter subgroup of M [that is, for x E R, i ( x ) = (exp icx, exp idx) E S1 x S1, where c/d is irrational]. Thus N is a submanifold of M , but i ( N ) is not even closed in M , and hence not locally compact. By altering the C" structure of an open submanifold we can get examples of subspaces (topological) which are not submanifolds, since the inclusion map need not be C", or, if it is C" it may not be regular at some points.

22

1. MANIFOLDS

Problem 26. Let t+b : N + M be C" with dim M < dim N , and let S be the set of measure zero defined in problem 11. Prove that if m 4 S then P = $-l(m) with the topology induced from N has a unique C" structure so that it becomes a submanifold of N under the inclusion map [88].

+

Problem 27. For $ = uZ3 u13 - 3u1u2 : R2 ---f R find what S is and show why +-l(m) is not a submanifold in the cases where m E S. When t+b is a real-valued function on R3 the submanifolds obtained as in problem 26 are the traditional surfaces in 3-space. Problem 28. Let f : P -+ M be C", i : N -+ M a submanifold, such that f ( P ) C i ( N ) . Let g = i-' o f : P 4N .

(a) If g is continuous show that g is C". (b) Find an example for which g is not continuous. Restrictions of functions. If i : N M , N a submanifold of M , f E F ( M ,m), m E i ( N ) , then f o i E F ( N ,i-l(m)) is the restriction o f f to N . Conversely, if g E F ( N ,n), then there is a neighborhood U of n in N and f E F ( M ,i(n)) such that f o i I u = g I u . However, this is --f

not true globally. For example, in the striped pants example the identity function u : R -+ R is not the restriction of any continuous real-valued function on the torus M .

If i : N -+ M , N a submanifold of M , X a C" vector field on M such that for all n E N , X ( i ( n ) )E di(N,), then by problem 15 there is a C" vector field Y on N such that di(Y(n)) = X ( i ( n ) ) . Y is the restriction of X to N . Note that by Restrictions of vector fields.

problem 17 the bracket of restrictions is the restriction of the bracket. Vector fields on N can be extended locally, that is, locally realized as restrictions, in the same way as functions. I n particular, if y is a C" curve in M with nonvanishing tangent y* , then locally y* can be extended to a vector field in M (cf. problem 21). 1.6 Distributions and Integrability

<

A p-dimensional distribution on a manifold M(p dim(M)) is a function 6' defined on M which assigns to each m E M a p-dimensional linear subspace O(m) of M , . A p-dimensional distribution 6' on M

23

1.6. Distributions and Integrability

is of class C" at m E M if there are C" vector fields X , , ..., X p defined in a neighborhood Uof m and such that for every n E U , X,(n), ..., X,(n) span 8(n). An integral manifold N of 8 is a submanifold of M such that di(N,) = 8(i(n))for every n E N . We say that a vector field X belongs to the distribution 0 and write X E 8, if for every m in the domain of X , X (m ) E 8(m). A distribution 8 is involutive if for all C" vector fields X , Y which belong to 8, we have [ X , Y ]E 8. A distribution 8 is integrable if for every m E M there is an integral manifold of 8 containing m. We sometimes write 8, for 8(m). That an integrable C" distribution is involutive is easily seen by problem 17 or by theorem 4. T h e latter also gives us an insight into the reason the converse is true. Going around a one-parameter family of rectangles having sides tangent to a distribution might give a curve which has a tangent not in the distribution, but in the case of an involutive distribution this cannot happen; thus when the distribution is involutive integral manifolds can be generated by going along the various integral curves of vector fields belonging to the distribution. Every one-dimensional C" distribution is both involutive and integrable, by the existence of integral curves. Problem 29. Let P,Q, R be C" functions on an open set U in R3 which do not all vanish simultaneously. Let O(m) be the linear subspace of Rm3which is orthogonal to (P(m),Q(m),R(m)).Show that 8 is involutive if and only if

p(D& - D$)

+Q ( 4 R

-

+ R(D$

- &Q) =

0.

An integral manifold is a solution of the equation

P dul + Q du, + R du,

=

0.

T h e following theorem, the converse mentioned above, is proved most easily by first getting the local form. T h e form stated here then follows by taking unions. T h e dual formulation in terms of differential forms is the classical theorem of Frobenius.

A C" involutive distribution 8 on M is integrable. Furthermore, through every m E M there passes a unique maximal connected integral manifold of 8 and every other connected integral manifold containing m is an open submanifold of this maximal one [25, p. 941.

Theorem 6.

1. MANIFOLDS

24

T h e local theorem gives more information as to how the integral manifolds are situated with respect to each other: If 6 is a C" involutive distribution on M , m E M , then there is a coordinate system (xl, ..., x d ) on a neighborhood of m, such that x,(m) = 0 and for every m' in the coordinate neighborhood the slice { p E M I x i @ ) = xi(m') for every i > e} ( e = dim 6) is an integral manifold of 6, when given the obvious manifold structure induced by the coordinate map. Theorem 7.

Outline of proof. It suffices to construct a coordinate system such that for every X belonging to 6, Xx, = 0 for i > e. Let Yl , ..., Ye be C" vector fields which span 6 in a neighborhood of m. Choose coordinates yi so that Ye = D V 1 .(This can be done by theorem 5 applied to one vector field.) Let Yi' = Y, - ( Y,yl) Ye, i < e, Ye' = Ye. Then since Y,'yl = 0, i < e, it follows that [Yi', Y,'] y1 = 0 for all i , j , and then that [Y,', Yj'] can be written as a linear combination (with real-valued C" functions as coefficients) of Yl', ..., Ye-l' . In particular, Y,', i < e, span an involutive ( e - 1)dimensional distribution. Repeating this process e - 2 more times gives vector fields Xl , ..., Xewhich span 6 and such that the first i, i e, have brackets expressible as a linear combination of the = Ye, Xe-l = Y,-,',.etc.). first i - 1 (X, Using the Xijust constructed we follow, step by step, the proof of theorem 5 except in the last paragraph, where we want to show less (Xixj = 0 only for j > e), and hence need less ( X k X i = X,Xk h e a r combination of the Xh with h < i).

<

+

Problem 30.

Let 6,,

..., 6 h

be complementary C" distributions on

M , that is, M , is the direct sum of the O,(m) for every m E M . Prove

that if all the 6, are integrable, then they are simultaneously integrable in the following sense: at every m E M there is a coordinate system (xl, ..., x d ) such that D x l , ..., Dxd1span 0 1 , ..., Dldh-l+l, ..., Dxdspan o h , where d, - d,-, = dim 6, (do = 0, dh = d ) . Problem 31.

Prove the following addition to problem 28:

(c) If N is an integral manifold of a distribution on M , then g is continuous.

CHAPTER 2

Lie Groups T h e basic definitions and theorems on Lie groups and Lie algebras are given, mostly without proof. In particular, the correspondence between subalgebras and subgroups is discussed, along with homomorphisms, the exponential map, and the adjoint representation [25, 27, 33, 42, 55, 721.

2.1 Lie Groups

A Lie group G is a set which is both a group and a manifold and such that the group operations are C", that is, the maps G x G -+ G

given by

(g, h) - t g h

G -+ G

given by

g +g-' are Cc.

Examples. Gl(d, R ) is a Lie group under the standard operations of multiplying matrices. T h e group operations are given by rational functions in the coordinate variables, so that they are not only C" but analytic. Since Gl(d, R ) may be considered as the group of nonsingular linear transformations on Rd, and every d-dimensional vector space V over R is isomorphic to Rd,the group of nonsingular linear transformations of V , G1(V ) , may be considered a Lie group isomorphic to Gl(d, R). Rd is a Lie group under addition. T he circle T 1= S1 is a Lie group under the usual multiplication. T e is then a Lie group by defining multiplication componentwise, and so also is Rd x Te. This is the most general Abelian Lie group. (See problem 13.) I n general, the product of Lie groups is a Lie group under componentwise multiplication, while a covering space of a connected 25

2. LIE GROUPS

26

Lie group admits a Lie group structure such that the covering map is C" and a homomorphism. In particular, every connected Lie group is locally isomorphic to a simply connected Lie group. A Lie subgroup of a Lie group is a subgroup which is also a Lie group and a submanifold.

2.2 Lie Algebras

A Lie algebra is a vector space L for which is given a bilinear function from L x L to L , called bracket, and denoted by [ , 3, which satisfies (1) [x, x ] = 0 for every x E L (2) the Jacobi identity: for every x, y , z E L

+

+

[x, [Y, 231 [x,[X,Y11 [Y, [z,211 = 0. A consequence of (1) is [x, y ] = - [ y , x] and this relation implies (1) if L is not over a field of characteristic 2. Examples

(1) If M is a C" manifold, the global C" vector fields form an infinite dimensional Lie algebra. (2) T h e set of all linear transformations of Rd, gI(d, R), is a Lie algebra under the bracket operation: [ A , B] = AB - BA. We use similar notation, gI( V ) , for the Lie algebra of linear transformations of a vector space V . (3) For any vector space V a bracket may be defined by setting all brackets equal to 0. In this way we obtain the Abelian Lie algebra on V . (4) If K and L are Lie algebras, then K @ L is a Lie algebra with the bracket [(x, Y ) , (XI, Y" = ([x, 4, [ Y , Y ' l ) . A subalgebra of a Lie algebra L is a subspace which is closed under the bracket operation of L. An ideal of a Lie algebra L is a subalgebra K such that for any x E L and y E K , [x, y ] E K . A homomorphism of one Lie algebra into another is a linear transformation which preserves brackets. An isomorphism of Lie algebras is a homomorphism which is one-to-one onto. T h e kernel of a homomorphism is an ideal. T h e image of a homomorphism is a subalgebra.

2.2. Lie Algebras

27

Let G be a Lie group. A left invariant vector Jield of G is a vector field which is fixed under the differentials of left translations, that is, if L , : G + G is defined by L,(h) = gh, then X is a left invariant vector field if dL,X(h) = X(gh) for every g , h E G. A left invariant vector field is globally defined and C". T h e sum of two left invariant vector fields, the product of a left invariant vector field by a real number, and the bracket of two left invariant vector fields are again left invariant vector fields. A left invariant vector field is uniquely determined by its value at the identity of the group (cf. [25], pp. 102 and 103). T h e Lie algebra of G, denoted by g, is the Lie algebra of left invariant vector fields. According to the above remark, we may identify the vector space of the Lie algebra of G with the tangent space at the identity, namely, X E g + X(e) E G,. T h e Lie algebra of Rd x T e can be identified with the Abelian Lie algebra on Rd+e. Locally isomorphic Lie groups have isomorphic Lie algebras. For example, R2 and T 2 both have R2 as Lie algebra, and similarly for any covering group of a Lie group. Also, the Lie algebra of the product of Lie groups is the direct sum of the Lie algebras. General linear group. If we consider Gl(d,R) as being the group of nonsingular linear transformations on Rd,then we get an isomorphism of the Lie algebra g of Gl(d, R ) and gr(d, R) as follows. Let { , ) be the usual inner product on Rd. For each v, w E Rd define a real-valued C" function f,,w on Gl(d, R) by

f,",,(T ) = (TZJ,w > ,

T E Gl(d,K).

Then the map J : g+gI(d, R), given by < J ( X )u, w) = X(e)fi.,w for every v, w , is the isomorphism. Equivalently, we may do as follows: every element v of Rd may be considered as a C" map Gl(d, R ) + Rd if we set v( 7') = T(v).Then for each X E g we get a linear transformation J ( X ) on Rd by defining J ( X )ZI = X(e)v. Then J is a Lie algebra isomorphism J : g + gI(d, R). From now on we make the identification J ( X ) = X , that is, we shall write X v for X ( e ) u . On the other hand, if we consider Gl(d, R ) as matrices, and g1(d, R ) as the space of all d x d matrices, then g may be identified with gr(d, R ) as follows: if { x i j } are the coordinate functions on Gl(d, R ) ,

28

2. LIE GROUPS

that is, xij(g)

=

ijth entry of g as a matrix, then for X

E

g,

Xij =

x(e>(xii>* Problem 1.

(b)

Show that: (a) xii o Lg = C,x,,(g)

dLg(DzhAl))

xpj.

xjk = xjh(g) sik.

(c) If Eii is the matrix with 1 in the i j position, 0 elsewhere, then the left invariant vector field Xij corresponding to Eij is

(d) Use this formula for Xij to verify directly that the brackets are preserved under the identification of g with gI(d, R ) [25]. 2.3 Lie Group-Lie

Algebra Correspondence

Theorem 1. Let G be a Lie group. Then there is a one-one correspondence between the connected Lie subgroups of G and the subalgebras of the Lie algebra of G.

Outline of proof. T h e correspondence is given as follows. If H is a Lie subgroup, then the left invariant vector fields on H can be extended uniquely to left invariant vector fields on G. T h e set of extensions form a subalgebra of the Lie algebra of G which is isomorphic to the Lie algebra of H . T h e existence of the subgroup corresponding to a subalgebra is established by considering maximal integral manifolds of the involutive distribution obtained from the subalgebra. T h e one containing the identity of G is easily seen to be an abstract subgroup of G. I t then follows trivially from problems 28 and 31 of Chapter 1 that it is a Lie subgroup (cf. [25], pp. 107-109). For every Lie algebra the one-dimensional subspace generated by a nonzero element is a subalgebra. Thus we have:

For every X E g, the Lie algebra of G, there is a onedimensional subgroup with Lie algebra generated by X ; that is, there is a curve y : R + G such that Corollary.

+

(1) r ( S t ) = Y ( 4 At) (2) X ( Y ( 4 ) = Y*(+

2.4. Homomorphisms

29

y ( R ) is called the one-parameter subgroup corresponding to X . It is the integral curve of X which passes through e. T h e group of all orthogonal transformations of Rd, O(d), is a Lie subgroup of GZ(d, R). Under the isomorphism J of 2.2 the Lie algebra of O(d) is identified with the subalgebra o(d) of gl(d, R ) consisting of all skew-symmetric transformations. T h e fact that O ( d ) is a Lie group follows from theorem 1 as applied to the component of the identity in O(d),that is, the rotation group R, or SO(d), as a subgroup of the component of the identity of GZ(d, R).

Orthogonal group.

Lemma 1.

Let J : G -+ G by J ( g ) = g-l. Then if t E G,

,

d ] ( t ) = -dR,-, o dL,-,(t).

(R,is right multiplication by g.) Proof. Let D : G -+ G x G by D(g) = (g,g),

/3 : G x

G + G by /3(g, h )

Then /3 o (1 x J ) o D Let t E G, , then

=

= gh.

constant, so d/? o d(1 x J ) o dD

=

0.

0 = dP o d(1 x ]) o d D ( t ) =

dP

0 (dl(t)

= dP(t =

Corollary.

+ dl(t))

dR,-,(t)

If X

+ dJ(t))

E g,

+ dL,(dJ(t))

(theorem 1.2)

QED

then d J X is right invariant.

There is a parallel formulation of the Lie algebra of a Lie group in terms of right invariant vector fields and d J gives the connection with our formulation. 2.4 Homomorphisms

A homomorphism of one Lie group into another is a mapping which is both a homomorphism of the underlying groups and a C" mapping of the underlying manifolds. We assume for the remainder of this section that our groups are connected.

30

2. LIE GROUPS

If j : G + H is a homomorphism and t is a tangent at the identity of G , then it is easily verified that the left invariant vector fields corresponding to t and dj(t), respectively, are j related. Thus j gives rise to a Lie algebra homomorphism dj : g -+ t,. T h e converse of this last result is not true, unless we replace the concept of homomorphism by that of “local homomorphism” in which case the correspondence is one-to-one. [25, p. 1121. However, if G is simply connected, a local homomorphism can be extended to a homomorphism, so we obtain: Let G and H be Lie groups with G simply connected. Then the correspondence j t)d j is one-to-one between homomorphisms of G into H and homomorphisms of g into b.

Theorem 2.

T h e kernel of a homomorphism is a closed normal subgroup, and the kernel of the corresponding Lie algebra homomorphism is an ideal, and it is easily seen that this ideal belongs to the subgroup. Conversely, if H is a closed normal subgroup of G, then the set of left cosets G / H can be given a natural manifold structure in such a way that the projection G -+ G / H is a homomorphism of Lie groups. More generally we have: If H is a Lie subgroup of G , a necessary and sufficient condition for H to be normal is that its Lie algebra f~ be an ideal in the Lie algebra g of G . I f , moreover, H i s closed, the Lie algebra of G / H is naturally isomorphic to the factor algebra g/t, [25, pp. 115 and 1241.

Theorem 3.

If G is simply connected and t, is an ideal of g, the Lie algebra of G , then by Ado’s theorem there is a Lie group having Lie algebra g / t , [42]. Then by theorem 2 there is a homomorphism of G corresponding to the projection g -+g/t,, from which it follows that the normal subgroup H belonging to t, is closed. Thus, if a Lie subgroup of a simply connected group is normal, then it is closed. It is known that a closed subgroup of a Lie group is a Lie group (Cartan’s criterion, [25, p. 1351).

2.5 Exponential Map

Let G be a Lie group, X E ~Let . y x be the integral curve of X starting at the identity. Then the exponential map Q + G is the map

31

2.5. Exponential Map which assigns yx(l) to X ; we write exp t + exp t X is just y x .

X = yx(l).

Clearly the map

FIG. 13.

Commutativity with homomorphisms.

then the diagram

I7

If j : G +H is a homomorphism,

dj

exp

exp

is commutative.

G-H

Proof. This follows immediately from the fact that y d j ( x ) = j o y x for any X E ~ . From this and the following theorem we see that the exponential map gives the correspondence between subalgebras of g and subgroups of G.

Theorem 4. T h e map exp is everywhere C" and in a neighborhood of 0 in g , it is a diffeomorphism.

32

2. LIE GROUPS

Proof. Assuming for the moment that exp is C" we first show that it is a diffeomorphism in a neighborhood of 0. I t is sufficient, according to the inverse function theorem, to prove that d exp is onto at 0. If s E G, , then there is a left invariant vector field X such that X(e) = s. By definition, exp takes the ray generated by X (that is, the curve in g given by r -+ r X ) into the integral curve of X through e . Hence d exp takes the tangent to this ray at 0 into the tangent to the integral curve of X at e, that is, into s. We blame the differentiability of exp on theorems in differential equations which say that solutions depend in a C" manner on parameters entering in a C" manner into the functions defining a system of differential equations (see appendix). I n this case we have the system of differential equations determined by a left invariant vector field X and X depends linearly on a system of linear coordinates of g. QED Let X , , ..., X, be a basis of g; then we identify g with Rd as a manifold by the correspondence CciXi tf (c, , ..., c,). I n a sufficiently small neighborhood U of 0 in g, exp is a diffeomorphism and hence can be taken as a coordinate map. T h e coordinates and coordinate neighborhood obtained in this way are designated as canonical.

For matrix groups there is another exponential map, which turns out to be the same as the one already defined if we identify the Lie algebra of the matrix group with the right subalgebra of the Lie algebra of all matrices of the given order; namely, if A is a d x d matrix we define

M a t r i x exponential.

" 1

eA = z g A k

(Ao= I ) .

0

This series can be shown to converge in the norm induced on the space of d x d matrices through identification with Rd2,and in fact, convergence is uniform on bounded sets. Furthermore, it is evident that if A B = BA, then eA+B= eAeD; in particular, e(s+t)A= esAeLA, s, t real numbers, and eAe-' = eo = I . Thus e A E Gl(d, R ) for every A, and the curve s -+ esAis a one-parameter subgroup; it is easy to see that the tangent at I of this one-parameter I n this subgroup is naturally identified with A = (d/du)(O)(eUA). way it is shown that the two exponential maps are essentially the same.

33

2.5. Exponential Map

The following commutative diagram for Gl(d, R ) with the identification indicated by " w'' will clarify relations.

G4d, R ) Problem 2. By showing that B = (-'/,'-:) has no square root, prove that exp does not map the Lie algebra of SZ(2, R)(= 2 x 2 real matrices with determinant 1) onto ,542, R). Orthogonal group. A canonical coordinate neighborhood of the identity in O(d) is obtained by taking the exponentials of skew-symmetric matrices lying in a sufficiently small neighborhood of 0. T h e coordinates for a skew-symmetric matrix may be taken as the elements of the matrix lying above the main diagonal. Thus the dimension of O(d) is +d(d - 1). For example, O(2) is the one-dimensional group having as neighborhood of the identity the matrices

and this correspondence is a diffeomorphism for 1 B 1

< 7.

Let A E gr(d, R), x E Rd, and define a C" curve u in (exp sA) x. Prove that u*(s) = A (exp sA) x, where tangents to Rd are identified with elements of Rd by the map C a,D,(y) -+ ( a , , ..., ad). Problem 3 .

Rd by

a(s) =

Let ( , ) be the usual inner product on Rd and let o l , be two C" curves in Rd. Then a C" function 5 is defined by U(S) = (ul(s), a,(s)). Using theorem 1.2 or otherwise, prove that Problem 4.

oz

o.'(s) = (01*(4

4)) +

(

4

)

1

u,*(s)>.

Problem 5. If A E gl(d, R ) is skew-symmetric, x E Rd, show that ((expsA) x, (exp $A)x) is a constant function of s, thus equal to (x, x), the value at s = 0; thus exp sA is orthogonal for every s.

Prove this directly using the map e above.

2. LIE GROUPS

34

Define a complex Lie group. Examples are Gl(d, C), and the complex torus P ( C ) = C d / D ,where D is a group generated by 2d real-linearly independent translations of Cd. Problem 6. Cd,

Problem 7. Show that a connected, compact, complex Lie group is Abelian (use a generalization of the maximum modulus theorem)

~31. Let U ( d ) = { g E Gl(d, C ) 1 ggl = I } . Show that U ( d ) is a compact Lie group (the unitary group) but not a complex Lie group. Problem 8.

2.6 Representations

A representation of a Lie group G is a homomorphism of G into a matrix group. A representation of a Lie algebra g is a Lie algebra homomorphism of g into a Lie algebra of matrices. If instead of matrices we use the group, GI(V ) ,[or Lie algebra gI( V ) ]of linear transformations of a vector space V , then we say we have a representation on V . I n neither case do we exclude matrices with complex entries or vector spaces over the complex numbers. A faithful representation is a representation which is an isomorphism into; if a group has a faithful representation it is thus isomorphic to a subgroup of a matrix group. Problem 9. (a) Show t h a t j : C-+ GZ(2,R ) , C = complex numbers, is a faithful representation of the Lie group by j(x i y ) = (;-:) C* of nonzero complex numbers with multiplication as operation. What is the subalgebra of g l ( 2 , R ) corresponding to j(C*) ? What are the one-parameter groups of C * ? (b) Show that the restriction of exp to the Lie algebra of j ( C * ) corresponds to the usual complex exponential function ex+ig = ex ( c o s y i sin y ) and that C with addition as its operation is the simply connected covering group of C* with e* as the covering map. (c) Construct a map 4 : C* -+ S1 x S1 such that &eis) = (cis, 1) and is a Lie group homomorphism which covers S1 x S’. (d) Taking 1, i as the basis of C as a vector space over R, show that j corresponds to “multiplication by,” that is, if $(a ib) = ( a , b ) E R2 then the matrixj(z) acts on x E R2 asj(z) x = $(z$-l(x)).

+

+

+

Problem 10.

Let Q*

=

nonzero quaternions. T h e regular left re-

2.6. Representations

35

presentation of Q* on Q is the representation q3 of Q* on the real vector space Q given by +(q) q’ = qq‘ (+(q) = “left multiplication by 4”).

(a) Show that with respect to basis 1, i, j , k (with the usual multixi y j zk, plication table) q3 has matrix form, for q = w

+ + +

(b) Show that +(q) is orthogonal if and only if IqI =

w2+xZ+y2+22=

1.

(c) Compute det +(q) by showing +(q)+ ( q ) I is a multiple of I and observing what the coefficient of w 4 is in det +(q). Automorphisms. An automorphism of a Lie group G is an isomorphism of G onto itself. The set of all automorphisms of G form a group A . For every j E A we have the automorphism dj of the Lie algebra g of G, and the diagram

is commutative. Since dj is a nonsingular linear transformation of g we have that the map j + dj takes A into the group of linear transformations of g, and it is evidently a homomorphism, since d ( j o k ) = dj o dk. Speaking loosely, we have a representation of A on g. In case G is connected this representation is faithful, a fact which follows from the remarks of 2.4. However, we prove this directly. Suppose j were in the kernel; this means that for every X E ~d j, X = X . However, since j (exp X ) = exp (djX ) , j must then leave the image of exp fixed, and since this contains a neighborhood of the identity, which generates G, j is the identity automorphism. T h e group A is actually a Lie group, but we do not show this here [25, pp. 137 and 1381.

36

2. LIE GROUPS

T h e set of inner automorphisms of G is a subgroup of A , that is, for every x E G we have the inner automorphism j , : y -+ xyx-l. Moreover, the map x - j , is a group homomorphism. The map Ad : G - + GZ(g) defined by Ad(x) = d j , is called the adjoint representation of G. Adjoint representation.

Proposition.

Ad is a representation of G on

g.

Proof. Ad is evidently a group homomorphism so that it suffices to show that it is C", and in fact, to show that it is C" in a canonical coordinate neighborhood. First we note that for a fixed y in G the map x -+j,(y) is C". In fact, it is the composition of maps involving group operations, which are obviously C": x + (x, x) --+ (x, xy-1)

--f

(x, yx-1)

xyx-1.

4

Now for y in a canonical coordinate neighborhood we have y = exp X, and by commutativity, j,(y) = exp(Ad(x) X ) . If we choose a basis X I , ..., X , of g, then Ad(x) is given in terms of a matrix (aii(x)): Ad(x) Xi = C aij(x) Xi. Now for y = exp( tXj) so that the canonical coordinates we get j,(y) = exp (C taij(x) Xi), of j,(y) are taij(x), i = 1, ..., d, this being defined for t sufficiently small. SincejJy) is C" in x this means that aij(x) is C" in x for all i, j , that is, Ad(x) is C". T h e center. T h e center of G is the set of all elements of G each of which commutes with every other element of G. It is clear that the kernel of the group homomorphism x + J , is just the center of G, and since j , -+ dj, is faithful if G is connected, the center is the kernel of the adjoint representation. As such it is a closed subgroup, and being obviously normal, we have that the adjoint induces a faithful representation of the factor group G/(center of G), whenever G is connected. As a corollary to theorem 3.1 we shall see that the differential of the adjoint representation is the adjoint representation of g, which is given by ad(X)Y = [X, yl,that is, d(Ad)(X) is the restriction of the Lie derivative with respect to X to g.

Problem 11. Show that Q* is the direct product of the positive real numbers and S3, giving a polar decomposition of Q* = R* x S 3 . Since R* is in the center of Q* the adjoint representation has R*

37

2.6. Representations

in its kernel and may be considered to have as representation space the tangent space to S3 at the identity 1. Identify that space with the subspace of Q spanned by i, j, k and show that the adjoint representation on S3is then given by $(q) P = qPq-l, P = xi yj zk. (For q E S3, q-l = 6.) Identify the rest of the center of Q* and the image of the adjoint representation on S3as a subgroup of GZ(3, R).

+ +

Problem 12. Let X I , X , , X , , X , be the left invariant vector fields on Q* which are equal to Di , i = 1, 2, 3, 4, at I = (1, 0, 0, 0). (Here we have identified Q with R4.) Show that

Xi

= UiDi

+

~ $ 2

+

~ $ 3

+u~D, 9

and find the corresponding formulas for X , , X , , X,. Problem 13. Show that an Abelian Lie group G has an Abelian Lie algebra (see problem 3.3) and hence by theorem 2, G M Rd/D, where D is a discrete subgroup. Therefore conclude that

G w Re x Tf, e

+ f = d [55, pp. 83-86)].

Problem 14. Show that a continuous homomorphism of a Lie group is C", and hence show that a Lie group structure is a topological group invariant [25, p. 1281. Problem 15. Prove that an integral curve of a left invariant vector field is also an integral curve of a right invariant vector field. Problem 16. Show that there is a neighborhood of the identity in a Lie group which contains no subgroups except the trivial one. Use this, problem 14, and the Peter-Weyl theorem [55, p. 991 to show that every compact Lie group has a faithful (C") representation. Remark on quotient spaces. I n 2.4 we considered quotients of Lie groups by closed normal subgroups. Now let H be a closed subgroup of G. T h e H has the induced topology and the left cosets G/H have a natural manifold structure such that the projection r : G -+ G/H is C", G acts as diffeomorphisms on G/H, and f: G/H -+ R is C" if and only if f o r is C".

CHAPTER 3

Fibre Bundles In the first section transformation groups are discussed and an interpretation of an important special case of the bracket operation is derived. T h e remainder of the chapter is devoted to principal and associated fibre bundles and reduction of the structural group, the treatment being from the point of view of transformation groups, although coordinate bundles are also defined [4, 55, 66, 8.51.

3.1 Transformation Groups

Let G be a Lie group and M a C" manifold. G acts (dzxerentiably) on M to the left if there is a C" map : G x M -+ M , and we write +(g, m ) = g m , satisfying the following conditions:

+

(a) For each g E G , the map g : M is a diffeomorphism. (b) For all g , h E G, m

-+

M , given by g ( m ) = gm,

M , (gh) m = g(hm). G is said to act effectively if g m = m, all m E M , implies that g = e = identity of G. G acts on M to the right if (b) is replaced by: (b)'

+: M

E

For all g , h E G, m E M , (gh) m x G + M in this case.

=

h(gm). We also shall write

Every Lie group acts on itself to the left by left translation and by inner automorphism, while it acts on itself to the right by right translation. If G acts on M , then to every m E M there corresponds a C" map, denoted by m also, of G into M defined by: m(g) = gm. 38

3.1. Transformation Groups

39

G acts transitively to the left if for every m, n E M , there is g E G such that g(m) = n. I n this case, fixing some m E M , let H = {gE G 1 g(m) = m},the isotropy group of m, then H is a closed subgroup of G and the map GIH (left cosets of H) --+M defined b y g H -+ g m is C“, one-to-one, onto. If GIH is compact, for example if G is compact, then this map is a homeomorphism. Example. Gl(d, R ) acts differentiably to the left on Rd and on Rd - (0). T h e action on Rd - (0) is transitive; the isotropy group g), where of (1, 0, ..., 0) consists of matrices of the form (i

B

E

GZ(d - 1, R), A

E Rd-l,

0 is a column of d - 1 zeros. T h e subgroup H may be identified as the semidirectproduct of Gl(d - 1, R ) and Rd-l, that is, the multiplication is given by ( B , A)(B’, A’) = (BB’, AB‘ + A’). [In general, if group G acts as homomorphisms to the right on a group H, then the semidirect product of G and H is given by defining the products as (g, h)(g’, h’) = (gg’, (hg’) h’).] Conversely, if H i s a closed subgroup of G, then G acts transitively on GIH byg(kH) = (gk)H. A space with a transitive group of operators is called homogeneous. If G acts on M , g the Lie algebra of G, then we define a Lie algebra homomorphism h of g into a Lie algebra of vector fields on M , denoted by g, as follows: if X E g, then (hX)(m) = dm(X(e)). We shall also write X = A X . Problem 1.

of

X is elx.

Prove that the one-parameter group of transformations

If G acts effectively, then h is one-to-one. G acts freely if the only element of G having a fixed point on M is the identity, that is, if for some g E G there is m E M such that g m = m, then g = e. If G acts freely, then the elements of are nonvanishing vector fields on M . Moreover, if N is the orbit of m, that is, N = {gm I g E G}, then for every t E N , there is a unique X E g such that 8(m) = t, since m : G 4 M is a diffeomorphism #J of G with N and hence we may take the XEg such that X(e) = d#J-’t. Problem 2 .

Prove that if X(m)

=

0 then etx(m) = m for every t.

3. FIBREBUNDLES

40

T h e differentials of the transformations making up a transformation group G, as they act on g, are given as follows: (a) If G acts to the left, then d g ( 8 ) = Adg X .

--

(h) If G acts to the right, then d g ( x ) = Adg-lX.

Proof of (a). First we compute the composition of g : M-+ M and g-lm : G -+ M : g c g-lm(h) = g(hg-lm) = ghg-lm = m(j,(h)). Thus (dg x ) ( m ) = dg(x(g-lm)) = dg o d(g-lm) X ( e ) = dm c dj,(X(e)) = dm(Adg X ) ( e ) )= A d g X(m).QED Let W be the linear space of C" vector fields on M , and let 9be a finite-dimensional subspace of W invariant under G, that is, for any g E G, d g ( 9 ) C 9. Hence, we have a representation? j of G in the that is, in GZ(9); group of nonsingular linear transformations of 9, and hence, if g I ( 9 ) , the algebra of all linear transformations of 2, is regarded as the Lie algebra of GZ(9), we have the commutative diagram

-1

G

-lexp i

GZ(Y)

T h e following result has many uses. Theorem 1.

particular,

For every X

[X,Y ]is in 9.

E g,

Y E9, dj(X) Y

=

-[x,yl. In

Momentarily assuming this, we state and prove the corollary on the differential of the adjoint representation: Corollary.

ad(X) Y

=

If X , Y ~ [X,Y].

g the ,

Lie algebra of a Lie group G, then

Proof. Let G act to the left on itself as follows: g E G, g : G + G is defined by g(h) = hg-l; that is, the transformation g is just R,-1, right multiplication by g-l. Since Ad g is dL, o dR,-1 , and X is t Here we are assuming the action is a left action. In the case of a right action, obvious modifications are necessary in the definitions, although the statement of the theorem is the same.

3.2. Principal Bundles

41

invariant under left translations dL, for every X E ~we, may apply the theorem with 9 = g and j = Ad. We have to determine, for X E g, the vector field on G. For m E G= M , we have that m : G+M as above is given by m(g) = g(m) = mg-l, so m =L, o J , where J is the inversing map. Thus x ( m ) = dL, o dJ(X(e)) = dL,(-X(e)) = - X ( m ) , by lemma 2.1. Therefore k = - X . Since dj = d(Ad) = ad, the theorem now gives the result. Problem 3 .

Prove as a corollary to this: if G is Abelian, then

g

is

Abelian. Proof of theorem. Assume the action is to the left. Let X Eg, so y : t+ exp t X is a one-parameter group of diffeomorphisms of M ,

so there is a vector field 2 on M which arises from differentiating functions along orbits. For a real-valued function f on M , and m E M we have d Zf(m)= ( O ) ( f @ 0 Y) = dm(r*(O))f,

so Z(m) = dm(X(e)) = X ( m ) . On the other hand, since the diagram above is commutative exp ( t d j ( X ) ) = exp (dj(tX))= jexp ( t X ) .

Thus, dj(X) Y is the derivative at 0 of the curve t -+ ( j exp ( t x ) ) Y = d(exp(tX)) Y in 2 '.But for m E M , (d exp tX(Y))(m) = d(exp(tX))(Y(exp ( - t X ) m)), that is, the curve in M , which we are differentiating is given by the values of Y along the curve t -+ exp ( t ( - X ) ) m pulled back to m by the action of the one-parameter group t + exp(t(-X)). Thus we are taking the Lie derivative with respect to -8 (see 1.4), so by theorem 1.3 the conclusion follows.

3.2 Principal Bundles A (Cm)principal fibre bundle is a set (P, G, M ) , where P, M are C" manifolds, G is a Lie group such that: (1) G acts freely (and differentiably) to the right on P, P x G -+ P. For g E G, we shall also write R, for the map g : P + P. (2) M is the quotient space of P by equivalence under G, and the projection 7~ : P+ M is C", so for m E M , G is simply transitive on r-l(m).

42

3. FIBREBUNDLES

(3) P i s locally trivial, that is, for any m E M , there is a neighborhood U of m and C" map F , : n-l( U )-+ G such that F , commutes with R, for every g E G and the map of n - I ( U ) + U x G given by p -+ ( ~ ( p F,(p)) ), is a diffeomorphism.

G

U

FIG.14.

P is called the bundle space, M the base space, and G the structural group. For m E M , d ( m ) is called the jibre over m. T h e fibres are diffeomorphic to G, in a special way via the map p : G +r - I ( n ( p ) ) C P, defined by p ( g ) = R,p. We note that in terms of the F , , the right action of G on P is given by right translation, that is, if p 4( ~ ( p , ) F,(p)), then pg + (n(p),F&) g ) . This follows from the fact that F d P d = F d P )g . We now give some examples of bundles. If G is a Lie group, M a manifold, then M x G provided with the right action of G on itself in the second factor, that is, ( m , g) h = ( m , gh), is the bundle space of a principal bundle, the trivial bundle. A bundle is isomorphic to a trivial bundle if and only if there is a C" cross section of n, that is, a C" map K : M - t P such that n o K is the identity on M (cf. [85], pp. 25 and 36). Trivial (product) bundle.

P

=

3.2. Principal Bundles

43

Bundle of bases. Let M be a C" manifold and B ( M ) the set of (df 1)tuples (m, el , ..., ed), where m E M and el , ..., ed is a basis of M,, and let r : B ( M ) + M be given by r ( m , e l , ..., ed) = rn. T h e n GZ(d, R ) acts to the right on B ( M ) by: let g E GZ(d, R), viewed as a matrix, g = (gij); let (m, el , ..., ed) E B ( M ) , and define R,(m, el, ..., ed) = ( m , C gilei , ..., C giaei). If m E M , (xl , ..., x d ) a coordinate system defined in a neighborhood U of m, then we define F , by: if m' E U , Fu(m',fl , ...,fa)= (dxj(fi)) = (gij) E GZ(d, R). Thus the functions yi = xi o 7-r and yii = xii o F , give a coordinate system on r - l ( U ) , where xii are the standard coordinates on Gl(d, R ) (see 2.2). Using the C" structure given to B ( M ) by the local product representation (IT, F,), we see that B ( M ) is the bundle space of a principal bundle, called the bundle of bases of M . It is sometimes convenient to view B ( M ) as the set of nonsingular linear transformations of Rd into the tangent spaces of M , that is, we identify p = (m, el, ..., ea) with the map p : ( r l , ..., r d ) -+c riei. When this is done it is natural to consider GZ(d, R ) as the nonsingular linear transformations of Rd, for we have: pg(rl, ..., r d ) =

that is, pg (as a map)

=p

(as a map) o g.

b = (m, e, , ..., ed) E B ( M ) is in the coordinate neighborhood r-l( U ) show that dr(D,$(b)) = Cj yii-l(b)ej , where (yii-l(b)) is the inverse of the matrix (yii(b)). Problem 4. If

Homogeneous spaces. If G is a Lie group, H a closed subgroup, then there is a principal bundle with base space G / H (left cosets), bundle space G, and structure group H such that IT : G + G / H is the canonical map and the right action is given by (g, h) -+gh (see [85], p. 33).

(1) Let R - (0) = R* act on Rd+l- (0) by scalar multiplication. Then this action is differentiable, free, and simply transitive on orbits. T h e orbit space is Pd, d-dimensional projective space, so is a principal bundle. ( R d f l- {0}, R*, P d ) Examples

3. FIBREBUNDLES

44

( 2 ) If we use the positive reals R+ instead of R* we get (Rd+l - { 0 } , R+,S d )

is a principal bundle.

(3) If we use C* and Cd+l - {0), C = complex numbers, then we get complex projective space CPd as base space: (Cd++'- {0}, C*, C P d )

is a principal bundle.

Problem 5. Let (P, G, M ) be a principal fibre bundle with P connected. Then if Go is the component of the identity in G, there is a unique principal fibre bundle (P, G o ,h?) such that the action of Go on P is the same and h? is a connected covering space of M (cf. examples 1 and 2 above). We give an alternate approach to principal fibre bundles. Principal Coordinate bundles. Let (P,G, M ) be a principal bundle, and be an open covering of M such that wl( U i ) can be represented let { Ui} ( i )-+ G. For i, j such that as a product space via the function Fi: ~ - l U Ui n U j # 0, we define a map Gji : U, n Uj-+ G as follows: if m E Ui n U j , let p E n-l(m), and put Gji(m) = Fj($)(Fi(p))-l. Gji measures how much the cross section over Uicorresponding to U, x {e} under the product structure given by Fi differs from the cross section over U j determined similarly by F j . We want to show that the definition of G j i ( m )is independent of the choice of p . If p' E .rr-l(m) also, then since r - l ( m ) is the orbit of p , there is g E G such that p'=pg, so we have

F,(P')FZ(P')F1= P,F;g) F,(Pg)-l = FAP) F i ( P Y ,

= F,(P) g(Fi(P)g)-l

as desired. T h e functions satisfy the further properties: G,,(m)

=

G,,(m) Gji(m)

for

mE

U in U j n U,.

(*I

T h e functions Gji are called the transition functions corresponding to the covering { Ui},and in fact, with this covering they define a principal coordinate bundle in the sense of Steenrod. Hence, our definition of bundle is an equivalence class of coordinate bundles, which is another way of saying that principal coordinate bundles are equivalent if and only if their right actions agree. Therefore, we have, by [85, p. 141, that any set of functions Gii defined for a covering { U i } ,

3.3. Associated Bundles

45

satisfying (*), uniquely determine a principal bundle whose transition functions relative to the covering { Ui}are the Gji. Problem 6 . If 4 : N - t M is a C" map, ( P , G, M ) a principal fibre bundle, then let P = {(n, p ) E N x P 1 +(n) = ~ ( p ) } . (a) Show that 1? is a submanifold of N x P under the inclusion map. (b) Show that ( P , G, N ) becomes a principal fibre bundle if we define right action by (n, p ) g = (n, p g ) . ( P , G, N ) is called the bundle induced by and (P, G, M ) .

P N

P

M

(c) Show that transition functions for (P, G, N ) may be taken as G,, o 4, so that we could define the coordinate bundle directly. The vector fields of 8. If (P, G, M ) is a principal bundle, then since G acts freely and effectively we have an isomorphism A : g + 8 = Lie algebra of vector fields on P, which consists of nonvanishing vector fields. By remark (6) in 3.1 we have dR,(h(X)) = h(Ad(g-l) X ) , where g E G, X E g.

For the real coordinates u l , ..., uZd+, on Cdfl - {O} ({z, = uzjPl iuZj}is the dual basis to the standard complex basis of Cd+') find the expression for h(aXl + bX,), where X , and X , are the left invariant vector fields on C*: X, = ulDl u2D2, X , = -u,D, ulD, and h is the isomorphism associated with (Cd+l - {O}, c*,CPd). Problem 7.

+

+

+

3.3 Associated Bundles

Let (P,G, M ) be a principal fibre bundle, and let F be a manifold on which G acts to the left, We define the jibre bundle associated to

46

3. FIBRE BUNDLES

(P, G, M ) with Jibre F (it also depends on the action) as follows. Let B’ = P x F , and consider the right action of G on B’ defined by ( p , f )g = (pg, g-lj), where p E P, f~ F, g E G. Let B = B‘/G, the quotient space under equivalence by G, then B is the bundle space of the associated fibre bundle. We have the following structure. . T h e projection 7r’: B -+ M is defined by: ~ ’ ( ( pf, ) G) = ~ ( p ) If m E M , take U a neighborhood of m as in 3.2(3), with F , : T-’( U )-+ G. Then we have F,’: r’-l(U )+F given by

so that 7 r t - l ( U ) is homeomorphic to the product U x F , and hence we define B as a manifold by requiring these homeomorphisms to be diffeomorphisms. Note that now 7r’ E C“, and also the natural projection B’ -+ B is C“. Associated coordinate bundle. If we define transition functions Gji‘ for the associated bundle ( B , F , G, M ) analogously to the definition in the principal bundle case, we have, for a covering { Ui} admitting functions Fit, for m E Ui n U j , (p,f)G E 7rt-l(m):

Gji’(m) = F j ( ~ ) ( F i ( p ) ) -= l Gji(m).

We therefore have that ( B , F , G, M ) is the equivalence class of the coordinate bundle associated to the principal coordinate bundle defined by the transition functions Gji, in the sense of Steenrod [85, Part I, $91. Examples

(1) Let (P, G, M ) be as above, and let G act on itself by left translation. Then (P, G, M ) is the bundle associated to itself with fibre G.

(2) Tangent bundle. Consider the bundle of bases B (M). (Note that we shall often denote a bundle by its bundle space alone.) Now by definition GZ(d, R ) is the group of nonsingular linear transformations of Rd, and hence acts on Rd to the left. The bundle space of the associated bundle with fibre Rd is denoted by T ( M ) , and the bundle is called the tangent bundle to M . T ( M ) can be identified with the space of all pairs ( m , t ) , where m E M , t E M , [the “m” in the pair

3.3. Associated Bundles

+

47

is actually redundant, as it is in the ( d 1)-tuples making up B ( M ) , but is inserted for convenience], as follows: ( ( m ,el

7 *'*I

4,( y 1 ... 1

1

Td))

G W ,R)

-

(m,

z

yiez),

or, if we regard B ( M ) as the set of maps p : Rd +M m ,then for x E Rd this identification is ( p , x ) GZ(d, R ) --t (m,p x ) , where m = ~ ( p ) . With this later formulation it is easy to see that the identification is well defined, for if (p', x') GZ(d, R ) = ( p , x) GZ(d, R), then there is g E Gl(d, R ) such that p' = pg, x' = g-lx, and hence, p'x' = (pg) (g-lx) = p x , since as a map, pg is the composition of p and g. Hence we may view the fibre of T ( M ) above m E M as the linear space of tangents at m, that is, as M m , and T ( M ) itself as the union of all the tangent spaces supplied with a manifold structure. Furthermore, under this identification, the coordinates of T ( M ) may be easily exhibited; namely, let U be a coordinate neighborhood in M , with coordinates x l , ..., xd. We define coordinates y l , ..., Y2d on rr'-l( U ) as follows: if (m, t ) E t-r-l( U ) ,

Y d S h

t ) = d%(t)

A C" vector field may then be regarded as a cross section of T ' . I n particular, the trivial vector field (values all 0) gives an imbedding of M as a submanifold of T ( M ) . Problem 8. Prove that if y is a C" curve in M , then y* is a C" curve in T ( M ) . (3) Tensor bundles. When we replace Rd in example ( 2 ) by a vector space constructed from Rd via multilinear algebra, that is, the tensor product of Rd and its dual with various multiplicities or an invariant subspace thereof, we get a tensor bundle. A cross section which is C" on an open set is called a C" tensor field, and is given type numbers according to the number of times Rd and its dual occur. T h e group of a tensor bundle is Gl(d, R ) ; it acts on the factors of the tensor product independently, and on Rd as with the tangent bundle, on the dual via the transpose of the inverse: if v E the dual, X E Rd, g E GZ(d, R), then g v ( x ) = v(g-'x). Frequently the bundle B ( M ) has its structural group reduced to a subgroup (see 3.4) in which case more subspaces of tensor products of Rd and its dual may become invariant, leading to tensor bundles

48

3. FIBREBUNDLES

of different sorts. For example, this is the case when M has an almost complex structure (cf. problem 11). (4) Vector bundles. These are bundles in which the fibre is a vector space and the structural group is a subgroup of the general linear group of that vector space. T h e tensor bundles are a special case. They are frequently defined with no explicit mention made of the structural group by giving the bundle space as the union of vector spaces, all of the same dimension, each associated to an element of the base space, and defining the manifold structure via smooth, linearly independent, spanning cross sections over a covering system of coordinate neighborhoods. For example, we essentially did this for T ( M ) when we exhibited the coordinates: in that case the cross sections were the coordinate vector fields D,*. Another example of this type is the quotient space bundle of an imbedding, which usually is considered in the case of Riemannian manifolds as the normal bundle (the Riemannian metric, defined in Chapter 7, is employed to get a uniform choice of representatives for the quotient spaces). This vector bundle may be defined as follows: Let i : N+ M be the imbedding of the submanifold N in M . T h e fibre over n E N is the quotient space Mi(Jdi(Nn), and the bundle space is the union of these fibres, so we may consider the bundle space di(N%)), where t E Mi(%). T o get as the collection of pairs (n, t coordinates we first take a coordinate system in M at i(n), say xl, ..., xd, and we may assume that xj o i = yi, j = 1, ..., d‘ give a coordinate system at n, and that Xj(n) = D,.i(n)) di(Nn) are 1, ..., d. Then for some neighborlinearly independent for j = d‘ 1, ..., d are still linearly independent, hood U of n, Xj(n’), j = d‘ so there is a dual basis Vi(n’). Then for (n‘, X ) E ~’-l( U ) , where n’ is the projection from the bundle space to N , we define coordinates z1, ..., zd by:

+

+

+

+

zj(n’, X ) = yj(n’)

if

j = 1,

zj(n’, X)= Vj(n’)(X)

if

j = d’

..., d‘

+ 1, ..., d.

T h e group of this bundle may be taken to be Gl(d - d‘, R). ( 5 ) Grassmann bundles. T h e set of e-dimensional subspaces of Rd can be given a manifold structure so that Gl(d, R ) acts in the obvious manner as a differentiable transformation group on the left. T h e bundles associated to B ( M ) by this action are called the (unoriented)

3.4. Reduction of the Structural Group

49

Grassmann bundles of M ; the bundle space may be regarded as the collection of e-planes in the tangent spaces of M . A C" e-dimensional distribution is a C" cross section of this bundle. The action of a principal bundle on an associated fibre. Let (P, G, M ) be a principal bundle and B an associated bundle with fibre F. Then the quotient projection P x F + B defines, by restriction of the first variable, for every p E P a C" map p :F -+ B, namely, p(f) = (p, f ) G. I t satisfies p(gf) = (pg) f for every g E G. We have already seen this in the case of B ( M ) and T ( M ) (cf. [85], Part I, $8.9). Remark. An associated bundle is trivial if its principal bundle is trivial. This is not equivalent to the existence of a cross section of the associated bundle, but implies the existence of a family of cross sections with pairwise disjoint ranges which fill the associated bundle. Problem 9 . Verify the above, and also show that for a vector bundle with fibre Re triviality is equivalent to the existence of e cross sections linearly independent at each point. It is well known [85] that the tangent bundle to a differentiable manifold admits a nonzero cross section if and only if the Euler characteristic is zero; for example, if the manifold is compact and odd-dimensional. Hence all odd-dimensional spheres have such cross sections, although it is a deep result of Milnor and Kervaire [I41 that only S1,S3,S7 have trivial tangent bundles (i. e., are parallelizable).

3.4 Reduction of the Structural Group

Let (P,G, M ) be a principal bundle. We assume G is separable. Let H be a subgroup of G, then in the sense of Steenrod, the structural group G is reducible to the subgroup H if and only if there exists a coordinate bundle in the equivalence class determined by ( P , G, M ) whose transition functions take their values in H , that is, if and only if there exists a covering { Ui}whose Gji satisfy Gji(Uin U j )C H . In terms of the right action P x G - t P, this definition can be formulated in the following way ([66], p. 20). Let (P, G, M ) , (P', G', M ' ) be principal bundles. A bundle map f : ( P , G, M ) -+(P', G', M ' ) is a set of C" maps (fp,fc,fM) between the obvious pairs, such that fc is a homomorphism, and the following

3. FIBREBUNDLES

50 relations are satisfied:

fiMcx=7r'0fp

f p o R,

=

RfG(g, of p

for every

g E G.

Now we may state the second definition as a theorem. If (P, G, M ) is a principal bundle, H a subgroup of G, then G is reducible to H if and only if there exists a principal bundle (P', H , M ) which admits a bundle map f : (P',H , M ) +(P, G , M ) such that f M is the identity on M , f p is one-to-one, and f G is the inclusion map H C G (Proof omitted.) Theorem 2.

Steenrod proves [85, p. 591 that if (P, G, M ) is a principal bundle, H a maximal compact subgroup of G, then G can be reduced to a

bundle with structural group N.In particular, every principal bundle with Gl(d,R ) as structural group [for example, B(M)]can be reduced to a bundle with structural group the orthogonal group O(d). We shall return to this when we consider Riemannian metrics on a manifold. T h e reduction of a principal bundle induces the reduction of associated bundles, in an obvious sense, since we have given the definition in terms of transition functions only. Problem 10.

Complex manifolds. Let M be a complex manifold,

9 = complex valued differentiable functions defined on a neighga= conborhood of m E M , F a = holomorphic functions in F, jugates of functions in flu (conjugate holomorphic functions),

F T= real valued functions in F, .Ym= complex linear derivations d mcomplex linear extensions of Mm , gm= the annihilator of 9, of F a in Tm. For t E F m , f E 9 , define if = if. Show that

+

(direct sum). (a) Y m= d m i d m (b) f E TwL for every t E Ym. (c) d m = all t E Y m such that tFTC R. (d) Sm= the annihilator of Fa. (e) t = f if and only if t E Am.

(f) Z mn iFm = 0. (g)

y m = *m

+ 2,.

3.4. Reduction of the Structural Group

51

(h) If t E d m, then the decomposition of t from (g) is of the form t = h h, where h E Z v l ,i.e., d m = {h I h €Zm}. This defines a one-to-one real linear map P : d m +Z m , t --t h.

+

+

Let j : Y m -+ Y mbe multiplication by i. Define J = P-l jP. (i) J z = -identity.

(j) Compute J in terms of real coordinate vector fields which come from the real and imaginary parts of a complex coordinate system.

(k) J is defined on T ( M ) and is a bundle map. An almost complex structure on a manifold M is a bundle map

J : T ( M ) + T ( M ) such that

(1) J ( M m ) = M , for every m E M . (2) J z

=

-identity

on each Mm.

T h e J obtained in the above problem is called the complex structure of the complex manifold M . An almost complex structure will be called a complex structure only if it is obtained in this way. Problem 11. (a) If M has an almost complex structure, then M has even dimension.

(b) M has an almost complex structure if and only if the group of the bundle of bases can be reduced to Gl(d/2, C) represented in (which corresponds to Gl(d, R) as matrices of the form (g ):( A iB)E Gl(d/2, C)). Every 2-dimensional orientable differentiable manifold admits a complex structure, so, of course, every 2- dimensional manifold which admits an almost complex structure admits a complex structure. T h e latter result is not true for higher dimensional manifolds [21]. If M has an almost complex structure then Mm iMnlhas a direct iJx for sum decomposition Z m Slnsuch that H m= all x x E Mm. We say that C" vector field X belongs to Z if X ( m ) E Z m for every m, and then write X E 2. I t is not in general true that if X E 8, Y E H then [X, Y ]E 8. However, if J is a complex structure the X v l agrees with our previous definition and [Z,Z]c 2 ;in fact, this latter condition has been shown to be also a sufficient condition that J be a complex structure [64].

+

+

=

+

T h e maximal compact subgroups of R* and C* are So and {eie} = S1,respectively. Show that the reduction

Problem 12.

(1, -1}

+

52

3. FIBREBUNDLES

of the principal bundles of examples (1) and (3) in 3.2 to these subgroups gives principal bundles: ( S d ,So,P d ) and (SZd+l, S1,CP). Carry out the same construction to get principal bundles over quaternionic projective spaces QPd: (S4d+3, S3,QPd). In the case d = 1 the bundle projections become the Hopf maps S3 -+ S 2 = CP1, S7+ S4 = QP'.

CHAPTER 4

Differential Forms Differential forms are defined via Grassmann algebras, and the intrinsic formula for the exterior derivative is derived. Frobenius' theorem, vector-valued forms, and forms on complex manifolds are also discussed [24,25,29,33,36,66,831. For other topics, particularly the use of differential forms in the study of topological invariants, the reader is referred to [12, 30, 781. 4.1 Introduction

In the last chapter the concept of a tensor was briefly mentioned, and differential forms, the subject of the present chapter are just special types of tensors. However, we shall initially introduce differential forms here by means of the more explicit formulation in terms of Grassmann algebras and shall then return briefly to the tensorial approach (4.5). I f f is a C" function on M , then we notice that to every m E M there corresponds the differential of f at m, 1.3, which is a linear functional on M , , and this correspondence is smooth in the following sense. Let X be a C" vector field on M , then df(X)(m)= df,X(m) =Xf(m) defines a C" function Xf on M . Such a smooth assignment of linear functions is called a differential 1-form, although not every differential I-form arises as the differential of a C" function. However, before pursuing this subject further, we must develop some machinery, namely, Grassmann algebras. 4.2 Classical Notion of Differential Form

A differential form at m is something which in terms of a coordinate system can be expressed as C ail,,,i, dxi, dxiy, where the summation 53

4. DIFFERENTIAL FORMS

54

is over all ordered subsets (il, ..., i,) of (1, ..., d } , and the ail,,,i, are real numbers. So a strict definition will involve some kind of multiplication of differentials and then linear combination of these products. Thus M,* will be imbedded in an algebraic system which has both multiplication and vector operations. Furthermore, this shall be done so that if (yl, ...,y d )is a second coordinate system at m then the expression C bil.,.ipdyi;.. dyiD,obtained from the other by the usual rules for change of variable in multiple integrals, will be just the expansion of the same algebraic object in terms of the new basis. T h e required algebraic system for this is a Grassmann algebra, which we now discuss.

4.3 Grassmann Algebras

Let F be a field and V a finite-dimensional vector space over F of dimension d . A Grassmann algebra over V is a set G such that (i) G is an associative algebra with identity e over F. (ii) G contains V . (iii) Every element v E V satisfies v2 = 0. (iv) G has dimension 2d. (v) G is generated by e and V , that is, every element of G is a sum of products of scalar multiples of e and elements of V . Notice that e is not in V because e2 = e # 0, while by (iii) v2 = 0 for all v E V . Also, if u, v E V , then uv = -vu. This is shown by the standard polarization trick: 0

=

(u

+ V)Z = u2 + + vu + v2 = uv + vu. 11v

Property (iv) is a short but poor way of stating that there are no more relations among the elements of G than those which follow from (iii). T o each basis e l , ..., ed of V there corresponds a basis of G . T h e elements of this basis of G are in one-to-one correspondence with the subsets of {I, ..., d } as follows: (a) If the subset is+ = empty set, we let e4 = e . (b) If the subset is s = {il , ..., i,} with i, e, = e, 1 ..- eig*

<

< ip,

let

4.3. Grassmann Algebras

55

Note that there are exactly 2d subsets, so that in order to show that the e,'s are a basis it is enough to prove the: T h e e,'s span G .

Lemma 1.

Proof.

By (v) above we have for any g g

E

G

= a,eb

+ 2 (products of elements of V )

= a,eC

+ 2 (coefficients in F ) (products of e,'s).

But by using the fact that elements of V anticommute we can write any product of e,'s in increasing order and then by (iii) all except the e,'s are 0. An element g E G is homogeneous of degree p if it can be written as the sum of products of exactly p elements of V , or, equivalently, g = CssPases, where a , E F , P = subsets of (1, ..., d } which have exactly p elements. T h e set of all such g E G form a linear subspace Gp of G which is of Clearly, GpGQC GP+'J. Moreover, by anticommutativity dimension it is easily seen that if g E GP, h E G'Jthen g h = (- 1)pqhg. An element g of G which is homogeneous of degree p is decomposible if there exist v l , ..., v p E V such that g = vl ... v p . Otherwise g is called indecomnposible.

(g).

Problem 1. If dim V 6 3, then every homogeneous element is decomposible. If dim V > 3, then if o l ,v 2 , v 3 , v4 are linearly independent, v1v2 v3v4 is indecomposible. I n general, the decomposible elements form an (nonlinear) algebraic variety in GP.

+

Problem 2 . Prove that for g E G2 there exist linearly independent vl , ..., vZkE V such that g = vlo2 v3v4 ... v2k-lv2k;that if the characteristic of F is not a prime 6 i d , then k is the largest integer such that g k # 0, and in this case g k is decomposible. We say that g has rank 2k.

+

+ +

Problem 3 . If dim V = 4 and the characteristic of F is not 2 then g is decomposible if and only if g 2 = 0 and g is homogeneous. I n

general, if dim V

=4

and g

E

Gd-l , then g is decomposible.

Problem 4. Let x # 0, x E V ,g there is h E G such that g = xh.

E

G . Prove xg = 0 if and only if

56

4. DIFFERENTIAL FORMS

(1) If vl , ..., vupE V , then ul ... up # 0 if and only if , ..., v p are linearly independent. Proof. If u1 , ..., up are linearly independent, then we may take them as part of a basis for V ; so that u1 ... up = e , # 0, s = (1, ..., p } . If ul , ..., up are linearly dependent, say, vl = C:=, u p + , then by the

Remarks.

ul

distributive law and anticommutativity we have

(2) Any two Grassmann algebras over V are essentially the same, that is, if G and H are two such then there is an isomorphism of G onto H which leaves each element of V fixed. In fact we need only choose a basis of V and map the e,’s for G into the corresponding ones for H . (3) If T is a linear transformation of a vector space V into a vector space W , and G ( V ) , G ( W ) are the Grassmann algebras over V , W , respectively, then there is a unique extension of T to a homomorphism of G ( V ) into G ( W ) , denoted by Th , and Th satisfies T,(G(V)P)C G( W)p. Proof. Uniqueness follows immediately from the facts that T(e) must equal e, and that G( V ) is generated by e and V . By linearity, Th can be extended to all of G( V ) after defining T,(q . * * v p )= T(v,) ... T(v,) for v1 , ..., v p E V ,

and such a Th is clearly a homomorphism. Problem 5. If T : V - t W , S : W-+ X are linear transformations of vector spaces, show that ( S T ) , = S,T,. Problem 6 . If T : V -+ V is a linear transformation, then it admits a unique extension to a derivation of G( V ) into G( V ) ,that is, a map

T, such that if g, h E G(V ) then T,(gh)

=

T,(g) h

+ gT,(h).

Problem 7. If T , S : V -+ V , show that [S, , T,] is a derivation and that [ S , TId = [ S , , T J , that is, ( S T ) , - ( T S ) , = S,T, - TdSd. Give an example to show that S,T, is not always a derivation. If V is a d-dimensional vector space and G its Grassmann algebra, then Gd is one dimensional. A linear transformation of Gd into itself is simply scalar multiplication by an element of F. If S is the transformation then we denote this scalar by k ( S ) .

4.4.Existence of Grassmann Algebras

57

Now let T be a linear transformation of V into itself. By (3) and problem 6 there are extensions Th and Td of T to G which are a homomorphism and a derivation, respectively. In particular, Th and Td are linear transformations of Gd. We define determinant of T

=

det T

trace of T

=

tr T

=

=

k(T,),

k(T,).

I t is easily checked that the mapping T -+ det T of gI( V ) into F is a homomorphism on GI(V )c g1( V ) , and that T E GI(V ) if and only if det T # 0. It is also easy to show that this definition of determinant coincides with the more usual ones.

+

+

Show (a) tr(S T ) = tr(S) tr(T) (b) tr(ST) = tr(TS). (Use problem 7 to do this without using coordinates.) Problem 8.

Problem 9. Let f be in the dual space of gI(V) and satisfy: (a) f(ST) = f(T S ) for every T, S E gl( V ) , (b) f(identity) = d = dimension V . Prove that f = tr and that [g1( V ) , gl( V ) ]has co-dimension 1 in gI(V) with a complementary space consisting of multiples of I = identity.

4.4 Existence of Grassmann Algebras

T h e particular Grassmann algebra over V which we wish to consider will consist of the space of multilinear alternating functions on the dual W of V . Thus we will have

Alternating Functions.

Go = F , the field of V ,

G1 = linear functions on W into F.

G1 is then canonically isomorphic to V in the usual way: v E V corresponds to the unique f E G1 such thatf(w) = w ( u ) for all w E W . This isomorphism will be our imbedding of V in G. Gp = the linear space of p-linear alternating functions from W x ... x W ( p factors) into F . More explicitely, i f f E GP, then (i)

f(wl , ..., aw,

+ bw,’, ..., w D )= uf(wl , ..., w , , ..., w p )

+ bf(w1

7

* * a ,

w’, .*.,

wp),

4. DIFFERENTIAL FORMS

58

that is, f is linear in each variable when the others are fixed, (ii)

f(Wl

,

* a * ,

w,) = 0

whenever two of the wi's are equal. If the characteristic of F is not 2, then we can use the standard polarization method to show that (i) and (ii) are equivalent to (i) and (ii)'

f(wl , ..., w i, ..., w ) , ..., w,) =

, ...)wj , ...)wi , ...)w,).

-f(wl

Moreover, since transpositions generate the symmetric group S, (ii)' is equivalent to (ii)"

f(wn1,

..*, Wno)

=

sgn

(4f(w1,

...I

,

w,)

for every 7r E S,. Even for characteristic 2, (i) and (ii) imply (ii)'. It will sometimes be convenient to regard W x -.. x W ( pfactors) as functions from { 1, p } into W, and then we would rewrite (ii)" as . . a ,

f(z o T) = sgn ( ~ ) f ( z ) for

z

E

W x ... x W.

Let S,,, be the permutation group on {l, ..., p + q}, S, the subgroup leaving { p 1, ...,p q} pointwise fixed, S,' the subgroup leaving {I, ..., p } pointwise fixed. By a cross section of S,S,' in S,,, we mean a subset K of S,,, having one and only one element from each left coset of S,S,'. A particular cross section which is often used is the set of shufle permutations, namely,

Multiplicotion.

K

=

{nE S,,,,

+

+

I nl < n2 < ... < ~p n(P

and

+ 1) < n ( p + 2) < . ' . < n(p + q)). +

T h e name is derived from the fact that if a deck of p q cards is cut into two stacks of p and q cards and then shuffled together with the usual technique, then the resulting permutation on the deck is a p , q shuffle permutation.

FIG. 15.

4.4. Existence of Grassmann Algebras

59

Iff E GP, g E Gq, K a cross section of S,S,' in S,+,, we let A, be the operation of adding p , and then for x E W p + q we define the product fg by fS(") = sgn ( 4 f k a 4 g(" 0 0 4).

2

neK

T o clarify notation, if x(i)

f(.

=

w inote that

0 n) = f(%l

d"0 n 0 A,)

, ..., %,)

= g('uIT(p+1)

.*.j

wncp+g)).

Using (ii)" it is easy to show that this multiplication is independent of K . Proposition 1. fg

E Gp+q.

Proof. ( p + q)-linearity of fg is obvious. Suppose K a cross section as above. Then jg(z o u) = Z s g n ( . r ) f ( zc ne K

=

sgn (0)

2 sgn

u

E

S,,,

,

o n)g(z o 0 o n o A,)

( P ) f ( Z 0 P) g b 0 P 0

4.

peoK

Then it suffices to show that a K is a cross section also. We have if K , (un)-l(un') = n-ln' which is not in S,Si unless n = T'. Thus the elements of uK are all in different cosets, and since the cardinality of this set is right, it must be a cross section. Notice that we have only shown that (ii)' holds forfg. T h e proposition is true for characteristic 2, but we do not prove it.

n,n' E

Proposition 2.

This multiplication is associative.

Proof. (Outline). Let f E G", g E GQ,h E G'. We consider S,,, as a subgroup of S,,,,, , and S," the permutations on { p q 1, ..., p q Y} and similar conventions for S, and Sq+,'. Then we show , that if K is a cross section of S,S,' in Si,+,,K' of S,+,S,'' in Spfq+? L of Sy'S,"in SQ+,', and L' of SpSqfr'in S,+,+, , then K'K and L'L are both cross sections of SpSy'Sr''in S,,,,,. Thus if we write out the products associated in both ways we see that we need only show that a sum

+ +

+ +

60

4. DIFFERENTIAL FORMS

is independent of the cross section of S,S,'S," we sum. This again is trivial by (ii)". Proposition 3.

in S,,,,,

over which

For q ,..., v p E V we have

T h e proof is just an extension of the arguments for the proof of proposition 2. Corollary.

If

ZI E

V , then v 2 = 0.

Finally, addition, scalar multiplication, and the identity are defined in the obvious ways. Dimension and Generation by { e , V } . We have already shown that G = CGi satisfies axioms (i)-(iii) for a Grassmann algebra. It remains to prove (iv) and (v).

Lemma 2.

D

=

Let e l ,

{I, ..., d}.

..., ed be a basis for V . Then e, # 0, where

Proof. Let w l , ..., wd be a dual basis to el , ..., ed. Then by proposition 3 above e , ( w l , ..., w d ) = 1 f 0.

Proposition 4.

T h e e,'s are linearly independent, so dim G 2 2d.

Proof. Since the sum C Giis direct we need only show that the e,'s, with s all of cardinality p , are independent. Let P be the set of all such s. Now if we have CsePuses = 0 and if so E P, then

since e; = 0 and the ei anticommute. Hence, by the lemma 2, 0. QED

aso=

Proposition 5.

T h e e,'s span G, so dim G

=

2d.

s = {il , ..., ip}, w, = (wi, , ..., win). Let el , ..., ed be the dual basis for V . Then by proposition 3, e,(w,) = I , e,(wt) = 0 if s # 2. Proof.

Let w l ,

..., wd be a basis for W. For

i, < i2 < ... < ip , let

61

4.4.Existence of Grassmann Algebras Now for f

E

G p we have

, i,) is the permutation needed to put i, where ~ ( i ,..., ascending order. Applying this formula with f = e, gives

Hence f

=

, ..., ip in

Xsspf(ws)e,.

Problem 10. Prove that the shuffle permutations are a cross section 2. and use them to write out all possible products for p , q

<

Problem 11. Suppose we define another product on the collection of multilinear alternating functions on W by: if f E Gp, g E G*, p , q 2 1, then f*g = apqfg, where a,,, is a nonzero element of F , fg is the old product. I n order that G = CGi be a Grassmann algebra over V with * as multiplication we must have upq = a and a condi?*. tion equivalent to the associative law. Derive this condition and use it to express up* in terms of a,, , q 2 , ..., u ~ , - ~Conversely, . show that aij may be assigned arbitrarily, and that when we let ulj = j 1 the formula for f*g is

+

f*g(4 =

c; sgn (4fb

0 n)g(z0

0

4).

neS!J+,

This formula may be used to define multiplication instead of the one we have given, but it does not work when the characteristic of F is a prime d. Our formula has the fewest possible terms.

<

Problem 12. Let e = {el, ..., e,} be a basis of V , 9 = gI(V), J : 9+ V d the isomorphism of 9 onto the d-fold Cartesian product

of V given by J( T) =; ( Tel , ..., Te,), f E Gd(W ) the unique element of G d ( W )such that f(e,, ..., ed) = 1 (f is an alternating d-linear function on V d ) .Prove that det = f 3 J : 2’+F.

4. DIFFERENTIAL FORMS

62

Problem 13. Let f E Gd(Fd*) be the unique alternating d-linear function on = d x d matrices such that f ( 1 ) = 1. Prove that f is the usual determinant of matrices. Problem 14. Describe a natural isomorphism of G(V*) and G(V)*. Use this to show that an inner product on V extends naturally to an inner product on G( V ) ,since an inner product leads to an isomorphism of V and V* which extends uniquely to an algebra isomorphism of G( V ) and G( V*).Also work out the expressions for the inner product on G( V ) in terms of a basis.

4.5 Differential Forms

Henceforth we will be concerned with the case V = M,*, mEM, a manifold, W = M,, G,= G(M,*). Thus if (xl, ..., xd) is a coordinate system at m, dx, , ..., dxd at m form a basis of M,*, so that every element f of G,p is uniquely expressible as f = C,, a, dx,. A diflerential p-form of M is a function 0 defined on some subset E of M , whose value at each m E E is an element of Gmd.0 is a C" p-form if, for each set of C" vector fields V , , ..., V , on M , the function 0( V , , ..., V,) defined on the intersection of the domains of 0 and of the Vl 9 *.*, V , by O(V1

9

***,

VD)(m) = em(V,(m),

***,

Vp(m))

is C". Here 8, denotes the value of 0 at m. When there is no confusion the "m" will be dropped. A O-form is simply a real valued function on M . We notice that 0 is C" if and only if for each coordinate system (x, , ..., xd) and (unique) expression 0 = C,, a, dx, we have a, E C".

We now relate our definition of forms to the brief discussion of tensors given in 3.3. First form the following tensor bundles over M : Gp = v G,P.

Alternative Approach.

mcM

The projection is the natural one, T : G,p+m. T h e differentiable structure is given by taking as coordinate neighborhoods sets of the form = ul,caY G,p, where V is a coordinate neighborhood in M with coordinates x1 , ..., xd , and has coordinates x, 3 rr , ..., xd o T

v

v

4.5. Differential Forms

63

(g)

plus the functions w, , s E P , dual to the basis {dx,} of GnLPat each point of V . Alternatively, if Rd* is the dual of Euclidean d-space Rd, then by dualizing the action of Gl(d, R ) on Rd we see that GZ(d,R ) acts on Rd* by taking the transpose of the inverse and hence also on G(Rd*)p by extending t o homomorphisms. Gp is then the bundle associated to the bundle of bases B ( M ) by this action [see 3.3(3)]. A dzflerential p-form 0 on B C M is then a cross section of Gp over E, that is, a map 0 : E - t Gp such that T O 0 = identity. 0~ C" if 0 is a C" map.

If E is a vector bundle over M we let E, = the set of nonzero vectors of E. E, is an open submanifold of E. If M is connected then Gd, is either connected or has exactly two components. This follows easily from the facts that any path in M can be lifted to a path in Gd, and each fibre of Gd, has exactly two components (d = dimension of M ) . M is orientable if Gdohas two components. Each component of Gd, is in this case called an orientation of M . M is nonorientable if Gd, is connected.

Orientation.

Lemma 3. A paracompact manifold M is orientable if and only if M admits a continuous, nonvanishing, globally defined d-form. Proof. Suppose 0 is a continuous, everywhere defined d-form with values in Gd,. Let y be any curve in Gd, which begins and ends in the same fibre GmdO= nonzero elements of Grnd.Since Grid is onedimensional for each n E M , we may write y ( t ) = a ( t ) O(T o y ( t ) ) , where a is a continuous real-valued function. Now a ( t ) is never zero, so a ( t ) is of constant sign. Hence, the initial and final points of y are in the same component of G,,,d,. Thus there is no curve connecting the two components of Grndo,so Gd, is not connected and M is orientable. be a locally Conversely, suppose that 11.1 is orientable. Let {Ui} finite covering of M by connected coordinate neighborhoods, {fi} an associated C" partition of unity. Choose an orientation of M . Then for any connected coordinate neighborhood U with associated coordinates xl,..., xd, = dx, is a continuous cross section of U in Gag, so either or -4 is in the chosen orientation. Thus we get +i defined on U , with values in the chosen orientation and 0 = C fi+i defines the desired nonzero C" d-form. Q E D

+

+

64

4. DIFFERENTIAL FORMS

For an orientable M such a 8 is called a volume element of M . If M is not connected, then we say that M is orientable if every component of M is orientable, and an orientation of M is a choice of orientations for every component. There are other conditions equivalent to orientability which are sometimes used as the definition. They are (1) M is covered by coordinate neighborhoods in such a way that any two systems are related by a system of equations having positive Jacobian determinant. (2) (When M is connected.) T h e bundle of bases B ( M ) is not connected. B ( M ) will then have just two components, each representing an orientation of M . Prove that the following manifolds are orientable: (a) T h e tangent bundle of any manifold. (b) A parallelizable manifold, hence, Lie groups. (c) Complex manifolds and almost complex manifolds.

Problem 15.

4.6 Exterior Derivative

T h e exterior derivative d is a map which assigns to each C" p-form 8 a C" (p + 1)-form d8 such that (i) if p = 0, d8 agrees with the definition of the differential of a C" function (see 4.1), (ii) domain of 8 = domain of d8, (iii) d is R-linear, (iv) if 8 is a p-form, 4 a q-form, then d ( @ ) = (do)(b (- l)W(d+), (v) d(d8) = 0 for all 8.

+

Theorem 1.

There is at most one map d satisfying (i)-(v) above.

If we write 8 = C,,, a, dx, on the intersection of the domains of 8 and the xi , we must have

Proof.

4.6. Exterior Derivative

65

Now by (iv) and (v)

2 (9

d(dx,)

=

1)j-1

dXi,

. . . d(dXij) ... axip

i=l

=

0.

Thus we get d0 = XEp Cj”=l DZja,d x j dx, as the unique expression for do. There is one point that has been passed over; namely, when we wrote (l), we should actually have put for the left-hand side 46

I(domaln o n domain { z i j ) )!

and what we claim is that this coincides with do

I(domaln 8 n domaln {xi)) *

A similar remark holds for the right-hand side of (1). Thus we still must show that if 0 and C$ agree on an open set U , then d0 and dC$ also agree on U . T o do this we let f be the function with domain U and value 1 at every point of U. Then f 0 and f + are equal and have the same domain U. By (i) df = 0, so by (iv) de

lu = f d e

=

d ( f e ) = q+) = f d + = d+

iU,

as desired. This concludes the proof of the theorem. T h u s we have demonstrated the form which d must take in a coordinate system. In order to show that d exists we would have to show consistency in overlapping coordinate systems. This is actually quite easy, since we can verify (i)-(v) for a coordinate neighborhood, and then the uniqueness and consistency under restriction to smaller domain will give the consistency on the intersections of coordinate neighborhoods. Now we develop an intrinsic formulation of d which will be used extensively later. Intrinsic Formula for d . Let 0 be a C“ p-form. Then we define an R-(p + I)-linear function 0 of ( p + 1)-tuples of Cm vector fields into C” functions on M by

B(vl , ..., v,,,)

=

2 (- 1)i-l v,e(v,, ..., vi-l , v,+~, ..., v9+,) a

66

4. DIFFERENTIAL FORMS

If t , , ..., t,,, E M , , , m E domain of 8, choose vector fields V , , ..., V,,, such that Vi(m)= t i . Define dO(t,, ..., tp+l)= 8(Vl, ..., V,,,)(m). We must show that this definition is independent of the way in which V l , ..., V,+, are chosen. This will be proved in a series of lemmas. Lemma 4.

i

=

1,

Proof.

..., p

If Vi and W icoincide in a neighborhood of m for

+ 1, then 8(V1 , ..., VP+,)(m)= 8(W,, ..., WP+,)(m).

This is clear from the definition of 8.

B(fV, , V , , ..., V,,,) (b) 8 is alternating, that is,

Lemma 5. (a)

8( v,, ..., vz , ... vj , ..., V,,,) )

(c) B(V1 , ...,fvi ,

a'.,

=

Vp+l)=

= f 8 ( ~ , ..., ,

vP+~).

-B(V, , ..., vj , ..., vi , ...)vD+l).

fQ( V , , ..., V,+,).

Proof. (a) follows from problem 1.14 and the definition of 8. (b) follows immediately from the definition of 8. (c) then follows from (a) and (b).

Lemma 6. d8 is well defined, that is, 8(Vl, ..., Vp+,)(m)is independent of the choice of Visuch that ti = Vi(m). Proof. Let xl, ..., xd be coordinates at m. Then Vi= CiaiiDx, in a neighborhood of m. If Wi(m)= Vi(m), then W i= CibiiDxj and aij(rn) = bij(m)for all i,j. By lemma 4 we need only to consider 8 in this coordinate system. By (c) of lemma 5 ,

=

B(w1 , ..., wp+l)(m).

QED

The map 8 + d8 defined via 8 above is the exterior derivative, that is, satisfies properties (i)-(v).

Theorem 2.

Properties (ii) and (iii) are obvious from the definition. We verify property (i) directly; namely, let 8 = f~ C", and we show d8 = 4f. Let EM,, V a vector field such that V(m) = t. But e ( V ) = (-I)l-lVf, the other terms being vacuous. Hence, at m Proof.

&(t)

=

8(V)(m) = V ( m ) f = tf

=

df(t),

as desired.

4.6. Exterior Derivative

67

T o show the other properties we establish that d coincides with the operation defined locally in the proof of theorem 1. Let d denote this operation, that is, if xl, ..., xd are coordinates, then for % =zasdx, SEP

d0

= SEP

2 D,, i

a , dxj dx,.

Now the assignments 0 -+ d%and 0 -+ & are both R-linear. Further, if 0 and q5 agree on a neighborhood then d0 and dq5 also agree on this neighborhood. Hence it is sufficient to show they are the same for 0 = f dx, , and by linearity of forms it is enough to show that d0 and 20 agree when applied to ( V , , ..., Vp+l),where Vi = 0, :

av1 ,

a*.,

V,+l) =

2 D,,f dx, dxs(V1 ... V9+1) 2 D,,f 2 s g n b ) dxk(T/,l)dxs(Vnv,, 9

'i

>

k

=

>

k

* * a 9

V37n(p+1,),

RGK

where K is a cross section of S,S, in S,+, . Using shuffle permutations < n ( p 1) we get 7r2 <

+

On the other hand, since [ V i, V j ]= 0,

and since dx,(Vl , ..., Vm-, , V?,#,, , ..., V,,,)

=

constant,

Problem 16. Write out in full the intrinsic expression for d0 when 0 is a 1-form and a 2-form. In the latter case use skew-symmetry to put it in the form of a sum over the cyclic permutations of 1, 2, 3.

68

4. DIFFERENTIAL FORMS

Problem 17. If U , V , W are constant vector fields (brackets all 0) on R3 and X is a C" vector field, set up correspondences between the various parts of the calculus of differential forms and ordinary vector analysis to obtain the following formulas:

from the intrinsic formula for d: gradf . U

=

Uf,

curl X . U x I/ = U ( X . V ) - V ( X . U )

(div X ) U . V x W = U ( X . V x W )

+ V(X . W x

U)

+ W(X . U x

V)

(These formulas could be used to define grad, curl, and div.) from axiom (iv) for d:

+

grad cfg) = g gradf fgrad g curl (fX) = (gradf) x X fcurl X div (fX)= (gradf) . X f div X div(X x Y ) = (curl X ) . Y - X . curl Y

+ +

from axiom (v) for d: curl gradf

= 0,

div curl X = 0.

4.7 Action of Maps

If M , N are manifolds, 4 : M 4N a C" map, then we have seen that 4 carries functions on N into functions on M by composition and carries tangents on M into tangents on N. We now define a map +* of forms on N into forms on M , namely, +* : 8-+ 8 o d+, that is, if 8 is a p-form on N , then

+*(W,, ..., b) = @(&t l , .*.,4 tn). Notice that on the space of 1-forms at a point dual linear transformation to dd.

+* is just the usual

4* is a Grassmann algebra homomorphism and it commutes with the exterior derivative.

Theorem 3.

Proof. That 4* is a Grassmann algebra homomorphism is automatic from the definition of multiplication.

4.7. Action of Maps

69

T o prove that 4* commutes with d we notice first that both 4* and dar e R-linear and that it is a local problem. Locally forms are generated by functions and differentials of functions. Using the fact that +* is a homomorphism and d is an antiderivation the problem is reduced to consideration of individual factors of terms, namely, to functions and differentials of functions. But we have for any function f on N and $* df = 4f o d+ = d ( f o 4) = d($*f) QED d(+* df) = d ( d ( f 0 4) = 0 = +*(O) = +*(d(df)).

+

=

+

If 8 = u2 du, u, du, u, du, and (sin u1 cos u 2 , sin u, sin u 2 , cos u,), then

Example.

4

+*O = sin u1 sin up d(sin u, cos i d 2 )

+ sin u, cos u pd cos u1

=

(sin u1 cos u, sin u2 cos u2

+ (-sin2

u, sin2 u2

rj

: R2 + R3 by

+ cos u1 d(sin u1 sin u2)

+ cos2 u1 sin u2 - sin2 u1 cos u2)du,

+ sin u1 cos u1 cos u2)du,.

Problem 18. Denote by X ( M ) the collection of all C“ forms defined on open subsets of M , X p ( M )the collection of C” p-forms; sometimes we shall omit “(M).” Let X be a vector field and define a linear function i ( X ) : X +3? satisfying the following conditions: (a) i ( X ) : 3 ? p + X p - - l , p 3 1, i ( X ) ( X o )= 0. ( b ) i ( X ) is an antiderivation, that is, if 6’ E Z p , 4 E 33, then i(X)(04)= ( i ( X >8) 4 (- l)”O(i(X) 4). (c) If 8 E X1,then i ( X ) 8 = 8 ( X ) .

+

Show that there is a unique function satisfying (a)-(c), and verify the formula i ( X ) 8(X,, ..., X,-,) = 8(X, X , , ..., XP-,), where O E X p , and X , , ..., X,-, are vector fields. Problem 19. Show that L,, the Lie derivative of X , restricted to X is a derivation. Show that i ( X ) d d i ( X ) is a derivation. Hence prove the formula L , = i ( X )d d i ( X )

+

+

by showing that it holds for functions and differentials of functions. Problem 20. Let G be a Lie group, X , , ..., X , a basis for the left invariant vector fields on G, ciik the constants (structural constants)

4. DIFFERENTIAL FORMS

70

such that [Xi, X j ] = C,cijkX,. Define dual l-forms w1 , ..., wd to the Xiby w i ( X j ) = Sij (constant function). Prove (a) wi is left invariant, that is, for every g E G, L,*wi = wi. (b) dwi = - ~ C j , , c j k i w j w= k C j < k c j k i w k w j . (This is called the equation of Maurer-Cartan.) 4.8 Frobenius' Theorem.

A K-dimensional codistribution E on a manifold M is a map which assigns to each m E M a k-dimensional subspace E9nof Gml. E is a C" codistribution if for each m E M there is a neighborhood on which are defined k C" 1-forms which span E a t each point of the neighborhood. A submanifold N of M is an integral submanifold of E if for each n E N , En = annihilator of dI(N,), where I : N -+M is the injection map. E is completely integrable if there is an integral submanifold of E through each point m E M. The distribution associated with E is the ( d - k)-distribution D defined by D, = annihilator of Ent.

Let W , be the ideal generated by Em in G,. Then W,. if and only if for every t , , ..., t , ED, we have w ( t l , ..., t,) = 0. Lemma 7.

wE

If 8,, ..., Ok is a basis for E m ,then if o E W,. Ci+i8i, +i E G,P-l. Thus

Proof. w =

4 1 ,

* * a ,

tv) =

C;+iW,,

a * * ,

we have

tP)

i

=

2 (- i ) q q t l , ...,

tj-l

, tj+l , ..., t P )ei(tj)

i.1

=o since t j E D,, = annihilator of En&. Conversely, if w ( t l , ..., t,) = 0 for every t , , ..., t, ED,, let 8k+l, ..., Od be a completion of e l , ..., 8, to a basis of G,,1, and let e l , ..., ed be a basis dual to 0, , ..., ed. Then e k + l , ..., ed span D,, and if we write w = CsEpa,8,,we have, if k < j, < <j , ,

4.9. Vector-Valued Forms and Operations Hence each term of

w

must have a Oi with i

71

< k, that is, w E W,.

Frobenius' Theorem [29]. Let E be a C" k-dimensional codistribution on M . Let W be the function which assigns to each rn E M the ideal W, generated by Em in G,. Then E is completely integrable if and only if W is invariant under the exterior derivative operator, that is, d( W )C W .

We shall show that this theorem is equivalent to theorem 1.6, which says that a C" distribution is completely integrable if and only C Wis equivalent if it is involutive. First notice that the condition d( to the assertion that for each m E M there is a local basis of 1-forms 8, , ..., 8, in E such that do, E W. We have E is completely integrable if and only if the associated ( d - k)-distribution D is completely integrable if and only if D is involutive if and only if Oi(Vj) = 0, i = 1, ..., k; j = 1, 2 implies Oi([V,, V,]) = 0, i = 1, ..., k if and only if Oi(Vj)= 0, i = 1, ..., k; j = 1, 2 implies dO,(V,, V,) = 0, i = 1, ..., k, (theorem 2) if and only if do, E W , i = 1, ..., k (lemma 7). QED

w>

Problem 21.

Derive the integrability condition given in problem Q du,

1.29 for the 1-codistribution spanned by the 1-form P du, R du,. (Hint: Use problem 4.)

+

+

<

Problem 22. Let w be a p-form with p d - k. Prove that w is in the ideal generated by linearly independent 1-forms el , ..., 8k if and only if we1 ... Ok = 0. On the other hand, if p > d - k, then w is always in the ideal generated by 8,, ..., $ k , so this shows: W is the space annihilated by multiplication by 8, ... $k.

4.9 Vector-Valued Forms and Operations

Let V be a vector space over R, M a manifold. A V-valued p-form on M is a map w which assigns to each t , , ..., t p E M , an element w ( t l , ..., t P )of V such that iff is any element of the dual space of V then f o w is a (real-valued) p-form on M . If V = Rd, then a V-valued p-form is simply a d-tuple of p-forms ( w , , ..., w d ) = (ul o w , ..., u d o w ) . (2) If V = gl(Rd), the Lie algebra of d x d matrices, then a V-valued p-form is just a matrix of p-forms (wii).

Examples. (1)

4. DIFFERENTIAL FORMS

72

+

If is a g-valued p-form, w a g-valued q-form, where g is a Lie algebra, then [+, w ] is the g-valued ( p q)-form given by

+

where K is a cross section of S,Sq‘ in S,,,. Note that [+, w ] = (- l)”q-l[w, +] since we get an additional factor of - 1 when we change the order of bracketing, besides the factor (- l)pq resulting from-reversing the order of the forms. If is a p-form with values in a space of linear transformations on a vector space V , and w is a V-valued q-form, then #w is defined as a V-valued ( p q)-form by

+

+

where K is again a cross section as above. Problem 23.

Let G be a Lie group, g its Lie algebra. Define g-valued , is, if x = X ( g ) 1-form w on G by w ( X ) = X for every X E Q that then ~ ( x = ) X. (a) Prove that w is left invariant. (b) Prove that dw = - 3 [w, w ] . (c) If X , , ..., X, is a basis for g, then we may write w(x) = Ciwi(x) X i , defining real-valued 1-forms w , , ..., w,. Prove that w , , ..., w , are the same as in problem 20, so that (b) is a coordinatefree form of the equations of Maurer-Cartan. If +, w are gl(d, R)-valued forms, define the product using matrix multiplication instead of bracket multiplication as above. (b) Show that [+, +l(X, Y ) = 2[+(X),+(Y)I = 2#+(X, Y ) when is a 1-form. Problem 24. (a)

+w

+

4.10 Forms on Complex Manifolds [23, 30, 50, 921

If M is a complex manifold, then the differential forms are defined in terms of Grassmann algebras over the complex field, and the operation J (problem 3.10) on the real tangent spaces induces a bigradation of the forms. As in problem 3.10, we have the various

73

4.10. Forms on Complex Manifolds

+

tangent spaces A,,,, F m, a n d z , , . Since F m = d m i A n L, J may be extended to be a complex linear endomorphism of F m ,still satisfying J z = -1. T h e dual of J extends to a derivation J* of the complex Grassmann algebra G(F,n*). w E G P + Q ( F m * ) is called an element of type ( p ,q) if J*w = ( p - q ) iw. By allowing m to move over M , we define complex differential forms of type ( p , q). We denote the set of these objects byXP,Q(M). Elements of %'pq0 are called holomorphic p-forms. Problem 25. For f a complex-valued C" function on M , define df and show that it is a complex differential I-form. Problem 26. For a complex coordinate system (z, , ..., zd), show that the dzi , dZi at m, with 1, generate G (F m * ) .Express a form of type ( p , q) in terms of these generators, and show that

G'(Fm*)

C

=

GP'q(Fm*).

p+q=r

Find dim

Gp*Q(Y,*).

Problem 27. Show that whenever X,E 2.

w

E X ~ PifOand only if w ( X , , ..., X,)

=

0

Problem 28. Show that d = d' + d", where d' : X P ~ q + X p + l , q and d" : X P - Q - - + X P , Q + l . Also verify that (d')2 = 0 = (d")2 and d'd" - - d'ld'.

Show that the algebras X P v Q may be defined for a manifold with an almost complex structure.

Problem 29.

Problem 30.

For any almost complex structure, dlyO,1

= d"0

+ d1.1 + d 0 . 2 ,

where d p , Q : X o J - + X P - Q . Show that the condition for the almost complex structure to be a complex structure (see problem 3.1 1 et al.) is equivalent to d2Yo = 0, and hence to the assertion of problem 28 above.

CHAPTER 5

Connexions T h e definition of a connexion on a principal bundle is given and the horizontal lift of a curve and parallel translation are established. T h e curvature form is defined and the structural equation is proved. Existence of connexions, connexions on associated bundles, and structural equations for horizontal forms are discussed, and holonomy groups are introduced via a sequence of problems [49, 50, 66, 831. 5.1 Definitions and First Properties

T h e appropriate vehicle for certain differential geometric structure on a manifold appears to be a connexion, which in a special case, treated in the next chapter, is equivalent to the formulations in terms of the more usual notion of parallel translation. Connexions are associated with principal bundles over a manifold, and for the most part we shall be considering only connexions on the bundle of bases. However, many fundamental properties of connexions hold on a general principal bundle, and these we develop in this chapter. Let P be the bundle space of a principal bundle over M with structural group G and projection map T : P + M . Let V be the dim(G)-dimensional distribution on P of vertical vectors, that is, for p E P, V,

= { t E P,

~

d7r t = O}.

Recall that there is a homomorphismh : 9 +. g defined by the right action of G on P (3.1): if X E 9,p E P,then ( h X ) ( p )= dp(X(e)),where p is here regarded as the injection of G into P given by p(g) = p g . = AX, then Then the elements of g are vertical, for if

d ~ ( X ( p )=) d ~ d p ( X ( e )= ) O 14

5.1. Definitions and First Properties

75

since rr o p is the constant map G 4rr(p). Further, if t E V, , then there exists an E fi such that x(p)= t , since p maps G onto rr-l ( ~ ( p ) ) Hence, . the elements of fi span the vertical space at every point, which shows that V is indeed a distribution with dimension equal to dim(g) = dim(G), and further, V E C". Indeed, if Uis a distinguished neighborhood of ~ ( p )F , the associated map into G, then the map p: G -+ n-l(m) may be factored as follows: p = (n x F,)-l(m, LFU(,)), from which follow the desired properties. (See remark near the beginning of 3.2). A connexion on the principal bundle (P, G, M ) is a d-dimensional distribution H on P such that (i) H EC", (ii) for every p E P, H, V , = P, , that is, H, is a linear complement to V , , (iii) for every p E P , g E G, dRgHp = HPg.

+

M FIG. 16.

+

If t E P, , we write t = V t H t , where V t E V , , H t E H,. If X is a vector field, then we write also (VX)(p) = V(X(p)), (HX)(p) = H(X(p)). Elements of H , are called horizontal. Problem 1. Prove that X

E

C" if and only if V X and HX E C".

Since dn lx, is one-to-one, drr: H , w Mn(p, by dimensionality. Hence, to every vector field X on M there corresponds a unique vector field 2 on P, called the (horizontal) lift of X, with the properties that for everyp E P, x ( p ) E H, and drr T ( p ) = X(rr(p)). From problem

76

5 . CONNEXIONS

1 it follows that if X

E

C", then dRJ=

E

C". We also have

X,

gEG,

R+P=(X+Y) 7 2

H [ X , PI

r--J

=

[ X ,Y].

All these properties are trivial to verify. We now give the dual formulation of a connexion. T h e 1-form of a connexion H is the Lie algebra valued 1-form 4 on P defined by: if p

E P,

t

E P,

, then + ( t ) = that X

E g

such that

x(p)= Vt.

A p-form w on P is called vertical (horizontal) if it vanishes when one or more of its entries is horizontal (vertical). If w has values in g, then it is called equivariant if, for every g E G, w o dR, = Ad g-l o w. Notice that the definition of a vertical form depends on the existence of a connexion, that is, a notion of horizontal tangents, whereas the horizontality of a form is a notion that is independent of a connexion. Lemma 1.

T h e 1-form

properties:

4

(i) C$ is vertical, (ii) If X E g, E 8, then (iii) 4 is equivariant, (iv) 4 E C".

x

of a connexion H has the following

+(x(p)) = X , all p E P,

(i) and (ii) are trivial. (iii) follows immediately from 3.l(b). Let X be a C" vector field on P, so there exist functionsfi such that

Proof.

xi

E g. Then the f i E C" [25, prop. 1 , p. 881, s o $ ( X ) = Cif,Xi where E C", proving (iv). T h e properties (ii)-(iv) characterize 1-forms of connexions in the following sense.

If 4 is a 1-form on P with values in g satisfying properties (ii)-(iv), then there exists a unique connexion H on P such that

Theorem 1.

5.2. Parallel Translation

77

4 is its

1-form. Hence, there is a one-to-one correspondence between such 1-forms on P and connexions on P.

{t E P, I +(t) = O}. Then for t E P,, let Vt Vt E H,, which proves property (ii) in the definition of a connexion. Further, for any g E G, 4(dR,t) = Adg-l4(t) = 0 if t E H , , so dR,t E HPg. Hence, dRgHpC H,, , which proves (iii). Further, if X is a C" vector field on P, then V X = + ( X )E C", so H X E C". From this it follows trivially that H E C", which establishes that H is a connexion. Because of this theorem we shall sometimes speak of q5 as being the connexion. Let H p

Proof. =

7b(t)(p),so t

=

-

Problem 2 . Let 4 and I) be two connexion forms on P and f a C" function on M . Show that: (a) ( f o T ) + + ( l - - f o T ) $ i s a c o n n e x i o n f o r m o n P. (b) If t E Pp and t = t , t , , t = ' t , ' t , are the horizontal and vertical decompositions of t with respect to the connexions of 4 and $, find the decomposition with respect to the connexion of (fo T ) 4 (1 - f o T ) $. [Hint: +('tl) = -$(tl).] In particular, when we take f = constant, this shows that the connexions of P form an affine space.

+

+

+

5.2 Parallel Translation

If y is a broken C" curve in M , then a (horizontal) lift of y is a broken C" curve 7 in P such that (i) 7 is horizontal, that is, 7* is horizontal, and (ii) T o 7 = y. By broken we mean y is continuous and piecewise C". Let y be a broken C" curve in M , y : [0, 11 -+ M . n-l(y(0)). Then there exists a unique lift 7 of y such that

Theorem 2.

Let p

E

Y(O> = P.

Proof. We may assume that y is C" since we may chain the lifts of pieces together, in fact, in only one way. Extend y to a C" map of ( -E, 1 + 6 ) = I into M . Then by problem 3.6, N = { ( r , q ) E Ix P I y ( r ) = n(q)} is the bundle space of a principal bundle ( N , G, T ' , I ) . The map 0 : N -+ P by B(r, q ) = q may

78

5. CONNEXIONS

be used to get a connexion on N ; the 1-form of this connexion is = 8*+ (see problem 3 below).

6

FIG. 17.

Let X be the unique horizontal lift of D on I to N , and let u" be the integral curve of X starting at (0, p ) . T h e domain of u" is all of I since it could be extended about a neighborhood of the upper limit otherwise. We define 7 = 8 o u". It is obvious that 9 is a lift of y , and since +(f*) = +(do o u"*) = +(X)= 0, 7 is horizontal. That 7 is unique follows easily from the fact that any lift can be factored through N via 0 and a lift of u, but the latter is obviously unique. QED Corollary 1. If H is a connexion on (P, G, M ) , y a curve in M as in the theorem, then there is defined a diffeomorphism T, of n-l(y(0)) onto r-l(y( l)), called parallel trailslution from y(0) to y( 1) along y. T,, is independent of the parametrization of y and satisfies T, o R, = R, o T, , all g E G. Further, if y and a are two such curves with o(0) = y(l), then T,, = T, o T,.

Let p E n-l(y(O)),f be the lift of y with P(0) = p , and define T,(p) = 7(1). We use the right invariance property, which is trivial, to prove T, E C" and T, is a diffeomorphism: if po is any fixed element of n-l(r(O)), Pl = T,(Po), then T,(Pog) = T,(Po)g, so T , = p , o pop1: n-'(y(O)) + G + r-l(y( 1)). Proof.

5.2. Parallel Translation

79

T h e other properties are immediate. We remark that a concept of parallel translation is equivalent to a connexion; namely, if we have parallel translation in P given and satisfying right invariance and certain smoothness conditions-specifically, tangent curves give coincident infinitesimal transformations-then the connexion may be recovered by differentiating as follows: let y be an appropriate curve in M , p E r-l(y(O)), and let t E P, such that d r t = y*(O). Then define Ht = T,(p),(O),where we view T,(p) as the curve t 4 parallel translate of p from y(0) to y ( t ) along y. T h e following result provides an interesting interpretation of the connexion form. Let y, 7 be as in theorem2, and let T be any other lift (not necessarily horizontal). Define 01 : I+ G by: a(r) is the unique element of G such that 7(r) = T ( ~ ) O I ( T ) . Then d R ( a ( r ) - ~ ) ~= * ( r- $) ( ~ * ( r ) ) . (We are identifying G, with g by left translation.)

Corollary 2.

P

FIG. 18.

Proof. Let I,: G ---f P by I,(g) = ~ ( rg.) It is trivial to verify that I , o L, = ~ ( rg): G + P. By theorem 1.2, where the map P x G + P is right action, we have

?*(r) == dIra*(r)

+ dKa(r)

(1)

5. CONNEXIONS

80 Now

) ) dLa(,t-l(a*(r)),recalling our identification Therefore, ~ $ ( d I ~ a * ( r= of G, and g. ) , the O n the other hand, 4(dRacr,7*(r))= Ad a(r)-l d ( ~ * ( r ) by equivariance of 4. Hence, since is horizontal, applying 4 to (1) gives

~L,c,t-+*(r))

=

-

Ad

‘Y(r)-l

+b*(r))

Problem 3 . Let ( f B ,f , ,f M ) : ( B , G, M)-+(B’, G’, M’) be a bundle map with df, : g + g’ an isomorphism onto. Show that any connexion on B’ induces in a natural way a connexion on B. In particular, if f:M -+ M’ and ( B , G’, M ) is the bundle induced by f over M , then this will be the case.

5.3 Curvature Form and the Structural Equation

Define, for a form w on P, the form Dw by Dw = dw o H .

where H is a connexion. More precisely, if w is a p-form, then for Note that t , , ..., tptl E P, , Dw(tl , ..., t,+,) = dw(Ht, , ..., Ht,,,). Dw is always horizontal. T h e curvature form @ of a connexion H with l-form4 is the horizontal g-valued 2-form D+.It is easy to verify that @ is equivariant. We need the following lemma for the proof of Cartan’s structural equation. Lemma 2.

horizontal.

If

X E g, V a horizontal vector field on P, then

[x,V ]is

5.3. Curvature Form and the Structural Equation

81

Proof. We cannot apply theorem 3.1 directly, since the horizontal vector fields do not form a finite-dimensional vector space. However, we may derive the result indirectly as follows. Let V, be a right invariant horizontal vector field, that is, a horizontal lift of a vector field on M . Then, taking 9 = {Vi> in theorem 3.1, we have V,] = 0. Now locally we may write V = Z i f i V i , where the Vi are horizontal and right invariant and the fi are C“ functions on P . Then V ] = X i ( z f i ) V , , by problem 1.14, which is certainly horizontal.

[x,

[x,

i(x)

Problem 4. Applying the formula L8 = d + di(x) to the connexion form and evaluating on the horizontal vector field V , show that +( VJ)= 0, thus giving an alternate proof of the lemma. [Hint: Notice that the one-parameter group of transformations associated with is right action by e“. Hence LR(+) is vertical.]

+ [x, x

(structural equation). If on P, @ its curvature form, then

Theorem 3

d+ =

-

4 is the

1-form of a connexion

Q [+,41 + 0-

Notice that if G is a matrix group and 9 is identified with a space of linear transformations, then - &[+,+] = +. We show that the above 2-forms applied to vector fields X , Y on P agree. Since forms are linear, we need only consider the cases for which the X,Y either belong to fi or are horizontal, in all comProof.

binations. (i) X , Y Eg, so there are elements X’, Y’ E g such that = X, P’ = Y , and hence +(X)= X’, +(Y)= Y’. Now by theorem 4.2 W X , Y ) = X + ( Y )- Y+V)- +([X,YI) =

X ( Y ’ ) - Y ( X ’ ) - [X’, Y’]

=

-

=

-

H+>+l(X,Y )

Q [+,+ l W , Y ) + @ ( X ,Y ) ,

(4.9)

as desired, since @ is horizontal. Note that X ( Y ’ ) = X(constant g-valued function on P ) = 0, and Y ( X ’ ) = 0 similarly. (ii) X E g, Y horizontal. Let X‘ E g be as in (i): &(X, Y ) = X + ( Y ) - Y+(W - N X ,YI)= 0,

5. CONNEXIONS

82

since + ( Y )= 0, + ( X ) = constant, and [X,Y ] is horizontal by the lemma. On the other hand, @ is horizontal, so @ ( X , Y ) = 0 as X is vertical, and [+,+ ] ( X ,Y ) = 0 since Y is horizontal. (iii) X,Y both horizontal. + ( X ) = 0, +( Y ) = 0, and @ ( X ,Y ) = d4(HX, H Y ) = d&Y, Y),

which is the structural equation in this case. QED Remark. T h e restriction of the structural equation to vertical vectors is essentially the equation of Maurer-Cartan. Another interpretation is that it says that d+ has only a horizontal and a vertical part with no mixed part.

(the Bianchi identity). If @ is the curvature form of a connexion on a principal bundle P, then

Theorem 4

D@ Proof.

=

0.

From the structural equation,

D@

=

D db, - D[+, 43.

Now Dd+(X,, X , , X,) = ddcj(HX,, H X , , HX,) = 0, since d2=0. Also, D[+,+] = 0, since [+,#I is a vertical 2-form, and so vanishes when one of its entries is horizontal. Hence, D@ = 0. Let H be a connexion on P, @ its curvature form. Then @ = 0 if and only if H is an involutive distribution, which in view of theorem 1.6, means that P admits local horizontal cross sections. In particular, if M is simply connected, then by a standard monodromy argument P must be the trivial bundle. A connexion with @ = 0 is called p a t . Theorem 5.

Proof.

If X , Y are horizontal vector fields on P, then

@(Z Y) = 4 ( X , Y ) = X W ) - Y 4 V ) - N X , Yl) =

-MX, Yl);

so H is involutive if and only if [ X , yl is horizontal if and only if @ ( X , Y ) = - + ( [ X , yl) = 0, which implies @ = 0, as asserted. Problem 5. Let H be a closed subgroup of a Lie group G and consider the principal bundle (G, H , GIH). Let C$ be a connexion and show

5.4. Existence of Connexions

83

that 4 o dR, = dR, o 4. Now assume 4 is invariant under dL, for every g E G. Show that (a) 4 defines a projection$: g -+ @ of the corresponding Lie algebras, (b) if m = ker$, then [m, $1 C m, (c) conversely, if g = m ij (direct sum) with [m, $1 C m, then the projection of g onto $ gives rise to an invariant connexion in the above sense. Hence, there is a one-to-one correspondence between invariant connexions on (G, H , GIH) and reductive complements m of $. An H admitting such an invariant complement m is called reductive in G (the name arises from the fact that the adjoint representation of G restricted to H is reducible to the adjoint representation of H plus the representation of H on m via Ad,, at least if H i s connected). (d) For the connexion 4, show that the curvature form may be considered as defined on g x g, and derive a formula for it.

+

5.4 Existence of Connexions and Connexions in Associated Bundles Existence of Connexions.

C" connexions exist in abundance. I n Chapter

7 we shall establish the existence of Riemannian connexions on B ( M ) . Here we show that any principal bundle (P, G, M ) , with M

paracompact, has a connexion. Let {Ui} be a covering of M such that n--l(Ui) is trivial. Let {fi}be a C" partition of unity subordinate to the covering { Ui}. Let +i be a flat connexion on n--'( Ui) and define 4 = X(fio n-) bi.4 is a not necessarily flat connexion form on P. Problem 6 .

Verify that 4 is a connexion form.

Remark. If (P, G, n-, M ) is a complex analytic principal bundle over a complex manifold M , it of course admits C" connexions, but in general it will not admit a complex analytic connexion. A necessary condition in a special case is given in [3]. However, real-analytic manifolds admit analytic connexions, although the proof is much more difficult. See the remark following theorem 7.2.

Let (P, G, M ) be a principal bundle with a connexion H , and let ( B , G, F , M ) be an associated bundle with fibre

Associated Bundles.

84

5. CONNEXIONS

F (see 3.3). Then in some sense H induces a “connexion” on B. To be precise, there is a distribution H‘ on B which at each point complements the vertical tangent space. Further, there is a notion of parallel translation of the fibres of B , which derives as before from the horizontal lifts of curves. Let y be a broken C“ curve in M , b E ~ ’ - l ( y ( O ) ) . We define a lifting 7 of y into B which will turn out to be horizontal in the sense below. Let f E F and p E P be such that ~ ( p= ) y(0) and pf = 6 , wherep is here the map defined in 3.3. By theorem 2 there is a horizontal lifting y of y into P with y(0) = p . Now define 7 by p(t) = y ( t )f . We then define parallel translation T, along y from n’-l(y(O>) to ~’-l(y(1)) as in P. Hence we have ?‘, = T,(p) o p - l , so parallel translation is a diffeomorphism. Parallel Translation.

The Distribution H’. Take b E B , p E P such that ~ ’ ( b= ) ~ ( p ) We . may view Pp as a subspace of ( P x F ) ( p , f ,, where f E F is such that pf = b. Let A : P x F + B be the natural map (3.3), and define H i = dA(H,). This definition is independent of p in view of the right invariance of H , while it is clear that the lift defined above is horizontal with respect to H ’ , if the definition of the map

p :F + r ’ - l ( r ( ~ ) ) is recalled. Problem 7. Let 4, y5 be connexion forms, H , K their connexions, H’, K’ the corresponding distributions on B. Show that if s, t E B, , H’s = s, K’t = t , and dn’(s)= dr’(t), then rs (1 - r ) t is in the (1 - r ) y5. distribution on B of the connexion belonging to r+

+

Problem 8.

to R.

+

Determine all connexions on T ( R ) ,the tangent bundle

Problem 9 . Show that there exist horizontal distributions on associated bundles which are not connexions. [Hint: Take T(R ) w R2 and define a distribution with slope eu.]

5.5 Structural Equations for Horizontal Forms

We first prove a basic lemma. Let G be a Lie group of diffeomorphisms of M , G x M - t M ,

5.5. Structural Equations for Horizontal Forms

85

and let 4 be a representation of G as nonsingular linear transformations on a vector space V . Then there is an associated representation of g as linear transformations on V , which may be defined as follows. If v E V , X E g, then set $(X) v = X ( e ) v . This makes sense since v is a vector-valued function on G, namely, v(g) = +(g) v ; and so is mapped into a vector by the tangent X ( e ) (see 1.4, 2.2, and 2.6).

4

Let X E and ~ w be a V-valued p-form, and let Yl , ..., Y , be invariant vector fields on M . (i) If w satisfies w o dg = +(g) w for all g E G, then

Lemma 3.

( 8 X ) w(Y1

9

.", Y,)

=

(AX) 4 Y l , *.., Y,),

where AX is the vector field on M defined in 3.1. (ii) If w satisfies w o dg = +(g-') w for all g E G, then -@X) w ( Y , , ..., Y,)

=

(AX) w(Y1

)

..., Y,).

Proof. We shall prove (ii). T h e proof of (i) is similar. I f f € M , then w ( Y l , ..., Y,) o f is a V-valued function on G, and in fact

4Y1,

..., Y,)of(g)

=

=

w(Y,(gf), *'., Y,(gf)

=

w(dgY,Cf),

=

W1)W ( Y l ( f ) )

.a*,

dgY,Cf)) Y,(f))

4 Y l ( f ) , -*.* YD(f)) #(P)l

where +(g) = g-l, since the Yiare invariant and w is equivariant with respect to the representation 4. Therefore we have

86

5. CONNEXIONS

Theorem 6. Let 4 be the 1-form of a connexion on P and let w be a Q-valued, horizontal, equivariant p-form on P. Then we have the structural equation for w : dw =

-[+ , wl

+ DfJJ.

+

p 1 vector fields Y , , ..., Yptl chosen from a set which locally spans the tangent space to P . For this set we choose vector fields (AX}, X E ~to, span the Proof. We prove this by applying both sides to

vertical tangent space. T h e remaining vector fields we choose in various ways. We consider several cases. (i) No Yiis vertical. We may then assume the Y , are all horizontal, and so [$, u](Y1, ..., Yp+,)= 0 since $ is vertical. Also, H Y , = Yi , so D w ( Y i , ..., Yp+,)= d w ( Y l , ..., Yp+,>,which proves the result in this case. (ii) One Yi is vertical. Assume Y,+, = AX. We may choose Y , , ..., Y p so that they are right invariant and so that [Yi, AX] = 0. T o do this in the neighborhood of a point f E P, we choose a coordinate system at f which derives from the local product structure of P. Then the partial derivatives with respect to the variables coming from M suffice, for they are clearly right invariant and they bracket correctly with AX, since AX depends only on the other coordinates. Now H(AX) = 0 implies that D w ( Y 1 , ..., Yp+,)= 0. We also have from theorem 4.2 dw(Y1 , ..., Y,+l) =

2 (- 1)i-1 Y,w(Y,, ..., Y&+Y,+l, ... Y,+l) )

i

since w is horizontal (-I)~Y,+,W(Y , ..., ~ Y,) (-l)P(hX) W(Yl , ..., Y,) = (-l)p+l ad X w ( Y l ,..., Y,)

=

=

4 = Ad and 2.6, since w is equivariant, = (-l)P+"X, W ( Y l ,..., Y,)]

by the lemma with

(-l)P++"X), W(Y, ..-,Y,)l = - 14,WI(Y1,.*.,y , , YV+I), =

using 4.9 and fact that prove.

I

w

is horizontal. This is what we wanted to

87

5.6 Holonomy (iii) Two or more Yivertical. From the fact that clear that everything vanishes. QED Corollary.

If

Q, is the

w

is horizontal it is

curvature form associated with 4, then we have d@ = -[+,

@I.

Proof. @ is horizontal and equivariant, and so the result follows from the theorem and Bianchi’s identity. We shall have several more occasions for employing the above lemma in the ensuing chapters.

5.6 Holonomy [2,5, 51, 66, 77, 821

We develop the material in this section as a series of problems. Let ( P , G, rr, M ) be a principal bundle with connexion 4. Let P E P and define K, C G by K , = {g E Glpg is a parallel translate of p}.

K, is a subgroup of G,the holonomy subgroup of rp at p. Problem 11. If p’ E P is a parallel translate of p, then K p = K,’. Problem 12. If p’ = pg, g E G,K p <= K P g = g-lK,g. (Hence, the holonomy group of 4 is defined up to isomorphism.) Problem 13. Let K,, = {g E K , I pg is a parallel translate of p along Problem 10.

a null homotopic curve}. Show that K,, is an arcwise connected normal subgroup of K p and hence an arcwise connected subgroup of G, called the restricted holonomy group of 4 at p. It then follows from a theorem of Yamabe [95] that K,, is a Lie subgroup of G. Problem 14. There is a natural homomorphism of the fundamental ) the quotient group K p / K p 0 . group of M based at ~ ( ponto Problem 15. If M is simply connected, then Kpo= K,. For example, if M = R, then K, = (1) since every connexion is integrable. It is known that K,, is the component of the identity of K, [66]so that K, is a Lie group. Let P,‘ = {p’ E P I p’ is a parallel translate of PIProblem 16. (P,’, K,, M , T) ; 1 is a principal bundle, and the inclusion ip: P,’+ P gives a reduction of G to K p . Furthermore, i,*(+) is a connexion form on P,’.

88

5 . CONNEXIONS

Problem 17. If G can be reduced to a subgroup K via a bundle map i : P' + P such that i*(+) is a connexion on (P', K , M ) , then for any p E P', K , C K .

Problem 18. If @ is the curvature form of +, then for every p E P, @(Pp, P,) C f, , the Lie algebra of K p . (Ambrose-Singer [2] prove even more, namely, if V = {@(P,, P,)lp is a parallel translate of a single p , E P}, then the Lie algebra generated by V is f,,. A later result [ 701 shows that V spans fpo.) Problem 19. Any discrete subgroup of the positive real numbers can be realized as the holonomy group of a connexion on the bundle of bases B(S1)of the circle, and no other subgroup can arise as a holonomy group.

+ If + is a connexion on (S4d+3,S3, QPd),then the holon-

S1, CPd),then the holonomy Problem 20. If is a connexion on (S2d+1, group at any point is S1. Problem 21. omy group is either S3or S1. I n the above problems it will be found necessary to show that any two broken C" loops which are homotopic are homotopic via a homotopy which is broken C" at every stage, and that any homotopy class of loops has a broken C" member.

CHAPTER 6

Afine Connexions T h e additional structure available for a connexion on the bundle of bases is defined, including the torsion form, basic vector fields, the torsion and curvature transformations, and geodesics. T h e additional structural equation is proved and difference forms are considered, particularly in their relation to torsion and the configuration of geodesics. T h e exponential map, completeness, and normal coordinates are defined. The chapter concludes with a treatment of covariant differentiation and the classical definitions of the above [33, 49, 50, 66, 831. 6.1 Definitions

Let M be a manifold, B ( M ) its bundle of bases. Then a connexion on B ( M ) is called an a@ne connexion. Since any connexion on a subbundle of B ( M ) can be extended to an affine connexion by the right action of the group GL(d, R), such a connexion is also called an affine connexion. From the notion of parallel translation in the bundle B ( M ) given by an a 6 n e connexion we obtain a notion of parallel translation in the tangent bundle, or parallel translation of tangents along curves. Since the tangent bundle T ( M ) is an associated bundle of B ( M ) , we can derive this property from 5.4, but it is simple enough to give an explicit definition in this case. If y is a curve in M and t E M,,, , m = y(O), then we obtain the parallel translate of t along y to n = y ( u ) as follows: choose any b E B ( M )with ~ ( b=) m,and let 7 be the unique horizontal lifting of y to b. Then if y(s) = (y(s), e,(s), ..., ed(s)) and t = C,a,e,(O), we define the parallel translate of t to be Caiei(u). I t is easy to check, a9

90

6. AFFINECONNEXIONP

using the invariance of the connexion under right action, that this transformation is independent of the choice of b over m.

6.1.1 The Solder Forms

B ( M ) always has defined on it certain horizontal l-forms, which are independent of any connexion on the bundle. T h e solder l-forms wi are defined as follows. Let t E B ( M ) , , where b = (m,e, , ..., ed). Then dx t =

w i ( t )e,.

Or these w ican be considered as a single Rd-valued l-form by 4) = ( w & ) , ..*,W d ( t ) ) .

w,

defined

T h e solder form satisfies the following properties: (i) w E C", (ii) w is horizontal, (iii) w is equivarient, that is, for every g E GZ(d, R),

Lemma 1.

R,*(w) = w o dR,

= g-l

o w,

where on the right-hand side g is viewed as acting to the left in Rd.

91

6.1. Definitions Proof.

(ii) is immediate. T o prove (iii), let t E B ( M ) , ,b

Then dR,t

E

=

(m, el , ..., e d ) , drr t

=

2 aiei.

B(M),, , bg = (m, Cigilei , ..., &gidei). Now dn

=

2

(g;luj)(gkiek),

i?i,k

so

wi(dRst)=

Z<&zj.

Hence, w(dR,t) = g-lw(t), as asserted. T h e proof of (i) is direct. Let yi , y j k be product coordinates on a coordinate neighborhood of B ( M ) (3.2), and we have to show that w(Dzl,,) and w(Dy,) are C". By (ii), W ( D ~ , =~0, ) so we need only consider w(Dy,). But d7r D J b )

=

(problem 3.4)

z y > ' ( b ) ej j

where b

=

(m, e l , ..., ed). Hence, w(D,J = (yi:, ...,yy:)

E

Cm.

QED

6.1.2 Fundamental and Basic Vector Fields

A vector field 8 on B ( M ) is called fundamental if E X(gI(d, R ) ) , that is, if there is an X E gI(d, R ) such that 8 = AX, (3.1). I n particular, the fundamental vector field corresponding to X i j E gl(d, R), where X i j is the matrix with 1 in the (i, j)th entry and 0 elsewhere, is denoted by Eij. T h e following lemma summarizes the properties of a fundamental vector field. Lemma 2. If X is a fundamental vector field on B ( M ) , then we have (i) X E Cm, (ii) X is vertical, (iii) if 8 = AX, then

dR,8

=

for g E Gl(d, R).

h(Adg-l X )

[3.l(b)]

92

6. AFFINECONNEXIONS

Let H be a connexion on B ( M ) , and let X E Rd, b E B ( M ) . Then there is a unique horizontal tangent E(x)(b) at b such that w(E(x)(b)) = x, since d n is an isomorphism on Hh. T h e vector field on B ( M ) whose value at b is E(x)(b) is called a basic vector Jield and denoted BfMl

p = fm,e,, ..., edl

FIG.

20.

by E(x). In particular, if St = (ali, ..., adi) E Rd, then we have basic vector fields Ei = E(Si) which give a basis of the horizontal tangent space at every point of B ( M ) . These E ( x ) are not the horizontal lifts of any vector fields on the base space M , not being right invariant by (iii) of this lemma. Let E(x) be a basic vector field on B ( M ) . Then we have (i) E(x) E C", (ii) E ( x ) is horizontal (by definition), (iii) dR,E(x) = E(g-lx), forgE Gl(d, R), wheregisviewedasactingonRd totheleft,asinlemmal. Lemma 3.

Parts (i) and (iii) follow from lemma 1 and the (defining) fact that w(E(x)) = X. Notice that the vector fields Ei , Ejk give a parallelization of B ( M ) , that is, for every b E B ( M ) ,Ei(b), Ej,(b) are a basis of B(M),. Further, they are dual to the 1-forms wi , +jk , where +jk is the 1-form defined by &(t) = (i, k)th entry of +(t), the 1-form of the connexion. E j k and w i do not depend on the connexion but are intrinsic on the bundle of bases. Ei and +jk on the other hand do depend on the connexion, and in fact the connexion H may be given by specifying either the {Ei}or the {&}. For the +ik we saw this in theorem 5.1.

6. I. Definitions

93

Moreover, if Ei are d linearly independent nonvanishing vector fields on B ( M ) satisfying u(E,J = ai then the distribution H defined by setting Hb = span of the {Ei(b)) is clearly a connexion on B ( M )with basic vector fields Ei. 6.1.3 Alternate Definition of the Solder Form

Sometimes it is convenient to regard each bER(M)as an isomorphism of Rd onto M,, , where b = ( m , e l , ..., ed), by defining

This corresponds to the fact that T ( M ) ,the tangent bundle to M , is an associated bundle to B ( M ) under the left action of Gl(d, R ) on Rd, so that we have b(gx) = bg(x) for g E G (see 3.3). Let b E B ( M ) , t

Lemma 4.

E

B ( M ) , , then

w(t) =

b-'(dn t ) ,

and this formula may be used to define Further, if x E Rd, then

w.

E(x)(b) = ( d n I H s ) - l ( W .

T h e proofs of these statements are immediate, and the advantage of using these formulas as definitions is that they tend to put these concepts and subsequent manipulations of them on a more intrinsic and subscript-free footing. For example, property (iii) of lemma 1 LRd'M, b might be proved as follows. T h e mapping bg : Rd has an inverse (bp)-'

= g-lb-l

b-1

: M,n +R

g-1

Rd,

and hence, w(dR,t) = (bg)-'(dr t ) = g-l(b-l(dn t ) ) = g-'w(t). 6.1.4 Torsion

T h e torsion form Q of an affine connexion H on B ( M ) is the Rdvalued 2-form S2 = Dw = dw o H.

94

6. AFFINECONNEXIONS

(Compare the curvature form.) It is easy to verify that Sz is C", horizontal, and equivariant, that is, 1;2 o dR, = g-' o 1;2.

We now relate torsion and curvature forms to the basic vector fields of the connexion. Let x, y

Lemma 5.

E

Rd. Then we have

[W,E(Y)l = --h@(E(x), W )- W

W x ) ,W ) ) ;

that is, the curvature and torsion give the vertical and horizontal components of the bracket of two basic vector fields. Proof. We show that w and 4 applied to each side yield the same function, and this will be sufficient since these forms are parallelizing, that is, are dual to a set of parallelizing vector fields. T h e right-hand side is easy to calculate since AX is vertical and E ( x ) is horizontal:

+(right side)

w(right side)

=

-+(A@(E(x), E ( y ) ) )

=

-@(W, E(y)),

=

--w(E(Q(E(x), E ( y ) ) )

=

-Qn(E(x), E(Y)).

T o apply 4 and w to the left-hand side we use the intrinsic formulas for the exterior derivatives d$,dw (see 4.6). -+(left side)

=

-+([E(x), E(y)])

=

E ( x )+(E(y))- W

=

~+(E(xE ),W )

=

d+(HE(x),HE(?))

=

W E W E(Y))

=

@(W, E(Y)).

)4 ( J W - + ( [ E ( x ) ,E(J91)

Similarly, --(left

side) = dw(E(x),E ( y ) ) = Q ( W ,W ) .

6.1. Definitions

95

We have used the facts that d ( E ( y ) )= d(E(x))= 0 = constant and w ( E ( x ) ) = x = constant, w(E(y))= y = constant, so that their derivatives in the directions E ( y ) and E(x)are 0. Q E D Let 4 be a connexion form on B ( M ) . Then the curvature and torsion forms of 4 vanish on B ( M ) if and only if the following condition on M is satisfied. Let m E M . There exists a coordinate system (xl, ..., xd) at m, with domain U , such that the image of the cross section x : U - t B ( M ) , defined by

Theorem 1.

x ( 4 = (n,&l(n),

.*.I

DXd(4),

is horizontal. Proof.

Notice first that f v X DXi(4)=

Ei(X(4).

Hence, if the image is horizontal we shall have on the image, and hence everywhere by equivariance, [Ei, E j ] = 0, which implies that curvature and torsion are zero, by the above lemma. Conversely, if these forms vanish, then by theorem 5.5 there is a horizontal manifold N , and E l , ..., Ed are vector fields with trivial brackets. Pulling them down to M , we have by theorem 1.5 that there is a coordinate system xl, ..., xd on M such that Dxi = d r E i

IN

This coordinate system will then have the required property. Problem 1. T h e property of having torsion zero is invariant under combinations of connexions. Using the existence proof in Chapter 5 show that every paracompact manifold has affine connexions with torsion zero.

6.1.5 Curvature and Torsion Transformations

T h e curvature and torsion forms give rise to tensors on M , which may be considered in the context of the tensor calculus on M , but we prefer to approach these concepts directly in terms of linear transformations on tangent spaces.

96

6. AFFINECONNEXIONS

T o every m E M and pair s, t E M , we make correspond a linear transformation R,, : M,,, + M , , called the curvature transformation, as follows. Let b E B ( M )be such that ~ ( b = ) m,f, S E B(M)bsuch that d T ( f ) = t , dT(S) = s. Then using the notation of 6.1.3, we define R,,(u) = --b@(S,

t) b-l(u)

for any u E MwL. Using the horizontality and equivariance of rP it is easy to verify that the above definition is independent of the various choices made, namely, the choices of b, S, and i. T h e definition of R,, may be rephrased in terms of matrices as follows. Let b = (m, e, , ..., ed), S, f be as before. Then R,, is the linear transformation of M,, whose matrix with respect to the basis e l , ...,ed is -@(S, f). For example, Rs,el = -ZirPij(f, f) ei. If X , Y , Z are vector fields on M , then we denote by RxyZ the vector field satisfying R X Y Z ( 4 = Rxcn,, Y(nL)z(4. T o each m E M and pair s, t E M , we make correspond a tangent T,, E M , , called the torsion translation, as follows. Let b E B ( M ) , S, f E B ( M ) , be as in the above definition of the curvature transformation. Then define

T S t= -bQ(S, t).

T h e proof that T,, is well defined is similar to the proof for R,,. Also, for vector fields X and Y we have a vector field T x y . 6.1.6 Geodesics

Let y be C" curve in M . A vectorfield along y is a cross section of T ( M ) ,the tangent bundle of M . For example, y* defines a vector field along y . A vector field X along y is said to be a parallel vector field along y if, for any u, v , X ( u ) is the parallel translate of X ( v ) along y from y ( u ) to y ( v ) . In terms of the connexion on the associated bundle T ( M ) (see 5.4) this says that X defines a horizontal curve in T ( M ) . A C" curve y in M is called a geodesic if its tangent vector field, y* , is parallel along y . Notice that a geodesic is a parametrized curve and not just a point set. And the only reparametrization of a geodesic which will again give a geodesic is a linear change in the parameter.

y in

6.1. Definitions

97

If y is a C“ curve in M , 7 its horizontal lift through b E ~ l ( y ( 0 )in) B ( M ) , then y is a geodesic if and only if there exists c E Rd such that 7 is an integral curve of E ( c ) if and only if ~ ( 7 , )is constant, that is, when we express

Theorem 2.

the fiare constant. T he proof of this is trivial. Applying the theorem that vector fields have unique integral curves we obtain: For every m E M , t E M m there exists a unique geodesic such that y ( 0 ) = m and y*(O) = t.

Corollary. y

6.1.7 Geometric interpretation of R,, and T,,

I n this section we relate R,, and T,, to parallel translation around an infinitesimal parallelogram with sides s and t. We generate a family of “parallelograms” in a manner similar to the construction in theorem 1.4, except that we use geodesics instead of integral curves of vector fields. As we go around the broken geodesic starting at m,we carry s and t along by parallel translation, generating vector fields S and T along the curve. T h e pieces of the curve are then in turn, the geodesic with tangent vector field S , T , -23, and - T and each is followed parameter distance u. T h e end point of this broken curve we call u(u), so (J is a C“ curve starting at m. Since the “parallelograms” are not generally closed, u is not necessarily the constant curve. Let A(u) be the linear transformation of M , given by parallel translation along the broken curve and then backward along u. T h e first order tangents of u and A are 0. T h e second order tangent (see problem 1.20) of u is 2T,, , and A”(0) = 2R,, , where A is viewed as a curve in the vector space gI(A4,). I n other words, the first order parts of the displacement given by traversing a parallelogram and parallel translation around such a parallelogram both vanish, and the second order parts are given by torsion and curvature, respectively.

Theorem 3.

’

Proof. If we lift the broken geodesic horizontally to B ( M ) starting at b then we get integral curves of a pair of basic vector fields X and Y

98

6. AFFINECONNEXIONS

in the way that we did in theorem 1.4. So by that theorem the curve y generated by the end of these lifts, which is above u, will have second order tangent 2 [ X , Y ] ( b ) , with the first order tangent 0. Thus, by lemma 5 the horizontal component will be a lift of 2 T s t ,which proves the first assertion. Similarly, the vertical component of 2 [ X , Y ] ( b )measures the second order deviation of y from parallel translation, so A has second derivative equal to the corresponding transformation on Mm , which is 2R,, by lemma 5 . Development of curves in M into M,. We compare curves in M and in flat afine space M,, by a process known as development. This is done

by making the tangent vectors in each case have the same relation to the parallel translates of a particular basis. Let y be a broken C" curve starting at m, b = (m, e l , ..., ed) a basis at m. Let u be the horizontal lift of y to b in B ( M ) .Then we may write y* = uf, where f = up1 y* is a curve in Rd. Then g ( t ) = j,,t f defines a curve in Rd which we call the development of y into Rd with respect to b, and bg is a curve in M , which we call the development of y into M,. bg is independent of the choice of basis b at m.

w'=Y*

.

+r ,---\

/'

Rd

FIG. 21.

Y

\ '

6.2. T h e Structural Equations of an Affine Connexion

99

T h e process can be reversed; for every broken C" curve T in Mm starting at 0, there is a corresponding curve y in M starting at m such that T is the development of y . To show this we get the horizontal lift cr of y first; what we know is that a,(t) = E(b-b'(t))(u(t))so the existence then follows from: Lemma 6. Let E be a linear map from R d into a linear space of C" vector fields on a manifold N . Then for every broken C" curve f in Rd and n E N there is a unique broken C" curve u in N such that

Proof. Define vector field X on U x N , U a neighborhood of 0 in R, by X ( t , n') = &(t) E(f(t))(n'). Then the integral curve of X starting at (0, n) is the graph of u, so u exists. Note that geodesics are developed into straight lines.

+

Problem 2. Lets, t E Mm and let T be a broken C" closed curve in the plane of s, t such that ~ ( 0 = ) 0. For every z) E R let 07 be the curve in Mnl defined by (ZIT)(.) = v ( T ( u ) ) . Let T(U) = p ( u ) s q(u) t, T parametrized on [0, 11,

+

the area enclosed by T relative to s, t. Let h be the map of [0, I] x R into M such that A(., v ) develops into ZIT. Let y ( v ) = h(1, z)), and let S(v) be the linear transformation of M,,, given by parallel translation around the closed curve consisting of h ( - ,z)) and a piece of y-l. Prove the following generalization of theorem 3: y*(O) = 0 and

the second order tangent is 2AT,,.

S'(0) = 0 and S"(0) = 2AR,,.

6.2 The Structural Equations of an Affine Connexion

Let H be an affine connexion on B ( M ) .Let +, W , @, SZ be the 1-form of the connexion, the solder form, the curvature form, and the torsion form, respectively.

100

6 . AFFINECONNEXIONS

Theorem 4.

We have

+ Q,

dw = - 4 w

4 = - 4[4,41 + @

These are called the first and second structural equations of the connexion. (See 4.9 for definitions of the forms &J, [+, 43.) Proof. T h e second structural equation is simply the structural equation for a connexion on a principal bundle (see theorem 5.3). T o prove the first structural equation we prove the following more general result.

Let 6 be an Rd-valued equivariant horizontal p-form on B ( M ) . Then d8 = -48 D8.

Theorem 5.

+

Proof. T h e proof is almost identical to that of theorem 5.6. We proceed by evaluating each side on vector fields Yl , ..., Y,+l which are drawn from a collection of vector fields which locally span the tangent space to B ( M ) . We consider the same cases as before. (i) No Yiis vertical. Hence we may assume all Yiare horizontal. But then the 46 term is 0, and the remainder is merely the definition of Dd. (ii) One Yi is vertical. Assume Up+, = AX, X E aI(d, R ) . As before we choose right invariant horizontal Yi’s such that [Yi, AX] = 0, i = 1, ...,p. Then by theorem 4.2,

do( Yl

, ..., Y,+l)

=

(- I)”(AX) O( Y , , ..., Y,).

Now applying the lemma 5.3 with the representation operation on Rd,we have (-l)pAX8(Y1

, ..., Y,)

=

(-i)p+l

On the other hand, Dd( Y , , ..., Y , ,AX) so the right-hand side gives -48( Yl , ..., Y , , AX) = -

2 (-

Y+l

=

matrix

xqy, , ..., Y,). =

0, since Dd is horizontal,

l)i-14(Yi)8(Yl, ..., Yi-l, Yi, 1, ..., Yp+l)

i=l

-(-l)’+(AX) 8(Y1 , ..., Y,) = (-i)p+lxe(yl,..., Y,).

=

4

6.2. T h e Structural Equations of an Affine Connexion

101

We have therefore reduced both sides to the same function, as desired. (iii) Two of the Yiare vertical. In this case everything is 0, and the equality holds automatically. QED Theorem 6. Let 8 be an equivariant horizontal p-form with values in either gr(d, R ) or Rd. Let denote the operation of gI(d, R) on either gI(d, R) or Rd by bracketing or matrix multiplication, respectively. Then 020 =

Proof.

a e.

8 satisfies the structural equation d8

=

-4

nB

+ DO,

either by theorem 5.6 or theorem 5 above. Applying D to both sides yields d20 o H = -d(+ n 0) o H + D28. But d 2 = 0, and d(+ no) 0 H

= d+O

H

0-40 H

DB

=@me.

and this gives the result. Combining this with the Bianchi identity (theorem 5.4) gives the afine Bianchi indentities: D@ = 0, DSZ = @u. 6.2.1 Dual Formulation of the Structural Equations

The first structural equation has a dual in terms of the bracket of a fundamental and a basic vector field. Theorem 7.

Let X

E gI(d,

R ) , x E Rd. Then

[AX, E ( x ) ] = E(Xx),

where

Xx is the action of the matrix X on the vector x.

Proof. We can prove that [AX,E ( x ) ] is horizontal by means of the second structural equation or notice that this is precisely what the lemma 5.2 says. Thus we need only show that W([AX,E ( x ) ] ) =

xx.

6. AFFINECONNEXIONS

102

But by the first structural equation w([AX,E ( x ) ] )= +XXW(E(X))- E ( x ) w(XX)- d ~ ( h XE, ( x ) ) = &(AX, =W

(4.6.)

E(x))

X )w ( E ( x ) )

=XX.

QED

For the vector fields E i ,Ejk this theorem gives the formula = 8ikEj. We remark that this formula also follows directly from theorem 3.1 and hence can in turn be used to prove the first structural equation.

[Ejk, Ei]

Problem 3. Give an alternate proof of theorem 7 by applying the formula LA, = i(hX) d &(AX) to the solder form w and evaluating on the vector field E ( x ) . (Compare with problem 5.4.)

+

6.2.2 Difference Forms

Let 4,$ be two connexion 1-forms on B(M). Then we define the diflerence form T by T = i+h - 4. T h e difference form T has the following properties: (i) T E C”, (ii) T is horizontal, (iii) T is equivariant. Conversely, if 4is a connexion form and T is a gI(d, R)-valued 1-form satisfying (i)-(iii), then 4 + T is again the 1-form of a connexion. Lemma 7.

These facts are immediate. A difference form T gives rise to a linear transformation T, : Mm-+Mm ) m we define for each s E M,; namely, if ~ ( b = T,t

=

h(S) b-lt,

where f E B(M), is such that dn S = s. Conversely, given a function T which assigns to each m E M , s EM, a linear transformation T,: M m + M n t such that T is linear in s and “differentiable” in m and s [that is, as a function from T ( M ) to the bundle of linear transformations of the tangent spaces to M , which is the bundle with fibre gI(d,R)and action the adjoint representa-

6.2. T h e Structural Equations of an Affine Connexion

103

tion of Gl(d, R ) associated to B ( M ) ] ,then we can define a gI(d, R)valued 1-form satisfying (i)-(iii) by T ( S ) = b-'T,,sb

for any iE B(M),. Also T,t = b ~ ( ib-lt ) holds, which shows there is a one-to-one correspondence between difference forms and certain functions which we shall call linear transformation j?elds. Let 4 be a connexion on B ( M ) .Then there is a difference T is the 1-form of a connexion whose form T such that II, = 4 torsion 2-form is identically zero (cf. problem 1). Further, if T' also has this property, then TW = T'W (see 4.9 for notation).

Theorem 8.

+

Proof. We assume T exists and see what conditions it must satisfy. Writing down the first structural equations for 4 and $, we have dw =

dw

+ s2

-+J

= -#w,

since II, has torsion zero. So (c$- II,)w the equation =

-TW

=

Q, and hence

T

must satisfy

s.

T h e last part of the theorem is now immediate. T o show that such a exists, we simply define, for b E B ( M ) , s E B ( M ) , , x E Rd,

T

-T(S)(X)

=

8 G(S, E(X)(b)).

I t is not hard to verify that T is a C", equivariant, horizontal 1-form. Further, for b E B ( M ) , s, t E B ( M ) , , -TU(s,

t)= = =

U(t)

-T(S)

8 Q(s, & Q(s,

= Q(s,

+ T(t)

W(s)

E(w(t))(b))t ) - Q(t, s)

4

8 w,E(w(s))(b))

t),

as desired.

Notice that the linear transformation field T associated with T is T,(t) = 8 T,, , where T,, is the torsion translation corresponding to s, t E M , (6.1.5).

6. AFFINECONNEXIONS

104

Before proceeding we give a convenient formulation of parallel translation. Lemma 8. Let p be any C” curve in M , t E M,,,, , b E B ( M ) such that ~ ( b = ) p(0). Let 4 be a connexion form on B ( M ) and /I the +horizontal lifting of p through b. Then the +parallel translate of t along p to p ( u ) is /3(u)b-’t.

T h e proof is immediate in view of the action of any C E B ( M )in mapping Rd isomorphically onto T h e following is a useful characterization of geodesics. Lemma 9. Let y be a curve in M , 7 a not necessarily horizontal lift of y to B ( M ) . Then y is a geodesic if and only if

(7* + $(?*)I 47*) = 0. Proof. Let P be the principal bundle induced by y over the domain U of y from B ( M ) . Thus

P

= ((24, b ) I u E

u,Tb = y(u)}.

Then the map a : P-+B(M), a(u,b) = b is a C” map, so we may pull back the solder and connexion forms, w’ = a * w , 4’ = a*+, SZ‘ = a*SZ = 0 since the horizontal space of P is one-dimensional. u : U - t P given by ~ ( u = ) (u, ? /(u ))is a C“ curve in P, and u* extends to a right invariant C“ vector field X on P. If we do the same thing using a horizontal lift /3 of y instead of 7, we get a C“ vector field Y on P such that +’(Y)= 0. Moreover, X and Y both project to D = djdu on U , so [ X , Y ] is vertical and w’(X) = w’(Y). By theorem 2, y is a geodesic if and only if w(/3*) is constant, or, P*w(/I,) = 0, which by applying a becomes Y w ’ ( Y ) = 0.

Now the first structural equation gives dw’(X, Y ) = X w ’ ( Y ) - Y w ’ ( X ) =

-+‘(X)CO’(Y),

or, since

Xw‘(X)+ + ‘ ( X )w ’ ( X ) = Yw’( Y ) .

w ‘ ( X ) = w’(Y),

6.2. T h e Structural Equations of an Affine Connexion

105

Thus the condition for y to be a geodesic is that the left side of this equation is 0, and this becomes, along points of cr, d

- W’(U*)

du

Inserting the meaning of

+ +yo*) w ’ ( u * ) = 0.

w ‘ , +‘we

have

This is the conclusion desired in one direction. T o go the other way we only need reverse the implication

+ +’(X) W’(X) = 0 + +’(u*) w ’ ( o * ) = 0;

XW’(X) d =* --‘(a*) du

this is possible by the right invariance of X . T h e following theorem is not relevant to Riemannian geometry, but has interesting consequences for the theory of connexions.

+

Let andJ! , I be two connexion forms on B ( M ) , T their difference form. Then these two connexions have the same geodesics in M if and only if for every b E B ( M ) ,s E , Theorem 9.

T(S) W(S)

0.

If the two connexions have the same geodesics, then given = s. (If s is vertical y = constant.) Then by lemma 9 Proof.

sE

B(M), there is a geodesic y such that a lift 7 of y satisfies 7,(0)

Y*(O) 47*) =

-+w 4)

= -#(s)

W(S)

Conversely, if T ( S ) w(s) = 0 for all s, then if Y*W(Y*)

-+(7*)-(?*I = -#(f*) -(7*),

as desired.

7 is a lift of a +-geodesic y ,

=

so y is a #-geodesic.

since

~ ( 7 *~ )( 7 *=) 0,

I06

6. AFFINECONNEXIONS

T h e following is a generalization of theorem 8 and a comparison of the proofs shows that the two connexions occurring there have the

same geodesics. Let C$ be a connexion and 0 an equivariant horizontal Rd-valued 2-form on B ( M ) . Then there is a unique connexion $ having 0 as its torsion form and the same geodesics as 4. In particular, if the geodesics and torsion forms of two connexions coincide, so do the connexions. Corollary.

Proof. T h e proof of uniqueness will give a formula for T = C$ - $, hence existence will follow from the trivial verification that T is horizontal and equivariant.

So let us suppose that C$ and $ have the same geodesics and that the first structural equations are dw

=

dw =

-r+

-#w

+a + 0.

Subtracting these gives Tw =

that is, for s, t

E

(4 - 4 ) w

=

a- e = 7,

B(M),, T(S) W ( t ) - T ( t ) W(S)

= 7(S, t ) .

By polarization, the condition for equality of geodesics becomes ~ ( t+) ~ ( tw(s) ) = 0. Adding the last two equations gives a formula for T , since w ( t ) is arbitrary: T(S)

T(S)

w(t) =

i q ( ~t ), .

QED

Problem 4. Show that if two connexions have the same geodesics then so does any combination of them in the sense of problem 5.2. Problem 5 . Connexions on parallelizable manifolds. Let p be an Rdvalued I-form on M which gives a parallelization of M . We define three connexions associated with p as follows. T h e direct connexion is the one for which the vector fields p-'(x), for a fixed x E Rd, are parallel along every curve. T h e torsion zero connexion is the one with the same geodesics as the direct connexion but with torsion zero. T h e opposite connexion is the one with connexion

6.2. T h e Structural Equations of an Affine Connexion

107

form 4 + 27, where 4 is the connexion form of the direct connexion and 4 + 7 is the form of the torsion zero connexion. Prove the following facts about these connexions: (a) T h e parallelization p leads naturally to a cross section of M in B ( M ) . This cross section is horizontal with respect to the direct connexion and by means of it the structural equations pull down to one equation on M : dp = P, where P is the torsion form pulled down. (b) Two parallelizations are related by a C" function from the manifold into Gl(d, R ) and the two corresponding direct connexions are the same if and only if the function is a constant ( M connected). (c) T h e geodesics of all three connexions are the same and consist of the integral curves of the vector fields p-'(x). (d) If p is constant on vector fields X , Y , 2, then the torsion and curvature of these connexions are given by the table: direct connexion torsion zero connexion opposite connexion

T x r = [X, UJ, RXY Tx = 0, RxvZ T x y = -[X,YJ, Rxy

,

=

0

=

i"x,Y l J I

=

0.

(e) For the opposite connexion R x , = 0 if and only if p is constant on rX, Y1. I i p is constant on all such [ X , Y ] then it is well known that a local Lie group structure can be given M such that the constant fields are the left invariant vector fields. More generally, the problem of local equivalence of direct connexions has been studied in the larger context of G-structures. A lucid account of this matter may be found in a book by Shlomo Sternberg, "Lectures on Differential Geometry," Prentice-Hall, Inc., Englewood Cliffs, New Jersey,

1964. Problem 6. Connexions on Lie groups. A Lie group has a parallelization by means of left invariant vector fields and also by right invariant vector fields. W e call the corresponding direct connexions the left connexion and the right connexion. Show that they are opposite connexions of each other, so the two torsion free connexions coincide. The first structural equation of the left connexion pulls down, as in problem 5(a), to the equation of Maurer-Cartan (problems 4.20,4.23). Hence express P in terms of the structural constants c z .

108

6 . AFFINECONNEXIONS

T h e three connexions coincide if and only if the group is Abelian, so the various products of Euclidean space and tori have flat affine connexions with torsion zero. 6.3 The Exponential Maps

T h e exponential map at m of 0 E M , into M ,

E

M is a mapping of a neighborhood U

exp, : U + M .

For those t E M , for which exp,,(t) is defined, it is given as follows. Let y be the unique geodesic in M such that y(0) = m and y*(O) = t. Then define e.Prn(t)

= ~(1).

Note that exp,,(ut) = y(u), u a real number, if y ( u ) exists. T h e domain of exp,, is an open set in M , which is star-shaped with respect to 0 E M , in the sense that if t is in the domain then so is the line segment from 0 to t. Besides this exponential map we also shall consider a certain lifting ) m, we define W , ( t ) = y(l), to B ( M ) . For 6 E B ( M ) , with ~ ( b = where 7 is the unique horizontal lift of y through b. Since y is a geodesic, 7 is an integral curve of E ( x ) , where bx = y*(O) = t (theorem 2).

FIG. 22.

6.3. The Exponential Maps

109

We will show that e x p b E c",and hence also exp, = rr o Fb E c". I t will immediately follow that exp, is a diffeomorphism onto a neighborhood of m, since d exp,, maps onto Mn,;in fact, if el , ..., ed is a basis for M , , u l , ..., u d the dual base, then d exp,D,,(O) = ei , and the dimensions of M,, and M are the same. Theorem 10.

-

expb E C".

Proof. We proceed in a slightly more general fashion. Consider the afine bundle A ( M ) over M , that is, the bundle with bundle space consisting of pairs (b, t ) , b E B ( M ) , t E h f n ( b ) , and projection map (b, t ) -+~ ( b ) This . is a manifold under the obvious definition of the differentiable structure. We define a mapping

F : B(M) x A(M)- T(B(M)) by F(b, c , t )

=

E(c-'t) (b).

For each (c, t ) E A ( M ) we have a vector field E(c-lt) on B ( M ) . F is clearly a C" mapping. By the theorems on differential equations in the appendix, there is a C" mapping G of a neighborhood of {0} x B ( M ) x A ( M ) into B ( M ) given by G(u, b , c, t ) = y(u), where y is the integral curve of E(c-lt) with y(0) = b. Then e G b t = G ( l ,b, b, t ) . Corollary.

T h e map Exp: T ( M )

--f

Exp(m, t )

QED

M , defined by

=

exp, t,

is defined on a neighborhood of the trivial cross section of T ( M ) and is C" there. Proof. For m E M choose a C" cross section x of a neighborhood of m into B ( M ) . Then on this neighborhood the mapping

J : (mt )

-

(1, x(m),x(m), t )

into { u } x B ( M ) x A ( M ) is C", and we have Exp

=n J

G c J.

QED

6. AFFINECONNEXIONS

110

Problem 7. (a) If P is a principal bundle over manifold M , M having an affine connexion, m E M , then exp, may be factored through P. (b) Every bundle over Rd is trivial.

6.3.1 Completeness An afine connexion is called complete if all geodesics can be infinitely extended, that is, if each exponential map is defined on the whole tangent space. This is equivalent to saying that the local group of transformations of B ( M ) generated by any basic vector field E ( x ) can be extended to a global one-parameter group of transformation of B ( M ) .We shall see in Chapter 8 that completeness of a Riemannian connexion is equivalent to the completeness of the Riemannian metric.

6.3.2 Normal Coordinates

A coordinate map : U-+ Rd, U C M , is called a normal coordinate map at m = &l(O) if the pre-images of rays through O E Rd are geodesics, where a ray is a straight line of the form u -+ux, x E Rd. If we choose a basis b e B ( M ) with n(b) = m, then we have an identification of lid with M,. Combining this with exp, and applying the theorem we see that the function exp, o b is the inverse of a normal coordinate system at m. A normal coordinate neighborhood N , the domain of a normal coordinate map 4,has the property that every n E N can be joined to #J-'(O) by a unique geodesic in N . We remark that if the curvature and torsion of our connexion both vanish, then by theorem 1 there are coordinate systems whose inverses send any lines in U C Rd into geodesics, that is, the coordinate system is normal with respect to each of its points. This is the affine version of the local isometry of a flat Riemannian manifold with Euclidean space (see corollary, theorem 9.3). #J

Problem 8. Show that the exponential map at the identity of a Lie group, for any of the connexions of problem 6, is the same as the exponential map of the Lie group when we identify the Lie algebra with the tangent space at the identity, so these connexions are all complete. Hence, in view of problem 2.2, the exponential map of a complete affine connexion need not be onto even though the manifold be connected.

6.4. Covariant Differentiation and Classical Forms

111

Problem 9. If G = 5’3, as a subgroup of the multiplicative quaternions, show that the geodesics of S3are great circles.

6.4 Covariant Differentiation and Classical Forms 6.4.1 Covariant Derivatives

We can use parallel translation in T ( M )to give a notion of directional derivatives of vector fields. More generally, this can be done in any vector bundle associated with B ( M ) , so that each representation of GZ(d, R ) on a vector space F gives rise to the concept of covariant differentiation of cross sections of the bundle with fibre F. We give several definitions of this notion, ignoring the problems of equivalence and independence of the choice of curve. We fix an afine connexion H with form 4. Let ( W, F , G, M ) be a vector bundle, associated with B ( M ) , with fibre F , and group G = Gl(d, R). Then each b E B ( M ) gives an ) that forf E F , isomorphism of F onto the fibre of W over ~ ( bsuch g E G we have b ( g f ) = ( b g ) f ( 3 . 3 ) . Let U C M be a neighborhood of m E M and let X : U -+ W be a cross section of U. Finally take t E M,. We shall give several definitions of V , X , the covariant derivative of X with respect to t. T h e notation D,X is also commonly used. In the first place, V , X will be an element of the fibre of W over m. If Y is a vector field on U , then V,X will denote the cross section over U given by V,X(n) = V Y ( n X ) , the covariant derivative of X in the direction Y(n). (i) V , X gives a measure of how much X fails to be horizontal in the direction t. Thus it is a comparsion between the lifting of t given by X,namely dX t , and the horizontal lifting oft, H’(dX t ) ,where H’ is the connexion on W arising from the affine connexion on B ( M ) (5.4). Since the fibres of W are vector spaces we may identify the vertical tangents with elements of the fibre, in the usual way in which a vector space is identified with its tangent space at any point (2.5). Using this identification we define

v,x = V(dX t ) = dX t

-

H’(dX t ) .

(ii) V , X is differentiation with respect to parallel translation. Let be a curve with y*(O) = t , let e,(u), ..., ek(u) be a basis of the fibre of F over y(u) such that each ei is a horizontal lift of y , that is, ei(u)

y

6. AFFINECONNEXIONS

112

is obtained by parallel translation of ei(0) along y to y(u) (5.4). Define real-valued, C" functions fi by X ( y ( u ) ) = Cifi(u) e,(u). Then we have V,X = ei(0).

Z~;(O) a

(iii) V,X corresponds to the derivative in a horizontal direction of a function on B ( M ) associated to X . For each b E B ( M )n n-l( U ) we define

2 ( h ) = b-'X(.rrb),

3 is a function from B ( M ) n +(U) into F. Note that X ( b g ) = g-lb-lX(rr(b)) = g-IX(b). Conversely, any C" function .$ : B ( M )-+F such that X(bg) = g-'X(b) gives rise to a cross section x : M -+W , namely, n(b) + b X ( b ) . I n this case it is just as easy to define V,X as VJ. Let 7 be the unique horizontal lift of Y to W , so that P(b) is the unique horizontal tangent such that d n y(b) = Y(rr(b)).Then we have so that

V,X

=

the cross section associated with the function px.

Problem 10. Show that the definitions of covariant derivative given are all the same. We give some examples. (i) If F = Rd,W = T ( M ) , then X is just a vector field. If X is parallel along y then V,*X = 0, and conversely. Note that in this case, the 2 of (iii) above is just w ( x ) , where 2 is any lift of X to B ( M ) . Using this and the structural equations allows us to get a convenient formulation for V,X. Theorem 11. Let X be a vector field on M , t E b E B ( M ) ,8 a lift of X.

Then V,X

=

b(h(2)

Mm , t a lift of t to

+ +(t)w ( x ( b ) ) ) .

Proof. Let Y be an extension of f to a right invariant vector field on B ( M ) .Then [Y - H Y , 2 1 is vertical, being rr-related to [0, XI, so

dw(Y

-

H Y , X ) = ( Y - H Y )w ( X )- X w ( Y = (Y - H Y )w(X)

H Y ) - w ( [ Y - H Y , XI)

H Y ) w ( X ) + +(X) w(Y - H Y )

=

-+(Y

=

-+(Y) w ( X ) .

-

~

6.4. Covariant Differentiation and Classical Forms

113

Evaluating at b we get H L ( X ) = L(X)

+ +(t)w(X(6)).

Applying b to H i w ( x ) gives V , X by (iii), which proves the formula. Comparing with the result of lemma 9 easily yields: Corollary.

y is a geodesic if and only if V , J ,

=

0.

(ii) If F = gr(d, R) and the action of G on F is simply the adjoint representation, then W is the bundle which has fibre above m equal to the set of all linear transformations of M , into itself. T h e cross sections correspond to horizontal, equivariant, Ramvalued 1-forms on B ( M ) . (iii) G acts on Rd* as follows (4.5). I f f €Rd*, g E G, v E Rd, then (&Y)(V)

= f(g-'v)*

T h e corresponding bundle W is just the Grassmann bundle G1, and cross sections are 1-forms. T h e action of G on Rd* can be extended uniquely to a group of homomorphisms of the Grassmann algebra over Rd*. This action leads to covariant differentiation of differential forms. (iv) By combining the actions of G on Rd and Rd*, we get an action on tensor products of copies of Rd and Rd*. T h e cross sections of the corresponding associated vector bundles are tensor fields. If there are r copies of Rd and s copies of Rd*, then the tensor fields are said to be of type (r, s). For example, the torsion translation is a tensor field of type (1, 2), while the curvature transformation is a tensor of type (1, 3). Example (ii) above is actually the case of tensors of type (1, 1). (v) If X is a cross section into W andf a real-valued function on M , then fX is again a cross-section over M . Its covariant derivative is related to that of X as follows. Let t E M , , then V t ( f - v =f ( m ) V J Proof.

+ (tf) X ( m ) .

By the third definition of covariant derivative, we have that fi(6)

=

b-l(f(.rrb) X ( n b ) )

= f(7Tb) b-lX(nb) =f(n6) X(b).

6 . AFFINECONNEXIONS

114 That is

fi

Hence

it@)

= (fon)X

+ f 0 n ( b ) ix

=

i ( f 0 n) X ( 6 )

=

(tf)X ( 6 ) + f(.rrb) iX,

and applying b to both sides yields V,(fX) = (tf)X ( 4

+f(4 VJ.

Problem 11. For parallelizable M show that cross sections of a tensor bundle are in one-to-one correspondence with functions of M into the fibre F and that covariant differentiation for the direct connexion corresponds to differentiation of these vector-valued functions. Problem 12. For parallelizable M and vector fields X,Y as in problem 5(d) derive the following formulas for covariant derivatives: direct connexion torsion zero connexion opposite connexion

VXY = 0 VXY = i[X,Y ] V,Y = [X,Y].

Problem 13. Let M be a d-dimensional manifold and let V be the covariant derivative of a torsion zero affine connexion over M . Let 6' be a differential form on M , and let Xl , ..., xd be a parallelkation of a neighborhood U of M , pl, ..., p d the dual 1-forms on U. Prove the following formula for the exterior derivative of 6' in U :

(Hint: Show that the operator on the right is an antiderivation and that it agrees with d on functions and 1-forms.) Hence conclude that a differential form is closed if there exists a torsion zero affine connexion with respect to which covariant derivatives of the form are zero; that is, the form is parallel along every curve. (A differential form 6' is closed if d6' = 0.) Problem 14.

Let 6' be a 2-form and X,Y , 2 vector fields. Show that

Generalize this.

6.4. Covariant Differentiation and Classical Forms

115

6.4.2 Covariant Derivative Definition of a Connexion

Covariant differentiation in the tangent bundle is sufficient to determine the connexion, since by the first example above it gives a differential equation V,*X = 0 for parallel translation along y . This differential equation is linear and hence has a unique solution for any initial condition. Thus an affine connexion is specified as soon as we give for each m E M , t E M T n ,and vector field X,an element V,XE M,,, such that (i) V,Xis linear in t , Vut+vsX

=

+ 4vm,

4VtX)

for u,v E R and t, s E M,; (ii) iff is a real-valued, C" function on M , then

V,(fX) = (tf)X ( m ) + f ( m ) vtx. Sometimes it is convenient to turn the covariant derivative around, that is, for each C" vector field X defined on an open set U of M we defined on each Mm with m E U , consider linear transformation T(X), by the formula 7(X) t = V , X . A connexion is then given by hypotheswhich satisfy the following condition, izing the existence of such T(X)'S t = f ( m ) T(X) t ( t f )X(m)(see [65]). corresponding to (ii): ~(fx) We now give a direct relation between covariant differentiation of vector fields and the connexion form on B ( M ) . It will depend on a local cross section of M in B ( M ) . Let X,,..., X, be vector fields defined on an open set U C M such that

+

x :m

-

(m,- J a m ) ,

...I

X&))

is a cross section U + B ( M ) . For any vector fields Y X on U we define L(XY)= (XfJ xi.

=

C, fixiand

2 1

Then the relation between the covariant derivative of Y in the direction of X and the connexion form q5 is (iii)

V X ( r n )y =

x(m) W

X a m ) )x h - l Y ( m )

+L(XY)(m),

where x ( m ) is now considered as a mapping of Rd to Mnl.

116

6. AFFINECONNEXIONS

Since 4 is known on vertical tangents and the vertical tangents along with those of the form dxt span B ( M ) x ( m )formula , (iii) is enough to define 4 on all of B ( M ) x ( n land ) , then by equivariance, on all of w l (U ) . Problem 15. Show that covariant derivatives of connexions combine in the same way as the connexions, that is, if V is covariant derivative for connexion H , is covariant derivative for connexion K , f a C" functiononM, then the covariant derivative for (fo T ) H (1 - f o T ) K is f V (1 - f ) T .

r

+

+

6.4.3 The Structural Equations [66]

T h e structural equations correspond to formulas for the torsion and curvature transformations in terms of covariant derivatives. T h e formulas are:

T h e first formula informs us that torsion tells how much V,Y-VyX fails to be [ X , Y ] , and the second that curvature tells how much V fails to be a Lie algebra homomorphism. Problem 16. Prove (i) and (ii). 6.4.4 Coordinates

I n any given coordinate system the connexion in terms of covariant derivatives can be expressed in terms of sufficiently many real-valued functions on M , which are indexed so as to make manipulations easier. These functions are called the coeficients of the connexion. Other functions give the coefficients of the torsion translation, the curvature transformation, and their covariant derivatives. T h e way in which these coefficients change when the coordinate system on M is changed can be found by using the chain rule for partial derivatives and the properties of the covariant derivative. We define these classical coefficient functions. Let x1 , ..., xd be the coordinate system on M , and put X i = Dz,.

6.4. Covariant Differentiation and Classical Forms

117

Then we have

vx,xi= 2 rijkxk, d

k=l

As a consequence of the equations of the last section we have

and

T h e condition for zero torsion then becomes ri,k = rjtk.

For this reason connexions with torsion zero are referred to as symmetric connexions. T h e notation used is that given by Nomizu [66], except for a change in the sign of T and R. Finally, we define the functions Ti: and Rimik directly in terms of the connexion. Consider the coordinate cross section x of B ( M ) defined by x(n) = (n,X1(n), .-.,X,(n)>. Then R i w i i k = @irn(dx Xj , dx X k ) , and rijk= + k j ( d x

where

Xi),

and QinL are the components of these gI(d, R)-valued forms.

6. AFFINECONNEXIONS

118

A knowledge of the rijk for each coordinate system of a family whose coordinate neighborhoods cover M determines the connexion, since 4 is determined on the cross sections [compare the remark following (iii) in 6.4.21. Problem 17. Let x be the cross section as above and let p, +, P , Y be the pulled down forms w o dx, 4 o dx, SZ o dx, and @ o dx. Obtain the following formulas for these forms in the coordinate neighborhood: p = (dx, , ..., dxd)

yirn=

2 Rirnjkdx, dXk. i,k

Thus derive the equation given for Rimikin terms of the rijk by using the pulled-down structural equation d , + $ - L2

[ A *I

+ YJ.

Problem 18. Pull down the formula d@ = -[4, @] (corollary, 5.5) to get the coordinate form of the Bianchi identity Rrnnij.k

Problem 19.

riik(m)

-k

Rmnjk,i

Rmnki,j

=

0.

Prove that if xi are normal coordinates at m, then

+ r j i k ( m )= 0.

Problem 20. Prove that torsion is zero if and only if for every m there is a coordinate system at m such that Tijk(m)= 0. Problem 21. Connexions and action of groups. Let G act on M to the left in such a way that if g E G and dg is the identity on some M , , then g is the identity of G. Choose b E B ( M ) , b = ( m , el , ..., ed) and define Ib : G -+ B ( M ) ,Z,(g) = (gm, dg el , ..., dg e d ) . Show that z b is an imbedding. Let G also act on B ( M )by gb = I&). Then the action is by bundle maps. An affine connexion on M is invariant under G if the connexion form 4 is invariant, 4 o dg = 4, for every g E G.

6.4. Covariant Differentiation and Classical Forms

119

If M is a homogeneous space of G, show that an invariant connexion on ( G , M , H ) (see problem 5.5) induces an invariant affine connexion 5, as before, then the imon M . I n this case, if we write g = m bedding takes into fundamental vector fields, m into all basic vector fields, restricted to I b ( G ) .

+

Problem 22. Product connexions. Let M’, M“ be manifolds with solder forms affine connexions having connexion forms +’, w ’ , w ” , etc. Let M = M’ x M“ and define the bundle of adapted bases over M to be the submanifold of B ( M ) : +It,

B(M’, M ” )

= {((m’, m”),e ,

and

, ..., e d ) I (m‘,e, ,. .., ed,) E R(M’)

(m”, eCtl , ..., ed) E B(M”)}.

T h e group of B(M’, M”) is Gl(d’) x G l ( d ” ) and it is clear that B(M’, MI’) may be identified with B(M’) x B(M”). Define a conon B(M‘, M“) and extend by equivariance to a nexion +‘@

+”

FIG.23.

+

connexion on B(M). M with this affine connexion is called the afine product of affinely connected manifolds M’ and M“. Show that the product connexion has the following properties: (a) If y’ and y” are curves in M’ and M”, X‘ and X” are parallel X” is parallel along y’ x y”, vector fields along y‘ and y ” , then X‘ and, conversely, a parallel vector field on M has this form. (b) T h e geodesics are products of geodesics on M‘ and M”. Hence the affine product of complete connexions is complete.

+

+ T“sp,,p,.

(4

Ts,+s..,t.+t..= T’s,,t,

(4

R,,,,,.,t,+p. = R’sf,tff R”,t,,t,..

6. AFFINECONNEXIONS

120

Problem 23. If i : iV+ M is a covering map, then there is an induced natural covering map i : B(N)+ B ( M ) . (The prolongation of i.) If M has an afine connexion and if y4 is the connexion form, then i*$ is a connexion form on B ( N ) .Describe this connexion in terms of parallel translation. Problem 24.

Connexions on the afine bundle. Let

A(d, R ) =

A x

1) E GZ(d

+ 1, R ) I A E Gf(d,R),x

E Rd

I.

( x is viewed as a column matrix.)

Show that this defines A ( M ) as a principal bundle over M . (See proof of theorem 10.) If we view Rd as the hyperplane of Rd+l which has final coordinate I , then A ( d , R ) acts to the left on Rd:

I)y = Ay + x,

X

for

y E Rd.

Hence there is an associated bundle S ( M ) which has fibres homeomorphic to the tangent spaces of M . Make this correspondence explicit. Define maps 7] : B ( M )+ A ( M ) T~ : GZ(d,R ) + A ( d , R )

by q(b) = (b, 0), qc(A)= (t :), and show that this gives a bundle map of B ( M ) into A ( M ) which induces the identity map on the base space M . Let w be the solder form on B ( M ) and consider a connexion c j on B ( M ) , with @ and 1;2 as the curvature and torsion forms. T h e Lie algebra of A ( d , R ) may be considered to be

6.4. Covariant Differentiation and Classical Forms

121

6

Define an a(d, R)-valued form on ? ( B ( M ) )with the property that (8 :), and show that may be extended by right translation to a connexion form on A ( M ) ,also denoted by+. If 3 is the corresponding curvature form, show that

6

y*$ =

A connexion on A ( M ) arising in this way from a connexion on B ( M ) is a special case of a Cartan connexion (see [35,48, 491). By considering other horizontal, @-valued equivariant forms in place of w , more general Cartan connexions may be defined, and in general all the connexions on A ( M ) whose distributions are disjoint from T ( q ( B ( M ) )arise in this way, since gI(d, R ) is reductive in a(d, R). Returning to the connexion 6,we note that the parallel translation induced in the associated bundle S ( M ) gives rise to affine transformations of the tangent spaces to M which depend on both the curvature and torsion of the connexion 4 on B ( M ) . Infinitesimally, a curvature transformation R may be defined in a way analogous to the curvature transformation of an afine connexion as follows. Let x, y , x E Mm , ( b , t ) E A ( M ) such that ~ ( b t,) = m, 2, jj lifts of x, y to (b, t). Then

R,, z

=

-(b, t ) @@, j q b , t)-'z,

where we are identifying the fibre of S ( M ) at rn with Mm, By choosing (b, t ) E q ( B ( M ) ) ,show that

Rr,z = R,, z + Tzy-

CHAPTER 7

Ri ern a n n i an M an i fo Ids T h e definition of a Riemannian structure on a manifold is given and the corresponding topological metric is shown to induce the same topology. T h e bundle of (orthonormal) frames is defined and the existence and uniqueness of the Riemannian connexion is established. T h e chapter concludes with a large selection of examples [33,50, 831. 7.1 Definitions and First Properties 7.1.1 Riemannian Metrics and Associated Topological Metrics

A Riemannian manifold is a manifold M for which is given at each m E M a positive definite symmetric bilinear form ( , ) on M , , and this assignment is C" in the sense that for any coordinate system (xl , ..., xd) the functions g..= ( D Z i, D,,)E C". Such an assignment 2.3 is called a Riemannian metrzc on M . If we let S y ( M ) be the bundle of symmetric positive definite tensors of type (2, O), then a more elegant version of the above definition is that a Riemannian manifold is one with a preferred C" cross section of this bundle Sy(M). Let M , N be Riemannian manifolds with metrics ( , )M and ( , ) N . Then a C" map f : M + N is an isometry if it is a homeomorphism and preserves the metrics, that is, for t , s E M , , (dft, dfs)N = ( t , s ) ~ .An isometry is a diffeomorphism. f is called a ZocaZ isometry if we relax the requirement that it be one-to-one. If M is an oriented Riemannian manifold then there is a unique d-form % which determines the orientation and such that % ( e l ,..., ed) = h l for every orthonormal basis e, , ..., ed of M m . % is called the Riemannian volume element of the oriented Riemannian manifold. 122

7.1. Definitions and First Properties

123

We defer examples in this chapter to the last section. If (xl, ..., x d ) is a coordinate system on any manifold M with domain 0, then there is a natural inner product on the tangent spaces to 0, namely, the Euclidean inner product (D, , D,,)= aij. We denote by 1 1 11' the Euclidean norm. We also let ( 1 I ( be the Riemannian norm, that is, 1 1 t 1 1 = ( t , t)1/2, t E T ( M ) , so that we have 1 1 Dz, 1 1 = (gii)lI2E C", and in fact for ai E C", 1 1 ZiaiDZ,1 1 E C" at each point at which not all the aivanish. If we require ( , ) to be only nondegenerate instead of positive definite, then M is called a semi-Riemannian manifold. T h e main result of this chapter, namely, the existence and uniqueness of a Riemannian connexion, holds in the semi-Riemannian case. Problem 1. T h e index of a symmetric quadratic form on a real vector space is the dimension of a maximal subspace on which the form is negative definite. Prove that for a connected semi-Riemannian manifold the index of the metric is the same on every tangent space. For nonconnected manifolds we also require that the index be constant for a semi-Riemannian metric. A manifold with index 1 or d - 1 is called a Lorentz manifold. T h e four-dimensional time-space universe of Einstein is a Lorentz manifold.

If y is a broken C" curve in M , then its arc length is defined by I Y I = f l l aY * I I l

where [a, b] is the interval of definition of y . Problem 2. Let y be a broken C" curve defined on [a, b]. Define nondecreasing continuous function f on [a, b] by

f(4= j-1/ I Y* II. (a) Show that f is C" at every x such that y,(x) exists and is nonzero. (b) Show that Y o f-l : [0, 1 y I] + M is a continuous well-defined function even though f-1 may not be a function, and that it is C" at every f ( x ) for which y,(x) # 0. (c) Let (x,y) be a coordinate system on a two-dimensional manifold and define C" curve y on an interval with 0 as an interior point by the equations

124 x(y(t)) =

st

7. RIEMANNIAN MANIFOLDS

1'' exp (- l/s2) sin l / s ds.

[exp (-l/sz) sin 1/sI2ds, y ( y ( t ) )=

0

' 0

Show that for this curve y o f-' is not a broken C" curve so that y cannot be reparametrized with respect to arc length so as to remain broken C". We define a function p : M x M

where

r = set of all broken C"

Lemma 1.

-+

R

v

{+ a} by

curves from m to n.

The function p is a metric on M .

Proof. It is trivial that p is symmetric and satisfies the triangle inequality, so the only thing remaining to be proved is that p(m, n) = 0 implies that m = n. Assume that m # n, and let (xl , ..., xd) be a coordinate system at m with domain U . Let 0 be a ball with respect to the xi such that 0 C U and n $ 0 .Define a function f : Rd x U -+ R by f ( a , .'*> a d 4 = aPl.,(m) ?

7

(1 x

I/

Then f I S d - l X L I is continuous and positive, and therefore there exists a k > Osuchthat 1 -
I I Z aiDZi(m)I I'

=

1 on Sd-l x 0, and hence we have

for (a,, ..., a d , m) E Sd-l x 0. Then by the fact that the expressions in (1) are homogeneous in the q ' s , we have that (1) holds for ( a l , ..., a d , m) E Rd x 0. Now let y be any broken C" curve from

FIG. 24.

7.1. Definitions and First Properties

125

m to n, and let y’ be the part of y from m to the first place where y

intersects the boundary of 0. Finally, let a be the radius of 0. We then have 1. 1 p(m, n) = inf I y I 2 inf I y’ I 2 - inf 1 y’ 1’ 2 - a > 0, k k which completes the proof of the lemma.

Problem 3 . I n the proof of the lemma we have assumed the theorem about Euclidean space that says 1 u I’ 2 o( for every curve u which goes from the origin to a point on the sphere of radius a. Prove this result, and also that equality occurs only if u is a broken C” reparametrization of a straight line. [Hint: Let r be the function ( Z xi2)1/2.Split u*(t) into two components, one, u** tangent to the ray from the origin, the other normal to that ray. Show that I d(r o u)/dt I = 11 u*= (I, and apply the fundamental theorem of calculus. Compare the similar theorem for geodesics in a Riemannian manifold in Section 8.1.1

T h e topology given by p is equivalent to the topology of M as a manifold. Hence, p is a continuous function on M x M .

Theorem 1.

Proof. It is sufficient to find for each m, E M a neighborhood P whose topology is given by p. Let x1 , ..., xd be a coordinate system at m, , 0 be an open ball with respect to the xi with center at m, , 0, similarly, with 0, C 0. Let p’ be the Euclidean metric on 0 defined via the x i , so that p‘ defines the topology on 0. Hence, we wish to find a neighborhood P of m, and a number c > 0 such that P C 0 and (l/c) p’ p cp‘ on P x P. From Eq. ( 1 ) above, we have that there exists c > 0 such that

< <

for t E M , , m E 0,. Hence for any broken C” curve in 0, , we have (I/c) 1 y I’ ( y I c 1 y 1’. So we need only worry about curves which leave 0,. This we do by cutting down again. Let 6, = p’-radius of 0, , and take /I S,/(2c2 1). Let P be the open ball of radius /I with respect to the xi about m, . Now from the above remark, it is clear that in order to show

<

< <

+

7. RIEMANNIAN MANIFOLDS

126

on P x P, we need only prove that in the calculation of p on P x P we can restrict ourselves to curves in 0, . That is, if (m, n) E P x P and y is a curve from m to n, then there exists a curve 7 such that 1 7 I Iy I and 7 is in 0, . Now let y, = part of y from m to the first intersection point of y with the boundary of 0, , and let p = the straight line from m to n with respect to the xi . We then have

<

1

1

I Y I AYII >,cIyiIf3,(6,--)32c193cIyj'>,Iy[.

QED

FIG.25.

Before discussing further properties of Riemannian manifolds, we shall show that a large class of manifolds admits Riemannian structures. If M is a paracompact manifold, then M admits a Riemannian metric.

Theorem 2.

Proof. Let { Ui} be a covering of M by coordinate systems, and let { f i } be a C" locally finite partition of unity subordinate to this covering. Let (xli, ..., xdi) be the coordinate system associated with U , , and denote by ( , )i the Euclidean inner product defined on Ui via the xji. We then define an inner product on M by ( , ) = E fi( , )i. Remark. There is a similar result for real-analytic manifolds arising from the solution of the analytic imbedding problem [31, 561.

7.1.2 Vector Bundles

Let ( B , R", M ) be an n-dimensional vector bundle over M (3.3(4)), with B the bundle space and Rn the fibre. A Riemannian metric on the

7.2. T h e Bundle of Frames

127

vector bundle B is a C" assignment of a symmetric, positive definite bilinear form to each fibre. By C" we mean the following. Let U C M , x1 and x z be C" cross sections of U in B. Then the function f : U + R given by f ( m ) = ( x l ( m ) ,x z ( m ) ) is C". By a proof similar to the above it can be shown that an n-dimensional vector bundle B over a paracompact base M admits a Riemannian metric. I n fact, let U C M be an open set such that n--l( U ) is diffeomorphic to Rn x U. Then by considering a basis of Rn it is easy to see that there exist n linearly independent C" cross sections x1 , ..., xn of U into n--l(U). We therefore can define a Riemannian metric on r-'( U ) by setting (xi , xj) = aij . Now using a covering of M by open sets over which B is trivial and an associated partition of unity, the proof goes through in the same way. 7.2 The Bundle of Frames

Let M be a Riemannian manifold. We have discussed the bundle of bases B ( M ) over M. Let F ( M ) = {(m,el , ...,ed) I m

E

M , el , ..., ed an orthonormal basis of Mm},

and let n-' : F ( M ) -+M be the obvious projection. F ( M ) C B(M). We shall put a local product structure on F ( M ) so that it becomes a manifold, a submanifold of B ( M ) , and a principal bundle over M , called the bundle of frames, which represents a reduction of the group of B ( M )to the orthogonal group. Let m E M , (xl, ..., xd) be a coordinate system at m with domain U. We define a function A, : U + Gl(d,R). Let Vl , ..., V , be vector fields defined on U with the property that for all n E U , Vl(n),..., Vd(n)is the Gram-Schmidt orthonormalization Then h,(n) is defined by of D, (n),..., DXd(n).

2 (Au(fi))ijDzj(n)i

vi(n) =

j=1

Now we define the following maps,

4, 4,'

: T-'( U )+ GZ(d,R ) , : ~'-l( U )

---f

O(d),

128

7. RIEMANNIANMANIFOLDS

as follows: ei =

z

( 4 U h

el

I

*..,ed))ijvj(n)

fi = 2 ( Q / ( n , f * > ..., f d ) ) i f v M (n, e l , ..., ed) E B ( M ) , (n,fl , ...,fd) E F ( M ) . Pro-

= 1, ..., d, where ceeding similarly for a covering of M by coordinate neighborhoods and defining right action exactly as in 3.2, we see that we have defined principal bundle structures on B ( M ) and F ( M ) , and the local product representations are compatible with the differential structure. In particular, F ( M ) is a manifold and has structural group O(d),while the A, being clearly C", show that the bundle structure defined on B ( M ) is the same as that introduced previously. Further, we have immediately from 3.4 that F ( M ) represents a reduction of the group GZ(d, R) of B ( M ) to O(d), and F ( M ) is a submanifold of B(M).

i

M

=

M

FIG. 26.

Since o(d) may be viewed as the set of all d x d skew-symmetric real matrices, it clearly admits {Xii - Xii I i <j} as a basis, where Xii is defined as in 6.1.2. Hence, Ao(d) = Z(d) admits as a basis , {F.a$ = E a3. . - Eii 1 i < j } . I n other words, at each , ~ E F ( M )the vertical tangent space to F ( M ) , namely, V,' = V , n F ( M ) , , where V, is the vertical tangent space to B ( M ) , is spanned by the tangents

Fdf)= Wf)- Wf)I i < j } . If H' is a connexion on F ( M ) , then it can clearly be extended to a connexion H on B ( M ) by right action. Hence, H gives rise to a parallel translation of tangents to M along curves in M . Further, it is clear that this parallel translation preserves scalar products, for it is defined via F ( M ) by sending one orthonormal base into another.

7.3. Riemannian Connexions

129

Conversely, if H is a connexion on B ( M ) such that parallel translation preserves scalar products, then H comes from a connexion H' on F ( M ) in the above manner. Indeed, let b = ( m , fl , ..., fd) E F ( M ) and let y be a horizontal curve in B ( M ) passing through (m,fl , ...,fd). Then every point on y must belong to F ( M ) , since parallel translation along T 3 y takes fi , ...,fd into orthonormal bases, by assumption. Therefore Hb C F ( M ) , , so we may define H' by H'b = Hb . Problem 4. Prove that a reduction of the group of B ( M ) to O(d) gives a unique Riemannian metric such that the reduced bundle is

F(M)* Problem 5. Extend the results of this section to apply to a

Riemannian metric on an arbitrary vector bundle.

Remark. In a similar way, a semi-Riemannian structure on M gives a reduction of B ( M ) to a subgroup of GZ(d, R ) which leaves invariant a non-degenerate symmetric bilinear form, and conversely.

7.3 Riemannian Connexions

Let M be a Riemannian manifold. A connexion on B ( M ) is called a Riemannian connexion if it satisfies the following properties: (i) parallel translation preserves inner products, (ii) the torsion form is zero. We note that in view of the above remarks, a connexion on B ( M ) is a Riemannian connexion if and only if it is the extension of a connexion of F ( M ) whose torsion form is zero. We defined the solder form w on B ( M ) in the last chapter. If i : F ( M ) 5 B ( M ) is the inclusion map, then we again denote by w the horizontal 1-form i*w defined on F ( M ) .If H' is a connexion onF(M), H its extension to B ( M ) , and if we denote the corresponding l-forms by $ and $', respectively, then the first structural equation (theorem 6.4) pulls back to F ( M ) ; namely, dw = - 4 ' w

+ i*Q.

Hence, we have that H is a Riemannian connexion if and only if it comes from a connexion on F ( M ) , also called a Riemannian connexion, whose 1-form 4' satisfies the relation dw = +'w.

130

7. RIEMANNIAN MANIFOLDS

Lemma 2. There is a one-to-one correspondence between horizontal o(d)-valued 1-forms T on F ( M ) and horizontal Rd-valued 2-forms, in such a way that the 2-form associated with T is T W . When T is not horizontal it is still determined by T W .

We first show that for a horizontal 2-form l2 there is a unique horizontal 1-form T such that SZ = T W . Let K(x, y , z ) = (Q(x, y ) , ~ ( x ) ) so , that K is 3-linear, horizontal, and skew-symmetric in the first two variables. Since w is onto Rd, there is a unique T ( X ) E gr(d, R), x E F ( M ) , , such that for every y , z E F ( M ) , Proof.

2 <+) 4 Y ) , 42))

=

K(x,y , z )

+ K(z,y , .x) - q x , z, y).

This does not overdetermine T ( x ) , because if we alter y and x by ) ~ ( x do ) not change, then the rightvertical vectors, so that ~ ( yand hand side will not change either. T is obviously horizontal. Since interchanging y and z on the right-hand side changes its sign, we must have T ( X ) E o(d). T o show that TW = SZ, it suffices to show that K(x,y , z )

= (.(X)

4 Y ) , 44)- (+)

w(x),4.)>.

This is true because the second plus the third term of the right-hand side above is invariant under interchange of x and y , while the first term changes sign. When T is not horizontal then using TW in place of SZ in defining K , it is an automatic verification that 2(T(X) ~ ( y )~, ( z )is) given as above in terms of K , so TW determines T. QED Remark. Lemma 2 may be proved by showing the map T + TW is one-to-one and then applying a dimensionality argument. T h e lemma also holds if the forms T are taking values in a subalgebra consisting of all linear transformations which are skew-symmetric with respect to some non-degenerate symmetric bilinear form in Rd.

There is a one-to-one correspondence between connexions on F ( M ) and horizontal equivariant 2-forms on F ( M ) . T h e 2-form may be taken to be the torsion form of the connexion, so two connexions are equal if and only if they have the same torsion.

Theorem 3.

+

Proof. Let 4 be a connexion form on F ( M ) . Fixing we get a one-toone correspondence between connexion forms $I on F(M) and the difference forms T = $I - +, which is an arbitrary o(d)-valued horizontal equivariant 1-form. T h e lemma shows that T is determined

7.3. Riemannian Connexions

131

by T W , and furthermore it is trivial to verify that T is equivariant if and only if TW is equivariant. This establishes a one-to-one correspondence between connexions on F ( M ) and horizontal equivariant 2-forms T W . Finally, by the first structural equation we have

+

that is, Ql = TW Q. Thus by choosing r w = -Q we get a connexion +o with torsion 0. If we started with +o instead of we would have Q, = T W , that is, the correspondence would then be between connexions and torsion forms. QED Problem 6 . Find a formula for +o in terms of above lemma and the fact that +ow = -dw.

+

w

and

dw

using the

There exists a unique connexion on F ( M ) having torsion 0; thus there exists a unique Riemannian connexion on B(M). (This follows immediately from the previous proof.)

Theorem 4.

Remark. By the previous remark it is clear that by the same method we may prove the existence and uniqueness of Riemannian connexions (same definition) for semi-Riemannian structures.

Proposition. T h e connexions on F ( M ) having the same geodesics as a given connexion are in one-to-one correspondence with horizontal equivariant 3-forms on F ( M ) . Proof. T h e condition on the difference form T in order that two connexions have the same geodesics is T ( X ) w ( y ) = - T ( Y ) W ( X ) (theorem 6.9). T h e 3-linear function ( T ( x ) w ( y ) , w ( z ) ) is a 3-form if and only if T satisfies this condition and is o(d)-valued. That it is horizontal and equivariant is easily checked. [Equivariance here means invariance under right action of O(d).] Problem 7. On a two-dimensional Riemannian manifold different connexions on F ( M ) have different geodesics.

two

Prove that an isometry of a connected Riemannian manifold is determined by its value and differential at one point as follows:

Problem 8.

7. RIEMANNIAN MANIFOLDS

132

(a) First reduce to the problem of showing that an isometry which leaves a point and the tangent space at that point fixed is the identity. (b) An isometry takes geodesics into geodesics. (c) Show that an isometry which leaves a point and tangent space at the point fixed leaves a neighborhood of the point fixed. (d) T h e collection of all fixed points of a continuous map is a closed set. 7.4

Examples and Problems

1. Euclidean space. For every m E Rd we define (Di(m), D j ( m ) )= &, thus defining the jlat Riemannian metric on Rd. Unless otherwise stated, a metric on Rd will be assumed to be the flat one. Problem 9. Show that the connexion on Rd given by its group structure as in problem 6.6 is the Riemannian connexion, so the Riemannian connexion is indeed flat. What are the geodesics ? 2. One-dimensional Riemannian manifolds. Since Riemannian manifolds are metrizable they must be paracompact, so the underlying manifold of a connected one-dimensional Riemannian manifold must be either R or S1.In either case there are only two vector fields having norm constantly 1, and an integral curve of such a vector field provides an isometry with an interval of R (as in example I ) or a covering map of S1 by R which is locally an isometry and periodic as a function on R. I n the latter case the smallest period is the circumference of S1, divided by 27r it is the radius of S1,since S1may be imbedded in R2 as a circle of that radius (see example 3). 3. Imbeddings. If i : N -+ M is a C" imbedding (or, more generally, an immersion) and M is a Riemannian manifold, then there is a unique Riemannian metric on N such that di preserves inner products; when N has this metric i is said to be an isometric imbedding (immersion). A very difficult theorem of Nash [63] says that every Riemannian manifold has an isometric imbedding in an arbitrarily small neighborhood of Euclidean space of dimension d(d 1) (3d 11)/2, and if it is compact, dimension d(3d ll)/2. One of the harder outstanding problems is to what extent this theorem can be improved in the allowable gap in dimensions and in the amount of uniqueness possible. (Immersions are defined and discussed in Chapter 10.)

+

+

+

7.4. Examples and Problems

133

Problem 10. Take a circle of radius r and center (s, 0, 0), r < s, and lying in the plane uz = 0 and rotate it about the u,-axis. Realize this surface as the range of an imbedding of T 2 ,the 2-torus. Compute the Riemannian metric. 4. The Riemannian d-sphere of radius r , Sd = {x E Rd+' I I I x j 1 = r } , is obtained by giving it the induced Riemannian metric from Rd+l. Rotations are all isornetries, so S d has O ( d 1) as a transitive group of isometries.

+

Problem 11. Let xo , ..., xdPl be linearly independent points of Rd+l all of which lie on S d .Show that there exists a unique isometric imbedding of Sk,the k-sphere with radius r, in Sd whose range contains xo , ..., xk , k = 1, ..., d - 1. This gives a chain S1 c S2 C ... C Sd-l C Sd. Prove that any such chain can be mapped on any other by an element of S O ( d 1).

+

Problem 12. Prove the following facts about Sd:

(a) For any m E S d ,there is a unique isometry j , leaving m fixed and such that dj, = - identity on Sdlll. (b) For each nonzero t E Sd, , there is a unique great circle F ( t ) tangent to t , and S1(t)is invariant under j , , that is, S1(t)= j , ( S 1 ( t ) ) = Sl(-t). (c) If X is a parallel vector field along S1(t),then a!jmX== - X . (d) If m, n, p are equally spaced along S 1 ( t )and X ( m ) = t , use j , to show that X ( p ) = a tangent to S 1 ( t )at p , and hence S 1 ( t )is a geodesic. 5 . Riemannian products. If M' and M" are Riemannian manifolds, then M = M f x M" becomes a Riemannian manifold when the inner products are defined as (s'

+ s", t' + t " ) = (s',

t')'

+ (S'',

t")'',

that is, tangents to one manifold are regarded as being perpendicular to those of the other manifold. M is then called the Riemannian product of M' and M". For example, Rd is the d-fold Riemannian product of R . Problem 13. Show that the Riemannian connexion of the Riemannian product of Riemannian manifolds is the affine product of the Riemannian connexions of the manifolds. (See problem 6.22.) 6. Flat tori. T h e Riemannian product of S1with itself d times is called the $at d-dimensional torus T d . No distinction is usually made

7. RIEMANNIAN MANIFOLDS

134

between S1of different radii, and in a given torus all of the radii need not be the same. Show that the flat d-dimensional torus T dis flat. 7. Covering manifolds. If i : N + M is a covering map, M a Riemannian manifold, then i may be considered as an immersion, and when N is given the induced Riemannian metric, (i, N ) is called a Riemannian covering of M . For example, the simply connected Riemannian covering space of a flat torus is R d , since this is true for d = 1, and the relation is preserved under products. Problem 14.

Prove the last remark, namely, that if i’ : N‘ + M ’ , are Riemannian coverings, then the Riemannian product N’ x N” is the Riemannian covering of M‘ x M ‘ with covering ma pi’ x i”. Problem 15.

i“ : N”

+

m’

Problem 16. There is a reverse process, also. If i : N + M is a covering map and if N has a Riemannian structure for which the deck transformations are isometries, then there is a natural induced Riemannian structure on M which makes (i, N ) into a Riemannian covering of M . For example, real projective space P a is covered by S d and can be given a structure from that on the Sd of example 4. 8. Parallelixable manifolds. If X I , ..., X, are parallelizing vector fields on a manifold M , then ( X iXi) , = (constant functions) defines a corresponding Riemannian metric on M . Rd and T d are special cases; more generally, any Lie group has such a metric. Problem 17. Show that the Riemannian connexion of this metric is the same as the direct or opposite connexions if and only if [Xi , Xi] = 0 for every i , j . Problem 18. T h e Riemannian connexion of a parallelization is the torsion free connexion (problem 6.5) if and only if for all constant linear combinations X,Y of the parallelizing fields ([X, Y], X) = 0, which is equivalent to the condition

+

([Xi Xjl, x,> <Xj [Xi > Xkl) I

7

=

0

for every i , j , k . Prove this by piecing together the following: (a) T h e Riemannian and torsion free connexions are the same if and only if the geodesics of the Riemannian connexion are the same as those of the direct connexion (corollary to theorem 6.9).

7.4. Examples and Problems

135

(b) Let 0 be the O(d)-valued form on M obtained by pulling down the Riemannian connexion form via the natural cross section of the parallelization, and p the parallelizing form. Show that the two connexions are the same if and only if O ( t ) p ( t ) = 0 for every t E T ( M ) . (Use theorem 6.9, noting that 0 is a difference form pulled down to M.) (c) Use the structural equation dp = -0, and the skew-symmetry of 0 to show the desired result. (Note ( p ( [ X ,Y ] ) p, ( X ) ) = ( [ X , Y ] ,X ) . ) Notice that an affine connexion gives a parallelization of B ( M ) and hence a Riemannian structure, and the same holds for a Riemannian connexion on F ( M ) . Show that in the latter case, the horizontal geodesics in F ( M ) are precisely the lifts of geodesics in M , while the vertical geodesics are integral curves of the fundamental vector fields F i j . However, integral curves of constant linear combinations of fundamental and basic vector fields are not in general geodesics. 9. Homometries. Iff : N + M is a diffeomorphism of Riemannian manifolds, then f is called a homometry (with expansion factor a ) if for every s, t E N , , any n, (df(s), df(t)) = a2(s, t ) . Iff is the identity map on a manifold, then for a given Riemannian metric on M (or N ) this formula defines another metric so that f is a homometry. We have already noted that S1 may have different radii, and this amounts to changes of S1by homometries. If N = M as a Riemannian manifold, then there may be no homometries which are not isometries ( a = 1). Indeed, this is the case when M is compact (more generally, has finite volume), for then a homometry would have to multiply volume by the factor ad. (We have not defined volume, but it can be done along fairly usual measure theoretic lines.) If M has infinite volume it may well have self-homometries. For example, on Rd multiplication by a is a homometry.

-

Problem 19. Show that a homometry f : M-+ N , when extended to a

B ( N ) ( J h , el , e d ) = ( f ( m ) ,df(e,), ... df(e,))), map J : B ( M ) called the prolongation of f to B ( M ) , preserves the Riemannian connexions of the homometrically related metrics on M and N . I n particular, multiplication of a metric by a positive scalar does not alter the Riemannian connexion. However, two metrics with the same connexions need not differ by a scalar, as can be seen by considering a Riemannian product. a * * ,

9

7. RIEMANNIAN MANIFOLDS

136

10. Conformal maps. If we allow a to be a positive C" function of n in example 9, we get a conformal map

f : Iv-+ M ,

(df(s),W)) = (.(W (s, t>,

s, t

E

Nn

T h e linear fractional transformations of the Riemann sphere of complex analysis are conformal maps of the Riemannian 2-sphere, and the homometries of Rd combined with stereographic projection give conformal maps of the Riemannian d-sphere. Problem 20. Prove that stereographic projection is a conformal map of Sd - {pt} onto Rd. 11. Action by compact groups. If G is a compact Lie group then G has a unique (Haar) measure which is invariant under left and right translations and assigns 1 to all of G. If G acts differentiably on M , then we can obtain a metric on M so that G acts as a group of isometries. For let { , ) be any Riemannian metric on M , and for s, t E M , , define

Problem 21. Show that G acts as isometries on ( M , ( , )'). Iff is any self-isometry of ( M , ( , )) which commutes with the action of G, that is, fg = gf for every g E G, show that f is an isometry of ( M , ( , )'). Problem 22. Show that the orthogonal complements of the vertical tangents with respect to a right invariant metric on a principal bundle form a connexion distribution. Hence, give an alternate proof that a paracompact principal bundle admits a connexion (compare 5.4). Problem 23. (a) Show that a left invariant metric on a Lie group G corresponds to an inner product on the Lie algebra g. (b) A left invariant metric on G is also right invariant (and so adjoint invariant) if and only if the corresponding inner product ( , ) on is invariant, that is, if X , Y , 2 E g, then

that is, a d X is skew-symmetric. (This is simply a restatement of the fact that the function (Ad elxY, Ad e"2) is constant if and only if its derivative is zero. Compare with problem 2.5.) (c) If G is compact, it always admits such a metric.

7.4. Examples and Problems

137

(d) T h e Riemannian connexions of such invariant metrics on G are all equal to the torsion free connexion of problem 6.6, and hence are complete (see problem 18). (e) T h e Killing f o r m of a Lie group G is a bilinear form k( , ) on g defined by: if X , Y Eg, then R(X, Y ) = tr(ad X o ad Y ) .

K ( , ) satisfies the invariance property of (b), but is not in general

definite or even nondegenerate, for example, if g has a nontrivial center. 12. Riemannian homogeneous spaces. If M is acted upon by a transitive Lie group of isometries, then M is a Riemannian homogeneous space. By example 11, a homogeneous space of a compact Lie group may be given a homogeneous metric. Problem 24. Let H be a closed reductive subgroup of a Lie group G (see problem 5.5), so that g = b + m, where [b, m] C m. Assume m admits an inner product which is invariant under A d H . Then show that G / H is a Riemannian homogeneous space. V d , ? the , Stiefel manifold of ordered sets of r orthonormal vectors in Rd,is a Riemannian homogeneous space of both O ( d ) and SO(d), namely, Vd,T= O(d)/O’(d- r ) = SO(d)/SO’(d- r ) ,

where O’(d - r ) and SO’(d - r ) are viewed as acting on the last d - r components in Rd. If we make the definition O(0) = SO(0) = {I}, then O ( d ) = V d , d S, O ( d ) = V d , d - l ,and Sd-’ = Vd , l are special cases. 13. Flag manifolds. If d , , ..., d, is a partition of d , then we define flag manifold Fl(d ; d, , ..., d,) as the set of n-tuples ( V , , ..., V,), where Vi is a subspace of Rd of dimension di and these subspaces are mutually orthogonal. Alternatively, Fl(d ; dl , ..., d,) may be considered to be the set of increasing sequences (0)

=

W , C W , C W , C ... C Wn-l C W,

of subspaces of Rd with di = dim

=

Rd

W i- dim Wi-l .

Problem 25. (a) Establish a natural one-to-one correspondence between these two sets. (b) T h e orthogonal group O ( d ) acts on n-tuples ( V , , ..., V,) and the general linear goup Gl(d, R ) acts on increasing sequences

138

7. RIEMANNIAN MANIFOLDS

W, c W, c c Wn-, c W , = R d . Find the isotropy group in each case, thus giving FZ(d ; d, , ..., d,) the structure of a homogeneous space in two ways.

FIG. 27.

(c) Since O ( d ) is compact, there is a Riemannian metric on Fl(d ; d, , ..., d,) on which O(d) acts transitively as isometries. For p = ( V , , ..., V J E F Z (;~d, , ..., dn) and 1 i < j n, define mij to be the subspace of FZ(d; d,, ..., d& spanned by tangents to curves on which only Viand V j are varied. Show that these mij must be mutually orthogonal by using the invariance of the adjoint action of the isotropy algebra. 14. Riemannian symmetric spaces [13, 18, 331. If a Riemannian manifold M has an isometry fm for every m E M which leaves m fixed and such that df,, l M m = - identity (fm is called the symmetry at m ) then M is a Riemannian symmetric space. Since an isometry must take geodesics into geodesics, it is easy to see by a step-by-step use of symmetries at points along a geodesic that geodesics are infinitely extendable, so M is complete. If M is connected, then any two points of M may be joined by a broken geodesic (see problem 8), and hence the composition of the symmetries about the midpoints of the geodesic segments is an isometry sending one of the points into the other. Thus the group of isornetries is transitive; and since this group is always a Lie group, M is a Riemannian homogeneous space [50]. Sd is a symmetric space.

<

<

Problem 26. Prove that for a Riemannian symmetric space any C" curve invariant under the symmetry about each of its points is a reparametrization of a geodesic. (Compare problem 12.) Problem 27. A Riemannian homogeneous space with a symmetry fm at one point is Riemannian symmetric.

Alternate de$nition of Riemannian symmetric spaces. Let M be a Riemannian homogeneous space, M = G / H . Then M is a Problem 28.

7.4.Examples and Problems

139

homogeneous Riemannian symmetric space if G admits an automorphism f with the properties (i) f 2 = identity, (ii) H is pointwise fixed under f and contains the greatest connected subgroup pointwise fixed under f . (a) Let T : G + M be the canonical projection, and define fo : M + M by: f o ( n ( g ) )= T ( f ( g ) ) . Show that fo is well-defined and df, = - identity on Mo , where 0 = x ( e ) . (b) Use the formulaf, o L, = Lfcg., o f o , where L denotes the leftaction of G on M , and the fact that L, is an isometry to show that f o is an isometry. Hence, prove that every homogeneous Riemannian symmetric space is a Riemannian symmetric space. (c) Let M be a connected Riemannian symmetric space. We have already seen that M is Riemannian homogeneous, M = G/ H, where H = { g E G I g ( m ) = m}, m fixed E M . Define f : G --t G by: f(g) = f,gf,-’. Show that f satisfies properties (i) and (ii) of a homogeneous Riemannian symmetric space. (See problem 8.) Problem 29. Let G be a Lie group with a two-sided invariant metric. Show that the inverse map ~!,t : G + G, defined by # ( g ) = g-l, is an isometry, and hence prove that G is a Riemannian symmetric space. G may also be exhibited as a homogeneous Riemannian space of the group G x G, with the automorphismfdefined byf(g, h) = (h,g ) . Find the isotropy group and make the identification of G with the homogeneous space. In particular, describe the action of G x G on G. Problem 30. Grassmann manifolds. G d , r ,the Grassmann manifold of r-planes in Rd, is a Riemannian homogeneous space of O(d),namely, Gd,r =

O ( d)

O(Y)x O’(d - Y)’

where O(r) is viewed as acting on the first r components in Rd and O’(d - r ) on the last d - r . Let g E O ( d ) , so that

where A is r x r , D is ( d

-

r ) x ( d - r ) , and define f by

7. RIEMANNIAN MANIFOLDS

140

Prove that f defines Ga,ras a homogeneous Riemannian symmetric space.

G , r = SO(Y)

SO(4

x SO'(d - r ) '

the Grassmann manifold of oriented r-planes in Rd, is a twofold covering of Gd,rand is also symmetric. Problem 31. Let M and N be Riemannian symmetric spaces, i : P - t M a covering map. Show that M x N and P with the

induced Riemannian structures are Riemannian symmetric spaces. Problem 32.

(z1

on

-

Lens spaces. Consider the map

, z2)

(z1

S3= {(zl , ~

exp 2 4 P , z2 exp 2 7 4 P )

I

2 )zi

E

C , I z1 I'

+ I z2 1'

z=

I},

where p and q are relatively prime integers. This map generates a discrete group of isometries of S3, and this group may be used as the deck transformations of a manifold M covered by S3. M is called a lens space and has a metric induced from S3. Prove that M is not Riemannian symmetric, even though its covering is. Remark. Notice that if, instead of connectedness, one assumes that the components of a Riemannian symmetric space M are isometric, then M is still a homogeneous Riemannian symmetric space, so the two notions are now equivalent. I t is not hard to think of examples of Riemannian manifolds for which this does not hold but which still admit symmetries f m , but they are rather artificial and will not be missed. 15. Complex projective spaces. Let S2d+1 be a Riemannian sphere of radius r . Then we have the principal fibre bundle (S2d+1,S1,C P d ) (problem 3.12), and S1acts as isometries, so the normal subspace to the vertical space at a point defines a connexion H in this principal bundle (problem 22). If we define dn- I x , to be an isometry between CP,d,,, and H , , this defines a Riemannian metric on CPd. T h e bundle maps which are isometries on S2d+1(this includes all of the unitary transformations of Cd+l restricted to S 2 d f 1 ) form a transitive group of isometries on CPd, so CPd is a Riemannian homogeneous space. A closer examination of the available bundle maps show that CPd is also symmetric.

7.4. Examples and Problems

141

16. Complex manifolds [23, 30, 921. Let M be a complex manifold of complex dimension d (see problems 1.7 and 3.10). M is called an Hermitian manifold if for each m E M there is given a positive definite Hermitian bilinear form ( , ) on Znt, and this assignment is C" in the sense that if zl,..., zd is a complex coordinate system at m, then the functions gij = (ajaz, , a/azj) are complex-valued C" functions, We cannot demand that ( , ) be holomorphic, since that is, gij E 9. g23. . = g3% .. Problem 33. Since M is complex, its bundle of bases B ( M ) is reduced to a bundle C B ( M )with group GL(d, C), and C B ( M ) may be viewed as the bundle of holomorphic bases of the holomorphic tangent spaces Z mto M . Show that the existence of a Hermitian structure on M is equivalent to the reduction of C B ( M ) to a bundle CF(M) with group U(d)C GL(d, C). Hence or otherwise show that a Hermitian structure always exists. Show that C F ( M ) may be taken to consist of holomorphic bases which are orthonormal with respect to the Hermitian forms. Problem 34. Establish that ( , ) can be uniquely extended to a positive definite Hermitian form on Y Vin n such a way that (s, t ) =

Problem 35.

-

(s, t)

if

s, t E=@,,, ,

and

(.*,,,,

s,,,) = 0.

Show that if s, t € A r n then , (s, t )

=

2R ( P s , P t )

=

Q R (S,

t),

where B = s - iJs, f = t - iJt and R = " real part of ". This then defines a Riemannian structure on the underlying real manifold of M , and so F ( M ) is defined and has a Riemannian connexion. Actually, we shall consider the F ( M ) defined by the metric ( >' = 2( , >. Show that J is an orthogonal transformation field with respect to this symmetric inner product. 9

+

Problem 36. Since F m = AnL i d n ,, any t € Z nmay X be decomposed as t = Rt iIt, with Rt, It ~ d ., Show , that ( R t , I t ) = 0 and hence show that the map

+

i : CE'(M)+F ( M )

142

7. RIEMANNIAN MANIFOLDS

given by i(m, t, , ..., td) = (m,Rt, , ..., Rtd , It, , ...,Itd) is a welldefined imbedding of the real manifold C F ( M ) and is consistent with the previously defined inclusion of C B ( M ) in B( M) . We shall write C F ( M ) for i(CF(M)). T h e following problems show that the Riemannian connexion on F ( M ) does not in general reduce to a connexion on CF(M). Prove that the following assertions are equivalent: (a) T h e Riemannian connexion on F ( M ) reduces to a connexion on CF(M),that is, if b E CF(M),then Hb is tangent to CF(M). (b) T h e parallel translate of a holomorphic tangent is holomorphic. (c) Parallel translation commutes with the projection P. (See p. 51.) (d) Parallel translation commutes with J . (e) Covariant derivatives of J vanish. (f) If X , Y are vector fields on M , then V,(JY) = JV,(Y). Problem 37.

Problem 38. For s, t E .A,$, define Q(s, t ) = -( Js, t ) . Show that Q is a differential 2-form on M and that Q(S,

t)

21(PS, P t ) = *I(?, t). =

M is said to be a Kahler manifold if dQ = 0. Prove that the following assertions are equivalent: (a) dQ = 0. (b) All covariant derivatives of Q are zero. (c) V,(JY) = JV,(Y), X , Y any vector fields.

-

[Hint: prove the following implications:

(4

(4 =- (b)

(4.

Recall problems 6.13, 6.14 and the vector field formula for the exterior derivative.] A direct proof that M is Kahler if and only if the Riemannian connexion on F ( M ) reduces to C F ( M ) is outlined in this problem. Let & , w!, be the components of the connexion and solder forms on F ( M ) . Let J?= Q o d r . Define a linear transformation field on Problem 39.

p

7.4. Examples and Problems

F(M) by:

following:

I y b = identity and

9 IHI,

=

( d r IHb)-l G J

143 G

d r . Prove the

(a) d c = 0 if and only if dQ = 0. (b) T h e vertical tangent space to C F ( M ) is spanned by the vector fields Fij + Fi+d,i+d and Fi,i+d - Fi+d,i . On CF(M) we have (c)

~i

G

=

- U . E+d

7

Wi+d

j

=

(1

Wi

< < d).

-

(f) t tangent to CF(M)implies Vt tangent to CF(M) 0 dsZ

=

0.

Problem 40. Show that Q is a form of type (1, 1) and that Qa # 0. If M is compact and Kahler, then show also that SZP is not exact for a n y p < d. (A form 6 is exact if there is a form i+h such that d$ = 6.) Problem 41. If ( , ) is a Riemannian form on a complex manifold with respect to which J is orthogonal, then construct a Hermitian form on M from which the Riemannian form is derived as in problem 35. Problem 42. Show that a submanifold (complex) of a Kahler manifold is again Kahler. Problem 48. Complex projective space. Let t o , ..., t, be homogeneous complex-valued coordinate functions on CPd. Then a basis of coordinate systems may be described as follows. Let

lJi = { p E CPd I ti@) # 0},

i = 0, ...,d ,

and define coordinate functions 4, : U , + Cd by z!3

=

x3.0 4 z. = t.p. 3 1 .

T h e 4, then define a complex structure on CPd. On each U iconsider z$$ . Notice that on ui n u k the function fi =

xy=o

fk

=fix:,$.

144

7. RIEMANNIAN MANIFOLDS

Use this to show that there exists a real closed form 1;2 of type (1, 1) on CPd such that on Ui Q = z d'd''jj. (Problem 4.28.) - '

Show that the symmetric form associated with 1;2 is positive definite and admits J as an orthogonal transformation, and hence show that CPd is a Kahler manifold with respect to the corresponding Hermitian structure. A result of the preceding two problems is that every nonsingular projective variety admits a Kahler structure.

CHAPTER 8

Geodesics and Complete Rie m a n n ia n M Q n if0 Ids In the first section the local minimizing properties of geodesics are established with the help of the Gauss lemma, while in the second the Hopf-Rinow theorem on complete Riemannian manifolds is proved. In particular, it is shown that in the complete case geodesics realize global distances. T h e arc length of continuous curves is also discussed [24, 33, 50, 831.

8.1 Geodesics

In 6.3 we defined for any m E M , b = (m,e, , ..., ed) E B ( M ) , maps exp, : M , -+ M , +b : M , -+ B ( M ) , depending on a connexion on B ( M ) and such that was a rather natural lifting of exp, . We noticed that these maps in general were only defined on a neighborhood of the origin of M , , they were C" on this neighborhood, and exp,, was a diffeomorphism of a possibly smaller neighborhood of the origin. I n order for them to be globally defined we must assume that geodesics from m are infinitely extendible. We now assume M is a Riemannian manifold and we obtain Z p , as follows: Let b = (m,fi , ...,fd) E F ( M ) .Let be the Riemannian connexion on F ( M ) , and let p = C p i f i E M , , and let 6 be the unique integral curve of CpiEi with G(0) = b. Then we define G , ( p ) = c?(1). Ei is the restriction of the vector field Ei on B ( M ) (6.1.2) to F ( M ) . exp, is C" and is in fact the radially horizontal lifting of exp, to F ( M ) .I n particular, the image of M , under eWb is inF(M). If @ is the curvature form of the Riemannian connexion, w the

+

145

146

8. GEODESICS AND COMPLETE RIEMANNIAN MANIFOLDS

solder formonF(M) as defined in 1.3, then we define forms 0, 0, yh on M , as follows: -

0 = exp,*+

@=E$+@ $

~

exp,*

=

W,

and the real-valued forms Oii, O i i , yhi are defined accordingly. T h e structural equations become d#

=

-0#

3 P, 01 + 0.

de = -

(In terms of components, the structural equations become

2 d

d#i

=

eki$k

k=l

d

doij =

-

k=l

+

OikOkj

@ij.)

If P E M , , p = ray from m to p , u = exp,o p, q b p = (m,fit, ..., fdt) (so fit is the parallel translate of fi along a), then if s, t E (W7')P 9

(4 (b)

d expms

j

=

2 #?(4,c

( d e ~ ~ rdexprnt) n~, = <#(s), #(t)>

I I d expms I l2

=

I I #(s) I l2 =

=

2 #i(s)

+i(t)

2 #N.

[(b) is an immediate consequence of (a), which in turn follows easily from the definition of 4.1 Thus, if xl, ..., xd is the dual base to the f i , s = Zs,D,((p), then &(s) - si is a measure of the difference between parallel translation in M and M , . If p = Zpifi E M , , s = c ZppiD,,(p)E ( M m ) p, that is, s is tangent to the ray p above, then (c)

(4

= CPi

qS)= 0.

8.1. Geodesics

147

This says that the lengths of radial vectors are preserved under d exp,, and that tangents to horizontal curves in F ( M ) are horizontal tangents. A C" rectangle in M is a map Q of a rectangle [a, b] x [c, d ] in R2 into M which can be extended to a C" map of a neighborhood of the rectangle. Q(a, c) is the initial corner of Q , while the base of Q is the (a, d )

/b,d)

FIG. 28.

curve T defined by Q o j c , where j c ( t ) = ( t , c). More generally, the curves T , = Q o j , are called longitudinal, while the curves = Q o 'j, ('jz(t)= (x, t ) ) are transversal. T h e "vector field" V defined along the base T by V(s) = r S * ( c )is the associated vectorjeld of Q (it is actually a curve in the tangent bundle of M ) , and Q is an associated rectangle of V . Canonical lifting. If Q is a C" rectangle in a Riemannian manifold M , and if F ( M ) has the Riemannian connexion 4, then for every f~ n-I(Q(a, c)) there exists a unique C" rectangle in F ( M ) with initial corner f such that (a) Q = n o Q,that is, Q is a lifting of Q , (b) +(Qo j,*) = 0, that is, the longitudinal curves of Q are horizontal, and (c) +(Qo 'ja*)= 0, that is, the initial transversal curve is horizontal. Problem 1. Prove the existence, uniqueness, and differentiability of the canonical lifting of a C" rectangle.

If Q is a C" rectangle in M , Q : [a, b] x [c, d ] -+ M , whose longitudinal curves are geodesics and the tangents to these geodesics are all of the same length, then if V is the associated vector field, the function (T,* , V ) is a constant. In par( t V) ( t ) for all t E [a, b]. ticular, if TC*(a)1 V ( a ) , then ~ ~ * 1

Theorem 1 (Gauss' lemma).

148

8. GEODESICS AND COMPLETE RIEMANNIAN MANIFOLDS

Proof.

C$Q

= dwQ =

Let Q be the canonical lifting to f E F ( M ) of Q. Also, write Q*$, wQ = & * w . T h e first structural equation becomes

-4QwQ, which when applied to (Dl, D,)

QwQ(D2) - D,UQ(D1)- wQ([D1 9

gives

Del) = -+Q(D,) wQ(Dz)

+ $Q(Dz)wQ(D,).

Now [Dl ,D,]= 0, and C$O(Dl)= 0 since the longitudinal curves of are horizontal; so we have, taking inner products with wQ(D,), (D1wQ(D2),w Q ( W - (DzwQ(D1), wQ(D1)> =

Q

(CQ(D2)W Q ( D l ) > w Q ( W

=o

since dQ(D2)is skew-symmetric. Now wQ(D,)oj, is constant since the curve T, is a geodesic, so

also,

Dl (wQ(D,), w

D,( w Q ( 4 ) ,

Q ( W =

wQ(D1)) =

PQm, wQ(D1)>;

(D

2 (D2wQ(D1), wQ(D1)>.

But (wQ(Dl), wQ(Dl))is constant, since we assumed the tangents to the longitudinal curves all had the same length, so Dz(wQ(D1), w Q ( W = 0.

Hence, the above equation becomes Dl(wQ(D2),wQ(Dl)> = 0. But clearly, along the base curve (wQ(D2),wQ(Dl)) = ( V , T,*). QED Corollary. Let p E M , , p = ray from 0 to p , cr = exp,, o p, and s E (M& . Then s Ip (in the Euclidean inner product) implies that d exp,s 1a.

8.1. Geodesics

149

This follows by applying Gauss’ lemma to a rectangle Q whose initial transversal is the degenerate curve m E M and the longitudinal curves are the images under exp, of the rays from 0 in M,. I n particular, the base is u. T h e exact construction is left to the reader. (This rectangle may be described briefly as “a piece of pie.”) T h e following result expresses the fact that locally geodesics minimize arc length among broken C“ curves. Let B C M , be a ball about 0 on which exp, is a diffeomorphism, let p E B , p = ray from 0 to p, a = exp, o p, and let T be any broken C“ curve from m to exp,,p in M . Then I T 1 3 I a 1, and equality holds only if T is a broken C“ reparametrization of a.

Theorem 2.

Proof. Let xl, ..., xa be dual to f l , ...,fa on M , and define the following objects on either B or B‘ = exp,B: f = r o expi]

T

=

2

=

d exp,T o expm-l, defined on B’ - {m}.

Xi%,

, the radial unit vector field defined on B - (0).

+

If q E M , , s E (M,,Jq, then we write s = sT sN where sT is a multiple of T(q) and sN 1T(q).For t E M,, , b E B‘, we write similarly t = tT t N ,where t T is a multiple of T ( b ) and t N _L T(b). We have d exp,,s = d expmsT d explnsN, and from the corollary above, d expl,,sN 1d exp,sT . Hence,

+

(i) d exp,sT

+

=

(ii) I I d expnis, I I

. Also, from (b), p. 146, I I ST I I .

(d exp,s), =

Let [a, b] be the interval on which T is defined, and let c E [a,b] be the smallest number such that P ( T ( c ) ) = r(p) = I a I. Now define a curve 7 on [a, c] by 7)=ao---

7

IIPII

Or.

150

8. GEODESICS AND COMPLETE RIEMANNIAN MANIFOLDS

Noting that

1 1 u*

-

jj =

1

__

II P II

/ I p 11, we have

d a o dr o d exp,-'(T*)

hence,

this is true on [a, c]. Therefore, if

T'

= T jLa,cl

, we have

~ ~ ~ > ~ ~ ' ~ > ~ ~ l > l ~ l ,

FIG. 30.

as asserted. If I T I = I u 1, then we clearly must have T * ( u ) = 0 for u > c and T*,, = 0, from which it follows that T and u have the same image. QED T h e square of the distance to m, p(m, Z)z, is a C" function on B'. Corollary 1. Proof.

It equals Z ( x i o exp,-1)2. (Z is the identity map on M.)

8.1. Geodesics Corollary 2.

for c

< radius

Corollary 3.

1TI

=

151

If B ( m , c ) denotes the ball of radius c about m, then of B, exp,(B(O, c)) = B ( m , c). If

T

p(m, n ) , then

is a broken C" curve from m to n such that T is a broken C" reparametrization of a geodesic.

Proof. T minimizes arc length from m to n, so T locally minimizes arc length and, by the theorem, is locally a broken C" reparametrization of a geodesic. This is enough. Q E D

Lemma 1. Let m E M , 0 a ball C M,, , such that exp, is a diffeomorphism : 0 -+ U , y : (a, b) + U a C" curve in U. Suppose that r = p(m, y ) has an argument t such that r'(t) = 0. Then the geodesic in U from m to y ( t ) is perpendicular to y J t ) at y ( t ) . Proof. This is an immediate consequence of Gauss' lemma applied to the lifting of the curves to M7,,and the comparable fact for Euclidean space.

Let N, P be submanifolds of M , u a geodesic from n E N to p E P such that I u I = p(P, N ) . Then u is perpendicular to both N and P. Theorem 3.

Proof. It is obvious that a piece of u minimizes arc length from N o r P to any point on u. For such points which are sufficiently close to N or P lemma 1 then shows that u is perpendicular to curves in N or P which pass through n or p , respectively, and hence u is perpendicular to N, and P, . Q E D If N is a submanifold of a Riemannian manifold M , let I ( N ) , the normal bundle to N , be defined by

I ( N ) = {(n,t ) E T ( M ) 1 t E M , for some n E N and t 1N,} [cf. 3.3(4)]. Show that I ( N ) is a submanifold of T ( M ) and that Exp I I , N ) is nonsingular on the trivial cross section of I ( N ) . T h e tubular neighborhood, L r ( N ) , of N with radius r in I ( N ) is the open set of I ( N ) which intersects each fibre in the open ball of radius r about the origin of the fibre. Problem 2 .

If N is a compact submanifold of the Riemannian manifold M , then there exists an r > 0 such that exp maps I , ( N )

Theorem 4.

152

8. GEODESICS AND COMPLETE RIEMANNIAN MANIFOLDS

diffeomorphically. T h e image of 1r ( N ) ,called a tubular neighborhood of N in M , has the property that all its points are joined to N by unique geodesics which minimize arc length to N . T h e proof is left as an exercise. A local version of this gives us the existence of normal coordinates for N in the following sense. Let n E N . A coordinate system at n in M is normal for N if the points of N correspond to part of a linear subspace of dimension equal to that of N and the straight lines perpendicular to this subspace correspond to geodesics perpendicular to N . Problem 3 . If i : N -+M is an immersion (diis an isomorphism into for every point), define a normal bundle of the immersion and tubular neighborhoods in it. When N is compact show that Exp : T ( M )-+ M gives rise to an immersion of l , ( N )into M , some r . Give an example to show that the property of having unique minimizing geodesics to i ( N )need not hold for points of the immersion of L r ( N ) . Problem 4. Consider the flat 2-dimensional torus obtained by identifying opposite sides of the unit square in R2 having opposite corners (0, 0) and (1, 1). Sketch the locus of points at distance 2/3 from the corner point. If we obtain a noncomplete (see below) manifold from this torus by removing the closed line segment from (1/4, 0) to (1/4, 1/2), how does this change the locus ? Problem 5. Show that a Riemannian covering map is distancedecreasing.

8.2 Complete Riemannian Manifolds

Assume M is a Riemannian manifold. We have a map E : T ( M )+ M x M , given by: E(m, t ) = (m, exp,t). For E to be defined on all of T ( M ) we must assume that all geodesics are infinitely extendible. But in any case, E is defined on a neighborhood of the zero cross section of T ( M )and is in fact C" there. We also have Lemma 2. For each m E M , dE is an isomorphism on T(M)cm,ol , so by the inverse function theorem, E is a diffeomorphism of a neighborhood of (m,0) onto a neighborhood of (m, m).

8.2. Complete Riemannian Manifolds

153

Proof. It is sufficient to prove that dE maps T(M),,,,,, onto M x M(,,,,,, , since the dimensions are the same. Let ni : M x M + M be the projection onto the ith factor i = 1, 2. Now we know that , which is given by E I n - ~ ( , n ) ( m , t ) = (m , exp,t), maps E (MvJo onto the tangent space to n;l(m) at (m,m). We conclude by showing that dE maps the tangents to the zero cross section of T ( M ) onto the tangent space to the diagonal of M x M , which suffices since M x M(,,,,,, is clearly spanned by tangents to ncl(m) and tangents to the diagonal. Let D : M + M x M be the diagonal map, D(m) = (m , m). Then we have E I(zero cross section) = D 0 I(zero cross section) . dn is onto, and dD is onto the tangent space to the diagonal. QED

Let C be any compact subset of M . Then there exists is defined on B(O(m),c) and maps it diffeomorphically onto B ( m , c), where O(m ) is the origin in Mm . Proof. We first note that if all geodesics are not infinitely extendible, exp, may only be defined in a neighborhood of O(m). However, by the theory of differential equations, it is clear that for each m E M exp, is defined on a ball whose radius is a continuous function of m, and hence we may take a c1 such that exp,n is defined on B (O(m ),cl) for every m E C. By corollary 2 to theorem 2, we need only show that there exists c > 0 such that, for each m E C, exp, is a diffeomorphism on B(O(m),c). Lemma 2 essentially says this locally, and hence by compactness the result follows. However, we must first translate lemma 2, which says precisely that for every m E C, there exists a neighborhood P, of (m, 0) which is mapped diffeomorphically onto a neighborhood of (m, m). Therefore, d exp, is regular (that is, 1 - 1 onto) and hence for every n E n (Pm), exp, is a diffeomorphism on the set {t E M , I ( n , t ) E PnL}= PnL,,;so we wish to show there is a c,, > 0 such that the ball, B(O(n),cm),in the Riemannian metric is contained in P,n,, , for all n E n(P,). Using the facts that T ( M ) has a local product structure with the topology of the fibre being given by the Euclidean metric, that the Riemannian and Euclidean metrics are equivalent on each tangent space (see lemma 7.1), and that the Riemannian metric is continuous, one finds that P,, contains a neighborhood of the form Lemma 3.

ac

> 0 such that, for each m E C, exp,

=

{(% t ) I n

E u r nI

II t II < C , J ,

154

8. GEODESICS AND COMPLETE RIEMANNIAN MANIFOLDS

where Urn is a neighborhood of m and c, > 0. Hence for n E U,,, , exp, maps B(O(n), ern) diffeomorphically onto B(n, cm). By compactness, C is covered by a finite number of U r n ,to each of which corresponds a c , ~ .Letting c = min c, gives the desired result. QED It now follows that the square of the distance function, p2, is C“ on a neighborhood of (m, m) in M x M . Theorem 5. (Hopf-Rinow [40, 77, 931). Consider the following

conditions on a connected Riemannian manifold M :

(a) M is complete. (b) All bounded closed subsets of M are compact. (c) For some point m of M , all geodesics from m are infinitely extendible. (d) All geodesics are infinitely extendible. (e) Any m, n E M can be joined by a geodesic whose arc length equals p(m, n). T h e conditions (a)-(d) are equivalent, and they imply (e). Problem 6. Find an example to show that (e) does not imply (a). Also find an example to show that a minimizing geodesic between two points need not be unique; in fact, there may be infinitely many. Notice that (c) can be stated “exp,, is defined on all of Mm”, and (d) can be stated “ T h e Riemannian connexion is complete’’ (6.3).

Proof. In any metric space it is true that (b) implies (a). That (d) implies (c) is trivial, and that (a) implies (d) follows from extendibility theorems in differential equations (see appendix). We therefore have only to prove the implications from (c) to (b) and from (b) to (e). We now fix m, E M , and define, for any real Y > 0,

8, = {m E &I I p(m, ,111)

< r } = B(m, , r ) ; -

E,. = {m E B, I m can be joined to m, by a geodesic of length p(m, m,)).

I t suffices, assuming (c) for m, , to prove (1) E, is compact and (2) E, = 8,;for any bounded subset is in a 8,, so by (1) and (2) a closed bounded subset is compact, which is (b); then since (b) implies (d), we can use any m for m, , so that (2) implies (e).

8.2. Complete Riemannian Manifolds

I55

which is clearly compact, since it is closed and bounded. Compactness of E , then follows from the fact that E , = exptnOErand the continuity of expm,. QED For the proof of (2) we need the following: Lemma 4. If E , = 8, for some r , and if p(m, , n) > r, then there p(m, n). exists m such that p(m, , m) = r and p(m, , n) = r

+

Proof. For K = 1, 2, ..., choose a, , a broken C" curve from mo to n with I a, I < p(m, , n) l/k (this is possible by the definition of p). Let mk be the last point on a, in B , , so p(m,, m,) = r . By com-

+

FIG. 31.

pactness of E , = 8,, the m k have a limit point m,and by passing to a subsequence we may assume {m,} converges to m. Now p(m0 7 n )

<

dm0

9

m)

+

f(H4

n)

=

r

+ p(m, 4.

On the other hand, p(m, , n) > 1 a , I - l/k = I part of ak from m, to m k I I part of ak from mk to n I - l / k r p(m, , a) - l/k, which has limit r p(m, n), so p(m, , n ) >, Y p(m, n), so equality holds. This proves lemma 4. We now prove (2) using the connectedness of the nonnegative real numbers; that is, we show:

+

(i) E, (ii) E , (iii) E,!

+

> + +

8,. = B , , r' < r , implies E,,! = 8,f . = 8,t for all r' < r implies E , = 8 , .

=

156

8. GEODESICS AND COMPLETE RIEMANNIAN MANIFOLDS

(iv) E , = 8, implies there exists c Er+c =

> 0 such that

Bwc-

(i) and (ii) are trivially true. Proof o f (iii). Let m E B , . If m E B,t for r' < r, then by hypothesis m E E,) C E , . Hence, we assume p(m, , m) = r . By lemma 4 we may choose a sequence (m,} which has limit m, where mk E 8, , r k < r . Then by hypothesis, mk E ErkC E, , so m E E , by compactness of E , , which proves (iii). Proof of (iv). By lemma 3, there exists c > 0 such that for each m E 8, , exp, maps B(O(m),2c) diffeomorphically onto B(m, 2c), since E, is compact. Let n E B,+, . We show n E E,+, . By lemma 4 there p(m, n), where p(m, , m) = r . exists m E 8,such that p(m, , n ) = r Therefore, p(m, n) c, so there exists a geodesic y from m to n with I y j = p(m, n). Let u be a geodesic from m, to m with I u 1 = p(m, , m) = r . Then u y is a broken C" curve from m, to n with I 0 y I = I 0 I I y I = r p(m,n) = p(m0, n), so by corollary 3 to theorem 2, u y can be reparametrized as a geodesic. Hence, n E E,,, . This completes the proof of the theorem. It has been shown by K. Nomizu and H. Ozeki [67] that every connected paracompact manifold admits a complete Riemannian metric; furthermore, they show that if every Riemannian metric is complete then the manifold is compact. T h e converse, that a compact metric space is complete, is well known.

+

<

+

++

+

+

Problem 7. Construct a noncomplete connected Riemannian manifold of infinite diameter such that no two points at distance greater than 1 from each other may be connected by a curve which minimizes arc length.

Let i : N + M be an isometric imbedding. Let p be the Riemannian distance on M , p' = p o i, p " the Riemannian distance on N . Give examples to show that all eight possibilities for these metrics to be complete or not may occur. Problem 8.

Problem 9. Show that for a connected Lie group which admits a left and a right invariant metric the exponential map is onto. Problem 10. Let M and N be complete Riemannian manifolds. Show that the Riemannian product M x N is complete. Completeness of affine connexions is also a property preserved under products.

8.2. Complete Riemannian Manifolds Problem 11.

157

Let M be a complete Riemannian manifold.

(a) Then any Riemannian covering of M is complete. (b) For every m E M there is a geodesic segment starting and ending at m in every homotopy class of loops at m. Problem 12. Give examples to show that the result in problem 1 l(b) may or may not hold if M is not complete. Problem 13. Let M be a complete Riemannian manifold which is not cannot be a C" function on all simply connected. Show that p(m, of M . Example. Consider a mechanical system with a finite number of degrees of freedom and no elements which dissipate energy. Then the conjiguration space is a manifold M which is a mathematical model of the totality of all positions of the system. A curve in M is generally thought of as occurring in time, so that a tangent vector is an assignment of velocities to the elements of the system consistent with the constraints of the system. T h e phase space of the system is T ( M ) . T h e kinetic energy given by an assignment of velocities is a positive definite quadratic form on each Mm , and thus gives a Riemannian metric on M . A force field on M is a 1-form 9; the integral of this 1-form on a curve is the amount of work done in traversing the curve. If it is a conservative force field then 9 = - d V , where V is the potential energy. If D is the covariant derivative symbol of the metric and X is the vector field defined by 2 ( X , Y ) = O( Y ) for every Y , then Newton's laws of motion for the system become

where y is a curve parametrized by time. I n particular, when the motion is free the path of the motion is a geodesic and the time is proportional to the length of the geodesic. T h e Riemannian manifold is thus complete if the system will coast indefinitely when given an arbitrary push. I n this case the system may be started in any one configuration with kinetic energy 1 in such a way that it attains another given configuration by free motion with an elapse of time equal to the distance between the two points on M . More specifically, the configuration space of a rigid body constrained

158

8. GEODESICS AND COMPLETE RIEMANNIANMANIFOLDS

to rotate about its center of gravity is homeomorphic to P3 = SO(3).

T h e kinetic energy metric is invariant if and only if the ellipsoid of inertia is spherical.

8.3 Continuous Curves

Let y be a continuous curve, y : [a, b] -+ M , and let a = to < t ,

< t , < .'. < t , < b

=

t,+,

be a sequence of numbers between a and b. Then we define the arc length of y by I y I = sup { p(y(ti), y(ti+,)) I all such sequences ti}. Proposition 1. If y : [a, b] -+ M is a continuous curve from m to n with I y I = p(m, n), then y is a continuous reparametrization of a geodesic. This implies that geodesics locally minimize arc length among continuous curves. Proof. Pick a c > 0 by lemma 3 with respect to the image of y . Take a < t, < t, < b such that p(y(tl), y(t,)) < c/2, so there exists a geodesic u from y ( t l ) to y(t,) with I u 1 = p(y(tl),y(t,)). We claim that for every t E (tl , t J , y ( t ) lies on u. For choose geodesics u1 , u, from r(t1) to y(t), y ( t ) to y(t,), respectively, with I 0 1 I = p(y(t,), y ( t ) ) and 1 u, 1 = p(y(t), y(tz)). Then since y clearly locally minimizes arc length, we have 1 0 1 0 2 1 = p(y(t1), y ( t ) ) p(y(t), y(t,)) = I y from r(t1) to y(t,) I I y from y ( t ) to y(t2) I = p(y(t,), y(t,)), so by the theorem 2, I u, u, I is a broken C" reparametrization of u, which proves y ( t ) lies on the image of u. Hence, y is locally a continuous reparametrization of a geodesic; so in particular it is a reparametrization of a broken C" curve. T h e result then follows from corollary 3 to theorem 2.

+ +

+

Proposition 2.

+

If y is a broken C" curve, then the two definitions of

arc length agree. Proof. Let { y } be the length of y as a continuous curve, retaining I y 1 as the notation for the integral of tangent lengths. Then it follows trivially from the definition of distance that { y } 1 y I. We

<

also have, easily, that both definitions are additive:

{r+d={r)S{4, I r + u l

=

IrlS

101.

8.3. Continuous Curves

159

+

Now suppose y is a curve for which { y } 12 = I y I, 12 > 0. Then by splitting y in half, we must have that the discrepancy on one of the halves is at least k / 2 . By repeatedly halving we get a nested sequence u, such that the discrepancy of y restricted of parameter values s, to [s,, u,] is at least k/2", and u, - s, = c/2,, where c = b - a, [a, b] is the interval of definition of y. Let t be the common limit of s, and u, , and let t , be a number in [s, , u,] such that

<

which exists by the mean value theorem. Then

Now reparametrize y so that t 2"/c, and take limits, getting

=

0, multiply this inequality by

Let xi be normal coordinates at y(O), and fi = xi o y . Then we have p(y(s), y(0)) = ( Zfi(s)2)1/2= ( Xfi(Ois)2)1/2 I s I, where 0 < Bi < 1, by the mean value theorem. Thus if we write g(4

=

P ( Y ( 4 , r(O))/ls I

and d u n ) un - As,) sn = g(Un)(un - s n )

+ ( d u n ) - d-4)sn,

we get

Since limn+mg(u,) = 1 1 y.+(O) 1 1 , this shows

II Y*(O> II 2 K i C + II Y*(O) 11, which contradicts 12

> 0. QED

Problem 14. Let y : [a, b] -+ M be a continuous curve with finite length in a Riemannian manifold. Show that y can be uniformly approximated by broken geodesics.

160

8. GEODESICS AND COMPLETE RIEMANNIAN MANIFOLDS

Problem 15. Let 4 : M ---t N be a map between Riemannian manifolds which is onto and preserves distances. Prove that 4 is an isometry, (See [50],p. 169.) This says that the topological metric of a Riemannian manifold determines both the Riemannian structure and the differential structure. Problem 16. Let 4 : M + N be a C" map of complete Riemannian manifold M onto Riemannian manifold N such that for every m E M , M,, decomposes orthogonally into subspaces V , and H,, V and H C" distributions, where V , = ker(d4,) and dr$, restricts to an isometry from H , onto NbC,,.Show that: (a) If y is a C" curve in N and +(m) = y(O), then there is a unique lift 7 o f y to M s u c h that Y(0) = m,y = 4 o 7, and I y I = I 7 I. (b) T h e lift, as in (a), of a geodesic is a geodesic.

CHAPTER 9

Riemannian Curvature T h e main properties of the Riemannian curvature are established, including a direct, but for the most part impractical, method of computing curvature. Following a selection of examples, the Jacobi equation is established for vector fields associated with rectangles with geodesic longitudinals and a number of local and global consequences are derived. In particular, it is shown that for a complete Riemannian manifold with nonpositive curvature, the exponential is a covering map [24,33,50,831. 9.1 Riemannian Curvature

Let M be a d-dimensional Riemannian manifold with metric ( ,) and curvature transformation R,, , s, t tangents to M . A plane section P at m E M is a 2-dimensional subspace of M,,, . Let P be a plane section at m, and let s, t E M , be two vectors spanning P. T h e Riemannian (or sectional) curvature of P, K ( P ) , is defined by

where A ( s , t ) = ( 1 1 s l l a 1 1 t I j a - (s, t ) 2 ) 1 / zis the area of the parallelogram spanned by s and t. T h e first aim is to prove that K ( P ) depends on P alone and not on the particular choice of s and t spanning P. Simultaneously, it will be proved that K ( P ) determines R,, , and so nothing is lost by considering the Riemannian curvature instead of the curvature form on F(M). 161

9. RIEMANNIAN CURVATURE

162

Note that, if dim M = 2, there is only one plane section at each m E M , and so K is a real-valued function on M , called the Gaussian curvature. Problem 1. Letf : M -+ N be a local isometry. Show thatfpreserves curvature. Prove that the d-dimensional Riemannian sphere has constant curvature, and hence also the d-dimensional real projective space. Lemma 1. For x, y, z,w E Adrn, the curvature tensor R satisfies the following properties:

(4 Rx, = -Rim

+ R,xy +

(c) Hxvz

(b) (RXYZ,w > = -<%

I

Rgzx

RZYWh

(4 (RxyZ,w > = ( R Z W X , Y ) .

= 0,

One way of interpreting these properties is to view R as a linear transformation of the second Grassmann space G2,L, that is, the space of bivectors. T o do this we define R,, to be the bivector which satisfies ( R z , , z w ) = (Rryx,w ) for all decomposible bivectors xw. Then (b) says that R,, is a well-defined bivector, (a) says that it depends only on the bivector xy and not on x and y individually, so that xy + R,, can be extended linearly to an endomorphism of all bivectors; (d) says that R is a symmetric transformation of bivectors, so the corresponding quadratic form determines the transformation. Furthermore, (c) says that the quadratic form is determined by its values on the decomposible elements alone. (See corollary 2 below.) If xl, ..., xd is a coordinate system at m, then the classical object Riikl is given by Ri3kl

=
7

Xl>

where X , = Dxt . T h e above formulas then correspond to the classical ones, namely: (a’)

Rtjkl

= -R,z/cZ

(c’)

R,jkl

f

Rkz,Z

(b’)

1

f R,k?l

=

O,

(d’)

Rwkl = Rt3kl

-Rzm

= Rklz~

I

*

Proof of lemma 1. Let x, y E MnL, b E F ( M ) ,x , y E F ( M ) ,such that d r r x = x,drrjj=y. T h e n b y 6 . 1 . 5 , i f z ~ M , ,

RXoz= -b@(n, 9)b - l ~ , where b is regarded as a map Rd -+ Mn, .

9.1. Riemannian Curvature

163

(a) then follows from the fact that 0 is a 2-form, and hence is alternating, while (b) follows since @ is o(d)-valued, o(d) consisting of skew-symmetric transformations of Rd. In order to prove (c), we first notice that for a, b, c E Rd,

Choosing 2,j j above in a particular way, we have

But then the Jacobi identity gives

which proves (c), since E and b are one-to-one. (d) follows from (a), (b), (c) by taking inner products of equation (c) with w, then cyclicly permuting x, y , x, w . T h e four equations thus obtained are then added, and proper use of (a) and (b) will yield (d). Details are left to the reader. Corollary 1. Proof.

out that

K(P)is well defined.

For x, y

E

Mm , let K ( x , y ) = ( R , , x , y ) / A ( ~ , y ) ~We . point

(i)

q x ,Y ) = q y ,4,

(ii)

K(ax, by) = K ( x,y ) ,

(iii)

K(x t- cy,y) = K ( x , y ) .

+

if ab # 0,

+

I t then follows that if x' = ax by, y' = cx dy, ad - bc # 0, then K(x',y ' ) = K ( x , y ) , since it is well known that the transformation from (x, y ) to (x', y ' ) can be obtained by a sequence of the types indicated in (i), (ii), and (iii).

9. RIEMANNIAN CURVATURE

164 Corollary 2.

T h e ( R z y x ,y ) determine the curvature transfor-

mations. Proof. More precisely, ( R Z y z , w ) is the only 4-linear function satisfying the properties of lemma 1 which restricts to ( R Z g x , y ) .So we assume that we have two 4-linear functions, f and f on M , which satisfy the conditions corresponding to (a)-(d) and such that f ( x , y , x, y ) = f ’ ( x , y , x, y ) , all x, y E M , . Letting g = f - f’,we see that g satisfies these same conditions corresponding to (a)-(d). z in g(x, y , x, y ) = 0 we get Replacing x by x I,

+

,.(g

+

y , x, Y )

+,.(g

y , z, Y ) 4-,.(g

g(., y , z, y )

+,.(g

y , x, Y) = 0.

y , x, Y) , .(g

and hence,

Replacing y by y

Y , z, Y ) = 0

+ w and following the same procedure gives g(x, w , z, y )

+ g(.,

y , z, w ) = 0.

By (d), then (a), we get

., ., w ) is invariant under cyclic permutation of the three entries. But the sum over such permutations is 0 by (c), so g = 0. QED

sog(.,

Remarks. (1) Sometimes it is more convenient to deal with curvature instead of curvature transformations and this corollary assures us that this will not lose information. (2) If M has two Riemannian structures such that at a single point the inner product and curvature are the same, then the curvature transformations are the same. (3) It is not correct to say that the curvature determines the curvature transformation, for two different Riemannian structures, with different curvature transformations, may give rise to the same curvature. For example, let f: S2 + S2be any diffeomorphism of the Riemannian 2-sphere. Viewing f as an isometry gives two Riemannian structures on S2 with different curvature transformations but the same (constant) curvature.

9.2. Computation of the Riemannian Curvature

Problem 3.

165

Use the following outline to prove Schur's theorem

[ I 71: If K is constant on every fibre of Gd,2(M), then K is constant on Gd,2(M),for d > 2. (a) This hypothesis is equivalent to: for every x, y E Rd, ( @ ( E ( x ) ,E ( y ) )x, y ) is constant on fibres of F ( M ) . (b) Since the functions depending on x, y in (a) determine the functions ( @ ( E ( x ) ,E(y))z , w ) , the hypothesis is equivalent to @(E(x),E(y)) is constant on fibres of F ( M ) .

( c ) If F@(E(x), E ( y ) ) = 0 for every vertical F , x, y E Rd, then E(z) @(E(x),E(y)) = 0 for every x, y , x E Rd, and hence @ ( E ( x ) ,E ( y ) ) is constant on F ( M ) , K constant on G d , 2 ( M ) . [Hint: Use the Bianchi identity D@(E(x),Q),E ( 4 ) = E ( x ) @(E(Y)Y E ( 4 ) and the fact that

[A,E(x)] = A E ( x ) - E ( x ) A so

E(x)A

=

+ E(y)@(E(z),E ( 4 )

+ E ( 4 @(E(x),E(y))

E(Ax)

for

A

E

=

0,

o(d),

+ E(Ax) = AE(x).]

9.2 Computation of the Riemannian Curvature

We indicate briefly how the Riemannian curvature can be computed in terms of the metric coefficients g i j . In particular, we show the connection between the curvature transformation and the metric. By 6.4.3, if X , Yare vector fields, then

9. RIEMANNIAN CURVATURE

166

where V , is the covariant derivative in the direction of X.However, by [66, p. 771 if X , Y , Z are vector fields, 2 < V ,Y ,Z)= X ( Y , Z ) + Y ( X , Z ) - Z ( X , Y )

+ ax, YI, z>+ w,XI, y >+ ( X , [Z, YI>.

These two formulas give the desired connection. The above formula depends on the following facts: (1) Torsion zero if and only if [ X , Y ] = V,Y - V,X, X , Y any C" vector fields. (2) Parallel translation preserves the inner product if and only if X ( Y , Z ) = (V,Y, 2 ) ( Y , V,Z). Prove these statements and the formula. Derive an explicit formula for K ( D x t ,D x j )in terms of the gii * Problem 5. Use this formula to obtain an alternate proof for problem 7.18. Problem 4.

+

9.3 Continuity of the Riemannian Curvature

K is not a function on M , the Riemannian manifold, but it is a function on the Grassmann bundle of 2-planes of M [3.3(5)],and in fact a continuous function. From this it will follow that the curvature on a compact subset of M is bounded. Let Gd,, be the Grassmann manifold of plane sections (twodimensional subspaces) of Rd.(See problem 7.30.) So Gd.2 =

O(d)/O(2)x O'(d

-

2).

We denote by G,,,(M) the bundle with fibre Gd,, associated to the frame bundle F ( M ) , where M is a Riemannian manifold. Thus Gas,(&') = F ( M ) x O ( d ) G d , If 2 . m E M , we write G,,,(m) for the ) m, then fibre of G,,,(M) over m. If b E F ( M ) such that ~ ( b = b : Gd,,3 G,,,(m) by : P -+ { ( b , P)} = ( b , P ) O ( d ) , the equivalence class of (b, P)in G,,,(M). But we know that b : RdZMm, s o b : Gd,2 {plane sections at m}, and the resulting identification of G,,,(m) with {plane sections at m} is independent of b. Hence, the Riemannian curvature K can be viewed as a real-valued function defined on Gd,,(M). for the space (We are here using the notation F ( M ) x O(d)Gd,z (F(&') x Gd*,)lO(4 of 3.3.)

9.3. Continuity of the Riemannian Curvature

167

Proposition 1. T h e function K : G d , 2 ( M-+ ) R is C", and hence, in particular, continuous. Proof.

Consider the diagram F(M) x O(4

p , 4 identification maps

It is only necessary to show that K o q o p is C". T o define the map p we must first choose an element, say P o , of G d , 2 .Then p(b, g ) = (b, gPo). Hence, K o q o p ( b , g ) = K(b(gPo)),remembering that b : Gd,2-+ {plane sections at m}. Let Po be spanned by orthonormal vectors x, y E Rd. Then K(bgpO)

=

(Rb(g%)b(gyl

=-

b(gx), b(gy))

(b@,(E(gx)(b),- m ) ( 4 ) g x , g y ) ,

which is clearly C" in b and g . QED Since Gd,2is compact, we have the: If C C M is compact, then there exist H , L E R such that for any plane section P at any point m E C , H K ( P ) L.

Corollary.

<

<

Remark. T h e curvature of a Riemannian manifold clearly depends on the particular Riemannian structure which the manifold is given. Thus, the flat torus has zero curvature everywhere, while the imbedded torus (doughnut) has points of both positive and negative curvature. However, a given manifold cannot admit arbitrary curvature. For example, it will be proved that a simply connected compact manifold cannot have everywhere nonpositive curvature (see corollary 2, theorem 4), while a noncompact complete manifold cannot have positive curvature bounded away from zero (Chapter 11). Moreover, there is a relationship between the curvature and the topological invariants of the manifold given by the Gauss-Bonnet theorem, which we do not study [19,22]. I n general, though, not much is known about this problem [ZI, 121.

I68

9. RIEMANNIAN CURVATURE

Problem 6. A complete Riemannian manifold is locally symmetric if the curvature of a plane section is invariant under parallel translation of the plane section along geodesics. Show that this is equivalent to the vanishing of the covariant derivatives of the curvature transformation. Problem 7. Let M be a Riemannian symmetric space. Show that the symmetry fm of M , carries a tangent into the negative of its parallel translate along the geodesic through m. Hence, show that M is locally symmetric. Conversely, it follows from a monodromy argument that a locally symmetric simply connected manifold is Riemannian symmetric. Let M be a Riemannian symmetric space as above. From previous results (7.4.14) we know that M is a homogeneous symmetric space, that is, M = G / H , where G admits an involution f. Further, let 0 = eH E M , b = { X E g I d f ( X ) = X } , m = { X Eg I d f ( X ) = - X } , then b is the Lie algebra of H . Problem 8.

Prove the relations [b,

$1 C b,

[Ij,

m] C m, [m, m] C 8.

Problem 9 . Prove that for X E ~e f,x 0 is a geodesic in M . [Hint: Show that if a is a geodesic, the curve f l ( t ) = f0f&) in G satisfies ( I ) fl is a one-parameter subgroup (of transvections) (2) fl corresponds to an element of m.] Show that there is hence an isomorphism between m and M , . Further, if the inner product ( , ) on M , is pulled back via this isomorphism to an inner product ( , ) on m, then show that it satisfies

([XYI,2 ) + (Y,[ X ,4) = 0, where X E b, Y , 2 e m . (A form satisfying this relation is called invariant with respect to b. Cf. problem 7.23.) Problem 10. Show that H is compact, using that it may be viewed as a closed subset of an orthogonal group. Hence, show that the form ( , ) may be extended to an inner product ( , ) on all of g (see 7.4.11) which is invariant under df and with respect to b. T h e Killing form k( , ) on g is also invariant under df, since ad(d!X) c ad(df Y ) = df c ad X o ad Y c df-' (problem 7.23). Hence, show that m and b are orthogonal with respect to each of these forms. Problem 11.

M is said to be an irreducible symmetric space if

9.3. Continuity of the Riemannian Curvature

169

ad b I m is real irreducible. I n general, there is a linear transformation S , of m such that for Y , Z E m, k(Y, Z ) = (sky,Z).

Show that m decomposes into the characteristic subspaces of S, and that these subspaces are invariant under ad b. I n particular, if M is irreducible there is a real number h such that on m

4 1= 7

, 1.

From problem 7.8 it follows that only the identity in H acts trivially on m,and, hence, if X E ~X, # 0, then ad XI, # 0. Using this and the fact that ad X is skew-symmetric with respect to ( , ) on g, show that k( , ) is negative definite on 5. When M is irreducible with h # 0 this shows that K ( , ) is nondegenerate on g, from which it easily follows that g has no proper Abelian ideals, that is, g is semisimple. I t can be shown that if k( , ) is negative definite, or h < 0 in this case, then G is compact (see [33], p. 122). Problem 12. Pick f E F ( M ) such that ~ ( f )= 0, and consider the corresponding imbedding of G as a closed submanifold of F ( M ) (see problem 6.21). Show that G is a subbundle of F ( M ) with group H , and hence that H may be viewed as a subgroup of O(d). Problem 13. G acts on F ( M ) and so it makes sense to speak of invariant vector fields on F ( M ) . Show that basic and fundamental vector fields are invariant. Problem 14. Let X E m, Y E b. Show that X is the restriction to G of a basic vector field, and hence that the Riemannian connexion on

F ( M ) reduces to a connexion on G. This proves that the holonomy group of M with respect to f is contained in the isotropy group H . T h e converse is also true [28].Show that Y is the restriction to G of a fundamental vector field, and in fact Y = hY, where is viewed as a subalgebra of o(d). If X = E ( x ) Ic , x E Rd,show that ad Y ( X )= E ( Y x ) IG. Problem 15. Let X , Y , Z E m transformation is given by

RXJ

=

M

M,

.

Show that the curvature

“ X , YI, ZI.

170

9. RIEMANNIAN CURVATURE

Use this to derive a formula for the curvature of a plane section of M,, in terms of the Lie algebra structure of g and the inner product ( , ). This then serves to determine the curvature everywhere on M . Problem 16. Assume that M is irreducible. Show that if X = 0, then curvature is 0 ; otherwise, curvature is nonnegative or nonpositive according as h is negative or positive, respectively. Problem 17. Calculate the curvature (constant) of the d-dimensional Riemannian sphere of radius r . Examples.

Grassmann Manifolds

Real case. If A is a matrix with real entries we let A* denote the transpose of A. T h e Grassmann manifold Gd+e,dof d-planes in Rd+e may be realized as the homogeneous space O(d e ) / O ( d ) x O(e). When split into corresponding sized blocks, the Lie algebra of O(d e ) may be considered to be the matrices of the form (fee" i), where A* = -A, B* = -B. It is easily checked that df(!c*$) = ($;c) defines a Lie algebra involution with fixed algebra o(d) o(e) which generates an involution of O(d e ) with fixed group O(d) x O(e). X ) to be the sum of the squares of entries of X,in If we define (X, e), (X,X) = -tr X 2 , then ( , ) is a positive this case X E o(d definite form on o(d e ) which is invariant under o(d e ) and df. Thus Gd+e,dbecomes a Riemannian symmetric space. T h e oriented Grassmann manifold

+

+

+

+

+

+

Gi+,,d= SO(d

+

+ e)/SO(d)x SO(e)

is a twofold covering of G d + e . dso , it is also a Riemannian symmetric space. Problem 18. Verify the unproved statements above. Find an explicit matrix formula for the involution f. Show that GA+e,dhas only one nontrivial isometry which corresponds to the identity on Gdte,d. What are the groups of isometries of Gd+e,dand G;+e,d? What are the isotropy groups of a point ? Complex and quaternion cases. We proceed as in the real case except that we let A* be the transpose conjugate of A. T h e same formula for df is an involution of the real Lie algebras, and extends to an

9.3. Continuity of the Riemannian Curvature

171

involution of the unitary or symplectic group to give the representations Hd+e,d= subspaces of dimension d in Cd+e = U(d + e ) / U ( d ) x U(e). Kd+e,d = subspaces of dimension d in Qd+e =

Sp(d

+ e)/SP(d) x W e ) .

(For the definition of the symplectic group Sp(d), see [25; p. 201.) T h e sum of norms of entries is again a positive definite quadratic form on the Lie algebra which is invariant under the whole Lie algebra (u(d e) or 5 4 d e)) and df. Thus each of these homogeneous spaces becomes a Riemannian symmetric space.

+

+

T h e tangent space at the basic point in each of these symmetric spaces may be identified with matrices of the form (”* :), and then with C itself, where the entries of C are in the appropriate field. As such it has the structure of a complex or quaternion vector space. I n the complex case this structure is invariant under the action of ad(u(d) u(e)), so that it may be induced on every other tangent space invariantly by using the action of U(d e); thus Hd+e,dhas the structure of an almost complex manifold. (It is true that “almost” may be dropped.) We call those plane sections consisting of complex multiples of a single vector holomorphic. I t is not true that the quater5 p ( e ) ) , so that the nion structure is invariant under ad(ep(d) quaternion structure cannot be induced invariantly on any tangent space of K d f e , d .

+

+

+

Problem 19. Show that the symmetric spaces Gd+e,d,Hd+e,d, Kd+e,d are irreducible with negative h (see problem 11). Furthermore,

they have nonnegative curvature, but not all positive curvature unless d o r e = 1. Problem 20. When d = 1 the Grassmann manifolds become projective spaces or spheres; show that the curvature may be described as follows, using problem 15: All sections have the same curvature. Real field: If X 1 Y , X , Y E m and 8 is the angle between Complex field: the holomorphic sections of X and Y , then K ( X , Y ) = 1 3 cos2 8. I n particular, only the holomorphic sections attain the maximal curvature of 4,and the so-called holomorphic curvature is constant.

+

9. RIEMANNIAN CURVATURE

172

If X 1 Y , X , Y E m, and 9 is the angle Quaternion field: between the 4-dimensional subspaces X Q and YQ, then K ( X , Y ) = 1 3 cos2 9.

+

T h e Lie algebra of matrices of the form ($* g), where -B, with entries in one of the above-mentioned fields, has an involution given by Opposite spaces.

A*

=

-A, B*

=

A 4 C *

C B)

A -C =

(4 B" )*

This involution extends to an involution of the connected subgroup G of Gl(d e, F ) corresponding to the Lie algebra, and the fixed subgroup H may be divided by to give a homogeneous space M . Those matrices { X } with A= B=O may be identified with the tangents to M at 0 = eH, and the sum of norms quadratic form, in this case tr X 2 , may be used to give M the structure of a Riemannian symmetric space. I n this case the inner product on m is invariant under only ad b, not under all of ad g as with the Grassmann spaces.

+

Problem 21. Show that these symmetric spaces are irreducible with positive A. Moreover, the map which takes (OC. 0") into (& 0") gives an isometry from the tangent space of the corresponding Grassmann space such that the curvature of corresponding plane sections under this isometry is the same in magnitude but opposite in sign. I n the case of the real field and d = 1 we obtain in this way hyperbolic e-space, Re provided with a metric of constant negative curvature. Show that the map exp : Re + M given by

0

exp(X) = exp (x*

1

H is one-to-one.

9.4 Rectangles and Jacobi Fields Most of the theorems of this chapter as well as those of Chapter I 1 are connected with the behavior of nearby geodesics or the infinitesimal analog thereof. Generally it is sufficient to consider a oneparameter family of such geodesics, and hence the natural setting for their study is a rectangle (8.1). Let Q be a C" rectangle in a Riemannian manifold M . We use the notation of 8.1 and the proof of therem 8.1.

9.4. Rectangles and Jacobi Fields

173

Lemma 2. If the longitudinal curves of Q are geodesics, then we have the following formulas:

D,wQ(Dz)- DzwQ(D1) = +Q(Dz>w Q ( W Dd"(D.2) = W D l , DZ), D,2wQ(D2)= W ( D , , D,)wQ(D1).

(a)

(b)

(4

(c) is a version of the Jacobi equation. [QQ was not used in 8.1, but is, of course, Q*Q, where Q is the canonical lifting of Q to f E F ( M ) . ] (a) has already been proved and used in theorem 8.1. T h e second structural equation, theorem 6.4, gives

Proof.

d+Q =

-

[+Q,

+Q]+ W .

Applying this to D , , D , gives: DAQ(D2)- Dz+Q(Dl)- +Q([Q D21) =

-

4([+Q(Dl>l# Q ( 4 ) l

But c # ~ ( D= ~ )0 and [Dl , D,]

=

-

+

[+Q(&), 4Q(D1)1) aQ(D1

Y

4).

0, so we have

DAQ(D?)= aQ(4 , a), which is (b). Now apply D,to both sides of (a), D12wQ(Dz)- DID,wQ(D1)= (WQ(Dz))w Q ( D 2 )

++Q(mDlwQ(m)*

But DlwQ(Dl)= 0, and so also D,D,wQ(D,) = D,D,wQ(D,) = 0 since [ D , , D,] = 0. Hence this becomes D12,Q(D2) = (~,+Q(Dz,) wQ(D1).

#

(c) now follows by substituting (b) into #. QED Jacobi fields provide the connection between the behavior of nearby curves and curvature. They are certain vector fields defined along a geodesic. Let u be a geodesic and let V be a vector field along u. [Actually V is a curve in T ( M ) over u.] V is a Jacobifield if V,*(VU*(V)) = -%,vo*.

Since V is defined only on 0,we shall often write V' for Vu*V.So the Jacobi equation becomes V" = R,*u* .

9. RIEMANNIANCURVATURE

174

We shall reduce this to the classical equation. Define functions Riikl on F ( M ) by: Rijkl = Q i j ( E k , E l ) . Choose b = ( ~ ( 0 e, ) ; , ..., ed) E F ( M ) such that ed = u,(O). Let 6 ( t ) = ( u ( t ) ; el(t), ..., e,(t)), P and Fa be the horizontal lifts of u, V , and ed respectively, and write V ( t ) = Z v i ( t ) ei(t). Then R V ( t ) U , ct,o*(t) = R V c t , e d c t , e d ( t )

= -

2

@id(Q),

G ( t ) )e i w

(6.1.5)

On the other hand, V”(t) = C vi”(t) ei(t), so the Jacobi equation becomes V i “ ( t ) = - Z k vk(t)Ridkd(6(t)). In particular vd”(t) = 0, so the component of V in the direction of u* is a linear function of t. Thus if V is perpendicular to u* at two points, it is perpendicular everywhere. In any case, the behavior of the other vi does not depend on vd since Ridad= 0. In the two-dimensional case we have vl” = -R 1 2 1 2 ( ~ q( t()t)) , or v“ Kv = 0, where K is the Gaussian curvature. Since the Jacobi equation is a linear second order equation we immediately obtain from the theory of differential equations:

+

Proposition 2. T h e Jacobi fields along u form a linear space of dimension 2d over R . For every x, y E M,,,,, there is a unique Jacobi field V such that V(0)= x, and V’(0) = y . T h e Jacobi fields which vanish at u(0) form a linear subspace of dimension d whose values for sufficiently small t consist of all of Mu,,,. If t is sufficiently small, for every x E Mu,,,and y E Mu(,,there is a unique Jacobi field V such that V(0)= x and V ( t )= y .

T h e following theorem characterizes Jacobi fields in a more geometric manner. A vector field V along a geodesic u is a Jacobi field if and only if there is a rectangle Q having base curve u, all longitudinal curves geodesics, and V its associated vector field.

Theorem 1.

Proof. If Q is such a rectangle, then the fact that V is a Jacobi field is really a corollary to lemma 2, for (c) is the Jacobi equation, only

9.4. Rectangles and Jacobi Fields

175

formulated in RZ.In fact, V ( t )= dQ(D,(t, c)), u*(t) = dQ(D,(t, c)), so regarding .(t) as a map : Rd + Mo,,,we have Rvo,a,

P, O*)O-’(o,)

=

-O@(

=

--o@Q(R01) WQ(D,)

=

OD12wQ(D,)@ j c

=

V’.

1

(6.1.5) 0i c

[ j c ( t ) = ( t , c)]

[(c), lemma 21

On the other hand, if V is a Jacobi field, then by proposition 2 it will be sufficient to find a rectangle with longitudinal curves geodesics and associated vector field (along u) W such that W(0) = V(0)and W’(0) = V’(O),for then Wwill be a Jacobi field by what we have just proved and so will agree with V everywhere by uniqueness. Let y be a curve such that y J 0 ) = V(O),and let 7 be the lift of y starting a t f = (‘(o);fl , ...,fd) E F ( M ) ,so y(t) = ( y ( t ) ; f l ( t ) *..,fd(t))* , Let U be a curve above y in T ( M ) , U ( t ) = Ch,(t)f,(t), such that U(0) = u,(O) and C h,’(O)f, = V’(0).Now we define the rectangle Q by a s ,t)

=

exP,(,,su(t).

I t is clear that the longitudinal curves of Q are geodesics since for fixed t we just get the exponential map applied to a ray. Furthermore, since U ( 0 ) = u,(O), the ray for t = 0 is the one belonging to u, so u is the base curve of Q. For s = 0 we get y , so the associated vector field Wsatisfies W(0)= y*(O) = V(0).Thus it only remains to show W’(0) = V’(0). Now Q satisfies the condition of lemma 2, so we have by (a) : DlwQ(D2) = D,wQ(Dl) +Q(Dz)uQ(Dl). But 7 is horizontal, so (bQ(D2(0, t ) ) = +(y*(t))= 0. Also f ( t ) wa(Dl(O, t ) )is the tangent to the longitudinal curve at height t , which from the definition of Q is U(t). Thus for t = 0 we get

+

w‘(0) = f ( 4 W Q ( W O ? 0 ) ) = f(&JQ(Dl)(0, 0 ) ) = =

f(h’(0)) V(0).

(h@)= (hl(t), * * . , hd(t)))

QED

We now apply this to the case where V(0)= 0 and Q is degenerate at s = 0, that is we take y to be the constant curve. Thus in this case Q will be factored through Mr,L(m= u(O)), that is, Q = exp, o S, where

176

9. RIEMANNIAN CURVATURE

S is a rectangle in M,, . Hence our Jacobi field will arise as d exp, applied to a vector field along the ray p in M , which goes into u under exp,, . We may taken U to be linear in t : U ( t ) = u*(O)

+ tV’(O),

and this corresponds to S(S, t ) = sa*(O)

+ stV‘(0).

A linear homogeneous vector field along a ray p in M,, is a curve X above p in T(M,) such that X(0) = 0 and X ” = 0 (differentiation is possible since M,, is a linear space). For every x E (& it is I, clear ), that , there is a unique linear homogeneous vector field such that X(0) = x. From the above definition of the rectangle S we thus have:

FIG. 32.

If V is a Jacobi field along u = exp, o p which vanishes at m and X is the linear homogeneous vector field along p such that V’(0) = d exp,,X’(O), then V = d exp,X.

Corollary.

I n the remainder of this chapter we obtain results by making comparisons between the growth of X in the “flat” space M , and the growth of V in M . This comparison will involve curvature, so

9.4. Rectangles and Jacobi Fields

177

according to theorem 1 we shall have a relation between curvature and the behavior of nearby geodesics. T h e metric on M,n is the one induced by the inner product, so M , is isometric to Rd under, for example, f E F ( M ) . We remark in passing that X is a Jacobi field along geodesic p in M,, . We introduce the notation:

w = p*

,

u = u* ,

K ( V )= K ( U , V )=

curvature of the section spanned by V and U when they are linearly independent. Because the behavior of the U-component of V has been completely determined (it is a linear function of U ) and because the part of V perpendicular to U is independent of the U-component, we assume for the remainder of the chapter that ( U , V ) = 0. We assume also, for convenience, that I / U I / = 1. Since 1 1 V Il’(0) = 1 1 V’ Il(0) = / I X 1 1 ’ (see (a) in the proof of Theorem 2. We are here, of course, assuming that V(0)= 0), we can base our comparison of growth on the second derivative, which is given by the:

[As in the definition of K(P),A(x,y) is the area of the parallelogram spanned by x and y.3 Notice that if K( V ) is negative, then 1 I V 11’ ’ is positive. Proof.

II VII”

II V II‘ = (Q)(K V>-””+ ( V , 0) = I1 VII-’ = =

((V”, V>+(V’,V’>)Il ~ l l - l - ( V ’ , V ) l l

(w,v’)(v,v) - (v’,

02)

A(V’, V ) z11 V

II

v

11-3

VII-2(IT’,

OII V1l-l

+ w, 1)’ I I v

11-1

- (R,,U, V ) / I V 11-l (V is a Jacobi field) = A(V’, V ) y Vl1-3- K ( V ) A ( U , V)*II VII-1.

=

But A( U , V ) = 1 1 V / / by the assumptions ( U , V ) = 0 and

1 1 U / I = 1. QED

Problem 22.

Determine the Jacobi fields on Rd (see problem 7.9).

9. RIEMANNIANCURVATURE

178

9.5 Theorems Involving Curvature

We use the notation of 9.4. Let p be a ray through 0 in &Ir,*. (i) If K ( V ) 0 along p , then I1 VII = I I dexp,X / I 3 I I X I / along p . Further, strict inequality is preserved in this implication. (ii) If K ( V ’ ) > 0 at 0, then 1 1 d exp,X 1 1 < 1 1 X I / at points of p near 0.

Theorem 2.

<

Proof. Let g = 1 1 V 1 1 - 1 1 X 11. Then since 1 1 X 1 1 ( t ) = t 1 1 X‘ 1 1 for t 3 0, it is clear that g‘ exists on the positive real numbers whenever 1 1 V 1 1 # 0. Thus it will suffice to show (a) g has right-hand derivative 0 at 0. (b) If K ( V ) 0, then g” 2 0, and the same implication for strict inequality. (c) If K(V‘)(O)> 0, then g“ < 0 in a neighborhood of 0 in the positive reals.

<

For (a) we have: since X(t)

=

tX’(t),

I I ( t ) = II dexp,,,tX’(t) II

II

=

Dividing by t and taking the limit to 0

/ I V II’P +)

=

t

I I dexprnX’(t) II.

+ we get

II dexprnX’(0 +) II = I I X’(0 +) II

since exp,, is an isometry on (M?J0. Since 1 1 X 11‘’ = 0, (b) is immediate from lemma 3. T o show (c) we write the formula for j 1 V I I” as

Since lim K( V ) = lim K( Vjt) = K ( V’(0))by the continuity of K , it suffices to show that the limit of A(V’, V ) z( V , V)-z is zero. T o do this we let Y = Vju, so Y = dexp,,X’, V’ = Y uY‘, V” = 2Y’ uY“ = -R,,U (u is the coordinate on R). By the uY’ span a parallelogram of the same area as fact that u Y and Y uY and uY’ and because A ( x , Y ) is~ quadratic in each variable, we have A( v‘,V)Z( V , V)-Z = A(uY, uY’)Z (UY, uY)-Z = A(Y, Y’)Z
+

+

+

9.5. Theorems Involving Curvature

179

Now the limit of Y at 0 is d exp,,X’(O) = V’(0) # 0, whereas the limit of Y’, from the equation for V”, is 0. Thus the limit of A( Y , Y’)2( Y , Y ) - z is 0. QED Corollary 1. If M has everywhere nonpositive curvature, then exp, cannot decrease lengths of curves. Proof. By (i), since we can take u to be any geodesic from m and X any linear homogeneous vector field perpendicular to p, it follows that exp,, cannot decrease lengths of vectors perpendicular to p. But 1 1 d exp, U I I = 1 1 U 11, so exp, cannot decrease the length of any vector.

<

Corollary 2. If K( V ) 0 for all Jacobi fields along u, then geodesics from m near u pull away from u as compared to the corresponding rays in M , .

m

&

m

6 K>O

Ks 0

FIG. 33.

If K ( V ) > 0 along u near m , then geodesics from m near u pull towards u as compared to the corresponding rays in M , . Proof. We need only make precise the words. Let p, T be rays in M , through 0. Let S be a sphere about 0, and let y be the shorter segment of the great circle in S intersecting p and T . Then the geodesics exp,, o p and exp, o T pull together if I y I >, I exp,, o y I and pull apart if I y 1 I exp,,, c) y 1. T h e corollary then follows from theorem 2.

<

Problem 23. Plane curvature can be used as follows to give a comparison between the lengths and areas of “circles” in M and those in R2. Let x, y E M , , x,y unit orthogonal. Define a C” rectangle Q by

Q(s, t )

0

< s, 0 < t < 2

of Q.

=

~ Let . K

exp, (s(x cos t =

K ( x ,y ) , V

+ y sin t ) ) , =

the transverse vector field

180

9. RIEMANNIAN CURVATURE

(a) Show that the length of V may be expressed as

I I V I I = s - 8 K.s3+ .@h(s,t ) , where h is C“ for s > 0, continuous at s = 0. (b) Let L(s) be the length of the “circle” Q,. Show that L(s) = 2 4 s - Ks3/6) s4f(s), where f is a continuous function of s; hence K = ( 3 / ~ ) ( ~ T S- L ( s ) ) / s ~ .

+

!s

(c) Define A(s) = J i L ( u )du, the “area” of the “circle,” and obtain the formula K = (12/T) ljz ( T S 2 - A(S))/S4. Remark. I?. Cartan gives a generalization which tells the area of small spheres, volume of small 3-spheres, etc. [ I 7, p. 2521. Problem 24. Let M = Sd of radius r . Using problem 7.12 show that the “circles” of “radius s in Sd” are circles of radius r sin sir in Rdfl. Hence, calculate L(s) and prove that K = l/r2 for all plane sections. Problem 25. Use the above to calculate explicitely the Jacobi fields on Sd as follows: let y(s) be a geodesic with s arc length. Let x, y E Sd,(,,, be tangents perpendicular to y*(O). Then identifying tangents to S d with tangents to Rd+l,we may write

Y

=

2 b,~,(y(O)).

Show that the formula for the Jacobi field X along y with X ( 0 ) = x, X(0) = y is X ( S )= cos S/Y

2 a i ~ , ( y ( s )+) sin s/r 2 ~JI~(~(s)).

Let M , ‘ M be d-dimensional Riemannian manifolds, m E M , ’m E ’ M , f = ( m , f i , ...,f d ) E F ( M ) ,’f = (’m, ’fi , ..., ’ f d ) EF(’M), and define as in 8.1 8

0

= =

-

expJ*+ -

expf*@

-

I#= expJ*w

’8

=

-

exp,,*‘q5

-

‘0 = exp,,*’@ =

-

expef*’w.

9.5. Theorems Involving Curvature

181

Let X E o(d); then f o (exp uX)3 f-' is a one-parameter group of rotations of M , and thus induces a vector field 3 on M , . Using 'f instead off we likewise get a vector field on . T h e vector field is linear homogeneous along and perpendicular to every ray p in Mn,; further (3o p)'(O) = d exp,n-l( fXf-'(p( 1)).

'x

x

FIG. 34.

Let T be the radial vector field defined on all of Mrn except 0 which satisfies T o p = U whenever p and U are as before. Similarly we define ' T on ' M , , , . Theorem 3 (fi. Cartan). If for every X E N ,a neighborhood of 0 in Rd, and for every X E o(d) we have

O(T,S ) ( f X )

=

'O('T,'X)(yX),

then there is a neighborhood U of m and an isometry J : U + ' M such that J(m) = ' m and d J o f = 'f. Proof. Let L be the linear map ;ff-' : M , -+ 'M,,,, . Let 0 be a neighborhood of 0 in Mr,, on which exp,, is a diffeomorphism and which is contained in f ( N ) .Then we set U = exp,,u and define J by J = exp,, o L e (exp, I D)-l.It is then clear that J ( m ) = ' m and d J == L on Mnl , so it only remains to show that J is an isometry.

9. RIEMANNIANCURVATURE

182

It is clear that dJ preserves lengths of radial vectors, that is, d](dexp,,T) = d exp,,‘T. It therefore suffices to show that d] preserves lengths of vectors normal to the radial ones. Now if t is such a tangent to U , then there is p E M , and X ~ o ( d )such that d exp,,X(p) = t , by the choice of U. Also, dJ(t) = d exp,,’X(Lp). II, It d](t) I / Therefore, by formula (b), p. 146, I1 t I I = I I #(x)(p) = 1 1 ’+(’X)(Lp)1 1 , and hence it will be sufficient to prove for every X .

#(X)= ’#(’X)o L

Now we make the observations: (a) # and f-’ o d exp, are the same on ( M , ) o , and similarly for ‘#. (b) # ( T ) = ’#(’ T ) 0 L (c) #(X)(O) = ’#(’X)(L(O)) For every ray p in M , ,

(4 (#(XI0 P)’(O)

= =

=

0.

#(WP)’(O))

=

Xf-lp(1)

(’#(’X) 0 L 0 p)’(O),

by (a) and the remark before the statement of the theorem describing

(xBy P)’(O). and ’+(’x) o L satisfy the same initial conditions (c) and (d) #(x) 0

along any ray. They will be the same if we can show that the second derivatives are the same. But rays are integral curves of T , so we must show that T 2 + ( X )and T*(’#(’X)o L) coincide. Deferring the proof #( T ) , we have that T 2 # ( 2 )= @(T ,

x)

TZ(’#(’X)0 L ) = ( ’ P ’ # ( ’ X ) )n L = (’O(’T, ’X)’#(’T))0 L = (’@(IT,’ X ) 0 L)(’#(’T)0 L ) by hypothesis and by (b) = @( T , X ) #( T ) =

PI,/ X). ,(

The following lemma will complete the proof: Lemma 4.

(a’) (b’)

T#(X) = X + ( T ) + e ( X ) # ( T ) , TO(X) = @ ( TX), , T2#(X) = @ (T ,2)# ( T ) .

(c’) Proof. Except for an obvious substitution of symbols, the proof is the same as that of lemma 2. T h e only fact we need to establish is

9.5. Theorems Involving Curvature

183

[ T , X] = 0. This follows easily from the geometric interpretation of bracket, theorem 1.4, for the same result is obtained from either order of the operations moving a given distance along a ray (following an integral curve of T) rotating about the origin (following an integral curve of

x).

Corollary. If M is a flat d-dimensional Riemannian manifold, that is, K ( P ) = 0 for all plane sections P on M , then M is locally isometric to Rd. I n the following, we assume our manifolds are connected. Theorem 4. Let M be a complete Riemannian manifold, and let m E M . Assume that dexp, is everywhere nonsingular. Then exp, is a covering map.

First define a new Riemannian metric on M , [that is, on T(M,)] by pulling back the metric on M by means of d exp,, . Since geodesics from 0 E M , are straight lines with linear parametrization it follows from theorem 8.5 that M,, is complete in this metric. Therefore, to prove the theorem it is sufficient to prove that if F : N + M is a local diffeomorphism with dF everywhere an isometry, N complete, then F is a covering map. For this we need to show that for any m E M , there is a neighborhood U of m which is evenly covered by F , that is, F-l( U ) is the disjoint union of sets U, , each of which is diffeomorphic to U under F. Fix m E M . Let 0 be a ball about m which is diffeomorphic to a ball 0’ of radius r about O(m) in M , . Let U be the ball of radius r / 2 about m, let F-l(m) = {ui}, and let Ui be the ball of radius r / 2 about ui . T o show that this is an even covering we must prove that Proof.

(1) F I ui is 1-1, (2) F-l( U ) is the union of the U , , (3) ui f uj implies Uiand Uiare disjoint. Proof of (1). Let U‘ be the ball of radius r / 2 about O(m) in M , , and let dFui map Nu
184

9. RIEMANNIAN CURVATURE

of length k < ~ j from 2 m' to m. Locally, cr can be lifted to a geodesic from u'. Since N is complete, 0 can be extended to length k, and since F is a local isometry, F o C? is a geodesic and hence is u again. But then the end point of 6,A say, goes into the end point of u, that is, F(A) = m. Hence, there exists an i such that u, = A, and so u' E U, . This proves (2). C?

Proof of (3).

Assume that u, # uj and u'

p(u', ui)< r / 2 , p(u',

zij)

< r/2

so

E

Ui n Ui. Then p(ui , I+)


But by reasoning similar to that of (l), F is 1-1 on a ball of radius Y about ui , which contradicts F(ui) = F(uj). This proves (3) and the theorem. Corollary 1. Let M be a complete simply connected d-dimensional flat Riemannian manifold. Then M is isometric to Rd. Proof. By the corollary to theorem 3, exp,, : M , + M is a local isometry, for any rn E M , and hence by theorem 4 is a covering map, which is 1-1 since M is simply connected.

Corollary 2. (Hadamard-Cartan). Let M be a complete simply connected d-dimensional Riemannian manifold with nonpositive curvature at all plane sections. Then M is diffeomorphic to Rd. Proof.

This follows immediately from theorems 2 and 4.

Problem 26. Assume N , M are Riemannian manifolds, N complete, q5 : N + M regular and locally distance-nondecreasing. Show that q5 is a covering map if dim N = dim M . Problem 27. Let M and N be complete and have the same constant curvature. (Such manifolds are called space forms.)

(a) Show that the curvature form may be written CP = kwwl, where o1is the transpose of o. (b) Show that M and N are locally isometric. (c) If M and N are simply connected and R 5 0, then they are isometric ( M and N connected). (d) If k = u2 > 0, then the sphere of radius -rr/ain M , is mapped to a point by exp,, , and exp, is regular within that sphere. Construct a Riemannian covering map from the sphere Sd of radius l/u to M , so that if M is simply connected it is isometric to Sd.

C H A P T E R 10

Immersions and the Second Fundamental Form I n this chapter we consider the immersion of a manifold in a Riemannian manifold and the resulting induced Riemannian connexion on the immersed manifold. T h e second fundamental form is defined and related to curvature and parallel translation, and Synge's theorem is proved. T h e problem of the existence of immersions is formulated, and the chapter concludes with a section on hypersurfaces [20, 38, 50, 831. 10.1 Definitions

Let N be an f-dimensional Riemannian manifold with metric = f - d > 0. An immersion of M into N is a C" map I : M -+N such that dI is one-to-one on every M,, . Recall that I is an imbedding if it is one-to-one (1.5). The induced Riemannian metric (or first fundamental form of the immersion) is ( , )' = ( d l , d l ) , which makes M a Riemannian manifold. We assume from now that I is an immersion and that M has the induced Riemannian structure, and so also the connexion. N and M being Riemannian manifolds, they have associated frame bundles F ( N ) and F ( M ) . We consider two more bundles. First, let F,(N) be the principal bundle on M induced by I and F ( N ) . Thus, ( , ) and M a d-dimensional manifold with e

we shall suppress the I ( m ) in b ( m , b) = ( m , el , el).

=

( I ( m ) ,e l , ..., er) and write instead

* * a ,

185

186 10. IMMERSIONS AND

THE

SECOND FUNDAMENTAL FORM

O ( f ) acts on Rf and is the group of the bundles F ( N ) and F,(N). Decompose Rf into Rd Re, and consider the following subgroups of O( f ) : O(d) = {g E O ( j )I gRd C Rd, g I R e = identity}, q e ) = {g E O(f) I g I R d = identity, gRe C Re}.

+

Let o(f), o(d), D(e) be the corresponding Lie algebras, so that o(f) = o(d) o(e) f, where f = { X E o(f) lXRd C Re and XRe C Rd}.

+

+

Notice that O ( d ) x O(e) C O ( f ) is a subgroup, and Ad (O(d) x O(e)) f C f. Let I ( M ) , T ( M ) be the normal and tangent bundles of M , respectively, with fibres MmL and M , . Now define F(N, M ) = { b EF,(N) I b(Rd)= d Z M , , m

= n(b)}

(3.3);

that is, F ( N , M ) = {(m,el , ..., e,) E F M W )I el , ..., ed E M,, , ed+l , ..., e, E JfmL}.

F(N, M ) is called the bundle of adapted frames over M . It realizes a reduction of the group O( f ) of the bundleF,(N) to O ( d ) x O(e). We now have the commutative diagram:

where s is the projection onto the first d tangents. Problem 1. Let S be the vector space of symmetric d x d matrices over R. Then O ( d ) acts on S by the restriction of the adjoint representation of GZ(d, R ) : for T E O(d), X E S , Ad T ( X ) = TXT-l. Let I , be the r x r identity matrix,

187

10.2. T h e Connexions

(a) Let M be the orbit of X,under O(d). Show that the isotropy group which leaves X , fixed is O(r) x O(d - r ) , so M is diffeomorphic to Gd.,. (b) If we introduce the metric ( , ) on S defined by ( X , X ’ ) = tr XX‘, then S is isometric to flat RD space, D = d(d 1)/2, and O(d) acts as isometries. T h e metric induced on M is thus invariant under O(d), so M , with this metric, is a Riemannian symmetric space.

+

Problem 2. If we let O(d) act on S as in problem 1, and X E S has n distinct characteristic values with multiplicities dl , ..., d, , show that the orbit is the flag manifold FZ(d; d, , ...,, d,). (See problem 7.25.) Establish the relation between the inner products on the blocks mij and the characteristic values of X .

10.2 The Connexions

We shall shortly wish to induce from a connexion on F,(M) one on the subbundle F(N, M ) . In general, the horizontal distribution on F,(N) will not be tangent to F(N, M ) , and hence it will have to be “projected” onto F ( N , M ) in some sense. We describe a procedure for carrying this out, using the dual formulation for connexion. Let (P, G, M ) be a principal bundle over a manifold M and let be a connexion form : T ( P )+ g. Let ( B , H , M ) be a subbundle of ( P , G, M ) , i : B C P, H C G, representing a reduction of G to H . Then i*+ is a I-form on B taking values in g, and hence is not in general a connexion form on B. However, if f is a vector complement 9, Ad H(f) C f , then the of b in g invariant under Ad H , that is, g = f projection of i*# into b under this decomposition of g is a connexion form on B (see problem 5.5).

+

+

+

Problem 3 .

Verify that under the condition stated, the projection of

i*+ into b is a connexion form.

+

Let be the Riemannian connexion form on F( N) , w the solder form, and denote by the same symbols these forms pulled back to F,(N) by I’, so is a connexion form on F,(N), and dw = -+w is the first structural equation. Replacing g, b, f by o(f), o(d) o(e), f in the above, we see that the projection of r*# onto o(d) + o(e) is a connexion form on F(N, M ) , denoted by #d + e , with the obvious meanings. This again gives a connexion form on F,(N), by equivariance; and so we may consider

+

+

+

188 10. IMMERSIONS AND

THE

SECOND FUNDAMENTAL FORM

+

the difference form - +, which is horizontal and equivariant. Thus T = r*+ +J is also horizontal and equivariant. T is essentially the second fundamental form of the immersion, although the formal definition will be in terms of a related object. We now use 4, to define a connexion on F(M). Notice that satisfies

+

(1) +a(ds-l(O)) = 0 (2) = ,for g E Id O(e). From (1) and (2) it follows that there exists a unique form +o on F ( M )such that +d

= +o 0 ds.

It is easy to verify that +o is a connexion form on F(M). We now study the structural equations of these various connexions. First notice that if b E F ( N ,M ) , then dT’(F(N, q

b )

c MY(*)

and Therefore, if we let w’

=

r*w,

w’(F(N,M)b) = b-l(dT’(F(N, M ) b ) )C Rd,

is the solder form on F ( N ) pulled back to F,(N). Hence, d(r*w) = Y * dw also takes values in Rd, and 4,w’ = 0. So the structural equation for onF,(N) gives

where

dw‘ =

w

+

dW’ = r * d w = =

-

*+w

- $dW‘

=

-($d

+ + ). de

w’

- TW’.

Now, dw’ and -+do’ are Rd-valued, while w’ is Rd-valued. Hence, dW‘

= -+dw’

TW‘

=

TW’

is Re-valued, since

0.

+

In particular, the connexion 4, has zero torsion. Further, since w’ = s * w o , where wo is the solder form of F ( M ) , the structural equation of +o is duo

= -+orno,

10.3. Curvature so that b0 has zero torsion. Therefore, connexion on F ( M ) . Problem 4.

Prove that

w’ = s*wo

189

4o is

the unique Riemannian

.

Problem 5. If we define a Riemannian metric on F ( N ) by letting the basic and fundamental vector fields, E, , Fij , be orthonormal, this induces a metric on F,(N). (a) Show that the subspace dl’(F(N, M),) C F M ( N ) bis orthogonal to the part of the vertical space which corresponds to f, that is, it is orthogonal to hf. (b) Show that the horizontal space of theb,z connexion at b EF,(N) is the orthogonal projection of the horizontal space of the 4 connexion onto dl’(F(N, M),). This result provides a more geometric description of the )I connexion: it is the closest connexion to the connexion which paralleltranslates adapted frames into adapted frames.

+

10.3 Curvature

We now employ the other structural equation of 4 to relate the curvature of the induced structure on M to the curvature of N . Note that

( l ) [bd 7 be1 = (2) [+d be,T] is f-valued ( 3 ) [bd ,bd] is o(d)-valued (4) [be,+,I is o(e)-valued ( 5 ) [ T , T] is O ( d ) $- o(e)-valued,

+

[T,T] = [ T ,

T]d f

[T, TI,.

These facts are all trivial. and write Now let @ be the curvature form of 4 onF,(N), @‘ = r*@ = @d Qe where takes values in f. Then the structural equation d + = - - ;[+>dl @

+ + r,

r

+

gives the equations d+d

=

-

dde = -

+

3

dT = - [ d d

[+d

!

ddl -

[+e

9

deI -

+

de

9

T]

fr ,.[

3

.Id

[ T , .Ie

+ r-

+ +

@d

@e

190 10. IMMERSIONS AND THE SECOND FUNDAMENTAL FORM In particular, if Qo is the curvature form of 4oonF(M), then Q0

0 dS =

@d -

[T,T ] d .

Thus, the curvatures are related by means of the difference form T. More precisely, let x,y be orthonormal tangents in M , ,b E F(N, M ) with ~ ‘ ( b=) m, ff,Y horizontal liftings of x, y to b, so that x” = ds(n), 9 = ds(jj) are do horizontal lifts of x, y to s(b) E F ( M ) . Denote by Ko(x,y ) and K(x,y) the curvatures of the section spanned by x and y in M and N , respectively. Then KO(X,Y) = (RorvX,Y) = - (s(b) @o(x”, = =

But (Qe(Z, j j )

(Rd, Re) = 0,

-

9)(@))-%r>

W) @o(ds(f), dW)(s(b))-’x, r> (bad(*, 9 ) b-lx, r) $- 8 ( b [ T , T I ( $ , 9 )b-lx, y ) .

+ T(Z,3))b-lx E Re, since b-lx E Rd. Therefore, since

(bod($, j j ) b-lx, y ) = ( b W ( f , P )b-lx, y ) = -K(x, so

Ko(x,y ) = K(x,y )

where [7,TI(%

Thus,

4

[T,T I ( * ,

+ 4( b b ,

TI(*,

y),

9 )b-lx, r>,

j j ) = 2[7(*), .(jj)l.

7 )6 - l ~= [ ~ ( f~) (, y )~’(2). ]

This last term will be given geometrical significance via the second fundamental form.

10.4 The Second Fundamental Form

Let x E MmL. T h e second fundamental form of z is a bilinear form H , on M , defined by H z ( x , y ) = - (z, Wf) 4

9 h

where x, y E M , , b E (r’)-l(m), and 2, are lifts to b. H,(x, y ) is clearly independent of the choices involved and is symmetric since

10.4. T h e Second Fundamental Form

191

= 0. Hence, there is a corresponding symmetric transformation S , on M , given by

TW’

Hz(x, Y ) =

(SAY).

Now let T be the linear transformation field associated with the difference form T, that is, if x E M , ,y E NIc,, , then T,y

= b ~ ( 3b-’(y), )

where b E (n’)-l(m) C F(N, M ) and 2 is a lift of x to b. I n particular, T,(M,) C T,(MmL) C M , . If y E M , , then T Z y = b ~ ( Z ) ~ ’ ( j j ) . Now (SP, Y > = H A x , Y ) =

- (Z,

&(a) W’(y))

=

- (Z,

T,Y)

=

(T,Z,Y)

since ~ ( 2is) skew-symmetric. Thus, for x E M , S,X

=

, x E M,,*,

T,z.

We have the following interpretation of T, in terms of parallel translation. I n fact, the connexions 4 and y!I in F,(N) give rise to distinct parallel translations of the fibre, that is, of NIc,, , m E M. T is an infinitesimal measure of the difference. Let x E M , , X a vector field on M defined in a neighborhood of m and such that X ( m ) = x. Let Y be a vector field of tangents to N defined on M , that is, Y(m’)E NIc,,,,. Finally, let D and E denote covariant differentiation with respect to the connexions 4 and y!I, respectively. Then we have Proposition 1.

TxY(m) = (D,Y

-

E,Y)(m).

Proof. As in 6.4.l(iii), define a function f Y : F,(N) -+Rf by: f y ( b ) = b-lY(n(b)). Let 8 and 2 be the 4 and $ horizontal lifts, respectively, of X to F,(N). Then from 6.4.l(iii),

( E x Y ) ( m )=

w3*)(~))*

( D x Y ) ( m )= w a Y ) ( b ) ) >

192 10. IMMERSIONS AND

THE

SECOND FUNDAMENTAL FORM

where ~ ( b= ) m. But

so

We give another interpretation of the transformation Txy.Let x E M , , y E MmL. Let y be a curve with y(0) = m and y*(O) = x. For fixed t, let Z(t) be the I) parallel translate of y along y to y(t), Z(t)E and then let Y ( t )be the # parallel translate of Z ( t ) back along I o y to I(m), so t + Y ( t )is a curve in N,(,,). Proposition 2.

space NICd Proof.

DxZ

=

-1

Y’(0) = Tzy [(’) denotes differentiation in the vector

Since Z is I) parallel along y , we have ExZ = 0. But Y’(O),so the result follows from proposition 1.

(a) If Yo is a vector field along curve yo in M and = I o yo in N , then Yo is parallel (with respect to #o) if and only if D,*Y is in the normal bundle I ( M ) . (b) Hence, yo is a geodesic in M if and only if D,*y* is in l ( M ) . Problem 6 .

Y = dI( Yo)is the corresponding vector field along y

10.5 Curvature and the Second Fundamental Form

We now apply this interpretation of the second fundamental form

to our previous expression for curvature K o ( x , y ) to draw some

immediate conclusions. Recall that for orthonormal x, y

E

Mm ,

10.5. Curvature and the Second Fundamental Form

193

+

where ~ ( b = ) rn and 2, 7 are (+d +J horizontal lifts of x, y to b EF(N, M). Now I T ( % ) , T ( j j ) ] U’(a) = T ( f ) ‘ T ( j j ) W ’ ( f ) - T(7).(a) W ’ ( 2 ) =

b-’(TXTyx

TYTxx).

~

Note that similar arguments give

where P is the orthogonal projection NJ(nL, M , . --f

Lemma 1.

Let xl, ..., x, be an orthonormal basis of M,,I

Hi= Hs, . Let x,y be an orthonormal pair in M , . Then KO(X>r) = K(x,Y )

+ c (HZb 4 HdY, Y )

-

and let

HZk Y)2).

2

Thus, the induced curvature is the curvature of N plus the sum of the “squares of areas” with respect to the second fundamental form. Proof.

(Tyx, za> = --Ha(x, Y ) ,

so

~

y

= x

-

2 ~ z ( xY, ) za* 1

Hence, TxT,x = -

2 H,(x,y ) T x z z . a

Now

(Txzz Y ) ?

=

Hz(x, Y ) .

Therefore, (TxT,x,y> =

-

2 Hz(x,Y Y . 2

Similarly, T,x

= -

H*(X,X )

~

i

,

2

(TyZa Y > = H ~ ( YY,) , 9

so -

( T , Txx,Y > =

2 a

K ( X ,

x ) HdY ,Y 1.

194 10. IMMERSIONS AND

THE

SECOND FUNDAMENTAL FORM

Theorem 1. Let M be immersed in N by I , let u be a curve of M such that I o u is a geodesic of N . Let P be any plane section of M

<

tangent to u. Then Ko(P) K(P). If we let Y be the c $ ~ parallel vector field along u such that P is spanned by x = u,(O) and y = Y(O),then equality holds between K,(P) and K ( P ) if and only if D,dI( Y ) = 0. In particular, KO(Y , u*) = K( Y , u*) if and only if dI( Y ) is parallel in N . Proof. We may assume that Y and u* have been normalized, (u* , u*) = 1, ( Y , u*) = 0. T h e local arc length minimizing properties of geodesics (or problem 6) imply that u is a geodesic in M . Hence, u* and dI(u*) are parallel in M and N , respectively; so

(Y, Y ) =

T,*u* = 0 by proposition 1. Let zi,Hibe as above, so Hi(u,, u*) = 0 for every i, and K,(Y, u*) - K ( Y , u*) = - r) Hi(Y, .*)Z

< 0.

z

This proves the first assertion. It also shows that Ko(P)= K (P ) if and only if Hi(y, x) = 0 for every i. But Hi(Y, 4

= - (Xi >

TXY),

so this is equivalent to Tzy = 0, or, by proposition 1,

D,Y

=

E,Y

=

0,

since Y is parallel in M along u. T h e case of the above theorem for dim M = 2 is known as Synge’s theorem [ 731. Let M be a submanifold of a Riemannian manifold N with the induced Riemannian structure. Then M is called a totally geodesic submanifold if every geodesic of M is a geodesic of N . Remark.

M is a totally geodesic submanifold of N if and only if its second fundamental form vanishes identically.

Theorem 2.

Proof. T h e vanishing of the second fundamental form H is clearly equivalent to the vanishing of the difference form 7,which means that the bundle F ( N , M ) is situated “horizontally” in the bundle F,(N). If this is true, then parallel translation with respect to N and M coincide, and hence every parallel vector field in M is parallel in N , which proves that geodesics in M are also geodesics in N . Hence, M is totally geodesic in N .

10.6. T h e Local Gauss Map

195

Conversely, if M is totally geodesic, then every x E M,,, is tangent to a curve u which is a geodesic in both M and N . Thus by the same argument as in the proof of theorem 1, Hi(x, x) = 0 for all i, x. Hence, since Hiis symmetric, Hi= 0, all i, that is, H = 0. QED Problem 7. Prove the following formula for the curvature transformations of an immersed manifold. Let zi, Hibe as above, Sithe symmetric linear transformation such that Hi(x,y ) = (Six, y ) , all x, y E M , . Then

If we view R, , R as maps of bivectors into bivectors, G 2 , --t G2, , this becomes (b)

R,(XY) = P,R(XY)

or R,

=

P,R

+ 2 (SiX)(SiY), i

+ 2 Si, . i

Here, P is the projection NIcm, --t M , , and if A is a linear transformation, A, is the extension to bivectors given by A,(xy) = (Ax) (AY). Problem 8. A submanifold M of N is geodesic at m if every geodesic of M through m is also a geodesic of N . (a) If N has constant curvature show that every submanifold which is geodesic at a point is totally geodesic. (b) Conversely, if every submanifold which is geodesic at a point is totally geodesic, then N satisfies the hypothesis of Schur’s theorem, problem 9.3, so has constant curvature (dim N > 2) [17, pp. 232-2331. 10.6 The Local Gauss Map

Let U be a Riemannian normal coordinate neighborhood in M at a point m E M (see 6.3.2). Let I( U ) be the restriction of the normal (U ) -+ Mm MmL is defined bundle of M to U. Then a map G, : I by: if z E i(U ) , then G,(z) = parallel translate in N of z back along the geodesic ray in U from m to the base point of z. G , is called the Gauss map with respect to U .

+

196 10. IMMERSIONS AND Remark.

If N

=

THE

SECOND FUNDAMENTAL FORM

R f , then parallel translation in N is independent

of path, so that G may be defined globally on 1 ( M ) , and it will

agree with G, on I ( U ) . More generally, this works whenever N has a trivial holonomy group.

FIG. 35.

Notice that i ( M ) , as a bundle associated to F(N, M ) , has a connexion corresponding to y5d & (5.4).

+

Proposition 3. Let x E M,, z E Mmi, and let x" be a horizontal U( ) . Then dG,(x") may be identified with an element lift of x to z in I y of M , Mmi, and y = Txz.

+

Proof. Let (J be the ray in U with tangent x at m. If 6 is the horizontal lift of (T toF(N, M ) , then the curve

t + 6 ( t )6(0)-1z

in I (U ) is tangent to 2. Composing G, with this curve gives us the curve t

--f

G(0) G(t)-16(t)6(0)-~x,

where 0 is the horizontal lift of (J to F,(N). By construction, the tangent to this curve is dG,(x") = y , but by proposition 2, y = Txz. QED Corollary 1. If x, y E M , , z E MmL, x" any lift of x to z in I( U), then HZ(x,y ) = (dG,(x"),y ) , where again we make an identification. Proof. If x" is a horizontal lift, then this follows from the proposition above. If x" is not horizontal, then let x' be the horizontal lift, and notice that dG,(x" - x') is tangent to Mnti, so that its inner product

10.7. Hessians of Normal Coordinates of N with y is zero. This follows since x" is the identity on M,I. QED

-

x' is tangent to

197

MTaLand G,

10.7 Hessians of Normal Coordinates of N

Let M be any manifold, f a real-valued C" function on M , m E M . f has a criticalpoint at n if df, = 0. If m is a critical point off, then the Hessian o f f at m, H f , is a bilinear function on M , defined as follows. If x, y E Mm , X a C" vector field such that X(m) = x, then H f ( X , Y ) = r(xf).

Since df, = 0, it is easy to verify that H f ( x ,y) does not depend on the choice of X and further that H , is symmetric. Verify these facts about H f without using coordinates. We now return to an immersion I : M -+ N. T h e second fundamental form will be related to the Hessians of certain functions on M (see theorem 3 below). Let V be a normal coordinate neighborhood in N at n = I ( m ) , and let v , , ..., vUf be normal coordinate functions on V so that V,(n) = (a/&,)(.) are an orthonormal basis of N , . Let ui= vi o 1. Then a linear combination u = Xi aiui has a critical point at m if and only if Problem 9.

that is, du = Xiai dui annihilates Mnt . For simplicity we shall assume that u = u1 has a critical point at m. We shall calculate H,(x, x) for all x E M , , hence determining H , . Let u be the geodesic with a,(O) = x, and we shall use u* in the definition of H,(x, x). Let X be an extension of u* to a neighborhood of m, and let Y be an extension of dI X to a neighborhood of n. Let the coordinate expression for Y be Y = XigiVi . But

so

wherey

Ifu(%, x) = x x u =

Y(n).

=

x(g, 0 I ) = dZ(x)g, = yg, ,

198 10. IMMERSIONS AND

THE

SECONDFUNDAMENTAL FORM

By proposition 1 we have T,x

D,X - E,X = D,X since 0 is a geodesic in M. =

=

D,Y

This last step, D, Z,g,(n) Vi= 0, is true because the vi are normal coordinates in N so that Zigi(n) Viis the tangent field to the geodesic in the direction of y. Thus fux,

4 = YE1 =

(Txx, Vdn)>

=

-H,(x, x),

z = V,(n).

We have proved: Let I : M + N be an immersion, m E M , V a normal coordinate neighborhood of N around n = I(m), u = Z u , u , , where ui = vi o I are the normal coordinates pulled back to M . Assume z = Z aiVi(n) E Mn:. Then u has a critical point at m and its Hessian form is the negative of the second fundamental form H , . Theorem 3.

If N = R f , then the normal coordinates may be taken on all of N , and u is a linear function plus a constant pulled back to M . T h e Gauss map, defined on all of J J M ) , may be considered as a map of I ( M ) into N , identifying N , with N , and G is independent of n. For any n E R f , we get a corresponding linear function u, , u,(n’) = (n’,n), on R f . If G(z) = n, z E MmL,then the normal coordinate combination corresponding to z is just a constant plus u, . Hence we have: Corollary 2. If I : M --+ Rf is an immersion, z then the following three forms are equivalent: (a) H , , the second fundamental form of z,

E

Mnbi,G(z) = n,

10.8. A Formulation of the Immersion Problem

199

(b) the negative of the Hessian of u, o I , (c) the form (x,y ) -+ (dG(Z), d l y ) , where f is a lift of x to I ( M ) , . Problem 10. Suppose that M is a submanifold in the neighborhood of n E N which is given as the locus of equations gi = 0, i = 1, ..., e, where the gi are real-valued C" functions on N such that gi(n) = 0 and dg,(n) are linearly independent. Let zli be normal coordinates at n as above, and let M' = V n exp,(M,), where exp, is the exponential function of N . Let

g

=

2 qg, , a

dg(n) =

z i

ba dVi(?2),

u =

z i

bpi ,

Show that H, = - H , , both being defined on M,

=

f = g lM!

.

M,'

Problem 11. If S, , ..., S, are symmetric d x d matrices, then show that Rd-+ Rf given by x + (x, -(S,x, x), ..., -(Sex,x)) is an imbedding such that S,, ..., S, are the matrices of the second fundamental form with respect to the basis D@), i d, of Rd,, and basis Di(0), i > d , of the normal space (only at this one point).

<

+

+

Problem 12. Let f = uluz u 3 u 4 ,g = u1u3 uzu4:R4--t R. Then the intersection of f-l(l) ng-l(O) with a neighborhood of (1, 1, 0, 0) is a 2-dimensional submanifold of R4.Find the curvature of this submanifold at the point (1, 1, 0, 0).

10.8 A Formulation of the Immersion Problem

In what follows we shall be concerned with giving sufficient conditions that a map of M into N be an isometric immersion, and also giving sufficient structures from which such maps can be obtained. These sufficient conditions are also some of the necessary conditions from Sections 10.2 and 10.3, which involve the bundle of adapted frames. Since we wish to give conditions which depend mainly on the intrinsic structures of M and N , we must somehow construct the bundle of adapted frames a priori. If M is diffeomorphic to Rd this is no problem because all bundles over M will then be trivial, and the bundle of adapted frames will be equivalent to the product bundle M x O ( d ) x O(e).If M is not so trivial, then the bundle of adapted frames will not necessarily be unique as a bundle; however, if there

200 10. IMMERSIONS AND

THE

SECOND FUNDAMENTAL FORM

is to be a solution to our problem we must certainly be able to immerse M differentiably into N . If we combine the normal frames of such an immersion with the frames of M from the given Riemannian metric (not the metric induced by the immersion) we get a bundle B with group G = O ( d ) x O(e) which can serve as a model for our bundle of adapted frames. We assume henceforth that we have this bundle B. If we had the isometric immersion I : M + N , then we would also have the corresponding immersion I' : B -+ F ( N ) .I' is a bundle map, in that it takes fibres into fibres and commutes with the action of elements of G. If we are given 1', we can recover I by projecting into M and N . T h e approach to the problem of finding a suitable I' will be stated in terms of its graph, which is a submanifold of P = B x F ( N ) . We note that G acts on P by means of the diagonal imbedding of G in G x O(f). Proposition 4. A submanifold Q of P is the graph of a bundle map I' : B -+ F ( N ) if and only if Q is invariant under the action of G and the projection of Q onto B is a diffeomorphism. Proof. If p , and p , are the projections of P onto B and F ( N ) ,then the formula for I' is I' = p,(p, lo)-'. T h e proposition is a simple verification.

T h e solder forms of F ( M ) and F ( N ) can be pulled back to forms on P, the former through the intermediate bundle B, the latter directly by p , . We shall denote them by 0 and w , respectively; w decomposes into its Rd valued and Re valued parts, w = w' $- w " . Let n-, : B -+ M and 7 ~ ,: F ( N ) .+N be the bundle projections. The properties of 0, w we shall need are that for t E P,

II d7hdPdt) II

=

II e ( t ) I I

and

II d7r2 dP&) II = II 4)I I *

Theorem 4. A submanifold j : Q + P, j the inclusion map, which is the graph of a bundle map 1', has an isometric immersion for the induced map I : M + N if and only ifj*(O - w ' ) = 0 and j*w" = 0. Proof.

Since by hypothesis p ,

is defined.

IQ is a diffeomorphism,

10.8. A Formulation of the Immersion Problem

201

Consider the following commutative diagram:

i

0-P

Suppose that j*(O - w ’ ) = 0, j * w ” lift of x to B : d7-r1(2)= x. Then

=

0. Let x E Mm , let 2 be a

I I d 4 x ) I I = I I d r , dZ’(f) I I = I I d r , dp2 4 d J ( f ) I I = I I W ’ dj d J ( f ) I I I I W” = I1 d d j d J ( f ) I1 0 = I I dnl 4 d J ( f ) II = II dr1(f) I I

+ +

=

dj dl(f) I I

IIXII.

T h a t j * ( 8 - w ’ ) = 0 andj*w” = 0 follow from I being isometric has already been proved in Section 10.2. We shall now give conditions under which such a submanifold Q can be constructed. Since these conditions will depend on having what is to be the second fundamental form and we wish to make our statement depend as much as possible on the intrinsic properties of M and N , it is desirable to make the hypothesized properties of the second fundamental form depend only on the curvature form of F ( M ) and not also on that of F ( N ) . But the curvature form of F ( N ) enters the formula for that of F ( M ) ,so the only way is to make N have curvature which is independent of position and direction. Thus we assume that N has constant curvature K and hence that the curvature form of F ( N ) pulls back by p , to @ = Kwwt on P, where w 1 is the transpose of the column vector w , so @$j = kwiwj (see problem 9.27). We adopt the convention that forms on B have a subscript 0, with the corresponding pullbacks to P having the subscript removed. Thus 8 = pl*OO. T h e connexion form onF(M) pulls back to B , by the

202 10. IMMERSIONS AND

SECOND FUNDAMENTAL FORM

THE

projection q : B +F ( M ) , to an o(d)-valued form + o . Thus we let = P,*+~.Similarly for curvature, so the structural equation on P becomes d+ = -+2 Y. T h e connexion form on F ( N ) pulls back by 9, to a form on P. I t decomposes into several blocks:

+

+

+

4‘

o(d)-valued

7

d x e matrix, €,-valued

-d

e x d matrix

4”

o(e)-valued.

and T h e structural equations on F ( N ) then give dw’ = dwtt=

-4‘w’

+

-Ttw’

+ +l‘wf‘

d+’

=

-4’2

+

d7

=

-+’T

-

&”

=

TW”

+k w ’ ~ ’ ~ 74’’ + K w ’ ~ ’ ’ ~ + kw ” w ” ~ ,

T T ~

- +”2

Let M be connected and simply connected, N complete.

+

Let uo and be 1-forms on B such that +o is a connexion form with the o(e)-valued part, uo is horizontal, equivariant €,-valued, and satisfying the conditions

Theorem 5.

Yo = uouOt+ k8,eOt

(1)

(2)

u,,Vo = 0

(3)

duo =

d$o”

(4)

=

-I//OU~

- ~ ~ $ 0 ”

-u

~ u ~ .

+

Then the codistribution on P spanned by 0 - w ’ , w ” , - +’, - 4’’ is involutive and every maximal connected integral manifold is a component of a manifold Q satisfying the hypothesis of theorem 4, and so gives an isometric immersion of M into N . u - 7, and

+”

10.8. A Formulation of the Immersion Problem

203

Proof. That this codistribution is involutive follows from the conditions (1)-(4) and the structural equations. For example,

d(+ - +’)

+ ?!’ +

~d - k ~ ’ w ’ ~

=

-+z

=

-(+ - 4’) # - +’(# - 4’)

uut

=

-(+ - 4’)

( 0 - T ) Ut

+ k(O

-

+

w’)

-

+’2

-

Ot

+‘(+-

+ 4’) +

+ kw’(et

-

+ met

- TTt

+

- kwtw’t

T(Ut - T t )

w’~).

T h e other four exterior derivatives are found to be in the ideal generated by the five I-forms in much the same way. (Actually it is the components of these vector-valued forms that should be considered, but the justification for using the whole forms in this way is trivial.) We first show that the dimension of the codistribution is constant and complementary to the dimension of B. This follows from the fact that on the subspace of tangents to the factor F ( N ) of P (that is, the kernel of dp,) the five forms become - w ’ , w ” , -I$’, -7, -4”, and these are a parallelization of that subspace, except for trivial duplications (I$ij’ = -I$ji’). Thus the 5-tuple (6 - w ’ , w”, $ -I$’, CI - T, $” - 4”) is a linear map of each tangent space P, onto Rf o ( f ) (more or less), so the kernel has constant dimension the same as B. By this same argument the kernel of the linear map is complementary to the kernel of dp, , so dp, is 1- 1 and onto on the distribution in question. Hence the integral submanifolds will be mapped locally diffeomorphically into B. Since - I$‘ and +‘‘ - 4“ annihilate the tangents to the orbits of the diagonal action of G , a maximal integral submanifold Qo will contain a component of such an orbit whenever it contains a point. Moreover, the equivariance of the forms in the distribution guarantees that an element of G takes an integral manifold into an integral manifold. Thus Q = Q0G will be an integral manifold invariant under G. T o show that r = p , IQ is a map onto B it suffices to prove that the horizontal lifts of geodesic segments in M can be lifted to Q: starting with a single point in B we can get all other points by moving along such curves and by the action of G. If yo is the horizontal lift of a geodesic of M we will have

+

$J

Bo(yo*) = x = constant,

+o(Yo*)

=

0,

+o”(ro*)

=

0,

204 10. IMMERSIONS AND and if uOij

SECOND FUNDAMENTAL FORM

THE

=

z

SOijkeOk

Y

k

then

the functions sOijko yo are continuous and hence bounded, since the domain of yo is a closed interval. T h e lift of yo to P, which we shall call y, must satisfy

Since it is only thecomponent of y which isin F ( N ) whoseexistence is in question, we project y* onto F ( N ) ,and we find that d$%Y* = E ( x )

+ 2

i,kd

xk(sijk

y

pZ)Fij

=

x-

Integral curves of such an X clearly exist by the condition of completeness of N , so the lift y exists. It remains to show that r is 1-1. To this end we note that Q is a principal bundle with group G and some base manifold M‘ and that r : Q -+ B is a bundle map. Since curves can be lifted from B t o Q they can also be lifted from M to M‘. Thus the induced map M’ + M is a covering map, is 1-1 because M is simply connected, M’ connected. This implies that r is also 1-1. QED Remarks. (a) T o find local imbeddings of M into N it suffices to construct forms such as uo , on a cross section of M into B, because they can then be extended by equivariance to the remaining part of the fibre; instead of a cross section we can also use M itself. (b) Equations (3) and (4) are the structural equations for uo and I,!I~’’.Equation (4) says that the o(e)-valued part of the curvature on B is -uoluo. T o interpret (3) as a structural equation we consider the adjoint action of G on its orthogonal complement in o ( f ) . Then (-:otp) has values in that G-module, is horizontal, and satisfies an

10.8. A Formulation of the Immersion Problem

205

invariance of the type in lemma 5.3, part (ii). Thus (3) may be restated as Duo = 0, (3') where D is as in 5.3 for the connexion 40

+

40".

(c) Equations (1) and (2) are algebraic at each point. Cartan [I61 has shown that they possess many solutions provided f 3 d(d l), so that (1) and (2) are solvable locally. T h e solutions which Cartan obtained are of a sort which he calls regular, and for analytic Riemannian manifolds this regularity implies the existence of local solutions of (3) and (4) also; this is the statement of the CartanKahler theorem [15, 431 in our context. Analyticity is definitely required because the Cartan-Kahler theorem leans on the CauchyKowalevsky theorem, which is true only in the analytic case [34, p. 81; 431.

4 +

Problem 13. T h e algebraic problem (1) and (2) is equivalent to finding second fundamental form matrices Si, i = 1, ..., e, such that the curvature K ( x , y ) of orthonormal vectors x, y E M , is given by

+

e

~ ( xy ), =

((sex, i=l

x> (siyt Y > - (Six,Y > ~ ) A.

Problem 14. Assuming Cartan's solution of (1) and (2), which only depends on the curvature identities, show that if R is a tensor (not a tensor field) satisfying those identities then there is a Riemannian manifold having R as curvature tensor at one point. (Hint: let k = 0 consider the imbedding of problem 1 1 .) T h e following problem gives an outline of the solution of (1) and (2), but withf larger by d than Cartan's solution. Problem 15. Let U = 4-linear functions on Rd satisfying the curvature identities, S = symmetric d x d matrices, P = symmetric 4-linear functions on Rd, and T = 4-linear functions on Rd satisfying the identities: for t E T , x , y , z, w E Rd, t ( x , y , z, w ) = t ( y , x, z , w ) = t ( z , w , x,y ) . P is a subspace of T ; T may be considered to be the space of symmetric bilinear forms on V = the symmetric part of Rd @ Rd. Let D = d(d - 1)/2, E = D d = dimension V . When we speak of t E T being positive definite we mean it as a symmetric bilinear form on V .

+

206 10. IMMERSIONS AND

THE

SECOND FUNDAMENTAL FORM

Let F : S ---t T be defined by

andG: T -

Uby G(t)(x,y , z, w)= t ( x , z, y , 4 - t ( x , w , y , 4.

Verify the following: (a) If sl, ..., sk E S, then R = XiG(F(si)) is the formula which expresses the curvature tensor R in terms of second fundamental forms si for an imbedding in Rd+k. (b) dim T = E ( E 1)/2, dim P = d(d3 2d2 3d 2)/24, dim U = d2(d2- 1)/12 = dim T - dim P, and G is linear with kernel P, hence G is onto U. (c) A symmetric bilinear form is nonnegative semidefinite and of rank one if and only if its matrix may be written X X , where X is a column matrix. (d) If we take as basis of V elements of the form i(e$ @ ej ei @ ei), i j , and index these basis elements by pairs I = (i, j ) , i j , then the matrix of F(s) is (sIsJ),where sI = sii is the i, j entry of the d x d matrix s. (e) Every nonnegative semidefinite symmetric bilinear form t on V may be written as a sum of k = rank t of those of rank 1, k E. Thus the set of sums Xi F(si) is exactly the nonnegative semidefinite part of T .

+

+

+ +

+

<

<

<

(f) If we order the double indices I so that the repeated ones come first, and let I , = the identity matrix of order r, K = the d x d matrix with all entries 1, then the bilinear form on V with matrix (IdfK ) is positive-definite and in P . ID (g) Every coset t P contains a positive-definite element.

+

Problem 16. T h e following abstracts part of the proof of theorem 5 . Let F : M --t N be a differentiable map, where N is parallelized by 1-forms wi , i = 1, ...,f.Let p , , p , be the projections of the product M x N . Then the codistribution spanned by pl*F*wi - p , * w i ,

1, ..., f i s integrable and the graph of F is an integral submanifold. Conversely, if forms B i , i = 1, ...,f, are given on M such that p,*Bi - p2*wi span an integrable codistribution, then an integral

i

=

10.9. Hypersurfaces

207

submanifold is locally the graph of maps of neighborhoods of M into N . 10.9 Hypersurfaces

A hypersurface is an immersed manifold in a space of one dimension higher; in the notation above e = 1. When the immersion is into Rf the curvature may be expressed in terms of one second fundamental form, which implies that each R,, is either 0 or of rank 2, so the curvature transformations of a hypersurface in Rf are very special. T h e possibilities for the bundle of adapted frames is also limited in the hypersurface case. In fact, the unit normal bundle must be either a connected double covering of M or two copies of M ; thus B is either a double covering of F ( M ) or two copies of F ( M ) . When M is simply connected only the two copy-case is possible. This makes the specialization of theorem 5 much simpler, since +’’ = 0 is determined. Let M be a simply connected Riemannian manifold with curvature form Yo = ooool KBoBot, where uo is an Rd-valued equivariant 1-form on F ( M ) such that Duo = 0, ooBol= 0. If N is a ( d + 1)-dimensional complete manifold with constant curvature k, then M may be immersed in N .

Theorem 6.

+

T h e immersion I is uniquely determined by the specification of the second fundamental form uo , dI, for one m E M , and a choice of unit normal at M. This yields the uniqueness theorem: Theorem 7. Let N be of constant curvature k and such that the group of isometries of N is transitive on F ( N ) (e.g., N = Sd+l or

N = Rd+l).If Ii : M + N , i = 1, 2, are isometric immersions of M as a hypersurface such that the second fundamental forms coincide, then there is an isometry J of N such that ]I1 = I , . Theorem 7 holds without the restriction that M be simply connected because we are given the existence of the immersions and thus know that the integral manifolds give the graphs of their prolongations uniquely. ] is produced by forcing it to satisfy the condition at some single point: d ] dIlm = dIZTnand dJ(zl) = z, , where ziis the chosen unit normal at m for Ii. Problem 17.

Let Id : M + N be isometric imbeddings as hyper-

208 10. IMMERSIONS AND

THE

SECOND FUNDAMENTAL FORM

surfaces, i = 1, 2, such that the second fundamental forms agree and for some m E M there is an isometry J : N -+ N such that d J d I , , = d I Z m , and if z is a normal at m, z' = d ] ( x ) , then SzS,= S,, # 0, where S, , S , are the second fundamental forms of

I , , I2 .

Prove that ]Il = I , . [Hint: Use problem 16. Alternately, if y is a curve in M starting at m, n = l l ( m ) , show that the development of Il o y into N , depends only on the second fundamental forms along y and the development of y into Mr,L.] Problem 18.

Let g : N M

-+

R be differentiable. Show that

= g-'(O) n {n

I dg(n) f 0)

is a hypersurface in N . Problem 19. Let g : R f -+ R be differentiable, and consider Rj provided with the Euclidean metric, giving the M of problem 18 the induced metric. Show: (a) At any m E M there is a unique unit normal z1 such that dg(z,) = a > 0. These normal vectors form a differentiable field in I(&') so ,M is orientable. (b) If M' is the linear hypersurface of Rf through m E M which is l z , , show that the second fundamental form Hl of M at m is given by H,(x, y ) = (lia) H&, Y ) , where g' = g IM' , x, Y E M7lL * (c) If x = Xaa,Di(m),y = Z biDi(m), and we let X = X a i D i , Y = X b,D, , then the curvature in M of the plane of x and y is KOb, Y ) = (l/a2) { x ( X g )Y ( W - ( 4 Y g ) Y I Problem 20. Use the formula for KOgiven in problem 19(c) to show that the curvature of the sphere of radius r, Sd = g-l(O), where g = X ui2 - r2 : Rd+l-+ R, is constant and equal to l/r2. Problem 21. A ruled surface is a surface in R3 such that through every point there is a line (a ruling) in R3 which is contained in the surface. Show that the curvature of a ruled surface is nonpositive. Problem 22. Let y be a ruling in a ruled surface, and assume that y is the base curve of a C" rectangle having rulings as longitudinal curves, parametrized by arc length. (a) Use the fact that the associated vector field V is a Jacobi field in R3 to show that it may be written V = U A B, where A, B are

+

10.9. Hypersurfaces

209

parallel along y in R3. Moreover, if the initial transverse curve is chosen at the right point then we may have A , B, and y * mutually perpendicular. (b) Use the fact that V is also a Jacobi field in the surface to conclude E2V = -KV, where K = curvature along y , E is covariant derivative in the surface with respect to y* . (c) Use the fact that EV, E2V are the projections of DV, & E l / ) onto the tangent plane to the surface (proposition 1) to find E2V explicitly in terms of ( A , A ) and ( B , B ) and thus find the formula for K along y. Problem 23. A surface in R3 is doubly ruled if it contains two lines in independent directions through every point. Prove that a doubly ruled flat surface ( K = 0) must be a plane. Problem 24. Let M be immersed in Rd,I : M -+ Rd, in such a way that the last coordinate vd = ud o I is always positive on M . Let I' denote the first d - I coordinates of I , so I = (I', vd). Considering Rd as the subspace of Rd+ehaving the last e coordinates 0, we get an immersion ] of M x Se, by defining

m,4

=

("(m),V d @ )

4.

We say that M x Se is immersed as a manifold of partial rotation. If M is a curve in R2 we say ] ( M x S e ) is a hypersurface of rotation. We give M the metric induced by I , M x Se the metric induced by I. (a) Show that M x {x} C M x Se is a totally geodesic submanifold for every x E Se and is isometric to M . (b) Show that { m } x Se C M x Seis a totally geodesic submanifold if and only if m is a critical point of vd , and in that case {m}x Se is isometric to an e-sphere of radius vd(m). (c) J is an imbedding if and only if I is also. Problem 25. Products of spheres as hypersurfaces. If we let M = Sd in problem 24, M imbedded in Rdfl as a translate of the usual Sd so as to have v d > 0, then J gives us an imbedding of S d x Se as a hypersurface. (a) Find the curvature K ( X , Y ) in the cases: (i) X , Y both tangent t o S d ;

210 10. IMMERSIONS

AND THE SECOND

FUNDAMENTAL FORM

(ii) X tangent to S d , Y tangent to Se, (iii) X , Y both tangent to Se. (b) Generalize this to an imbedding of a product of any number of spheres as a hypersurface, and find what can be said about totally geodesic submanifolds and curvature. A special case is the usual imbedding of a torus as the surface of a bagel. Problem 26. Let M be an orientable hypersurface in Rd+l, N a submanifold contained in a bounded region of Re+l. By considering Rd+1 c Rd+e+l we get a normal bundle to M which has fibres of 1, and because M is orientable this normal bundle has dimension e e 1 independent global cross sections. (a) Show that there is a C" positive function r on M such that the exponential map of Rd+e+lis a diffeomorphism on

+

+

I,(W = {(m4 E I(WI II x II < r(m)>* (b) Show that M x N may be imbedded in Rdfe+l. (c) If M is compact show that r may be taken constant and hence the imbedding of M x N constructed in such a way as to have induced metric with formula (s

+ 4 s' + t'>

= (s,

o n + c (4 f > z ,

where s, s' E M , , t , t' E N , , ( , ), is a metric on M which is a differentiable function of n E N , ( , )2 is the metric on N induced from Re+l,and c is constant. (d) If M , N are only immersed, then the procedure given will yield an immersion of M x N . However, find an example where M is not imbedded, N is imbedded, and by taking r nonconstant we get an imbedding of M x N . Remark When M is Sd the procedure in part (c) gives N x Sd as a manifold of partial rotation. Problem 27. If M is a hypersurface in Rd+l show that the curvature transformation, when viewed as a symmetric linear transformation of bivectors, is diagonal with respect to a basis of decomposible bivectors and has characteristic values hihi,i < j , where Xi are the characteristic values of the second fundamental form. T h e hi are classically the principal curvatures.

10.9. Hypersurfaces

21 1

Problem 28. If M is a hypersurface of N , N having constant curvature, show the following relations between the second fundamental form and curvature at a point m:

(a) T h e curvature at m is constant and the same as that of N if and only if the second fundamental form has rank 0 or 1. (b) T h e difference between the curvature transformations of M and N (restricted to M,) has its range in a single two-dimensional subspace of Mm if and only if the second fundamental form has rank 2. (c) If neither (a) nor (b), then the curvature determines the second fundamental form. Problem 29. Classical rigidity theorem. Let N be homogeneous of constant curvature k , and let It : M + N , i = 1, 2 be immersions of M as a hypersurface. T h e type number t(m) of m E M is the rank of the second fundamental form at m ; by problem 28, if t ( m ) 3 2 then t ( m ) is determined by the curvature independently of the immersion, but is only defined with respect to an immersion at points where M has constant curvature k . Prove that if t ( m ) > 2 for every m, then there is an isometry J of N such that ]I1 = I , . Problem 30. If M is a hypersurface in R d f l , then we define the Gaussian curvature of M to be the real-valued function k on M given by k(m) = det S , , where S , is the second fundamental form transformation for a normal z at m.

(a) Show that k is well defined when d is even but only up to sign when d is odd. (b) Show that k(m) depends only on the metric on A4 and find a formula for it in terms of curvature. I n particular, when d = 2, k coincides with the curvature. Problem 31. Show that the curvature transformations at a point of a 3-dimensional Riemannian manifold may be realized as the curvature transformations at a point of a hypersurface in R4. Problem 32. Let M be a complete hypersurface of Rd+lsuch that the holonomy group of M is reducible, that is, there is a distribution V on M of dimension e, e # 0, d, which is invariant under parallel translation on any curve. T h e orthogonal complement V1 is then another such distribution.

212 10. IMMERSIONS AND

THE

SECOND FUNDAMENTAL FORM

Assume that the set of points at which there is some nonzero curvature is dense in M . Show that: (a) the second fundamental form is 0 on V or VL, let us say V. (b) the parallel translate of elements of V in M coincides with their parallel translate in Rd+l. (c) the linear submanifold in Rd+l of dimension e with tangent space V , , at any m E M , is contained in M and the straight lines in this linear submanifold are geodesics in M . (d) if Rd-e+l is the linear subspace of Rd+l perpendicular to V , then M n Rd-e+l= N is a hypersurface of Rd-e+l and M is imbedded in Rd+l as a Riemannian product N x Re C Rd-e+l x Re.

CHAPTER 11

Second Variation of Arc Length T h e first and second variations of arc length are considered and Synge's formula for the unintegrated second variation and its specializations are proved. T h e index form for general end points is defined, and after a treatment of the elementary properties of focal and conjugate points, the Morse index theorem for one fixed endpoint is proved. Minimum points and closed geodesics are discussed as well as various formulations of convexity. T h e chapter closes with a version of Rauch's comparison theorem and a number of consequences, including relations between curvature and volume [38, 50, 831. 11 .I First and Second Variation of Arc Length

We now take u p the study of the relation between the lengths of the longitudinal curves of a rectangle and the associated vector field. If the base curve is a geodesic and the initial and final angles of the associated vector field satisfy certain reasonable conditions, then the derivative of the length with respect to the transverse parameter is zero. Hence, in this case it is the second derivative which determines how nearby curves compare in length with the base curve. In general, these first and second derivatives of the lengths of longitudinal curves are given by differentiating the length integral under the integral sign with respect to the transverse parameter. Thus we get what are called the Jirst and second variations of arc length along the curve of a rectangle in the form of integrals. A broken C" rectangle Q is a map [a, b] x [c, d ] -+ M such that there is a finite partition u,, = a < u1 < ... < u,-~ < u, = b of [a, b] for which the restriction of Q to ui] x [c, d ] is a C" rectangle, i = 1, ,.., n. 213

214

11. SECOND VARIATION OF ARC LENGTH

(a) Show that a broken C" rectangle is continuous. (b) If Q is a broken C" rectangle show that dQ(D,(u, v ) ) is well defined for every (u, v ) in the domain of Q and defines a broken C" rectangle in T ( M ) . (c) On the other hand, dQ(D,(ui , v ) )need not be well defined. (d) If M has a Riemannian structure, then a broken C" rectangle has a canonical lifting to F ( M ) as defined in Section 8.1. (e) If T is a broken C" curve in M and V is a broken C" lift of T to T ( M ) ,then there exists a broken C" rectangle Q having T as base curve and V(u) = dQ(Dz(u,0 ) ) for u E [a, b], that is, V is the vector field associated to Q. (f) If Q is a broken C" rectangle, w a form on M , then Q * u is not defined on the vertical lines u = u i , although Q*w(D2) may be defined continuously while Q*w(D1) will have right and left limits, Q*w(D,)(ui+, v ) and Q*w(D,)(ui-, v), respectively. Now let Q be a broken C" rectangle in M defined on [a, b] x [0, 11 with associated vector field V and longitudinal curves T~ , y E [0, 11. Then the length of T~ is Problem 1.

so

and

are the first and second variations. T h e integrands D, j I dQ(Dl) I I (u, 0) and DZ2I I dQ(D,) I I (u, 0) are called the unintegrated jirst and second variations of arc length. be a canonical lifting of Q to F ( M ) . Using the notation of Let Section 8.1, uQ(D,),wQ(D2),c,bQ(Dl),etc., are functions with domains in R2 and values in Rd or o(d). wQ(DZ)and c,bQ(Dz)are defined and broken C" on all of [a, b] x [0, 11, while uQ(D,)may be discontinuous on vertical lines u = ui , but in any case has left and right limits and = 0 because the longitudinal curves derivatives at these points. +"(Ill) of 8 are horizontal.

1 1.1. First and Second Variation of Arc Length

215

T h e structural equations applied to D,, D,give (1)

D1wQ(D2)- D2wQ(D1)= -4"(D1) =

(2)

wQ(02)

+

d Q P 2 ) WQ(Dl)

4Q(WwQ(D1);

DAQ(D2)- D24Q(Dl)= DAQ(D2)

+ OQ(D1 D2)

=

-[CQ(D1),4Q(D2)l

=

@Q(D,, D2).

9

T h e inner products of the longitudinal and transverse fields of Q are given by the corresponding inner products of wQ(D,)and wQ(D2). Lemma 1 . Let Q be a C" rectangle, and assume that the tangent to the base curve has constant length C. Then 1 b l'(0) = - - (wQ(D2),DlwQ(Dl)>(u,0 ) du C

ja

+ c1 {<wQ(D2),wQ(D1)>(b,0) - (wQ(D2),w Q ( 4 ) > ( a ,0)). Proof.

According to the last remark above,

D2 I I d W 1 ) I I (u, 0) = ~ 2 < w Q ( D 1 ) ,wQ(Dl)>""%0 ) 1 = (u,0 ) =

=

=

c c1 wQ(D1)>(u, 0 ) c1 (u,0 ) c1 D, (wQ(D2), 0) 1 - c (wQ(D2), GwQ(D1)>(u, 0 ); -

wQ(D1)>(.,

we have used (1) and the fact that +O(D,) is skew-symmetric. Therefore, 1 b Z'(0) = - - (wQ(D2),D1wQ(D1))(u, 0 ) du

J

C a

+ c1 (wQ(D2),wQ(D1)>(.,0)li which is the desired formula.

9

11. SECOND VARIATION OF ARC LENGTH

216

If Q is a broken C" rectangle having the tangent to the base curve a constant C in length, then Corollary 1.

If the base curve of broken C" rectangle Q is a geodesic (unbroken) and (wQ(D2),wQ(Dl))(b,0) = (wQ(D,), wQ(Dl))(a,0), then 1'(0) = 0. I n particular, l'(0) = 0 if the transverse curves are all perpendicular to the base geodesic. Corollary 2.

Proof. T h e base curve is a geodesic if and only if wQ(D,)is C" and DlwQ(Dl)(u,0) = 0 for every u. Under the hypothesis given the integrand in corollary 1 is 0 and the sum telescopes to give

(wQ(U,), WQ(Dl))(b, 0)

- (W0(D2),

w Q ( 4 ) > ( a , 0) =

0.

Corollary 3. Let N, P be submanifolds of M , and let us consider only broken C" rectangles whose initial and final transversals are in

FIG. 36.

N and P. Then a curve T has the property that for all such rectangles with base curve T, Z'(0) = 0 if and only if T is a geodesic from N to P which is perpendicular to both N a n d P. Proof. T h e idea is that we are able to get rectangles with sufficiently arbitrary wQ(D,) that the formula for Z'(0) = 0 will yield DlwQ(Dl)(u,0) = 0, and then that T is smooth at the breaks, perpendicular to N a n d P at the ends.

11.1. First and Second Variation of Arc Length Let

T

217

be a horizontal lift to F ( M ) . If r # ui , define a curve y in

Rdby y(u) = f ( u ) Dw(?*)(u),wheref(u) is a non-negative C" function which is positive at r and 0 at every ui . Then V = f y is a lift of T to

T ( M ) ,so we may define Q(u, v ) for this Q, <W"(D,),

=

exp,,,,vV(u). It is easy to see that

4 w Q ( W ( u ,0 ) =

f(4I I D I W Q ( D 1 ) I l P

(% 01,

and that the terms in the sum of corollary 1 are all 0. Thus I'(0) = 0 gives S:f(u)

/ I DIWQ(D1)1 1 2

( u , 0) du =

0,

which implies D,wQ(D,)(r,0) = 0. T h e terms in the sum are treated similarly. We show that ( t , '*(ui+) - T * ( z L - ) ) = 0 for every t E M T ( u i )0, < i < n, by parallel translating t along T , multiplying the field so generated by a non-negative function positive near ui but 0 outside a neighborhood of ui , and defining Q as above. T h e rectangles to show normality at the ends of T may be taken to be those having longitudinal curves consisting of a short geodesic e ) and the segment T I r a + r , b l , where y segment from y(v) to T ( U is a curve in N with y(0) = T(u);the other end is treated likewise.

+

Problem 2. Show (independently of the arguments in Chapter 8) that a curve which minimizes distance between two points is a geodesic. (Classically, a geodesic was defined as a solution of the calculus of variations problem of finding minimal curves, with self-parallel condition Vv,y* = 0 following as a consequence.)

Our applications of variational theory will only involve variation of a geodesic in a direction perpendicular t o the geodesic at every point, that is, we shall only be using rectangles having a geodesic as base curve and the associated vector fields perpendicular to the base. As further justification for this assumption we prove the following remark which shows that for C" rectangles satisfying the end conditions the assumption entails no loss at all. T h e remark is not true for broken C" rectangles, however, unless the base geodesic is allowed broken linear reparametrizations. Remark. If Q is a C" rectangle having a geodesic base curve and with the initial and final transversals perpendicular to the base curve,

11. SECOND VARIATION OF ARC LENGTH

21 8

then there is a partial reparametrization of Q for which the longitudinal curves are reparametrizations of those of Q and the associated vector field is perpendicular to the base. Let I'be the associated vector field of Q, T the base curve of ( V , T * ) , s o f is a real-valued function on [a, b]. By hypothesis f ( a ) =f(b) = 0. Define map F : [a, b] x [c, c E] + [a, b] x [c, d ] by Proof.

Q, f

=

+

F(u, 71)

= u

i

1

-

- wf(u), a),

k

where

E

>0

is such that the range of F is in [a, b] x [c, d ] , and k Then Q o F is the desired rectangle. Problem 3.

= (T*

,T * ) .

Complete the proof that the vector field associated to

Q o F is perpendicular to T .

Henceforth we shall assume that all broken C" rectangles have geodesic base and associated vector field perpendicular to the base. For convenience we also assume that the base curve is normalized so as to have unit tangent vectors, initial parameter value a = 0, and hence length b = b - a. Lemma 2 (Synge's variation is given by

DZ2 I I w Q P 1 ) I I

=

formula

I I D1wQ(Dz) I- ,1

[86]). T h e unintegrated second &%?(Dl),

dQ(D2)) I I W Q ( D , )

I l2

+ W ( w Q ( D 1 ) , DzwQ(Dz)>+ <wQ(D1),+Q(Dz)wQ(Dz)>h

where all these functions are restricted to [0, b]

x (0).

The proof is by direct computation, always recalling that we are dropping the argument (u, 0):

Proof.

Dz2 I I w Q ( W I I

= D,(~,(wQ(D1)7 w Q ( 4 ) > ?'l =

-(D,wQ(m,

wQ(4)

>, + D , (DzwQ(D1),

wQ(4)>,

since the factors appearing in the denominators are powers of <wQ(D1), wQ(D1)>(u,

0) = 1.

Now using the first structural equation ( I ) and the fact that dQ(D2) is skew-symmetric gives (a)

D',

/ I w Q ( W II

= -<4wQ(4),

wQ(DIY

f D2(D,wQ(D,), w Q ( 4 ) > .

11.1. First and Second Variation of Arc Length

219

T h e base curve T is a geodesic, so DlwQ(D,)= 0, hence, z= (Dl)2

=

0,

because the associated field is perpendicular to T.

Let V be the associated vector field along base geodesic V' the covariant derivative with respect to T* , and let 7 be the transverse vector field of Q. Then Corollary 1.

T,

I"(())

=

s" ( 1 1 0

V'

Il'(U)

- K ( V ) 11

/I2( .)

du

+

(T*

9

v,p>],b.

( K (v) = K( V,T*), Cf. 9.4.)

This follows from the observation V ( u ) = ? ( D 1 ~ Q ( D 2 )0)) (u, and 0,v = Q(D2 +"(D2))wo(D2) (cf. theorem 6.11).

+

Corollary 2. Let N and P be submanifolds, Q a rectangle with base geodesic T perpendicular to N and P and with initial and final transversals in N and P. Then

220

11. SECOND VARIATION OF ARC LENGTH

where S is the second fundamental form of the appropriate manifold and V is the associated vector field as before. Proof. T h e integral is the same as in corollary 1. By proposition 10.1, if T V o ,is the difference transformation of MT,,, given by N as a submanifold of M , then for vector field W on N TVO,

W ( 0 ) )= &Yo, W - E ” ( 0 , w.

But E V o , Wis perpendicular to T,(O), so for W =
D Y(01 P)

=
=

P,

T”(01 V(0D

-
= -<~r*(o,w9,

W)) W)).

Similarly for the other term. Note that when N a n d P a r e points, the above formula for the second variation reduces to

This follows from the Jacobi equation and integration by parts.

11.2 The Index Form

Let N a n d P be submanifolds of M , and let T be a geodesic segment perpendicular to N and P at its ends ~ ( 0 )and T(b). Let 9 be the linear space of broken C“ vector fields along T which are perpendicular to T and have their initial and final vectors tangent to N and P. T h e index form at T is a bilinear form on 9 defined by: if V , W E 9, nb

11.2. T h e Index Form

22 1

Roughly speaking, if we consider the length function defined on curves from N to P, then corollary 3 of lemma 1 says that the critical points of this function are geodesics such as I-.T h e index form may be viewed as a natural generalization of the Hessian of a function at a critical point, since it is the symmetric bilinear form associated with the quadratic form Z”(0) of V in corollary 2 to lemma 2. Morse theory is concerned with this generalization. If we observe that, except for points where V is not differentiable (u0 = 0, ..., U, == b), (V’, w,)= ( I f ’ , W)’

-

(V”, W ) ,

then we get another formula for I( V , W ) ,

From this formula and application of techniques similar to that used in the proof of corollary 3 to lemma 1 it is clear what the null space of I is, that is, what conditions V must satisfy in order that I( V , W ) = 0 for all W E9. We single out the properties satisfied at each end. Definition.

A Jacobi field V is an N - Jacobijeld if it satisfies

(i) I/ is perpendicular to the geodesic

T;

(ii) V(0)E NTc0,; (iii) S,*(o,V(0) - V’(0)is perpendicular to N,,,, . If N is a single point, N = 1-(0),then (i), (ii), and (iii) reduce to V ( 0 )= 0, V’(0)is perpendicular to 1-,(0). Problem 4.

dimension d

Show that the N-Jacobi fields form a linear space of 1, where d = dimension M .

-

T h e null space of I consists of the intersection of the spaces of N- Jacobi fields and P- Jacobi fields.

Theorem 1.

222

11. SECOND VARIATIONOF ARC LENGTH

We have already seen that a Jacobi field is characterized by the fact that it is the field associated to a rectangle having geodesics for longitudinal curves. T h e next theorem gives a similar characterization of N - Jacobi fields.

V is an N - Jacobi field if and only if V is associated to a rectangle Q such that all of the longitudinal curves of Q are geodesics starting perpendicularly from N and parametrized by arc length.

Theorem 2.

FIG. 37.

Proof. By problem 4 it is sufficient to show that the Jacobi fields associated to such rectangles are N-Jacobi fields and have values at some E > 0 which equal the space normal to T * ( E ) . T h e rectangles in question can be factored through I ( N ) by means of the exponential map l ( N ) -+ M . In fact the longitudinal vector field dQ(Dl) = X to such a rectangle Q gives a curve y , y(v) = X(0, v ) , in I 1 ( N ) , the unit normal bundle, and Q may be expressed as Q(u, = exp ~ ( v ) . Since V(0)= dQ(D,(O, 0)) is the tangent to the projection of y into N , (ii) is satisfied.

by the first structural equation (1). But E is covariant derivative on N and the normal bundle, so the derivative of normal field X(0, * ) is again a normal vector, which shows (iii). Moreover, to show (i) we

11.2. T h e Index Form

223

need only show that V and V' are perpendicular to T * at one point, which we have already for V(O),hence for S,*o,V(0).Thus it suffices to show E,,,,X is perpendicular to T,(O) = X(0). But

D,( X , XXO, 0 ) = 2 (Evco,X, X ( 0 ) ) = 0 since ( X , X ) = 1. We know that exp : I ( N ) -+ M is nonsingular on the 0 cross section and thus also at x = ET,(O) for some E > 0. However, the tangents at x to rectangles in l ( N ) of the form (u, v ) -P uy(v), y as above, fill up the tangent space l ( N ) , . Consequently, the tangents at T ( E ) of rectangles Q fill up the tangent space M T ( (,) and the normal , up the space normal to T * ( E ) . parts, V ( E )fill Corollary. = T(u), is the

T h e range of d exp, : l ( N ) , -+M , , where p = exp x space spanned by T * ( u ) and the values at u of N - Jacobi

fields, u # 0.

Any tangent at x may be decomposed into a vector tangent to I , ( N ) and a vector tangent to the ray through x. T h e component tangent to l , ( N ) maps by d exp to the value of an N - Jacobi field, the component along the ray maps to a multiple of T*(u). ( l , ( N ) = { Y E I ( N )I / I Y I / = u } . ) We shall denote the quadratic form associated with I by I also, thus the second variation writing I(V ) = I( V , V ) . Since I( V )is, for V E 9, Z"(0) of a rectangle attached to V , we have immediately: Proof.

If V E 2'is such that I( V ) < 0 then every neighborhood of T contains shorter curves from a neighborhood of ~ ( 0 in ) N to a neighborhood of ~ ( bin) P.

Theorem 3.

More generally, the dimension of a maximal subspace of 9on which I is negative definite, the index of I, tells in how many independent directions T can be pushed so as to shorten length, still obtaining a curve from N to P. It is shown below that the index is finite. Problem 5. Let M = Rd with the Euclidean metric and suppose that submanifolds N and P do not intersect. Let K ( N , P ) = broken C" curves from N to P, H ( N , P ) = straight lines from N to P. Suppose that these curves are parametrized by reduced arc length, that is, they all have domain [Q, I] and constant-length tangents. We make

224

11. SECOND VARIATION OF ARC LENGTH

K ( N , P ) into a metric space by defining the distance between curves u, T to be p(.,

7) =

max p(.(u),

+))

+ !I

I

-

1 I I. 7

Let 4 : K ( N , P ) + H(N, P ) be the map which assigns to u the line segment with the same end points as u. Show that (a) 4 is homotopic to the identity via a homotopy which leaves H ( N , P ) fixed. Thus H ( N , P ) is a deformation retract of K ( N , P ) and they have the same ordinary topological invariants. (b) H(N, P) is topologically the same as N x P and the length function L is differentiable when viewed as a function on N x P. (c) Vector fields V which are associated to rectangles with range in H ( N , P ) are Jacobi fields in Rd and may be identified with tangents to N x P. (d) T h e critical points of L as a differentiable function on N x P are the line segments which are perpendicular to N and P at the ends. (e) T h e Hessian of L is essentially the index form restricted to Jacobi fields in 9. 11.3 Focal Points and Conjugate Points [37,57, 59,60, 61, 731

Let N be a submanifold of M , I ( N ) the normal bundle. T h e exponential map of M by restriction gives a map exp : I ( N ) ---t M , which we have already seen to be a diffeomorphism in a neighborhood of the zero cross section. For n E N , let I ( N ) ( n )be the fibre of l ( N )

FIG. 38.

over n. Then t E I ( N ) ( n )is a focal point of N if d exp is singular at t. If p is the ray from 0 to t in l ( N ) ( n ) ,then exp(t) is called a focal point of N along exp o p , which is, of course, a geodesic perpendicular to N . When N is a single point, say m, so I ( N ) = M,,, , then a focal

11.3. Focal Points and Conjugate Points

225

point is called a conjugate point to m. T h e order of a focal point is the dimension of the linear space annihilated by d exp. Theorem 2 and its corollary show that it is equivalent to define focal points in terms of Jacobi fields as follows. If 7 is a geodesic which starts perpendicular to N , then ~ ( 6 )is a focal point of N along 7 if and only if there is an N- Jacobi field which vanishes at b. T h e order O f T ( b ) is the dimension of the space of such Jacobi fields. By theorem 1, the order is also the nullity of the index form at 7 with end manifolds N and the single point ~ ( 6 ) . T h e last statement seems to indicate how to generalize the concept of focal point to something which might be called a “focal submanifold.” Presumably one would then aim to prove an “index theorem,’’ as we shall below for the case of one end-manifold and a point, for the two end-manifold case. The purpose of such a theorem is to express the index of the index form in terms of orders of focal points (or focal submanifolds) in between the end manifolds. Such theorems have been formulated by both Morse [57] and Ambrose [ I ] for the two end-manifold case, but the difficulties in statement and proof are far greater than for the case we give, for which the formulation given by Morse is clearly the right one. Problem 6. Show that conjugacy is a symmetric relation, that is, if m is conjugate to n along geodesic 7,then n is conjugate to m along -7, where -7 is the geodesic with initial tangent -T,(O). Problem 7. Show that if 7 is a geodesic segment perpendicular to N on which there are no focal points of N , then there is a neighborhood U of 7 in M and a neighborhood V of ~ ( 0in ) N such that 7 minimizes distance among curves in U which go from a point of V to ~ ( 6 ) (cf. theorem 8.2). Problem 8. Show that if M is complete and contains a point which has no conjugate points, then M is covered by lid. Problem 9. Show that if M has nonpositive curvature, then there are no conjugate points. Problem 10. Determine the conjugate points and their orders for a point on a d-sphere of constant curvature.

Let CPd = SZd+l/S1 have the metric induced by the metric on SZd+lhaving curvature 1. Show that the conjugate points to a 1) n/2 and nn, n an integer, with point in CPd occur at distances (2n

Problem 11.

+

226

11. SECOND VARIATION OF ARC LENGTH

orders 1 and 2d - 1, respectively. Use the fact that a Jacobi field attached to a family of horizontal geodesics in S2d+1must project to a Jacobi field on CPd. Problem 12. In a way similar to that of problem 11, show that the S4d+3= unit sphere, conjugate points to a point in QPd = S4df3/S3, occur at distances (2n 1) n-12 and n7r with orders 3 and 4d - 1, respectively.

+

11.4 The Infinitesimal Deformations

A useful technique for analyzing the index form is to break it into the sum of several index forms obtained by inserting intermediate manifolds of dimension d - 1 normal to T . Since we will be dealing with several index forms, but all along segments of the same geodesic T , we introduce the notation I( Tl , T,) to mean the index form with end manifolds T , and T , . We shall also omit restriction symbols, the domain of I ( N , P), T , and T , are normal to T that is, if V E 3, at T ( u ~and ) T(u,), and I , = I ( T , , T,), then we shall write Il(V) instead of 11(V ~[ul,u,l). If we choose the intermediate manifolds close enough together then any curve in some neighborhood of T [by neighborhood of T we shall mean curves which go from a neighborhood of ~ ( 0in) N to a neighborhood of ~ ( bin) P, as well as lying in some open set containing T ] can be replaced by a unique shorter broken geodesic with breaks only at the intermediate manifolds. If Ti is placed at T ( u ~ ) i, = 1, ..., n, ui < u ~ + then ~ , the requirement is that there be no focal point of N on T I l o , u l , that there be no conjugate point of T ( u ~on ) T I ui,ui+l , i = 1, ..., n - 1, and that there be no focal point of P on T I[u, bl .A curve CJ sufficiently close to T will intersect each T i , and we let the first intersection be pi;requiring u to be even closer if necessary, there will be a unique shortest geodesic segment fromp, to the neighborhood of ~ ( 0in ) N , a unique shortest geodesic segment from pi to pi.kl, and a unique shortest geodesic segment from p , to the neighborhood of T(b)in P. Chaining these geodesic segments together gives a broken geodesic y having breaks only on the intermediate manifolds T i . The map 4 : CJ -+ y is a length-nonincreasing deformation of curves nearby T into the much smaller collection of broken geodesics; in fact, the range of 4 may be viewed as the product manifold

11.4. T h e Infinitesimal Deformations

227

T, x T, x x T , = T , and the length function on curves as a differentiable function on T . T is a critical point of the length function and the index form is the Hessian at T.

FIG. 39.

Detailed proofs of these facts are more proper to an exposition of Morse theory, so we content ourselves here with their infinitesimal versions. Lemma 3. Under the conditions prescribed on the choice of ~ is a unique vector the ui , for every choice of vectors yi E T * ( U ~ )there = yi , Y I[u, u,+l is a Jacobi field, and field Y E2’ such that Y(ui) Y Ilo,u,l , Y I [ u , b l are N-and P-Jacobi fields, respectively. G(y,, ..., yn) = Y , is a linear T h e map G : E:=l 7 * ( u i ) l -+ 9, isomorphism.

T h e linear transformation which assigns to an N- Jacobi field V its value V(ul)is 1-1 onto 7*(ul)l because T(u,) is not a focal point of N . Thus Y l[o,ul exists and is unique; similarly for Y Iru, bl . For the same reason there is a unique 7(ui)-Jacobi field Vi and a unique 7(ui+J-Jacobi field Wi such that V,(u,+J = yi+, , Wi(ui)= yi . Then Vi Wi is a Jacobi field having values yi at ui , yi+l at ui+, , which shows existence of Y ~ I U t , l L t + l l . Moreover, if Y is given, y I[u*,ut+Il - Wi = Vi IruI,u(+Il b Y the uniqueness of Vi , which shows . uniqueness of Y I r u , u , + l T h e linearity and one-to-one-ness of G follow easily from the uniqueness of Y and the linearity of the Jacobi equation. Proof.

+

We denote the range of G by X . Thus dim X = ( d - 1) n. By evaluation of V E 2’ at u1 , ..., u, we get a linear transformation E : 2’ -+ C7*(ui)l, E ( V ) = ( V ( u l ) ,..., V(u,)). T h e composition

228 F

11. SECOND VARIATION OF ARC LENGTH

G o E : 9 -+ Y is the infinitesimal version of the deformation y , so we call F an injinitesimal deformation. Corresponding to the fact that 4 is length-nonincreasing is the fact that F does not increase the index form I , which is proved below. =

4 : u -+

If Q is a broken C" rectangle with the transverse curves lying in Ti , then we get another rectangle +(Q)by applying 4 to the longitudinal curves of Q. If V is the vector field associated to Q, show that F( V )is the field associated to +(Q). Problem 13.

Lemma 4. If Y and 2 are Jacobi fields, then ( Y , 2') - ( Y ' , 2 ) is constant. If Y and 2 are N- Jacobi fields the constant is 0. Proof.

( ( Y ,2')

-

( Y ' ,2))' = ( Y , 2")

-

( Y " ,2 )

=

-
=

0,

+ (R,*;y7*,2 )

by a curvature identity,

<&*(")Z(O)- - V O ) , Y(0))= 0,

(b)

so subtracting (a) and (b) and using the symmetry of ST*(o)gives

-
q o ) ) + (Z'(O),

Y(0))= 0.

Problem 14. Let # be a subspace of the Jacobi fields along dimension d - 1 and such that for every Y , 2 E .X,

( Y , 2') - ( Y ' , 2 )

=

Show that there is a submanifold N normal to the space of N- Jacobi fields. Theorem 4.

T

having

0. T

at ~ ( 0such ) that

9 is

The basic inequality. Suppose there is no focal point of

N on ~ ( ( 0b, ] ) . For V E 9there is a unique N - Jacobi field Y such that Y ( b ) = V(b),by lemma 3 . Then I( V ) I( Y )and equality occurs if and only if V = Y . Proof.

Let Y , , ..., Ydp1 be a basis of N - Jacobi fields. Then there is a

unique expression V

=

Zi"r,'fiYi

valid on (0, b], where the fi are

11.4. T h e Infinitesimal Deformations

229

continuous broken C“ €unctions. We leave as an exercise the fact that such an expression exists and is unique on [0, b] as well. (See problem 15.) At points where V‘ exists we let W = Xifi’Yi and 2 = XifiYi’, so that V’ = W 2. Then

+

)

11. SECOND VARIATION OF ARC LENGTH

230

T h e N-Jacobi field which coincides with V at b is Y = Ci ciYi , where ci = fi(b). Letting W , = Ci ci‘Yi = 0, 2, = Xi ciYi’, the same computation (a)-(e) gives, since Z,(b) = Z(b), Y(b) = V(b), I ( Y ) =
(f)

Hence I( V ) - I( Y ) = J: ( W , W ) du 0. Equality occurs if and only if W = 0, which in turn is equivalent tof,‘ = 0 for all i , f i are constant for all i, and, finally, Y = V . Corollaries 1 and 2, which follow, have the same hypothesis on N and ~ ( ( 0b]), , but not corollary 3. Corollary 1.

only if V

=

0.

If V(b)= 0, then I( V ) 2 0, and equality occurs if and

Corollary 2. Let Y E 9. Then Y is an N- Jacobi field if and only if I( V , Y ) = 0 for all V E 9such that V(b)= 0. Proof.

V,’

=

(I

By polarization, (e) in the above proof gives, for V , , V , E 9, 2, Wl , V,’ = 2, W, , etc.,

+

~

1 ~ >2

+

= )

< ~ , ( b)

sr . ( b ) vl(b),~ / , ( b )+) jb < ~

1

w,> , du.

If Y is an N-Jacobi field and V(b) = 0, letting V , = Y , V , = V gives W , = 0, V,(b) = 0, so I( V , , V,) = I( Y , V ) = 0. Conversely, suppose I( V , Y ) = 0 for all V E 3 such that V(b)= 0. Let Y , be the N-Jacobi field such that Y(b) = Y,(b). Then I ( Y , Y - Y,) = 0 by hypothesis, I ( - Y l , Y - Y,) = 0 as just proved, so Z(Y - Y1)= Z(Y, Y

Hence by corollary 1, Y

-

-

Yl)

+ Z(-Y1, Y

-

Y l ) = 0.

Y , = 0.

Corollary 3. Suppose that there is no conjugate point of ~ ( 0 on ) ~ ( ( 0 b]). , For V E there ~ is a unique Jacobi field Y such that Y(0) = V(O), Y(b) = V(b), by lemma 3. Then I ( V ) >, I ( Y ) and equality occurs if and only if V = Y . Proof. Let T , and T, be ( d - 1)-dimensional transverse manifolds at ~ ( 0and ) ~ ( b )respectively, , such that their second fundamental forms

23 1

11.4. T h e Infinitesimal Deformations

and hence I,( V ) - I( V) = I,( Y ) - I(Y ) , so it suffices to prove the result for I , instead of I. Let I2 = I(T(O), T,), I3 = I ( T , , ~ ( b ) )so, that I , , I , , and I3 are the same except for their domains. Y , , where Yl is the T(O)-Jacobi field From lemma 3, Y = Y , such that Y,(O) = V(0). Since V - Y is 0 at both ~ ( 0 )and ~ ( b ) , Il(V - Y , Y,) = I,(V - Y , Y,) = 0 and I,(V - Y , Y,) = 13( - Y , Y,) = 0, by corollary 2 applied to I , and I3 , respectively. Adding these two equations gives

+

11(v - Y , Yl

+ Y,) = I,(V

-

Y , Y) = 0,

or,

Il(V, Y)

=

Il(Y, Y).

But by corollary 1,

= Il(K

V ) - Zl(K Y),

and equality occurs if and only if V

-

Y

=

0.

Problem 15. If V , , ..., V , are C" vector fields along T and are independent except at T(O), and V is a broken C" vector field along T , with V = ZfiVi the expression for V in terms of the Vivalid on (0, b ] , show by an example that the fi need not have continuous extensions to [0, b] even if V(0) = ZuiVi(0). However, if the Viare Jacobi fields then there are unique continuous extensions of the fi .

An infinitesimal deformation F is I-nonincreasing; that is, I ( V ) 2 I ( F ( V ) ) . Moreover, equality is obtained only if V = F( V ) .

Theorem 5.

Problem 16.

Prove theorem 5 by applying theorem 4 and corollary 3.

(For N = point this is due to Jacobi.) minimize distance to N beyond the first focal point.

Corollary 1.

T

does not

Proof. Suppose T ( C ) is a focal point of N , c E (0, b), and let Y be a nonzero N - Jacobi field which vanishes at c. Let V E 2z be given by

11. SECOND VARIATION OF ARC LENGTH

232

w,

V l [ O , C l = y I[O.Cl , I‘ I[C,bl = 0. Let I = I ( N , T ( b ) ) , I1 = T(C)), I2 = ~ ( T ( c ) ~, ( b ) )Then . I( V ) = 11( V ) I,( V ) = 11(Y ) I,(O) = 0 by corollary 2 to theorem 4. Now choose intermediate manifolds for an infinitesimal deformation F so that c is not one of the ui . Then F( V ) is smooth at c, while V has a break at c, hence F( V ) # V and I(F( V ) ) < I( V ) = 0. By theorem 3 there are shorter curves from ~ ( bto) N .

+

+

FIG. 40.

Corollary 2.

T h e first focal point of N is that point

T(C)

such that

T([O,b ] ) fails to minimize arc length to N for b > c, but T([O,b’]) minimizes arc length to N among curves in a neighborhood of T([O, b‘])for b’ < c.

T h e augmented index of a quadratic form is the dimension of a maximal subspace on which the form is negative semidefinite. Corollary 3. T h e index and augmented index of I are the same as the index and augmented index of I Ix , hence are finite. Proof.

of I

Since it is obvious that the index of I is not less than that of

Ix it suffices to prove the inequality the other way, and similarly

for the augmented index. Let 8 be a subspace of 2 on which I is negative semidefinite. is in the kernel of F, then Then F IJy is an isomorphism. For if V € 2 F( V ) = 0, I(F( V ) )= 0 I( V ) = 0, so from equality, V = F( V ) = 0. Since I is negative semidefinite on F ( X ) this proves the desired inequality for the augmented index. For the index the argument is the same except for using “definite” in place of “semidefinite.”

<

Now let N have focal points at distances c, along 7 , i = 1, ..., h- , where ci ci+l < b and each ci is included a number of times equal to the multiplicity of T ( c ~ )as a focal point; we do not exclude the possibility that (ci} is a proper subset of the set of focal values in (0, b ) ,

<

233

11.5. T h e Morse Index Theorem

and we do not know a priori that the number of focal points is finite. Thus there are fields YiE 9, i = 1, ..., h_ , such that Yil i o , c , l is an N-Jacobi field, YiIl c , b l = 0, and for each c E (0, b), {Yi1 c = ci} are independent. Let %- be the linear space spanned by these Yi, and let X = 2- 3,, where Z ois the null space of I .

+

Prove that the Yiare independent, so the dimension of 3- is h- . Moreover, the sum%%,, is direct. Problem 17.

Lemma 5.

+

T h e restriction of I to ,P is identically 0.

= 0 it suffices to show Z(%- , %-) Proof. Since I(%, Z 0 ) that I ( Y i , Y j ) = 0 for every i, j . We may assume ci I , = I ( N , 7(ci)), I , = Z(7(cj), P). Then

=

0, or

< c j . Let

I(Yi , Yj) = I,(Yi , Yj)

+ 12(Yf, Yj) = I , ( Y i , Yj) = 0

since Y j is in the null space of I , by theorem 1. Let a ( I ) ,i ( I ) , and n(Z) denote the augmented index, the index, and the nullity of I , respectively. I t is well known that on a finite-dimensional space the sum of the index and nullity of a quadratic form equals the augmented index. Because we have been able to reduce I to a finite-dimensional subspace X without altering the index and augmented index, this result is also true for I , so a(I) = i ( I ) n(Z). Henceforth, when we speak of the number of focal points we shall mean the sum of their orders.

+

The number of focal points of N on ~ ((0,b ) ) is finite and is a lower bound for i ( I ) .

Theorem 6. Proof.

h-

By problem 17 and lemma 5 and the fact that a(Z) is finite,

+ n ( I ) = dim% < a ( I ) = i ( I ) + n ( I ) ,so h- < i(Z).

11.5 The Morse Index Theorem [57, 831

In this section we restrict to the case where P = ~ ( b ) Then . the Morse index theorem says that the inequality of theorem 6 is an equation. Let I = Z(N, 4 6 ) ) . Then a ( I ) is the number of focal points of N on ~ ( ( 0b]). ,

Theorem 7.

234

11. SECOND VARIATION OF ARC LENGTH

Proof. The idea is to examine I , = I ( N , ~ ( t and ) ) the integer-valued functions a ( t ) = a(Il), i(t) = i(l,),and n(t) = .(I,) as t passes from 0 to b. Because the domain of I , may be regarded as an increasing function of t , a, and i are nondecreasing; for small t, I , is positive-definite, so a and i are initially 0. We have already noted that a = i n, and that n(t) = 0 except for a finite number of t, where n(t) is the order of the focal point. Thus a must have jumps which at least add up to the sum of the n(t) (theorem 6 ) .

+

I FIG. 41.

On the other hand, because I , is continuous in t, in order for some of the positive definite part to go into the negative definite part it must pass through the null part. This causes the jumps of a and i to occur separately, on opposite sides of the focal values, and with magnitude n(t). As a value t is entered from the left the jump of a is n(t), i is n(t), i(t) = i(t-). continuous from the left; formally, a ( t ) = a(t-) When we pass on from t a is continuous from the right, n drops to 0, and i must take up the slack in the equation a = i n; a(t+) = a(t), i(t+) = i(t) n(t).So what we need to establish in detail is, for every t , a(t+) = a ( t ) and i(t-) = i(t); the theorem then follows by a simple computation. Let F , be an infinitesimal deformation for I , having intermediate i = 1, ..., n. Then the same intermediate manifolds manifolds at T(u~), will serve to give an infinitesimal deformation F, for I , when u E U , a neighborhood of t. This infinitesimal deformation is given by the

+

+

+

11.5. T h e Morse Index Theorem

235

evaluation map E and an isomorphism G,: 9 -+ X u , where 9 = Z l ( u i ) is the direct sum of the tangent spaces of the intermediate manifolds. T h e map H , = G,G;l : X , -+ X , is an isomorphism, and it alters only that part of Y E X t defined on (urL,t ] ;the part defined on [0, u,] is unchanged. T h e value of I , on Y is a sum of terms of which only one, -( Y’(u,+), Y(u,)), is altered by H , , and even here Y(u,) remains fixed. Let Y , = H,(Y). In order to prove continuity of Iu(Yu)as a function of u it is sufficient to prove that Yu’(un+)is continuous in u. By its definition Y , , restricted to [un , u ] , is the Jacobi field which = y . For y , Z E l ( u , ) let satisfies Y,(u) = 0 and Y,(u,) = Y(uYL) Y,,z be the unique Jacobi field such that Yu,g(u,)= y and Y;,Z(un)= z ; the map ( y , x) + Y,,z is linear i n y and x,and so is evaluation at u, so we may write Yy,s(u) = A,(y) B,(z), where

+

4 > Bu

:I

( U J

-

I(u)

are linear transformations. A , and B, are continuous in u since YY,#is continuous. However, there is a unique solution for x of A,(y) B,(x) = 0, so that Bil exists and is continuous. Thus the solution Yu’(un+)= x = -B;’A,(y) is continuous in u, and finally so is I,( Y,). Now let .X+ be a maximal subspace of X , on which I , is positive definite and let S+ be the unit sphere of X + with respect to I! . Then a(t) = codim X+.Then f : ( u , Y ) -+I,(Y,) is a continuous function on U x S+ which is positive on {t} x S+. Since S+ is compact, there is a neighborhood U‘ of t such that f is positive on U‘ x S+ (see lemma 6 below). Thus for u E U‘, I , is positive-definite on the space H,(X+) spanned by H,(S+), and a(.) a(t). But a is nondecreasing, so a(t+) = a(t). T h e same argument applies to a maximal negative definite subspace of I , on X , to show that i(u) 2 i(t) for u in some neighborhood of t . Thus i(t-) = i(t).

+

<

Lemma 6. Let f be a continuous real-valued function on a fibre bundle B with compact fibre F and base M ; denote by F, the fibre over p E M . Define fr,p,fM : M -+ R by fm(P) =

min W

fM@) =

max

)I b EF,)

{f(b)I

T h e n f,,, and fM are continuous.

EFD).

236

11. SECONDVARIATION OF ARC LENGTH

Since this is a local theorem, we may assume that B = F x M . For E > 0 we can find a finite covering W i= Vi x W o f F x {p}, W a neighborhood of p , such that I f ( b ) -f(b’) I < E for every 6, b’ E W i. Then for x E F and p‘ E W , If(x, p’) - f(x, p ) I < E. Now if b, = (xl,p ) is a point where f(b,) = f,&) and b, = (x2,p’) is a similar point for fM(p’),then Proof.

fM(P’)

-6 =

f(%

,P’)

-


?

IfM(p’) - fM(p) 1 < E, which concludes the proof for proof forf,,l follows by considering -f.

so

fM

. The

Problem 18. Let J be the inner product on 9, the domain of I ( N , P ) , given by J( Y,2) = J: ( Y , 2 ) du. Also let J( Y ) = J( Y,Y). A characteristic value of I with respect to J is a number X such that n(I - X J ) # 0; a characteristic vector of I belonging to X is a vector field in the null space of I - X J . Show that: (a) For X sufficiently negative I - XJ is positive-definite. (b) Y is a characteristic vector of I if and only if Y satisfies the end conditions at N and P (S,*Y - Y’is perpendicular to N , P ) and is a solution of the second order equation Y” R,*A-* + XY = 0. (c) T h e augmented index of I - X J is finite and is equal to the

+

number of independent characteristic vectors belonging to values A. (d) Let P = r(b) and let Jibe the restriction of J to the domain of I , . Number the characteristic values of I , , counting multiplicities, starting with the least, so as to obtain a sequence of functions {hi} o f t . Then the Xi are continuous and nonincreasing on (0, b] and n(I,)is the number of i such that Xi(t) = 0.

<

Problem 19. With each X E I ( N ) , x not in the 0 cross section, associate the index form I , = I ( N , exp x) at the geodesic exp ux ILo, l . Show that: (a) If n(I,) = 0, there is a neighborhood U of x such that for every x’E U , i(Iz,) = i(Iz). (b) If M is complete and N is closed in M , the set {m I n(Iz) = 0 for every nonzero x E exp-’ m} is everywhere dense in M (see problem 1.11).

11.6. T h e Minimum Locus

237

Those interested in studying Morse theory and its applications are referred to [37,38, 54, 57, 71, 811. 11.6 The Minimum Locus [46, 60, 62, 931

A minimal segment is a geodesic segment which minimizes arc length between its ends. A minimum point m of p along a geodesic y is a point on y such that the segment of y f r o m p to m is minimal but no larger segment from p is minimal. T h e set of all minimum points of p is called the minimum (or cut) locus of p. I t follows immediately from the fact that geodesics do not minimize arc length beyond the first conjugate point that if m is the first conjugate point of p along y , then there is a minimum point of p along y which is not beyond m. A geodesic ray from p has at most one minimum point of p, but there may be none. If t E M p is such that exp,t is the minimum point of p along the geodesic exp,ut, then we also say that t is a minimum point of p in M p . (a) If m is not a minimum point of p, then there is at most one minimal segment y fro mp to m. If {ui}is a sequence of curves from p to m such that lim I ui I = p(p, m ) , then ui converges to y (if properly parametrized). Theorem 8.

(b) If there is a minimal segment from p to m on which m is a conjugate point of p , then m is a minimum point of p . (c) If M is complete the converse is true: if m is a minimum point of p, then either there are two minimal segments or m is a conjugate point of p along the unique one. Proof. (a) T h e first statement follows from the second, because if y and u were both minimal segments letting uf = u for all i would give limui = u = y . We suppose that y and each ui are parametrized by arc length. By

compactness of a normal coordinate ball of some radius E at m there will be convergent subsequences of {ui(Li -E)}, where Li = I ui j . If one of these subsequences did not converge to y ( L - e), where L = p(p, m), then the corresponding curves would eventually all form a corner with the extension of y beyond m . By cutting across this corner we would get shorter curves from p to y ( L + E) than L E,

+

238

11. SECOND VARIATION OF ARC LENGTH

contradicting the minimality of y beyond m. By using triangle inequalities and local minimizing of geodesics only, the argument can be made precise, and shows even more, that ui converges to y within a regular neighborhood of m . By covering y with a finite number of regular neighborhoods the convergence can be shown inductively down y from m top. By “regular” we mean that every two points in the neighborhood can be joined by a unique minimal segment. T h e details are left as an exercise. (b) follows immediately from the fact that geodesics do not minimize arc length beyond the first conjugate point. (c) Suppose M is complete, and m is a minimum point of p such that there is a unique minimal segment y . Then we must show that m is a conjugate point along y. Let L = p(p, m ) and let ui be a minimal segment from p to mi = y ( L l/i), parametrized by arc length, and with length Li . Then {ui*(0)} must have limit points, but the geodesic in the direction of such a limit point would give a minimal segment from p to m; hence, lim ui*(0) = y*(O). But then

+

exp,(liui*(o))

=

and limLiui,(0)

=

lim

mi

=

exp,

(L + y1a j y*(o)

(L +

so exp, is not one-to-one in a neighborhood of Ly,(O). Thus d exp, is singular there and m is a conjugate point. Problem 20.

In (a) show that the convergence of

ui

to y is uniform.

Problem 21. Show that the relation “is a minimum point of” is symmetric. Thus if y is a minimal segment from p to minimum point m there are shorter curves than y from m to the points on the extension of y from p away from m. Problem 22. Assume M is connected. Define m to be between p and q if all three points are distinct and p(p, m ) p(m, q) = p(p, 4); this relation is denoted by [p, m, q ] . Show: (a) If [p, m,q ] , [ p , n, q ] , and [p, m , n ] , then [m,n, q] and a minimal segment y from m to n is unique. I n case y exists let u be the largest (open) geodesic extension of y which does not include p or q. Then

+

239

11.6. T h e Minimum Locus

<

every point on u is between p and q. Hence I u I p(p, q) and u has no minimum points of any of its points. (b) If M is not complete then for every p E M there is a geodesic ray f r o m p which has no minimum point o f p on it. (c) If M is complete then [p, m,q] if and only if m is in an open minimal segment with ends p and q. Let Q ( M ) = {t E T ( M ) I t is a unit vector and c,t is a minimum point of n’t for some C, > O}. Define f : Q ( M ) 4 M by f(t) = exp c,t; that is,fassigns to a tangent t the minimum point of the base point o f t along the geodesic in the direction oft, if there is one. Theorem 9. f is continuous. Proof. For {ti}, a convergent sequence in & ( M ) ,we wish to show that limf(ti) = f(1im ti). Let yi = exp uti ,pi = yi(0), mi= f(ti) = yi(ci), t = lim ti , y = exp ut, p = y(O), and m = f(t) = y(c). I t will be sufficient to show lim ci = c, for then by continuity of exp,

lim m, = lirn exp citi

=

exp lim c,t,

=

exp (lim ci lim ti) = exp

ct = m.

Suppose limsupci > c. Then there are E > 0, a subsequence {di}of {ci}, and k such that for i > k, di > c E and y(c E ) is defined. Then for the corresponding subsequences {ui} of {yi}, {ni} of {mi},and {qi} of {pi},ui is a minimal segment from qi to

+

“i(C

+ €1,so

c

+

E

=

lim p(qi , ai(c

=p =

(lim qi , lim

P(P,

+

E))

U,(C

r(c + El),

+

+

E))

which contradicts the fact that y does not minimize arc length beyond y(c) = m.

T o show lim inf ci = c we consider convergent subsequences; thus we may suppose that lim ci = c’ < c and reach a contradiction. Let E = (c - c’)/2 > 0. Then p i = yi(ci E ) is beyond the minimum point of pi on yi , so there is a shorter curve T~ from pi to qi;by adding E ) we get a to T~ short segments f r o m p to pi and from qi to q = y(c’ curve ui from p to q such that lim 1 ui I = p(p, q). By theorem 8(a), uiconverges to y , and hence T~ converges t o y also.

+

+

240

11. SECOND VARIATION OF ARC LENGTH

Let E = T' x exp : T ( M )-+ M x M . Then E is nonsingular on the compact set {ut I 0 u c' E}, so by problem 1.12 there is a neighborhood U on which E is a diffeomorphism. Let V = E( U ) , so V is a neighborhood of E(ut) = ( p , y(u)). For sufficiently large i both (pi,yi(u)) and (pi,T ~ ( u )will ) be in V . However, E-l(pi, qi) must E) ti and on the other hand have length ci E because qi = exp,i(ci cannot have length any greater than the integral of the radial lengths of r i , which is less than ci E. QED

< < +

+

+

+

T h e distance from a fixed point p to its minimum point in the direction t E M p is a continuous function of t where defined. (This is immediate from the fact that the distance function is continuous.) Assume now that M is connected.

Corollary 1.

Corollary 2. A Riemannian manifold M is compact if and only if for some point p there is a minimum point in every direction from p . Proof. If M is compact then M is complete and bounded. Thus every geodesic ray can be extended indefinitely but cannot minimize arc length beyond the bound on M . Conversely, if p is a point such that every geodesic ray from p has a minimum point, then the function g : S -+ R, where S is the unit sphere in M , , g(t) = p ( p , exp c,t), is continuous by corollary 1. Thus the set = { t E M , I I I t I I <&ill t I l l or t = 01

is closed and bounded in M , , hence compact. But then M is compact. (See problem 22(b).)

=

exp B

If every geodesic ray from a point p has a conjugate point of p , then M is compact. Corollary 3.

Corollary 4. If a complete M has a covering space which is not compact, then for every p E M there is a geodesic ray on which there are no points conjugate top.

Proof. We may assume the covering is a Riemannian covering. Hence, the projection preserves geodesics and so also conjugate points. By corollary 3 there will be a geodesic from a point above p which has no

11.7. Closed Geodesics

24 1

conjugate points and then the projection of this geodesic will be without conjugate points. If M is complete, then the distance to the minimum locus is a continuous function of the point. Corollary 5. Proof.

Let f ( p ) = distance from p to its minimum locus, )e-J(p)

d P ) = (0

if p has a minimum point otherwise.

It suffices to show that g is continuous. Let h(p, t ) be the distance from to the minimum point ofp on the geodesic in the direction of t E M p . Then h is continuous on Q ( M ) and

p

e-h(p*t)

dPl

t)

=

lo

if h(p, t ) is defined

otherwise

is continuous on all of the unit tangent bundle, since if ( p , t ) E T ( M ) is a limit point of Q ( M ) ,it follows from the first part of the proof of theorem 9 that limr+(p,l)h(x) = +m. Since it is clear that g is the maximum-on-fibres of 1 the result now follows from lemma 6 . Remark. Corollary 1 shows that the complement of the minimum locus of a point is topologically a cell when M is complete, and in fact the exponential map gives the homeomorphism. Hence, much of the topological interest of a manifold lies in its minimum locus.

Problem 23. Extend the results of this section, except for theorem 9 to the case where p is replaced by a closed or a compact submanifold N , using which ever the result requires. I n particular, show by examples that the extensions of corollaries 2, 3, and 4 require N to be compact.

11.7 Closed Geodesics

A closed geodesic is a geodesic segment for which the initial and final points coincide; a smooth closed geodesic is one for which the initial and final tangents coincide. I t is intuitively clear that a homotopy class of loops based at p should have a closed geodesic for its minimal-length representative,

242

1 1 . SECOND VARIATION OF ARC LENGTH

provided the manifold is complete. Indeed, such a class is represented in the simply connected covering by the class of all curves from p , to p , , where p , and p , are in the fibre over p . A minimal segment from

FIG.42.

p , to p , projects

to a closed geodesic in the class, and is clearly a minimal-length curve in the class. T o make the identity in the fundamental group nonexceptional we must consider the constant curve as being a closed geodesic. To get smooth closed geodesics we consider free homotopy classes of loops. These classes are in one-to-one correspondence with the conjugacy classes of the fundamental group, because, roughly, the isomorphism between the fundamental groups based at different points, which free homotopy ignores, is determined only up to an inner automorphism, or conjugation. It is not necessarily true that such a class will have a minimal-length member, even if M is complete, because tightening up a loop might force it to go to “infinity.” For example, this is the case for the surface obtained by revolving the curve xx = 1 around the x-axis. However, if M is compact there is a minimal member in each free homotopy class. For such a class A will be represented by the lifts of its members to the simply connected covering A?. Let

b = inf{p(p, q) I if

u

is a curve from p to q then

7r

o u E A},

where 7r : i@ + M is the covering projection. By taking a sequence {(pi, qi)} such that lim p(pi , pi) = b, and a sequence of minimal segments yi from pi to qi , a convergent subsequence of the closed geodesics 7r o yi can be extracted, using the compactness of M . T h e

11.7. Closed Geodesics

243

limit will be a minimal closed geodesic in A , and it will be smooth because it is a minimal member of the homotopy class based at each of its points. If M is a compact Riemannian manifold, then every free homotopy class of loops has a minimal-length member which is a smooth closed geodesic.

Theorem 10.

Problem 24. Let u be a loop in free homotopy class A. Then u may be approximated uniformly by a broken geodesic loop yo E A . Let M be compact. Then we may assume that the breaks of yo are closer together than the distance from any point to its minimum locus. Construct a sequence of broken geodesics inductively by letting yi be the broken geodesic loop which has as its segments the minimal segments between the midpoints of the segments of yi-l. Show that yi E A and that a subsequence converges to a smooth closed geodesic in A.

(Synge’s theorem [87]). If M is compact, orientable, even-dimensional, and has positive sectional curvatures, then M is simply connected.

Theorem 11

Proof. T h e idea is to use second variation to show that a nontrivial smooth closed geodesic cannot be minimal, from which it follows by theorem 10 that there is only the trivial homotopy class of loops, M is simply connected.

FIG.43

Let y be a smooth closed geodesic. Then parallel translation once around y is an orthogonal transformation T of the odd-dimensional normal space to y . Since M is orientable, T has determinant 1, and hence must have at least one characteristic value equal to 1. (The characteristic values have absolute value 1 and the nonreal ones occur in conjugate pairs.) T h e characteristic vectors for the value 1 are fixed points of T , so there is a smooth parallel field V along y . Letting N

244

11. SECOND VARIATION OF ARC LENGTH

be any transverse manifold, I and V' = 0, so I ( V )=

-

= I ( N , N ) , the

J'I

q y ,

end terms of I( V ) cancel

, V )( V , V >du.

Since curvature is positive this is negative and there are shorter nearby curves; y cannot be minimal. QED Problem 25. Let M be compact, even-dimensional, nonorientable, and have positive curvature. Show that the fundamental group of MisZ,. Problem 26. Let M be compact, odd-dimensional, and have positive sectional curvatures. Show that M is orientable. Problem 27. Let M be a compact Kahler manifold which has positive holomorphic curvature. Show that M is simply connected. [Hint: if y is a geodesic and J is the complex structure operator, then J ( y * ) is parallel along y and y* , J ( y * ) span a holomorphic section.]

By using properties of the minimum locus it is sometimes possible to show the existence of closed geodesics with a method of Klingenberg

14.51. Theorem 12. Let M be complete, p E M such that the minimum locus of p is nonempty, and let m be a point on the minimum locus of p which is closest t o p . If m is not a conjugate point of p, then there is a unique closed geodesic with ends at p and passing through m such that both segments are minimal. Proof. By theorem 8, if m is not a conjugate point, then there are at least two minimal segments from p to m. We show that there are exactly two and that they match smoothly at m. Let y1 and y, be any two. If they do not match smoothly at m, then there is a geodesic u starting at m which makes an acute angle with each of y1 and y z .

FIG. 44.

1 1.7. Closed Geodesics

24 5

There will be minimal segments near to y1 , f r o m p to points on u near m,and shorter than y l ; similarly, there will be such minimal segments near to y z . Since y1 and y 2 are distinct these minimal segments will also be distinct when the points on u are sufficiently close to m. Then the points on u will be minimum points of p, by theorem 8, but this contradicts the fact that m is the closest minimum point t o p . Corollary 1. Let M be compact and let (p, m) be a pair which realizes the minimum of the distances of points to their minimum locus. Then either p and m are conjugate to each other or there is a unique smooth closed geodesic through p and m such that both segments are minimal.

Let M be compact, even-dimensional, orientable, with positive curvature, and let p, m be as in corollary 1. Then p and m are conjugate.

Corollary 2.

Proof. Assume p, m are not conjugate, so by corollary 1, there is a unique smooth closed minimal geodesic loop y through p and m, y ( 0 ) = p. Using Synge's trick, we have a one-parameter family of smooth loops y u such that yo = y and 1 yu 1 < I y I for u # 0. T h e n the

Y FIG.45.

unique minimal segments from y J 0 ) to the other points of yu form all possible angles with yu at yu(0). Those which form a fixed angle 0 have a convergent subsequence, as a function of u, to a minimal segment from p to a point m' on y . By the uniqueness of rn as the minimum point of p on y , m' = m. This contradicts the fact that there can only be two minimal segments from p to m. Remark. This corollary shows that under these conditions there is a point at which the conjugate locus and cut locus intersect. This can be used to derive lower bounds on diameter from upper bounds on

246

11. SECONDVARIATION OF ARC LENGTH

curvature. Under the much stronger assumption that M is a simply connected Riemannian symmetric space the minimum and first conjugate locus coincide [28, 761. 11.8 Convex Neighborhoods [24; 33, p. 53; 931

A set B in a Riemannian manifold M is convex if for every m, n E B, there is a unique minimal segment from m to n and this segment is in B. T h e open ball B(m, ro) of radius ro about m is locally convex if each

FIG.46.

sphere S(m, r ) of radius r < r,, about m satisfies the convexity coildition: if y is a geodesic tangent to S(m, r ) at n = y(O), then for sufficiently small u, p(m, y(u)) 3 r . If B(m, ro) is locally convex then exp, must be one-to-one on B(0, y o ) c M n l ,for otherwise there would be in B(m, ro) points on the minimum locus of m ; if y were perpendicular to a geodesic T from m at a point ~ ( r beyond ) the minimum point, then p(m, y(u)) < r for small u, since p(m, ~ ( r )< ) r. T h e relation between the concepts of convexity and local convexity is not as simple as it is in Euclidean spaces. For example, on a flat cylinder a normal coordinate ball with diameter greater than half the circumference of the cylinder will be locally convex but will not be convex because it will contain opposite points, which have two minimal segments. On the other hand, if the convexity condition fails for S ( m , r ) , then B(m, r ) is not convex. T o show this let y be a geodesic tangent to S(m, r ) at n = y(0) and having p = y(u) near n inside S(m, r ) . Then a Jacobi field along y which points outward at p and vanishes at n will have a corresponding rectangle having longitudinal geodesics, of which only y is tangent to S ( m , r ) . T h u s there will be

11.8. Convex Neighborhoods

247

segments which start near p , pass outside S(m, r ) , and return inside S(m, r ) at n. However, it might happen that B(m, r’) is convex for some r’ > r . T h e example of a flat cylinder shows that the following is the best possible result of its kind. Proposition 1. Let B(m, 2r0) be locally convex. Then every minimal segment between a pair in B(m, ro) is entirely within B(m, ro). Proof. If p , q E B(m, yo), then a minimal segment between them cannot go outside B(m, 2r0). If p(m, y ) does not take on its maximum at an end of y , then the least parameter value of y for which p(m, y ) is maximum would give a point of tangency of y with a sphere S(m, r), r < 2r,, and one end of y from that point of tangency would be inside S(m, r ) , contradicting local convexity. Thus the maximum of p(m, y ) occurs at an end, so all of y lies inside B(m, ro).

If T is a geodesic from m to n = T ( Y ) , then N = exp,(T*(r)’) n U , where U is a neighborhood of n, is a submanifold containing all small geodesic segments tangent to S(m, r ) at n. Thus the index form I = I(m, N ) will determine largely whether S(m, r ) satisfies the convexity condition at n. If I has nonzero index, then there will be points on N which are closer to m than n is. Thus if the convexity condition on S(m, r ) is satisfied I will be positive-semidefinite; if I is positive-definite we need only the requirement that n be before the minimum point of m on T to obtain the convexity condition at n for S(m, r ) . I n turn, whether or not I is positive (semi-)definite is determined by the behavior of m-Jacobi fields along 7.For if V is an m-Jacobi field the end terms in I( V )are 0, because the second fundamental form of N at n is 0, so I ( V ) = (V’(r), V ( r ) ) , by corollary 3 to lemma 2. We summarize the result as follows: Let % be the space of m- Jacobi fields along geodesic from m, parametrized by arc length. Let b, be the quadratic form on X defined by b,(V) = ( V ’ ( r ) , V ( r ) ) . (a) If B(m, ro) is locally convex, then b, is positive-semidefinite for r E (0, ro). (b) If B(m, ro) is a normal coordinate ball and b, is positivedefinite for all such T and for all r E (0, ro), then B(m, r o ) is locally convex. Proposition 2.

T

248

11. SECOND VARIATION OF ARC LENGTH

When all such b, are positive definite we call B(m,r,,) strongly locally

convex, abbreviated SLC.

Problem 28. Let B(m,ro) be a normal coordinate ball, 7 a geodesic , r E (0, r o ) .Then the values of m-Jacobi from m,and z = ~ * ( r )where . fields { V ( r )I V E X } form the tangent space to S(m, r ) at ~ ( r )Show that:

(a) X is the space of S(m, r)-Jacobi fields along T , so m is a focal point of S(m, Y ) of order d - 1. (b) T h e second fundamental form H , of S(m, r ) is essentially the same as 6 , . Problem 29. Show by continuity considerations that B(m,r ) is SLC for sufficiently small r . If B(m,r ) is locally convex but not SLC, then B(m,rl) is not S L C for rl > r . Find examples to show that B(m,r ) can be normal for all r , SLC for Y E (0, a), locally convex for r E [a, b ] , and not locally convex for r > b, where a, b are arbitrary except for the 6 03. restriction a

< <

Problem 30. A minimal submanifold is one for which the second fundamental forms all have trace zero, that is, for every normal z, tr S , = 0. Show that a compact minimal submanifold cannot be immersed in a strongly locally convex ball. Problem 31. Let B(m,ro) be a strongly locally convex ball and let N be a compact manifold which is immersed in B(m,ro), giving N the induced metric. Suppose 1 2 d i m N > d = dim M . Use the following theorem of Otsuki [68,69] to show that there is a plane section P such that K,(P) > K,(P). Let V be a real vector space of dimension n. Suppose Q1, ..., Qk , k < n, are symmetric bilinear real-valued forms on V such that

+

(Pi(%4 QkJ, 4 Qh, 4z) <0 -

9

for all u, ZI E V . Then there is at least one nonzero vector Qi(u, u ) =: 0 for allj. Prove this theorem in the case k = 1.

U E

V such that

Problem 32. Show that every ball in Euclidean d-space is strongly locally convex. Then apply the preceding problem to show that a flat

249

11.8. Convex Neighborhoods

e-dimensional torus cannot be imbedded isometrically in Ed unless d 3 2e. T o get more specific results on the size of S L C balls the following theorem can be used. Let H be a positive upper bound on sectional curvatures of M and let ro = ~ / 2 H l /Then ~ . every normal coordinate ball having radius r ro is SLC. If sectional curvatures are nonpositive, then every normal coordinate ball is SLC.

Theorem 13.

<

Proof.

We have for every m-Jacobi field V , I b,( V ) = I( V ) =

(( V‘, V’)

JT

=

I ( m , N ) as above,

-

K(T* , V )( V , V ) )du

-

H ( V , V ) )du.

0

>,

jl(( V’, V ’ )

This last expression looks like the second variation of a vector V on a Riemannian manifold P which has constant curvature H . It is not difficult to see that it may be considered as such, and hence we may apply the basic inequality provided that r < n-/H1l2 = the distance to the first conjugate points on P. T h e m-Jacobi fields on P have the form W = sin ( H l k ) E, where E is parallel. Letting E ( r ) = V ( r ) ,the basic inequality on P gives b,(V) 2 <W‘(.), W(4> =

H 1 j 2sin ( H 1 % ) cos ( H 1 / 2 r() E , E ) .

Thus b, is positive definite if r Problem 33. Y

< ro , as desired.

Let C be a compact subset of M . Show that there is

> 0 such that for every m E C, B(m,r ) is convex and SLC.

Problem 34. If M is complete, simply connected, and has nonpositive curvature, show that every ball in M is convex. Problem 35. Let M have nonpositive curvature and let B be a convex ball containing a geodesic triangle with lengths of sides a, b, c and opposite angles a , /3, y . Show that u2

+ b2

-

< c2 +B +Y <

2ab cosy

[33,P. 731

250

11. SECOND VARIATION OF ARC LENGTH

A generalization of problem 35 for arbitrary curvature is due to Toponogov (see [9]and [90]). Remark. T h e argument employed above involving the index forms for M and P is one step in the proof of Rauch’s comparison theorem. The theorey of the index form may be broadened to include the integral encountered without reference to P.This is done by replacing the Ricci transformation R, : V + R,,X by an arbitrary smooth field S of symmetric linear transformations of the normal spaces to 7.This is general enough for our purposes, but Morse considered even more general forms [57]. If N and P are end manifolds we retain the end terms, so we get a quadratic form I , given by

Is(V’)= <&*(O)V(Oh V(0))- <&*WV(Q7 V(4)

+

s”

((V’, V ’ ) - ( S V , V ) )du.

0

An N-S-jield V is a field along 7 which satisfies the end conditions SV = 0. An S-focal point of N is a at N and the equation V” point on 7 where a nonzero N-S-field vanishes. T h e basic inequality is valid and takes the form: Suppose there are no S-focal points of N on 7. For V E there ~ is a unique N-S-field Y such that Y(b) = V(b).Then I,( V ) >, I,( Y ) and equality holds only if V = Y . T h e most important application will be the case where S is derived from the Ricci transformation of some Riemannian manifold. T h e above proof, where S = identity, is typical (cf. problem 18).

+

11.9 Rauch’s Comparison Theorem [9, 74, 7.51

We have already considered a special case, theorem 9.2, of Rauch’s theorem, and some of the same ideas are involved in the proof of theorem 13. In the case d = 2, the analytic content of Rauch’s theorem is essentially the same as Sturm’s comparison theorem for second order ordinary differential equations. Let M and N be Riemannian manifolds, u and geodesic segments parametrized by arc length on [0, b] and starting at m E M and n E N, respectively. Suppose that there is no conjugate point of n on 7. Finally, assume that for all plane sections P and Q

Theorem 14. 7

11.9. Rauch’s Comparison Theorem

25 1

tangent to D and i- at ~ ( r and ) ~ ( r )respectively, , for all r E [O, b], KM(P) KN(Q). Let x = bu.+.(O), y = b ~ , ( 0 ) ,s E ( M , , J Z t, E ( N J U. Then if s and t have the same length I / dexprnr I I 2 / I dexpnt II.

<

FIG. 47

Proof. Let V be the m-Jacobi field such that V(b) = d expms, W the n- Jacobi field such that W(b)= d exp,t, f = ( V , V ) , andg = ( W , W ) . We wish to show thatf(b) 3 g(b). It will be sufficient to prove

(a) ( f i g ) ( O + ) = 1, (b) f‘/f2 g‘ig on (0, b], because then fig will be nondecreasing. T o prove (a) we note that there are constant vector fields A and B on the rays to x a n d y in Mr,Land N , such that A ( x ) = s, B ( y ) = t , V = u X , and W = u Y , where X = d expmA, Y = d exp,B. Then fig = ( X , X > / ( Y , y>.But ( X , X)(O) = ( A , A ) = (s, s) = ( t , t ) =

( Y , Y)(O).

T o prove (b), suppose first that there is no conjugate point of m on o((0,Y]). Thenf(r) f 0, so we may let X = V/f(~)l/~; similarly, let Y = W/g(r)1/2.Then X and Y are Jacobi fields, so

2 52

11. SECOND VARIATION OF ARC LENGTH

where X , is any nonzero vector field normal to T . By choosing parallel bases Ei and Fi along u and T such that X ( r ) = El(r-), Y ( r )= Fl(r), and using the coefficients of X with respect to Ei as the coefficients of X , with respect to Fi , we get an X , which coincides with Y at r and for which ( X I , X ’ ) = (X”, X,‘) and ( X , X) = ( X , , X,). By the basic inequality for the index form along T we then have

> 2 f((Y’7 Y’>

(f’if)(7)

-

, Y )( Y , Y > )du

0

=

( Y , Y>’(r)

=

(g‘/g)(r)s

It now follows that f ( r ) 3 g ( r ) as long as there is no conjugate point of m on ~ ( ( 0r ,] ) .However, this was only needed to enable us to divide byf(r)li2, so the inequality and continuity now give that f ( r ) >, g ( r ) for r E (0, b]. Under the above hypothesis, the first conjugate point of n must occur before that of m. Corollary 1.

Corollary 2. (Bonnet). Let M have all curvatures of plane sections tangent to a geodesic y starting at m satisfy the inequalities 0 < L K ( P ) H , L and H constants. Then if s is the distance along y to the first conjugate point of m on y ,

<

<

< s < n/L’iz.

.rr/H112

Proof. This follows immediately from corollary 1 and by comparison with spheres of constant curvature L and H . Remark. Another consequence is that the hypothesis of Synge’s theorem, theorem 11, may be weakened to completeness and strictly positive curvature instead of compactness. Problem 36. Generalize Rauch’s theorem by replacing the points m and n by totally geodesic submanifolds of equal dimensions.

Rauch’s comparison theorem and Klingenberg’s results on closed geodesics are among techniques used in the study of “pinched” manifolds as well as for the proof of Topogonov’s theorem on geodesic triangles [6-9, 45-47, 74-76, 89-91].

11.10. Curvature and Volume

253

11.10 Curvature and Volume [ l o , ZZ, 321

T h e Ricci transformations R, : y + RXvxextend to derivations of the Grassmann algebra; since R, is symmetric with respect to ( , ), the extensions are symmetric with respect to the natural extension of ( , ) (problem 4.14). Let y1 , ..., y p , x be orthonormal vectors and P thep-plane spanned by y1 , ..., yr, . T h e p-mean curvature of x and P is the inner product K ( x , P ) = (R,(yl ...y,), y1 ..*y,). In particular, there is a unique d -- 1-plane orthogonal to x, and its d - 1 mean curvature is called simply the mean (or Ricci) curvature of x, K(x, XI). Problem 37. Show that K(x, P ) = X K ( x , y i ) , and hence that such sums depend only on the plane the yi span. Moreover, K(x, x') = tr R, .

Let y be a geodesic starting at m, y

=

exp,,, o p, where p is a ray in

M , , parametrized by arc length. Let J p ( t )be the maximum of all the

factors by which exp, multiplies lengths of decomposible p-vectors normal to p at p(t), that is, J U ( t )is the maximum of the ratios

I I dexpms1 ... dexPm%JI111I s1 ... s p Ill where si are linearly independent tangents to M,, normal to p at p(t). Similarly, let j p ( t ) be the minimum of such ratios. In particular, ]d-l(t) = jd-.l(t)is the Jacobian determinant of exp, at p(t). Note that J d O ) = i,(O) = 1. Theorem 15.

Suppose m has no conjugate point on y((0, c]). Let

( p ) be the condition: For every t and every p-plane P normal to y at y(t), K ( y * ( t ) ,P) 3 Pa2.

<

(1) If (p) then J p ( t ) (sin at/at)P for t E (0, c]. (2) Suppose that for every t and every vector y normal to y at y ( t ) , K ( y * ( t ) ,y ) b2. Thenj,(t) 2 (sin bt/bt)P for t E (0,c]. In case p = d - 1 we can assert more, namely, in (1) that J,-,(t)(at/sin is a nonincreasing function of t , and in (2) that J,-,(t)(bt/sin bt)d-l is a nondecreasing function of t.

<

We do not assume that a, b are real, but use the complex extension of sin in case a2 or b2 is negative; if a or b = 0 we replace at/sin at or &/sin bt by 1. Proof.

Let sl, ..., sP be independent vectors normal to p at p(t).

254

11. SECOND VARIATION OF ARC LENGTH

T h e si generate linear homogeneous fields along p, which project under d exp, to p independent m-Jacobi fields. Thus we have a smooth field of p-planes, P, spanned by these m-Jacobi fields at the points along y. Let f : [0, t] -+ R be the function which gives the ratios corresponding to P , s o f ( 0 ) = I , and f(t) is a typical ratio of which J,(t) is the maximum and j,(t) the minimum. In particular, if p = d - 1 then P i s unique and f = Jd-, = jd-,. If Y , , ..., Y , are m-Jacobi fields which span P at one value in (0, t ] , then they will span P at every value. Letting Yi= d exp,uAi, where Ai are constant fields along p in M , , then f = 1 1 Y , Y , I I/u”A, where A = 1 1 A, A, 1 1 is constant. Now suppose Y,(r),..., Y,(r) are orthonormal; for each Y E (0, t] there is such a set of Yi. Then ( Y , ... Y , , Yl . . * Y,)’(Y)

which follows from the fact that the basis of #-vectors generated by an orthonormal basis of vectors is orthonormal. [Express Yi’(r)in terms of an orthonormal basis which includes Y l ( r ) ,..., Y,(r) and expand.] Using this to differentiatef2 yields the relation

(4

f’(r)if(r) =

2 (Y,‘, Yi>W

-

p/r.

By the basic inequality, if Wi is a vector field along y such that Wi(0)= 0 and Wi(r)= Yi(r),then

(Yi’,y i > ( r ) = Z r ( Y i ) < IJWi), where I , = I(m, N7) with intermediate submanifold N , having null second fundamental form. I n particular, if Ei is the parallel field generated by Yi(r)and g is a broken C” function such that g(0) = 0, g(r) = 1, taking W i= gEi gives (yi’,yi> ( r ) <

f

((g’)2 - ~ ( y >* E i ) g 2 ) du.

0

Adding these inequalities and using the hypothesis

(PI:

Q* Ez) 2 Pa2, 9

11.10. Curvature and Volume

255

it follows from (a) that

The integral in this inequality is the second variation of a vector field gE, where E is parallel and unitary, in a space of constant curvature a2. It follows that the best choice for g is that which will make gE a Jacobi field, by the basic inequality. Thus we let g = sin au/sin ar, and the integral becomes (g’E, p E ) ( 7 )

(If a

=

0 let g

=

from q to t , q

E

a cos arjsin a7.

u/r.) Now we integrate the resulting inequality:

f’(7)if(r)

(b)

=

< p ( a cos ar/sin a7

-

l/r)

(0, t ) and take exponentials, obtaining f(t)(at/sin at)”


In case p = d - 1 this yields the desired monotonicity of Otherwise we take the limit as q --t 0, , using J d p l ( t(atlsin ) the fact thatf(0) = 1, so f(t)

< (sin atjut)”.

This is true for all such f(t), hence also for their maximum Jp(t). This completes the proof of (1). T o prove (2) we return to the inequality (a) and use the hypothesis W y , Yi) b2: 9

<

where hE, is a “Jacobi field” for a space of constant curvature b2, the last step following from the basic inequality; thus h = sin bu/sin br. Now we proceed as before, obtaining (b) with the inequality reversed and a replaced by b, and so on to the conclusion of (2).

256 Corollary 1.

11. SECOND VARIATION OF ARC LENGTH If ( p ) and u2 > 0, then the first conjugate point of m on

y occurs at least within distance 7r/u along y .

[ p ( t ) is a conjugate point of m if and only if a nonzero p-vector is annihilated by d exp, .]

Corollary 2. (Myers' theorem [62]). If M is a complete Riemannian manifold with mean curvature bounded away from 0 by ( d - l)u2 > 0, then M is compact, the diameter of M is n / a , and the fundamental group of M is finite.

<

Proof. T h e first two assertions follow from corollary 1. T h e simply connected Riemannian covering of M has the same local properties, so must also be compact. But the fibres of a covering space are discrete, so in a compact covering they are finite, and hence the fundamental group is finite.

Corollary 3. Let v(m, r ) , for r sufficiently small, denote the. volume of the sphere S(m, r ) contained in a normal coordinate neighborhood. If ( d - 1) u2 is a lower bound for mean curvature on M , and b2 is an upper bound for curvature, then v(m, r ) (u/sin ~ r ) ~ - -isl a nonincreasing function of r, v(m, r ) (b/sin b7)d-l is a nondecreasing function of r . Proof. Since we want to consider all geodesics from m, we let J(r, x) denote JdPl(r)from above when x = y*(O). J(r, x) is the Jacobian of the restriction of exp, to the sphere S , of radius r in M , . Combining exp, with the map r -+rx we have a map of S, onto S(m, r ) with Jacobian rd-lJ(r, x). Thus

so the result now follows from the fact that J ( r , x) (urisin ~ r ) ~ is- l monotonic in r , and similarly for the other case with b. Remark. T h e same conclusion obtains if curvature has range [a2,b2], which is a result of Gunther [32].

Corollary 4. If M is complete and ( d - 1) u2 is a lower bound for mean curvature, then the volume of a normal coordinate ball B(m, r ) is the volume of a normal coordinate ball of the same size in the

<

11.10. Curvature and Volume

257

simple spaceform (that is, sphere, Euclidean space, or hyperbolic space) of constant curvature u2. If u2 > 0 the volume of M is the volume of a sphere of radius l/a, and equality obtains only if M is isometric to such a sphere.

<

Proof. T h e volume of B(m, r ) is obtained by integrating the Jacobian of exp, on a ball of radius r in M,, . But the bound (sin ur/ur)d-l is the Jacobian of an exponential map of the space form. (This follows from the proof of theorem 15, since constant curvature gives equality throughout in that proof.) If u2 > 0, then M is compact and the volume is given by integrating the Jacobian of exp,, on the open set of Mm which is within the minimum locus. T h e integral of the bound for that Jacobian, (sin ur/ur)d-l on as great an open set, the ball of radius T / U , will be no less, but gives the volume of a sphere of radius l/u.

If the volume of M equals that of a sphere of radius l/u, then all of the inequalities in the proof of theorem 15(1) must be equations; in particular, the Jacobi fields of M are in the same form as those on a sphere of radius l / u , so M has constant curvature u2 and is locally isometric to a sphere of radius lju. But the minimum locus in M,,L must be the sphere S of radius r / u , and since d exp, annihilates all vectors tangent to S , exp,,(S) is a point. Thus by factoring through the exponential maps a global isometry is obtained. Problem 38. Show that the conditions (p) are monotone in strength, 1) (with the same u ). that is, (p) implies (p

+

Problem 39. Let S be the simple space form of curvature u2, M a manifold satisfying (p) along all geodesics from m, N either (1) a p-dimensional submanifold of Mm such that it is contained in a sphere with center 0 and within a normal coordinate ball, or (2) a ( p 1)-dimensional submanifold of M , within a normal coordinate ball. Let $ : M,, -+ S,,,, be a linear isometry. Show that

+

volume (exp,(N))

< volume (exp,?

04(N)).

+

[Hint: For (2) an extension of theorem 15 to arbitrary (p 1)-vectors is needed. Use the fact that if x = y * ( r ) , an arbitrary decomposible (p 1)-vector may be written (x y 1 ) y 2. * * Y ~ +where ~ , yi are all normal to x.]

+

+

Appendix. Theorems on Diferential Equations T h e reader may find it instructive to translate (i) and (ii) below into their coordinate forms and verify that essentially the same theorems are found in standard references [26]. Let F be a C" map, F : Un x Urn-+ T(Rn),where Un and Urnare open subsets of Rn and Rm,respectively, T(Rn)the tangent bundle to Rn, and such that F(u, u') E Runfor every (u, u') E U n x Urn.Thus F gives a C" vector field on Un for every u' E Urn; or, in classical terminology, a system of n first order differential equations in n unknowns, depending in addition on a parameter u' E Urn. (i) Existence and Uniqueness

There is a unique C" map 4 : W - t Rn, where W is a neighborhood {0} x Un x Urn in R x Un x Urn, such that for every p = ( t , u, u') E of

w

(a) d&(D,(p)) = F($(p), u'), where D, is the partial derivative operator corresponding to R and its coordinate u1 in R x Un x Urn. (b) +(O, U , u')

= U.

(ii) Continuation of Solutions

Suppose that F is bounded in the Euclidean metric on Rn. Then the neighborhood W above can be taken so that its intersection with the fibres of the projection R x Un x Urn-+ Un x Urnare of the form (a, b) x {u} x {u'} and either b = 03 or limt+b-+(t, u, u') exists and is outside Un, and similarly for the other end. a and b depend on u and u'. This says that integral curves can be continued in either direction until either the parameter becomes infinite or the curve runs out of un. (iii) Extension t o Manifolds

T h e above results are true if Un is an open subset of a manifold N , 258

Theorems on Differential Equations

259

Urnan open subset of a manifold M , F : U“ x UnL-+ T ( N ) . (iv) The Local Group Associated with F , u’

For each u’

E

E

Urn

Urn, t, s E R such that the following are defined:

(a) rj i ( j ) x U n X ~ u is ~ ~ a diffeomorphism. (b) rj(&rj(4 u, 4, u’) = 4(s t , u , 4.

+

Bibliography 1. Ambrose, W. The index theorem in Riemannian geometry. Ann. of Math. 7 3 (1961), 49-86. 2. Ambrose, W., and Singer, I. M. A theorem on holonomy. Trans. Amer. Math. SOC. 75 (1953), 428-443. 3. Atiyah, M. Complex analytic connections in fibre bundles. Trans. Amer. Math. SOC.85 (1957), 181-207. 4. Auslander, L., and MacKenzie, R. E. “Introduction to Differentiable Manifolds.” McGraw-Hill, New York, 1963. 5. Berger, M. Sur les groupes d’holonomie homogine des variktks B connexion affine et des variktks Riemanniennes. Bull. SOC.Math. France 83 (1955), 279-330. 6. Berger, M. Les variktks riemanniennes dont la courbure satisfait B certaines conditions, Proc. Znternat. Congr. of Mathematicians Stockholm, 1962. 447-456. 7. Berger, M. Pincement riemannien et pincement holomorphe. Ann. Scuola Norm. Sup. Pisa [Ser. 1111, 14 (1960), 151-159. 8. Berger, M. Les variktks riemanniennes (1/4)-pincCes. Ann. Scuola Norm. Sup. Pisa [Ser. 1111, 14 (1960), 161-170. 9. Berger, M. An extension of Rauch’s metric comparison theorem and some applications. Illinois J . Math. 6 (1962), 700-712. 10. Bishop, R. A relation between volume, mean curvature, and diameter. Amer. Math. SOC.Not. 10 (1963), 364. 11. Bishop, R., and Goldberg, S. Some implications of the generalized Gauss-Bonnet Theorem. Trans. Amer. Math. SOC.T o appear. 12. Bochner, S., and Yano, K. “Curvature and Betti Numbers,” Ann. Math. Studies No. 32. Princeton Univ. Press, Princeton, New Jersey, 1953. 13. Borel, A. Lectures on symmetric spaces. M.I.T. lecture notes (1958). 14. Bott, R., and Milnor, J. On the parallelizability of the spheres. Bull. Amer. Math. SOC.64 (1958), 87-89. 15. Cartan, E. Sur l’intkgration des systhmes d’kquations aux diffkrentielles totales. Ann. Sci. Ecole Norm. Sup. 18 (1901), 241-311. 16. Cartan, 8. Sur la possibilitk de plonger un espace riemannien donnk dans un espace euclidien. Ann. SOC.Polon. Math. 6 (1927), 1-7. 17. Cartan, 8. “Leqons sur la gkometrie des espaces de Riemann.” Gauthier-Villars, Paris, 1946. 18. Cartan, 8. Sur une classe remarquable d’espaces de Riemann, I ; 11. Bull. SOC. Math. France 54 (1926), 214-264; 55 (1927), 114-134. 19. Chern, S.-S. A simple intuitive proof of the Gauss-Bonnet formula for closed Riemannian manifolds. Ann. of Math. 45 (1944), 747-752. 20. Chern, S.-S. Topics in differential geometry. Inst. Advanced Study lecture notes (1951). 260

BIBLIOGRAPHY

26 1

21. Chern, S.-S. An elementary proof of the existence of isothermal parameters on a surface. Proc. Amer. Math. Soc. 6 (1955), 771-782. 22. Chern, S.-S. On curvature and characteristic classes of a Riemann manifold. Abh. Hamburger Univ. Math. Sem. 20-21 (1955), 117-126. 23. Chern, S . 3 . Complex manifolds. Univ. of Chicago lecture notes (1956). 24. Chern, S.-S. Differentiable manifolds. Univ. of Chicago lecture notes (1959). 25. Chevalley, C. “Theory of Lie Groups.” Princeton Univ. Press, Princeton, New Jersey, 1946 (2nd printing, 1960). 26. Coddington, E., and Levinson, N. “Theory of Ordinary Differential Equations.” McGraw-Hill, New York, 1955. 27. Cohn, P. M. “Lie Groups.” Cambridge Univ. Press, London and New York, 1957. 28. Crittenden, R. Minimum and conjugate points in symmetric spaces. Canad. j’. Math. 14 (1962), 320-328. 29. Flanders, H. “Differential forms”, Academic Press, New York, 1963. 30. Goldberg, S. I. “Curvature and Homology.” Academic Press, New York, 1962. 31. Grauert, H. On Levi’s problem and the imbedding of real-analytic manifolds. Ann. of Math. 68 (1958), 460-472. 32. Gunther, P. Einige Satze uber das Volumenelement eines Riemannschen Raumes. Publ. Math. Debrecen 7 (1960), 78-93. 33. Helgason, S. “Differential Geometry and Symmetric Spaces.” Academic Press, New York, 1962. 34. Hellwig, G. “Partielle Differentialgleichungen.” Teubner, Stuttgart, 1960. 35. Hermann, R. Cartan connexions and the equivalence problem for geometric structures. Univ. of California (Berkeley), N.S.F. report (1962). 36. Hermann, R. E. Cartan’s theory of exterior differential systems. Univ. of California (Berkeley) N.S.F. report (1963). 37. Hermann, R. Focal points of closed submanifolds of Riemannian spaces. Nederl. Akad. Wetensch. Proc. Ser. A 66 (1963), 613-628. 38. Hermann, R. “Dynamical Systems and the Calculus of Variations.” Academic Press, New York, in preparation. 39. Hocking, J. G., and Young, G. S. “Topology.” Addison-Wesley, Reading, Massachusetts, 1961. 40. Hopf, H., and Rinow, W. Uber den Begriff der vollstandigen differentialgeometrischen Flache. Comment. Math. Helv. 3 (1931), 209-225. 41. Hu, S. T. “Homotopy Theory.” Academic Press, New York, 1959. 42. Jacobson, N. “Lie Algebras.” Wiley (Interscience), New York, 1962. 43. Kahler, E. Einfuhrung in die Theorie der Systeme von Differentialgleichungen. Hamburger Math. Einzelschr. 16 (1934). 44. Kervaire, M. A manifold which does not admit any differentiable structure. Comment. Math. Helv. 35 (1961), 1-14. 45. Klingenberg, W. Contributions to Riemannian Geometry in the large. Ann. of Math. 69 (1959), 654-666. 46. Klingenberg, W. Uber kompakte Riemannsche Mannigfaltiglteiten. Math. Ann. 137 (1959), 351-361. 47. Klingenberg, W. Uber Riemannsche Mannigfaltigkeiten mit positiver Krummung. Comment. Math. Helv. 35 (1961), 47-54. 48. Kobayashi, S. On connections of Cartan. Canad. j’. Math. 8 (1956), 145-156. 49. Kobayashi, S. Theory of connections. Ann. Mat. Pura Appl. 43 (1957), 119-194.

262

BIBLIOGRAPHY

50. Kobayashi, S., and Nomizu, K. “Foundations of Differential Geometry.” Wiley (Interscience), New York, Vol. I, 1963; Vol. 11, in preparation. 51. Lichnerowicz, A. “Thkorie globale des connexions et des groupes d’holonomie.” Edizione Cremonese, Rome, 1955. 52. Milnor, J. On manifolds homeomorphic to the 7-sphere. Ann. of Math. 64 (1956), 399-405. 53. Milnor, J. Differential topology. Princeton Univ. lecture notes (1958). 54. Milnor, J. “Lectures on Morse Theory,” Ann. Math. Studies No. 51, Princeton Univ. Press, Princeton, New Jersey, 1963. 55. Montgomery, D., and Zippin, L. “Topological Transformation Groups.” Wiley (Interscience), New York, 1955. 56. Morrey, C. B. The analytic embedding of abstract real-analytic manifolds. Ann. of Math. 68 (1958), 159-201. 57. Morse, M. “The Calculus of Variations in the Large.” Amer. Math. SOC.,New York, 1934. 58. Munkres, J. R. “Elementary Differential Topology,” Ann. Math. Studies No. 54, Princeton Univ. Press, Princeton, New Jersey, 1963. 59. Myers, S. Riemannian manifolds in the large. Duke Math. J. 1 (1935), 39-49. 60. Myers, S. Connections between differential geometry and topology, I. Simply connected surfaces. Duke Math. J. I (1935), 376-391. 61. Myers, S. Connections between differential geometry and topology, 11. Closed surfaces. Duke Math. J . 2 (1936), 95-102. 62. Myers, S. Riemannian manifolds with positive mean curvature. Duke Math. J . 8 (1941), 401-404. 63. Nash, J. The imbedding problem for Riemannian manifolds. Ann. of Math. 63 (1956), 20-63. 64. Newlander, A . , and Nirenberg, L. Complex analytic coordinates in almost complex manifolds. Ann. of Math. 65 (1957), 391-404. 64A. Newns, W.F. and Walker, A.G., Tangent planes to a differentiable manifold, J . London Math. SOC.31 (1956), 400-407. 65. Nomizu, K. Invariant affine connections on homogeneous spaces. Amer. J. Math. 76 (1954), 33-65. 66. Nomizu, K. “Lie Groups and Differential Geometry.” Math. SOC.of Japan, Tokyo, 1956. 67. Nomizu, K., and Ozeki, H. The existence of complete Riemannian metrics. Proc. Amer. Math. SOC.12 (1961), 889-891. 68. Otsuki, T. On the existence of solutions of a system of quadratic equations and its geometric applications. Proc. Japan Acad. 29 (1953), 99-100. 69. Otsuki, T. Isometric imbedding of Riemann manifolds in a Riemann manifold. J. Math. SOC.Japan 6 (1954), 221-234. 70. Ozeki, H. Infinitesimal holonomy groups of bundle connections. Nagoya Math. J. 10 (1956), 105-123. 71. Pitcher, E . Inequalities of critical point theory. Bull. Amer. Math. SOC.64 (1958), 1-30. 72. Pontrjagin, L. “Topological Groups.” Princeton Univ. Press, Princeton, New Jersey, 1946. 73. Preissmann, A. Quelques propriktks globales des espaces de Riemann. Comment. Math. Helv. 15 (1942-3), 175-216.

BIBLIOGRAPHY

263

74. Rauch, H. A contribution to differential geometry in the large. Ann. of Math. 54 (1951), 38-55. 75. Rauch, H. “Geodesics and Curvature in Differential Geometry in the Large.” Yeshiva Univ. Press, New York, 1959. 76. Rauch, H. T h e global study of geodesics in symmetric and nearly symmetric Riemannian manifolds. Comment. Math. Helv. 35 (1961), 111-125. 77. de Rham, G. Sur la rCductibilitC d’un espace de Riemann. Comment. Math. Helv. 26 (1952), 328-344. 78. de Rham, G. “VariCtCs diffbrentiables.” Hermann, Paris, 1955. 79. Sard, A. The measure of the critical values of differentiable maps. Bull. Amer. Math. SOC.48 (1942), 883-890. 80. Seifert, H., and Threlfall, W. “Lehrbuch der Topologie.” Teubner, Stuttgart, 1934. 81. Seifert, H., and Threlfall, W. “Variationsrechnung im Grossen.” Chelsea, New York, 1951. 82. Simons, J. On the transitivity of holonomy systems. Ann. of Math. 76 (1962), 213-234. 83. Singer, I. Differential geometry. M.I.T. lecture notes (1962). 84. Smale, S. A survey of some recent developments in differential topology. Bull. Amer. Math. SOC.69 (1963), 131-145. 85. Steenrod, N. “The Topology of Fibre Bundles.” Princeton Univ. Press, Princeton, New Jersey, 1951. 86. Synge, J. L. T h e first and second variations of the length in Riemannian space. Proc. London Math. SOC.25 (1926), 241-264. 87. Synge, J. L. On the connectivity of spaces of positive curvature. Quart. J . Math. Oxford Ser. (l), 7 (1936), 316-320. 88. Thom, R. Quelques propriCtCs globales des variCtCs diffkrentiables. Comment. Math. Helv. 29 (1954), 17-85. 89. Tsukamoto, Y. On certain Riemannian manifolds with positive sectional curvature. Mem. Fac. Sci. Kyushu Univ. Ser. A , 13 (1959), 140-145. 90. Tsukamoto, Y. On a theorem of A. D. Alexandrov. Mem. Fac. Sci. Kyushu Univ. Ser. A , 15 (1962), 83-89. 91. Tsukamoto, Y. On Riemannian manifolds with positive curvature. Mem. Fac. Sci. Kyushu Univ. Ser. A , 15 (1962), 90-96. 92. Weil, A. “VariCtCs kahltriennes.” Hermann, Paris, 1958. 93. Whitehead, J. H. C. On the covering of a complete space by the geodesics through a point. Ann. of Math. 36 (1935), 679-704. 94. Willmore, T. “An Introduction to Differential Geometry.” Oxford, Univ. Press, London and New York, 1959. 95. Yamabe, H. On an arc-wise connected subgroup of a Lie group. Osaka Math. J., 2 (1950), 13-14.

This Page Intentionally Left Blank

Subject Index torsion translation, 96, 97 torsion zero, 95, 103, 106, 117, 118, 166 Affine product connexion, 119 Almost complex manifold, 64 structure, 51, 73 Alternating multilinear functions, 57 Arc length first variation of, 213, 214 minimizing, 149, 151, 154 of broken Cm curves, 123 of continuons curves, 158 reduced, 223 second variation of, 213, 214 Synge's formula, 218 Associated bundle, 45 connexions on, 83 coordinate bundle, 46 tangent bundle, 46 vector bundles, 48 Associated codistribution, 70 Associated distribution, 70 Associated vector field to a rectangle, 147 Atlas, same as basis for Cm structure, 3 Augmented index, 232 Automorphism of Lie algebra, 35 of Lie group, 35

A

Abelian Lie algebra, 26 Lie group, 25, 37 Action by compact group, induced Riemannian metric, 136 Action of Cm maps on forms, 68 Action of Lie groups, 38 effective, 38 free, 39 on connexion, 118 prolongation, I 18 to right, 38 transitive, 39 Action of principal bundle, 49 Adjoint representation of Lie algebra, 36 of Lie group, 36 of quaternions, 36 Ado's theorem, 30 Affine Bianchi identities, 101 Affine bundle, 109 connexion on, 120 Affine connexion, 89 classical definition, I 1 7 complete, 110 curvature and torsion zero, 95 curvature transformation, 96, 97 determined by geodesics and torsion, 106 on Lie group, 107 on parallelizable manifold, 106 symmetric connexion, see Torsion zero torsion form, 93

0 Bagel, 210 Base space of bundle, 42 Basic inequality for index form, 228 generalized, 250 265

266

SUBJECT INDEX

Basic vector fields, 92 Basis for Cw structure, 3 Bianchi identity, 82 affine identities, 101 Bonnet’s theorem, 252 Bracket of vector fields, 13 geometrical interpretation, 18 Broken Cm curve, 77 Broken Cm rectangle, 21 3 Bundle, see also Associated bundle, Principal bundle of adapted bases of product, 119 of adapted frames of immersion, 186 of bases, 43 connexion on, see affine connexion parallelization of, 92 reduction to O(d), 129 of frames, 127 as reduced bundle, 129 connexion on, see Riemannian connexion bundle map, 49 bundle space, 42 C Cw,see class Cm Cm-related, 2 Canonical lifting of Cw rectangle, 147 Cartan equations and forms of Maurer-, 70,72, 82, 107 theorem of Hadamard-, 184 Cartan connexion, 121 Cartan-Kahler theorem, 205 Cartan’s criterion for closed subgroups, 30 Cartan’s local isometry theorem, 181 Cauchy-Kowalevsky theorem, 205 Center, 36 Chain rule, 10 Characteristic value and vectors of generalized index form, 236 9 Class Ck, Class Cw,of action, 38 bundles, 41 codistribution, 70 covering map, 6

curve, 7 differential form, 62 distribution, 23 function, 2, 3 manifold, 3 partition of unity, 4 rectangle, 147 structure, 3 basis for, 3 tensor field, 47 Urysohn function, 2 vector field, 13, 47 Classical covariant derivative, 117 Classical coordinate definition of affine connexion, 118 Classical notion of differential form, 53 Classical rigidity theorem, 21 1 Closed form, 114 Closed geodesic, 241 smooth, 241 Codistribution, 70 Coefficients of connexion, 1 16 Comparison theorem, Rauch’s, 250 Complete connexion, 110 Riemannian manifold, 154 negative curvature, 184 zero curvature, 183 Complex differential forms of type (p, q), 73 Complex Grassmann manifold, 170 Complex Lie group, 34 compact, 34 Complex manifold, 7, 50, 64 complex structure, 51 Hermitian structure, 141 Kahler structure, 142 Complex projective space, 140 Complex structure, 51 Configuration space of mechanical system, 157 Conformal map, 136 Conjugate point, 225 complex projective space, 225 order of, 225 quaternion projective space, 226 sphere 225 Connexion, see also Affine connexion, Riemannian connexion

267

SUBJECT INDEX

curvature form, 80 existence of, 83 flat, 82 induced, 80 of Cartan, 121 on affine bundle, 120 on associated bundle, 83 on principal bundle, 75 one form of, 76 Conservative force field, 157 Continuous curve, 158 arc length, 158 Convexity, 246 local, 246 strong local, 248 Coordinate bundle associated, 46 principal, 44 transition functions of, 44 Coordinate neighborhood, 2 Coordinate system, 2 normal, 110 Covariant differentiation, 111 examples, 112 of Riemannian connexion, 166 parallelizable manifold, 114 structural equation, 116 Covering map, Cm, 6, 183 Riemannian, 134 Critical point of function, 197 Hessian, 197 Cross-sections of bundles, 42, 48, 49 Curvature Gauss, 162, 211 mean, 253 p-mean, 253 Ricci, 253 Riemannian or sectional, see Riemannian curvature Curvature form of connexion, 80 of immersion, 190 vanishing of, 82 Curvature transformations, 96 geometric interpretation of, 97 symmetries of, 162 Curve arc length, 123, 158

broken Cw, 77 cm, 7 continuous, 158 development of, 98 horizontal lift, 77 Cut locus, 237

D Decomposable elements of Grassmann algebra, 55 Deformation, infinitesimal, 226 Derivative, Lie, 16, 69 partial, 8 Derivation of Grassmann algebra, as extension of linear endomorphism, 56 Determinant of linear endomorphism, 57, 61, 62 Development of curve, 98 Diffeomorphism, 11 Difference forms, 102 and geodesics, 105 and torsion, 103 Differentiable, see Class C"0 Differential of map, 9, 12 chain rule, 10 Differential form, 62 action of Cm maps on, 68 action of Lie derivative on, 69 as cross-section, 63 classical notion, 53 closed, 114 complex, of type ( p , q), 73 difference, 102 equivariant, 75, 85, 100 exact, 143 holomorphic, 73 vector valued, 71 Differentiation, covariant, see Covariant differentiation Direct connexion on parallelizable manifold, 106, 114 Distance on Riemannian manifold, 124 Distribution, 22 associated codistribution, 70 Cm,23 Frobenius' theorem, 23

268

SUBJECT INDEX

integrable, 23 integral manifold of, 23 involutive, 23 Doubly ruled surface, 209 Dual forms of structural equations, 80, 101 E

Effective action, 38 Embedding, see Imbedding Endomorphism, linear determinant, 57, 61, 62 extension as derivation to Grassmann algebra, 56 extension as homomorphism to Grassmann algebra, 56 trace, 57 Equivariant forms, 76, 85, 86, 90, 100, 101 Euclidean space, 2, 108, 132 Exact form, 143 Exponential map affine connexion into M , 108 into B ( M ) , 108 Lie group, 30 matrices, 32 Extension of function on submanifold, 22 vector field on submanifold, 22 Exterior derivative, 64 intrinsic formula for, 65 F

Faithful representation, 34 Fibre, of associated bundle, 46 action of principal bundle on, 49 Fibre bundle, see Associated bundle, Principal bundle First variation of arc length 213, 214 unintegrated, 214 Flag manifold, 137, 187 Flat affine connexion, see Curvature and Torsion zero connexion, 82

complete Riemannian manifold, 183 tori, 133 Focal point, 224 order of, 225 Focal values, 232 Force field of mechanical system, 157 Form, see Differential form, Index form, Fundamental form Frames, bundle of, see Bundle of frames Free action, 39 Frobenius’ theorem, 23, 71 Function, see also class Cm alternating, 57 differential of, 9, 12 extension of, 22 restriction of, 22 Urysohn, 2 Fundamental forms of immersion first, 185 second, 190 Fundamental vector fields, 91 G

117 Gauss curvature, 162, 21 1 Gauss’ lemma, 147 Gauss map, 195 Gaussian curvature, see Gauss curvature General linear group, 4, 25, 27 complex, 7, 34 Geodesics, 96, 104, 113 and curvature, 179 closed, 241 connexions with same, 105 existence of, 97 global minimizing property on complete Riemannian manifold, 154 local minimizing property on Riemannian manifold, 149, 217 on sphere, 133 Grassmann algebra, 54 decomposable elements, 55 derivation of, 56 existence of, 57-62 homogeneous elements, 55 homomorphisms, 56 Grassmann manifold, 139, 170, 186 rljk,

SUBJECT INDEX

complex, 170 hyperbolic space, 172 opposite spaces, 172 oriented, 170 quaternion, 170 real, 170 Group, -, see - group H

Hadamard-Cartan, theorem of, 184 Hermitian manifold, 141 Hessian of function, 197 of normal coordinate function, 198 Holomorphic curvature, 171 forms, 73 section, 171 Holonomy group of connexion, 87 restricted, 87 Homogeneous elements of Grassmann algebra, 55 rank, 55 Homogeneous space, 39, 43 connexions on, 82 Riemannian, 137 Riemannian symmetric, 138 reductive subgroup, 83 Homometry, 135 Homomorphism of Grassmann algebra, 56 Lie algebra, 26, 30 Lie group, 29 continuous, 37 Hopf maps, 52 Hopf-Rinow theorem, 154 Horizontal differential form, 76 vector field, 75 Horizontal lift of curve, 77 of vector field, 75 Hyperbolic space, 172 Hypersurface, 207 of rotation, 209 products of spheres as, 209 rigidity, 21 1 type number of, 21 1

269 I

Ideal of Lie algebra, 26 Imbedding, see also Immersion, 21 isometric, 132 normal bundle, 48 Immersion, 132, 185 curvature of, 190, 193 first fundamental form, 132 Gauss map, 195 hypersurface, 207 Riemannian connexion, 188 second fundamental form, 190 totally geodesic, 194 Indecomposable elements of Grassmann algebra, 55 Index form, 220 augmented index of, 232 basic inequality, 228, 250 generalized, 250 index of, 223 null space, 221 nullity of, 225 Index of symmetric quadratic form, 123 Index theorem, 233 Induced bundle, 45 connexion on, 80 Infinitesimal deformations, 226 Integrable distribution, 23 Integral curve, 14 manifold of codistribution, 70 manifold of distribution, 23 Interior product i ( X )with vector field, 69 Invariant metric on Lie group, 136 Inverse function theorem, 11 Involutive distribution, 23 Isometry, 122, 160 local, 122 theorem of Cartan, 181 Isomorphism of Lie algebras, 26 Isotropy group, 39, 169

1 Jacobi equation, 174 Jacobi fields, 173

SUBJECT INDEX N-Jacobi fields, 221 focal points and, 225 generalized, 250 on Riemannian sphere, 180 rectangles associated to, 174 Jacobi identity, 14 Jacobi theorem, see Basic inequality for index form Jacobian matrix, 10

K Kahler, theorem of Cartan-, 205 Kahler manifold, 142 Killing form, 137 Kowalevsky, theorem of Cauchy-, 205

L Lens space, 140 Lie algebra, 26 abelian, 26 of Lie group, 27 Lie algebra- Lie group correspondence, 28 Lie derivative, 16 on differential form, 69 on vector field, 17 Lie group, 25 abelian, 25, 37 complex, 34 Lie algebra of, 27 Lie subgroup, 26 Lift, canonical, of rectangle, 147 horizontal, of curve, 77 of vector field, 75 Linear endomorphism determinant of, 57, 61, 62 trace of, 57 Linear homogeneous vector fields, 176 Linear transformation fields, 103 Local one parameter group of local diffeomorphisms, 15 associated vector field, 15 Locally symmetric space, I68 Longitudinal curves of rectangle, 147 Lorentz manifold, 123

M

Manifold, 2 almost complex, 64 complex, 7, 50, 64 product, 5 submanifold, 21 open, 4 topological, 2 Matrix, exponential of, 32 Jabobian, 10 Maurer-Cartan, equations and forms of, 70, 72, 82, 107 Mean curvature, 253 Myers’ theorem on, 256 p-mean curvature, 253 Measure zero, 12 Mechanical system, I57 Metric, Riemannian, 122 complete, 154 existence of, 126 Minimal segment, 237 submanifold, 248 Minimum locus, 237 point, 237 Morse index theorem, 233 Myers’ theorem, 256 N

Negative curvature, 179, 184 Non-orientable, 63 Normal bundle, 48, 151, 196 Normal coordinates, 110, 152 Null space of index form, 221 Nullity of index form, 225

0 One-dimensional Riemannian manifold, 132 Operations on vector valued forms, 72 Opposite connexion, 106 Opposite spaces to Grassmann manifolds, 172 hyperbolic space, 172

27 1

SUBJECT INDEX

Orbit, 39 Order of focal point, 225 Orientability, 63 Oriented Grassmann manifold, 170 Orthogonal group, 29, 33

quaternion, 52 conjugate points, 226 real, 5 , 43 Prolongation of covering map, 120 homometry, 135

P

Parallel translation in associated bundle, 84 in principal bundle, 78 of tangents, 89 Parallel vector field along curve, 96 Parallelizable manifold, 49, 64 connexions (direct, opposite, torsion zero), 106 covariant derivative, 114 Lie group, 107 Riemannian connexion, 134 Parallelization of bundle of bases, 92 Partial derivative, 8 Partition of unity, 4 Permutation groups, 58 cross-section, 58 shuffle permutations, 58, 61 Plane section, 161 Polar decomposition of quaternions, 36 Principal bundle, 41 affine bundle, 109 associated to, see Associated bundle bundle map, 49 bundle of bases, 43 bundle of frames, 127 coordinate bundle, 44 induced, 45 product (trivial), 42 reduction of structure group, 49, 87 Product bundle, 42 Product connexion, 119 Riemannian, 133 Product manifold, 5 bundle of adapted bases, 119 tangents to, 10 Product of spheres as hypersurface, 209 Projective space complex, 44, 52 conjugate points, 225 Kahler structure, 142

Q Quaternion Grassmann manifold, 170 Quaternion projective space, 52 conjugate points, 226 Quaternions, 34, 37 adjoint representation, 36 polar decomposition, 36 regular left representation, 34 Quotient space bundle of imbedding, 48 normal bundle, 48, 151, 196 R

Rank of homogeneous element of order two, 55 Rauch’s comparison theorem, 250, 252 Real projective space, 5, 43 Rectangle, Cm, 147 associated vector field, 147 broken Cm, 213 canonical lifting, 147 longitudinal curve, 147 transverse curve, 147 Reduced arc length, 223 Reduction of structural group, 49, 87 Riemannian structure as reduction of bundle of bases, 129 Reductive subgroup, 83, 187 Regular map, 14 Representation, 34 adjoint, 36 faithful, 34 of quaternions, 34, 36 Restricted holonomy group, 87 Restriction of function, 22 of vector field, 22 Ricci transformation, 250, 253 Ricci curvature, 253 Riemannian connexion, 129

272

SUBJECT INDEX

existence of, 131 of immersion, 188 uniqueness, 131 Riemannian curvature, 161 and geodesics, 179 computation of, 165 constant, 165, 184 continuity, 166 holomorphic, 171 negative curvature, 184 of immersed manifold, 190, 193 ofsphere, 162, 170, 180 of symmetric space, 170 zero curvature, 183 Riemannian manifold, 122 complete, 154 distance function, I24 examples, 132-144 imbedding of, 132 semi-, 123 Riemannian metric or structure, 122 existence, 126 induced by action of compact group, 136 on vector bundle, 127 Riemannian product, 133 Riemannian covering, 134 Riemannian homogeneous space, 137 Riemannian homogeneous symmetric space, 139 Riemannian symmetric space, 138, 168 Right action of Lie group, 38 Right connexion of Lie group, 107 Rigidity theorem, classical, 21 1 Rinow, theorem of Hopf-, 154 Rotation manifold of partial, 209 hypersurface of, 209 Ruled surface, 208 doubly, 209

s Sard’s theorem, 12 Schur’s theorem, 165 Second fundamental form, 190 and curvature, 193 interpretations of, 191, 192

Second variation of arc length, 213 Synge’s formula, 21 8 unintegrated, 214 Section, plane, 161 holomorphic, 171 Sectional curvature, see Riemannian curvature Semi-direct product, 39 Semi-Riemannian manifold, 123 index of, 123 Lorentz manifold, 123 Shuffle permutations, 58, 61 Solder form, 90, 93 Space form, 184 simple, 257 Special linear group, 33 Sphere, 4 conjugate points, 225 curvature, 162, 170, 180 Jacobi fields, 180 product of spheres as hypersurface, 209 Riemannian, 133 geodesics, 133 Stereographic projection, 4 Stiefel manifold, 137 Striped pants, 21 Structural constants of Lie algebra, 69 Structural equations dual formulation, 80, 101 for covariant differentiation, 116 of affine connexion, 100 of connexion, 81 of horizontal form, 86 Structure group, 42 reduction of, 49 to complex general linear group, 51 to orthogonal group, 129 Subalgebra, 26 Subgroup, 26 Submanifold, 21 minimal, 248 open, 4 totally geodesic, 194 Surface, 5 doubly ruled, 209 hyper-, 207 ruled, 208

213

SUBJECT INDEX

Symmetric affine connexion, see Torsion zero Synge’s formula for unintegrated second variation, 218 Synge’s theorem on imbeddings, 194 Synge’s theorem on connectivity, 243 T Tangent, 8 bundle, 46, 64 of product, 10 Tensor bundle, 47 field, 47 Topological manifold, 2 Torsion form, 93 structural equations, 100 Torsion translation, 96 geometric interpretation, 97 Torsion zero, 95, 103, 106, 117, 118, 166 Torus, 5 complex, 34 flat, 133 Totally geodesic submanifold, 194 Trace of linear endomorphism, 57 Transformations local one parameter group of, 15 group of, 38 Transition functions, 44 Transitive action by group, 39 Transvections, 168 Transverse curves of rectangle, 147 Trivial bundle, 42 Tubular neighborhood, 151 Type number of hypersurface, 2 1 1

U Unintegrated variations of arc length, 214

Synge’s formula, 21 8 Unitary group, 34 Urysohn function, 2

V Variations of arc length, 213 Synge’s formula, 21 8 unintegrated, 214 Vector analysis in R3, 68 Vector bundle, 48 Riemannien metric on, 126 Vector field, 13 along Cm curve, 96 parallel, 96 as cross-section, 47 associated to Cm rectangle, 147 broken Cm rectangle, 214 basic, 92 Cm, 13, 47 extension of, 22 fundamental, 9 1 Jacobi, 173 N-Jacobi, 221 left invariant, 27 associated one parameter subgroup, 29 linear homogeneous, 176 local one parameter group of, 15 Vector valued forms, 71 operations on, 72 Vertical forms, 76 vectors, 74 Volume element, 64 Riemannian, 122

This Page Intentionally Left Blank