METHODS OF DIFFERENTIAL GEOMETRY IN ANALYTICAL MECHANICS
NORTH-HOLLAND MATHEMATICS STUDIES 158 (Continuation of the N...
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METHODS OF DIFFERENTIAL GEOMETRY IN ANALYTICAL MECHANICS
NORTH-HOLLAND MATHEMATICS STUDIES 158 (Continuation of the Notas de Matematica)
Editor: Leopoldo NACHBIN Centro Brasileiro de Pesquisas Fisicas Rio de Janeiro, Brazil and University of Rochester New York, U.S.A.
NORTH-HOLLAND -AMSTERDAM
NEW YORK
OXFORD
TOKYO
METHODS OF DIFFERENTIAL GEOMETRY IN ANALYTICAL MECHANICS
M a n u e l de LEON CECIME Consejo Superior de lnvestigaciones Cientificas Madrid, Spain
Paulo R. RODRIGUES Departamento de Geometria lnstituto de Ma tema tica Universidade Federal Fluminense Niteroi, Brazil
1989
NORTH-HOLLAND -AMSTERDAM
NEW VORK
OXFORD TOKYO
ELSEVIER SCIENCE PUBLISHERS B.V. Sara Burgerhartstraat 25 P.O. Box 211, 1000 AE Amsterdam, The Netherlands Distributors for the U.S.A. and Canada: ELSEVIER SCIENCE PUBLISHING COMPANY, INC. 655 Avenue of the Americas New York, N.Y. 10010, U.S.A.
ISBN: 0 444 88017 8
0 ELSEVIER SCIENCE PUBLISHERS B.V.. 1989 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the publisher, Elsevier Science Publishers B.V. / Physical Sciences and Engineering Division, P.O. Box 103, 1000 AC Amsterdam, The Netherlands. Special regulations for readers in the U.S.A. - This publication has been registered with the Copyright Clearance Center Inc. (CCC), Salem, Massachusetts. Information can be obtained from the CCC about conditions under which photocopies of parts of this publication may be made in the U.S.A. All other copyright questions, including photocopying outside of the U.S.A., should be referred to the publisher.
No responsibility is assumed by the publisher for any injury and/or damage to persons or property a s a matter of products liability, negligence or otherwise, or f r o m any use or operation of any methods, products, instructions or ideas contained in the material herein. Printed in the Netherlands
To our parents
This Page Intentionally Left Blank
vii
Contents Preface
1
3 1 Differential Geometry 1.1 Some main results in Calculus on Rn . . . . . . . . . . . . . . 3 5 1.2 Differentiable manifolds . . . . . . . . . . . . . . . . . . . . . 1.3 Differentiable mappings . Rank Theorem . . . . . . . . . . . . 8 9 1.4 Partitions of unity . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Immersions and submanifold . . . . . . . . . . . . . . . . . . 11 1.6 Submersions and quotient manifolds . . . . . . . . . . . . . . 13 1.7 Tangent spaces . Vector fields . . . . . . . . . . . . . . . . . . 15 1.8 Fibred manifolds . Vector bundles . . . . . . . . . . . . . . . . 22 1.9 Tangent and cotangent bundles . . . . . . . . . . . . . . . . . 26 1.10 Tensor fields. The tensorial algebra . Riemannian metrics . . 30 1.11 Differential forms . The exterior algebra . . . . . . . . . . . . 38 1.12 Exterior differentiation . . . . . . . . . . . . . . . . . . . . . . 47 51 1.13 Interior product . . . . . . . . . . . . . . . . . . . . . . . . . . 1.14 The Lie derivative . . . . . . . . . . . . . . . . . . . . . . . . 52 1.15 Distributions . Frobenius theorem . . . . . . . . . . . . . . . . 55 1.16 Orientable manifolds . Integration . Stokes theorem . . . . . . 61 1.17 de Rham cohomology. PoincarC lemma . . . . . . . . . . . . . 71 1.18 Linear connections . Riemannian connections . . . . . . . . . 75 1.19 Lie groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 1.20 Principal bundles . Frame bundles . . . . . . . . . . . . . . . . 91 1.21 G-structures . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 1.22 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 2 Almost tangent structures and tangent bundles 2.1 Almost tangent structures on manifolds . . . . .
111
. . . . . . . 111
Con tents
viii
2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11
Examples . The canonical almost tangent structure of the tangent bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . Integrability . . . . . . . . . . . . . . . . . . . . . . . . . . . . Almost tangent connections . . . . . . . . . . . . . . . . . . . Vertical and complete lifts of tensor fields t o the tangent bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Complete lifts of linear connections to the tangent bundle . . Horizontal lifts of tensor fields and connections . . . . . . . . Sasaki metric on the tangent bundle . . . . . . . . . . . . . . Affine bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . Integrable almost tangent structures which define fibrations . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
114 116 119 120 126 129 135 138 139 144
147 3 Structures on manifolds 3.1 Almost product structures . . . . . . . . . . . . . . . . . . . . 147 3.2 Almost complex manifolds . . . . . . . . . . . . . . . . . . . . 151 3.3 Almost complex connections . . . . . . . . . . . . . . . . . . . 156 161 3.4 Kahler manifolds . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Almost complex structures on tangent bundles (I) . . . . . . 165 3.5.1 Complete lifts . . . . . . . . . . . . . . . . . . . . . . . 165 166 3.5.2 Horizontal lifts . . . . . . . . . . . . . . . . . . . . . . 3.5.3 Almost complex structure on the tangent bundle of a Riemannian manifold . . . . . . . . . . . . . . . . . . 167 3.6 Almost contact structures . . . . . . . . . . . . . . . . . . . . 169 176 3.7 f-structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 3.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4 Connections in tangent bundles 181 4.1 Differential calculus on TM . . . . . . . . . . . . . . . . . . . 181 183 4.1.1 Vertical derivation . . . . . . . . . . . . . . . . . . . . 4.1.2 Vertical differentiation . . . . . . . . . . . . . . . . . . 184 4.2 Homogeneous and semibasic forms . . . . . . . . . . . . . . . 186 4.2.1 Homogeneous forms . . . . . . . . . . . . . . . . . . . 186 4.2.2 Semibasic forms . . . . . . . . . . . . . . . . . . . . . 190 4.3 Semisprays. Sprays. Potentials . . . . . . . . . . . . . . . . . 193 4.4 Connections in fibred manifolds . . . . . . . . . . . . . . . . . 197 4.5 Connections in tangent bundles . . . . . . . . . . . . . . . . . 199 4.6 Semisprays and connections . . . . . . . . . . . . . . . . . . . 206 4.7 Weak and strong torsion . . . . . . . . . . . . . . . . . . . . . 211
Con tents 4.8 4.9
4.10 4.11 4.12 5
Decomposition theorem . . . . . . . . . . . . . . . . . . . . . Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Almost complex structures on tangent bundles (11) . . . . . Connection in principal bundles . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ix
. .
213 216 218 221 224
Symplectic manifolds and cotangent bundles 227 5.1 Symplectic vector spaces . . . . . . . . . . . . . . . . . . . . . 227 234 5.2 Symplectic manifolds . . . . . . . . . . . . . . . . . . . . . . . 5.3 The canonical symplectic structure . . . . . . . . . . . . . . . 237 5.4 Lifts of tensor fields to the cotangent bundle . . . . . . . . . . 240 5.5 Almost product and almost complex structures . . . . . . . . 245 5.6 Darboux Theorem . . . . . . . . . . . . . . . . . . . . . . . . 249 5.7 Almost cotangent structures . . . . . . . . . . . . . . . . . . . 253 5.8 Integrable almost cotangent structures which define fibrations 258 261 5.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6 Hamiltonian systems 263 6.1 Hamiltonian vector fields . . . . . . . . . . . . . . . . . . . . 263 267 6.2 Poisson brackets . . . . . . . . . . . . . . . . . . . . . . . . . 272 6.3 First integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Lagrangian submanifolds . . . . . . . . . . . . . . . . . . . . . 275 282 6.5 Poisson manifolds . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Generalized Liouville dynamics and Poisson brackets . . . . . 287 6.7 Contact manifolds and non-autonomous Hamiltonian systems 289 6.8 Hamiltonian systems with constraints . . . . . . . . . . . . . 295 297 6.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7 Lagrangian systems 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10
301 Lagrangian systems and almost tangent geometry . . . . . . . 301 306 Homogeneous Lagrangians . . . . . . . . . . . . . . . . . . . . Connection and Lagrangian systems . . . . . . . . . . . . . . 308 Semisprays and Lagrangian systems . . . . . . . . . . . . . . 317 A geometrical version of the inverse problem . . . . . . . . . 323 The Legendre transformation . . . . . . . . . . . . . . . . . . 326 Non-autonomous Lagrangians . . . . . . . . . . . . . . . . . . 330 336 Dynamical connections . . . . . . . . . . . . . . . . . . . . . . Dynamical connections and non-autonomous Lagrangians . . 344 The variational approach . . . . . . . . . . . . . . . . . . . . 347
Contents
X
7.11 Special symplectic manifolds . . . . . . . . . . . . . . . . . . 357 7.12 Noether’s theorem . Symmetries . . . . . . . . . . . . . . . . . 362 7.13 Lagrangian and Hamiltonian mechanical systems with constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367 7.14 Euler-Lagrange equations on T*M @ TM . . . . . . . . . . . 370 7.15 More about semisprays . . . . . . . . . . . . . . . . . . . . . . 376 7.16 Generalized Caplygin systems . . . . . . . . . . . . . . . . . . 391 7.17 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395 8 Presymplectic mechanical systems 8.1 The first-order problem and the Hamiltonian formalism . . . 8.1.1 The presymplectic constraint algorithm . . . . . . . . 8.1.2 Relation to the Dirac-Bergmann theory of constraints 8.2 The second-order problem and the Lagrangian formalism . . 8.2.1 The constraint algorithm and the Legendre transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2 Almost tangent geometry and degenerate Lagrangians 8.2.3 Other approaches . . . . . . . . . . . . . . . . . . . . . 8.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
399 399 400 404 409 409 413 428 436
A A brief summary of particle mechanics in local coordinates439 A.l Newtonian Mechanics . . . . . . . . . . . . . . . . . . . . . . A.l.l Elementary principles . . . . . . . . . . . . . . . . . . A.1.2 Energies . . . . . . . . . . . . . . . . . . . . . . . . . . A.2 Classical Mechanics: Lagrangian and Hamiltonian formalisms A.2.1 Generalized coordinates . . . . . . . . . . . . . . . . . A.2.2 Euler-Lagrange and Hamilton equations . . . . . . . .
.
B Higher order tangent bundles Generalities B.l B.2 B.3 B.4
Jets of mappings (in one independent variable) . . . . . . . . Higher order tangent bundles . . . . . . . . . . . . . . . . . . The canonical almost tangent structure of order k . . . . . . The higher-order PoincarB-Cartan form . . . . . . . . . . . .
439 439 441 443 443 445
45 1 451 452 454 454
Bibliography
457
Index
471
1
Preface The purpose of this book is to make a contribution to the modern development of Lagrangian and Hamiltonian formalisms of Classical Mechanics in terms of differential-geometric methods on differentiable manifolds. The text is addressed to mathematicians, mathematical physicists concerned with differential geometry and its applications, and graduate students. Chapter 1 is a review of some topics in Differential Geometry. It. is included in the text to state its main properties and to help the reader in subsequent chapters. Chapters 2 and 3 are devoted to the study of several geometric structures which are closely related to Lagrangian mechanics. Almost tangent structures and tangent bundles are examined in Chapter 2. The theory of vertical, complete and horizontal lifts of tensor fields and connections to tangent bundles are also included. In Chapter 4 we study the differential calculus on the tangent bundle of a manifold given by its canonical almost tangent structure. Connections in tangent bundles, in the sense of Grifone, are examined and other approaches to connections are briefly considered. In Chapter 5 we study symplectic structures and cotangent bundles. In fact, the canonical symplectic structure of the cotangent bundle of a manifold is the (local) model for symplectic structures (Darboux theorem). Lifts of tensor fields and connections to cotangent bundles are also included. In Chapter 6 we examine Hamiltonian systems. As there are many specialized books where this topic is extensively dealt with we decided to reduce the material to some essential results. This chapter may be considered as an introduction to the subject. Chapter 7 is devoted to Lagrangian systems on manifolds. We apply the main results of our previous chapters to Lagrangian systems. It is usual to find in the literature regular Lagrangian systems obtained by pulling back to the tangent bundle the canonical symplectic form of the cotangent bun-
2
Preface
dle of a given manifold, using for this the fiber derivative of the Lagrangian function. In this vein we do not need t o use the tangent bundle geometry. Nevertheless there is an alternative approach for Lagrangian systems which consists of using the structures directly underlying the tangent bundle manifold. This gives an independent approach, i.e., an independent formulation of the Hamiltonian theory. This point of view is that of J. Klein which was adopted in the French book of C. Godbillon (1969). More recently some points which use this kind of geometric formulation have also been presented in the book of G. Marmo et al. (1985). We think that this viewpoint gives a more powerful and elegant exposition of the subject. In fact we may say that almost tangent geometry has a similar role in Lagrangian theories to the role of symplectic geometry in Hamiltonian theories. Chapter 8 is concerned with presymplectic structures. As the reader will see in Chapter 7 the almost tangent formulation of classical lagrangian systems does not require regularity conditions on the Lagrangian functions. Thus, in general, if we wish the Euler-Lagrange equations t o define a vector field describing the dynamics (as it occurs in the regular case) we are lead into constrained Lagrangians. Presymplectic forms also appear in the Hamiltonian formalism, originated, for example, by degenerate Lagrangians, and lead to the so-called Dirac-Bergmann constraint theory. In this chapter we describe the geometric tools for such situations which have been inspired by many authors. We conclude the book with two Appendices. One is concerned with Particle Mechanics in local coordinates and is addressed t o students who are not very familiar with the classical approach. The other is devoted t o a brief summary on the theory of Jet-bundles, an important topic in modern differential geometry. We would like to express our gratitude t o the Conselho Nacional de Desenvolvimento Cientifico e Tecnologico, CNPq (Brazil) Proc. 31.1115/79, the FundaCao de Amparo a Pesquisa do Rio de Janeiro (FAPERJ), Proc. E29/170.662/88 and the Consejo Superior de Investigaciones Cientificas, CSIC (Spain) for their financial support during the preparation of the manuscript. We thank Pilar Criado for her very careful typing of the text on a microcomputer using Our thanks are also due t o Luis A. Corder0 and Alfred Gray who helped us to use this typesetting system and to John Butterfield for his valuable suggestions. To the Editor of Notas de Matematica, Leopoldo Nachbin and to the Mathematics Acquisitions Editor of Elsevier Science Publishers B.V./Physical Sciences and Engineering Division, Drs. Arjen Sevenster, our thanks for including this volume in their series.
m.
3
Chapter 1
Differential Geometry 1.1
Some main results in Calculus on R"
-
In this section we review briefly some facts about partial derivatives from advanced calculus. Let f : U c R" R be a function defined on an open subset U of R". Then f (x)= f(z', . . . , q , x= (21,. . . ,z")E
u.
At each point zo E U ,we define the partial derivative (af/axi),, off with respect t o xi as the following limit (if it exists):
(2)
= lim
20
h-0
f (z', . . . ,zi
+ h, . . . ,x")
-
f(d, . , . ,xi,. . . $2")
h
If a f /ad is defined at each point of U ,then a f /axi is a new function on U. When the n functions a f /ax',. , . ,a f /axn are continous on U ,we say that f is differentiable of class C'. Now, we define inductively the notion of differentiability of class C k: f is of class Ck on U if its first derivatives a f /axi, 1 2 i 5 n, are of class Ck-'.If f is of class Ckfor every k, then f is said to be Coo(or simply differentiable). Then we have the partial derivatives of order k defined on U by
Chapter 1. Differential Geometry
4
We can easily prove that the value of the derivatives of order k is independent of the order of differentiation, that is, if (jl, . . . , j k ) is a permutation of ( i l , . . . , i k ) , , then
akf
-
axil . . . a x j k
akf
-
axil
. .. a z i k *
-
Next, let F : U c R" R"' be a mapping (or map). If 7ra : Rm R, 1 5 a 5 m denotes the canonical projection ma(xl,. . . ,P ) = P,then we have m functions Fa : U c Rn R given by Fa = 7ra o F. We say that F is differentiable of class C',Ck or Cooif each Fa is C',Ck or Coo,respectively. We may sometimes call a CO" map F smooth or differentiable. If F is differentiable on U ,we have the m x n Jacobian matrix
at each point a U. LetF :Uc R"-RmandG:Vc defined on U .
-
R"-RPsothatH=GoFis
Theorem 1.1.1 (1) H i s differentiable; (2) J ( H ) = J ( G ) J ( F ) , that is,
Let F : U c R" --iV c R" be a mapping. We say that F is a diffeomorphism if (1) F is a homeomorphism and (2) F and F-' are differentiable. Obviously, if F is a diffeomorphism, then F-' is a diffeomorphism too.
-
Theorem 1.1.2 (Inverse Function Theorem) Let U be an open subset of R" and F : U R" a differentiable mapping. If J ( F ) at zo E U i s n o n - s i n g ~ l a r then ~ there exists an open neighborhood V of x0,V c U ,such that F(V) i s open and F : V + F(V) i s a diffeomorphism. (See Boothby [9] for a proof). Let F : U c Rn --+ Rm be a differentiable mapping. The rank of F at xo E U is defined as the rank of the Jacobian matrix J ( F ) at 20. Obviously,
1.2. Differentiable manifolds
5
rank ( F ) at 20 5 i n f ( m , n ) . Then, if the rank of F at 20 is k, we deduce that the rank of F is greater or equal t o k on some open neighborhood V of 20. In particular, if F : U c R" + F ( U ) c R" is a diffeomorphism, then F has constant rank n.
Theorem 1.1.3 (Rank Theorem) Let UO c R",Vo c Rm be open sets, F : UO +VO a differentiable mapping and suppose the rank of F to be equal to k on Uo. I f 2 0 E Uo and yo = F ( x 0 ) E Vo, then there exist open sets U c Uo and V c Vo with 20 E U and yo E V, and there exist diffeomorphisms G : U +G ( U ) c Rn,H : V + H ( V ) c Rm such that ( H o F o G - ' ) ( G ( U ) ) c V and
( H o F o G-')(zl,
. . . ,z")= ( 21, . . . ,zk ,o,. . .
(see Boothby 191 for a proof).
Remark 1.1.4 We can easily check that Theorems 1.1.2 and 1.1.3 are equivalent.
1.2
Differentiable manifolds
Definition 1.2.1 A topological manifold M of dimension m is a Hausdorff space with a countable basis of open sets such that for each point
of
M there is a neighborhood homeomorphic to a n open set of Rm. Each pair (U,+)where 4 : U + 4 ( U ) c R"' is called a coordinate neighborhood. If z E U , then 4(z) = (~'(z),. . . ,zm(z)) E Rm;zi(z),1 < i < rn, is called the ith coordinate of z, and the function8 zl, . . . ,zm, are called the coordinate functions Corresponding to (U, 4) (or local coordinate
system). Now, let (U,~#J), (V, $J) be two coordinate neighborhoods of M. Then
+-'.
In local coordinates, if ( z i ) ,(y') is a homeomorphism with inverse 4 o are the local coordinates corresponding to (U, 4), (V, CC)), respectively, then we have (21,.
. . ,Zm)
-
(yl(xi),
. . . ,y"(z')).
Chapter 1. Differential Geometry
6
Definition 1.2.2 (U,+), (V,+) are said to be C"-compatible if and I$ o + - l are Coomappings.
+
o q5-l
Definition 1.2.3 A differentiable or (C") structure on a topological manifold M i s a family U = {(U,,I$,}of coordinate neighborhoods such that: (1) the U, cover M; (2) for any a,fI (U,,&) and (Up,4@) are Cw-compatible; (9) any coordinate neighborhood (V,$J) C"-compatible with every (U,,4,) E U belong to U. A Coo(differentiable) manifold i s a topological manifold endowed with a C"-structure. Remark 1.2.4 Suppose that M is a topological manifold. If U = {(U,,4,)) is a family of C"-compatible coordinate neigborhoods which cover M , we define a set U by U = { (U, +)/ (U,4) is a coordinate neighborhood Coocompatible with any (U,,&) E U}. Obviously, U c U and U is the unique C" structure on M which contains U ;U is called a C" atlas and a maximal Cooatlas. Remark 1.2.5 Let (U, +) be a coordinate neighborhood on a C" manifold M . If V c U is an open set of M , then ( V , I $ p )is a new coordinate neighborhood (its coordinate functions are the restriction of the coordinate functions corresponding to (U,+)). If z E U, then we may choose V c U such that z E V and +(V)is an open ball B ( + ( X ) , c )with radius c or a cube C(+(z),c) of side 2c,c > 0, in R"'.Moreover, we may compose 4 with a translation such that $J(tj(z))= 0 E R"'.
+
Examples (1) The Euclidean space Rm. In fact, the canonical Cartesian coordinates define a Cw structure on P with a single coordinate neighborhood. (2) Furthermore, let V be an m-dimensional vector space over R. If { e i } is a basis of V ,then V may be identified with R"'. By means of the identification w = z1 el
+ . . . + xmem
-
( z l ,. . . ,zm) E Rm,
V becomes a C" manifold of dimension m and this C" structure is independent of the choice of the basis { e i } . ( 3 ) Let gl(m,R) be the set of m x m matrices A = (a:) over R. Then gZ(rn, R) is a vector space over R of dimension m2.With the identification
7
1.2. Differentiable manifolds
(a:)
--t
1 (al,. .., a ? , .
. .a,1 . . .a:)
E Rm',
then gl(m, R) becomes a CO" manifold of dimension m2. (4) Open submanifolds.
Let U be an open set of a differentiable manifold M of dimension m. Then U is a CO" manifold of the same dimension. To see this, it is sufficient to restrict the coordinate neighborhoods of M to U. The manifold U is called an open submanifold of M. (5) The general linear group Gl(m, R). A particular case of (4) is the following. Let Gl(m, R) be the group of all non-singular m x rn matrices over R. Then Gl(m,R) is an open set of gl(m, R). In fact, let det : gl(m, R)
-
R
be the determinant map. Then Gl(m, R ) = gl(m, R) - (det)-'(O).
Thus, Gl(m, R) is an open submanifold of gl(m, R). (6) The sphere S". The sphere S" is the set S" = {z = (zl,, . . ,z"+l)E R"+'/
n+ 1
C(zi)'= l}
i= 1
Let N = (0,. . . , O , l ) and S = (0,. . . ,0, -1). Then the standard CO" structure on S" is obtained by taking the Cooatlas
u = {(s" - N,PN),(s" - s,PS)), where PN and ps are stereographic projections from N and S, respectively. (7) Product manifolds. Let M, N be two Coomanifolds of dimension m, n respectively. We consider the product space M x N. If (U,+),(V,+) are coordinate neighborhoods of M , N ,respectively, we may define a coordinate neighborhood (U x V , + x +) on M x N by
(4 x
$)(Z,Y)
= ( 4 b ) A Y ) ) ) z E U,Y E v*
+
Then M x N becomes a Coomanifold of dimension m n. A particular case is the m-torus Tm = S' x . . . x S', the m-fold product of circles S1.
Chapter 1. Differential Geometry
8
Remark 1.2.6 In the sequel, we will say simply manifold for Coomanifold; we may also sometimes say differentiable manifold.
-
1.3
Differentiable mappings. Rank Theorem
Definition 1.3.1 Let F : M N be a mapping. F i s said to be (C"") differentiable if for every x E M there ezist coordinate neighborhoods (U, 4) of z and (V,$) of y = F ( z ) with F ( U ) c V such that
i s a diferentiable mapping.
-
This means that Flu : U V may be written in local coordinates d,. . . xm and ,'y . . . ,y" as follows: )
-
F : (xl,. . . P) )
(yl(x1,. . . ,X r n ) ) .
..
)
y"(x1,.
..
where each yo = ya(zl,. . . ,P ) ,1 5 o 5 n, is Cooon 4 ( U ) .
-
Remark 1.3.2 Obviously, every differentiable mapping is continous. Remark 1.3.3 Let f : M R be a function on M. Then f is differentiable if there exists, for each x E M a coordinate neighborhood ( V , 4 ) of 2 such that f o 4-l : 4 ( U ) 4 R is differentiable. Here, we consider the canonical differentiable structure on R. We denote by C o o ( M )the set of all differentiable functions on M. Obviously, all the definitions rest valid for mappings and functions defined on open sets of M.
-
Definition 1.3.4 A diferentiable mapping F : M N i s a diffeomorphism if it i s a homeomorphism and F-l i s diferentiable. In such a case, M and N are said to be diffeomorphic. A difeomorphism F : M M is said to be a transformation of M.
-
Let F : M + N be a differentiable mapping and let x E M. If (U,4) and (V, $) are coordinate neighborhoods of x and F ( x ) , respectively, with F(U)c V, then F is locally expressed by
Definition 1.3.5 The rank of F at z is defined to be rank of F at +(x).
1.4. Partitions of unity
9
Hence the rank of F at x is the rank at +(x) of the Jacobian matrix
One can easily prove that this definition is independent of the choice of coordinates. The most important case for us will be that in which the rank is constant. In fact, from the Rank Theorem in Section 1.1,we have the following.
-
Theorem 1.3.6 (Rank Theorem).-Let F : M N be as above and suppose that F has constant rank k at every point of M . If x E M then there ezist coordinate neighborhoods (U,+) and (If,$) as above such that +(x) = 0 E R m , $ ( F ( z ) ) = 0 E R" and P i s given by P(X1,.
. . ,Xrn) = (21,. . . ,xk ,o,.
. . ,O).
Furthermore, we may suppose that ( U , 4) and (V, 9 ) are cubic neighborhoods centered at x and F ( x ) , respectively.
Corollary 1.3.7 A necessary condition for F t o be a diffeomorphism i s that dim M = dim N = r a n k F .
1.4
Partitions of unity
Partitions of unity will be very useful in the sequel, for instance, in order t o construct Riemannian metrics on an arbitrary manifold. First, let us recall some definitions and results. Definition 1.4.1 A covering {Ua}of a topological space M i s said t o be locally finite if each x E M has a neighborhood U which intersects only a f i n i t e number of sets U,. If {Ua} and {Vp} are covering of M such that if Vp c U, for some a, then {V'} i s called a refinement of {Ua}. Definition 1.4.2 A topological space M i s called paracompact if every open covering has a locally f i n i t e refinement.
Now, let M be a manifold of dimension m. Then M is locally compact (in fact, M is locally Euclidean; so, M has all the local properties of P). A standard result of general topology shows that every locally compact Hausdorff space with a countable basis of open sets is paracompact (see Willard [127], for instance). Then we have.
Chapter 1. Differential Geometry
10
Proposition 1.4.3 Every manifold is a paracompact space. Definition 1.4.4 Let f E Ca,(M). The support off is the closure of the set on which f d o e s not vanishes, that is, SUPP
( f ) = c l { z E M l f (4# 01.
We say that f has compact support i f supp (f) is compact in M.
Definition 1.4.5 A partition of unity on a manifold M is a set {(U,, f;)}, where (1) {U;}is a locally finite open covering of M; (2) f i E Ca,(M),f; 2 0 , f ; has compact support, and supp ( f ; ) c U;for all a; 2 E M , C i f ; ( z )= 1. (Note that b y virtue of (1) the sum is a well-defined function on M). A partition of unity is said to be subordinate to an atlas {Ua}of M i f {U;} is a refinement of { U,} .
(9) for each
Lemma 1.4.6 Let U;= B(0,l ) , U2 = B(O,2) in Rm. Then there is a Coo function g : R"' + R such that g is 1 on U1 and 0 outside U2. We call g a bump function. Proof: Let 8 : R
-
R be given by
Now, we put
J --M
J-a,
Then 81 is a Coofunction such that 81(s) = 0, if s < -1, and O1(s) = 1 if s > 1. Let
e2(.) Thus,
82
= 41(-8 - 2).
is a Coofunction such that &(s) = 1 if s > 1, and 8,(s) = 0 if
s > 2. Finally, let
Then g is the required function. 0
1.5. Immersions and su bmanifolds
11
Lemma 1.4.7 Let U1,U2 be open sets of an m-dimensional manifold M such that cl(U1) c U2. Then there ezists g E C"(M) such that g is 1 on U1 and is 0 outside U2. Proof The proof is a direct consequence of the Lemma 1.4.6. 0 Proposition 1.4.8 If {Va} is an atlas of an m-dimensional manifold MI there is a partition of unity subordinate t o {Va}. Proof: Let {Wx} be an open covering. Since M is paracompact, then there is a locally finite refinement consisting of coordinate neighborhoods { ( U ; , 4 ; ) } such that &(U;)is the open ball centered at 0 and of radius 3 in P ,and such that (4;)-'(B(O,1))cover M . Now, let {Va}be an atlas of M and let {(K, 4;)) be a locally finite refinement with these properties. From the Lemma 1.4.7, there is a function g; E C m ( M )such that supp (9;) c V, and g; 2 0. We now put
Then { f;} are the required functions.
1.5
Immersions and submanifolds
In this section we shall consider some special kinds of differentiable mappings with constant rank.
-
Definition 1.5.1 Let F : N M be a differentiable mapping with n = d i m N 5 m = d i m M . F is said to be an immersion i f rank F = n at every point of N . If an immersion F is injective, then N (or its image F(N)), endowed with the topology and differentiable structure which makes F :N F ( N ) a diffeomorphism, is called an (immersed) submanifold of M.
-
-
From the theorem of rank, we deduce that, if F : N M is an immersion, then, for each x E N , there exist cubical coordinate neighborhoods (U,+),(V,$) centered at z and F ( z ) , respectively, such that F is locally given by
-
F : (2,. . . ,Zn) Hence F is locally injective.
( d , . . ,zn,o,. . . )O).
Chapter 1. Differential Geometry
12
Remark 1.5.2 We note that an immersion need not be injective. For instance, the mapping
given by
F ( t ) = (cos2nt,sin 27rt) is a immersion, but F(t
+ 27r) = F ( t ) .
Definition 1.5.3 An embedding is an injective immersion F : N + M which is a homeomorphism of N onto its image F(N), with its topology as a subspace of M . Then N (or F(N)) is said to be an (embedded) submanifold of M .
-
Remark 1.5.4 We note that an injective immersion need not be an embedding. For instance, let F : R R2 be given by 1 2
F ( t ) = (2cos(t - -7r),sin2(t -
1 2
-7r)).
The image of F is a figure eight denoted by E; the image point making a complet circuit starting at (0,O)E R2 as t goes from 0 to 27r. E = F ( R ) is compact considered as subspace of R2,but R is the real line. Then E and R are not homeomorphic. Let M be a differentiable manifold of dimension m. D e f i n i t i o n 1.5.5 A subset N of M is said to have the n-submanifold property if, for each 2 E N, there ezists a coordinate neighborhood (V,4) with local coordinates (z', . . . ,P)such that
Now, we consider the subspace topology on N. We put U' = U n N and
.
+'(z) = ( 2 ' ) . . ,z") E R",z E
U'.
Let (U, 4)) (V, $) be coordinate neighborhoods as above. Then $' o (4')-' : +'(U') + $'(V') is a Coomapping. Then N is a n-dimensional manifold and the natural inclusion i : N + M is an embedding. Thus, N is an
1.6. Submersions and quotient manifolds
13
-
embedded submanifold of M. It is not hard to prove the converse, that is, if F :N M is an embedding, then F ( N ) has the n-submanifold property. We leave the proof to the reader as an exercise. To end this section, we shall describe a useful method of finding examples of manifolds.
-
Proposition 1.5.6 Let F : N M be a differentiable mapping, w i t h d i m N = n , d i m M = m,n 1 m. Suppose that F h a s constant rank k o n N and let y E F ( N ) . T h e n F - ' ( y ) i s a closed embedded submanifold of N. Proof: First, F - ' ( y ) is closed since F is continous. Furthermore, let z E F-'(y). By the theorem of rank, there exist coordinate neighborhoods (U, +), (V, $) of z and y , respectively, such that F is locally given by (21,.
-
. . ,z")
(21,.
. . ,zn ,o,. . . ,o).
Hence we have
u nF - ~ ( Y = ) (2 E u/zl = . . . = 2 = 0).
-
Therefore, F - l ( y ) has the (n - k)-submanifold property. 0
Corollary 1.5.7 Let F : N M be a s above. If rank F = m at every point of F - ' ( y ) , t h e n F - ' ( y ) i s a closed embedded submanifold of N . Proof: In fact, if rank F = m at every point of F - ' ( y ) , then F has rank m on an neighborhood of F - l ( y ) . Then we apply the Proposition 1.5.6. 0 Example.- Let F : R" + R be the mapping defined by n
F ( z ' , . . . ,zn)=
C
(z')'.
i=l
Then F has rank 1 on R" - (0). But S"-' c R" - (0). Thus, S"-' is a closed embedded submanifold of R". It is not hard to prove that this structure of manifold on S"-' coincides with the one given in Section 1.2.
1.6
-
Submersions and quotient manifolds
Definition 1.6.1 Let F : A4 N be a differentiable mapping with m = d i m M 2 n = d i m N . F i s said t o be a submersion i f rank F = n at every point of M.
14
If F : M
-
Chapter 1. Differential Geometry
-
N is a submersion, then F is locally given by
F
: (21).
..
)
zrn)
(21).
..
)
2").
This fact is a direct consequence from the Rank Theorem. Hence, a submersion is locally surjective.
-
N be a submersion. Let y E N . Then Definition 1.6.2 Let F : M F-'(y) i s called a fibre of the submersion F. From Proposition 1.5.6, we deduce that, if y E F ( M ) , then the fibre F-'(y) is a closed embedded submanifold of dimension m - n of M . Now, let M be a topological space and an equivalence relation on M . We denote by M / the quotient space of M relative to -. Let T : M M / be the canonical projection. It is easy to prove (see Willard [127]) the following.
-
-
-
N
- --
Proposition 1.6.3 (1) If T : M M/ i s an open mapping and M has a countable basis of open sets, then M / has a countable basis also. (2) Put R = {(x,y)/z y}. Then M / i s Hausdorfl if and only if R i s a closed subset of M x M .
-
-
-
Next, let M be a differentiable manifold of dimension m. Proposition 1.6.3 is useful in determining those equivalence relations on M whose quotient space is again a manifold. If M / is a manifold such that T : M M / is a submersion, then M / is said t o be a quotient manifold of M . It is not hard t o prove that, if such a manifold structure exists, then it is unique. Example (Real projective space RP").- Let M = R"+l- ( 0 ) . We define z y if there is t E R - ( 0 ) such that y = t z , that is,
-
- -
-
-
y'=tz',
l 0 is an open interval in R) then the t a n g e n t vector of u at t is defined by d u / d t = b ( t ) = i ( O ) , r ( s )= u ( s t ) .
+
16
Chapter 1. Differential Geometry
Definition 1.7.2 The tangent space T z M o j M at z is the set of equivalence classes of curves at x. Now, we define a map
4' : T z M -+
Rm
by
It is clear that
.
4'
is injective. Furthermore, it is surjective. In fact, let u = ( u ' , . . . ,urn) E Rm and consider the curve given by
+
.(t) = +-l(+(z) t v ) .
)I.[('+
Hence = u. Then we consider a vector space structure on T,M such that 4' is a linear isomorphism. One can easily prove that this vector structure is independent of the choice of (U,+). Thus T,M is a vector space of dimension m. Let u' be the curve at x defined by
where ( e l , . . . ,ern) is the canonical basis of Rm. Then the tangent vectors (b'(O),. . . ,bm(0))form a basis for T,M. In the sequel, we shall use the notation
(a/az'),
= bi(0),l 5 i
5
m.
Obviously,
Now, let F : M linear mapping
-
+ ' ( ( a / a ~ ' )=~e,. ) N be a differentiable mapping. Then we define a
d F ( x ) : T z M + TF(,)N
.I.
d F ( z ) ( [ a ]= ) [Fo
17
1.7. Tangent spaces. Vector fields
Definition 1.7.3 The linear mapping d F ( z ) (also denoted b y F,) will be called the differential of F at x. Now, let z E M and (V,4), (V, $) coordinate neighborhoods of x and F ( x ) with local coordinates (zl, . . . ,zm) and (y', . . , ,y"), respectively. A direct computation shows that n
d F ( x ) (( a / a z ' ) z ) = C(aYa/ax')(a/Yo)F(z). o=l
Hence d F ( x ) is represented by the Jacobian matrix of F . We deduce that rank F at x = rank d F ( x ) Consequently, we have
Proposition 1.7.4 (1) F i s an immersion i f dF(x) i s an injective linear mapping, for each x E M . (2) F is a submersion i f dF(x) is a surjective linear mapping, for each x E M . (3) If F is a diffeomorphism, then dF(x) i s a linear isomorphism, for each x E M . Conversely, i f dF(z) is a linear isomorphism, then F i s a local diffeomorphism o n a neighborhood of z.
-
-
The following result generalizes the well-known chain rule.
Theorem 1.7.5 Let F : M N and G : N mappings. T h e n G o F i s differentiable and
P be differentiable
d ( G o F ) ( x ) = d G ( F ( x ) )o dF(x),for each z E M .
Proof: Obviously, if F and G are Coo,then G o F is Cooalso. Now, let o be a curve at x E M; then d ( G o F ) ( s ) ( [ o ]= ) [(Go F ) oo]. On the other hand, we have
d G ( F ( x ) ) ( d F ( z ) ( [ o= ] ) d G ( F ( z ) ) ( [ Fo o])= [G o ( F o o)]. This ends the proof. 0
Definition 1.7.6 A vector field X o n a manifold M i s a function assigning to each point x E M a tangent vector X ( z ) of M at x. (in Section 1.9 we shall precise the word function employed here).
Chapter 1. Differential Geometry
18
Let (U, 4) be a coordinate neighborhood with local coordinates ( x i ) , 1 5 i 5 m = dim M . For each x E U ,we have m
X(x) = Cx'(x)(a/axi).. i=l
Then we have m
x = Exi a p X i i=l
on U.
Definition 1.7.7 A vector field X i s said to be Coo if the functions Xi are Coofor each coordinate neighborhood (U, 4). Let X be a vector field on a manifold M . A curve u : I -+ M in M is called an integral curve of X if, for every t E I , the tangent vector X ( u ( t ) ) is the tangent vector to the curve 0 at a ( t ) .
Proposition 1.7.8 For any point xo E M , there is a unique integral curve of X, defined on (-E,E) for some E > 0, such that xo = a(0). Proof In fact, let (U, 4) be a coordinate neighborhood of xo with local coordinates ( x i ) and let m
x = Ex' a p x ' i=l
on U. Then an integral curve of X is a solution of the following system of ordinary differential equation
Now, the result follows from the fundamental theorem for systems of ordinary differential equations.
Remark 1.7.9 From Proposition 1.7.8 a vector field is also called a first order differential equation.
1.7. Tangent spaces. Vector fields
19
Definition 1.7.10 A l-parameter group of transformations of M i s a mapping @:
-
RxM
-
M,
such that (1) f o r each t E R, @t : x @(t,x) is a transformation of M; (2) f o r all s,t E R and E M , @ a + t ( X ) = @ t ( @ a ( Z ) ) . Each l-parameter group of transformations @ = (at)induces a vector field X as follows. Let x E M . Then X ( x ) is the tangent vector to the curve t at(.) (called the orbit of z)at x = @ o ( x ) . Hence the orbit @ t ( x ) is an integral curve of X starting at x. X is called the infinitesimal generator of @:. A local l-parameter group of local transformations can be defined in the same way. Actually, a:(.) is defined only for t in a neighborhood of 0 and z in an open set of M . More precisely, a local l-parameter group of local transformations defined on ( - 6 , 6 ) x U ,where 6 > 0 and U is an open set of M , is a mapping
-
such that (1) for each t E (-c, c), @t : x -+ @(t,x) is a diffeomorphism of U onto the open set @:(U); (2) If s , t , s + t E ( - 6 , ~ )and if .,@a(.) E U ,then
As above, @: induces a vector field X defined on U . Now, we prove the converse.
Proposition 1.7.11 Let X be a vector field o n M. T h e n , f o r each xo E M I there exists a neigborhood U of 20, a positive real number 6 and a local 1parameter group of local transformations @ : ( - 6 , c) x U M which induces X o n U.
-
Proof Let (V, 4) a coordinate neighborhood of zo with local coordinates (zi) such that zi(zo) = 0 , l 5 i 5 m. Consider the following system of ordinary differential equations:
20
Chapter 1. Differential Geometry
df'/dt = X ' ( f ' ( t ) , .
..
)
f"(t)),
(1.1)
where X = X' a/az' on V . From the fundamental theorem for systems of ordinary differential equation, there is a unique set of functions ( f ' ( t , 2)).. . , fm(t,z))defined for z with xi E (-6,6) and for t E (-X,X) such that (f') is a solution of (1.1) for each fixed x satisfying the initial condition f'(0,z) = z', 1 5 i 5 rn.
We set
and
W = {x/z' E ( - 6 , 6 ) } . Now, if s , t , s + t E (-X,X) and x, at(,) E W ,then the functions g i ( t ) = f'(s + t , x) form a solution of (1.1) with initial conditions
-
From the uniqueness of the solution, we deduce that g'(t) = f ' ( t , Q 8 ( x ) ) . This proves that Qt(Q8(z)) = @ 8 + t ( z ) . Now, since Qpo : W W is the identity, there exist p > 0 and v > 0 such that, if U = {./xi E (-p,p)) then @ t ( U ) c W ,if t E (-v,v). Consequently, we have
for every x E U,t E (-v, v). Hence CPt is a local l-parameter group of local transformations defined on (-v,v) x U which induces X on U.0
Definition 1.7.12 A vector field X on M is called complete i f X generates a global 1-parameter group of transformations on M . Proposition 1.7.13 O n a compact manifold M, every vector field is complete.
21
1.7. Tangent spaces. Vector fields
We leave the proof to the reader as an exercise. To end this section, we interpret a vector field as an operator on functions. Let u E T,M be a tangent vector to M at z E M. If f is a differentiable function defined on a neighborhood U of z,then we can define a real number uf by
where o E u. We have the following properties: (1) 9 ) = of + ug, (2) 4 4 ) = a ( u f ) , aE R, (3) (Leibniz rule) u(fg) = f(z)(ug) (uf)g(z). Now, let C""(z) be the set of differentiable functions defined on a neighborhood of z. Two functions f and g of C""(z) are related if they agree on some neighborhood of z,that is, if f and g define the same germ at z. The quotient set is denoted by C'"(z). Hence C'"(z) is a real algebra. By a derivation on C'"(z) we mean a linear operator
4f +
+
D : C"(z)
--t
P ( z )
such that D ( f g ) = ( D f ) g + f(Dg), f,g E C'"(z). Then each tangent vector v of M at x is a derivation on C"(z). We prove the converse. First, if D is a derivation on C'"(z), then Da = 0, for each constant function a. Now let f E Cm(z). The Taylor expansion of f, with respect to a local coordinate system (y') at z,is
where zi = y'(z),z' = y'(z) and w;, we have
(d2f/ayiayj) when y
--i
-
z. Then
Df = (af/ay')(Dy'). Hence, we deduce that
D = Xi(d/dy'),, where X i = Dy', 1 5 i 5 m. Now, let X be a vector field on M . I f f E CM(M),then we can define a new CM function X f by
Chapter 1. Differential Geometry
22
Xf(4 =X(4f.
+
(Obviously, if f is constant, then X f = 0). Then we have X ( f g ) = ( X f ) g f ( X g ) . Thus, X acts as a derivation on the algebra C " ( M ) . Denote by x ( M ) the set of all vector fields on M . Obviously, x ( M ) is a vector space over R and a C"(M)-module. Now, let X,Y E x ( M ) . Then we can define a new vector field [X,Y] as follows:
[X,Yl(z)(f) = X(.)(Yf)- Y(4(Xf),
= E M , f E C"(4.
Then [X,Y] is a vector field on M , which is called the Lie bracket of and Y.A simple computation shows that
X
(1) [KYI= -[Y,XI; (2) [ f x , g Y l = f ( X g ) Y - N f ) X
(3) (Jacobi identity)
+ ( f g ) [ X YI; [[X,Y],Z] + [[Y,Z],X] + [[Z,X],Y] = 0.
Remark 1.7.14 This properties show that ( x ( M ) , [, (see Section 1.19).
I)
is a Lie algebra
In terms of local coordinates, we have
[X,Y] = ( X ' ( a Y J / a z ' )- Y'(axj/az')) a/azJ, where
1.8
x = xi a/aZi,y = Y',Y = Y' a / a d Fibred manifolds. Vector bundles
-
Definition 1.8.1 A bundle is a triple ( E , p , M ) , where p : E M is a surjective submersion. The manifold E is called the total space, the manifold M is called the base space, p is called the projection of the bundle. For each z E M I the submanifold p - ' ( z ) = E, is called the fibre over x. We also say that E is a fibred manifold over M .
-
Example.- Let M, F be manifolds. Then (M x F, p , M ) is a bundle, where p :M x F M is the canonical projection on the first factor. This bundle is called a trivial bundle.
-
Definition 1.8.2 Let (El p , M ) be a bundle. A mapping s : M -+ E such that p o s = i d is called a (global) section of E. If s i s defined on an (open) subset U of MI then s : U E is called a local section of E over U.
1.8. Fibred manifolds. Vector bundles
23
Note that there always exist local sections since p is a surjective submersion.
-
Definition 1.8.3 Let (E, p , M) and ( E ( , p ' , M') be t w o bundles. A bundle morphism ( H , h) : ( E , p ,M) ( E ' , p ' , M') ie a pair of differentiable m a p s H :E E' and h : M -+ M' such that p'o H = h o p . (Roughly speaking, a bundle m o r p h i s m i s a fibre preserving map).
-
From Definition 1.8.3 one easily deduces that H maps the fibre of E over
-
x into the fiber of E' over h ( x ) .
-
Definition 1.8.4 A bundle m o r p h i s m ( H , h ) : ( E , p ,M) (E',p',M') i s a n isomorphism i f there ezists a bundle m o r p h i s m ( H ' , h') : ( E ' ,p', M') ( E , p ,M) s u c h that H' o H = idE and h' o h = idM. T h e n ( E , p ,M) and (E',p',M') are said t o be isomorphic. Now, we consider bundles (or fibred manifolds) with an additional vector space structure on each fibre.
Definition 1.8.5 Let M be a diflerentiable m-dimensional manifold. A real vector bundle E of rank n over M i s a bundle ( E , p ,M) such that: (1) For each z E M, E, has the structure of a real vector space of d i m e n s i o n n; (2) Local triviality) For each x E M there ezists a neighborhood U of x and a diffeomorphism H : U x R" - p - ' ( V )
-
such that, f o r each y E U , the correspondence w H ( y , w ) defines a n i s o m o r p h i s m between the vector space R" a n d the vector space E,,.
Examples.- (1) Let M be a differentiable manifold. Then M x Rn is a (trivial) vector bundle of rank n over M. (2) The tangent and cotangent bundles of M (see Section 1.9). Definition 1.8.6 Let (E,p,M), (fl,p',M") be vector bundles. T h e n a vector bundle homomorphism i s a bundle m o r p h i s m (H,h) s u c h that the restriction H , : E, --+ EL(,) i s linear f o r each x E M. (H,h) i s called a vector bundle isomorphism z f there ezists a vector bundle h o m o m o r p h i s m (H',h') : (E',p', M') ( E , p , M )such that H ' o H = idE a n d h'oh = idM. I n such a case, E and E' are said t o be isomorphic. If M'=M, a M-vector bundle homomorphism (or vector bundle homomorphism over M) i s defined by a vector bundle h o m o m o r p h i s m of the f o r m ( H , i d ~ ) .
-
24
Chapter 1. Differential Geometry
-
-
(E',p',M') is a vector bundle isomorphism, If ( H , h ) : ( E , p , M ) then: (1) H and h are diffeomorphisms; (2) the restriction H , : E, E&zI is a linear isomorphism, for each x E M. The converse is true for vector bundles with the same base. Proposition 1.8.7 Let H : E + E' be a vector bundle homomorphism over M. If for each x E M , Hz : E, ---t EL i s a linear isomorphism then H i s a vector bundle isomorphism.
- -
We leave the proof to the reader as an exercise.
Definition 1.8.8 A vector bundle p : E if it i s isomorphic to M x R" M.
M of rank n i s called trivial
Remark 1.8.9 Hence the local triviality property means that p-'(U) is a trivial vector bundle over U isomorphic to U x Rn. Next, we will describe a number of basic constructions involving vector bundles (see Godbillon [63], Milnor and Stasheff [96]). (1) Restricting a bundle to a subset of the base space. Let ( E , p , M) be a vector bundle over M and N c M a submanifold of M. We set E = p - ' ( N ) and denote by p : E + N the restriction of p to E . Then one obtains a new vector bundle (,!?,p, N ) called the restriction of E t o N. Each fiber E,, x E N , is equal to the corresponding fiber Ez. (2) Induced bundles. Let N be an arbitrary manifold and ( E , p , M ) a vector bundle. For any map f : N M we can construct the induced bundle f * ( E ) = (E,p, N) over N as follows. The total space E c N x E consists of all pairs ( z , e ) such that f (x) = p(e). The projection p : E N is defined by p(x, e) = z. Then one obtains a commutative diagram
-
-
E
-f E
where f ( x , e ) = e. The vector space structure in Ez is defined by a(x,e)
Thus
+ P(x,e') = ( z , a e + Pe'),
f is a vector bundle homomorphism over f .
a,P E R.
1.8. Fibred manifolds. Vector bundles
25
(3) Cartesian products. Given two vector bundles ( E l , p l , M l ) and (E2,p2,M2) the Cartesian product is the vector bundle (El x E2,pl x p 2 , A41 x M2). Obviously, each fiber (El x E2)(21,2a) is identified in a natural way with x ( E ~ ) , ~ ,Ez l
-
M1,22 E M2. (4) Whitney sums. Let ( E l , p l ,M ) , (E2,p2,M ) be twovector bundles over M. Let A : M M x M be the diagonal mapping defined by A(z) = (z,z). The vector bundle A*(& x E2) over M is denoted by El 6j E2, and called the Whitney sum of El and E2. Each fiber ( E l @ E2), is canonically identified with the direct sum ( E l ) Z I e3 ( E 2 ) Z 2 . ( 5 ) In general, the algebraic opeations on vector spaces can be extended in a natural way to vector bundles. Details of the corresponding constructions are left to the reader (see Godbillon [63]). Definition 1.8.10 Let ( E , p , M ) and ( E ' , p ' , M ) be two vector bundles over M and H : E' + E a vector bundle homomorphism (over M ) such that the E, of H to any fiber E!! is injective. W e say that restriction H , : EL ( E ' , p ' , M ) is a vector subbundle of ( E , p , M ) (Obviously, we may identity E' with H ( E ' ) ) .
-
Definition 1.8.11 Let ( E , p ,M ) ,(E',p',M ) be two vector bundles over M and H : E + E' a vector bundle homomorphism over M. Then
Ker H =
u ker H , ,EM
is a vector subbundle of E which will be called the kernel of H and
is a vector subbundle of E' which will be called the image of H . Moreover, if E' is a vector subbundle of E , then we can define a new vector bundle Errover M by setting
E" is called the quotient vector bundle of E by E'.
Chapter 1. Differential Geometry
26
-
-
Let now ( E , p , M ) ,(EI,p', M) and (E",p", M ) be vector bundles and b,G: G G' vector bundle homomorphisms over M . The sequence
H :E
is said t o be exact if for each z E M the sequence of vector spaces
is exact. In such a case, we writte 0-
E
5 E'- G
,??'-O.
For instance, if E is a vector subbundle of E' and E" is the quotient vector bundle of E' by E, then the sequence
is exact, where a' is the canonical inclusion and p the canonical projection.
1.9
Tangent and cotangent bundles
Let M be an m-dimensional manifold. We set
Let
TM
:T M
-
TM=
U TzM. ZEM
M be the canonical projection defined by
= T U , for each open set U of M. Let (U,q5) be acoordinate Hence (TM)-~(U) neighborhood on M with local coordinates (zl, . . . ,zm). Then we can define a mapping : U x Rm
-
TU
given by
@(z,a ) = a'(a/az'),,
27
1.9. Tangent and cotangent bundles
where a = (a', . . . ,am) E R"',+(z) = (z',. . . ,zm).@is a bijective mapping, since, if u E T,M, z E U ,then u = u'(a/az'),.
Consequently, @(z', mapping
. . . ,zm,u',. . . ,urn)
= u. Hence @ defines a bijective
a' : 4 ( U ) x Rm -+ TU given by @'(z',, . . , zm,al,.. . ,am) = @(z,a). Now, it is clear that there is a unique topology on T M such that for each coordinate neighborhood (U,q5) of M , the set TU is an open set of T M and @ : U x Rm T U , defined as above, is a homeomorphism. Thus we have local coordinates ( z i ,u') on T U called the induced coordinates in T M . Next, we prove that, in fact, T M has the structure of manifold of dimension 2m. Let ( U , d ) , (V,$) be two coordinate neighborhoods on M such that U n V # 0;then TU nTV # 0. Let u E T,M,z E U n V . Then, if (zi),(yi) are local coordinates corresponding to (U, 4)) (V,$), respectively, we have
-
u = u'(a/az'),
where wi = uj(ay'/azJ),.
= w'(a/ay'),,
Hence
is given by
Hence the neighborhoods (TU,(a')-') determine a C"-structure on T M of dimension 2m relative to which TM is a submersion. In fact, TM is locally given by TM(zi,Ui)
(z').
Moreover, ( T M ,T M , M ) is a vector bundle of rank m, which will be called the tangent bundle of M . Actually, it is clear that a vector field X on M defines a section of T M , and conversely. It is easy to prove that a vector field X : M T M is C" if and only if X is Cooas a mapping from M into T M .
-
28
Let F : M map T F : T M
-
Chapter 1. Differential Geometry
N be a differentiable mapping. Then we can define a
+T N
as follows:
TF(w) = d F ( z ) ( u ) , for u E T,M, z E M . Thus T F is a vector bundle homomorphism such that the following diagram
TM
2
TN
is commutative. Sometimes, we shall employ the notation T F ( u )for d F ( z ) ( u ) if there are no danger of confusion. Now,let 2 be a point of M . We set
T,*M = H(T, M )*, i.e., T,*M is the dual vector space of T , M ; T,*M is called the cotangent vector s p a c e of M at z and an element a E T,fM is called a t a n g e n t covector (or l-form) of M at z. Let f E Cm(M).Then the differential df(z) of f at z E M is a linear mapping
Since T f ( , ) R may be canonically identified with R , we may consider df(z) as a tangent covector at 2. Let u E T,M and u a curve in M such that u ( 0 ) = z and b(0) = u. Then we have
On the other hand, we have
But [f o u] is the tangent vector of R at f(z)defined by the curve f 0 u. If t denotes the coordinate of R , we obtain
29
1.9. Tangent and cotangent bundles
since a (illat),(,) E Tj(,)R is identified with a E R . Hence we deduce that
df(z)(u) = 4f). Now, let (U, z') be a local system of coordinates at z and consider the 1-forms (dz')(z) at z,1 5 i
+
.
Now, let V be an m-dimensional vector space. For a positive integer r, we call
T'V = V @ . . . @ V ( r - times) the contravariant tensor space of degree r . An element of T'V is called a contravariant tensor of degree r. T'V is V itself, and ToV is defined to be R. Similarly, T,V = V* @ . . . @ V * ( s times) is called the covariant tensor space of degree s. An element of T,V is called a covariant tensor of degree s. Then T1V = V * and ToV is defined to be R. Let { e l , . . . , e m } be a basis for V and { e l , . . . , e m } the dual basis for V * . Then {ei,
@ . . . @ e , , ; I 5 i l , ...,i, 2 m}
(resp. {& @ . . . a ei*;1 5 j 1 , . . . ,j, L m}) is a basis for T'V (resp. T,V). Then, if
K
E T'V (resp. L E T d V )we have
Chapter 1. Differential Geometry
34
where Kil-.ir(resp. Ljl...j,) are the components of K (resp. L). We define the (mixed) tensor space of type (r,s), or tensor space of contravariant degree r and covariant degree s as the tensor product
T,'V = T'V @ T,V = V 8 . .. @ V 8 V * @ . . . @ V * (V r-times and V* s-times). In particular, we have T,'V = T'V, T,OV = T,V, T,OV = TQV= TQV= R. It is obvious that the set {eil 8 . . . @ ei,
21 . . . a 2.;1 5 i l , . . . ,i,,jI,.. . ,j,5 m)
is a basis for T,'V. Then dim T,'V = mr+'. An element K E T,'V is called a tensor of type (r,s) or tensor of contravariant degree r and covariant degree 8 . We have . . K = K',l...treil 8 . . . @ e;, @ 2 1 8 . .. @ 31...I*
2*,
where Kfl...fr 11...3. are the components of K . For a change of basis we easily obtain
(1.2) E!; =
Aiej,
TV = @TiV. Then an element of TV is of the form K = Cr,,KJ, where KJ E T,'V are zero except for a finite number of them. If we define the product K @ L E T:::V of two tensors K E T,'V and L E T:V as follows:
1.10. Tensor fields. The tensorial algebra. Riemannian rnetrics
...*r+p ( K @ Q;:...;.,+q
-
K!i . . . I r Li;+1...lr+p J1...3.
Jm+l.,.j.+q'
35
(1.4)
a simple computation from (1.3) shows that (1.4) is independent on the choice of the basis { e i } . Then TV becomes an associate algebra over R which is called the tensor algebra on V . In TV we introduce the operation called contraction. Let K E T,'V given by (1.2) and (i,j)a pair of integers such that 1 5 i 5 r , 1 5 j 5 8 . We define the contraction operator Ci as follows: K is the tensor of type ( r - 1, s - 1) whose components are given by i l ...k . . . i , - l
( c ~ K ) ~ {= : : : ~K j l~. . .~k . . . j . - l
7
(1.5)
k
where the superscript k appears at the i-th position and the subscript k appears at the j-th position. (As above, (1.5) does not depends on the choice of the basis). Next we shall interpret tensors as multilinear mappings.
Proposition 1.10.5 T,V i s canonically isomorphic to the vector space of all s-linear mappings of V x . . . x V into R. Proof By generalizing Proposition 1.10.4, we see that T,V = V * @. . .@ V* is the dual vector space of T"V = V @ . . .@ V ,the isomorphism given by (u; 8 . . . @ a : ) ( b l @ . . . @ b a ) =
. . . < ba,a: > .
Now, from the universal factorization property, it follows that (T"V)*is isomorphic to the space of s-linear mappings of V x . . . x V into R. 0 If K = Kjl...j,ejl @ . . . @ ej, E T,V, then K corresponds to an s-linear mapping of V x . . . x V into R such that
Proposition 1.10.6 TiV i s canonically isomorphic to the vector space of s-linear mappings ojV x . . . x V into V . Proof We have TiV = V @ T,V. From Proposition 1.10.2, V @ T,V @ V . But T,V @ V CY Hom((T,V)*,V)CY Hom(T"V,V) by Proposition 1.10.3. By the universal factorization property, Hom(T"V,V ) can be identified with the space of s-linear mappings of V x . . . x V into V .0
T,V
Chapter 1. Differential Geometry
36
8.. .@eJ*E T,'V, then K corresponds to an s-linear If K = Kjl...j,eiQDejl mapping of V x . . . x V into V such that K ( e j , , . . . , eI .. ) = Kjl...j,e;. Now, let K E T,V (or T,'V). We say that K is s y m m e t r i c if for each pair 1 5 i ,j 5 s, we have
K ( o 1 , .. . ,
..
..
~ i , . , ~ j , . ,u8) =
K ( v 1 , .. . ,
.
.
~ j , . . , ~ i , .. ,u,).
Similarly, if interchanging the i-th and j-th variables, changes the sign:
then we say that K is skew-symmetric. We can easily prove that K is symmetric (resp. skew-symmetric) if
where u is a permutation of (1,.. . ,s) and cu denotes the sign of u.
Tensor fields on manifolds Definition 1.10.7 A tensor field K of type ( r , s) on a manifold assignement of a tensor K ( x ) E T,'(T,M) to each point x of M.
M is
an
Let ( U , z i ) be a local coordinate system on M. Then a tensor field K of type ( r , s ) on M may be expressed on U by
. . K = K:l-.s.ra/axil 31...3. a . . . QD a / a x i r c31 dxjl
. . . c31 dzj.,
where Kii:::i;are functions on U which will be called the c o m p o n e n t s of K with respect to (V,x'). We say that K is C" if its components are functions of class Coowith respect to any local coordinate system. The change of components is given by (1.3))where A$ = ( a Z i / a x J )is the Jacobian matrix between two local coordinate systems. From now on, we shall mean by a tensor field that of class C" unless otherwise stated.
1.10. Tensor fields. The tensorial algebra. Riemannian metrics
37
From Propositions 1.10.5 and 1.10.6, we can interpret a tensor field K of type (0,s) (resp. (1,s)) as a s-linear mapping
K :X(M) x ... x X ( M ) (resp.
-
Crn(M)
K : X ( M )x . . . x x ( M )
+
x(M))
defined by
We denote by r ( M ) the vector space of all tensor fields of type ( r , s ) on M. We note that r ( M ) is a CM(M)-module. Given two tensor fields A and B on M we may construct a new tensor field [ A ,B] given by
[ A , B ] ( X , Y= ) [AX,BY]
+ [ B X , A Y ]+ A B [ X , Y )
+ B A [ X , Y ]- A [ X ,B Y ] - A [ B X , Y ]- B [ X , A Y ] - B [ A X , Y ] ,
Then [ A , B ] is a tensor field of type (1,2)satisfying [ A , B ] = - [ B , A ] . We call [ A , B ]the Nijenhuis torsion of A and B (see Nijenhuis [loll).
Remark 1.10.8 A tensor field K of type (1,p) on a manifold M, p 2 1, is sometimes called a vector p f o r m on M.
Riemannian metrics Definition 1.10.9 A Riemannian metric o n M i s a c o v a r i a n t t e n s o r f i e l d g of degree 2 which satisfies: (1) g ( X , X) 2 0 and g ( X ,X ) = 0 if and only if X = 0 , and (2) g i s s y m m e t r i c , i.e., g ( X ,Y ) = g(Y,X ) , for all vector fields X,Y o n M . If g is a symmetric covariant tensor field of degree 2 which satisfies
(1)’ g ( X , Y ) = o for all Y implies X = 0 ,
38
Chapter 1 . Differential Geometry
then g is called a pseudo-Riemannian metric on M . In other words, g assigns an inner product g z in each tangent space T , M , z E M . If ( U , z i ) is a local coordinate system then the components of g are given by gij
= g(a/az',
a/axJ),
or equivalently,
We shall give an application which illustrates the utility of the partitions of unity.
Proposition 1.10.10 On any manifold there ezists a Riemannian metric. Proof: Let M be an m-dimensional manifold and { (U,,p,)} an atlas on M . There exists a partition of unity { ( U i ,f i ) } subordinate to {U,}. Since each Ui is contained in some U,, we set p i = p a / U i . Then we define a covariant tensor field g i of degree 2 on M by
for all X , Y E T z M , where < , > in the standard inner product on Rm. Hence g = Cigi is a Riemannian metric on M . 0
1.11
Differential forms. The exterior algebra
Algebraic preliminaries Let V be an m-dimensional vector space. We denote by APV (resp. SPV) the subspace of TpV which consists of all skew-symmetric (resp. symmetric) covariant tensors on V . Obviously, AoV = SoV = R , A'V = S'V = V * . We now define two linear transformations on TpV:
--
alternating mapping A : TpV symmetrizing mapping S : TpV
TpV TpV
39
1.11. Differential forms. The exterior algebra as follows:
where the summation is taken over all permutations a of ( 1 , 2 , . . . ,p). One can easily check that AK (resp. S K ) is skew-symmetric (resp. symmetric) and that K is skew-symmetric (resp. symmetric) if and only if AK = K (resp. S K = K). If w E APV and T E AqV, we define the exterior product w A r E Ap+qV by
Similarly, one can define the symmetric product of w E SPV and r E SqV as given by w 0 T = ( ( p + q ) ! / p ! q ! ) S (8 w 7)).
The proofs of the following propositions are left to the reader as an exercise.
Proposition 1.11.1 We have (w A a w l , .
a
a
9
%+q)
= E'eu
wbu(l),*
* *
,%(p))
,
+ J u ( p + l )*
* *
,%(p+q)),
where C' denotes the sum ower all shuffles, i.e . , permutations a of ( 1 , . . . ,p+ q ) such that a(1) < . . . < a ( p ) and a ( p 1) < . . . < a ( p q ) .
+
+
Proposition 1.11.2 The exterior product is bilinear and associatiwe, i.e.,
40
~1
Chapter 1. Differential Geometry From Proposition 1.11.2, we can writk w A A .. . Aw,. Let
tA
q , or, more generally,
AV = @ APV = R CEI A'V CEI A ~ CB V ... p=o
Then AV becomes an associate algebra over R , which will be called the exterior or Grassman algebra over R.
Proposition 1.11.3 If w E APV and wA
t
=
t
E AqV, then t A W.
Proof: This is equivalent to prove that A(w @ T ) = (- l)'*A(t @ w ) . To prove this we note that
-
c
1 (P+ Q)!
T(%(p+l),
- ..)u,(p+q)) w(uu(l), . . . ,vu(p)),
since (w @ T ) ( w ~.,. . ,wp+q) = w(w1,. . . , wp)t(wp+l,. . . , wp+q). Let a be the permutation given by
( 1 ) . .. , p + q ) Then we have
-
( P + 1 , . . . , P + Q, I , . . - ,PI*
1.11. Differential forms. The exterior algebra
=
A(' 63 W ) ( V l , .
41
. . ,up+q)
since E , = (- 1)PQand c, = cue,. Next we shall determine a basis for AV.
Proposition 1.11.4 If p > m, then APV = 0. For 0 5 p 5 m, d i m APV =
( ).
Let { e l ,... , e m } be a basis for V and { e l , . .. , e m } the dual
basis for V * . Then the set {e'l
A
.. . A e'p/1 5 il < i 2 < . . . < ip 5
m}
is a basis for APV.
Proof: If p > m, then p ( e i l , . . . ,ei,) = 0 for any set of basic elements; thus APV = 0. Suppose that 0 5 p 5 m. Since A maps TpV onto APV, the image of the basis { e i l 63 . . . 63 eip} for TpV spans APV. We have 1 ' A(e" 63 . . . €3 e'p) = -e'l
k!
A
. . . A eiP.
Permuting the order of il, . . . ,i, leaves the right side unchanging, except for a possible change of sign according to Proposition 1.11.3. Then the set {e'l A ... A e'P/1 5 i l < iz < . . . < ip 5 m} spans APV. On the other hand, they are independent. In fact, suppose that some linear combination of them is zero, namely
c
i l < ... 0, where ~i= Aie,. It is easy to check that there are exactly two equivalence classes. An orientation on V is a choice of one of these classes and then V is called an oriented vector space. We next show that this concept is related to the choice of a basis of AmV.
Proposition 1.16.1 Let w E A m V be a n m-form on V and let { e l , . . . ,em} be a basis of V . If { v l , . . . ,urn} i s a set of vectors with vi = Aie,, then w(v1,.
. . , v m ) = det(Ai) w(e1,. ..,ern).
The proof is left to the reader as an exercise.
Corollary 1.16.2 A non-zero w E ArnV has the same sign (or opposite sign) on two bases if they have the same (resp. opposite) orientation. Proof: It is straightforward from Proposition 1.16.1.0 Hence the choice of an m-form w # 0 (i.e., a volume form on V) determines an orientation of V given by all the bases {ei} of V such that
Chapter 1. Differential Geometry
62
w(e1,. . . ,em) > 0. In such a case {ei} is called a positively oriented basis. Moreover, two such forms w l and w2 determine the same orientation if and only if w2 = Awl, where X is a positive real number. To extend the concept of orientation to a manifold M we must try to orient each tangent space T,M in such a way that orientation of nearly tangent spaces agree.
Definition 1.16.3 W e say that a n m-dimensional manifold M is orientable i f there i s a n m-form w o n A4 such that w(x) # 0 f o r all x € M ; w i s called a volume form. Then any such w orients each tangent space. If w' = Xu, where X is a positive function on M , then w' gives the same orientation on M . Example R"' with the form w = dx' A . . . A dzm is an orientable manifold. The form w determines the standard orientation of Rm.
Definition 1.16.4 Let M I and M2 be orientable m-dimensional manifolds with volume f o r m s w1 and w2, respectively. W e say that a difleomorphism F : MI M2 preserves (resp. reverses) orientations i f F*w2 = Awl, where X > 0 (resp. X < 0 ) i s a f u n c t i o n o n M I . If F*w2 = w1, we say that F preserves volume forms.
-
Now, we interpret the orientability of a manifold in terms of local coordinates.
Proposition 1.16.5 Let M be a connected manifold. T h e n M i s orientable if and only i f A4 has a n atlas {(U,,p,)} such that the Jacobian (i.e., the determinant of the Jacobian matrix) of p g o pi' i s positive. Proof In fact, suppose that M is orientable with volume form w . We choose any atlas { (U,, pa)} by connected coordinate neighborhoods U,, with local coordinates x i , . . . ,x r such that w is locally expressed on U, by w = X,dx,
1
A
.. . A d x r , with A, > 0.
We may easily choose coordinates such that A, is positive on U,, since by (-zi, z i , . . . , changes the sign of A,. replacing ( z i , . . . , If U , A Up # 0, then we obtain
zr)
X p = A, det(ax:/dxi)
zr)
(see Proposition 1.16.1).
1.16. Orien table manifolds. integration. Stokes theorem
63
Since A, > 0, Ag > 0, we deduce that d e t ( a z h / a G ) > 0.
(1.8)
Conversely, suppose that M has an atlas as above. Let { U a , f a ) } be a subordinate partition of unity with respect to {(U,,pa)}. Since each Ua is contained on some U,, then {(Ua,pa)}, where pa = p , / U a is a new atlas of M satisfying (1.8). Define w E A m M by w=
C f a d e : A .. .A d z r , a
extending each summand to all of M by defining it to be zero outside the support of f a . Let z E M. Then we may choose a coordinate neighborhood (V, $J) of z with local coordinates (z', . . . ,zm) such that
for all a. Hence we have ~ ( z=)
C fa(z)dzAA .. . A d z r ( z ) a
=
1
fa(Z)
det(dz:/azJ)dz'
A
. . . A dzm(z).
a
Now, each f a ( z ) 2 0 and at least one of them is positive at z. Moreover det(azh/azj) > 0. Hence ~ ( z #) 0.0
Definition 1.16.6 Let M be an orientable m-dimensional manifold with volume form w . A coordinate neighborhood (U,z') is called positively oriented if dz' A .. . A d z m and W / ( I give the aame orientation. Clearly, (U, z') is positively oriented if and only if w / v = f dz' A . , .Adzm, where f > 0 is a function on U. Now, let M be an orientable m-dimensional Riemannian manifold with volume form w . We define a natural volume form R on M. Let {XI,. . . , X m } be an orthonormal basis of T,M such that u(X1,. . . , X m ) > 0. We define an m-form fl by
R(z)(Xt,.. . ,Xm) = 1.
64
Chapter 1. Differential Geometry
If {Yl, . . . ,Y}, is another orthonormal frame at z such that w(Y1,.. . ,Ym) > 0, then y( = AiXj. Hence, from Proposition 1.16.1, we have
R(z)(Yl,.. , ,I'm) = det(Ai)R(XI,. . . , X m ) = 1, since det(Ai) = 1 (in fact, (A:) E O(m) and then I det(A;) I= 1, but since {XI,..., Xm} and { Y l , . , . Y,} , have the same orientation, one has det(Ai) > 0). Thus, n(z) is independent of the orthonormal frame chosen. Moreover, if ( z ' , . . . ,zm) is a positively oriented coordinate system, we have
so that
Hence
Since det B > 0, we obtain
fi= det B and thus
n(a/azl,. . . ,a / a x m ) =
a,
i.e.,
Next, we shall define the integral of an m-form on an rn-dimensional manifold. Let us recall that if f : Rm + R is continuous and has compact support then
1,16. Orien table manifolds. Integration. Stokes theorem
65
is defined as the Riemann integral over any rectangle containing the support of f. Moreover, suppose that G : Rm + R"' is a diffeomorphism given by G(z',
. . . , z")
-
. . ,y"(z')).
= (y'(zi),.
Suppose that f ' : Rm R is the function defined by f = f'o G. Then f ' has compact support and
1
f'(y',
=
/
f (z', . . .
)
2")
. . . ,ym)dy'
...dym
I det(ay'/ad) I dz'
. . . dxm
(1.9)
(1.9) is known as the change of variables rule. Now, suppose that (U, z') is a positively oriented coordinate neighborhood of an orientable m-dimensional manifold M with volume form w .
Definition 1.16.7 Let a be an m-form on M with cornpact support. If a is locally given by a = fdz' A
.. . A dz"
on U, we define
(We notice that f has also compact support).
Remark 1.16.8 If (V, y') is another positively oriented coordinate neighborhood such that supp a c U n V , and a
then from (1.9) we have
= gdy' A . . . A dy",
Chapter 1 . Differential Geometry
66
Now, suppose that a! is an arbitrary m-form on M. Let A = {(U,,pa)} be an atlas of positively oriented coordinate neighborhoods and { ( U i , hi)} a partition of unity subordinate to A. Define ai = hia. So a!i has compact support contained in some U,. Then we define (1.10) It is not hard to prove that' (1.10) does not depend on the choice of A or the partition of unity {hi}.
Definition 1.16.9
JM Q
i s called the integral of
Proposition 1.16.10 (1) If - M with opposite orientation, then
Q
on M.
denotes the same underlying manifold
(2)
aa
+ bP = a IM + b I M P , a!
-
for all a, b E R and a,P E A ~ Mwhere , (Y and P have compact support. (3) If F : MI M2 i s a diffeomorphism and a i s a m-form on M2 with compact support, then
with sign depending on whether F preserves or reserves orientations.
The proof is left to the reader as an exercise. We now introduce the concept of manifold with boundary. Let Hm = {z = ( d , .. . ,zm) E Rm/zm 2 0 ) with the relative topology of Rm and denote by aHm the subspace defined by aHm = {x E Hm/xm = 0 ) ; aHm is called the boundary of Hm. Obviously, aHm is homeomorphic to Rm-l by the map (xl,. . . ,zm-l) ( d , . . ,zm-l,O). We notice that differentiability can be defined for maps to R" of arbitrary subsets of Rm in the obvious way. If F : A c Rm R" is a map defined on a subset A of Rm, we say that F is C" on A if, for each point x E A, there exists a C" map F, on an open neighborhood U, of
-
-
1.16. Orientable manifolds. Integration. Stokes theorem
67
z such that F = F, on A n U,. Then the notion of diffeomorphism a p plies at one to open subsets U,V of H"; namely U,V are diffeomorphic if there exists a bijective map F : U V such that F and F-' are both Coo.If U , V c Rm - a H m , then U and V are actually open subsets of Rm and this definition coincides with our previous one. On the other hand, if U n a H m # 0, then V n dHm# 0 and F(U n a H m ) c V n a H m . Similarly F-'(V n a H m ) c U n a H m . In other words, F maps boundary points to boundary points and interior points to interior points. We also notice that U n a H m and V n dH" are open subsets of aH", a submanifold of Rm diffeomorphic to P-',and F, F-' restricted to U n a H " and VnaH" are diffeomorphisms. Moreover, F and F-' can be extended to open subsets U',V' of Rm such that U = U'n Hm and V = V' n H". These extensions will not be unique nor are the extensions in general inverses. However, the differentials of F and F-' on U and V are independent of the extensions chosen and we may suppose that even on the extended domains the Jacobians are of rank m.
-
Definition 1.16.11 A C" manifold with boundary i s a Hausdorffspace M with a contable basis of open sets and a C" differentiable structure A = { ( U a , p a ) }in the following sense: U, i s an open subset of M and pa i s a homeomorphism from U, onto an open subset of H m such that: (1) the U, cover M; (2) for any a,@ the maps p p o pi' and pa o (pa' are difleomorphisms of P a ( k n Up) and Pp(U, n Up); (3) A i s maximal with respect to properties (1) and (2). (Compare with Definition 1.2.3).
The (U,,pa)are coordinate neighborhoods on M . From the remarks above we see that if p(z) E 8 H m in one coordinate system, then this holds for all coordinate systems. The collection of such points is called the boundary of M , denoted a M , and M - a M is an m-dimensional manifold (in the ordinary sense), which we denote by Int M . If dM = 8, then M is a manifold in the ordinary sense; we call it a manifold without boundary when it is necessary to make the distinction.
Proposition 1.16.12 If M i s a manifold with boundary, then the differen-
-
tiable structure of M determines a differentiable structure of dimension m- 1 on aM such that the inclusion i : a M M i s an embedding.
68
Chapter 1. Differential Geometry
Proof: In fact, the differentiable structure A on d M is determined by the coordinate neighborhoods (0,p), where D = U n d M , 8 = ' p / v n a ~for any coordinate neighborhood (U, p) of M which contains points of d M . 0 Hence, if M is a manifold with boundary, then Int M = M - d M and a M are manifolds of dimension m and m - 1, respectively. Next we define orientation on a manifold with boundary. Definition 1.16.3 extends in a natural way to the case of manifolds with boundary. Namely, a manifold with boundary M is orientable if there exists an mform w on M such that ~ ( z #) 0 for all z E M . The reader may prove that Proposition 1.16.5 holds for manifolds with boundary. Let us recall that an orientation on M is a choice of orientations of all the tangent spaces in such a way that for all positively oriented coordinate neighborhoods, the maps ' p p op-'a : pa(U,nUp) ----t p p (U,nUp) "preserves the natural orientation" of H m ,i.e., 'pg op-' has positive Jacobian. Thus, we can define an induced orientation on d M as follows. At every z E d M , T , ( d M ) is an (m 1)-dimensional vector subspace of T,M so that there are, in a coordinate neighborhood intersecting d M , exactly two vectors perpendicular to zm = 0; one points inward, the other outward. We say that a basis { e l , . . . , e r n - l } for T , ( a M ) is positively oriented if { - d / d z m , e l , . . . ,e m - l } is positively oriented with respect to the orientation on T,M.
-
Theorem 1.16.13 (Stokes' Theorem) Let M be a manifold with boundary and a E hm-lM with compact support. Let i : a M M be the inclusion map and i*a E A m - l ( a M ) . Then
or for sake of simplicity
Proof: Since both sides of the equation are linear, we may assume without loss of generality that a is a form with compact support contained in some coordinate neighborhood U with local coordinates z l , .. . ,zm. There are two cases: U n a M = 0 and U n a M # 0 . We set m
a = C(-l)ia,dz'A
. . . A dZ'
A
. . .A d x m ,
i=l
where d35i means that this differential is omitted in the expression. Then
1.16. Orien table manifolds. Integration. Stokes theorem
m
d a =C(aai/ax')dz' A i=l
69
. .. A d z m
Thus
k
m
da = i=l
1
( a a ; / a x ' ) d x ' . . . dxm.
Rm
If U n a M = 0,we have
I,,
a = 0.
On the other hand, the integration of the ith term in the sum ocurring in JIM d a is
/,-,
[IR(2)
dx'] dx'
. . . d3' . . .dxm (no sum!)
(1.11)
But since a;has compact support we have
Thus Stokes' theorem holds for this case. If U n 8 M # 0, then we again have all the integrals in (1.11) equal to zero, except the one corresponding to i = m, which is
Lm-l[/R (2)
dxm] dx'
since amhas compact support. Thus
. . . dzm-l
Chapter 1. Differential Geometry
70
On the other hand, the local expression of i*a in the local coordinates zl,. . . ,zm-l, obtained by restriction of zl,. . . ,zm t o U n a M is
i*a = ( - l ) m - l ~ m ( z l.,..,zm-',O)dzl
A
.. . A dzm-',
since i*dxm = 0. Now, the basis {a/azl,. . . ,a/az"-'} is not positively oriented, since the outward unit normal vector is -a/azm in each point of U n a M . Thus L M
am(z1 ...zm-l ,0)dz'
;** = (-1)2m-l
. . . dzm-l
Lm-1
a/dz1,.
because the sign of {-a/az", the proof. 0
. . ,a/az"-')
is (-l)m.This ends
Remark 1.16.14 From the proof of Stokes theorem, we deduce that if M is a manifold without boundary, i.e., a M = 0, then we have da = 0. /M
Example (Green's Theorem) Let M be the closure of a bounded open subset of R2 bounded by simple closed curves (for example, let M be a circular disk or annulus). Then d M is the union of these curves (in the examples mentioned above, a M is a circle or a pair of concentric circles). If a is a 1-form on M, then LY
where
(2,
= f dz
+ gdy,
y ) are the canonical coordinates on R 2 . Then d a = [ ( d g / a z ) - ( a f / a y ) ] d zA d y .
By Stokes' theorem, we have
But if we cover a M with positively oriented coordinate neighborhoods, we deduce that the right-hand of (1.12) is the usual line integral along a curve C (or curves Ci) oriented so that as we transverse the curve the region is on the left. Thus (1.12) becomes
which is the Green's theorem.
1.17. de Rham cohomology. Poincarh lemma
1.17
71
de Rham cohomology. Poincare lemma
Definition 1.17.1 A p-form CY on a manifold M is said to be closed if d a = 0. It i s said to be exact if there i s a (p-1)-form /3 such that a = d p . Since d2 = 0 then every exact form is closed. Let Z k ( M ) denote the closed p f o r m s on M; since Z k ( M ) is the kernel of the linear map d : ApM + A p + l M , it is a vector subspace of APM. Similarly the exact p-forms Bp(M) are the image of d : AP-'M + A ~ M and then a vector subspace of APM. Since B p ( M ) c Z P ( M ) , we can form the quotient space
HP(M) = ZP(M)/BP(M), which is called the pthde Rham cohomology group of M. If dim M = m, then
H P ( M )= 0 when p > m. If we set m
H*(M) = @HP(M), p=o
then H * ( M ) is a vector space which becomes an algebra over R with the multiplication being that naturally induced by the exterior product of forms, i.e., [ w ] - [ T ] = [w A
HP(M), [r]E H q ( M ) .
T ] , [w] E
H * ( M ) is called the de Rham algebra of M is called the cup-product). Now, let F : M + N be a Cm map. Then the algebra homomorphism F* : AN AM commutes with d and hence maps closed forms to closed
-
( v
forms and exact forms to exact forms. Thus it induces a linear map
F* : H P ( N )
-+
HP(M)
given by
F * [ a ]= [ F * a ] ,[a]E H P ( N ) .
-
Moreover, we have an algebra homomorphism
F* : H * ( N )
H*(M).
The reader can easily check the following:
72
-
Chapter 1. Differential Geometry
Proposition 1.17.2 (1) If F : M M is the identity map, then it induces the identity on the de Rham cohomology, i . e . , F* = i d . (2) Under composition of maps we have (G o F ) * = F* o G * .
Corollary 1.17.3 A diffeomorphism F : M on the de R h a m cohomology.
-
N induces isomorphisms
Thus two diffeomorphic manifolds have the same de Rham cohomology groups. In other words, the de Rham cohomology is a differentiable invariant of a differentiable manifold M . In fact, the de Rham theorem proves that the de Rham cohomology is actually a topological invariant, i.e., the de Rham cohomology groups depend only on the underlying topological structure of M and do not depend on the differentiable structure. The reader is referred to Warner [123] for a proof of the de Rham theorem. Furthermore, if M is compact then the de Rham cohomology groups are vector spaces of finite dimension. The dimension bp of H p ( M ) is called the p t h Betti number of M and it is a topological invariant from the de Rham theorem.
Proposition 1.17.4 Let M be a connected manifold. Then H o ( M )= R . Proof A0M consists of Coo-function on M and Z o ( M ) of those functions f for which df = 0. But df = 0 if and only i f f is constant (see Exercise 1.22.10). Since B o ( M ) = 0, then H o ( M )N Z o ( M ) + R . 0 Next we shall prove the Poincark lemma.
-
Proposition 1.17.5 (Poincard Lemma) For each p 2 h there is a linear AP-lRm such that map h, : APRm
+ d o h, = i d .
hp+l o d
Proof: Let ( d ,, . . ,z m ) be the canonical coordinates in Rm. Consider the vector field m
x = 1s'(a/az') i=l
on Rm. We define a linear map A, : APRm
+ AP
Rm by
1.1 7. de Rham cohomology. Poincar6 lemma
A,(fdz"
A
. . . A d z i p ) ( z )=
73
(11
tP-'f(tz)dt) dzil A
.. . A d z i p ( z )
for all z E Rm and then we extend it linearly to all M R m . We have
(A, o L x ) ( f d z i lA
= A,[(pf
+ f Lx(dz''
.. . A dZ'P
= A,[(Lxf)dz" A
.. . A d z i P ) ( z ) A
...A dzip)](z)
+ x z i ( a f / a z i ) ) d z i Al . . . A dz'P](z) i
(since Lxdz' = dz')
=
[11
tP-'(pf(tz)
1
+ x z ' ( t z ) ( a f / a z i ) , z ) d t d z i l A .. . A d z i P ( z ) i
=
[11
1
$ ( t P f ( t z ) d t dzil
= f(z)dz" A
A
.. . A dziP(z)
. . . A dZip(z).
Hence
A,
0
LX = idhpRm
(1.13)
Moreover, A commutes with d , i.e.,
A,
0
d = d o Ap-l.
In fact,
((A, o d)(fdz"
A
.. . A d & - ' ) ) ( z )
(1.14)
Chapter 1. Differential Geometry
74
dz'
=d
(l1
tp-2
A
dz"
A
. ..A dziP-l(z)
. . . A dz'p-l(z)
f (tz)dt)d d l A
(since d and J commutes) = ( d 0 A p - l ) (f dZil A
Since L x = i x d
. . . A d&"')(z).
+ d i x , from (1.13) and (1.14))we obtain
= A,ixd
+ dA,-lix.
Now we set
h, = Ap-l
0
ix
Thus we obtain
h,+l
o
d + d o h, = A ,
Corollary 1.17.6 If a is i s exact. Proof We set
Hence
oix o
d
a p-form, p
+d
0
2 1,
A,-1 on
0
i x = idApRm.0
R" which is closed, then a
I .18. Linear connections. Riemannian connections
75
Corollary 1.17.7 The de Rham cohomology groups of R" are all zero for P > 1.
Since H o ( R m ) = R , we have computed all the de Rham cohomology groups of R". Moreover, if U is the open unit ball in R", since U and Rm are diffeomorphic, we deduce, from Corollary 1.17.3, that
H P ( U )= O for p 2 1. From Corollary 1.17.6, we deduce that every closed p f o r m on a manifold M is locally exact. In fact, let a be a p f o r m on an m-dimensional manifold M such that da = 0. For each point x E M there exists an open neighborhood ( U , p) such that p ( U ) is the open unit ball in Rm. Then (p-')*cr is a closed p-form on p(U) and hence (p-')*c~ = dp, where p is a ( p - 1)-form on p ( U ) . Therefore a = d(p*P) on U.
1.18
Linear connections. Riemannian connections
In this section we introduce the concept of linear connection. Further, we shall see that any Riemannian manifold possesses a unique linear connection satisfying certain conditions. In chapter 4, we shall generalize the notion of linear connection. Definition 1.18.1 A linear connection on an m-dimensional manifold M is a map that assigns to each pair of vector fields X and Y o n M another oector field V x Y such that: (1) VXl+X2Y= v x , y v x , y ,
+
+
(2) VX(Y1 Y2) = VXYl+ vxy2, (3) VfXY = f ( V x Y ) , (4) V X ( f Y )= f (VXY).
+
Remark 1.18.2 We notice that, if X1,X2,Y1 and Y2 are vector fields on M and if X1 = X2 and Y1 = Y2 in a neighborhood of a point x E M I then (VxlY1)2= ( V X , Y ~ This ) ~ . implies that V induces a map x ( U ) x x ( U ) x ( U ) satisfying (1)-(4)) where U is an open set of M .
-
Now, let U be a coordinate neighborhood of M with local coordinates ( x l , . . . ,x"). Then we define m3 functions I'fj on U by
Chapter 1. Differential Geometry
76
I'fjare called the Christoffel components of V. Let U be another coordinate neighborhood with local coordinates (3.'). On U n U we have
a/az'
= (ad/az')(a/axJ).
Hence, the transformation rule for the r's are:
+(a zk/a%") (a2% p / a 3' a 31). Then the I"s are not the components of a tensor field on M . (This is a consequence of (4)). Now, let X , Y E x ( M ) . Then we locally have
vXy= ( x j ( a y i / a x j+) r:kxjyk)(a/ad), where X = X'(a/ax') and
Y
(1.15)
= Y'(a/dx').
Remark 1.18.3 From (1.15), we deduce that ( V x Y ) ( x )depends only on X(4. Definition 1.18.4 Let u : R + M be a curve and X a vector field on M. We define the covariant derivative of X along u b y
DX/dt = V & ( t ) X . (From the Remark 1.18.3, we deduce that DX/dt is well-defined). We say that X i s parallel along u i f DX/dt = 0. We say that u ia a geodesic of V i f u ( t ) i s parallel along u, i.e., vau = 0.
(As above, V b u is well-defined, since DX/dt depends only on the values of X along a; then it is suficient to extend k ( t ) t o an arbitrary vector field on an open neighborhood of u ) .
1.18. Linear connections. Riemannian connections
77
In local coordinates we have
+
D x / d t = ( ( d x k ( t ) / d t ) r,:(t)x'(t)(dzi/dt)(t))(a/azL), and
where a ( t ) = (z'(t)),t5(t) = ( d z ' / d t ) ( a / a z ' ) . Hence, a is a geodesic of V if and only if it satisfies the following system of linear differential equations:
d2zk/dt2
+ I ' ! j ( d z ' / d t ) ( d z j / d t ) = 0 , 1 , X , Y E T,M,
-
where u is a linear frame at z, u E B q , ) ( M ) (here u is considered as a linear isomorphism u : Rm T,M). The invariance of < , > by O(m) implies that g,(X,Y) is independent of the choice of u E B o ( , ) ( M ) . To prove that g is Ccoit is sufficient to consider local sections of Bo(,)(M).
1.21. G-structures
103
Conversely, let M be a Riemannian manifold with Riemannian metric g. We set
We notice that a linear frame u = (XI,. . . ,Xm)at x belongs to O ( M ) if and only if { X I ,. . . ,Xm} is an orthonormal basis of T,M with respect to g z . It easily follows that a ( O ( M ) )= M and ( T / C I ( M ) ) - ' ( Z ) = uO(m), z E M , z = ~(u).Moreover, for each x E M , we can choose a neighborhood U of z and a frame field s = (XI,. . . ,Xm)on U such that { X , ( y ) , . . . , X m ( y ) } is an orthonormal basis of T,M for all y E U . In fact, we start with an arbitrary frame field {YI,. .. ,Ym} on a neighborhood W of x and, by the usual Gramm-Schmidt argument, we obtain {Xi, ...,Xm}on U ,U C W . From Proposition 1.21.3 we deduce that O ( M ) is an O(rn)-structure on M . O ( M ) is called the orthonormal frame bundle of M and an element u E O ( M ) is called an orthonormal frame. Thus giving an O(m)-structure on M is the same as giving a Riemannian metric on M . ( 3 ) In the next chapters we consider more examples of G-structures: Almost tangent structures, almost product structures, almost Hermitian structures, almost contact structures and almost sy mplect ic structures.
-
Definition 1.21.7 (1) Let f : M MI be a local difleomorphism. T h e n f induces a m a p F f : F M FM' as follows. If u = (Xi,. . . ,Xm),d i m M = d i m MI = m, i s a linear f r a m e at z E M I t h e n F f ( u ) i s the linear f r a m e at f (x) E M' given by F f (u) = (df (.)Xi,. . . ,df (.)Xm). F f i s called the natural lift of f . O n e can easily checks that F f is a principal bundle homomorphism. (2) L e t B G ( M ) and &(MI) be G-structures o n M and MI, respectively. MI be a difleomorphism. W e say that f i s a n isomorphism Let f : M of & ( M ) o n t o &(MI) if
-
If M = MI and & ( M ) = &(MI), t h e n f i s called a n automorphism of B G ( M ) . (3) Let & ( M ) and &(MI) be G-structures on M and MI, respectively. W e s a y that B G ( M ) and BG(M') are locally isomorphic i f for each pair (z,~') E M x MI, there are open neighborhoods U of x and U' of X I a n d a U' such that ( F f ) ( & ( M ) / v )= B G ( M ' ) / ~ . local difleomorphism f : U
-
Chapter 1. Differential Geometry
104
called a local isomorphism o f B c ( M ) onto BG(M'). I f M = M' and & ( M ) = &(MI), then f i s called a local automorphism.
f is
Examples: (1) If G = O(m), then a diffeomorphism f : M and only if
-
MI is an isomorphism if
g;(,)(~f(z)X,df(~)y) = g z ( K Y),
X,Y
E T,M, where g and g' are the corresponding Riemannian metrics on M and M ' , respectively. f is called an isometry. (2) If G = Sp(m), then a diffeomorphism f : M M' is an isomorphism if and only if
-
f *wI = w , where w and w' are the almost symplectic forms on M and MI, respectively. If w and w' are symplectic, then f is an isomorphism if and only if f is a symplectomorphism (see Section 5.2). Now, let FR'" be the frame bundle of the Euclidean space R" and (z', . . . ,z") the canonical coordinate system of Rm. Hence R" possesses an {e}-structure given by the global frame field s : 2 E R"
-
s(2) = ((a/az'),,
. . . (a/az"),). )
Moreover, we obtain a principal bundle isomorphism
FR"
-% Rm x Gl(m, R)
defined by
$(4= (2,(Xi')), where u = ( X I , .. . ,Xm) is a linear frame at x and Xi = X:(a/azi),. Thus if G is an arbitrary subgroup of Gl(m, R ) we obtain a G-structure &(Rm) on R" by setting
Bc(R") = +-'(R" x G). In fact,
&(P) is obtained by the group enlarging
{ e } + G , i.e.,
&(Rm) = { B ( z ) u / x E R", a E G } . This G-structure & ( Pis ) called the standard G-structure on
Rm.
I . 22. Exercises
105
Definition 1.21.8 A G-structure B c ( M ) on an m-dimensional manifold M is said to be integrable if it is locally isomorphic to the standard Gstructure BG(R") on Rm. It is easy to check the following.
Proposition 1.21.9 A G-structure B c ( M ) on M is integrable if and only if there is an atlas { (Ua) xi).. . x:)} such that f o r each U, the local frame ( ( d / d ~ k ). .~. , ( d / d ~ r ) takes ~ ) its values in B c ( M ) . field z
-
)
Examples (1) If a Riemannian structure is integrable then the Riemannian connection is flat. The converse is also true (see Fujimoto [57]). (2) In the next chapters we obtain necessary and sufficient conditions for integrability of many examples of G-structures.
1.22
Exercises
1.22.1 Let F : N + M be a C" map and suppose that F ( N ) c A , A being an embedded submanifold of M . Prove that F is C" as a map from N to A. 1.22.2 Let X be a vector field on a manifold M and pt a l-parameter group of local transformations generated by X. Prove that if pt(z) is defined on ( - 6 ) 6 ) x M for some 6 > 0, then X is complete. 1.22.3 Define a (global) l-parameter group of transformations pt on R2 by p t ( z , y ) = ( z e t , y e - ' ) , t E R.
-
Determine the infinitesimal generator. 1.22.4 (1) Let p be a transformation of M and T p : T M T M the vector bundle isomorphism defined by Tp(u) = dp(z)(u), u E T Z M .Show that if X is a vector field on M , then Tp(X) defined by Tp(X)(p(z))= dp(z)(X(z)) is a vector field on M . (2) Suppose that X generates a local l-parameter group of local transformations pot. Prove that the vector field Tp(X) generates p o pt o p-'. (3) Prove that X is invariant by p, i.e., Tp(X) = X, if and only if p commutes with pt. 1.22.5 Let p : E M be a fibred manifold. Prove that there exists a global section s of E over M ( H i n t : use partitions of unity).
-
Chapter 1. Differential Geometry
106
1.22.6 Let p : E + M be a vector bundle. A metric g in E is an assignement of an inner product g , on each fiber E,, z E M , such that, if s1 and 5 2 are local sections over an open set U of M , then the function g(s1,s2) : U R defined by
-
is differentiable. Prove that a metric in T M induces a Riemannian metric on M , and conversely. 1.22.7 Let X and Y be two vector fields on a manifold M and pt the 1parameter group of local transformations generated by X . Show that
K Y I ( 4 = !$p/t)[Y(4
-
(TPt)Y))(41,
for all z E M . (The right-hand of this formula is called the Lie derivative
of Y with respect to X and denoted by L x Y ) . 1.22.8 Let F : M --t N be a Coomap. Two vector fields X on M and Y on N are called F-related if d F ( z ) X ( z ) = Y ( F ( z ) )for all z E M . Prove that if X1 and X2 are vector fields on M F-related to vector fields Y1 and Y2 on N , respectively, then [ X I X , z ] and [Yl,Yz] are F-related. 1.22.9 Let X be a vector field on a manifold M of dimension m and z E M . Prove that if X ( z ) # 0, then there exists a coordinate neighborhood U of z with local coordinates xl,.. . ,zm such that X = a/dzl on U. 1.22.10 Let M be a connected m-dimensional manifold and f a function on M . Prove that df = 0 if and only if f is constant on M . 1.22.11 Consider the product manifold MI x M2 with the canonical projections 9r1 : M I x M2 M I and T Z : M I x M2 M2. Prove that the map
-
-
-
T(Sl,S2)(M1 x Mz)
-
T~~M @ Tz9M2 I
defined by u ( d 9 r 1 ( ~ 1 , 2 2 ) ud9r2(q,z2)u) , is a linear isomorphism. 1.22.12 Let a and p closed forms on a manifold M . Prove that a A closed. If, in addition, p is exact, prove that a A p is exact. 1.22.13 Let a=-
p is
1 zdy-ydz
29r
2 2 + y2
*
Prove that a is a closed l-form on R2 - {(O,O)}. of a over S1 and prove that a is not exact.
Compute the integral
1.22. Exercises
107
-
1.22.14 (1) Let V be an oriented vector space of dimension m and F : V V a linear map. Prove that there is a unique constant det F , called determinant of F such that F*w = ( d e t F ) w , for all w E AmV. (2) Show that this definition of determinant is the usual one in Linear Algebra. (3) Let M I and M2 be two orientable m-dimensional manifolds with volume forms w1 and w2, respectively. Show that if F : M I M2 is a Coomap, then there exists a unique Coofunction det F on M I , called determinant of F (with respect t o w1 and w2) such that F*w2 = ( d e t F ) w l . (4) Let M be an orientable m-dimensional manifold with volume form w. Prove the following assertions: (i) If F , G : M + M are CM maps, then
-
det(FoG)= [(detF)oG](detG). (ii) If H = i d M , then det H = 1. (iii) If F is a diffeomorphism, then
det F-' =
1 (det F ) o F-'
(Here, all the determinants are defined with respect to w ) . 1.22.15 Prove that the de Rham cohomology groups of S' are: H o ( S ' ) = R, H ' ( S ' ) = R and H P ( S ' ) = 0 f o r p > 1.
1.22.16 Let M be a compact orientable m- dimensional manifold without boundary. Prove that H m ( M )# 0. 1.22.17 Let M be a Riemannian manifold with Riemannian metric g. Prove that an arbitrary submanifold N of M becomes a Riemannian manifold with the induced Riemannian metric g' defined by g',(X,Y) = g,(X,Y), for all z E N and X, Y E T, N c T,M. (Thus Sm is a Riemannian manifold with the induced Riemannian metric from Rm+'). 1.22.18 Let M be an m-dimensional Riemannian manifold with Riemannian metric g and curvature tensor R. The Riemannian curvature tensor, denoted also by R, is the tensor field of type (0,4) on M defined by
Prove that R satisfies the following identities: (1) R(X1, x2 ,XS, x4) = -R(X2, x1,x3,x4)-
108
Chapter 1. Differential Geometry
(2) R(X1)x2 x3 x4) = -R(Xl>x2 9 x4 x3)* (3) R(Xl)X2,X3,X4) R(XI,X3,X4,X2) R(Xl,X4,X2,X3)= o * (4) R(Xl,XZ,X3,X4)= R(X3,X4,Xl,X2). Now, let A be a plane in the tangent space T,M, i.e., R is a 2-dimensional vector subspace of T,M. We define the sectional c u r v a t u r e K ( A )of R by )
j
+
+
K(.lr) = R(Xl,X2,Xl,X2), where {X1,X2} is an orthonormal basis of R. Proves that K ( R )is independent of the choice of this orthonormal basis, and that the set of values of K ( A )for all planes R in T,M determine the Riemannian curvature tensor at z. 1.22.19 Let M be a Riemannian manifold with Riemannian metric g and curvature tensor R. We define the Ricci tensor S of M by m
S(X)Y ) =
C R(ei) y,ei, x), i=l
where { e l , . . . ,em} is an orthonormal frame at z. (1) Prove that S(X, Y )does not depends on the choice of the orthonormal frame {ei}. (2) Prove that S is a symmetric tensor field of type (0,2)on M . (3) Prove that in local coordinates we have m ..-
R,j
=
C Rkj, k=l
where R,, denotes the components of S. (4) Prove that 7(z) = S(e1,e l ) . . . S(e,,e,) does not depends on the choice of the orthonormal frame {ei} at z;7(z) is called the scalar c u r v a t u r e at z. ( 5 ) Prove that in local coordinates we have
+
+
-
where (9'') is the inverse matrix of (gij). 1.22.20 Let GI be a Lie subgroup of a Lie group G and F : A4 -+ G a C" map such that F take values in GI. Prove that F : M G' is also C". 1.22.21 Let G be a Lie group with Lie algebra g. We define a canonical g-valued l-form 8 on G by
I .22. Exercises
109
e,(x)= C ~ - ~ ( T I ~ - ~ ( X ) ) , for all X E T,G. Prove that O(A) = A for all A E g . 1.22.22 (1) Show that each element of GZ(m, R) has a polar decomposition, that is, each matrix K E Gl(m, R) can be expressed in the form
K = RJ, where R is a positive definite symmetric matrix and J E O(m). (Recall that a symmetric matrix R is positive definite if each of its (real) eigenvalues is strictly positive). (2) Let E be a real vector space of dimension m and any inner product on E. For each R E Aut(E) we define the transpose Rt of R by < @x,y >=< x, Ry >. We say that R is symmetric if @ = R. If R is symmetric then all the eigenvalues of R are real, and that R is positive definite if each of its eigenvalues is strictly positive. An element J of Aut(E) is orthogonal if < Jx,J y >=< x,y >. (Equivalently, if {el,. . . ,em} is a basis for E and is the matrix given by Re, = then R is symmetric (resp., positive definite, orthogonal) if and only if (4)is symmetric (resp., positive definite, orthogonal)). Prove that each element K of Aut(E) has a polar decomposition K = RJ,where R is positive definite symmetric and J is orthogonal. 1.22.23 Let G be a Lie group with Lie algebra g. For a fixed A E g , define expA : R G by expA(t) = exptA. Prove that expA is a Lie group homomorphism such that the tangent vector to the curve expA at t = 0 is precisely A( e) . 1.22.24 Let G be a Lie group with Lie algebra g and X,Y E g. Prove that
(q.)
q.ej,
-
(exptX)(exptY) = exp{t(X
t2 + Y )+ -[x,Y] + o(t3)}. 2
1.22.25 Prove that the unitary group U(m) is compact (Hint : U(m) is closed and also it is bounded in Gl(m, R)). It follows that SU(m), O(m) and SO(m) are also compact. 1.22.26 Prove that if F : G H is a Lie group homomorphism then: (i) F has constant rank; (ii) the kernel of F is a Lie subgroup; and (iii) dim Ker F = dimGI - rank F. 1.22.27 Let P ( M , G ) be a principal bundle over M with structure group G and projection A . Assume that G acts on R" on the left. Then G acts on the product manifold P x R" on the right as follows:
-
Chapter 1. Differential Geometry
110
(.,()a = (ua,a-'(), u E P, ( E R", u E G .
- -
This action determines an equivalence relation on P x F. We denote by E the quotient space and by p : E M the map defined by p[u, (1 = ~ ( u ) . (1) Prove that p : E M is a vector bundle of rank n which is said to be associated with P(Hint : If U is an open set of M such that rl, : T - ' ( U ) U x G is a local trivialization of P , then we define J :p-'(U) U x R" as follows. Given [ u , o E p-'(U), then we have $(u) = ( 2 , ~ ) .Thus we define $[u, (1 = (.,a() and : U x R" p-'(U) is a local trivialization of E ) . (2) Let {U,} be an open covering of M and, for each U,, let rl,, : x-'(Ua) U, x G be a local trivialization of P. If {$aa} are the transition functions corresponding to {ua}, prove that
-
4-l
-
- -
(The maps rl,~, are also called the transition functions of E corresponding to {Ua}). (3) If P = F M and the action of Gl(m, R), m = d i m M , on P is the natural one, i.e., a ( E Rm is the image of ( E Rm by the linear isomorphism a : Rm --+ Rm, then prove that the associated vector bundle with FM is precisely T M , the tangent bundle of M . (4) Conversely, let p : E M be a vector bundle of M . Let P be the set of all linear isomorphisms u : R" E, in all points x of M . Define T : P M by ~ ( u = ) x. Prove that: (i) P is a principal bundle over M with structure group Gl(n, R) and projection T ; and (ii) E is associated with P. 1.22.28 Let G' be a Lie subgroup of a Lie group G. Prove that if M possesses a G'-structure then it possesses also a G-structure. 1.22.29 Let M be a Riemannian manifold with Riemannian metric g. A vector field X on M is called a Killing vector field if Lxg = 0. Prove that X is a Killing vector field if and only if the local l-parameter group of local transformations generated by X consists of local isometries. 1.22.30 Let M be an m-dimensional manifold. Prove that giving an S l ( m , R)structure on M is the same as giving a volume form on M . Hence if M possesses an Sl(m, R)-structure, then M is orientable.=
-
- -
111
Chapter 2
Almost tangent structures and tangent bundles 2.1
Almost tangent structures on manifolds
In this section we introduce a geometric structure which is essential in the Lagrangian formulation of Classical Mechanics.
Definition 2.1.1 Let M be a digerentiable manifold of dimension 2n. An almost tangent structure J on M is a tensor field J of type (1,1) on M with constant rank n and satisfying J 2 = 0. In this case, M is called an almost tangent manifold. Let z E M . Then J , : T,M
-
T,M
is a linear endomorphism. Since J: = 0, we have ImJ, c KerJ, . Furthermore, because rank J , 1n, we deduce that ImJ, = KerJ,. Then
ImJ=
u
ImJ,,KerJ=
u
KerJ,
ZEM
ZEM
are vector subbundles of T M of rank n. Now, let H , be a complement in T,M of KerJ,. Then
J, : H,
-
KerJ, = ImJ,
112
Chapter 2. Almost tangent structures and tangent bundles
is a linear isomorphism. Hence, if { e i } is a basis of H,, then {e;,i?i = J e i } is a basis of T,M, that is, a linear frame a t x, which is called an adapted frame to J . Let H i be another complement t o K e r J , and {ei} a basis of H i . Therefore, we have
where A , B are n x n matrices, with A non-singular. Then the two adapted frames are related by the 2n x 2n matrix
where A E Gl(n, R). Now, let G be the set of such matrices; G is a closed subgroup of G1(2n, R) and therefore a Lie subgroup of Gl(2n, R ) . We put
BG = {adapted frames at all points of M } .
-
We shall prove that BG defines a G-structure on M . To do this, it is sufficient to find, for each x E M , a local section u : U F M of F M over a neighborhood U of x such that .(U) c BG. From the local triviality of K e r J and T M , there exists a neighborhood U of x and a frame local field {XI,.. . ,X,,, XI,. . . ,Xn} on U such that {&(y),Xi(y)} is an adapted frame a t y , y E U ,that is, X;(y) = J,X;(y). If we define
then 0 is the required local section. We remark that, with respect to an adapted frame, J is represented by the matrix
where I,, is the n x n identity matrix. In fact, the group G can be described as the invariance group of the matrix Jo, that is, a E G if and only if aJ0a-l = Jo.
2.1. Almost tangent structures on manifolds
113
Suppose now given a G-structure BG on M . Then we may define a tensor field J of type ( 1 , l ) on M as follows. We set
J d X ) =P(Jo(P-'(x))), where X E T , M , x E M and p E BG is a linear frame at x . From the definition of G, J , ( X ) is independent of the choice of p. In other words, J , is defined as the linear endomorphism of T,M which has at x the matrix representation Jo with respect to one of the linear frames determined at z by B G , and hence with respect to any other. Obviously, we have rank J = n, J 2 = 0. Thus, J is an almost tangent structure on M . Summing up, we have proved the following.
Proposition 2.1.2 Giving an almost tangent structure is the same as giving a G- structure on M . Now, let g be a Riemannian metric on M . Then g , determines an inner product on each tangent space T , M . Let H , be an orthogonal complement in T,M to K e r J , with respect to g , . If { e ; } is an orthonormal basis of H,, then {e;,i?i = J,e,} is an orthonormal basis of T , M , that is, an orthonormal frame at x, since
J , : H , :-
KerJ,
is an isometry. If { e i , i $ } is another orthonormal frame at x obtained from
a different orthonormal basis { e ; } of H,, then the two orthonormal frames are related by the (2n) x (2n) matrix
where e: = A i e i , A E O(n). Let B be the set of all orthonormal frames obtained as above in all points of M . Then B defines a (O(n) x O(n))structure on M . In fact, given a local frame field on a neighborhood of each point z E M , we obtain a local section of FM taking values in B by the usual Gramm-Schmidt argument. Conversely, given a ( O ( n )x O(n))structure B on M , we obtain an almost tangent structure on M , since (O(n) x O ( n ) )c G. Summing up, we have the following.
Chapter 2. Almost tangent structures and tangent bundles
114
Proposition 2.1.3 Giving an almost tangent structure i s the same as giving a (O(n) x O(n))-structure on
M.
Since O ( n ) x O(n) c SO(n), we have
Proposition 2.1.4 E v e r y almost tangent manifold i s orientable.
2.2
Examples. The canonical almost tangent structure of the tangent bundle
In this section we shall prove that the tangent bundle of any manifold carries a canonically defined almost tangent structure (hence the name). Let N be an n-dimensional differentiable manifold and T N its tangent bundle. We denote by TN : T N N the canonical projection. For each y E T z N , let
-
-
V,,= Ker{drN(y) : T u ( T N )
T,N}.
Then V,, is an n-dimensional vector subspace of T , ( T N ) and
-
is a vector bundle over T N of rank n (in fact, a vector subbundle of TTN : TTN T N ) . V (sometimes denoted by V ( T N ) )is called the vertical bundle. A tangent vector w of T N at y such that w E V, is called vertical. A vertical vector field X is a vector field X on T N such that X ( y ) E V,, for each y E T N (that is, X is a section of V). We remark that the vertical tangent vectors are tangent to the fibres of the projection T N . Now, let y E T , N , x E N . Then we may define a linear map
-
called the vertical lift as follows: for u E T,N, its vertical lift u" to T N at y is the tangent vector at t = 0 to the curve t y + t u . Furthermore, if X is a vector field on N , then we may define its vertical lift as the vector field Xuon T N such that
2.2. Examples
115
If X is locally given by X = X ' ( a / a z ' ) in a coordinate neighborhood U with local coordinates (z'), then Xu is locally given by
xu= x'(a/au') with respect to the induced coordinates (z',u') on T U . Next, we define a tensor field J of type ( 1 , l ) on T N as follows: for each y E T N , J is given by
Then J is locally given by J ( a / d z ' ) = a/du', J(a/au') = 0, or, equivalently,
J = ( a / a o ' ) @ (dz'). Consequently, J has constant rank n and J 2 = 0. Thus, J is an almost tangent structure on T N which is called the canonical almost tangent structure on T N (in Section 2.5, we shall give an alternative definition of J ) . We can easily prove that K e r J = I m J = V. To end this section we describe a family of almost tangent structures on the 2-torus T 2 . The 2-torus T 2 = S' x S1 may be considered as a quotient manifold R 2 / Z 2 ,where Z2 is the integral lattice of R 2 . So, the canonical global coordinates ( z , y ) of R2 may be taking as local coordinates on T 2 . Let a be any real number then
+ sin2a ( a / a y )8 ( d z )
-
cos a sin a ( a / d y ) 8 (dy)
determines an almost tangent structure on T 2 . The vertical distribution V, is tangent to the spiral which is the image of the line y = ztga under the canonical projection R2 T2.
-
116
2.3
Chapter 2. Almost tangent structures and tangent bundles
Integrability
The fundamental problem of the theory of G-structures is to decide whether a given G-structure is equivalent to the standard G-structure on R2".In this section we establish a necessary and sufficient condition for an almost tangent structure J on a 2n-dimensional manifold to be integrable, that is, locally equivalent to the standard almost tangent structure on R2" (see Section 1.23).
Definition 2.3.1 Let J be an almost tangent structure on a &n-dimensional manifold M. The Nijenhuis tensor N J of J is a tensor field of type (1,2) given by
N J ( X , Y )= [ J X , J Y ]- J ( J X , Y ]- J [ X , J Y ] , X , YE X ( M ) . Now, let Jo be the standard almost tangent structure on R2".Then JO is given by
Jo(a/az') = spy', Jo(a/ay') = 0,
(2.1)
where (xi,y') are the canonical coordinates on R2",1 5 i 5 n. If J is integrable, then there are local coordinates (z',y') on a neighborhood U of each point x of M such that J is locally given by (2.1). Hence, if J is integrable, then the Nijenhuis tensor NJ vanishes. Next, we prove the converse. Suppose that N J = 0. Therefore, we have
+
[ J X ,J Y ] = J [ J X , Y ] J [ X , J Y ] . Thus, the distribution V = ImJ = K e r J is integrable. From the Frobenius theorem, we may find local coordinates (z',~') on a neighborhood of each point of M such that the leaves of the corresponding foliation are given by 2' = constant, 1 5 i 5 n. Then the local vector fields
determine a basis of V. Hence we have
where (A!)is a non-singular matrix of functions, since J has rank n . Let H be a complement of V in T M ,that is,
2.3. Integrability
117
T M = H (33 V(Whitney sum) Then J : H -+ V is a vector bundle isomorphism. Thus, there exists a local basis (2;;1 5 i 5 n } of H such that
JZi = a/azi. We have
We set
Then (2;;1 5 i 5 n } is a set of linearly independent local vector fields on M such that
we deduce that =.:5
Hence (a:) is the inverse matrix of ( A ! ) . Because N J = 0, we have 0 = N J ( Z ~Z,j ) = - J [ d / a z ' , Z j ] - J [ Z ; , a / d z ' ]
= ((da:/az')
-
(da$/az'))A~(a/az').
Since (A:) is a non-singular matrix, we obtain
118
Chapter 2. Almost tangent structures and tangent bundles
From the compatibility conditions (2.2), we deduce that there exist local functions f k = f k ( z i , z i )such that =
afk/azi.
Now, we make the following coordinate transformation: 2 '
'
= z ' , y i = f (z1 , z i ), l . Since V U = VV = 0, and g ( U , V ) is constant, we deduce that V is the Riemann connection for g. Hence, p - ' ( z ) is a geodesically complete Riemannian manifold. Then, if y and I are two points of p - ' ( z ) , there exists, from the Hopf-Rinow theorem, a geodesic a such that a(0) = y and a(1) = t . Since the tangent vector &(O) is vertical, then b(0) = u", where u E T , N . Therefore, u is the integral curve of U trough y and I
= 4 u ( l , Y ) = PZ(Y,.).
This proves the transitivity of p,. Now, let r ( y ) be the isotropy group of p z , that is
The map @ : T z N follows:
--+ p-l(z)
given by @(u) = p , ( y , u ) may be factored as
where cr is the vertical lift map from T,N to the tangent space to p - ' ( z ) at y and e z p : TU(p-'(z))---t p - ' ( z ) is the exponential map of V restricted to p-' (z). Since a is a linear isomorphism and exp a local diffeomorphism, then /3 is a local diffeomorphism. Thus, r ( y ) is a discrete subgroup of the additive group T , N . Therefore, r ( y ) must consists of integer linear combinations of some k linearly independent vector v l , . . . ,Uk,O 5 k 5 n. Moreover, since T,N acts transitively on p - ' ( z ) , then this space is diffeomorphic to the coset space T , N / r ( y ) . Then we deduce that p - ' ( z ) is the product of a k-torus T k and RnWk.Thus, r ( y ) must be trivial, since p - ' ( z ) is simply connected. Therefore, the action p, is free. This ends the proof. 0
144
Chapter 2. Almost tangent structures and tangent bundles
-
Corollary 2.10.5 If (M,J) verifies all the hypotheses of the theorem and in addition p : M N admits a global section, then M i s isomorphic (as a vector bundle) to T N . (This isomorphism depends on the choice of the section). Corollary 2.10.6 If (M,J ) verifies all the hypotheses of the theorem ezcept the hypotheses that the leaves of the foliation defined b y V are simply connected, then T N is a covering space of M and the leaves of V are of the form
T kx Rn-k, where T k is the k-dimensional torus, 0 I k 5 n. Moreover, i f it is assumed that the leaves of V are compact, then T N is a covering space of M and the fibres are diffeomorphic to Tn.
Remark 2.10.7 In de L e h , MBndez and Salgado [33], [34], we introduce the concept of a palmost tangent structure and prove similar results for integrable p a l m o s t tangent structures which define fibrations.
2.11
Exercises
2.11.1 Prove Proposition 2.1.4. 2.11.2 (1) Let g be a symmetric tensor of type (0,2) on an n-dimensional vector space E . Prove that, if g has rank r , then there exists a basis { e l , . . . ,en} of E with dual basis { e l , , . . ,en} such that r i=l
where a; = f l , or, equivalently, the matrix of g is
(2) If g satisfies g(v, w ) = 0 for all w implies v = 0,
2.11. Exercises
145
then g has rank n and we have
i=l
where ai = k l , 1 5 i 5 n. We say that g has signature ( p , q ) with p + q = n if a1 = ... = ap = 1 and ap+l = . . . = a,, = -1. (3) Prove that if g is a pseudo-Riemannian metric on an n-dimensional manifold M then gz has the same signature (p, q ) , p q = n, for all x E M. We say that g has signature (p, q ) . (4) Prove Corollary 2.5.9. 2.11.3 Prove that if M is complete with respect to a linear connection V then TM is complete with respect to Vc, and conversely. 2.11.4 Prove Proposition 2.7.4. 2.11.5 Prove that V H is of zero curvature if and only if V is of zero curvature. 2.11.6 Prove (2.30).
+
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147
Chapter 3
Structures on manifolds 3.1
Almost product structures
In this section, we introduce some definitions and basic facts about almost product structures. For more details, we remit to Fujimoto [57], Walker [121], [122], Willmore [128]. Definition 3.1.1 Let M be a diflerentiable m-dimensional manifold. A n almost product structure on M i s a tensor field F of type (1,l) on M such that F 2 = I d . M, endowed with an almost product structure F is said t o be a n almost product manifold. We set P = (1/2)(Id+ F ) , Q = (1/2)(Id- F ) .
Then we have P 2 = P, PQ = Q P = 0, Q 2 = Q .
(3.1)
Conversely, if (P,Q) is a pair of tensor fields of type (1,l)on M satisfying (3.1), then we put F=P-Q,
and F is an almost product structure on M . We set
P = I m p , Q = ImQ.
Chapter 3. Structures on manifolds
148
Then P and Q are complementary distributions on M , i.e.,
T,M = P, @ Q,,x E M If P has constant rank p and Q has constant rank q , respectively then P is a pdimensional distribution and Q a q-dimensional distribution on M , respectively, and p + q = m. Conversely, if there exist on M two complementary distributions P and Q, then P and Q are defined to be the corresponding projectors
P, : T,M
-
P,,Qz : T,M
-
Qz,xE M
Let e l , . . . ,ep be a basis of P, and e p + l , . . . , e m a basis of Q,,x E M . Hence { e l , . . . ,ep,e p + l , .. . ,em} is a basis of T,M which is called an adapted frame at x. Let { e l , . . . ,em}, {ei, . . . ,e),} be two adapted frames at x. Therefore, we have ei = A i e , , l
5 i ,j 5 p,
where A E Gl(p, R ) ,B E Gl(q, R). Then 'are related by the m x m matrix a=
I0" 8" J
{ei; 1
5 i 5 m} and {ei; 1 5 i 5 m}
E Gl(m,R).
Clearly, a E Gl(p, R) x Gl(q, R ) , where Gl(p, R) x Gl(q,R) is identified to the Lie subgroup G of Gl(m, R ) given by
We note that, with respect to an adapted frame, P , Q and F have the following matricial representations
We set
3.1. Almost product structures
149
B = {adapted frames at all points of M}. One can easily proves that B defines a (Gl(p,R) x Gl(q,R))-structure on M. Conversely, if B is a ( G l ( p , R )x Gl(q,R))-structure on M, then we define P and Q t o be the tensor fields of type (1,l) on M which have matricial representations Pi and Q o with respect to any frame of B at z,for each x E M. Summing up, we have proved the following.
Proposition 3.1.2 Giving an almost product structure i s the same as giving a (Gl(p, R) x Gl(q,R))-structure.
We say that an almost product structure F is integrable if there exists a local coordinate neighborhood (U, zl, . . . ,z m ) a t each point of M such that the local frame field a :zE
u
-
a ( z ) = ((a/az')z,. . . (a/az")z) )
is a section of B , that is, a ( z ) is an adapted frame at z,for each z E U. Therefore we have
Proposition 3.1.3 F is integrable if and only ifP and Q are integrable. Next, we shall give a characterization of the integrability of an almost product structure F in terms of the Nijenhuis tensors Np,NQ and NF.
Proposition 3.1.4 The following four assertions are equivalent: (1) The almost product structure F i s integrable.
NF = 0. (3) Np = 0. (4) NQ = O. (2)
Proof First, we note that
Np
= ( ~ / ~ ) N F ,-(1/2)N~. NQ
Hence (2), (3) and (4) are equivalent. Next, we shall prove that (1) and (2) are equivalent. Let us recall that Np and NQ are given by
Np(X,Y )= [PX, P Y ] - P [ P X ,Y ] - P [ X ,P Y ] + P[X, Y],
Chapter 3. Structures on manifolds
150
since P2= P and Q 2 = Q . If F is integrable, then P and Q are integrable. Hence
N p ( X , Y ) = [ P X ,P Y ] - P [ P X ,Y ] - P [ X ,P Y ]
+ PIX, Y ]
= [PX,PY]-P[PX,PY+QY]-P[PX+QX, PY]+P[PX+QX,PY+QY] (since 2 = PZ
+ Q Z , for any vector field 2 on M )
= Q [ P X ,P Y ]
+P[QX,QY].
But [ P X , P Y ]E P and [ Q X , Q Y ]E Q, since P and Q are integrable. Thus N p = 0 and, therefore, NF = 0. Conversely, suppose that NF = 0. Then N p = NQ = 0, and thus
Consequently, P and Q are integrable and, so, F is integrable, by Proposition 3.1.3.0 Let V be a linear connection on M. Since
V F = 2(VP)= 2(VQ), we have
Proposition 3.1.5 The following three assertions are equivalent: (1) V F = 0; (2) V P = 0; (3) V Q = 0. Definition 3.1.6 A linear connection V on M such that V F = 0 is said to be an almost product connection. Proposition 3.1.7 There exists an almost product connection on every almost product manifold. Proof: Let V be an arbitrary linear connection on M. We define a tensor field of type ( 1 , 2 ) on M by
3.2. Almost complex manifolds
151
Since
(VxF)F= -F(VxF), we can easily prove, from a straightforward computation, that V = V - S is an almost product connection on M . 0 Now, let V be a symmetric linear connection on M . Then we obtain
N P ( X , Y )= ( V P X P ) Y - ( V P , P ) X
-
+
(P(VxP))Y (P(VyP))X,
From (3.2),we easily deduce the following.
Theorem 3.1.8 If there ezists a symmetric almost product connection on M then the almost product structure F is integrable. (See Ezercise 9.8.9). The converse is also true (see Fujimoto [57]).
3.2
Almost complex manifolds
Definition 3.2.1 An almost complex structure on a differentiable manifold M i s a tensor field J of type (1,l) such that J 2 = - I d . A manifold M with an almost complex structure J is called an almost complex manifold. Let J be an almost complex structure on M . Then, for each point z of M , J, is an endomorphism of the tangent space T,M such that J: = - I d . Hence T,M may be turned into a complex vector space by defining scalar multiplication by complex numbers as follows:
Therefore the real dimension of T,M must be even, namely 2n. We deduce that every almost complex manifold M has even dimension 2n. In fact, let { X I , .. . ,X , } be a basis for T,M as a complex vector space. Then { X I , .. . ,X , , J X 1 , . . . ,J X , } is a basis for T,M as a real vector space. In
Chapter 3. Structures on manifolds
152
fact, {XI,. . . ,X,,J X 1 , . . . ,JXn} is a set of linearly independent vectors, since, if
C(a'X;
+ b'(JX;)) = 0,
then we have
0 = C(a'Xi
+ b'(JX;)) = C(a' + a b ' ) x ' )
+
which implies ai a b ' = 0 , l 5 i 5 n. Thus, ai = b' = 0 , l 5 i 5 n. Moreover, if X E T , M , then
X = C(ai + f l b i ) X ' = Ca'X; + Cb'(JX;). Thus, { X i , JX;}span T , M . This basis is called an adapted (or complex) frame at x. Let now {Xi, J X i } , {Xi,JX:}be two bases as above. Therefore, we have
and, consequently,
JX;= A! ( J X j ) + Bf (J 2 X j ) = - B i X j
+ A! ( J X j ) ,
where A , B are n x n matrices. Then the two complex frames are related by the 2n x 2n matrix
A - = [ B
-B A ] '
Clearly, a! E G1(2n,R). Now, let G be the set of such matrices; G is a closed subgroup of G1(2n,R) and therefore a Lie subgroup of G1(2n, R). If Gl(n, C) is the complex linear general group, we have a real representation of Gl(n, C) into G1(2n,R) given by p : Gl(n,C) + G1(2n, R )
Q
=A
+
-B B A
A -----+
3.2. Almost complex manifolds
153
In fact, p is a Lie group monomorphism. Hence Gl(n,C) may be identified with p(Gl(n,C)) = G. We note that, with respect to a complex frame, J is represented by the matrix
Jo=
[ Yn
-I;],
where I , is the n x n identity matrix. It is easy t o prove that Gl(n,C) can be described as the invariance group of the matrix Jo, that is, Gl(n,C) = {a E GZ(2n,R)/aJo = Joa}. Now, we set
C M = {complex frames at all points of M } . We shall prove that C M defines a Gl(n,C)-structure on M . In order to do this, we note that the tangent bundle T M becomes a complex vector bundle of rank n. Then, for each x E M , there are n local vector fields X I ,..., X,,on a neighborhood U of x such that {X,(y) ,...,Xn(y)} is a basis for T,M as a complex vector space for any y E U. If we define
then u is a local section of FA4 over U such that a ( U ) c C M . Thus, C M is a GZ(n,C)-structure on M . Conversely, let B be a Gl(n, C)-structure on M . We define a tensor field of type (1J) on M as follows. We set
where X E T,M, z E M and p E C M is a linear frame at x . Obviously, J,(X) is independent of the choice of p and J: = - I d . Then J defines an almost complex structure on M . Summing up, we have proved the following.
Proposition 3.2.2 Giving an almost complex structure is the same as giving a Gl(n, C)-structure on M .
154
Chapter 3. Structures on manifolds
Definition 3.2.3 Let M be a topological space such that each point has a neighborhood U homeomorphic to an subset of C". Each pair (U,+), where U i s an open set of M and 4 is a homeomorphism of U to a open subset #(U)of C" is called a coordinate neighborhood; to z E U we assign the n complex coordinates zl(z), . . .,z"(z) of 4(z) E C". Two coordinate neighborhoods ( U , 4)) (V,+) are said to be compatible i f the mappings o and $J o 4-l are holomorphic. A complex structure on M i s a family U = {(U,,&)} of coordinate neighborhoods such that (1) the U, cover M; q5p) are compatible; (2) f o r any a, the neighborhoods (U,,4,) and (Up, (8) U is maximal (in the obvious sense). M, endowed with a complez structure, is said to be a complex manifold of complez dimension n.
+
+-'
Let M be a complex manifold of complex dimension n . Then M becomes a CW-manifold of real dimension 2n. In fact, each coordinate neighborhood U with complex coordinates z ' , . . . ,Z" gives real coordinates zl,. . . ,z",y', . . . ,y" by setting
We shall prove that every complex manifold carries a natural almost complex structure. Let ( z ' , . . . ,z") be a complex local coordinate system on a neighborhood U. We define an endomorphism J, : T,M
-
T,M,z E U,
We prove that the definition of J does not depend on the choice of the complex local coordinate system. If ( w l , . . . ,w " ) is another complex local coordinate system on a neihgborhood V ,U n V # 0 and
then the change of coordinates wi = w ' ( z j ) is a holomorphic function. Hence the following Cauchy-Riemann conditions hold: ( d u k / d z ' ) = ( a w k p y ' ) , ( a u k l a y ' ) = -(awk/az').
(3.3)
3.2. Almost complex manifolds
155
On the other hand, we have
+ (au"aZ')(a/a"k) a p y ' = (auk/ay')(a/auk) + (auk/ay')(a/auk). a/az' = (auk/az')(a/auk)
Let JL : T,M
-
(3.4)
T , M , x E U n V , defined by
J;(a/au') = a/&', J;(a/au') = -(a/du').
From (3.4), we have J;(a/dz') = (i3uk/az')J;(a/auk)
+ (auk/az')Ji(a/auk)
= (auk/az')(a/auk) - (auk/az')(a/duk)
= a/ayi
Similarly, we deduce that JL(a/dy') = -(a/az').
Hence JL = J, and, therefore, J is well-defined. To end this section, we shall give a characterization of the integrability of almost complex structures.
Definition 3.2.4 A n almost complex structure J on a 2n-dimensional manifold M is said t o be integrable if it is integrable as a Gl(n,C)-structure. Therefore, if J is integrable, for each point z E M , there exists a local coordinate system ( d ,. . . ,z", y', . . . ,y") such that J(a/az') = a/dy', J(a/ay') = - ( d / a z ' ) , 1
In fact, the local section
5i5
n.
156
Chapter 3. Structures on manifolds
of F M takes values in C M . Hence, if J is integrable, M becomes a complex manifold; it is sufficient to set
as complex local coordinates (details are left to the reader as an exercise). Hence, an integrable almost complex structure J is called a complex structure. If we denote by Jo the canonical complex structure on Cn = R2n, then an almost complex structure is integrable if and only if the corresponding Gl(n, C)-structure is locally isomorphic t o Jo.
Definition 3.2.5 Let J be an almost complex structure on M. The Nijenhuis tensor NJ of J is a tensor field of type (1,2) on M given by
N j ( X , Y ) = [ J X ,J Y ] - J [ J X , Y ]- J [ X ,J Y ]- [ X , Y ] ,
Obviously, if J is integrable, then the Nijenhuis tensor NJ vanishes. The converse is true; it is the theorem of Newlander and Niremberg [loo]. It is beyond the scope of this book to give a proof of this theorem.
Theorem 3.2.6 (Newlander-Niremberg) An almost complex structure J is integrable if and only if its Nijenhuis tensor NJ vanishes.
3.3
Almost complex connections
Let A4 be an almost complex manifold of dimension 2n with almost complex structure J .
Definition 3.3.1 A linear connection V on M is said to be an almost complex connection if V J = 0. We shall prove the existence of an almost complex connection on M. We need the following lemma.
Lemma 3.3.2 Let V be a symmetric linear connection on M. Then
N J ( X , Y )= ( V J X J ) ~( V J Y J ) X + J ( ( V y J ) X - ( V x J ) Y ) .
157
3.3. Almost complex connections
Proof: Since V is symmetric, we have
[ X , Y ]= VXY
-
vyx.
Then we obtain
N J ( X , Y )= [JX, JY] - J [ J X , Y ]- J [ X , J Y ]- [ X , Y ] = V J X ( J Y )- V J Y ( J X )- J(VJXY - V y ( J X ) )
- J ( V x ( J Y ) - V J Y X )- (VXY
-
VYX)
+ v y x - J ( V x ( J Y ) )- VXY
+J(Vy(JX))
+
= ( V J X J ) Y- ( V J Y J ) X J ( ( V Y J ) X ) J((VXJ)Y).O
Proposition 3.3.3 There ezists an almost complex connection V on M such that its torsion tensor T ia given b y
T =(1/4)N~, where
NJ is
the Nijenhuis tensor of
J.
Proof: Let V be an arbitrary symmetric linear connection on M. We define a tensor field Q of type ( 1 , 2 ) by
+
Q(x,Y) = ( 1 / 4 ) { ( v ~ y JJ) (x( V Y J ) X+) 2 J ( ( V x J ) Y ) ) , for any vector fields X, Y on M. Consider the linear connection V given by
VxY = VxY
- Q(X,Y).
First, we prove that V is, in fact, an almost complex connection. We have
Chapter 3. Structures on manifolds
158
J ( Q ( X , Y )= ) ( 1 / 4 ) { J ( ( v ~ y J )-x( V Y J ) X- 2 ( V x J ) Y ) . On the other hand, since
(VXJ)J = - J ( V x J ) , we obtain
J ( ( V x J ) ( J Y )= ) -(VxJ)(JZY)= (VXJ)Y. Hence, we deduce
Q ( X , J Y ) - J Q ( X , Y )= ( 1 / 2 ) J ( ( V x J ) ( J Y+) )( 1 / 2 ) ( V x J ) Y= ( V X J ) ~ . Consequently, we have
V x ( J Y )= V x ( J Y )- Q ( X ,J Y ) = (VxJ)Y
+ J ( V x Y )- Q ( X ,J Y )
and, then, VJ = 0. The torsion T of V is given by
T ( X , Y )= VXY
-
Vlyx - [ X , Y ]
3.3. Almost complex connections
= -Q(X,
159
Y )+ Q ( y , X ) ,
since
T ( X , Y )= VXY
-
vyx - [ X , Y ]= 0 .
Hence
= 4 N J ( x ,Y ) (by Lemma 3.3.2).0 Corollary 3.3.4 An almost complex structure J on M i s integrable if and only if M admits a symmetric almost complez linear connection. Proof If J is integrable, then the torsion T of the connection V constructed in Proposition 3.3.3 vanishes. Conversely, suppose that there exists a symmetric almost complex connection V on M. From V J = 0, we deduce that Q = 0. Hence V = V and N J = 4T = 0 . 0 The following result gives some properties of the torsion and curvature tensors of an almost complex connection. Proposition 3.3.5 Let M be an almost complex manifold with almost complex structure J and V an almost complex connection on M. Then the torsion tensor
T and
the curvature tensor
R
satisfy the following identities:
(1) T ( J X ,JY)- J(T,J x , Y ) ) - J ( T ( X ,JY))- T ( X , Y ) = - N J ( X , Y ) ; (2) R ( X , Y ) J = J R ( X ,Y ) .
Chapter 3. Structures on manifolds
160
Proof: (1) We have T ( J X , J Y ) = VJX(JY)- VJY(JX)- [JX, JY],
T(X,JY)= VX(JY)- V J Y X- [X, JY], T(X,Y) = VXY - v
y x
- [X,Y].
Hence,
T ( J X ,JY)- J ( T ( J X , Y ) )- J ( T ( X ,JY))- T(X,Y)
+
-[JX, JY]+ J [ J X ,Y ] J[X, JY]+ [x, Y]
since
VJ = 0.
(2) is proved by a similar device.
161
3.4. Kiihler manifolds
3.4
Kiihler manifolds
In this section, we introduce an important class of almost complex manifolds.
Definition 3.4.1 A Hermitian metric on an almost complex manifold with almost complex structure J is a Riemannian metric g on M such that
for any vector fields X, Y on M. Hence, a Hermitian metric g defines a Hermitian inner product g, on T,M for each z E M with respect to its structure of complex vector space given by J,, that is,
-
Then J, : T,M T,M is an isometry. An almost complex manifold M with a Hermitian metric is called an almost Hermitian manifold. If M is a complex manifold, then M is called a Hermitian manifold.
Proposition 3.4.2 Every almost complex manifold M admits a Hermitian metric. Proof: Let h be an arbitrary Riemannian metric on M . We set
+
g ( X , Y )= h ( X , Y ) h(JX, J Y ) . Then g is a Hermitian metric. 0 Now, let M be a 2n-dimensional almost Hermitian manifold with almost complex structure J and Hermitian metric g. The triple ( M ,J , g ) is called an almost Hermit ian structure. Before proceeding further, we prove the following lemma.
Lemma 3.4.3 Let V be a hn-dimensional real vector space with complex structure J (that is, J is a linear endomorphism of V satisfying J2= -Id) and a Hermitian inner product (i.e., < JX, J Y >=< X,Y > , X , Y E V). Then there exists an orthonormal basis {XI,. . . ,Xn, JX1,. . . ,
Jxn).
162
Chapter 3. Structures on manifolds Let X1 be a unit vector.
Proof: We use induction in dim V. { X I ,J X l } is orthonormal, since
Then
' Now, if W is the subspace spanned by { X I ,J X l } , we denote by W the orthonormal complement so that V = W @ W '. The subspace ' W is invariant by J . In fact, if X E W', we have
< J X 1 , J X >=< X i , X >= 0 , and, hence, J X E W '. By the induction assumption, W ' has an orthonorma1 basis of the form ( X 2 , . . . ,X n , J X 2 , . . . ,J X n } . Therefore {XI,. . . ,X n , JX1, . . . ,JX,,} is the required basis. Let (M, J , g ) be an almost Hermitian structure. For each point z E M, by the lemma, there exists an orthonormal basis {XI,. . . ,X n , J X 1 , . . . ,J X n } of T , M . This basis is called an adapted (or unitary) frame at z. Let now { X i , J X i } , {Xi, J X i } be two unitary frames at z. Therefore we have
X: = A!Xj
+Bf(JXj),
J X : = - Bf X j
+ A! (J X j ) ,
where A , B E G f ( n ,R ) . Then the matrix
belongs to GZ(n, C) n O(2n). It is easy to see that Gl(n, C) n O(2n) = U ( n ) , where G l ( n , C ) and O(2n) are considered as subgroups of GZ(2n, R). In fact, U ( n ) consists of elements of GZ(n,C) whose real representation (by p ) are in O ( 2 n ) . If we set
UM = {unitary frames at all points of M},
163
3.4. Kahler manifolds
then we can easily prove that UM is a U(n)-structure on M. Proceeding as in Section 3 . 2 , we deduce that giving an almost Hermitian structure is the same ae giving a U(n)-structure. Let (M, J,g ) be an almost Hermitian structure. We define on M a 2-form by
for any vector fields X and Y on M. R is called the f u n d a m e n t a l or Kahler f o r m of (M, J,9 ) .
Proposition 3.4.4 R ia invariant by J, that is,
n ( J x , JY)= R(X,Y).
Proof: In fact n ( J x , J Y ) = g ( J X , P Y ) = - g ( J X , Y ) = g ( X , J Y ) = n(x,Y).o In general, J is not parallel with respect to the Riemannian connection V defined by g .
Proposition 3.4.5 We have 2S((VXJ)Y,Z)= 3d0(X,JY,JZ) - 3dfl(X,Y,Z)+ g ( N , J ( Y , Z ) , J X ) , for any vector fields X , Y and
2 on M .
Proof: We have
Since
+
2g(VxY,JZ)= X g ( Y , JZ) Y g ( X ,JZ)- ( J Z ) g ( X ,Y )
Chapter 3. Structures on manifolds
164
3dR(X,Y,Z) = XR(Y,Z) + Y R ( Z , X ) + Z R ( X , Y )
+
+
3dR(X, J Y , JZ)= XR(JY, JZ) ( J Y ) R ( J Z , X ) (JZ)R(X,J Y )
-a ( [X, JY] ,JZ)- R ( [ JZ,XI,JY ) - R ( [JY, JZ],X), and
Nj(Y,Z)= [ J Y , J Z ] - J [ J Y , Z ] - J [ Y , J Z ] - [ Y , Z ] , we deduce our proposition by a direct computation.
Corollary 3.4.6 Let (M,J,g) be an almost Hermitian structure. Then the following conditions are equivalent: (1) The Riemannian connection V defined by g is an almost complex connection; (2) NJ = 0 and the Krihler form R is closed, i.e., dR = 0. Proof: If NJ = 0 and dR = 0, then V J = 0 by Proposition 3.4.5. Conversely, suppose that V J = 0. Since V is symmetric, we deduce that J is integrable (by Corollary 3.3.4). Moreover, since V J = 0 and Vg = 0, we easily deduce that V R = 0. Then dR = 0. 0
Corollary 3.4.7 If M is a Hermitian manifold, then the following conditions are equivalent: (I) V i s an almost complex connection; (2) R i s closed.
Proof It is a direct consequence of Corollary 3.4.6, since NJ = 0. 0 Next, we introduce two important classes of almost Hermitian manifolds.
3.5. Almost complex structures on tangent bundles (I)
165
Definition 3.4.8 An almost Hermitian manifold i.9 called almost Kahler ifits Kihler form iz is closed. ' I moreover, M i s Hermitian, then M is called a Kahler manifold. From Corollary 3.4.7, we deduce that a Hermitian manifold is Kahler if and only if V is an almost complex connection (for an exhaustive classification of almost Hermitian structures, see Gray and Hervella [ 7 0 ] ) .
Remark 3.4.9 It is easy to prove that the Kahler form of an almost Hermitian manifold satisfy
Rn = R A . . . A R # 0 (a times, where dim M = 2n) Then we deduce: (1) iz" is a volume form and, hence, every almost complex manifold is orientable; (2) if M is an almost Kahler manifold, then iz defines a symplectic structure on M (see Chapter 5 ) .
3.5
Almost complex structures on tangent bundles (I)
In this section, we shall prove that the tangent bundle T M of a given manifold A4 carries interesting examples of almost complex structures.
3.5.1
Complete lifts
Let M be an almost complex manifold of dimension 2n and almost complex structure F. Let T M be its tangent bundle and FC the complete lift of F to TM defined by F C X c= (FX)'. From Proposition 2.5.6 and Corollary 2.5.7, we obtain
(F')' = - I d . Hence Fc defines an almost complez structure on T M . Now, if N is the Nijenhuis tensor of F c , we have
+
N(Xc,YC= ) [ F C X c , F C Y c-] F C [ F c X c , Y c-] F C [ X c , F c Y c ] [ X c , Y c ] = ( [ F X ,FYI - F [ F X ,Y ] - F[X, FYI
+ [ X ,Y])'
Chapter 3. Structures on manifolds
166
= (NF(X, Y))", where
NF is the Nijenhuis tensor of F . Hence, N = (NF)".
Therefore, we have
Proposition 3.5.1 F" i s integrable if and only if F i s integrable. 3.5.2
Horizontal lifts
Let F be an almost complex structure on M. Consider the horizontal lift F H of F to TM with respect to a linear connection V on M. Let V be the opposite connection with curvature tensor R. From Proposition 2.7.3, we have
( F H ) 2= - I d , and so, F H in an almost complex structure on TM. Let N be the Nijenhuis tensor of F H . From a straightforward computation, we obtain
N ( X U , Y U )= ) 0,
N(XH,YH= ) (NF(X,Y))~
Hence we have
Proposition 3.5.2 If F H i s integrable, then F i s integrable. Conversely, suppose that V i s an almost complex connection (i.e. V F = 0); then, if F i s integrable and $' has zero curvature, F H i s integrable. Particularly, let F be a complez structure on M and V is a symmetric almost complex connection, then F H is integrable if V has zero curvature.
3.5. Almost complex structures on tangent bundles (I)
3.5.3
167
Almost complex structure on the tangent bundle of a Riemannian manifold
Let M be a differentiable manifold with a linear connection V. Let T and R be the torsion and curvature tensors of V. We denote by V the opposite connection with curvature tensor R. We define a tensor field of type (1,l) onTMby
F X H = -Xu, FX" = X H , (3.5) for any vector field X on M . From (3.5), we deduce that F 2 = - I d , and, so F i s an almost complez structure on T M . With respect to the adapted frame we have
FD; = -5, FV; = D;. Next, we study the integrability of F . Let F . We obtain
N be the Nijenhuis tensor of
N ( X " , Y " )= ( T ( X , Y ) ) H- y R ( X , Y ) ,
N ( X UY , H )= ( T ( X ,Y ) ) "+ F y k ( X , Y ) , N ( X H , Y H ) = (T(X, Y ) ) H - 7 R ( X ,Y ) , for any vector fields X, Y on M . Proposition 3.5.3 F i s
integrable
(3.6)
if and only if T = 0 and R = 0.
Proof Suppose that F is integrable. From (3.6), we deduce that T = 0 and R = 0. Since T = 0, V is symmetric and, hence V = V. Then R = R = 0. Conversely, suppose that T = 0 and R = 0. Therefore V = V and, then R = R = 0. Consequently, N = 0, and, thus, F is integrable. 0 Now, suppose that ( M , g ) is a Riemannian manifold and V the Riemannian connection defined by g . Since V has zero torsion, (3.6)becomes
NF(X",Y") = -yR(X, Y ) , NF(X",Y~= ) FyR(X,Y),
NF(XH,YH)= 7 R ( X , Y ) ,
(3.7)
where F is defined by V according to (3.5). From (3.7), we easily deduce the following
Chapter 3. Structures on manifolds
168
Proposition 3.5.4 F i s integrable if and only if (MIS) i s pat, i.e., R = 0. Consider the Sasaki metric
on TM determined by g. Then we have
Proposition 3.5.5 i j ia an Hermitian metric for F. Proof: We must check that
g( FX, F P = i j ( X ,P),
(3.8)
for any vector fields X , a on TM. It is sufficient t o prove (3.8) when X , Y are horizontal and vertical lifts of vector fields on M. Thus we have ij(FX",FY") = i j ( X H , Y H= ) (g(X,Y))"= i j ( X " , Y " ) ,
ij(FX", F Y H )= - i j ( X H , Y u )= 0 = g ( X u , Y H ) ,
i j ( F X H ,FYH) = - g ( X " , Y " ) = ( g ( X , Y ) ) "= B ( X H , Y H ) , for any vector fields X,Y on M . 0 Therefore (TM, F, j j ) is an almost Hermitian structure. Let us consider the Kahler form n associated to (TM, F, 8). We recall that n is given by
fl(X-,P)= i j ( X ,Fa), for any vector fields, X,Y on TM. Hence we have
n(x",Y")= R ( X H , Y H ) = O , R ( X " , Y H ) = - ( g ( X , Y ) ) " ,
(3.9)
for any vector fields X,Y on M. If we compute dfl acting on horizontal and vertical lifts, we deduce, by a straightforward computation from (3.9) that d n = 0. Therefore, by using Propositions 3.4.5 and 3.5.5, we obtain
Theorem 3.5.6 ( T M ,F, 8) i s an almost Ktihler structure. Furthermore, ( T M ,Fij) i s a Kihler structure if and only if (MIS) i s pat, i.e., R = 0. Remark 3.5.7 Since (TM, F,ij) is an almost Kahler structure, then !2 is always a symplectic form (see Chapter 5).
3.6. Almost contact structures
3.6
169
Almost contact structures
In this section we shall give alternative definitions of almost contact structures. Roughly speaking, almost contact structures are the odd-dimensional counterpart t o almost complex structures. We remit to Blair [8] for an extensive study of such a type of structures.
Definition 3.6.1 Let M be a (2n + 1)-dimensional manifold. If M carries a I-form 9 such that
then M is said to be a contact manifold or to have a contact structure. We call a contact form.
Example.- We set 9 =dt
+
y'dz', i
where (z',y',z; 1 5 i 5 n) are the canonical coordinates in R2"+'. Then 9 is a contact form on R2"+'. Moreover, a contact form on a (2n 1)dimensional manifold M can be locally expressed in this way (see Chapter 6). If 9 is a contact form on M , then there exists a unique vector field { on M such that
+
(See Chapter 6 for a proof). We call [ the Reeb vector field. Next, we generalize the notion of contact structure.
+
Definition 3.6.2 (Blair 1811.- An almost contact structure on a (2n 1)-dimensional manifold M i s a triple (4, E, 9 ) where 4 i s a tensor field of type (1,1), { a vector field and 9 a 1-form on M such that
42=-Id+(@3q,9(€)=1.
From (3.10)) it follows that
4(()
= 0.94 = 0 , rank
(4) = 2n.
(3.10)
Chapter 3. Structures on manifolds
170
If there exists a Riemannian metric g on M such that g(4X,4
v = 9(X, Y ) - rl(X)rl(Y),
(3.11)
for any vector fields X, Y on M , then g is said t o be a compatible or adapted metric and ( 4 , < , 9 , g ) is called an almost contact metric structure. From (3.10) and (3.11), we deduce that
Proposition 3.6.3 Let (4, (,q ) be an almost contact structure M admits a compatible metric.
on
M . Then
Proof Let g' be an arbitrary Riemannian metric on M. We set g"(X,Y) = g'(42x, 4")
+ rl( XI9 ( Y ) *
Then g" is a Riemannian metric satisfying g"(E,X) = dX).
Now, we define a Riemannian metric g on M by
Let now (4, €,q,g) be an almost contact metric structure on M. Let z E M. We choose a unit tangent vector XI E T,M orthogonal to E., Then
4x1 is also a unit
tangent vector orthogonal t o both
EZ and X I ;in fact,
3.6. Almost contact structures
171
(3) 9(dX1,4Xl) = g(X1,Xl)- q(4Xl)q(4Xl) = S(X1,Xl) = 1. Now, take X2 E T,M t o be a unit tangent vector orthogonal to (=,XI and 4x1;then 4x2 is a unit tangent vector orthogonal to t2,X1,4X1 and X2. Proceeding in this way we obtain an orthonormal basis { X i )#X;, c2} on T 2 M , that is, a frame at z,which is called a +baais or adapted frame. With respect to an adapted frame 4, (,q and g are represented by the matrices
go=[!
s
0 0 ;],
respectively. Let { X ; , d X ; ,C2}, {X;)q5X;,&} be two adapted frames at z. Then we have
Xi = A{Xj + B : ( 4 X j ) , 4X;= - B :xj
+ A{($x~))
where A, B E G l ( n ,R). Hence the two frames are related by the (2n (2n 1) matrix
+
Obviously a E U(n) x 1. We set
B = {adapted frames at all points of M}.
+ 1)
X
172
Chapter 3. Structures on manifolds
One can easily proves that B is a ( U ( n ) x 1)-structure on M (it is sufficient to repeat the above construction t o obtain a local frame field {Xi, +Xi,(} on a neighborhood of each point of M). Conversely, suppose that B is a ( U ( n ) x 1)-structure on M. Then we define an almost contact metric structure ( 4 , ( , q , g ) on M as follows. With respect to a frame of B at ~ , & ( ~ and , q ~gz are given by the matrices q50,(0,qo and go, respectively, for each x E M. Summing up, we have proved the following.
Proposition 3.6.4 Giving a n almost contact metric structure is the same as giving a (U(n) x 1)-structure. Let (+,(,q,g) be an almost contact structure on M. We define the fundamental 2-form R on M by
Then we deduce that q A R"
# 0, that is, q A R" is a volume form on M.
Definition 3.6.5 An almost contact metric structure be a contact metric structure if R = dq.
(4, (, q , g)
is said to
There exists in the literature an alternative definition of almost contact structure which generalizes Definition 3.6.1.
Definition 3.6.6 (Libermann /87]). A n almost contact structure or almost coaymplectic structure on a ( 2 n + 1)-dimensional manifold M is a pair ( q ,R), where q is a l-form and R a 2-form on M such that q A R" # 0. The following result relate these definitions.
Proposition 3.6.7 Let M be a (2n +l)-dimensional manifold. We have: (1) If M admits an almost cosymplectic structure ( q , R ) then M admits a n almost contact metric structure. (2) If M admits a contact form q , then there is a n almost contact metric structure (4, q , g ) such that the fundamental form R is, precisely, d q .
c,
Proof: (1) Since q A R" # 0, then M is orientable and, then, there exists a non-vanishing vector field (' on M such that = 0 (see Blair [S]). Let g' be a Riemannian metric on M and define a vector field ( by
3.6. Almost contact structures
173
Thus ( is a unit vector field on M . We now define a l-form q' by
Let D be the orthogonal complement of (, i.e., T M = D@ < >. Then R is a symplectic form on the vector bundle D (we also denote by D the corresponding distribution of sections of D). We consider a metric g" on D and an endomorphism 4' of D such that
g1yx)4'Y) = n(X)Y),p = -Id,
(3.12)
for any vector field X E D (see Exercise 3.8.1). Next, we define a Riemannian metric g by (3.13) g(X,Y) = grr(x,y),g(x, 0 = O,g(€,€)= 1 9 for any vector fields X,Y E D, and a tensor field 4 of type (1,1) by
for any vector field X E D. Thus (4, E , q') is an almost contact structure on M. (2) Let ( be the Reeb vector field, i.e., 9 ( € ) = 1 and i t ( d q ) = 0. Let h be a Riemannian metric on M and define a Riemannian metric g 1 by
Hence
Let gN be a metric on D and 4 an endomorphism of D such that (3.12) holds and define g by (3.13). Then (4, (, q,g) is an almost contact metric structure whose fundamental form is d q . Next, we study the integrability of almost contact metric structures. Let (4,
(3.14)
for any vector field X on M and any Coofunction f on M x R, t being the canonical coordinate in R . From (3.14) we deduce that J2 = - I d , and, thus, J is an almost complex structure on M x R.
Definition 3.6.8
(4, C, q ) i s said to
Proposition 3.6.9
(4, (,q ) i s
be normal if J i s integrable.
normal if and only if
Proof: Let J be the almost complex structure defined by (3.14) and Nijenhuis tensor. A straightforward computation shows that
N((O,d/dt), (O,d/dt)) = 0.
From (3.15), it follows that the vanishing of N implies that
N#
+ 2 € @ ( d q ) = 0.
Conversely, suppose that Nd
+ 2€ @I ( d q ) = 0. Then we have
0 = W X , €1
+ 2(drl)(X,€I€
N its
(3.15)
3.6. Almost contact structures
175
Applying now q , we obtain
Thus, N = 0 . 0 To end this section, we introduce an important class of almost contact metric structures. Let q be a contact form on M . By Proposition 3.6.7,there exists a contact metric structure (4, t,q,g) on M .
Definition 3.6.10 A normal contact metric structure ($,c,q,g) i s said to be Sasakian. A Sasakian structure is the odd-dimensional counterpart to Kahler manifold. In fact, we may prove that, if ($, €,q , g ) is a Sasakian structure and V is the Riemannian connection defined by g, then we have
(see Blair [8]).
Remark 3.6.11 For a classification of almost contact metric structures we remit to Oubiiia [1O2].
Chapter 3. Structures on manifolds
176
3.7
f-s t ruct ures
In this section we study f-structures on manifolds and give integrability conditions of an f-structure. D e f i n i t i o n 3.7.1 (Yano 11291) A non-null tensor field f of constant rank, say r, on an m-dimensional manifold M satisfying
is called a n f a t r u c t u r e (or f($,l)-i3tructure).
If m = t , then an f-structure gives an almost complex structure on M and m = r is even. If M is orientable and m - 1 = r, then an f- structure gives an almost contact structure on M and m is odd (see Yano [129]). We set
t=
-f2,
m = f 2 +Id,
where Id is the identity tensor field on M . Then we have
ft = tf = f , mf = f m = 0. Therefore we obtain two complementary distributions L = I m t and M = I m m corresponding t o the projection tensor t and m, respectively. If rank f = r, then L is r-dimensional and M is (rn - r)-dimensional. For each point z E M we have
T,M = L, EIM,. Since f : ( t X ) = - t X for all X E T,M, then L , is a vector space with complex structure f,/L,. Hence t must be even, say r = 2n. Now let {el,. . . , e n , f e l , . . . , f e n } be a basis of L , and {e2,,+1,. . . ,em} a basis of M,. Hence {el,. . . ,em, f e l , . . . , f e n , e2n+1,. . . ,em} is a basis of T,M which is called an adapted frame (or f-basis) at z. If {ei,. . . ,e;, fe:, . . . , fe;, ein+l,. . . ,e h } is another f-basis a t z we have
3.7. f-structures
177
f e i = -Bfe,
+ Ai(fe,),
eb = Caeb, b
+
1 5 a , b 5 m, where A, B are n x n matrices and 1 5 i , j 5 n, 2n C E Gl(m - r,R). Then the two f-basis at 2 are related by the m x m matrix ff=
q
; iB o o c
Clearly a E Gl(m, R). Now let G be the set of such matrices; G is a closed subgroup of Gl(m, R) and therefore a Lie subgroup of Gl(m, R ) which may be canonically identified to Gl(m,C) x Gl(m - 2n, R). If we set
B = {f-basis at all points of M } it is not hard to prove that B is a (Gl(n,C) x Gl(m - 2n, R))-structure on M . The converse is also true, i.e., if B is a (Gl(n,C) x Gl(m - 2n,R))structure on M then it determines an f-structure on M (details are left to the reader as an exercise). Thus we have:
Proposition 3.7.2 Giving an f-structure is the s a n e as giving a (Gl(n, C )x G l ( m - 2n, R))-structure on M.
Remark 3.7.3 With respect to an f-basis at z f , f! and m are represented by the matrices
0
respectively.
0
0
0
0
178
Chapter 3. Structures on manifolds
Definition 3.7.4 A Riemannian metric g on M is said t o be adapted t o an f-structure f i f (1) g ( t X , m Y )= 0, i.e., L and M are orthogonal with respect to g ; (2) g ( f ex,f e y )= s ( 4 In such a case ( M ,f , g ) is called a metric f-structure.
w.
Proposition 3.7.5 There always exists an adapted metric. Proof Let h be an arbitrary Riemannian metric on M . We define h' by
h ' ( X , Y ) = h ( t X , t ~+) h ( m X , m Y ) Then h' is a Riemannian metric on M such that L and Hence an adapted metric g t o f is given by
M
are orthogonal.
Now, proceeding as in Section 3.4, it is not hard t o prove the following.
Proposition 3.7.6 Giving a metric f-structure is the same as giving a (U(n) x O ( m - 2n))-structure on M . Now, suppose that L is integrable. Then f operates as an almost complex structure on each integral manifold of L.
Definition 3.7.7 When L is integrable and the induced almost complex structure is integrable on each integral manifold of L, we say that the f structure f is partially integrable. Proposition 3.7.8 f is partially integrable i f and only if N f ( t X , l Y ) = 0, where N , is the Nijenhuis torsion o f f . The proof is a direct consequence of Theorem 3.2.6.
Definition 3.7.9 An f-structure f on M is integrable if it is integrable as a ( G l ( n , C )x G l ( m - 2 n , R))-structure, i.e., for each x E M there ezists a coordinate neighborhood U with local coordinates (xl,. . . , xn, xn+' , . . . ,x2n, z~~+',. . . ,z m ) such that f is locally given in U b y f =
[ I ::] -1,
0
0
179
3.7. f-structures Theorem 3.7.10 f is integrable i f and only i f N j = 0. We remit t o Yano and Kon [131]for a proof. In a similar way, we can consider f ( 3 ,-1)-structures
on manifolds.
Definition 3.7.11 A n f ( 3 , -1)atructure on an m-dimensional manifold M is given b y a non-null tensor field f of type ( 1 , l ) on M of constant rank r satisfying 13 -
f = 0.
If we set
t = f 2, m = - f 2 + I d we have
! + m = l d , t2 = t , m 2 = m , h = m t = O .
f t = tf = f , f m = mf = 0. Then L = Irn t and M = I m m are complementary distributions on M of rank r and rn - r , respectively. When m = r then f is an almost product structure on M . Since f , ” ( t X ) = tX,for all X E T,M, then fi acts on L , as an almost product structure operator. If we set
1 2
1
P = - ( I + f)!, Q = i ( 1 - f)! then P 2 = P , Q 2 = Q and PQ = QP = 0. Hence P = IrnP and Q = I m Q determines two distributions on M of dimension p and q, respectively, such that p q = r and
+
T z M = Pz @ Qz
@Mz,
since L, = P, CB Q,. Now, proceeding as above, we have
Proposition 3.7.12 The following three assertions are equivalent: (1) M possesses an f ( 3 , -1)-structure of rank r; (2) M possesses a ( G l ( p , R ) x G l ( q , R )x Gl(m - r,R))-structure; (8) M possesses a ( O ( p )x O(q) x O ( m - r))-structure. Proposition 3.7.13 A n f(3,-1)-structure is integrable if and only i f N , = 0.
Chapter 3. Structures on manifolds
180
3.8
Exercises
3.8.1 (1) Let ( E , w ) be a symplectic vector space (see Chapter 5). Prove that there exists a complex structure J and a Hermitian inner product g on E such that the 2-form f2 given by n(z,y ) = g(z, J y ) is precisely w . (Hint: Choose any inner product on E. Then we define a linear isomorphism k : E + E by w ( z , y ) =< z , k y >. If k2 = -Id, we are done. Otherwise we consider the polar decomposition k = R J , where R is positive definite symmetric, J is orthogonal and J R = R J . Since kt = - k , we deduce J t = - J and then J 2 = -Id. Also, w ( J z , J y ) = w ( z , y ) . Now, we define a Hermitian inner product g by g ( z , y ) = w ( z , J y ) ) . (2) Let ( S , w ) be a symplectic manifold. Prove that there exists an almost Hermitian structure ( J ,g ) on S such that its Kahler form is precisely w. 3.8.2 Prove that, if V is an involutive distribution on a manifold M, then there exists a symmetric linear connection V on M such that V x Y E V for all Y E V (see Walker (1958)). 3.8.3 If F is an integrable almost product structure on a manifold M, then prove that any linear connection V on M is an almost product connection. Hence there exists a symmetric almost product linear connection on M. 3.8.4 (1) Let f be an f-structure on M of rank r. Prove that f c is an f-structure of rank 2r on T M . (2) Prove that f c is partially integrable (resp. integrable) if and only if f is partially integrable (resp. integrable). (3) Let V be a linear connection on M . Prove that the horizontal lift f H of f to T M with respect to V is also an f-structure of rank 2r on T M .
181
Chapter 4
Connections in tangent bundles 4.1
Differential calculus on TM: Vertical derivat ion and vertical differentiation
In this section we develop a differential calculus on tangent bundles determined by the canonical almost tangent structure and the Liouville vector field. Let M be a differentiable m-dimensional manifold and T M its tangent bundle. We define a canonical vector field C on T M as follows:
C is called the Liouville vector field on T M (sometimes we use the notation CM). We locally have
c = v'(d/dv')
(44
Let J be the canonical almost tangent structure on T M . From (4.1)) we easily deduce that JC = 0.
Remark 4.1.1 In Section 4.2, we shall give an alternative definition of C. We now consider the adjoint operator J* of J ; J* is defined by
Chapter 4. Connections in tangent bundles
182
(J*W)(Xi,.. . , X p )= w ( J X 1 , .. . , J X p ) , X i , . .., X p E x ( T M ) , w E A ~ ( T M ) .
(44
From (4.2), we deduce that J* is locally characterized by J * ( f )=
f, f E C Y T M ) ,
J*(dz') = 0, J*(dv') = dz', where ( z i , v i ) are the induced coordinates in T M .Then J* does not commute with the exterior derivative d o n T M .
Proposition 4.1.2 We have i x J * = J* 0 i j x .
Proof In fact, ( i x J * ) ( f )= i x ( J * f )= ixf = 0. On the other hand,
( J * 0 i j x ) ( f ) = J * ( i j x f )= J * ( o ) = 0. Moreover, if w E r\P(TM), we have
( ( i x J * ) ( w ) ) ( X i..., , Xp-1) = ( i x ( J * w ) ) ( X i.,. . , X p - 1 ) = ( J * W ) ( X , X i , .. . , X p - l ) = w ( J X , J X i , .. ., JXp-1) = ( J * W ) ( X , X I ,.. . , X p - l ) = ( i x ( J * w ) ) ( & , .. . , X P - 1 )
= ( ( i X J * ) ( W ) ) ( X*l., , X p - l ) ,
X i , ...,X p - l E x ( T M ) . U
Corollary 4.1.3 We have icJ* = 0.
Proof: In fact, JC = 0 . 0
4.1. Differential calculus on TM
183
4.1.I Vertical derivation We define the vertical derivation i J as follows:
i J f = 0, f E Cm(TM), P
( i J w ) ( X l ,... , X p ) = C W ( X 1 , .. . ,J X i , . . . ,Xp), (4.3) i=l w E A P ( T M ) , X 1..., , xpEX(TM). Then i J is a derivation of degree 0 of A ( T M ) (see Section 1.13) and a derivation of degree 0 and type i, in the sense of Frolicher and Nijenhuis. From (4.3), we deduce
Then we have
ij(dX') = O,ij(dV') = dx'.
(4.4)
From (4.2) and (4.4), we easily deduce the following.
Proof: We only prove (2); (1) and (3) are left to the reader as an exercise. If f E C m ( T M ) ,we have
and
If w E AP(TM),we obtain
Chapter 4. Connections in tangent bundles
184
= w(JX, XI,. . . ,XP-1)
+ c W(X,Xl,. . .,JX;, . . . ,x p - 1 ) P-1
i=l
c
P- 1
-
W
(X,x1,. . - ,JXi , . - * ,XP-1)
i=l
=w(JX,&,..~,xp-l)
XI,. . . ,xp-lE X ( T M ) .0 From (4.3), we deduce by a straightforward computation, the following:
Proposition 4.1.5 We have iJ(W
A
r)= (ijW) A r
+
W
A
(ijT),
w,r E A(TM).
4.1.2
Vertical differentiation
We define the vertical differentiation dJ on
T M by
d j = [ i j , d ]= i j d - d i j .
(4.5)
Then
and d j is a skew-derivation of degree 1 of A ( T M ) and a derivation of degree 1 and type d, in the sense of Frolicher and Nijenhuis. From (4.5) we easily deduce that
4.1. Differential calculus on TM
185
d j f = J * ( d f ) ,d j ( d f )= - d ( J * ( d f ) ) , f E C m ( T M ) . Then in local coordinates we have
d j ( d z i ) = d j ( d v i ) = 0. Proposition 4.1.6 We have
Proof In fact,
Proposition 4.1.7 We have (I) d: = 0, (2) d j ( w A 7 ) = ( d j w ) A 7 -k ( - l ) p w A ( d j . ) , if
(d
E AP(TM).
Proof (2) is a direct consequence of Theorem 1.12.1, and Proposition 4.1.5. To prove ( l ) ,it is sufficient to check that d$ f = 0 and d:(df) = 0, for any function f on TM. Let f E C m ( T M ) . Hence
= C [ d j ( a f / a v ' )A d d
+ ( a f / a ~ ~ ) d j ( d ~(by ' ) ]( 1 ) )
:
=
x(a2
f / a v ' d v J ) dxj A dx'
i<j
+ C(a2 f / a v i a v i ) dz' i>j
A
dz'
Chapter 4. Connections in tangent bundles
186
=
C(a2 f/ad&'))(dzJ
A
dd
+ dx'
A
hi)= 0 ,
i<j
since dxj A dx' = -dxi A d d . Moreover, if w = d f , f E C m ( T M ) ,we obtain
Proposition 4.1.8 W e have (1) [ i j , d j ] = i j d j - d j i j = 0, (2) [ ; c , d j ]= i c d j d j i c = i j , (3) [ d J ,LC] = dJLC - LCdJ = dJ, (4) J*dJ = 0, (5) d j J* = J*d.
+
The proof is left to the reader as an exercise.
4.2
Homogeneous and semibasic forms
In this section, we shall introduce two important classes of differential forms on tangent bundles.
4.2.1
-
Homogeneous forms
Let ht : T M
T M be the homothetia of ratio
e t , that is,
h t ( y ) = e'y, y E T,M, x E M .
Then ht is a vector bundle isomorphism (in fact, its inverse is h - t ) . An easy computation shows that ht is a (global) 1-parameter group on T M . Moreover, we have the following.
Proposition 4.2.1 The Liouville vector field C o n T M generates the 1parameter group ht .
4.2. Homogeneous and semibasic forms
187
Proof In fact, ht is locally given by h t ( z i ,u') = ( z i ,e'u').
Then, if X is the infinitesimal generator of ht, we have
X ( z i )= (d/dt)/t,o(zi ~ ( 0 ' )=
(d/dt)/,,o(u
i 0 ht)
o ht) = 0,
and
= ( d / d t ) p o ( e t v i ) - ui .
Thus, we obtain
x = u'(a/du'). Hence X = C.0
Definition 4.2.2 A differentiable function f on TM is said t o be homogeneous of degree r if L c f = rf.
(4.7)
Hence f is homogeneous of degree r if and only if rf(zi, u') =
C ui(af/dui). i
Proposition 4.2.3 A differentiable function f on TM is homogeneous of degree r if and only if
h;f = ertf, t E R.
Proof Suppose that f is homogeneous of degree r. Hence L c f = r f . Then, for each point y E T M ,(h;f)(y) = (f o ht)(y) is a solution of the differential equation ( d u l d t ) = ru with initial condition u(0) = f (y). From the uniqueness of solutions of differential equations, we deduce that h t f = ertf . Conversely, if h; f = ertf , we have
= lim [(er' f - f ) / t ] (by Proposition 4.2.1) t h o
im [(ert -( L o
l)/t])f = r f . 0
Next, we extend the notion of homogeneity to vector fields and forms.
188
Chapter 4. Connections in tangent bundles
Definition 4.2.4 A vector field X on T M i s said to be homogeneous of degree r if [C,X] = (r-1)X. The following result can be proved in a similar way as the Proposition 4.2.3.
Proposition 4.2.5 A vector field X on T M i s homogeneous of degree r if and only if the following diagram e('-')'Tht TTM * TTM
ht
TM
*
TM
i s commutative, i.e.,
Suppose that
x = x'(a/az') + Y'(a/av'). Then, from (4.7)) we easily deduce that X is homogeneous of degree r if and only if
C
(r - 1 ) X t = C d ( a ~ ' / a v j )rY' , = vJ(ay'/avj), i i
1 5 i I: m = dim M . Hence X is homogeneous of degree r if and only if geneous of degree r-1 (resp. r).
X' (resp. Y') are homo-
Proposition 4.2.6 Let X and Y be homogeneous vector fields on T M of degree r and 8, respectively. Then [XIY] i s homogeneous of degree r+s-1. Proof: We have
+
Lc [ X ,Y ]= [ [C,XI,Y ] [ X , [C, Y]] (by Jacobi identity)
+
= (r - 1 ) [ X , Y ] (s - 1 ) [ X , Y ]
4.2. Homogeneous and semibasic forms
= (r
189
+8 - 2)[X,Y].o
Now, let x ( T M ) be the set of all homogeneous vector fields of degree 1 on T M . From Proposition 4.2.6, we deduce that R ( T M ) is a Lie subalgebra of X(TM).
Definition 4.2.7 Let w be a difierential f o r m on T M ; w i s said to be homogeneous of degree r i f LCW = rw. As above, we deduce that w is homogeneous of degree r if and only if
h;'w = e"w, t E R. Moreover, if w is a Pfaff form locally expressed by
w = a,dz'
+ bidv',
then w is homogeneous of degree r if and only if a, (resp. b;) is homogeneous of degree r (resp. (r-1))) 1 5 i 5 m. Some properties of homogeneous vector fields and forms follow.
Proposition 4.2.8 (1) If w and T are homogeneous forms of degree r and s, respectively, then w A r i s homogeneous of degree r t s . (2) Let X be an homogeneous vector field of degree r and f an homogeneous function of degree 8. Then X f i s an homogeneous function of degree r+s-1. (3) Let w be an homogeneous p-form of degree r and XI,. . .,X,p homogeneous vector fields of degree s. Then w ( X 1 , . . . ,X,) i s an homogeneous function of degree r+p(s-1). Next, we study the behaviour of d , i c , i j and d j acting on homogeneous forms.
Proposition 4.2.9 Let w be an homogeneous p-form of degree r on T M . Then d w , i c w , i j w and d j w are homogeneous forms of degree r, r, r - 1 and r-1, respectively. Proof: In fact, we have
Chapter 4. Connections in tangent bundles
190
LC(iJW)
= ic(Lcw)
(since [ L c ,i c ] = Lcic - icLc = 0 )
= r(icw),
L c ( i ~ w=) i ~ ( L c w-) i ~ (by w Proposition ( 4 . 1 . 4 ) )
= r(iJw) - i j w = (r - l ) ( i j w ) ,
L c ( d ~ w= ) d ~ ( L c w-) d J w (by Proposition (4.1.8))
= r(dJw) - dJw = (r - l ) ( d j w ) . U
Finally, we introduce the notion of homogeneity of pvector forms (i.e., tensor fields of type ( 1 , p ) ) . Definition 4.2.10 A p-vector form L on of degree r if
L c L = (r
-
TM is said
t o be homogeneous
1)L.
Remark 4.2.11 Sometimes, the functions, vector fields and forms are supposed to be defined only on T M = T M - {zero section}. All the definitions and results in this section hold in such a case.
4.2.2
Semibasic forms
Definition 4.2.12 Let a! be a differentiable form on TM; semibasic if a! E I m P .
a!
is said t o be
4.2. Homogeneous and semibasic forms
191
Let us denote by SPB the set of all semibasic p-forms on T M , p 2 0 . (Obviously, SOB = C m ( T M ) ) .Since J*(dx') = 0, J*(dv') = dx', then SpB is locally spanned by
(hence the name). If we set m
S B = @SPB,rn = dim M, p=o
then S B is called the ( g r a d u a t e d ) algebra of semibasic f o r m s on T M . Next, we study the behaviour of d , i c , i j and dJ acting on semibasic forms.
Proposition 4.2.13 If a E SPB, then (1) i c a = i ~ c y= 0; (2) d J a E SP+'B. P r o o f (1) follows from ic J* = i j J * = 0 (by Corollary 4.1.3 and Proposition 4.1.4). (2) follows from dJ J* = J*d (by Proposition 4.1.8).0 Corollary 4.2.14 Iff i s a diferentiable function on T M , then d J f i s a semibasic P f a f form. Now, let f be a differentiable function on TM. Since
(af / a z ' ) d z '
df =
+ (af/ao')du',
i
we deduce that df is not, in general, a semibasic form on T M . Next, we stablish some important properties of semibasic Pfaff forms.
Proposition 4.2.15 Let cy be a Pfaf form on T M . Then a i s semibasic if and only ifcy(X) = 0, for any vertical vector field X on T M . Proof In fact, if a = J*p, then a ( J X ) = 0, since J 2 = 0. Conversely, if a ( X ) = 0, for any vertical vector field X on T M , then a is locally given by a = a i ( x , u)dx'. Hence cy is semibasic. Let a be a semibasic Pfaff form on T M . Then we can define a mapping
Chapter 4. Connections in tangent bundles
192
D :T M
-
T*M
as follows. Let y E T,M, z E M ; then D ( y ) E T,*M is given by
D ( Y ) ( X )= a!,(X), where X E T,M and X E T , ( T M ) is an arbitrary tangent vector such that ( r M ) * ( X ) = X . In fact, if P E T , ( T M ) is any other tangent vector such that ( r M ) * ( v ) = X , then ( ? M ) * ( X = 0 and so X - is vertical. Hence a,(X - a) = 0 by Proposition 4.2.15. We note that the diagram D
v)
v
-
TM
T*M
-
is commutative, where ?M : T M M and X M : T M + M are the canonical projections. Hence D is a bundle morphism, but it is not, in general, a vector bundle homomorphism. Conversely, suppose that D : T M + T*M is a mapping such that R M o D = r M . Then D determines a semibasic Pfaff form a! on T M as follows:
a y ( X ) =< ( W ) * X , D ( Y )>, where y E T , M , X E T , ( T M ) . Clearly a! is semibasic, since, if X is vertical, then (rM)*X = 0. Summing up, we have proved the following.
Theorem 4.2.16 The correspondence a ( X ) =< (rM)*X,D ( r M ( X ) )>, X E T T M , determines a bijection between the semibasic Pfaf forms o n TM and the mappings D : T M -+ T * M such that X M o D = ? M .
Let us remark that if the semibasic Pfaff form a!
= a!&,
v)dz',
a!
is locally expressed by
4.3. Sernisprays. Sprays. Potentials
193
then we easily see that corresponding mapping
D is locally given by
. . D(z',v') = ( Z $ q ) .
(4.8)
Let X be the Liouville form on T * M . We recall that X is locally given by
where ( z i , p i ) are the induced coordinates in T * M . Then we have the following.
Corollary 4.2.17 D*X = a.
Proof: Using (4.8), we obtain D*X = D*(p&)
= (pi
0
D)d(z'
0
0 ) = a&'
= cY.0
To end this section, we introduce the notion of semibasic vector forms.
Definition 4.2.18 A vector p-form L on TM is said to be semibasic if (1) L(X1,. . . ,X,) i s a vertical vector field, for any vector fields XI,. .., X p ; (2) L(X1,. . . ,X,) = 0, when some Xi i s a vertical vector field. If L i s skew-symmetric, then (1) and (2) become: (1)' J L = 0; (2)' i J X L = 0,for any vector field X on TM.
4.3
Semisprays. Sprays. Potentials
We know that a vector field on a manifold M is the geometrical interpretation of a system of ordinary differential equations (see Section 1.7). The aim of this section is to introduce a class of vector fields on the tangent bundle TM which interprets geometrically a system of second order differential equations.
Definition 4.3.1 A semispray (or second order differential equation) on M is a vector field on TM (that is, a section of the tangent bundle of T M , T T M : T T M + T M ) , C M on T M , which is also a section of the vector bundle TTM: T T M ----t T M .
Chapter 4. Connections in tangent bundles
194
Let ( be a semispray on M . Then ( is locally given by
< = ?(a/az') + C ( a / a v ' ) . Since TM is locally given by
T M ( z ~ ,=~(zi), )
we deduce that
Hence we have T r M ( ( ( z i ,v i ) ) = T T & ,
Yi,
p , (i) = (zi, p ) = ( 2 ,v i ) ,
because ( is a section of T T M . Thus, we obtain semispray is locally given by ( = vi(d/dzi)
where
p
= v i . Therefore, a
+ C'(a/av'),
(4.9)
= c ( z , v ) are Cooon T M .
From (4.9))we easily deduce the following.
Proposition 4.3.2 A vector field and only if J( = C .
< on TM, Cooon T M , i s a semispray if
Definition 4.3.3 Let ( be a semispray on M. A curve u on M i s called a path (or solution) of ( if u i s an integral curve of (, that is, U(t) = ( ( b ( t ) ) .
In local coordinates, if u ( t ) = ( z i ( t ) ) ,then we deduce that u is a path of ( if u satisfies the following system of second order differential equations: ( d 2 z i / d t 2 ) = is also isotropic.0 It is clear that if K is Lagrangian in ( V , w ) then its complement K' in V is also Lagrangian. Also from the proof of Proposition 5.1.12 one has that every finite dimensional symplectic vector space has a Lagrangian subspace.
Proposition 5.1.13 Suppose that K is an isotropic subspace of a vector symplectic space ( V , w ) . K is a Lagrangian subspace if and only if dim K = (1/2) dim V.
Proof If V is symplectic then dim V = 2n. If K is Lagrangian then K = K'. Thus dim V = 2 dim K . Conversely, suppose dim K = (1/2) dim V , that is, dim K = n. Then n = dim K' (since dim V = dim K+ dim K'), and so K K L because K is isotropic.0 Now, let K be a vector space of dimension n . Consider V = K @ K*, where K* is the dual space of K . We may define a symplectic form on V as follows: w ( u + c Y , u + ~ )= C Y ( O )
-P(u),
U,U€
K , C Y , ~K*. E
One easily proves that K is a Lagrangian subspace of ( V , w ) and its Lagrangian complement is, precisely, K*. Conversely, let (V, o)be a symplectic
232
Chapter 5. Symplectic manifolds and cotangent bundles
vector space and K a Lagrangian subspace in ( V , w ) . By K' we represent the complement of K and consider the isomorphism from K' to K* given by &,(u) = ivw, i.e., 3 , is the restriction of S, composed with the canonical projection of V * onto K*. Then, for all u,u E K , ii, ij E K' w(u
+ i i , v + a) = w(u,a) + W ( i i ) t J ) = SW(ii)U- S,(S)U.
Set S W ( G ) = a, S,(V) = P. Then a 2-form WK(U
WK
+ a, + P ) = a(.)
on K @ K* is defined by - P(U).
Moreover, as the mapping 1 @ % is an isomorphism from K @ K' onto K @ K*, WK is a symplectic form on K @ K* such that
( I @ .Sw)*wK = w , i.e. the following diagram
( K @ K') x ( K @K ' ) l ( 1 @ S W x) (1 @ S , )
( K @ K*) x ( K @ K * )
\
R
-4/
-
is commutative. Hence 1 @ % : V = K @ K' K @ K* is a symplectic isomorphism and V may be identified to K @ K*. Next, we extend these definition to vector bundles.
Definition 5.1.14 Let ( E , p , M ) be a vector bundle. Suppose that, for each x E M ,there ezists a syrnplectic form W ( X ) on the vector space E, = p-'(x) such that the assignernent x w(x) i s C" (i.e., if s1 a n d s 2 are two Cm sections of E, then w ( s 1 , s 2 ) defined by W ( S I , S ~ ) ( Z ) = o(z)(q(x),s2(z)) i s a C" function on M). Then E i s said to be symplectic. Now, let (E,p,M) be a symplectic vector bundle a n d K a subbundle of E. We define a new subbundle K' of E by
-
( K ' ) , = {e E EZ/w(x)(e,e') = 0, f o r all e' E Ez}, x E M. Then K is said to be isotropic, resp. coisotropic, resp. Lagrangian, resp. symplectic if K , i s isotropic, resp. coisotropic, resp. Lagrangian, resp. symplectic, for every x E M.
233
5.1. Syrnplectic vector spaces
Suppose now that V is a vector space and w a 2-form on V of rank 2s but not necessarily of maximal rank. Let dim V = 2s r, r 2 0. Then (V,w)is said to be a presymplectic vector space (if r = 0, then (V,w) is syrnplectic). The form w is said to be presymplectic. If we consider the linear map S, : V V * then S, is not necessarily an isomorphism.
+
-
Definition 5.1.15 A linear mapping P : V -----t V is a projector o n V if P2 = P . If P is a projector then V = I m P @ k e r P .
If P is a projector then we may define its complement Q = I d - P . Then = Q P = 0 . Also, I m P = ker Q , ker P = I m Q . Let us
Q 2= Q and PQ set
Vp = I m p , VQ = I m Q . Then V = Vp @ VQ. If P* and Q* are the adjoint operators on V * we also have V * = vpt @ V Q ~where , P*a = a o P, Q*a = a o Q , a E V * . Definition 5.1.16 W e say that the projector P is adapted to the presymplectic f o r m w on V if
K e r P = VQ = ker S,. Remark 5.1.17 If Vis a vector space with an inner product and w is a presymplectic form on V then we may take Vp as being the orthonormal complement of ker S, with respect to . This gives an adapted projector on V.
-
Proposition 5.1.18 Let ( V , w ) be a presymplectic vector space with an adapted projector P : V V . If a E V*,then there exists a unique vector u € Vp such that i,w = P*a.
In particular, if a E VpC, then i,w = a. Proof As ker S, = VQ one has that the restriction of S, to the subspace Vp is injective, and so
Chapter 5. Symplectic manifolds and cotangent bundles
234
as P*S,(v) = S,(o) for all v = P(w) E V p . Let us show this assertion. If u E V, since ker S, = VQ,one has (S,(U))U = W ( U , U) = ~ ( vPu,
+ Qu) = w ( v , Pu)+ w ( v ,Qu)
= w ( u , P u ) = (S,(v))(Pu) = P*(S,(v))(u)
and so S,(v) = P*(S,(v))(recall that Q u E K e r S w ) . Therefore S, induces an isomorphism from V p t o V p .
5.2
Symplectic manifolds
Let S be a CM manifold of dimension rn, T S (resp. T * S ) its tangent (resp. cotangent) bundle with canonical projections rs : T S + S (resp. r s : T*S -+ S ) . Let w be a 2-form on S. The rank (resp. corank) of w at a point z of S is the rank (resp. corank) of the form ~ ( zE) A 2 ( T 2 S ) . We say that w is non-degenerate or of maximal rank if for every point z E S, ~ ( zis)non-degenerate.
Definition 5.2.1 An almost symplectic form (or almost symplectic structure) on a manifold S i s a non-degenerate 2-form w on S. The pair ( S , w ) i s called an almost symplectic manifold. Then S has even dirnension, say 2n.
Let (S,w ) be an almost symplectic manifold of dimension 2n. Then, for each z E S, ( T 2 S , w ( z ) )is a symplectic vector space. Thus there exists a symplectic basis {el, . . . ,ezn} for T 2 S ,which is called a symplectic frame at z. Let B be the set of all symplectic frames at all the points of S. If { e i } , { e i } are two symplectic frames at z,then they are related by a matrix A E Sp(n). Further, by using an argument as in section 1, we can find a local section of F M over a neighborhood of each point of M which takes values in B . Hence B is a Sp(n)-structure on S. Conversely, let B s ~ (be~ ) a Sp(n)-structure on S. Then we can define a 2-form w on S as follows:
w ( z ) ( X , Y )= w o ( ~ - ~ X , t - ~ X Y ), Y, E T2M, where J E B s ~ (is~a )linear frame at z. (Obviously, ~ ( zis)independent on the choice of the linear frame z E B s ~ (at~z). ) Since wo is non-degenerate, then w is an almost symplectic form on S. Summing up, we have proved the following.
5.2. Symplectic manifolds
235
Proposition 5.2.2 Giving a symplectic structure i s the same as giving a S p ( n )-structure.
Let (S,w) be an almost symplectic manifold of dimension 2n. Then
wn = w A
.. . A w (n times)
is a volume form on S. Thus we have
Proposition 5.2.3 Every almost symplectic manifold is orientable.
-
Next we define a vector bundle homomorphism S, : T S
T*S
by
S,(X)= i x ( ~ ( ~ X) )E, T,S,
E M.
Proposition 5.2.4 S, i s a vector bundle isomorphism. Proof: Let ( u , z i ) be a coordinate neighborhood of S. Then we have induced local coordinates ( x i , v i ) , ( z i , p i ) on T U , T * U , respectively. Suppose that
C
w =
wijdz' A d z ' ,
lSi,jl2n
where
wij
= - wj,. Hence we have S,(d/dzi) = W i j d d .
Thus the map S, is locally given by . .
S,(z',w') = ( z i ) v % . J i j ) .
Then S, is Cooand rank S, = rank w = 2n. Therefore we have the required result. Furthermore, w defines a linear mapping (also denoted by S,)
s,
: x(S)
-
A'S
given by
S,(X)= ixw. An easy computation shows that S, is, in fact, an isomorphismof CM(S)modules.
236
Chapter 5. Symplectic manifolds and cotangent bundles
Definition 5.2.5 An almost symplectic form (or structure) w on a manifold S i s said to be symplectic if it i s closed, i.e., dw = 0 . Then the pair ( S , w ) is called a symplectic manifold. Remark 5.2.6 If ( S , w ) is an almost symplectic manifold, then TS is a symplectic vector bundle. If, in addition, w is closed, then TS is a symplectic vector bundle such that dw = 0. The reader may take notice of the study developed in the preceding section for vector spaces to reobtain some results in terms of the vector bundle structure of the tangent bundle of a given Coofinite dimensional manifold. For example, a submanifold K of a symplectic manifold (S,w ) is called isotropic, resp. coisotropic, resp. Lagrangian, resp. symplectic in ( S , w ) if T,K c ( T Z K ) * ,resp. (TzK)' c T , K , resp. if it is a maximal isotropic submanifold of S, resp. if ( T , K ) n ( T z K ) * = 0 for each z E K . We have dim K 5 n, resp. dim K 2 n, resp. d i m K = n, if d i m S = 2n. We will return to Lagrangian submanifolds in the next chapter.
Remark 5.2.7 Obviously, K is an isotropic, resp. coisotropic, resp. Lagrangian, resp. symplectic submanifold of (S,w ) if and only if the tangent bundle T K is an isotropic, resp. coisotropic, resp. Lagrangian, resp. symplectic, vector subbundle of TS.
-
Definition 5.2.8 Let ( S , w ) and ( W , a ) be symplectic manifolds of same dimension, say 2n. A diferentiable mapping h : S W i s called symplectic transformation if h*a = w , i . e . ,
f o r all z E S and X I , X2 E T,S.
-
-
For a symplectic mapping h : S W one has that d h ( z ) : T,S Th(,)W is a symplectic isomorphism. Thus h is a local diffeomorphism. If h is a global diffeomorphism then h is said to be C%ymplectic diffeomorphism (or symplectomorphim). In particular, when S = W then a symplectic map h : S W preserves the symplectic form w on S , i.e., h*w = w . In such a case h is said to be a canonical transformation. This definition is more general than the ones adopted in Classical Mechanics (which states that a transformation is canonical if it preserves the Hamilton equations, see Arnold [4]).
-
5.3. The canonical symplectic structure
237
Definition 5.2.9 Let ( S , w ) be a symplectic manifold. A vector field X o n S i s called a symplectic vector field (or a n infinitesimal symplectic transformation) i f i t s f l o w consists of symplectic transformations. Proposition 5.2.10 T h e following assertions are equivalent: (1) X i s a symplectic vector field; (2) the Lie derivative L x w = 0; (8) i x w = df (locally) f o r s o m e f u n c t i o n f , i.e., d ( i x w ) = 0. Proof: The equivalence of (1) and (2) follows from the definition of Lie derivative and from the fact that p t , the flow of X , is symplectic:
The equivalence of (2) and (3) follows from the H. Cartan formula Lxw = (ixd
+dix)w = dixw
and the PoincarC lemma. -
5.3 The canonical symplectic structure on the cotan-
gent bundle In this section we shall prove that the cotangent bundle of a manifold carries a natural symplectic structure.
-
Let M be an n-dimensional manifold, T*M its cotangent bundle and T M : T*M M the canonical projection. We define a canonical l-form AM on T*M as follows:
M P ) ( X ) = P ( Z ) ( d W ( P ) X ) ,x E T , ( T * W , P E T i W . If ( q i ) are coordinates in M and (qi,p;) are the induced coordinates in T * M , we obtain
Chapter 5. Symplectic manifolds and cotangent bundles
238
Then AM is locally expressed by
Definition 5.3.1 XM is called the Liouville form
on
T*M.
The following proposition gives an important property of the Liouville form.
Proposition 5.3.2 The Liouville form AM on T * M such that
P* for any I-form
/3
on
on
T*M i s the unique 1-form
AM = P ,
(5.3)
M.
Proof: Suppose that P is locally given by
P = Pidq', i . e . ,
Then we have
Furthermore, let X be a 1-form on T * M such that (5.3)holds for A. If X is locally given by X=
aidqi
+
i
and
P = xi &dqi is an arbitrary 1-form
bidpi,
i
on M . We obtain
5.3. The canonical symplectic structure
239
Hence
which implies a, = p;,
b; = 0.
Thus X = XM.O
If we now set
we locally have WM
=Cdp.'Adp;
(5.4)
i
From (5.4) one easily deduces that WM is a symplectic form on T * M , which is called the canonical symplectic form on T * M . Remark 5.3.3 Since T*M carries a symplectic structure that it is orientable.
Now, let morphism
F :M
WM
we deduce
+ A4 be a diffeomorphism. We may define a diffeo-
T*F : T*M
-+
T*M
as follows:
( T * F ) ( a ) ( X= ) a ( d F ( s ) X ) ,(I! E T,*M, X E TF-I(,.M Since F is a diffeomorphism, we may choose coordinates in M such that F is locally given by the identity map, i.e.,
F : (q')
-
(6)
Chapter 5. Symplectic manifolds and cotangent bundles
240
Then T*F is also given by the identity map:
T*F : ( 4 , p i )
+
(8,pi).
Thus, we have
(T*F)*AM = AM and then (T*F)*WM = W M . Hence we have,
Proposition 5.3.4 T*F
is a
symplectomorphism.
We will return to symplectic manifolds in Section 5.6.
5.4
Lifts of tensor fields to the cotangent bundle
Let M be an m-dimensional manifold, T*M its cotangent bundle and T*M + M the canonical projection.
TM :
Vertical lifts If f is a function on M , then the vertical lift of f to T * M is the function f" on T*M defined by
f " =f O T M . In local coordinates (qi,p;) we have
f " (8,Pi) = f (d)
(5.5) As in the case of the tangent bundle we can consider the vertical bundle V ( T * M )defined by
V ( T * M )= Ker{TnM : T T * M
-
TM},
i.e.,
V(T*M)=
u
zET'M
Vz(T*M),
-
where V Z ( T * M = ) Ker{dnM(e) : T z ( T * M ) TrM(z) M } , for all t E T * M . A tangent vector v to T * M a t J such that u E V,(T*M) is called vertical. A
5.4. Lifts of tensor fields to the cotangent bundle
24 1
vertical vector field X is a vector field on T * M such that X ( z ) E V Z ( T * M ) for all z E T * M . Now, let a be a 1-form on M . The vertical lift of a to T * M is the vertical vector field a" on T*M defined by au = -(S U M )-1(&a ).
If a is locally given by a = cY;(q)dq', then we have au = a;(q)8/8p;
From (5.5) and (5.6) we obtain
(fa)"= fUaU,[aU,PU] = 0,
for all function f and all 1-forms
a , P on M .
The operator i If X is a vector field on M , we define a function iX on T*M by
for all a E T,*M. If X = X i 8/8q', then we have
(iX)(qi, p;)=pix;
(5.7)
Now, let F be a tensor field of type (1,l)on M . We define a 1-form i F on T * M by
where a E T * M , 2 E Ta(T*M). If F = F! 8 / 8 q j @ dq', then we have
i F = p; Fidq' (5.8) If S is a tensor field of type (1,s) on M , we define a tensor field is of type ( 0 , s ) on T * M by
242
Chapter 5. Symplectic manifolds and cotangent bundles
where a E T * M , 21,.. . If
,zzT,(T*M). E
then we have
i s = p,Sjl...j,dfl 8 . . . @ dpl".
(5.10)
Thus we obtain an operator
The operator 7 As we have seen, the canonical symplectic structure O M on T*M induces an isomorphism
This isomorphism may be extended to an isomorphism (also denoted by SW,
1
as follows:
(SLJ,K)(%,.
. . , ? a ) =< K(21,
a
.
,*s-l),
SUM(%)
>,
for all 21,.. . ,g8E X ( T * M ) . Hence, if S is a tensor field of type (1,s) on M , we define a tensor field 7s of type (1,s - 1) on T*M given by
7s = -(sUM)-l(;s).
If S is locally given by (5.9), then we obtain (5.11)
5.4. Lifts of tensor fields to the cotangent bundle
Hence we have an operator 7 : T,l(M)
-
243
T,'_,(T*M)
A direct computation from (5.11) shows that 7(S
+ 2') = 7s + 7 T .
If F (resp. S) is a tensor field of type (1,l) (resp. (1,2)) then (5.11) becomes (5.12)
7s = p,$ka/dp,
@ dqk)
(5.13)
since (5.8) and (5.10). From (5.6), (5.12) and (5.13) we easily deduce the following
Proposition 5.4.1 Let be a E A'M, F,G E q l ( M ) and S E T,'(M). Then we have
[ ( Y " , ~ F=] ((YO F ) " ,
[ 7 F ,7G] = 7[F,GI, where a o F is a I-form on M defined b y
((YO
F ) ( X )= a ( F ( X ) ) .
Complete lifts of vector fields Let X be a vector field on M . Then the complete lift of X to T * M is the vector field X" on T * M defined by
X" = ( S W M ) - ' ( d ( i X ) ) If X = Xic3/aqi, then we have
xc= xia/ad - Pj(axj/aqi)a/api
(5.14)
From (5.5), (5.6), (5.12), (5.13) and (5.14), we obtain the following
Chapter 5. Syrnplectic manifolds and cotangent bundles
244
Proposition 5.4.2 Let be f E Cm(M), X,Y E x ( M ) , a E A'M, F E ql(M) and S E q'(A4).Then we have
(X
+ Y)' = X c + Y c ,( f x ) =" f " x c- ( ; X ) ( d f ) ' ,
where S x i s the tensor field of type ( l J l )on M defined b y S x ( 2 ) = S ( X , 2) for any Z E x ( M ) .
Complete lifts of tensor fields of type (1,l) Now, let F be a tensor field of type (1,l) on M. Then the complete lift of F to T*M is the tensor field F C of type ( 1 , l ) on T*M given by
Fc = (SuM)-'(d(iF)). If F is locally given by F = F i a/aqi 18 dqj, then we have
(5.15)
+Fj' a l a p i I8 dpj , since (5.8). From (5.6)) (5.12) and (5.15) we obtain the following
Proposition 5.4.3 Let be a E A'MJ X E x ( M ) and F,G E T,'(M). Then we have F e d = ( a 0 F)', F c ( 7 G ) = 7 ( G F ) , F c X c = (FX)'
+~ ( L X F ) .
5.5. Almost product and almost complex structures
245
Complete lifts of tensor fields of type (1,2) Suppose now that S is a skew-symmetric tensor field of type (1,2) on M . Then it is not hard to prove that is is a 2-form on T * M . Then we define the complete lift of S to T*M by
Thus S" is a tensor field of type (1,2) on T * M . By a straightforward computation from (5.10) we obtain the following
Proposition 5.4.4 Let be X , Y 6 x ( M ) , cr,p E A'M, F , G E T,'(M) and S E T;(M). Then we have S"(a',p') = 0 , S C ( a " , 7 G )= 0 ,
S C ( 7 F , 7 G )= 0 ,
S c ( ~ ' , Y c=) - ( a o Sy)',
5.5
Almost product and almost complex structures on the cotangent bundle
In this section we apply the constructions of lifts of tensor fields to obtain some interesting structures on T*M. Before proceeding further we prove the following lemma.
Lemma 5.5.1 Let T*M such that
51 and
52
be tensor fields of type ( 0 , s ) ( o r (1,s)) on
&x;,. . . , X S )
j;,(SE,. . . ,X,") = I
*
for any X i , . . . ,X,E x ( M ) . Then Si = S2.
Chapter 5. Symplectic manifolds and cotangent bundles
246
Proof: It is sufficient to show that if g(Xf,. . . ,XS)= 0 for any XI,. . . ,X, E x ( M ) , then = 0. We only prove the case of tensor fields of type (1,l). The general cme may be proved in a similar way. Let F be a tensor field of type (1,l) on T*M such that FXe = 0 for any vector field X on M. Then
implies
Now suppose that X is a vector field on M locally given by X = Xia/aqi. From (5.14) we have
which implies
Hence
F:
M ?
= F;' = 0 except on the zero-section. Since
F:
continous it follows that F i =
F;! = 0 at all T * M . o
Proposition 5.5.2 Let F be
a tensor field of t y p e ( 1 , l ) on
M
.
have
+
( F c ) 2= ( F 2 ) " ~ N F , where NF is the Nijenhuis tensor of F .
and
F:
-
are
M . Then we
5.5. Almost product and almost complex structures
247
Proof: In fact, we have
(Fc)2xc = F"(Fx)c
+7 ( L x F ) )
+
+(LxF)F},
= ( F 2 W C 7{LFXF since Proposition 5.4.3. On the other hand we obtain
+
= (F2x)c 7{LxF2
+ (NF)X)
since Propositions 5.4.2 and 5.4.3.
Now we have
(LFXF + ( L x F ) F ) ( Y )= [ F X ,FYI - F [ F X , Y ]
+[X, F2Y]- FIX,FYI =
[X, F2Y]- F 2 ( X , Y ]+ [FX, F Y I - F ( F X , Y ]- F(X, FYI
+ F2(X,Y]
+
= (LxF2)Y N F ( X , Y ) = ( LXF2
for any
+ (NF)X)y,
Y E x ( M ) . Hence (FC)2XC = ((F2)C + 7 i V F ) X C ,
for any X E x ( M ) , from which we have the required result since Lemma 5.5.1.0 By a straightforward computation from Propositions 5.4.1, 5.4.2, 5.4.3, 5.4.4 and 5.5.2, we easily obtain the following.
Proposition 5.5.3 Let F be a tensor field of type (1,l) on M . Then we have
Chapter 5. Symplectic manifolds and cotangent bundles
248
Complete lifts of almost product structures
+
Let F be an almost product structure on M . Since ( F c ) 2= (F2)" 7 l v ~ = I d ~ N Fwe,have
+
Proposition 5.5.4 FC i s an almost product structure on T * M if and only if F i s integrable. From Proposition 5.5.3, we obtain
Proposition 5.5.5 If F i s an integrable almost product structure on M then F" i s an integrable almost product structure on T * M . Now let F be an integrable almost product structure on M with projection operators P and Q , i.e., 1 1 P=-(I+F), Q=-(I-F). 2
Since P 2 = P and
2
Q2 = Q , we
have
+
(P"2 = (P")" 7Np = (P")" = P", (Qc)2
+
= (Q2)" 7NQ = (Q2)"= Q",
since Proposition 3.1.4. Hence P" and Q" are the projection operators corresponding to F", i.e., 1 1 P" = : ( I F " ) , Q" = - ( I - F " ) .
+
2
+
Suppose that rank P = r , rank Q = s, r s = m. Since F is integrable, then exists for each point of M a coordinate neighborhood U with local coordinates (8)such that
I m P =< Im Q =
,
a/aqr+l,.. . ,d/aqrn > .
An easy computation shows that
a/aql,.. . ,a/aqr, a/apl,. . . ,a/ap, >, ImQC=< (a/aqr + l )c ,. . . ,(a/aqrn)c,(dqr + l )u ,. . . ,(dqrnlu> =
c (TZK)*.
(6.9)
Now, suppose that K is coisotropic. Then (T,K)* c T,K for all z E K . So Xfi is tangent to K and thus X , , ( f , ) = {fj, f;} = 0 for any 1 5 i , j 5 k. Conversely, if {f;,f,} = 0 for any 1 5 i , j 5 k, then Xfi is tangent to K for all i and thus (6.9) holds. Hence we deduce that (T,K)* c T,K, for all z E K.0 Coisotropic submanifolds are also known as first class constrained manifolds in Dirac terminology (see Chapter 8). Let us recall that a submanifold N of ( S , w ) is called symplectic if (T,N)n (T,N)' = 0 for all z E N .
Lemma 6.4.2 Let KO be a submanifold of codimension k i n K c ( S , w ) . Then KO is symplectic if and only if
(T,Ko) n ( T z K ) * = 0,
5
E
Proof If KO is symplectic then
&KO) n (T,Ko)l = 0.
KO
(6.10)
Chapter 6. Hamiltonian systems
276
As T,Ko
c (T,Ko)'.
T,K, we have (T,K)'
Then (6.10) holds. Conversely, if (6.10) holds then taking into account exercise (2) of the present chapter, we have C
((T,Ko) n (TzK)')'
=' 0 = T z S ==+ (TZKo)'
+dim [(T,Ko)' (T,Ko)'
n (T,K)] = k
n ( T z K )= (T,K)*,
+ T z K = TZS
=+
since dim (T,K)'
= k,
and so
(T,Ko) n (T,Ko)'
= 0.
Theorem 6.4.3 (Jacobi's Theorem). Suppose that C = { f r , . . . ,f k } are Coofunctions on a neighborhood of a point x E S, where (S,w) is a 2ndimensional symplectic manifold. If they are i n involution, then k 5 n and there exists a neighborhood of x o n which there is defined a set of C" functions f k + l , . . .,fn such that C = { f l , . . , fn} is i n involution.
.
Proof (See Duistermaat [50],p. 1OO).U Now, we show an important result (we follow Duistermaat [50]). Theorem 6.4.4 Let K be a submanifold of codimension k i n a 2n-dimensional symplectic manifold ( S , w ) . Through each point x E K there passes a Lagrangian submanifold L c K if and only if K is coisotropic.
Proof: If through each x E K passes a Lagrangian submanifold L
C
K
then
T,K
2
T,L = (T,L)'
3
(T,K)'.
(6.11)
Thus K is coisotropic. Let us see the converse. As K is coisotropic, from Lemma 6.4.1 { f i , f j } = 0 . Thus (see Corollary 6.2.7) [ X , , , X f j ] = 0, 1 5 i , j 5 k. Then D : x + D ( x ) = (T,K)I gives an integrable distribution according to Frobenius Theorem. Let KObe a submanifold of codimension k in K transversal to the integral manifolds of D,i.e.,
6.4. Lagrangian submanifolds
277
where N is an integral manifold of D through z. Then (TzKo)n (T,K)* = 0 and so from Lemma 6.4.2 KO is a symplectic manifold. Let LO be a Lagrangian submanifold of KO. Let zo E Lo and U a sufficiently small neighborhood of 20 on which K is defined by the above independent functions f'. Let (cp;)t be the flow of the vector field X f i . Define
where
11
is a small neighborhood of 0 E R. Then we have
and so
( T ~ L ~ )=' (T,L~)*n < x,~(z)>*
.
But TzLo C T z K implies ( T Z K ) * c (TZLo)*= T,Lo. Thus
(T,Ll)*
3
(%Lo)+ < X,&)
> = TzL1,
i.e., T,L1 is isotropic if z E U n LO. As the map d ( c p d t ( 4 : (TzS,w(z))
-
( T ( p l ) t ( z ) s ((cpOl)t)*+)) ,
is symplectic then d(pl)t(z)(T,L1)is also isotropic showing that L1 is isotropic with dimension n - k 1. If we repeat the argument for
+
Li = PI)^; 0 * * *
0
( ~ ~ ) t l ) ( zE) u / zn LO, ( t l , * * , t iE)
11
x
* *
+
x Ii}
we find that L; is an isotropic submanifold of dimension n - k i . Thus, taking i = k, one obtains a Lagrangian submanifold L = Lk through zo E Lo. If for such zo there is (locally) only one Lagrangian submanifold then the assertion of the theorem is proved. So let us now show the (local) unicity of L. If is any Lagrangian submanifold of K then from (6.11) the integral manifolds of D are tangent to L and, if in addition LO c i then i contains (locally) the integral manifolds passing through points of LO. Now, L was defined by
and as the differential of the mapping
278
Chapter 6. Hamiltonian systems
( t l , - - * , t k , z)
-
((pk)tk
-
*
.
(pl)t,)(z)
has rank m (T,Lo and (T,K)* are transversal) one deduces that for sufficiently small neighborhoods U and 11x . . .x I k , the mapping is an embedding and L has dimension n, U n LO c L c (locally). Thus i = L since i is Lagrangian and d i m L = n . 0
-
Proposition 6.4.5 If (S,W)is a symplectic manifold then a submanifold K of S i s Lagrangian if and only if there is a fiber bundle E such that T K @E = T S / K with T,K and E, being isotropic subspaces of T,S. Proof One direction is obvious. The other direction is obtained by using the fact that on every symplectic manifold there is an almost complex structure J . Thus, for each z E S, J,(T,K) = E, is a Lagrangian complement of the Lagrangian subspace T , K . 0
Proposition 6.4.6 Let a be a 1-form on M and L c T*M its graph. Then L i s Lagrangian if and only if a i s closed.
-
Proof: Let a : M T*M be a l-form on M locally given by a(q') = aidq'. Let ( q ' , p ; ) be the induced coordinates in T * M . We recall that the ,canonical symplectic structure on T*M is given by WM = - d X M , where AM is the Lioville form. Then we have WM
= dq' A dp;.
Hence f f * ( w ~= ) a*(dqi A dpi) = d(q' o a ) A d(pi o a )
= dq' A d a ; = - d a
Since the graph L of a is given by
L = { ( z , f f ( 4 ) /Ez MI we have dim L = dim ( T * M ) . Furthermore, since a is closed if and only if CY*WM = 0, we deduce that L is Lagrangian if and only if a is closed. 0 The following result is given as an exercise.
6.4. Lagrangian submanifolds
279
-
Proposition 6.4.7 Let f : S W be a symplectomorphism f r o m a s y m plectic manifold ( S , w ) t o the symplectic manifold (W,fl). T h e n S x W is a symplectic manifold with symplectic f o r m
p = TTW - T
p )
where the T ' S are t h e obvious canonical projections. Lagrangian submanifold of S x W .
T h e graph o f f is a
Definition 6.4.8 L e t (S,w ) be a symplectic manifold and L a Lagrangian submanifold. If L i s the graph of a closed 1-form a t h e n locally there exists a f u n c t i o n F such that a = d F . W e call F the generating function of L.
-
Let ( q , p ) be symplectic coordinates in S (we are omitting the index i for q,p for simplicity), f : S S a symplectomorphism and set
f ( q , P) = (q(q, P),F h , PI)
*
Then (q, p) are symplectic coordinates and w = dg A dp. Let us now s u p pose that ( q , q) are independent coordinates, that is, the matrix ( a ( q ,g ) / a ( q , p ) ) is non-singular. Then the graph of f is given by (( P,Iq(q,P),P(q,P)) )
and may be expressed as the image of a closed l-form a = d F where (q,q) has the form
The Hamilton - J a c o b i method for solving the Hamiltonian equations consists in showing that H is independent of p by the use of F . In fact, the generating function F satisfies the equation
a)= G .
H(Q, (aF/dB)(q> The Hamiltonian equations are
dq - 0 ' -d p= - aG dt dt aq and a solution is given by q ( t ) = q ( O ) , p ( t ) = p(0)
+ t(aG/aq).
280
Chapter 6. Hamiltonian systems
-
Proposition 6.4.9 A
neccessary and suficient condition for the (autonomous) change of coordinates (q',p,) ( # , p i ) be canonical is that a9 -
aq
aP -
aq
ap
2--aq -
d p ) ag
__ ag
ap)
2 -- aq
aq9 ap
aq'
Proof: We want a transformation from the set of variables (q,p) to another set ( q , g ) such that H(9,P) = G(q(q>PMq>P))
The symplectic form of the Hamilton equations are (for each Hamiltonian) :
ixHw= dH, ixGg = dG, where a is w expressed in the new coordinates. Let us suppose that the change of coordinates is canonical. Then X H = X G and
xH = (aH/ap) a/aq - (aH/aq) a p p = Q a/aq + r; a/ap (the dot means derivative with respect t o t ) . But (6.12) (6.13)
On the other hand
The substitution of (6.12) and (6.13) into X H and the comparison with the above expression gives the desired result. We leave to the reader the proof of the other direction.^
6.4. Lagrangian submanifolds
281
Corollary 6.4.10 The above change of coordinates is canonical if and only
if
Proof We have (F,G}q,p =
aFaG
-- -
aq a p
aFaG
_-
a p aq
If the change of coordinates is canonical, from Proposition 6.4.9, we deduce
The converse is proved by a similar procedure. 0 Now, let us consider the generating function F = F ( q , q ) . One has
and F is the generating function of a canonical transformation. The use of Proposition 6.1.2 and Jacobi's Theorem gives the following result (see also Weber [124]).
Proposition 6.4.11 Let ( S ,w ) be a symplectic manifold of dimension 2n a n d C = { fi, . . . , f,,} a s e t of CO" independent functions i n involution o n n neighborhood U of a point x E. X . Suppose that for every 1 5 i ,j 5 n we have rank ( d f i l d p i ) = n, where ( $ , p i ) are canonical coordinates f o r 5'. Then there is a local canonical transformation g : U g ( U ) c S such that
(Tg)X,; = xp;.
-
Chapter 6. Hamiltonian systems
282
6.5
Poisson manifolds
Let ( S , w ) be a symplectic manifold. Then to (S,w) corresponds an operator { , } on the algebra of CM functions on S, such that { , } is a skewsymmetric bilinear mapping defined by { F , G } = W ( X , , X G ) , where XF and XG are vector fields on S defined by i x p o = dF, i x c w = dG. This result suggested Lichnerowicz [go] to study manifolds on which is defined an operator { , } giving a structure of Lie algebra on the space of Coofunctions on such manifolds.
Definition 6.5.1 A Poisson structure on a manifold P i s defined b y a bilinear map
coop) xCyP)
-
-
CM(P))
where C m ( P )i s the space of Coofunctions on P, noted b y ( F , G ) { F , G} such that the following properties are verified: (1) { F , G } = - { G , F } (skew-symmetry) (2) { F , { G ,H } } + { G , { H ,F } } { H , { F , G } } = 0 (Jacob; identity) (3) { F , G H } = G{F, H } H { F , G } , { F G , H } = { F , H}G F { G , H } . W e call { , } Poisson bracket and the pair (P,{ , }) Poisson manifold.
+
+
+
So, on every symplectic manifold ( S , w ) there is defined a Poisson struc-
ture, canonically associated to the symplectic structure defined by w . Let { , } be a Poisson structure on a manifold P. From (3) of Definition 6.5.1 we see that the map
{F, } :P ( P ) G
-
-
Coo(P)
{F,G}
is a derivation. Therefore there is a unique vector field XF on P such that X F ( G )= { F , G } ; XF is called the Hamiltonian vector field of F . One can easily check that the Hamiltonian vector field X{F,G)of the function { F, G } is [XF X G ] * Suppose that F, G E C M ( P )and x is a point of P. Then we have )
U w ( 4= ( X F ( G ) ) ( 4= d G ( 2 W F ( z ) ) * Therefore, { F , G } ( x )is a function depending on d G ( z ) ,for F fixed. The same reasoning gives that, for each G fixed, { F, G } ( z )is a function depending
6.5. Poisson manifolds
283
on dF(z) (one has {G, F)(z) = d F ( z ) ( X c ( z ) ) ) .So, for each z E P , there is a bilinear map
R(z) : T,*P x T,*P
-
R
such that
R ( z ) ( d F ( z ) ,d G ( 4 ) = { F , G ) ( z ) with n ( x ) being skew-symmetric. Then z + R(z) defines a skew-symmetric tensor field of type (2,O) on P. Therefore one has (Libermann and Marle
18811
Theorem 6.5.2 Let (P,{ , }) be a Poisson manifold. Then there i s a unique 2-form R on P such for all F , G E C m ( P ) and x E P ,
We call R the "Poisson tensor field".
Remark 6.5.3 We may try to show that if it is given a skew-symmetric tensor field R of type (2,O) on P, then is defined on P a Poisson structure. Lichnerowicz showed that this is only possible if s2 verifies the identity
where [ , ] is the Schouten bracket. These bracket are characterized by the following properties. Suppose that A (resp. B ) is a skew-symmetric tensor field of type (a,O) (resp. (b,O)). The skew-symmetric tensor field [n,n] of type ( u b - 1,O) is defined by
+
i ( A , B ] P ZZ
where
P is a closed
(-l)"b+biAdiBP
+ (-l)'iBdiAP,
( a + b - 1)-form. If a = 1, then [ A ,B ] = LAB. One has
[A, B] = (-l)"*[B, A ] .
If C is a skew-symmetric tensor field of type (c,O) then (-l)"b[[R,C],A]
(Jacobi identity)
+ ( - l ) b c [ [ C , A ] , B ]+ (-l)C'[[A,B],C] = 0
Chapter 6. Harniltonian systems
284
Let us consider the particular case where a , b = 2. Take
Xi A K , B = C Zj A Wj.
A =C Then
-(diuY,.)X;
A
Zj
A
Wj
+ ( d i u Zj)X; A Y, A Wj
- ( d i u W j ) X ;A yi A Zj)
Thus, [n,!2]= 0 if and only if Poisson brackets satisfy Jacobi identity. From these comments we may say now that a “Poisson manifold” is a pair (P,0) where P is a manifold of dimension m and R is a skew-symmetric tensor field of type (2,O) on P of rank 2n 5 m verifying [n,n]= 0” (If 2n = m, then P is a symplectic manifold). From the above Theorem we see that if (P,n) is a Poisson manifold, then for all z E P and all 1-form Q on P , there is a map
such that P(Pfl(Z)(4)
-
= n(.)(%P), for a w E T,*P.
Therefore, for every point z E P one obtains a mapping T P given by
T*P
fi
= pn :
P ( q 4 )=n ( 4 , for all cr,P E A’P. It is clear now that the Hamilton vector field X, is given by XF = fi(dF) and
6.5. Poisson manifolds
285
{F,G} = n(dF)G = -n(dG)F = n(dF,dG).
(6.14)
Therefore, we have an analogous construction t o the symplectic case. We observe that if (S,w ) is a symplectic manifold one obtains trivially a Poisson structure R on S , defining fi = (S,)-', where S, : T S + T*S. In such a case fi is an isomorphism. Let z be an arbitrary point of P . Then Imn(z) (resp. ker n(z))is a vector subspace of T,P and its dimension is called the rank (resp corank) of R(z). The rank (corank) depends on z. As R is of type (2,0), the rank of R(z) is an even number. If it is constant and equal t o the dimension of P , then R is said to be non-degenerate.
Proposition 6.5.4 Let ( P , n ) be a Poisson manifold such that the induced map fi : T * P + T P is an isomorphism. For all z E P and X,Y E T,P, we put
Then w i s a closed 2-form.
Proof Let us first recall the following facts: Suppose that X F ~ , X are F~ Hamiltonian vector fields, that is,
Then [ X F ~ , X F a ]= X { F l , F z } = fi(d{Fl,F2)).
So, if F3 is a third function such that fi(dF3(z)) = XF,(~), one has, from the definition of w ,
= {{Fl, Fz), F3)(2),
where in the last equality we have used again (6.14). Also, we have
Chapter 6. Harniltonian systems
286
= n ( d F l ) n ( d F z ,dF3) = { F l , {F2, F3H,
where in the last equality we have again used (6.14). From these equalities we may prove that
-w([xF,
2
xF3] x F l )
Therefore, we have
+{F2, {F3, F l } }
+ {Fs, { F l , F2}}] = 0
(from Jacobi identity).
Then w is closed.^
Theorem 6.5.5 Suppose that ( P , R ) i s a Poisson manifold of even dimension 2p. If R i s non-degenerate then there i s defined a symplectic structure on P . Proof In fact, if R is non-degenerate, then w is a non-degenerate closed 2-form on P . 0 &om this theorem we see that the non-degenerate Poisson structure on a manifold of even dimension is equivalent to the symplectic structure. In the general situation, a Poisson structure defines a morphism from T*P to T P and, in the symplectic case, an isomorphism from T S t o T*S.We remark that we may also consider situations where rank of n is constant equal to 2n < dimP, which generates a study similar t o presymplectic manifolds. We suggest the paper of S. Benenti (51 in this direction.
6.6. Generalized Liouville dynamics and Poisson brackets
6.6
287
Generalized Liouville dynamics and Poisson brackets
Marmo et al. [94] proposed an extension of volume forms preserving vector fields to arbitrary manifolds having, as a particular case, some results obtained from the symplectic formalism. This generalization offers a possibility of re-obtaining Poisson brackets in a different way as presented up to now.
Definition 6.6.1 Suppose that N is a manifold of dimension n (even or odd) and 0 a volume form on N . W e say that a vector field X on N has a Liouville property with respect to 0 if the Lie derivative LxO vanishes, i.e., if 0 is invariant under X . The motivation of such definition is clear: suppose that (S,w), is a symplectic manifold of dimension 2n. Then w" is a volume form and if XF is a Hamiltonian vector field with energy F , then we have seen that L x p w n = 0. If we develop LxO = 0, then one has dixO = 0. So, we shall say that X is locally Liouville, if for each z E N ,there is a neighborhood U of z and a (n - 2)-form X on U such that
We shall say that X is Liouville if X is globally defined. For a local Liouville vector field X one has that dX is invariant under X. Furthermore, if the (n - 3)-form ixX is closed then X is also invariant since
LxX = ixdX
+ dixX = ixixO= 0 .
Therefore we may say that X plays the role that a function H plays in the Hamiltonian formalism. We will see now some extensions previously presented for the present generalization. For example, suppose that
Then a necessary and sufficient condition to vector field is a to be closed:
X be
a locally Liouville
LxO = 0 a dixO = 0 . F'rom this, one obtains the following simple result. Suppose that
Fn-l are Cm functions on N such that
F1,.
.. ,
Chapter 6. Hamiltonian systems
288
Then X is closed and
X is locally Liouville. Therefore each Fi,
1Ii
I
n - 1, is a constant of motion, since ixX = 0 if and only if X ( F ; ) = 0, 1 5 i 5 n - 1. The dynamics here is therefore characterized by a set of
Hamiltonian constants of motion. This kind of generalization goes t o a type of Mechanics called Nambu Mechanics (Marmo et al. 1941). We show now the relation of Liouville vector fields and Poisson brackets has an equivalent form (see Flanders [55], p. 180):
{ F , G } W" = n ( d F A d G ) A w"-l, where w is a symplectic form and w" the corresponding volume form. In such a case, if XH is a Hamiltonian vector field, i x H w= d H , then
(LxHF ) wn = ( d F ) A ( ~ x ~ w " ) = n ( d F ) A ( i x , w ) A w"-l
= n dF
A
dH
= {F, H }
A w"-l
W"
(and so L x , F = { F , H } ) . We observe that the first equality in the above expression is obtained from 0 = i x ( d F A w") = ( i x d F ) W" - dF A ixw"
( L x F ) w n - dF
A
~xw".
Suppose now that X is Liouville on a manifold N. Then X is defined (at least locally) by a (n - 2)-form X through the equation L x 0 = 0, i.e., i X 0 = dX. Hence 0 = i x ( d F A 0) = ( L x F ) 0 - dF A dX,
6.7. Con tact manifolds and non-au tonornous Harniltonian systems
289
that is,
(LxF)O = dF
A
dX,
and if we define the Poisson bracket of a Coofunction F on N with respect to X by
{ F , X}e = d F h dX, then obviously we have an analogous result for the Lie derivative:
LxF = {F,X}. We may extend the definition of Poisson brackets for ( n - 2)-forms as we did for 1-forms (see section 6.2). For this, suppose that X and Y are Liouville vector fields (at least locally). Then there are (n- 1)-forms Ax, such that
Xy
ixe = x X , i y e = x Y , with d X x = d X y = 0. P u t
Then it is not difficult to see that this brackets verify the same properties of Poisson brackets. Moreover, we have
L
~ =XL x i~y e = i [ x , y l= ~{x,,xY}.
6.7 Contact manifolds and non-autonomous Hamiltonian systems
+
Let M be a (2n 1)-dimensional manifold and w a closed 2-form on M of rank 2n. From the Generalized Darboux Theorem (with r = 1) there is a coordinate system ( x i , y i , u ) , 1 5 i 5 n, on each point of M such that n
w = x d x i A dy'. i=l
In particular, let (Y be a contact form on such that a A (da)" # 0. We set
M ,i.e.,
ct
is a 1-form on M
Chapter 6. Hamiltonian systems
290
Since w has rank 2n, there exist on each point of M a coordinate system
(zi,yi,u) such that n
w =Xdz'A
dy'.
i= 1
Then
0 = da
+
c
m
i=l
i= 1
n
yidz').
dz' A dy' = d ( a -
Therefore, there exists a locally defined function z such that n
a-
C y'dz' = dz, i=l
and, so, a = dz
+
c n
y'dz'.
i=l
If we put Y = d / a z , then we have i y a = a ( Y )= 1, n
iyda =
-iy(Cdz' A dy') = 0. i=l
This shows the existence and unicity of such vector field Y which is called the Reeb vector field (see Godbillon [63], Marle [93]). Let us return to the general situation, i.e., M is a (2n 1)-dimensional manifold and w is a closed 2-form of rank 2n on M. We denote by A, the 1-dimensional distribution on M defined by
+
We notice that A, is involutive. In fact, for two vector fields X,Y E A,, we have
6.7. Contact manifolds and non-autonomous Hamiltonian systems
i [ X , Y ] W= L x i y w
291
- i yL x w
= - i y L x w (since i y w = 0 ) = -iy((ixd+ d i x ) ~ )
= - i y d i x w (since w is closed) = 0 (since i x w = 0).
Alternatively, Aw may be viewed as a vector bundle over M of rank 1; in fact, Aw is a vector subbundle of T M . A vector field X such that X E A w (i.e., i x w = 0) is called a characteristic vector field. Let now (S,w ) be a symplectic manifold of dimension 2n and consider the product manifold R x S. Let ps : R x S S be the canonical projection on the second factor, i.e.,
-
p s ( t , z ) = z. Set w' = (ps)*w. Then w' is a closed 2-form on R x S of rank 2n. Consider the distribution Aut and define a vector field X on R x S by
x(t,z)= a/at E T ( ~ , %x~s( R) N T ~ eR T,S. We have
w ' ( X ( t , z),2 ) = w ' ( a / a t , 2 )
= w(0,Z) = 0,
Chapter 6. Hamiltonian systems
292
for all 2 E T(+)(R x S). This shows that X E Awl. Furthermore, if Y E AUl, i.e., iyw' = 0, we have
a p t + u , b a p t + u')
0 = w'(Y, 2)= W ' ( .
= (p*sw)(aa p t
+ u, b a p t + u')
= w(u,u'),
+
+
where Y = a d / a t u, 2 = b a / d t u , a , b E Cm(R x S ) , u , u E x(S). Hence, u = 0 and so Y = a d / a t . Therefore, Awl is globally spanned by
apt. Now, let H : R x S define Ht : S + R by
-
R be a function on R x S. For each t E R, we
H t ( z ) = H ( t , z). We consider the Hamiltonian vector field x H t on S with energy H t , i.e., iXHtW = d H t .
For simplicity, we set Xt = x H t . Define a mapping X : R x S
TS
-t
by
X ( t , z ) = Xt(z)E T z S , t E R, z E S. Then there is a vector field X H on R x S given by
X H ( z) ~ ,= d / a t
+ X ( t ,z), i.e.,
XH(t,z) = d/at
+ xt(z).
If ( q i , p i ) are canonical coordinates in S , i.e., w = is locally given by the same expression:
dq' A dp;, then w'
n
Let u : I = Then we have
(-E,E)
-
W'
dq' A dp;.
= i= 1
R x S be an integral curve of X H , with
c
> 0.
6.7. Contact manifolds and non-autonomous Hamiltonian systems
293
.(t) = X H ( O ( t ) ) Thus d q / d t = 1 , i.e., a ( t ) = t . Therefore
4 t ) = ( t , 9'(t),Pi(t)). So we obtain
-
Therefore,
B
a~ a a H a api aqi aqi api
a
is an integral curve of X H if and only if
dq' dt
aH
dp; dt
-- -
--
ap;'
_ _a _H aqi
'
l= (< Y , Q >,< Y , p >) then we have < Y,( a , ~>= ) ((~(~;)+~;)Y'+~;€(Y'),~;Y'+~;€(Y~)), for all ( c . , ~ )E A : , ~ and Y E xc. As in the conservative situation, we deduce that if the above local expression holds for all ( ~ , / 3 ) E A i J then Y must be in xt. Next, let us examine symmetries in the non-conservative case.
389
7.15. More about semisprays
P r o p o s i t i o n 7.15.18 If Y is a dynamical symmetry of ( and
(&,a)E
satisfies
J*R;a = J * ( d f ) ,
Py@ = dg,
for some f,g E C " ( T M ) then
b r > P rE) A;& where a' = Lya - d((f) and
p' = L y p - d(-lift of F to T k M is the unique tensor field of type ( 1 , l ) F<j' on T k M such that
F<j>X