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
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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
--
Hence the equivalence class [z] of z is the line trough the origin and 2. We denote M / by RP", the real projective space of dimension n. Let T :M RP" be the canonical projection. For each t E R - {0}, we define +t:M-Mby
+*(z)= t z . Then
+* is a diffeomorphism with ++' =
+1/t.
If U c M is an open set, then
15
1.7. Tangent spaces. Vector fields
u
.-'(.(W =
4t(U).
t E R- ( 0 )
Then r ( U ) is an open set of RP". Hence T is open. Now, we define a real continous function f on M x M by
f
( 5 ,y)
= C ( x ' y ' - 29y')? i#i
Then R = f - ' ( O ) is a closed set of M x M . Hence RP" is Hausdorff. We define n 1 coordinate neighborhoods (Ui, 4;) as follows:
+
u; = .(Ui'), where U;= { x E M / x '
# 0 } , and
-
4; 1 U;
Rn
by
& ( x ) = ( X l / X i , . . . ,x y x i , xi+' / x i , . . . ,X"+l/Xi). It is easy to prove that of M of dimension n.
1.7
4j o
(4;)-' is C". Then RP" is a quotient manifold
Tangent spaces. Vector fields
In this section we introduce the tangent space of an m-dimensional manifold
-
M.
Definition 1.7.1 Let x E M . A curve at z is a differentiable m a p u : I M from an open interval I c R into M with 0 E I and u ( 0 ) = x . Let u and T be curves at z. W e say that u and r are t a n g e n t a t z if there ezists a coordinate neighborhood ( U , 4 ) of z with focal coordinates ( x i ) such that (dx'
0
a / d t ) p o = (dx'
0
r/dt)lt=o, 1 5 i 5
m.
-
One can easily check that the definition is independent of the choice of local coordinates. The equivalence class [u] of u is called the t a n g e n t vector of u at x ; sometimes, it is denoted by ci(0). If u : I M is a curve in M (where I = ( - E , E ) , E > 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 <_ rn. Then we have
(dzi)(z)(a/azi), = (a/ad),(z') = 6;. Therefore {(dz')(z), . . . ,( d z m ) ( z ) }is a basis for TZM. In fact, this is the dual basis of { (d/az'),, . . . ,( a / a z m ) , ) . Next, we compute the components of df(z) with respect to the dual basis {(dzl)(z),. . . ,(dzm)(z)}. We have
Hence we obtain m
(af / a z i ) (dz' ) (Z).
df (z)= i=l
Definition 1.9.1 A 1-form a on M i s a function assigning t o each point z E M a tangent couector a(.) E T,*M. Let (U, z') be a local system of coordinates in M , 5 i <_ m. For each z E U we have m
a(.)
= &(z)(dz')(z). i=l
Then we have
i=l
on U ,where ai : U + R. We say that a is Cw if the functions for each coordinate neighborhood U . Now we set
T*M = and define a mapping
U
x E M
T,*M
ai
are Coo
30
CY
Chapter 1. Differential Geometry
We can introduce local coordinates in E T,*M,x E U ,we have
TG'(U)
= T*U as follows. If
m ...
CY
= CCY;(x)(dz')(z),x = ( x i ) i=l
Then the coordinates of CY are (xi, CY,(Z)). These coordinates are usually denoted by ( z ' , p i ) and called the induced coordinates in T * M . Thus T * M is a 2m-dimensional manifold. Moreover, we can prove that ( T * M ,T M , M ) is a vector bundle over M of rank rn,called the cotangent bundle of M . Obviously, the canonical projection T M is locally given by
We note that a 1-form on M may be interpreted as a section of T*M over M.
1.10
Tensor fields. The tensorial algebra. Riemannian metrics
In this section we study the basic material on tensor fields on manifolds which is used in the succeding sections.
Algebraic preliminaries Let A and B be finite-dimensional vector spaces over R (or C). We denote by M ( A , B ) the vector space generated by the Cartesian product A x B, i.e., the free vector space generated by the pairs ( a , b ) , a E A , b E B. Let N ( A , B ) be the vector subspace of M ( A , B ) generated by elements of the form
1.lo. Tensor fields. The tensorial algebra. Riemannian metrics (a1 + a z , b )
+ a,)
31
- (4) - (az,b), - (.,a,)
- (.,a,)
-
a E R (or C), a , a l , a z E A , b , b r , b z E B.
We set A @ B = M ( A , B ) / N ( A , B )and define p : A x B A @ B by p ( a , b) = a @ b; A @ B is called the tensor product of A and B. Let C be a vector space and q : A x B C a bilinear mapping. We say that the couple (C, q) has the universal factorization property of A x B if for every space D and every bilinear mapping r : A x B D there exists a unique linear mapping 4 : C D such that the following diagram
-
-
AxB
9
-
*
C
is commutative.
Proposition 1.10.1 The couple ( A @ B , p ) has the universal factorization property of A x B and the vector space A @ B is unique up to an isomorphism.
-
-
Proof It is clear that p : A x B A @ B is bilinear. Now, let x B D be a bilinear mapping. Then r can be uniquely extended to an homomorphism f : M ( A , B) D such that r'(N(A, B)) = 0. Thus r' induces a mapping : A @ B + D such that r :A
-
The proof of the uniqueness is left to the reader as an exercise.0
-
Proposition 1.10.2 We have canonical isomorphisms: (1) A @ B E B @ A , a @ b b @ a; (2) ( A @B ) @ C A @ ( B @C),( a @ b ) 8 c a @ ( b 8 c); a@a a. (3) R @ A A @ R 2 A , a @ a
-- -
Proof (1) Let p : A x B B @ A be the bilinear mapping defined by p(a,b) = b @ a . From Proposition 1.10.1 we deduce that there exists a unique linear mapping (I, : A @ B --t B @ A such that $ ( a @ b) = b @ a. Similarly, there exists a unique linear mapping 4' : B @ A A @ B such that $'(b @ a) = a @ b. Hence (I,' o $J = i d A B B , (I, o $J'= i d B B A . so (I, is
-
32
Chapter 1. Differential Geometry
the desired isomorphism. The proofs of (2) and (3) are similar and hence omitted. 0 From (2) of Proposition 1.10.2, we can writte A @ B @ C and a @ b €3 c . Furthermore, if A1,. . . ,A, are vector spaces we may inductively define its tensor product A1 @ . . . @ A,. Since M(A, B)/N(A, B) is the quotient vector space, we have for any element (al, 0 2 ) @ (bl b2) E A @ B the distribution law
+
Hence, let Then
..
( ~ 1 , . ,a,}
be a basis for A and {bl, . . . ,b,} a basis for B.
is a basis for A@B. In fact, they generate A@B and are linearly independent. Hence
dim ( A @ B) = (dim A)(dim B) = mn.
-
Now, let A* be the dual vector space of A. For a E A and a* E A* we denote by < o,a* > the value of a* : A R on a, i.e.,
< a,a* >= a * ( a ) . Proposition 1.10.3 Let Hom(A*, B) be the vector space of linear mappings of A* into B. Then there ezists a canonical isomorphism :A8B + Hom(A*, B) such that
+
( $ ( a @ b))(a*) =< a , a*
-
> b.
Proof: Consider the bilinear mapping (o
:A x B
Hom(A*, B)
defined by
B
-
((o(a,b))(a*)=< a,a*
> b.
From Proposition 1.10.1, there exists a unique homomorphism $ : A @ Hom(A*, B) such that
1.10. Tensor fields. The tensorial algebra. Riemannian metrics
( $ ( a @ b))(a*)=< a, a*
33
> b.
Finally we prove that $ is an isomorphism. In fact, let ( 0 1 , . . . ,am} be a basis for A, {a;, . . . , a h } the dual basis for A* and { b l , . .. ,a,} a basis for B. It is sufficient to prove that { $ ( q@ b j ) } are linearly independent. If C Xij$(ai @ b j ) = 0, then 0=
Hence phism. 0
Xkj
(c
Xij$(ai @ b j ) ) ( a ; ) =
c
Xkjbj.
= 0. Since dim (A @ B) = dim Hom(A*, B), $J is ' an isomor-
The following result can be proved in a similar way.
Proposition 1.10.4 There exists a unique isomorphism $ : A* @ B* (A @ B)* such that ($(a* @ b*))(a
b) =<
a , a*
>< b , b* >
+
.
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 ) =<
bl,
> . . . < 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 < ...
Hence, if jl <
. . . < jp,we have
0=(
c...
il<
A;l,,,;peilA
. . . A e i P ) ( e j l ,. . , e j p )
Chapter 1. Differential Geometry
42
- X j l ...jp
Therefore, each coefficient Xil...i, must be zero. This ends the proof.0 From Proposition 1.11.4, we have m
Av = @ A p v . p=o
(;)=
Then AV is a finite dimensional vector space of dimension C,"=, 2m.
-
We next examine what happens when we map one vector space to another. Let F : V W be a linear mapping. Then it induces a linear mapping
F* : Tp*W
-
Tp*V
given by
( F * V ) ( U ~. .,. , u p ) = ~ ( F u I .,. ., F v ~ ) , (EPTPV,vl,.. . u p E V. If p = 0, we set F*X = A, for every X E R. Obviously, if (o is skewsymmetric (resp. symmetric) then F*(o is skew-symmetric (resp. symmetric). Thus we have two canonical induced homomorphisms
F" : ApW
-
ApV, F* : S p W
-+
spv.
Proposition 1.11.5 We have F*(w A r ) = (F*w) A (F * r ) , for every w E ApW,r E A q w
is a permutation of (1,. . . ,s), we denote by the covariant tensor of degree s given by
Proof If (ou
(o
E AaW and
Q
With this notation, we have
1.1 1. Differential forms. The exterior algebra
1 Acp = s!
43
C€aY3,.
Now, let w E APW, r E AqW. Then we obtain
-
( P + q ) l A ( ( F * w )8 ( F * r ) ) p!q!
-
= (F* w) A ( F * r ) .0
Corollary 1.11.6 F* : AW
AV is an algebra homomorphism.
Differential forms We introduce the following terminology.
Definition 1.11.7 A skew-symmetric couariant tensor field of degree p on a manifold M is called a differential form of degree p (or sometimes simply p-form).
Chapter 1. Differential Geometry
44
The set APM of all such forms is a subspace of T;(M) (in fact, a C"(M)submodule). If w E ApM and r E A ~ Mwe , define the exterior product w A r E Ap+qM by
( w A T ) ( x ) = ~ ( xA)r ( x ) , x E M . (It is obvious that w A T is differentiable). Hence, if we set AM = @,"=-, APM, where dim M = m, we deduce that A M is an associative algebra such that w A T = ( - l ) p q rA w , w E A ~ Mr, E A ~ M ;
AM is called the algebra of differential forms or exterior algebra on M . We also have (fw) A r = f ( w A T) = w A ( f r ) , for every f E C M ( M ) .
If ( U , s ' ) is a local coordinate system, then {dz'' A . . . A d & / 1 5 il < i2 < . . . < i, 5 m} is a basis of APU. Therefore, a p f o r m w on M can be locally expressed by
Now, let F : M homomorphism
+N
be a CM mapping. Then F induces an algebra
F* : A N
-
AM
given by
( F * w ) ( z ) = ( d F ( z ) ) * ( w ( F ( z ) ) ) ,z E M , w E A ~ N .
Definition 1.11.8 Letw be up-form on M. The support o f w i s the closure of the set on which w does not vanishes, i.e., s u p p ( w ) = c l { z E M / w ( x#) 0 ) .
We say that w has compact support i s s u p p ( w ) i s compact in M.
1.11. Differential forms. The exterior algebra
45
Derivations of AM Let AM be the algebra of differential forms on M .
-
Definition 1.11.9 A derivation (resp. skew-derivation) of A M is a A M which satisfies. linear mapping D : AM
D(w A r) = Dw (resp. D(w A
T)
= Dw A r
Ar
+w
A
Dr for w , r E A M
+ (- l ) p w A Dr for w E APM, r E A M ) .
A derivation or a skew-derivation D o f ~ M i s said t o be of degree k i f
it m a p s APM i n t o AP+kM f o r every k . Proposition 1.11.10 (1) If D and D' are derivations of degree k and k' respectively, t h e n the bracket product defined by [ D ,D'] = DD' - D'D is a derivation of degree k k'. (2) If D is a derivation of degree k and D' i s a skew-derivation of degree k', t h e n the bracket product defined by [ D ,D'] = DD'- D'D i s a skew-derivation of degree k k'. (3) If D and D' are skew-derivations of degree k and k' respectively, t h e n the bracket product defined by [ D ,D'] = DD' + D'D i s a derivation of degree
+
+
k+P. Proposition 1.11.11 A derivation or a skew-derivation D i s completely determined by i t s action o n CM(M)and A'M. Proof: We prove the result for derivations of AM. The proof for skewderivations of A M is similar and it is left to the reader as an exercise. First, we notice that a derivation D can be localized, i.e., if w E AM vanishes on an open set U ,then Dw vanishes on U. In fact, for each z E U , let f be a function such that f (z) = 0 and f = 1 outside U. Hence w = f w and we have
+
Dw = D ( f w ) = ( D f ) w f ( D w ) Since f ( z ) = 0 and ~ ( z=) 0, we deduce that ( D w ) ( z )= 0. Then, if w and r coincide on U ,then D w / u = 07/11. Now let D1 and D2 be two derivations of A M and set D = D1 - D2. Suppose that D is zero on C m ( M ) and A ~ M Let . be w E APM and z an
Chapter I . Differential Geometry
46
arbitrary point of M . Choose a coordinate neighborhood V of x with local coordinates xl,. . . ,x m . Hence w =
C
i 1 <...
~ i ~ , . . i ,Ad .~. .~A' dxiP
(1.6)
on V. We may extend wil...i, and dxi to M and assume that (1.6) holds in a smaller neighborhood U of x. Therefore we have
C
DW =
il<
+ C il
D ( ~ i ~ , . . i , d AX ~ . .'. A d&)
...< i ,
o i l . . . i p ( D d x iAl ) dxi2 A . .
. A dxiP
< ...< i ,
+. . . + C ~ <...
i ~ , . .A i..~. Ad (Ddz'P) ~ ~ ~ =0.0
il
Remark 1.11.12 There is another approach to the theory of derivations due to Frolicher and Nijenhuis [56]. They define a derivation of AM of degree r as a linear mapping D : A M --t AM such that: (1) degree (Do)= degree w r ( 2 ) D(w A r ) = D o A r (-l)pro A D r , where w E ApM.
+
+
The commutator of two derivation D and D' of degree r and r' is defined by
[ D ,D'] = DD'
-
+
(-1)"'D'D
and it is a new derivation of degree r r'. A derivation D is said to be of type i , (resp. d , ) if it acts trivially on AOM (resp. [ D , d ]= 0). Frolicher and Nijenhuis prove that a derivation D of AM can be decomposed in a unique way as a sum of a derivation of type i , and a derivation of type d,.
1.12. Exterior differentiation
1.12
47
Exterior differentiation
Let f E C m ( M ) be a differentiable function on M . Define the exterior derivative df of f by
df : x E M
-+
d f ( z ) E TZM.
Then df is a 1-form on M locally given by
df = (af lc9x')dx'. Now we shall extend the operator d to an arbitrary p-form.
Theorem 1.12.1 There ezists a unique linear operator d : A M such that ( 1 ) d : APM -+ Ap+'M, (2) d2 = 0, (9) df i s the differential o f f , for each f E C m ( M ) , (4) if w E APM, then d(w A
T)
= dw A r
+ (-l)pw
A
-+
AM
dr.
Proof: EXISTENCE. Let x E M . Consider a local coordinate system (xi) on a neighborhood U of x. If w E A P M , then w / U = ~ j ~ . . . ; A~ .. d ~. A"dx'p, where the summation is over 1 5 i l < i2 < by setting
. . . < i p 5 m.
dw(Z) = ( d ~ j l , , , j p )A( d~z)" ( z )
A
We define dw at x
. , . A dx'P(x).
Next we show that the definition of d w ( x ) is independent of the choice of coordinates. First we prove the following properties: (a) d w ( x ) E @'(T,*M). (b) dw(x) depends only on the germ of w at x. (c) If w , r E ApM and a , b E R,then
d(aw
+ br)(x) = a ( d w ) ( x )+ b ( d r ) ( x ) .
(d) If w E APM,r E AqM, then
48
Chapter 1, Differential Geometry
d(w A
T)(z)
= ( d w ) ( z )A
T(Z)
+ ( - ~ ) ' w ( z )A ( d r ) ( z ) .
From the linearity of d , to check (d) it is enough to consider forms w = a dzil A
. . . A dzip and r = b dzil A . . . A dzjq.
Then we have d(w A r)(z) = d ( ( a b ) ( z ) ( ( d z "A
+
.. . A dz'p) A (dzil A . . . A d z J p ) ) ( z )
= [ ( d a ) ( z ) b ( z ) a ( z ) ( d b ) ( z ) ] ( ( d z "A
= ( ( d a ) ( z )A (dz" A
+ ( - l ) P ( ( a ( z ) ( d z i lA
.. . A dz'p) A ( d z j l A . . . A d & ) ) ( z )
.. . A d z i p ) ( z ) )A (b(z)(dz" A . . . A d z i q ) ( z ) )
.. . A d z i p ) ( z ) )A ( ( d b ( z )A (dzj' A . . . A d & ) ( z ) ) ,
which completes the proof. The ( - l ) P is due to the fact that d b ( z )A ( d z " ) ( z ) A . . . A ( d z i p ) ( z )
= (-l)p(dzil)(zA )
. . . A ( d z i P ) ( z )A d b ( z ) .
(e) I f f E C'(M), then d ( d f ) ( z )= d ( a f / a z ' ) ( z )A ( d z ' ) ( z ) = (a'f/azJaz')(z)(dz')(z) A ( d z ' ) ( z ) = 0,
since (a2f/azJazi)(z)= (a'f/az'azJ)(z) and ( d d ) ( z ) A (dz')(z) = - ( d z ' ) (z)A ( d z i ) ( z ) . Now, we prove the independence of the choice of coordinates. In fact, let (z"')be another local coordinate system on a neighborhood U' of z. Then, since d' satisfy (a)-(e), we have
49
1.12. Exterior differen tiation
= (dWil...ip)(Z) A (dz")(z) A
.. . A ( d z i p ) ( z )
P
+C(-l)rWil...ip(Z)(d2i1)(Z)
A
,
. A d ( d z i r ) ( z ) A . . . A (dziP)(z)
r=l
= (dwil...ip)(z)A ( d z " ) ( z )
A
.. . A ( d z i p ) ( z )
= dw(z). Now, if w E APM, we define dw E Ap+'M by dw : 2 E
M
+ dw(z) E
AP++'(T,*M).
It follows that d satisfy (1)-(4). UNIQUENESS.Let d' be another linear operator on AM satisfying (1)(4). First, we prove that if w E ApM vanishes on a neighborhood U of z, then d'w(z) = 0. To see this we choose a function f on M which is 0 on a neighborhood of z and 1 on a neighborhood of M - U . Then f w = w and d'w(z) = d ' ( f w ) ( z ) =
+
(d'f)(z)A ~ ( z ) f(z)(d(w)(z)= 0.
Now let ( z i ) be a local coordinate system on a coordinate neighborhood AU as follows. If x € M , let f be a function on M which is 1 on a neighborhood V c U of z and has support in U. If w E APU, then f w is a p f o r m on M which is 0 outside of U and it agrees with w on V . We define d'w by
U . We define a new linear operator d' on
d'w(z) = d ' ( f w ) ( z ) .
From the above remark d'w is independent of the choice of f . It is easy to prove that d' verifies (a)-(.). Consequently we have d'w(z)
= dw(z),
for each w E APM and z E M . So d' = d . 0
50
Chapter 1. Differential Geometry
Remark 1.12.2 It is clear from the above proof that
for all open set U in M. From Theorem 1.12.1 we deduce that the exterior differentiation d is a skew-derivation of degree 1 of AM and a derivation of degree 1 and type d , in the sense of Frolicher and Nijenhuis. The following result will be very useful in the sequel.
+
c
O l i <j
(- 1)i+jw( [Xi,Xj],xo,. . . ,x i , . . . ,xj,. . . ,X,),
where the symbol" means that the term is omitted. Particularly, if w is a 1-form, then
We leave the proof as an exercise. Now, let F : M -+ N be a differentiable map and F* : A N induced algebra homomorphism. Then we have.
-
AM the
Proposition 1.12.4 F* and d commute, that is F*d = d F * . Proof: For forms f of degree 0 , that is, functions, the results holds: F*(df) = d ( F * f ) . In fact, if u E T,M,z E M, we have
51
1.13. Interior product
Next, we shall prove the result for an arbitrary p f o r m . Suppose that the proposition is true for all forms of degree less than p . It is enough to prove it for p-forms locally given by w = a dxil A
. . . A dxip.
Then we have
d(F*w) = d ( F * ( ( a dx")
A
(dxi2 A
...A ddp)))
= d ( ( F * ( a dx") A F*(dxia A . . . A dxip))
= d ( F * ( a dx")) A F*(dxi2 A
- ( F * ( a dz")
A
. . . A dz'p)
d(F*(dxi2A . . . A d x i p ) )
-$'*(a dx") A F*(d(dxi2 A . . . A dxiP))
= F*(d(a dx") A . . , A dxip)) = F*(dw),
since ddx'r = 0, 1
1.13
5r5
m.0
Interior product
For each vector field X on a manifold M , we define the interior product i x w of a p f o r m w by X as follows: (1) i x w = 0, when p = 0; ( 2 ) i x w = w ( X ) , when p = 1 ; and ( 3 ) ( i x w ) ( Y l ,..., Yp-l) = w ( X , Y l , ...,Y p - l ) , Y l , . ..,Yp-l E x ( M ) . Then ixw E A P - ~ M . In a similar way we can define the interior product i x G of a tensor field G of type ( O , P ) , P 1 0. The proof of the following proposition is left to the reader as an exercise.
Chapter 1. Differential Geometry
52
Proposition 1.13.1 We have ( I ) ( i ~= )0; ~and (2) i x ( A~T ) = ( ; X U ) A 7
+
A
( i x T ) , w E APM.
R o m Proposition 1.13.1 we deduce that i x is a skew-derivation of degree -1 of A M and a derivation of degree -1 and type i , in the sense of Frolicher and Nijenhuis. Now, let F be a tensor field of type (1,l)on M . We define the interior w a p f o r m w by F as follows: product i ~ of (1) iFw = 0, when p = 0; ( 2 ) (iFW)(Yl, ...,Yp) = C;='=,w(Y1,. . ., FYi ,... , Y p ) ,for all Y1,... ,Yp E X(M)* Then i ~ Ew APM. We can easily check that i~ is a derivation of degree 0 of AM and a derivation of degree 0 and type i , in the sense of Frolicher and Nijenhuis.
1.14
The Lie derivative
Let X be a vector field on M and $t a local 1-parameter group of local transformations generated by X. We shall define the Lie derivative Lxw of a pform w with respect to X as follows. For the sake of simplicity, we suppose that $t is a global 1- parameter group of transformations of M. Hence, for each t E R, 4: : A M + AM is an automorphism of the exterior algebra A M . Then we set
and Lxw is a p-form on M .
Proposition 1.14.1 Let X be a vector field on M . Then ( I ) L x f = X f , for every function f; (2) i f w E APM,T E A ~ Mthen ,
L x ( w T~) = LXw A (8) L x commutes with d, that is,
T
+w A LXr;
1.14. The Lie derivative
53
Proof: (1) in fact,
( L x f ) ( 4= !;mo(l/t)[f(4
-
f(4-t(41
4r1
But 4-t = is a local 1-parameter group of local transformations generated by -X. Hence
L x f = -((-X)f) = Xf (2) Let w E A p M , r E A ~ MThen . we have
= lim (l/t)[w(z) A r ( z ) - (4?tw)(z) A r ( z ) t-0
+ (drtw)A r(z)
(3) From (2) it is enough to check (3) when applied t o functions. Let f E C@'(M).If u E T,M,zE M , we have
Chapter 1. Differential Geometry
54
where f o 4-t can be considered as a differentiable function on R x M . On the other hand, we have
= lim (l/t)[uf - u(f o 4 + ) ] . t-0
If we extend u to a vector field Y on M , then we obtain two vector fields d/dt and Y on R x M such that [d/dt,Y] = 0. This fact implies (3).0 From Proposition 1.14.1, we deduce that L x is a derivation of degree 0 of A M and a derivation of degree 0 and type d, in the sense of Frolicher and Nijenhuis. As relation among d, Lx,and ix,we have the following (sometimes called
H. Cartan formula): Proposition 1.14.2 Lx = i x d
+ dix
Proof: First, observe that both Lx and i x d + dix are derivations of A M and commute with d because d2 = 0. Hence we need only t o check the identity when both sides act on functions. But, for every function f on M , we have
(ixd + dix)f = i x ( d f ) = d f ( X ) = X f = Lx f. This ends the proof.0
Proposition 1.14.3 W e have
w E APM,
X , X 1 , .. . , X p E x ( M )
1.15. Distributions. Frobenius theorem
55
Proof: The result follows directly from Propositions 1.12.3 and 1 . 1 4 . 2 . 0 Proposition 1.14.4 We have [ L x> i Y 1 = i [ X , Y ] The proof is left to the reader aa an exercise.
Remark 1.14.5 In a similar way we may define the Lie derivative L x F of a tensor field F of type (r,s) with respect to a vector field X on M. For instance, if F is a tensor field of type ( 1 , l ) then L x F is the tensor field of same type given by ( L x F ) ( Y )= [ X ) F Y ]- F [ X , Y ] . If G is a tensor field of type (0,2),then L x G is a tensor field of same type given by
1.15
Distributions. Frobenius theorem
The concept of vector fields on a manifold can be used to give an intrinsical treatment of certain first-order linear partial differential equations. First, we introduce some definitions.
Definition 1.15.1 Let M be a manifold of dimension m. A k-dimensional distribution D on M i s a choice of a k-dimensional subspace D(x) of T,M f o r each x i n M . D is said to be Cooi f for each x E M there is a neighborhood U of x and there are k linearly independent Coovector fields X I , . . . ,Xk on U which form a basis of D(y) for every y E U . Then we say that X I , . . . Xk i s a local basis of D. A vector field X on M is said to belong to D i f X ( z ) E D ( z ) for each z E M . D i s said to be involutive i f [ X , Y ]E D for every vector fields X, Y belonging to D. )
-
Definition 1.15.2 Let 9 : N M an immersed submanifold of M . Then N is an integral manifold of D i f d4(Y)(T”N)= D ( 9 ( Y ) )f o r each Y E N . D is said to be completely integrable i f there exists an integral manifold of D through each point of M .
Chapter 1. Differential Geometry
56
Lemma 1.15.3 Let D be a completely integrable k-dimensional distribution on a manifold M of dimension m. Then for each x E M there exists a local coordinate system ( x i ) on a neighborhood U of z such that xi(.) =0,l 5 i 5 m and the submanifold xi = constant, k 1 5 i 5 m, are integral manifolds
+
of
D.
Proof: Since each immersed submanifold is locally an embedded submanifold the result follows directly from the definitions. 0 Theorem 1.15.4 (Frobenius Theorem) A distribution D on a manifold M is completely integrable if and only if it i s involutive. Proof We shall prove that a completely integrable distribution is involutive. In fact, let X ,Y be vector fields belonging to D. We must prove that [X,Y](x)E D ( x ) for each x E M . Let 4 : N M be an integral manifold of D trough x, and suppose that # ( y ) = x. Since d+(y') : T,,N D(r#~(y')) is a linear isomorphism for each y' E N , then there exist vector fields X', Y' on N such that X',X (resp. Y',Y) are +related. Therefore [Xl,YI],[X,Y] are +related (see Exercise 1. ). Consequently, we have
-
-
Hence [X,Y] E D. In order to prove the converse we proceed by induction on the dimension of D. Suppose that dim D = k and dim M = m. When k = 1 then D is a field of line elements and a local basis is given by a nonvanishing vector field X which belongs to D at any point. Therefore there exists a local coordinate system (z') such that
x = a/axl (See Exercise 1.22.9). Hence D is completely integrable. Furthermore, an integral curve of X is an integral manifold of D. Now assume that the theorem holds for involutive distributions of dimensions 1 , 2 , . . . ,k - 1; we prove it for an involutive distribution of dimension k. Since D is smooth, there exists a local basis XI,. . . ,Xk of D on a neighborhood V' of x. Furthermore, we can find a local coordinate system ( y ' ) on a neighborhood V c V' of x such that y'(x) = 0 , l 5 i 5 m and
x1= a/ayl.
1.15. Distributions. Frobenius theorem
57
Since D is involutive we have k
[Xi,Xj]= cC:jX1, 1
< i ,j < k.
I='
Define a new local basis of
D on V by
Y;: = Xi - Xi(y')Xi, 2
< i < k.
Since Y2,. . . ,Yk are tangent to the manifolds y' = constant then it follows that [K,yi], 2 5 i , j k, must be tangent to the submanifolds y' = constant also. Therefore the distribution on V defined by Y2,. . . ,Yk is involutive on V and on each submanifold y' = constant of V including S c V defined by y' = 0. Let
<
Then 2 2 , . . . ,zk is a local basis of a C""(k - 1)- dimensional involutive distribution on S. By the induction hypothesis, there exists a coordinate system t 2 , ... ,zm on some neighborhood of x in S such that
+
and the submanifolds defined by t' = constant, k 1 5 i 5 m, are precisely the integral manifolds of the distribution defined by 2 2 , . . . ,2, on this neighborhood. Now, let p : V + S be the canonical projection in the y coordinate system. Then the functions 2
xi
-
- 2
i
1-
1
- Y ,
0p,2
are defined on some neighborhood of z in M and are independent at x. Furthermore we have xi(.) = 0 , l i m. Hence there exists a neighborhood U of x in M with the coordinate functions x', . . . ,xm, on U. Now we prove that
< <
Y;:(xk+j)= 0 on U, 1 5 i
< k,1 <_ j 5 m - k.
Chapter 1. Differential Geometry
58
In fact, we have
axj/ayl =
1, if j = 1, 0, if 2 5 j I m
on U. Consequently, we deduce that y1= a/axl on U.
Hence Y1(xk+J) = 0, 1 I j 5 m - k . Now when 2 5 i 5 k, 1 5 j 5 m - k , we have
But since D is involutive we know that
Thus we deduce that k
(a/axl)(qxk+j))=
c
cii~(~k+j).
(1.7)
1=2
Let us consider the submanifold N of U defined by z2 = constant, . . ., zm = constant. On N , (1.7) becomes a system of k - 1 homogeneous linear differential equations with respect to zl. The functions Y,(xk+j) are solutions of (1.7) satisfying initial conditions K(zk+j)= Z,(zk+J) - 0, when z1= 0. By the uniqueness of solutions we deduce that Y,(xk+J) must be identically zero on U. It follows that Yz,. . . ,Yk are linear combination of the vector fields a/dz2,. . . ,d/dzk. Then a / a z l , . . . , a / a z k is a local basis of D and the submanifolds x' = constant, k 1 5 i 5 m, are integral manifolds of
+
D. 0 A k-dimensional distribution D on a manifold M of dimension m is locally defined by k linearly independent vector fields. Alternatively, we may suppose D is locally defined by m - k linearly independent l-forms W k + l , . . . ,wm.
Proposition 1.15.5 The distribution D is involutive if and only if for each z E M there ezist a neighborhood U and m-k linearly independent 1-forms wk+', . . . , w m on U which vanish on D and satisfy the condition
1.15. Distributions. Fkobenius theorem
c
59
m
dwo=
w;Awb, k + l i a L m ,
b=k+1 for suitable I-forms w;.
Proof: First, we note that, in a neighborhood U of each point z,a local basis X I , . . . ,xk of D may be completed to a local field of frames X I , .. . ,xk, Xk+1,. . . ,xmon M . If {w', . . . w k ,w k + l ,. . . ,w m } is the dual field of coframes, then wk+', . . . , w m vanish on X I , .. . ,xk and hence on D. We have m
[ x r , ~=, ]Cc;,xa, 1 L r,s L mo=l
We know that D is involutive if and only if C:, = 0 for 1 5 r,s 5 k and a 5 m. Now, since w ' ( X , ) is constant, we deduce that
k+1 5
d w ' ( X r , X , ) = - w ' [ X r , X , ] = -Cf,, 1 5 i , r , s 5 m.
On the other hand, dw' may be uniquely expressed by
where
4q= -4p.Therefore 1 . d w ' ( X r , X , ) = -(4, - 4r)= 4,. 2
Hence $, = -Cia. Thus, if
dwa=
D is involutive, we have 1
-
2
c
dp"qd A w q
Ilp,qlm
m
= for k
+ 1 5 a 5 m. If we set k
w8 =
m C diq wp + 5l C p= 1
p=k+1
dp"qw P ,
60
Chapter 1. Differential Geometry
we deduce that m
q=k+l
The converse is proved by a similar procedure. CI We now introduce the concept of an ideal of AM.
Definition 1.15.6 A n ideal I J of AM ia a vector subapace which satisfy the following property: If w E y and r E A M , then w A r E y. Let have
y = {w E A I M / w vanish on D}. Then IJ
is an ideal of AM and we
Theorem 1.15.7 D is involutive if and only if d y = {dw/w E
y} c y
(i.e.,
IJ is a differential ideal). The proof follows directly from Proposition 1.15.5. To end this section, we introduce the concept of foliation (see M o h o [97] for more details). A foliation F (of class C"")of dimension k on an m-dimensional manifold M is a decomposition of M into disjoint connected subsets F = { L a / a E A} called the leaves of the foliation, such that each point of M has a coordinate neighborhood (U, z') such that for each leaf La, the components of L, n U are locally given by the equations z k + l = constant
, . . . ,x m =
constant.
These coordinates (zi) are said to be distinguished. Then each leaf La is a connected immersed submanifold of dimension k. The topology of La is given by the basis formed by the sets {z E V / z k + ' ( z )= constant, . . . ,zm(z) = constant, V open in U}. Hence the topology of L , does not necessarily coincide with the induced topology on L, from M. Thus, in general, La is not an embedding submanifold.
Definition 1.15.8 A maximal integral manifold N of a distribution D on a manifold M i s a connected integral manifold which i s not a proper subset of any other connected integral manifold of D. The following theorem shows that, globalizing Frobenius theorem, an integrable distribution determines a foliation.
1.16. Orientable manifolds. Integration. Stokes theorem
61
Theorem 1.15.9 Let D be a k-dimensional involutive distribution on M and x E M. Then through x there passes a unique mazimal connected integral manifold of D and every connected integral of D through x i s contained i n the rnazimal one.
An outline of proof: Let N be the set of all those points y in M for which there is a piecewise Coocurve joining y t o x, whose Cooportions are integral curves of D , i.e., their tangent vectors belong t o D. Then we can prove that N is the desired maximal integral manifold (see Sternberg [114], Warner [ 1231). From Theorem 1.15.9, we deduce that the maximal integral manifolds of D determine a foliation on M.
1.16
Orientable manifolds. Integration. Stokes theorem
First, we discuss orientation on vector spaces. Let V be an m-dimensional vector space. We introduce an equivalence relation in the set of all bases of V as follows. Two bases { e l , . . . ,em} and (21,. . . , E m } are said to have the same orientation if det(A!) > 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 <_ k <_ m.
(1.16)
From the existence theorem for ordinary differential equations, we easily deduce the following.
Theorem 1.18.5 For a n y point x E M and for any tangent vector X E T,M, there is a unique geodesic with initial condition ( z , X ) , i.e., a unique ) x and geodesic o(t) defined o n s o m e interval (-cy c), c > 0, such that ~ ( 0 = b(0) =
x.
Definition 1.18.6 A linear connection V o n M i s said t o be geodesically complete if every geodesic a ( t ) can be extended so as t o be defined for all t E R. Example.-Define on R"' a linear connection V by
v,,,,ia/ad
= 0,
for all i ,j = 1 , . . . ,m. Then I'fj = 0 and (1.16) becomes
d 2 x t / d t 2 = 0 , 1 5 k 5 m. Thus the geodesics are straight lines and V is complete. Next, we define the covariant derivative of a tensor field K of type (0,r) or ( l , r ) , r 2 1, as follows. First, if K is a tensor field of type ( 0 , r ) (resp. (1,r)) and X a vector field on M , we define a tensor field V xK of the same type by
Chapter 1. Differential Geometry
78
(resp.
r
-C K ( X 1 , .. . , V x X i , . . . ,Xr)). i= 1
V xK is called the covariant derivative of K along X . Then we define V K to be the tensor field of type ( 0 , r + 1) (resp. ( 1 , r + 1)) given by ( V K )( X I )- - . X r )X ) = ( V X K )( X I ,. . - xr). )
)
V K is called the covariant derivative of K. We say that K is parallel if V K = 0. Definition 1.18.7 Let V be a linear connection on M . Then the torsion tensor of V is the tensor field T of type (1,2) on M defined b y
T ( X , Y )= V X Y
- vyx - [X,Y].
Clearly, T is skew-symmetric, i.e., T ( X , Y )= - T ( Y , X ) . V is said to be symmetric if its torsion tensor T vanishes. In local coordinates we have
T ( a / a z ' , a / a z J )= T&(a/azk), where
T; = rk. - rfi. $1 Hence V is symmetric if and only if borhood.
rfj = I'ji
in any coordinate neigh-
Definition 1.18.8 Let V be a linear connection on M . Then the curvature tensor of V is the tensor field R of type (1,3) on M defined b y
R ( X ,Y ,Z ) = vxvyz
-
vyvxz - V[X,Y]Z.
I . 18. Linear connections. Riemannian connections
79
Clearly, R ( X , Y,2) = -R(Y, X,2). V is said to be flat if its curvature tensor R vanishes. In local coordinates we have
and we easily obtain
Sometimes we shall consider the tensor field R(X,Y ) of type (1,l)defined by
R(X,Y)Z= R(X,Y,Z). Theorem 1.18.9 (Bianchi’s identities). For a linear connection V on M, we have Bianchi’s Id*identity
Bianchi’s 2”d identity
where 5 denotes the cyclic sum with respect to X , Y and 2 . In particular, i f T = 0, then Bianchi’s l s t identity: 5 { R ( X ,Y ,2 ) )= 0; Bianshi’s Pd identity: $ { ( V x R ) ( Y2, ) )= 0.
(For a proof see Spivak [113] or Kobayashi and Nomizu [Sl]). Next we shall obtain a specific connection for a Riemannian manifold.
Theorem 1.18.10 Let M be a Riemannian manifold with Riemannian metric g. Then there is a unique symmetric linear connection V on M such that Vg = 0. This connection V is called the Riemannian connection (sometimes called the Levi-Civita connection).
Chapter 1. Differential Geometry
80
Proof: EXISTENCE. Given two vector fields X and Y on M , we define V x Y by the following equation:
+ g ( [ X ,Y1,Z) + g([z,XI,Y )+
Y1, X),
(1.17)
for all vector field 2 on M . It is a straightforward computation that the map ( X , Y )---t V x Y , satisfies the four conditions of Definition 1.18.1. Similarly, we prove that V has no torsion. To show that Vg = 0, it is sufficient to prove that
which follows directly from (1.17). UNIQUENESS. It is a straightforward verification that if V satisfies V x Y - VyX - [ X , Y ] = 0 and 6 9 = 0 , then it satisfies (1.17). Then 6=V.0
Remark 1.18.11 It is obvious that Theorem 1.18.10 holds for pseudoRiemannian manifolds. From (1.17)) one easily deduces that the Christoffel components I':j of the Riemannian connection V are given by
Let M be a Riemannian manifold with Riemannian metric 9 . Then the arc length of a C" curve c7 : [ u , b ] ----+ M is defined by L(a) =
/.*
In local coordinates we have
where a(t) = ( z i ( t ) ) .
( 9( b( t ) b( t ) )' k t . )
1.19. Lie groups
81
This definition can be generalized to a piecewise Coocurve of the obvious manner. Now, suppose that M is connected. Then we define the distance function d ( z , y ) on it4 as follows. The distance d ( z , y ) between two points z and y is the infimum of the lengths of all piecewise Coocurves joining z and y . Then we have: (1) d ( 2 , Y ) L 0 ; (2) d ( z , Y ) = 4 Y , (3) 4 2 , Y ) I d(z, 4 d ( z , Y ) ; (4) d ( z , y ) = 0 if and only if 2 = y . (See Boothby [9] for a proof of (4)). Thus d is a metric on M and we can prove that the topology defined by d is the same as the manifold topology of M (see Boothby [9] or Kobayashi and Nomizu [Sl]). A Riemannian manifold admits convex neighborhoods. Precisely, for each z E M , there is a neighborhood U of z such that any two points z1 and 2 2 of U may be joined by one and only one geodesic B lying in U ; this geodesic is minimal if for any other curve 7 from z1 to 2 2 , we have L(a) 5 L ( 7 ) . (See Kobayashi and Nomizu [81] for a proof).
4; +
Theorem 1.18.12 (Hopf-Rinow). Let M be a connected R i e m a n n i a n manifold. T h e n the following properties are equivalent: (1) M is geodesically complete. (2) M is a complete metric space with respect t o the distance f u n c t i o n d. (3) Every closed and bounded subset of M (with respect t o d) i s c o m p a c t . 0 We shall refer to Kobayashi and Nomizu [81] for a proof. Since every compact metric space is complete, we have the following.
Corollary 1.18.13 If a connected R i e m a n n i a n manifold i s compact, t h e n a n y pair of points z , y E M m a y be joined b y a geodesic whose length is precisely d ( z , y ) . 0
1.19
Lie groups
Lie groups are the most important special class of differentiable manifolds. Lie groups are differentiable manifolds which are also groups and in which the group operations are C'.
-
Definition 1.19.1 A Lie group G i s a differentiable manifold which is at the s a m e t i m e a group such that the m a p G x G G defined by ( a ,b ) ---+ ab-' is Coo.
a2
Chapter 1. Differential Geometry We use e to denote the identity element of a Lie group G .
-
- -
-
Remark 1.19.2 Let G be a Lie group. Then the map a a-1 is C" since it is the composition a (e,a) 0-l of C" maps. Also, the map (a,b) ab is C" since it is the composition ( a , b ) (a,b-') ab of C" maps.
- -
Examples of Lie groups. (1) R m is a Lie group under vector addition. (2) As we have seen in Section 1.2, the general linear group G l ( m , R )is an open submanifold of g l ( m , R ) . Moreover Gl(rn,R) is a group with respect to matrix multiplication. It is easy to check that the map ( A ,B ) AB-', A , B E G l ( m ,R) is C". (3) Let C* be the non-zero complex numbers. Then C* is a group with respect to the multiplication of complex numbers. Moreover C* is a (real) 2-dimensional manifold covered by a single coordinate neighborhood U = C*= R2- ((0,O)) with coordinates
-
1=2
+ J-ly
-
-
(2,y).
One easily check that the map ( 1 1 , ~ ~ ) zlz;' is C". (4) The circle S1 may be identified with the complex numbers of absolute value +l. Since I 1 1 2 2 I=I t l 11 1 2 I, then S' is a group with respect to the multiplication of complex numbers. Moreover the map (q, 12) zlz;' is C" (see Exercise 1.22.1).
-
Proposition 1.19.3 If G1 and G2 are Lie groups, then the direct product G1 x G2 of these groups (with the multiplication given b y ( a l , a 2 ) ( b l , b 2 ) = ( a l b l , azb2)), with the Cw structure of product manifolds is a Lie group.
The proof is left to the reader as an exercise. Corollary 1.19.4 The m-torus Tm is a Lie group.
-
Definition 1.19.5 A Lie algebra over R is a real vector space g with a bilinear map [ , ] : g x g g (called the bracket) such that: (1) [ X ,YI = - [Y,XI, (2) [ [ X , Y ] , Z ] [ [ Y , Z ] , X ] [ [ Z , X ] , Y ]= 0, (Jacoby identity), f o r all X,Y,Z E g .
+
+
1.19. Lie groups
83
Examples of Lie algebras (1) Let V be an rn-dimensional vector space. Then V becomes a Lie algebra by setting [ X , Y ]= 0, for all vector X , Y E V. (A Lie algebra g such that [ X ,Y ] = 0 for all X , Y E g is called Abelian). (2) Let M be an m-dimensional manifold. Then the vector space x ( M ) of all vector fields on M is a Lie algebra with respect to the Lie bracket of vector fields. (3) The set g l (m, R ) is a real vector space of dimension m2. If we set
[ A ,B] zz A B
-
BA,
then gl(rn, R ) becomes a Lie algebra.
Definition 1.19.6 Let G be a Lie group and a E G . W e denote by & (resp. r,) the left translation (resp. right translation) of G by a, i.e., & ( b ) = ab (resp. ra(b) = ba). Clearly ! ,is a difleornorphisrn with inverse la-,.A vector field X on G is called left invariant if it is invariant by all left invariant translations fa, i.e., d f , ( b ) X ( b ) = X ( a b ) , a,b E G . A p f o r m w on G is said to be left invariant if it is invariant by all
e,,
i.e., t?;w = w , a E
G.
Next we shall prove that there is a finite dimensional Lie algebra associated with each finite dimensional Lie group. Let G be a Lie group and g its set of left invariant vector fields. Clearly g is a real vector space. Define a linear map a:g-T,G
a ( X )=X(e). Then a is a linear isomorphism. In fact, a is injective, since if a ( X ) =
a(Y)then for each a E G we have X ( a ) = dl,(e)X(e) = dt,(e)Y(e) = Y ( a ) .
Chapter 1. Differential Geometry
a4
Moreover, a is surjective, since, given u E T,G, we set X ( a ) = dt,(e)u and X E g because d&(b)X(b) = d&(b)&(e)u = dt,b(e)u = X(ab). Hence dimg = dimT,G = dimG. Now, the Lie bracket [ X , Y ]of two left invariant vector fields X and Y is a left invariant vector field (see Exercise 1.22.8). Then g is a Lie algebra which is called the Lie algebra of G . Let X be a left invariant vector field on G. If p: is a local 1-parameter group of local transformations of G generated by X and pt(e) is defined for t E (--c,c), c > 0, then pt(a) can be defined for t E (--c,c) for every a E G. Furthermore we have pt(a) = t,(p:(e)) since p: commutes with t, for every a E G (see Exercise 1.22.4). Hence p:(a) is defined for t E R (see Exercise 1.22.2), and X is complete. We set pl(e) = exp X.
--
Now = pst is the 1-parameter group of transformations generated by tX. Indeed, we have that the tangent vector to the curve s $ $ ( a ) at s = 0 is equal to t times the tangent vector t o the curve s pst(a) at s = 0. Hence we deduce exptX = t+!Jl(e) = pt(e), for all t E R. The map have
X
--+ e x p X
exp(s
of g into G is called the exponential map. We
+ t ) X = (exp sX)(exptX),
exp(-tX) = (exp tX)-'.
-
-
We call exptX the 1-parameter subgroup of G generated by X. Proposition 1.19.7 exp : g G i s Cooand dexp(0) : Tog T,G is the identity map (with the usual identifications Tog li g and T,G 'Y g), s o exp gives a difleomorphism of a neighborhood of 0 i n g onto a neighborhood of e in G (for a proof see Warner [lSS]).
85
1.19. Lie groups
Example. Let Gl(m,R) be the general linear group. Since gl(m,R) is a vector space then we have a canonical identification g W g l ( m , R ) ) -L
h R),
where I is the identity matrix. Moreover, we have a canonical identification TI(Gl(m, R ) ) C'L TI(gl(m, R ) ) . Now, if g is the Lie algebra of Gl(m, R ) we define a canonical linear isomorphism
7 ( X )= P ( X ( I ) ) . One easily check that 7 is in fact a Lie algebra isomorphism (i.e., 7 [ X ,Y] = [ 7 X ,yY])and then gl(m, R ) is the Lie algebra of Gl(m, R ) . It is known that the exponential map exp : gl(m,R) Gl(m,R) coincides with the usual exponential map
-
00
A i / i != I
exp A =
+ A + A 2 / 2 !+ . . .
i=o
(see Warner I1231 for more details).
Remark 1.19.8 (1) Let V be an m-dimensional vector space. Let E n d ( V ) denote the group of endomorphisms of V , and let A u t ( V ) denote the vector space of automorphisms of V. Then A u t ( V ) is a Lie group with Lie algebra E n d ( V ) . A basis of V determines a diffeomorphism of E n d ( V ) with g/( m,R ) sending Aut(V) onto Gl(m, R ) . (2) Let G l ( m , C ) be the complex general linear group. We can easily prove that Gl(m,C) is a 2m2-dimensional Lie group with Lie algebra gl(m,C). If V is a complex m- dimensional vector space, and if E n d ( V ) denotes the group of complex linear transformations of V, and A u t ( V ) denotes the set of complex automorphisms of V, then A u t ( V ) is a 2m2-dimensional Lie group with Lie algebra E n d ( V ) . Definition 1.19.9 (1) Let G be a Lie group. By a Lie subgroup H of G we mean a subgroup which i s at the same time a submanifold of G such that H itself is a Lie group with respect to this differentiable structure. (2) Let g be a Lie algebra. By a Lie subalgebra of g we mean a vector subspace h of g such that [ X ,Y ] E h whenever X , Y E h.
Chapter 1. Differential Geometry
86
Let H be a Lie subgroup of G. If X E h (Lie algebra of H ) then X is determined by its value at e, X ( e ) , and X ( e ) E TeH C TeG determines a left invariant vector field 2 on G , i.e., 2 E g (Lie algebra of G). It follows that h can be identified with a Lie subalgebra iof 9. Conversely, every subalgebra iof g is the Lie algebra of a unique connected Lie subgroup H of G . To see this, we first define an involutive distribution U on G as follows: U(u) = { X ( u ) / X E
i}.
Then the maximal integral submanifold through e of U is a Lie subgroup H of G and the Lie algebra h of H can be identified t o i. Hence we have Proposition 1.19.10 There is a 1-1 correspondence between connected Lie subgroups of G and Lie subalgebras of the Lie algebra g of G.
Let H be a Lie subgroup of a Lie group G. Then H is an immersed submanifold of G , but H is not necessarily an embedded submanifold (see Example below). We have the following result about Lie subgroups which are embedded submanifolds. Theorem 1.19.11 A Lie subgroup H of a Lie group G is an embedded submanifold if and only if H is closed (See Warner [123] for a proof). A further result is the following.
Theorem 1.19.12 (Cartan Theorem). Let G be a Lie group, and let A be a closed abstract subgroup of G. Then A has a unique Coostructure which makes A into a Lie subgroup of G. We omit the proof (see Spivak [113], Warner [123]). Examples The following Lie groups are closed subgroups of the complex general linear group Gl(m,C) (if A E Gl(m,C), then we denote by A' the transpose of A and by A the complex conjugate of A ) : (1) Unitary group U ( m ) = { A E Gl(m,C)/ A'A = I }
Its Lie algebra is
1.19. Lie groups
u(m) = { A E gl(m,C)/ At
+ A = 0)
A matrix A E U(m) is called skew-Hermitian. (2) Special linear group S l ( m , C ) = {A E Gl(m,C)/ det A = 1) Its Lie algebra is
sl(m, C) = { A E gl(m,C)/ trace A = 0) (3) Complex orthogonal group
O(m,C) = { A E Gl(m,C)/ A'A = I} Its Lie algebra is
o(m,C) = { A E gl(m,C)/ A'
+A =0)
A matrix A E O(m, C) is called skew-symmetric. (4) Special unitary group
SU(m) = U(m) n S l ( m , C ) Its Lie algebra is
su(m) = u(m) n sl(m,C). (5) Real special linear group
Sl(m, R) = Sl(rn,C) n Gl(m, R) = {A E Gl(rn, R)/ det A = 1) Its Lie algebra is sl(rn, R ) = { A E gl(m, R)/ trace A = 0) (6) Orthogonal group
O(m) = U(m) n Gl(m, R) = { A E Gl(m, R)/ A'A = I)
Chapter 1. Differential Geometry
88
Its Lie algebra is o(m) = o ( m , C ) n gl(m, R) = {A E gl(m, R)/ A'
+ A = 0).
(7) Special orthogonal group SO(m) = O(m) n Gl(m, R).
Its Lie algebra is also o(m). The dimensions of these Lie groups are easily computed from their Lie algebras:
dim v ( m ) = m2, dim Sl(m,C) = 2m2- 2, dim O(m,C) = m(m - I ) ,
dim S U ( m ) = m2 - 1, d i m SI(m,R) = m2 - 1,
-
dim O(m)= dim SO(m) = (m(m - 1))/2. Definition 1.19.13 ( I ) A map p : G H is a Lie group homomorphism if p is both Cm and Q group homomorphism. We calI p a Lie group isomorphism if, in addition, p is a difeomorphism. A Lie group isomorphism of G into itself is called a Lie group automorphism. If H = Aut(V) for some vector space V, or if H = G l ( m , C ) or G l ( m , R ) , then a homomorphism 'p : G -+ H as called a representation of the Lie group G. (2) A linear map t,b : g h is a Lie algebra homomorphism ift,b[X,Y] = [t,bX,t,bY],for all X,Y E g. If t,b is, in addition, a linear isomorphism, then is calledd a Lie algebra isomorphism. An isomorphism of g into itself is called a n automorphism. If h = End(V) for some vector space V, or if h = g l ( m , C ) or gl(m, R), then a homomorphism 1/, : g h is called a representation of the Lie algebra 9.
-
-
1.19. Lie groups
89
Example Let T 2 = S' x S2 and let p : R2
-
T 2 be given by
Then p is a Coomap of rank 2 everywhere and is a Lie group homomorphism. Now, let a be an irrational number and define F : R -+ R2 by F ( t ) = ( t , a t ) . Obviously F is an embedding with image being the line through (0,O) of slope a. Hence p' = p o F is an injective immersion of R into T 2 and p'(R) is an immersed submanifold of T 2 . Furthermore p' is a Lie group homomorphism so that p'(R) is a Lie subgroup of T2.But p'(R) is dense in T 2 and, thus, it is not an embedded submanifold.
-
Now, let p : G H be a Lie group homomorphism. Since p maps the identity of G to the identity of H , then we have
d p ( e ) : T,G
-
T,H
-
and hence we can define a Lie algebra homomorphism @ : g h of g (Lie algebra of G ) into h (Lie algebra of H ) as follows: If X E g , then @ ( X )is the unique left invariant vector field on H such that
In particular, for every a E G, we can define an inner automorphism i, : G + G by i,(b) = ah-'. Then i, induces an automorphism of g (Lie algebra of G ) , denoted by Ad(a). The representation
Ad : G given by
Ad : a
-
-
Aut(g)
Ad(a)
is called the adjoint representation of G. For every a E G and X E g we have
= dt,-1 (a)df!,(e)(X(e))
Chapter 1. Differential Geometry
90
= (dra-l(a))X(a)
= (Tra-l)(x)(e),
since X is left invariant and i a = ra-l o &. Thus (Ad(a))X = (Tra-l)X.
Proposition 1.19.14 Let X , Y E 9. Then
[Y,X] = lim( l/t)[Ad(exp( - t X ) ) Y t-+O
-
Y]
P r o o f Let X , Y E g and pt the l-parameter group of transformations of G generated by X. Since pt(a) = f?,(pt(e)) = f?,(exptX) = a(exptX) = rexptX(a), we deduce that pt = rexptX. By Exercise 1.22.7, we have [Y,X] = lim(l/t)[(Tpt)Y - Y ] t-10
= lim(l/t)[Ad(exp(-tX))Y - Y].o t-10
-
Now, the adjoint representation Ad : G Aut(g) induces a Lie algebra homomorphism ad : g -+ End(g). It is not hard to prove that ad(X)(Y) = adxY = [ X , Y ] (see Spivak [113], Warner (1231). The representation ad : g -+ End(g) is called the adjoint representation of 9. Let G be a Lie group. If w is a left invariant l-form and X a left invariant vector field on G , then
= (f?:u(a))(X(e)) = w(e)X(e) = w(X)(e)
1.20. Principal bundles. Frame bundles
91
and thus w ( X ) is constant on G. Furthermore, if w is a left invariant form, then so is dw, since d commutes with for every a E G. Hence, if w is a left invariant l-form and X, Y E g , we obtain
c,
d w ( X , Y )= - w [ X , Y ] ,
(1.18)
-
which is called the Maurer-Cartan equation. Clearly, the vector space of left invariant l-forms on G can be identified with the dual space g* of G : w Li, where G(X)= w ( e ) X ( e ) . Thus if XI,. . . ,Xmis a basis for g and O’, . . . ,Om is the dual basis for g*, then B’, . . . ,Om are left invariant l-forms on G. We set
[xi,xj]= Chxk, where the Ct.’s are called the structure constants of G with respect t o the . . Xm. From (1.18) it is easy t o check that the Maurer-Cartan basis XI,. equation is given by )
m
dok = -1/2
C ChOi A O’,
15
k5
m,
i,j=1
since C& = -cji. k
1.20
Principal bundles. Frame bundles
Before introducing principal bundles, we need some generalities about Lie groups acting on manifolds.
-
Definition 1.20.1 Let M be a diflerentiable manifold and G a Lie group. A C” map 4 : M x G M such that (1) 4(z, ab) = 4(+, 4, b), (2) + , e l = 2, for all a , b E G and x E M i s called an action of G on M on the right. If G acts on M on the right, then for a fixed a E G, the map 4, : M --t M , &(z) = 4(z, a ) is a transformation of M (for this reason we call G a Lie A4 transformation group). For a fixed z E M , we denote by q5z : G
-
the C” map defined by q5z(a) = ~ $ ( z , a ) . We say that G acts effectively (resp. freely) if q5,(z) = z for all z E M = id^. (resp. for some z E M ) implies that a = e , i.e., For the sake of simplicity we sometimes set +(z, a ) = za.
92
Chapter 1. Differential Geometry
-
Remark 1.20.2 Similarly, we can define an action of G on M on the left as a Co3map 4 : G x M M such that (1) + ( a h 4 = 4 h 4(h 4, (2) + ( e , 4 = 2, for all a,b E G and z E M. Now, let us suppose that G acts on the right on M and let g be the Lie algebra of G. Then to each A E g we assign a vector field XA on M as follows: XA is the infinitesimal generator of the l-parameter group of transformations of M given by ( t , z )E R x
M
-
+(z,ezptA)
Proposition 1.20.3 W e have
Proof: In fact, ddz (e)Ae = d+z (0)( d ezpA ( 0 ) )( d / d t ( O ) )
(see Exercise 1.22.23).
since
-
Proposition 1.20.4 Assume that G act8 on M on the right. Then ( I ) The map X : g x ( M ) is linear. (2) X[A,B ] = [XA,XB]. (3) If G acts effectively then X is injective. (4) If G acts freely, then for each non-zero A E g , XA never vanishes.
1.20. Principal bundles. Frame bundles
93
Proof: (1) follows from Proposition 1.20.3. To prove (2) we note that [ A ,B] = lim(l/t)[B - A d ( e z p ( - t A ) ) ] , t-+O
by Proposition 1.19.14. On the other hand, we have 4ezptA
4 z ( e z p ( - t A ) ) ( a )= z [ e z p( - t A ) a ( e z p t A ) l ,
for all a E G. Since r$( z,e zpt A ) = z ( e z p t A ) is the l-parameter group of transformations generated by XA, we have
-
from Proposition 1.19.14. Hence X : g x ( M ) is a Lie algebra homomorphism. To prove ( 3 ) , suppose that XA = 0 everywhere in M. This implies that, for every z E M, the l-parameter group of transformations z ( e z p t A ) must be trivial, i.e., z ( e z p t A ) = z. If G acts effectively, then e z p t A = 0 for every t and hence A = 0. To prove (4), suppose ( X A ) ( z )= 0 for some z E M. Then z ( e z p t A ) = z for every t. !f G acts freely, then e z p t A = 0 for every t and, hence A = 0 . 0
Definition 1.20.5 Let M be a differentiable manifold and G a Lie group. A principal bundle over M, with group G, consists of a fibred manifold P over M with projection 7~ : P + M and a n action of G o n P o n the right satisfying the following conditions: (1) ~ ( u a=) .(a) f o r a11 u E P and a E G ;
Chapter 1. Differential Geometry
94
-
(2) P i s locally trivial, i.e., for each z E M there i s a neighborhood U of z and a difeomorphism rl) : T - ' ( U ) U x G such that $(u) = (n(u),p(u)) where p : T-'(U) G is a map satisfying p(ua) = (p(u))a for all u E .-'(U) and a E G . A principal bundle will be denoted by P ( M , G ,T ) , P ( M ,G ) , or simply P. We call P the total space or the bundle space, M the base space and G the structure group.
-
Since ~ ( u a = ) T ( U ) we deduce that { u a / a E G} c T - ' ( T ( u ) ) . Moreover, since p(ua) = (p(u))a, we easily see that { u a / u E G } = ( T - ' ( T ( u ) ) . Notice also that if ua = u for some u E P , then a = e. Hence G acts freely on P . Each fibre T - ' ( z ) , x E M , of P is diffeomorphic t o G . If x = T ( U ) for some u E P , then T - ' ( z ) is an embedded submanifold of P and the tangent space to ~ - ' ( z ) at u is a vector subspace V, of T,P which is called the vertical subspace at u. Clearly we have
V, = Ker{dn(u) : T,P
-
T,M}.
A tangent vector in V, is called vertical.
Examples of principal bundles (1) Trivial principal bundles Let M be a differentiable manifold and G a Lie group. We set P = M x G and define an action P x G P by
-
-
(x,a)b = (z,ab).
If we define T : M x G M by T ( z , ~=) x, then P is a principal bundle over M with structure group G and projection T . (2) Frame bundles Let M be an m-dimensional manifold. A linear frame u at a point z E M is an ordered basis XI,. . . ,X, of the tangent space T,M. Let F M be the set of all linear frames at all points of M and let T be the map of FM onto M which maps a linear frame u a t x into x. We define an action of the general linear group Gl(m,R) on F M on the right as follows. If a = ( a ; ) E G l ( m ,R) and u = (XI,. . . , X m )is a linear frame at x, then ua is the linear frame ( Y I ., . . , Ym)at z defined by m
j=l
1.20. Principal bundles. Frame bundles
95
In order to introduce a differentiable structure in FM we proceed as follows. Let (zl, . . . ,zm)be a local coordinate system in a coordinate neighborhood U of M. If z € U then (8/~3z')~, . . . ,( 8 / 8 z m ) , is a linear frame at x. Hence every linear frame u at z can be expressed uniquely in the form u = ( X I , .. . ,Xm)with m
j=l
where (X,!')E G l ( m ,R). Thus we have a bijective map
$ : r-'(U)
-
U x Gl(m,R)
given by
$b)= (z,(x;)). We can make FM into a differentiable manifold by taking (zi,Xi) as a local coordinate system in r-'(U). It is clear that the action FM x Gl(m,R) FM, u ua, is C*. Thus, if we set p(u) = (X,!') E G l ( m ,R), where u = ( X I , .. . ,Xm),then it is easy to check that FM is a principal bundle over M with structure group G l ( m , R ) and projection r. We call FM the f r a m e bundle (or bundle of linear frames) of M . We notice that if { e l , . . . ,em} is the canonical basis for R"' and u = ( X I , .. . ,Xm) is a linear frame at z,then u may be considered as a linear isomorphism u : R"' T,M defined by u(e;) = X;,1 5 i 5 m . On the other hand, a non-singular matrix a E G l ( m ,R) may be considered as the automorphism a : Rm R" given by a(e;) = afej. Hence the action of G l ( m ,R) in FM is given by the composition R"' 5 Rm 5T,M. Given a principal bundle P ( M , G ) , the action of G on P induces a Lie algebra homomorphism X of the Lie algebra g of G into x ( P ) by Proposition 1.20.4. For each A E g , XA is called the f u n d a m e n t a l v e c t o r field corresponding to A. Since the action of G sends each fibre into itself, (XA), E V,. As G acts freely on P , if A # 0 then XA never vanishes. Then the linear map
- -
-
defined by
Chapter 1. Differential Geometry
96
is injective. Since d i m g = dimV,,, we deduce that this map is a linear isomorphism. For each u E G, let R, : P P be the diffeomorphism defined by R,(u) = uu. We have the following.
-
Proposition 1.20.6 If XA is the fundamental vector field corresponding to A E g , then (TR,)(XA) is the fundamental vector field corresponding to (Ad(a-'))A E g .
Proof Since XA is induced by the l-parameter group of transformations R e z p t ~of P, then the vector field (TR,)(XA) is induced by the l-parameter group of transformations RaRczp~ A R , - I = R,-I(~~,, tA)o (see Exercise 1.22.4). But a-l(ezptA)a is the 1- parameter group of transformations of G generated by Ad(u-')A. Thus R , - I ( ~tA)a ~ ~is generated by x(A~(~)A). Definition 1.20.7 Let P ( M , G ) be a principal bundle with projection T . A section s of P over an open set U c M is a section of the fibred manifold x :P M , i.e., a Coomap s : U + P such that A o s = idu. If P is the frame bundle FM of a manifold M, then a section s of FM over U is called a local frame field of U. If x E U ,hence s(z) = ( s l ( x ) ,. . . ,srn(z)), m = dim M , is a basis of T z M . Thus a local frame field s on U determines m vector fields 9 1 , . , , , sm on U which are linearly independent at each point of u.
-
Remark 1.20.8 Given a principal bundle P ( M ,G ) with projection T , there always exist local sections. In fact, since P is locally trivial, for each x E M , there exists an open set U c M and a diffeomorphism $ : T-'(U) UxG such that +(u) = ( ~ ( u ) , ( p ( u )where ), (p : T - ' ( u ) G satisfies (p(uu) = (p(u))a. Hence we can define a local section s : U P by
--
- -
-
-
Definition 1.20.9 (1) A principal bundle homomorphism f : P'(M', P ( M , G ) consists of a map f l : P' P and a Lie group homomorG') phism f 2 : G' G such that fl(u'a') = f l ( u ' )fZ(a') f o r all u' E P' and u' E
1.20. Principal bundles. Frame bundles
-
97
-
Thus f i maps each fibre of P' into a fibre of P and then we can define map f s : M' M b y f 3 ( 2 ' ) = .(fl(u')), z' E MI, u' E P', ~ ' ( u ' )= x' where d : P' M' and .rr : P M are the correeponding projections. Hence the following diagram
GI.
a
-
is commutative. For the sake of simplicity, we shall denote
f.
f l , fi
and
f3
by
(2) f is called an embedding i f f : P' P is an embedding and f : G' -+G is injective. By identifying P' with f (P'), G' with f (GI) and M' with f ( M I ) , we s a y that P'(M',G') is a principal subbundle of P ( M , G ) . (3) If, moreover, M' = M and the induced map f : M M is the identity transformation of M , f : P ' ( M , G ' ) P ( M , G ) is called a reduction of G to G I . The subbundle P'(M,G') is called a reduced bundle and we say that the structure group of G is reducible to the Lie subgroup GI.
-
-
-
Let P ( M , G ) be a principal bundle with projection ?r : P M . Since P is locally trivial, it is possible to choose an open covering {U,} of M and a diffeomorphism $, : r - ' ( U a ) U, x G for each a such that t+!~~(u) = ( ~ ( u ) ,pa(u)), where p, : n-'(u,) G is a map satisfying pa(ua) = (pa(u))u for all ti E r-'(ua) and a E G . Hence we can define a map $pa : 27, n Up -+ G by
--
$pa(.)
= Pp(u)(Pa(u))-l, u E
.-'(4.
(1.19)
In fact, (1.19) depends only on ~ ( u not ) on u, since if a E G , then we have
Definition 1.20.10 The family of maps $pa are called transition functions of P corresponding to the open covering {U,}.
Chapter 1 . Differential Geometry
98
The principal bundle P can be reconstructed from the transition functions:
Proposition 1.20.12 Let M be a manifold, {U,} an open covering of M and G a Lie group. Assume that there exist a family of functions $pa : U, n Up G satisfying (1)-(3) of Proposition 1.20.11. Then there is a principal bundle P ( M ,G) with transition functions $pa.
-
Proof We set
X = U({a}x U, x G). a
Then each element of X is a triple ( a , z , a )where a is some index, z E E G. Clearly X is a differentiable manifold. We introduce an equivalence relation on X as follows. We say that (a, z,a) is equivalent to (p,y, b) if and only if z = y E U, n Up and b = $pa (.)a. Then P = X/ -. The projection 7r : P M is defined by ~ [ a , z , a= ] z, where [ a , z , a ] denotes the equivalence class of ( a , x , a ) . The action of G on P is given by
U, and a
-
-
[a, z,a]b= [a, z,ab].
In order to make P into a differentiable manifold we note that, by the natural projection X -+ P = X/ -, each {a}x U, x G is mapped one-toone onto rIr-'(Ua). Hence we introduce a differentiable structure on P by requiring that the map X P induces a diffeomorphism of {a}x U, x G onto 7 r - l ( U Q ) . Now we can easily check that P ( M , G ) is a principal bundle over M with structure group G and projection K. Moreover if we define
-
$, : 7r-'(U,)
$,[a,29.1 then the transition functions are precisely $pa.
$b,
-
U, x G
: U, n Up
-
4,
= (z,
G corresponding to {U,}
Theorem 1.20.13 The structure group G of P ( M , G ) is reducible to a subgroup G' if and only if there exists an open covering {U,} of M with transition functions $pa which take their values in G'.
1.20. Principal bundles. Frame bundles
99
-
Proof: Suppose that G is reducible t o G'. Then there exists a reduced bundle P'(M,G') with projection x' = x / P ' , where : P M is the projection of P over M . Let {U,} be an open covering of M with diffeomorphisms : (d)-'(U,) U, x G', and let : U, n Up G' be the corresponding transition functions. We extend $: to T-~(U,) as follows. If by u E n-'(Va), then u = u'a, where u' E (x')-'(U,) and a E G. Then we define
-
+:
$ha
(pa!: x-'(U,) + G
by &(U)
= (Oa(u'a) = Ph(+
and
+,
: x-'(U,)
-
U, x G
by +a(4
=
(44, P a
w
-
Then the corresponding transition funtions $0, : U, n Up G are given by $0, (z) = E G', for all z E U, n Up. Conversely, suppose that there is an open covering {U,} of A4 with transition functions +pa taking values in a Lie subgroup G' of G. From Exercise 1.22.20, the map +pa : U, n Up G' is Coowith respect t o the differentiable structure of G'. Hence, proceeding as in Proposition 1.20.12, we construct a principal bundle P ' ( M ,G') with transition functions $pa. Now, it is easy to prove that P' is a subbundle of P (details are left t o the reader as an exercise). Example Let F M be the frame bundle of an rn-dimensional manifold M . Let {U,} be an open covering of M by coordinate neighborhoods U, (i.e., an atlas of M ) with coordinate functions z i , . . . ,zz. Then the transition functions $pa of F M corresponding to {Ua}are given by
$ha(.)
-
$p,(z) = (azi/azk)(z), z E
U, n up.
(1.20)
In fact,
+a&)
= W(4(Pa(u))-l,
where u is an arbitrary linear frame at z. If we take u = (a/dzk, ..., a/azF), then we obtain (1.20).
Chapter 1. Differential Geometry
100
1.21
G-struct ures
This section concerns t o G-structures. G-structures on manifolds has been introduced by Chern in 1953. For a general theory of G-structures see Bernard [7], Chern [14] and Fujimoto [57].
Definition 1.21.1 Let M be a diferentiable manifold of d i m e n s i o n m and G a Lie subgroup of Gl(m, R). A G - s t r u c t u r e o n M is a reduction of the f r a m e bundle F M t o G. Thus, a G-structure on M is defined by a principal bundle & ( M ) over M with structure group G satisfying the following conditions:
-
1. B G ( M )is a submanifold of F M ; 2. The projection x' : & ( M )
x:FM-M;
-
-
M is the restriction of the projection
3. The map Rh : B G ( M ) & ( M ) is the restriction of the map R, : FM F M for all a E G.
In the sequel we will denote x' and R, by x and R,, respectively. From Theorem 1.20.13, we have,
Proposition 1.21.2 A manifold M of d i m e n s i o n m a d m i t e s a G-structure i f and only i f there i s a n atlas {Ua}of M with coordinate f u n c t i o n s xi,. . . ,xz, such that the Jacobian matrices (ax~/ax~)(x), f o r all x E U, n Up, belong to G. The following result will be very useful in the next chapters.
Proposition 1.21.3 Let M be a n m-dimensional manifold, G a L i e subgroup of G l ( m , R), and B a subset of F M . T h e n B is a G-structure o n M i f and only i f the following conditions are satisfied: (1) x ( B ) = M ; (2) (x/B)-'(x) = UG = {ua/a E G}, x E M , x = x(u); (8) f o r each x E M there ezists a neighborhood U of x and a C" section s :U F M of F M over U s u c h that s ( U ) c B .
-
Proof It is clear that if B is a G-structure on M , then (l),(2) and (3) hold. Let us prove the converse. From (3) we can choose an open covering
1 21. G-struct ures
101
{U,} of M such that, for each U,, there exists a local section s, of F M over U, taking values in B. Since n(s,(z)) = n(sp(z)),for each z E U, n Up, then, from (2)) there exists a unique element $p,(z) E G such that
-
sa
(4=
SB (.)$pa
(4
-
*
Hence $pa : U, n Up Gl(rn,R) is a CM map taking values in G. By Exercise 1. , it follows that $pa : U, n Up G is Coowith respect t o the differentiable structure on G. Now, we define a map
a, : U, x G
-
(=/B)-'(U,)
@ a ( z , a )= s a ( z ) a ,
for all x E U, and a E G. We can easily check that a, is bijective. Moreover, we have ('Pj'o Q a ) ( v 4 = ( W f + h ( z ) a ) .
We set
$a =
Q;'.
(1.21)
Then (1.21) becomes (1.22)
Hence we introduce a differentiable structure on B such that $, becomes a diffeomorphism. (From (1.22) this differentiable structure is well-defined). Moreover, if we define the action of G on B by the restriction of the action of GI(rn,R) on FM, B becomes a principal bundle over M with structure group G and projection n / B . It easily follows that the inclusion i : B + F M is an embedding and thus B is a G- structure.0
-
Remark 1.21.4 We notice that the function $pa : U, n UB G are precisely the transition functions of B corresponding to the open covering {Val-
Definition 1.21.5 Let & ( M ) be a G-structure on a diflerentiable manifold M . A linear frame u at x E M which belongs to B c ( M ) ia called an adapted frame at x. A local section s of F M taking values in B c ( M ) is called an adapted frame field. From Proposition 1.21.3 we easily deduce the following.
Chapter I . Differential Geometry
102
Corollary 1.21.6 M possesses a G-structure if and only if there exists an open covering {U,} of M and local frame fields s, on U, such that if we set for z E U, n Up
then $a,(.)
u,
E G.
Now we set s, (z) = (Xp(z),. . . ,XE) and define the m l-forms 6; on by
e;(z)(x,.(z)) = b; Then {6:,. {X,",. . . , X $ } .
. . ,Or} is
called the (adapted) c o f r a m e field dual to
Examples of G-structures (1) {e}-structures Let e be the identity matrix of Gl(m,R). From Proposition 1.21.3, it follows that M possesses an {e}-structure if and only if there exists a (global) section s : M + F M , i.e., a global frame field. Hence giving an { e } structure is the same as giving a global frame field. Then M possesses an {e}-structure if and only if M is a parallelizable manifold. (2) O(m)-structures (Riemannian s t r u c t u r e s ) Let < , > be the natural inner product in R"' for which the canonical basis {el,. . . ,em} is orthonormal. Then the orthogonal group O(m) may be described as follows: O(m) = { A E Gl(m, R ) /
< A ( , A q >=< c , q >, for all ( , q E Rm}
Suppose that M possesses an O(m)-structure Bo(,)(M). Then we can define a Riemannian metric g on M as follows. For each z E M we set
g2(X,Y) =< u - ~ X , U - ' Y >, 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 <-i <-n .
Then we have
apz' apz' =
=a/ai
+ (af j / a z ' ) ( a / a y j )
(af i / a z ' ) ( a / a y i )= ( r i ( a / a y j )
Thus, we deduce
J(a/az') = a p y ' ,
J(a/ay') = 0.
Hence J is integrable. Summing up, we have proved the following theorem due t o J. LehmanLejeune [861).
Theorem 2.3.2 An almost tangent structure J is integrable if and only if its Nijenhuis tensor N J vanishes identically. Corollary 2.3.3 The canonical almost tangent structure on the tangent manifold T N of any manifold N is integrable. Corollary 2.3.4 Every almost tangent structure on a 2-dimensional manifold M is integrable. Proof Let H be an orthogonal complement of V with respect to any Riemannian metric on M . Let { X , J X } be an adapted local field frame with X E H . Then we have
N j ( X , JX) 0. Hence N J = 0 vanishes.0
Remark 2.3.5 From Corollary 2.3.4, we deduce that the almost tangent structures J , defined on the 2-torus T 2are always integrable. Since the 2torus is compact then in neither case J , is globally equivalent t o the standard almost tangent structure on R2.Thus, the integrability of an almost tangent structure is a purely local matter.
2.4. Almost tangent connections
2.4
119
Almost tangent connections
In this section we give a characterization of the integrability of an almost tangent structure in terms of a symmetric linear connection.
Definition 2.4.1 Let J be an almost tangent structure o n M. A linear connection V on M is said to be an almost t a n g e n t connection if V J = 0, that is, V&Y)
= J(VxY),X,YE X(M).
Proposition 2.4.2 On each almost tangent manifold there always exists an almost tangent connection.
Proof: Let V be any symmetric connection on M , for instance, the Riemannian connection of some Riemannian metric on M . Choose an orthogonal complement H in T M of V = K e r J = ImJ. Now, we define a tensor field Q of type (1,2) on M as follows:
+
Q ( J X ,JY)= ( V J X J ) ( Y ) ( J ( V y J ) ( X ) ) , for X , Y E H . Let V = V - Q . Then V is a linear connection on M such that V J = 0 . 0 Now, let V be a symmetric linear connection on an almost tangent manifold M with almost tangent structure J . Since V is symmetric, we have
[ X , Y ]= VXY
-
vyx.
Hence the Nijenhuis tensor NJ of J may be written as follows:
+
N J ( X , Y )= ( V J X J ) ( Y )- ( V J Y J ) ( X ) ( J ( V Y J ) ) ( X-) J ( ( V X J ) ) ( Y ) . Therefore, we have
120
Chapter 2. Almost tangent structures and tangent bundles
Proposition 2.4.3 If there is a symmetric almost tangent connection V on M then J i s integrable. Proof: In fact, suppose that V is a symmetric almost tangent connection on M . Then V J = 0, and consequently, we deduce NJ = 0. Thus, J is integrable. We now prove the converse. Proposition 2.4.4 If J i s integrable, then there ezists a symmetric almost tangent connection on
M.
Proof: Let V be the linear connection on M constructed as in Proposition 2.4.2. Since J is integrable and V is symmetric, we have
Consequently, Q is symmetric. Then we have
+
= - Q ( X , Y ) Q(Y,X ) = 0 , where T denotes the torsion tensor of V. Then is the required symmetric almost tangent connection.
2.5
Vertical and complete lifts of tensor fields to the tangent bundle
In Section 2.2, we have introduced the vertical lifts of vector fields on a manifold N to its tangent bundle T N . Next, we shall define the vertical and complete lifts of tensor fields. Let F be a tensor field of type ( l , r ) , r 2 1, on N . Then the vertical lift of F to T N is the tensor field of type (1, r) on T N defined by
where XI,. . . , X r E T , ( T N ) , y E T , N , x E N . If
2.5. Vertical and complete lifts of tensor fields to the tangent bundle
121
then we have
where (z', d ) are local coordinates for T N . We can easily prove that
F"(X,V,.. . ,X,") 0, for any X i , . . . ,Xr E x ( N ) . Moreover, if I i s the identity tensor on N , then J = I" i s the canonical almost tangent structure on T N . Now, let G be a tensor field of type ( O , r ) , r 1 1, on N . The vertical lift of G t o T N is the tensor field of type (0,r) on T N defined by
where X I , .. . ,Xr E T y ( T N ) , yE T , N , z E N , and f" = f vertical lift t o T N of the function f E C " ( N ) . If
0 TN
denotes the
then we have G" = Gj1,..jr(d~") €3.. . €3 ( d z ' l ) .
(2.4)
We easily deduce that
G"(X,V,.. . ,X,") = 0, for any X i , . . . ,Xr E x ( N ) Moreover, if a = a;(dz') is a 1-form on N , then a" = a;(&'). Thus, a" is precisely the pullback of a to T N , that is
Next, we define the complete lift of tensor fields on N to T N . Let f be a function on N . Then the complete lift of f to T N is the function f" on T N given by
Then we have
122
Chapter 2. Almost tangent structures and tangent bundles
fC
= y'(af/az').
(2.5)
Now, let X be a vector field on N. Then X generates a local 1-parameter group of local transformations q5t on N. Let Tq5t be the (local) 1-parameter group of local transformations of 2" determined by q5t. The infinitesimal generator of Tq5t is called the complete lift of X to T N and denoted by xc.If
x = x'(a/az'), then we have
X C= x'(a/az')
+ uj(aX'/dz~)(a/au').
Thus, we obtain
From (2.3), (2.4) and (2.6), we easily deduce
F"(Xf,.. . ,X,C) = (F(X1,. . . ,X,))", (resp.G"(X;,.
. . , X i ) = (G(X1,. . . ,Xr))")
for any XI,. . . ,X, E x(N), where F (resp. G) is a tensor field of type (1,t) (resp. ( O , r ) ) on N. From (2.5) and (2.6), we easily deduce the following.
Proposition 2.5.1 For
any
X E x ( N ) and f E C m ( N )
X"f" = 0,X"f" = XCf" = (Xf)",XCfC = (xf)c. For the Lie bracket, we have
Proposition 2.5.2 For any X, Y E x ( N ) [X",Y"] = O,[X",Y"] = [X,Y]",[XC,YC] = [X,Y]". In order to define the complete lifts of tensor fields to TN, we first prove the following.
2.5. Vertical and complete lifts of tensor fields to the tangent bundle
123
Proposition 2.5.3 Let F , F ' ( r e s p . G , G ' ) be tensor fields of type (1,r) ( r e s p . ( O , r ) o n T N such that
q x : ,. . . )X,C)= E ' ( X i ) .. . ,q, ( r e s p . G ( X : , .. . for any XI,.
. . ,X , E x ( N ) .
,x;)= G ' ( X ; ,. . . ,X:))
Then
P = F'(re8p.G = G').
Proof: We only prove the case ( 1 , l ) (the general cases (1,r) and ( 0 , r ) may be proved in a similar way). It is sufficient to prove that if F(XC) =0 for any vector field X on N , then P = 0. Suppose that
P = Af(a/azj) @ (dz') + B,t'(a/azJ)@ (du')
+c;(a/auJ)
@ (dz')
+ q (a/auJ) @ ( d u ' ) .
If X = a / a z k ,we obtain
which implies A; = C,t' = 0. Then
E
+
= Bij(a/azJ)@ (dd) o ; ' ( d / d v i ) @ (do').
Now, let X = X k ( a / a z k ) .Since
xc = x k ( a / a z k )+ u3(ax'/az"(a/au'), we obtain
+
0 = F ( X " ) = w " ( a x i / a z 3 ) ( B ~ ( a / a z JDI:(a/auj)). )
Since X i and u k are arbitrary, we easily deduce that
B! = D! = 0. Hence F = 0 . 0
124
Chapter 2. Almost tangent structures and tangent bundles
Definition 2.5.4 Let F (reap. G ) be a tensor field of type (1,r) (resp. (Olr)) on N . The complete lift of F to T N i s the tensor field F e ( r e s p . G C )of type (llr) (resp. (0,r)) defined b y
F C ( X l , . .,X,C)= ( F ( X 1 , .. . ,xr))c, ( r e s p . G C ( X ;., . . ,X,C) = ( G ( X 1 , .. . ,X,))') for any X i , . . . ,X,E x ( N ) .
(Obviously, if I denotes the identity tensor on N , then Ic = Identity tensor on T N ) . Then, if F = F : ( a / a z j ) 8 (dz') is the local expression of a tensor field of type (1,l) on N, we have
+ F / ( a / a v J )8 (dv'). For a l-form a = a;&', we obtain a" = uk(da'/dzk)(dz')
+ cYI(dd).
For a tensor field G of type (0,2)we have
+ G'j(dU') €3 ( d z j ) . From (2.7), (2.8) and (2.9)) we easily deduce the following.
Proposition 2.5.5 For any X , Y E x ( N )
F C ( X " )= ( F ( X ) ) " a, C ( X " )= ( a ( X ) ) " , G C ( X " , Y c= ) G c ( X c , Y "= ) (G(X,Y))", G C ( X " , Y u=) 0 .
(2.9)
2.5. Vertical and complete lifts of tensor fields to the tangent bundle
125
Next, we consider the particular cases of tensor fields type ( 1 , l ) and
(W. Complete lifts of tensor fields of type (1,l). Proposition 2.5.6 (1) If F is a tensor field of type ( 1 , l ) on N of rank t , then FC has rank 2r; (2) if F, G are tensor fields of type ( 1 , l ) on N then we have (FG)" = F'G". Proof: (1) is a direct consequence of (2.7). To prove (2), we have
( F G ) " ( X "= ) ( ( F G ) ) ( X ) ) '= ( F ( G ( X ) ) ) ' = F C ( ( G X ) " ) = F C ( G C ( X "= ) ) ( F e G C ) ( X c0 ). Corollary 2.5.7 If P(t) is a polynomial i n one variable t, then
we) = (W))", for any tensor field F of type ( 1 , l ) on N.
Complete lifts of tensor fields of type (0,2). Proposition 2.5.8 (1) Let G be a tensor field of type (0,2) on N . If G has rank r, then G Chas rank 2r; (2) if G is symmetric (resp. skew-symmetric), then G c is symmetric (resp. skew- symmetric). Proof: (1) follows from (2.9), (2) is a direct consequence of the definition of G C . o Corollary 2.5.9 If G is a Riemannian metric on N, then G C i s a pseudo Riemannian metric on TN of signature (n,n). (For the definition of signature of pseudo-Riemannian metrics see Exercise 2.11.2). Corollary 2.5.10 If w is a &-form on N, then w c is a 2-form on TN and we have dwc = (dw)'.
To end this section, we define the complete lifts of distributions on N . Let D be a Ic-dimensional distribution on N. We define the complete lift of D to 7 "to be the 2k-dimensional distribution Dc spanned by the vector fields X u and X " , X being an arbitrary vector field belonging to D. Obviously, if D is locally spanned by { X I , .. . ,X k } , then Dc is locally spanned by {X,", . . . ,Xj!,X i , . . . , X i } .
126
Chapter 2. Almost tangent structures and tangent bundles
2.6
Complete lifts of linear connections to the tangent bundle
Let N be an n-dimensional manifold with a linear connection V. Then we define the complete lift of V to T N as the unique linear connection V" on T N given by V>,YC = (V,Y)", for any X,Y E x ( N ) . This assertion may be verified by an easy computation using Christoffel components. In fact, we obtain
V,/,,;a/azj
+ u~(ar;j/az~)(a/auk),
= rt.(a/azk) 'J
(2.10)
v;/,u;a/ad = F:+i,n+j(a/axk) + r;::,ntj
(a/auk>,
we deduce from (2.10) that
F:,
= rk. -k = Fk+i,n+j - p n++ki , n + j = 0 , '3 ' Ff,ntj = rn+i,j rn+k = '3
ul(arfj/azl),ri,n+j n+k = F;gj
= rfj.
(2.11)
From (2.11), we easily prove that V" defines a linear connection on 7".
2.6. Complete lifts of linear connections to the tangent bundle
127
Proposition 2.6.1 If T and R are respectively the torsion and curvature tensors of V , then T C and RE are respectively the torsion and curvature tensors of V " . Proof: In fact, we have
VXCYC- Ve,,XC- [ X C , Y = C ]( V x Y ) c- ( V y X ) c- ( [ x , Y ] ) c = (VxY
-
VyX
-
[X,Y])'= (T(X,Y))'= T c ( X C , Y c ) ,
for any X,Y E x( N ) . By a similar procedure, we prove that the curvature tensor of V cis, precisely, R C . o
Corollary 2.6.2 (1) V is symmetric if and only if V c is symmetric; (2) V i s flat i f and only if V c i s flat. Proof: In fact, let S be a tensor field on N of type (1,r) or (0,r). Then S = 0 if and only if S c = 0.0 From (2.11)) we can easily prove the following
Proposition 2.6.3 We have
V$.YU = (VXY)",V$,YC= (VXlqC, Vk.YC= V5,Y" = (VXY)". Proposition 2.6.4 For a tensor field S of type (1,r) or (0,r) on N we have
vcsu= (VS)",VCSC= (VS)C. Proof: We only prove the proposition for tensor fields of type ( 1 , l ) . The general case can be proved by a similar procedure. Let F be a tensor field of type ( 1 , l ) on N. Then we have (VcF")(YC,Xc) = (V$cF")(Yc) = V>c(F"YC) - F"(VkcYc)
= (VX(FY)- F(VxY))"= ((VF)(Y, X))" = (VF)"(X", Y".
Then VcF"= (VF)".In a similar way, we prove that VcFC= (VF)'.U
Chapter 2. Almost tangent structures and tangent bundles
128
Corollary 2.6.5 Let J be the canonical almost tangent structure on TN. Then
Vc i s
an almost tangent connection.
Proof In fact, we have
V c J = V c ( I " )= (VI)"= 0 , since I is the identity tensor field on N . 0
Corollary 2.6.6 Let V be the Riemannian connection with respect to a Riemannian metric g on respect to gc.
N. Then Vc i s the pseudo-Riemannian
connection with
Proof In fact, from Corollary 2.6.2, we deduce that V" is symmetric. Furthermore, from Proposition 2.6.4, we have Vcgc = (Vg)' = 0. Then the result follows. Next, we establish some properties about the geodesics of V and V". First, let u be a curve in N and 6 its canonical lift to 2". Then we have Proposition 2.6.7 If u is a geodesic with respect to Vc,then u i s with respect t o Vc. Proof Let CY be a geodesic of of differential equations:
a geodesic
V. Then u satisfies the following system
+
( d 2 2 / d t 2 ) r f j ( d z i / d t ) ( d d / d t ) = 0.
Since b(t) = ( z ' ( t ) , ( d z ' / d t ) ) , we deduce by a direct computation from (2.11), that b is a geodesic of V c . O
Now, let 8 be a curve in T N . In local coordinates, we have 8(t)= ( d ( t ) , d ( t ) ) .
We put u = rN o 8. Then u is a curve in 8( t )E
N and
To(t)N, for any t .
Thus 8 defines a vector field X ( t ) on
N along u ; X ( t )is locally given by
X ( t ) = v'(t)(a/az').
2.7. Horizontal lifts of tensor fields and connections
129
Proposition 2.6.8 Let B be a geodesic in T N with respect to Vc. Then its projection u onto N is a geodesic with respect to V and X ( t ) is a Jacobifield along u . Proof: Let 8 be a geodesic in T N with respect to Vc. From (2.11), we deduce
+
( d 2 z k / d t 2 ) I ' f j ( d z ' / d t ) ( d d / d t ) = 0,
(2.12)
+ 2 I ' ~ j ( d v ' / d t ) ( d z j / d t )= 0 .
If we set
then (2.13) becomes
+
( b 2 v k / d t 2 ) R%jve(dx'/dt)(dzj/dt)= 0,
(2.14)
where Rfij are the components of the curvature tensor of V. Now (2.12) implies that t~ is a geodesic and (2.14) implies that X ( t ) is a Jacobi vector field along u.
Remark 2.6.9 For a discussion of Jacobi fields see Kobayashi and Nomizu 1811 *
2.7
Horizontal lifts of tensor fields and connect ions
Let V be a linear connection on N with local components Ffj. We denote by R its curvature tensor, by T its torsion tensor and by R:, and Ti their local components. By 6 we denote the opposite connection of V defined by V,Y
=
vyx + [ X , Y ] .
Then we have = I?;'. If V is symmetric, then V = V. For every local coordinate system (V,z') in N ,we set
130
Chapter 2. Almost tangent structures and tangent bundles
Di = a/&'
-
dl-'fi(a/awk),1 5 i 5 n = dimN,
where (xi,w') are the induced coordinates in T U . Then ( 0 ; )is a set of linearly independent local vector fields on T N . We define the horizontal subspace H , at y E T N as the vector subspace of T , ( T N ) spanned by {Di(y)}. It is a straightforward computation to prove that H , is independent on the choice of the local coordinate system. Then we obtain an n-dimensional distribution H on T N given by
y E TN
-
H, c T,(TN).
Since T,,(TN)= H , $V,, then H is a complementary distribution of the vertical distribution V and we have
T T N = H $ V (Whitney sum)
H is called the horizontal distribution defined by V. Since V, = Ker d r ~ ( y )we , deduce that drN(y) restricted t o H , gives a linear isomorphism
Consequently, if X is a vector field on N , we can define the horizontal lift of X to T N to be the vector field X H on T N such that
Obviously, we have
(a/aZi)H=
(2.15)
and
x H= xi(a/axi)- wjxir!i(a/awk)), where
X = X'(a/dxi). From
(2.15) and (2.16), we deduce that
H H = Xi D;. Moreover, X H f "= (Xf)", for any function f on N .
(2.16)
2.7. Horizontal lifts of tensor fields and connections
131
The set of local vector fields { D i , h = d / d v ' } is called the adapted frame to V . The dual coframe {O', q'} is given by .
.
8' = dz', q' = vjI'i.kdzk + dv'.
(2.17)
{ O i , q i } is called the adapted coframe to V . A vector field X on T N is called horizontal if
X ( Y ) E H ~for, every y E T N . Now, let F be a tensor field of type ( 1 , l ) on N . We define a vertical vector field 7 F on T N by
If F = F / ( d / d z j ) @ (dz'), we have
7 F = v'F/(a/avj)*
(2.18)
From (2.6) and (2.18), we obtain
xc- X H = 7 ( V X ) ,
(2.19)
where V X is the tensor field of type ( 1 , l ) given by
( V X ) ( Y )= vyx. By a straightforward computation in local coordinates, we obtain the following.
Proposition 2.7.1
[ X " , Y H ]= [ X , Y ] "- (V,Y)" = - ( V y X ) " , [ X H , Y H ]= [ X , Y ] * - 7 k ( X , Y ) , where k is the curvature tensor of V and k ( X ,Y ) is defined by k ( X ,Y ) ( Z )= R ( X ,Y ,2). Let w be a l-form on to T N by
N. Then we define the horizontal lift wH of
w
Chapter 2. Almost tangent structures and tangent bundles
132
WH(XH)= O,WH(X")= (W(X))".
If w = widxi, we obtain
Hence,
(dz')H = dI$dxk
+ dv'.
With respect to the adapted coframe {O', q ' } , we have WH
=
wie'.
(2.20)
Next, we shall define the horizontal lifts of tensor fields of type ( 1 , l ) and (0,2) (For a general theory of horizontal lifts see Yano and Ishihara [130]). Let F be a tensor field on N with local components F:. We define the horizontal lift of F t o 2" with respect t o V t o be a tensor field F H of type ( 1 , l ) on 2" determined by
F H X H= ( F X ) H F , HX" = (FX)".
(2.21)
Then we have
F H = F / ( a / a d )@ (dz') + v'(I'iiF,k
-
I'&<')(LJ/avk)@ (dx')
+ F / ( a / a v ' ) @ (dv'). With respect to the adapted frame, we have
From (2.21), we easily deduce the following.
Proposition 2.7.2 Let F, G be tensor fields of type (1,l) o n N . Then ( I ) ( F G ) =~ F ~ G ~ , (2) ( I d ) H = I d . As a direct consequence of Proposition 2.7.2, we have
2.7. Horizontal lifts of tensor fields and connections
133
Proposition 2.7.3 If P(t) is a polynomial i n t, then ( P (F ) ) H = P( F H ) . Moreover, if
F has constant rank r, then F H has constant rank 2r.
We may extend these definitions for tensor fields of type ( 1 , r ) ) r 2 2 2 2, on N. Then the horizontal lift of S to T N is the tensor field SH of the same type on T N given by as follows. Let S be a tensor field of type (l,s), s
S H ( X f , .. . ,X,") = 0) S H ( X F , . .. )x;-l,x;)x;+l). H H . . ,X,") = (S(X1,. ..,
S"(X,",.
. . ,xp> = (S(X1,. . . xu))H. )
In a similar way, we define the horizontal lift to T N of a tensor field G of type (0,2)on N to be the tensor field GH of type (0,2)on T N given by G H ( X u , Y u )= ) 0, G H ( X U , Y H= ) G H ( X H , Y u= ) (G(X,Y))",
G H ( X H , Y H )= 0.
If G = G,j(dz')
@I
( d d ) , then
with respect to the adapted coframe. We can easily obtain the following.
Proposition 2.7.4 Let g be a Riemannian metric and V a linear connection on N. Then gH i s a pseudo-Riemannian metric on TN of signature (n,n). Moreover, gH and gc coincide if and only if Vg = 0.
The proof is left to the reader as an exercise. (Obviously, we may extend the above definition for tensor fields of type (O,s),s 1 3 ) .
134
Chapter 2. Almost tangent structures and tangent bundles
Now, let D be a distribution on N. Then the horizontal lift of D to T N is the distribution DH on T N spanned by Xuand X H , where X is an arbitrary vector field on N belonging to D . If D is a k-dimensional distribution, then DH is a 2k-dimensional distribution. In fact, if D is locally spanned by {XI,. . . ,xk},then DH is locally spanned by {Xp,. . . ,Xi,XB,.. . ,XB}. To end this section, we define the horizontal lift of linear connections. Let V be a linear connection on N. The horizontal lift of V to 7" is the linear connection VH on 2" given by
= 0, V ; . P
V$Y"
= 0,
v$Hyu = (VXY)", VCHYH = (VXy)H.
(2.23)
From (2.23), we deduce that VH has local components
= rk. 13' p n+i,j + k = rk.. $1
(2.24)
(Here, we use the same notation as in Section 2.6). From (2.23) or (2.24), we easily obtain
x,Y)),
V$YC = ( v x Y ) c - 7 ( R ( ,
(2.25)
where R( , X , Y ) is the tensor field of type (1,l) on N defined by
R( , X , Y ) Z = R ( Z , X , Y ) . Proposition 2.7.5 Vc and VH = 0.
coincide if a n d only if V i s
Proof: Directly from (2.25). 0
flat, that is, R
2.8. Sasaki metric on the tangent bundle
2.8
135
Sasaki metric on the tangent bundle
Let g be a Riemannian metric on N with local components g;, and Riemannian connection V. In [108],Sasaki defined a Riemannian metric j on T N given by
& ( X " , Y H ) = 0,
Iu(XH,YH) = 92(X,Y),
(2.26)
where X , Y E T , N , y E T,N and the horizontal lifts are taken with respect to V. It is easy to check that j defines a Riemannian metric on T N , called the Sasaki metric induced by g. If H and V denotes the horizontal and vertical distributions, respectively, we deduce that H and V are orthogonal with respect to j . From (2.26), we see that the local components of g are given by the following matrix: (2.27)
with respect to the induced coordinates (zi, u'). With respect to the adapted coframe to V,we have
or, equivalently,
3 = gijei 8 e j + gijrli B rli. Let V be the Riemannian connection determined by i . A long but straightforward computation from (2.26) or (2.27), permit us t o obtain the local components of 0:
136
Chapter 2. Almost tangent structures and tangent bundles
n+k - r n + i , j - C:f,n+j
= 0,
(2.28)
with respect t o the adapted frame, where Rbk are the components of the curvature tensor of V (here, we use the notations introduced in Sections 2.6 and 2.7). From (2.28), we obtain the following formulas (see Kowalski [84]): VX" Y" = 0,
(VXHYH)" = (VXY); - ( 1 / 2 ) ( ~ z ( ~ z , ~ z , Y ) ) ; ,
(2.29)
for all vector fields X ,Y on N ,y E T N and z = TN(Y). Moreover, Kowalski [84] proved that the curvature tensor R of V is completely determined by the following formulas:
R ( X " , Y " , Z " )= 0,
2.8. Sasaki metric on the tangent bundle
137
for any X , Y , Z E T,N, y E T N ,z = qq(y). The proof of (2.30) is left to the reader as an exercise. It is sufficient to use (2.29) and the definition of the curvature tensor. From (2.30), we have
Proposition 2.8.1 (TN, ?jis ) f l a t if and only if (N,g) is pat.
Proof: Obviously, if R = 0, then R = 0 . Then
= 0. Conversely, suppose that
R ( X H , Y H Z, H ) ,
= -((1/2)R(Y,Z,X)
+ (1/4)R(Y,Y,R(Y,Z,X)))yH = 0
If we set Y = y, we obtain R ( Y , Z , X ) = 0) for any X,Y,2. Hence R = 0.
138
2.9
Chapter 2. Almost tangent structures and tangent bundles
Affine bundles
In this section, we introduce the notion of affine bundles. The results obtained here will be used in the next section.
-
Definition 2.9.1 An affine space is a triple (A,V,r), where A is a set, V a finite dimensional real vector space, and r : A XV V is a free transitive action of the additive group V on A . We put r(a, u) = a v and say that the afine space A is modeled on V .
+
-
-
Definition 2.9.2 An affine morphism of the afine spaces (A,V,r) and (A',V',r') is a pair ( A , X ) where A : A A' is a map and X : V V' is a linear map such that r'(A(a),X(u))
= A ( r ( a , v ) ) , a E A , v E V,that is
A(a)
,
+ X(u) = A ( a + u).
Example.- Let V be a finite dimensional real vector space. Then (V, V, +) is an affine space modeled on V. Remark 2.9.3 Let A and A' be affine spaces modeled on the same real vector space V. Then A and A' are isomorphic. In fact, A and A' are isomorphic to (V, V ,+).
-
-
-
Definition 2.9.4 An affine bundle consists of a fibred manifold R : A N and a vector bundle p : E N of rank m, together with a morphism r :A X N E A of fibred manifolds over idN, such that f o r each x E N , r, : A , x E,
-
A,
is a free transitive action of the vector space E, on the set A,. Hence, each fibre A , i s an afine space modeled on E,. Now, let A be an affine bundle aa in Definition 2.9.4. Let U be an open set of N over which A admits a local section s and over which E is locally trivial (let us remark that A always admits local sections since R is a subjective submersion). Then there is a diffeomorphism
H : U x Rm
-
p-'(U)
2.10. Integrable almost tangent structures which define fibrations
over the idu. Let a E .-'(U) with .(a) = x E element +(a) of z-'(z) such that a
= s(x)
N. There exists
139 a unique
+ +(a).
-
Hence, we can define a diffeomorphism
4 : .-'(u) given by a
+-'
-
p-'(U)
+(a).
Then o H defines a trivialization of A over U. Thus, A is a locally trivial fibre bundle over N with standard fibre Rm. Finally. let T : A --+ N be an affine bundle modeled on a vector bundle p :E N of rank m. Since N is paracompact, there is a global section s of A. Then we can define an isomorphism of fibred manifolds
-
4:A-E given as follows. Let a E A,. Therefore, there exists a unique +(a) E E, such that a
=s(x)
+ 4(a).
Thus, we can define a vector bundle structure on A of rank rn such that A becomes a vector bundle isomorphic to E. Obviously, this vector bundle isomorphism depends on the choice of s.
+
2.10
Integrable almost tangent structures which define fibrations
In this section, we prove that an integrable almost tangent manifold which defines a fibration is the tangent bundle of some manifold. This result is due to Crampin and Thompson (see (241). Let J be an integrable almost tangent structure on a 2n-dimensional manifold M . Since the distribution V = I m J = Ker J is integrable, then it defines a foliation on M . Now, we define an equivalence relation on M as follows: two points of M are equivalent if they lie on the same leaf of the foliation defined by V .
Chapter 2. Almost tangent structures and tangent bundles
140
Definition 2.10.1 We say that J defines a fibration i f the quotient of M b y this equivalence relation (that is, the space of leaves) has the structure of a diferentiable manifold. This will be the case if for every leaf one can find an embedded local submanifold of M of dimension n trough a point of the leaf which intersects each leaf which it does in only one point. In this case, the space of leaves N is an n-dimensional manifold and the canonical projection p : M N is a subjective submersion (that is, N is a quotient manifold of M). Then p :M N is a fibred manifold and
-
-
for each y E M . Bearing in mind the definitions of the Section 2.2, we may define the vertical lift of tangent vectors on N to M as follows. If u E T,N and y E p-' ( z ) we define uv E T,M by
where ii E T,M and p*(u) = u. Since
v, = Ker{p, : T u M
-
T,N}
and
JJ" = 0 ) then u" is well-defined. Moreover, we have
-
uv E v,,
and the map u u" is a linear isomorphism of T,N with V,. If X is a vector field on N , then we may define its vertical lift to M given by
X" = J X , where
X is any vector field on M which is prelated to X. Obviously,
xvE v. Proposition 2.10.2 Let X, Y be two vector fields on N. Then we have ( I ) [X",Y"]= 0, (2) Lx. J = 0.
2.10. Integrable almost tangent structures which define fibrations
141
Proof: (1) Let X , Y be vector fields on M prelated to X ,Y. Then
[XU, Y " ]= [JX, JP] = J [ J X ,P]
+ J [ XJ,Y ] (since NJ = 0 )
+
= J[X",P]J [ X , Y " ] . But (Tp)[X",P] = [(Tp)X", (Tp)F] = 0,since X " , P are prelated to 0, Y , respectively. Similarly, we obtain (Tp)[X, Y " ]= 0. Hence [ X " , P ] [X, , Y"] are both vertical and, thus,
J[X",Y] = J [ X , Y " ]= 0. (2) It is sufficient to prove that
(LXUJ)(Z") = O,(LX"J)(Z) = 0, where 2 is a vector field on N and 2 is a vector field on M prelated to 2. But this is a direct consequence of (1).0 Now, let V be a symmetric almost tangent linear connection on M. Then we have
Proposition 2.10.3 V induces by restriction a connection o n each leaf of V which i s flat. Proof In fact, we have
VxvY" = V p ( J Y )= J ( V p P ) (since V J = 0)
+
= J ( V ~ X " [X",P]) (since V is symmetric)
= J(V,X")
+J[X",Y]
+
= Vp(JX") J [ X " , Y ]= 0,
where Y is any vector field on M p-related to Y . 0 Now, we prove the main result of this section.
142
Chapter 2. Almost tangent structures and tangent bundles
-
Theorem 2.10.4 (Crampin and Thompson 1241) Let (MI J ) be an integrable almost tangent structure which defines a fibration p : M N . Let V be any symmetric almost tangent connection on M and suppose that with respect to the fiat connection induced on i t by V, each leaf of the foliation defined by V i s geodesically complete. Suppose further that each leaf of this foliation (that is, the fibres o f p : M N ) is simply connected. Then M is an afine bundle modeled on TN.
-
Proof: We define a morphism p:
M X N T N ---t M
as follows. Let u E T , N , z E N . Then we define a vertical vector field U on P - W by
u, = uu E v,,y
E p-'(z).
Then VuU = 0 by Proposition 2.10.3. Thus U is a geodesic vector field and therefore, complete as a vector field. Let & ( t , y ) be the integral curve of U such that
where 4,, : R x p-'(z) U. We define p by
-
4"(0,Y) = Y ,
p-'(z) is the one-parameter group generated by
Next, we shall prove that pz defines a transitive, free action of T , N on p-'(z). In fact, for any u,u E T z N , the corresponding vector fields U ,V satisfy [ U , V ] = 0 (see Proposition 2.10.2). Consequently, their oneparameter groups commute, that is
Since U and V commute and are complete, then the composition of their one-parameter groups is a one-parameter group whose infinitesimal generator is U V. Thus we have
+
4u(t,4u(t,y)) = 4u(t,4u(t,y)) = 4u+u(t,Y).
From (2.31), we deduce that
(2.31)
2.10. Integrable almost tangent structures which define fibrations
PZ(PZ(Y,~),U= ) PZ(PZ(Y,U),4 = PZ(Y,U+
Hence pz : p - ' ( z ) x T,N
-
143
4.
p-'(z)
defines an action of T,N on p - ' ( z ) . Next, let <,> be an arbitrary scalar product on T,N. We define a Riemannian metric g on p-l(z) by g ( U , V ) =< u,u >. 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, <,q ) be an almost contact structure on a (2n l)-dimensional manifold M . Consider the product manifold M x R and define there a tensor field J of type ( 1 9 1 ) by
+
Chapter 3. Structures on manifolds
174
J(X, f ( d / W = ((4X - f t , rl(X)(d/W>
(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 ) = <'(z,(dz/dt),1
5 i 5 m = dim M
We shall express the non-homogeneity of a semispray.
Definition 4.3.4 Let ( be a semispray on M. We call deviation of ( the vector field <* = [C, (1 - (. A simple computation shows that J[C,<] = C. Then (* is a vertical vector field.
195
4.3. Sernisprays. Sprays. Potentials
Definition 4.3.5 A semispray ( is called a spray if (* = 0 and ( is C1 on the zero section. If, moreover, ( i s C 2on the zero section, then ( is called a quadratic spray. Remark 4.3.6 Let ( be a semispray. Since (* = 0 if and only if [C,(1 = (, we deduce that ( is a spray if the functions are homogeneous of degree 2 and C1 on the zero section (if, moreover, the functions are quadratic on vi, then ( is a quadratic spray). Thus, a spray is the geometrical interpretation of a system of second order differential equations homogeneous of degree 2 with respect t o the first derivatives.
I'
Proposition 4.3.7 Let ( be a semispray on M. Then we have
J[(,JX] = -(JX), x E X(TM) Proof In fact, since N J = 0, we deduce 0 = NJ((,X) = [C,JX] - J[C,X] - J[(,JX].
On the other hand,
-(JX)= ( L C J ) ( X ) = [C,JX] - J[C,X] Then we obtain
J [ ( ,JX]= -(JX).o Next, we introduce the potential of a semibasic form. Definition 4.3.0 Let a (resp. L) be a semibasic scalar p-form (resp. a vector 1-form) on TM. Then the potential ao of a (resp. Lo of L) is the scalar (p-1)-form (resp. vector (1-1)-form) given by ao = i p (resp. LO = i ( ~ ) ,
where
is a n arbitrary semispray on M.
Obviously, ao (resp. Lo) is independent of the choice of (. In fact, if is another semispray, then ( - (' is vertical. Proposition 4.3.9 ao and Lo are semibasic.
('
196
Chapter 4. Connections in tangent bundles (Let us remark that the scalar p-form a is not necessarily skew-symmetric).
Proposition 4.3.10 Let a be a semibasic scalar p-form on TM which i s homogeneous of degree r, with p r # 0. Then we have
+
a = (l/(p+ r))((dJa)'+ dJao). Hence if a i s dJ-closed, we deduce
a = (l/(p
+ r ) ) d J a0
3
that is, a i s ezpressed as the derivative of its potential.
Proof: From Exercise 4.12.4, we have
(dja)'
+djao = iedja +djiea = Lca - i[{,J]cr.
= -pa(&,
. . . ,XP)
Thus,
Lca - i[c,J ] a= ra + pa = (p + r ) a , which proves the Proposition.
4.4. Connections in fibred manifolds
197
Remark 4.3.11 A similar result holds for vector forms (see Klein and Voutier [80] and Grifone [73]).
4.4
Connections in fibred manifolds
Let p : E + M be a fibred manifold such that dim A4 = rn and dim E = rn n. We denote by V ( E )the vertical bundle of E , that is,
+
V ( E )=
-
u Vc(E),
cEE
where V c ( E )= K e r { p , : TcE Tp(,.) }. Then V ( E ) is a vector bundle over E of rank n. In fact, V ( E ) is vector subbundle of T E . Furthermore, we have
VC(E) = T c ( E p( e) ) , where Ep(c)is the fibre over p ( e ) .
Definition 4.4.1 A connection in E i s a vector bundle H over E (or a distribution H o n E) such that
T E = V ( E )@ H , that is,
TcE = V c ( E )@ H e , e E E . H i s called t h e horizontal bundle (or horizontal distribution) and He (resp. V c ( E ) )i s called the horizontal (resp. vertical) subspace at e. T h e tangent vectors t o E which belong t o H are called the horizontal tangent vectors. If a vector field X verifies that X ( e ) E H e , f o r a n y e E E , t h e n X is culled a horizontal vector field. If H is a connection in E , then a tangent vector X of E at e E E may be decomposed as follows:
X=vX+hX; v X (resp. h X ) is called the vertical (resp. horizontal) component of X . Then we have two canonical projectors v and h which will be called the vertical and horizontal projectors, respectively.
Chapter 4. Connections in tangent bundles
198
Now, we consider the induced bundle E
E
XM
XM
T M . Let us recall that
T M = { ( e , y ) E E x T M / p ( e )= rM(Y))
is a vector bundle over E such that the following diagram
E XMTM p1
I
E
P2
P
c
c
TM
I
rM
M
is commutative, where p1 and p2 are the canonical projections given by p l ( e , y ) = e and p 2 ( e , y ) = y . Consider the following commutative diagram:
TE
\ d
Then there exists a unique vector bundle homorphism S :
T E +E X M T M
such that
s ( x )= ( r E ( X ) , T p ( X ) )x , E TE, Hence we obtain a sequence of vector bundle homomorphisms
o - V E .LT E 4 E X M T M - 0 ,
(4.10)
where i : V E + T E is the canonical injection. One can easily prove that (4.10) is, in fact, exact. The following result is left to the reader as an exercise.
4.5. Connections in tangent bundles
199
Proposition 4.4.2 (see Vilms [119]).- A connection in E determines a left splitting of (4.10) and, conversely, each left splitting of (4.10) determines a connection i n E.
4.5
Connections in tangent bundles
In this section, we characterize the connections in tangent bundles by means of the canonical almost tangent structure. This theory is due to J. Grifone
-
1731. Let T M the tangent bundle of a manifold M and denote by TM : T M M the canonical projection. Suppose that a connection H in TM is given with horizontal and vertical projectors h and u, respectively. Since V = V ( T M )= Ker J = I m J , where J is the canonical almost tangent structure on T M , we can easily prove that J h = J, h J = 0, Ju = 0, v J = J. We now put
= 2h - I d T M . Then
is a tensor field of type (1,1) on
T M (or a vector l-form on T M ) such that J I ' = J , I'J = -J.
(4.11)
Conversely, if r is a tensor field of type ( 1 , l ) on T M satisfying (4.11), then we easily deduce that
In fact,
Hence, we have A: =
From (4.12) and Put
4,that is,
r(a/azi) = a/aZi + ~f(a/avl). (4.13), we easily deduce that r2 = I d T M .
(4.13) Now, if we
Chapter 4. Connections in tangent bundles
200
then Im u = V, and H = Im h defines a connection in T M . Thus, we give the following definition.
Definition 4.5.1 A connection I' in TM is a tensor field of type ( 1 , l )
on
TM,CM on T M , such that
Jr = J, rJ = -J. Proposition 4.5.2 A connection I' in TM defines an almost product structure on TM, Coo on T M , such that, for each y E T M , the eigenspace corresponding to the eigenvalue -1 is V, and conversely.
Proof In fact,
r2= I d T M . Now, let r X = -X.Then we have
Then J X = 0, and, hence, X is vertical. On the other hand, if X is vertical, then X = JY,for some tangent vector Y. Then r X = -X. Conversely, let r be an almost product structure on T M such that the eigenspace corresponding to the eigenvalue - 1 is V,, for any y E T M . Then r J = -J. Furthermore, let X E T , ( T M ) ,y E T M . Then we have
r(rx - x)= r2x- rx = -(rx - x), since
r2= I d T M .
This implies that
r X - X E V , ( T M ) = ImJ,. Thus I'X - X = JY,for some Y E T , ( T M ) . Hence, we have
and, consequently, we deduce that J r = J . 0 Let
h = (1/2)(1dTM
+ r), u = ( 1 / 2 ) ( 1 d T M - r).
4.5. Connections in tangent bundles
201
Then V = I m v and H = I m h defines a connection in TM (in the sense of the previous section). We easily deduce that
h2 = h, v 2 = v, J h = J , hJ = 0 , J v = 0 , v J = J ,
T ( T M )= v
@ H.
From (4.12) and (4.13)) we deduce that
r is locally given by
r(a/avi) = a / a d .
(4.14)
Since
defines a linear isomorphism
we can define the horizontal lift of a vector field I' to be the vector field X H on T M such that
X on M with respect to
We locally have
xH = xi(a/azi)- x'r{(a/avi),
(4.15)
where X = X ' ( a / d z ' ) . The functions I'i are Cooon T M and will be called the Christoffel components of r. From (4.14)) we deduce that h is locally given by
h(a/av') = 0. Next, we express the nonhomogeneity of a connection.
(4.16)
202
Chapter 4. Connections in tangent bundles
Definition 4.5.3 The tension of a connection I' in T M is the tensor field H of type ( 1 , l ) on T M given b y
Obviously,
H(X)= ( L c h ) ( X )= [C,hX]- h[C,X], where h = (1/2)(1+ I') is the horizontal projector of I'.
Proposition 4.5.4 H is semibasic. Proof: In fact, we have
H(a/dv') = 0.
(4.17)
From (4.17), we easily deduce that H is semibasic.0
Definition 4.5.5 A connection I' in T M is said to be homogeneous i f I' is homogeneous of degree 1, that is, its tension H vanishes. Obviously, 'I is homogeneous if the functions I?: are homogeneous of degree 1.
Definition 4.5.6 An homogeneous connection I' is said to be a linear connection i f r is c1on the zero section. If I' is a linear connection in T M , then the functions I'! are linear on v k . Then, we may writte
I'i (2,v) = v k r ,jk , where
4.5. Connections in tangent bundles
203
rih= (ari/avk). Next, we define the covariant derivative determined by a connection in TM. Before proceeding further, we consider the linear isomorphism
TzM
+
Vu, y E TzM,
given by the vertical lift operator
We denote by
the inverse linear isomorphism. With respect to the induced coordinates, we have (4.18) i
i
Let r be a connection in T M . We define the covariant derivative of a vector field Y on M with respect to a vector field X on M to be the vector field V x Y on M given by
( V X Y ) , = 4 Y ( , ) ( ~ ( d Y ( z ) ( X z )2) E , M.
(4.19)
Suppose that X = X ' ( a / a z ' ) and Y = Y'(a/az'). From (4.18) and (4.19), we obtain
vXy= x i ( ( a ~ j / a z+i )ri(z,u))a/azj.
(4.20)
From (4.20), one easily deduces that
v x 1 + x 2 y = VX,Y
+ vx2y,
VfXY = f ( V X Y ) ,
(4.21)
for any f E C " ( M ) , X i , X z , X , Y E x ( M ) . Moreover, (4.20) shows that ( V x Y ) , depends only on the value of X at z. Thus we can define the
Chapter 4. Connections in tangent bundles
204
covariant derivative of Y with respect to a tangent vector z E T,M as follows:
where X is an arbitrary vector field on M such that X, = z . If u ( t ) is a curve in M and r ( t ) is a vector field along u ( t ) ,that is,
r ( t ) E T J ( , ) M t, E R , then the covariant derivative of r ( t ) along u ( t ) is defined by
( V d ) ( t )= VU(t)Y, where Y is an arbitrary vector field on M which extends r ( t ) . Then ( V , r ) ( t ) is a new vector field along u ( t ) . We say that r ( t ) is parallel along u ( t ) if
v,r
= 0.
Definition 4.5.7 Let I' be a connection i n T M . A curve u in M i s said to be a path of I' if the vector field b ( t ) i s parallel along u ( t ) , i.e.,
V&
= 0.
If I' i s an homogeneous or linear connection, then the paths of I' are called geodesics. Hence a curve u in M is a path of I' if and only if its canonical lift b to TM is an horizontal curve. Suppose that u ( t ) = ( z ' ( t ) ) . From (4.20), we obtain
+
V,b = ( ( d 2 z ' / d t 2 ) I':.(z, d z / d t ) ( d z ' / d t ) ) a / a z ' . Hence u is a path of I' if and only if it satisfies the following system of second order differential equations:
d2z'/dt2 If
+ I ' j ( z , d z / d t ) ( d d / d t )= 0 , 1 5 i 5 rn = dim M.
'I is a linear connection, then d2z'/dt2
(4.22) becomes
+ r ~ , ( z ) ( d z ' / d t ) ( d z j / d t=) 0 ,
(4.22)
4.5. Connections in tangent bundles
205
which are the usual differential equations for geodesics of linear connections V on M (see below and Section 1.18). Now, let I' be a linear connection in T M . Then (4.20) becomes
VX(Yl+ Y2) = VXYl
+ VXY2,
(4.24) V x ( f Y )= ( X f ) Y+ f ( V x Y ) , for any f E C'(M),X,Y1,Y2,Y E x ( M ) . From (4.21) and (4.24)) we deduce that V is a linear connection on M. Then a linear connection in T M defines a linear connection on M (in the sense of Section 1.18). Moreover, the horizontal lifts of vector fields on M to T M with respect to a linear connection I' in T M coincide with the corresponding ones defined in
Chapter 2. Let I' be a connection in TM with Christoffel components I?:.
Di = h ( a / d z i ) = d/az' - r{(d/auj),
K
We set
= a/awi.
(4.25)
Then { D ; , K } is a local frame field which is called the adapted frame with respect t o r. The dual coframe field is given by {Oi, q i } , where 0' = d Z i , qi = r i d $
+ dvi,
(4.26)
We have
hD; = D;, hVi = 0 , OD; = 0 , OK = 0 , J D i = K, J K = 0. From (4.25)) we obtain
pi,oil = ((ar;pzk)- (arjk/azi)+ rf(arf/ad)
[ K , V j ]= 0.
(4.27)
If r is linear, then (4.25) and (4.26) are, precisely, the adapted frame and coframe defined by V (see Chapter 2).
Chapter 4. Connections in tangent bundles
206
Remark 4.5.8 In Definition 4.5.1 we restrict ourselves t o the tangent bundle without the zero section. So we can distinguish between linear and strictly homogeneous connections. In fact, if r is homogeneous and CM on all TM then r is a linear connection (see Grifone [73] and Vilms [119]). Remark 4.5.9 From Section 4.4, we obtain an exact sequence of vector bundle homomorphisms 0 -V(TM)
-
TTM
--+T
M X M T M+ 0 ,
(4.28)
and, by Proposition 4.4.2, we deduce that each connection in TM determines a left splitting of (4.28) and, conversely, each left splitting of (4.28) defines a connection in TM.
Remark 4.5.10 If p : E M is a vector bundle, we may consider homogeneous and linear connections in E. In this situation, we can define a covariant derivative of a section of E with respect to a vector field on M (see Vilms [119]). If the connection is linear, then we obtain a connection in E in the same sense of Koszul [83]). Remark 4.5.11 The extension of the theory for connections in tangent bundles of higher order may be founded in de And& et al. (see [27]).
4.6
Semisprays and connections
In this section, we study the relations between semisprays on TM and connections in T M . Let r be a connection in T M , (I an arbitrary semispray on T M and h = (1/2)(1+ r) the horizontal projector of I?. We set ( = h('.
If
(I'
(4.29)
is any other semispray on T M , we have
h(('
-
(") = h(' - h(" = 0 ,
since (' - (" is a vertical vector field. Hence the vector field defined by (4.29) does not depends on the choice of (I. Moreover, ( is a semispray; in fact,
since J h = J . The semispray ( is called the associated semispray to
r.
4.6. Semisprays and connections
207
Proposition 4.6.1 IfH i s the tension of
Ho = where
r, then we have
c*,
c is the associated semispray to r.
Proof In fact,
Ho = H(<)= [C,he] - h[C,(1. But h(E) =
c, since E is an horizontal vector field, and W*) = h[C,r1 he = h[C,El c, -
since
-
c* is vertical. Hence Ho = [ C , ( ]- ( =
(*.U
A simple computation from (4.16) shows that
c is locally given by
c = vi(a/azi)- dri(a/avi).
(4.30)
If I' is a linear connection, then (4.30) becomes
Corollary 4.6.2 If a quadratic spray).
'I is homogeneous (resp. linear) then ( is a spray (resp.
Remark 4.6.3 If I' is a linear connection defined by a Riemannian metric g on M, then ( is the geodesic spray defined by g (see Boothby [9]).
Now, let ( be an arbitrary semispray on T M . We set
r = -L€J, that is,
r ( X )= (+J)(X) for any vector field X on T M .
= -[E, JXI
+ J [ € ,XI,
Chapter 4. Connections in tangent bundles
208
Proposition 4.6.4 I' i s a connection i n T M whose associated semispray is € +(1/2)C Proof: We have
by Proposition 4.3.7. Hence I? is a connection in T M . On the other hand, if E' is the associated semispray to r, we have
= €+(1/2)€*.0
If
+ p(a/av'), then I' = -LtJ is given by r(a/azi) = a / a d + (a(J/avi),r(a/ad) = - ( a / a d ) .
€ = o'(d/dz')
(4.31)
Then ri = -(1/2)(8(j/i3ui). From (4.9)) (4.22) and (4.31)) we obtain the following:
Proposition 4.6.5 A connection I? and its associated semispray have the same paths.
Corollary 4.6.6 (1) If ( i s a spray, then
r
-LtJ i s an homogeneous connection whose associated semispray is, precisely, (. (2) If ( i s a quadratic spray, then I' = - Lt J i s a linear connection. =
= 0. So the associated semispray Proof: Clearly, if € is a spray, then to r is, precisely, t, by Proposition 4.6.4. Moreover, if H is the tension of I', we have
209
4.6. Semisprays and connections
= -(1/2)((a(J/au*) - uk(a2(j/auiauk)).
Since ( is a spray, we know that the functions degree 2, and so,
are homogeneous of
(J
Hence
which implies
Therefore,
H(a/azi) = 0, and then H = 0, since H is semibasic. (2) follows by a similar devide in local coordinates. 0
Example Let us consider a semispray f on ( = ua/az
R4 = TR2 given by
+ ua/ay + f(z,y)a/au + g(z, Y)a/au,
where (z,y) (resp. (z,y,u,u)) are the canonical coordinates on R2 (resp. P)and f , g are Coofunctions on R4. The Liouville vector field on TR2 is given by
c = ua/au + u a / a u . Then we have
and so
Chapter 4. Connections in tangent bundles
2 10
(* = [C, €1 - ( = -2f(z, y)d/du
-
2g(z, y)d/dv.
We set I' = - L t J . Then we obtain
h(d/dy) = q a y , h ( a / a v ) = 0. Then the associated semispray to I' is
Since the Christoffel components of I' are all zero, we deduce that the tension of r vanishes and so, I' is a linear connection in T R 2 . If we compute the covariant derivative associated to I' we obtain
V X Y = [x'(aY'/az)
+ x"aY'/ay)]a/az
+[x1(ay2/az) +~ ~ ( ( a ~ ~ / / a ~ ) ] a / a ~ ,
+
+
where X = X'a/az X2d/ay, Y = Y1a/az Y 2 a / a y are vector fields on R2. Hence V is the canonical flat linear connection on R 2 . In fact, V is the Riemannian connection determined by the standard Riemannian metric g on R2 given by
g(a/az, spy) = 0.
4.7. Weak and strong torsion
4.7
211
W e a k and strong torsion
In this section, the torsion forms of a connection in T M is defined and related with the torsion tensor for linear connection. Definition 4.7.1 Let The weak torsion of on T M given b y
IT be a connection i n T M with horizontal projector h. is the vector 2-form (or tensor field of type (1,2))
that is,
[J,h](X,Y) = [ J X , h Y ]+ [hX,JY] + J[X,Y] - J[X,hY] - J [ h X , Y ] - h[X, JY]- h[JX, Y], for any vector fields X,Y on T M .
+
+ v i ( a / a v i ) .Then a
Let X = X'(a/az') x ' ( a / a v ' ) , Y = Y i ( a / d z i ) simple computation shows that
t ( x , Y ) = x ' y j ( ( a r t / d v j )- ( a r f / a v i ) ) ( a / a v k ) .
(4.32)
From (4.32), we see that t is a skew-symmetric semibasic form. Then we define the strong torsion T of I' to be the vector l-form (or tensor field of type ( 1 , l ) ) on T M given by
T = tQ - H, where H is the tension of
r. We locally have
T = ( v i ( a r f / a v j )- rf)(a/avk) B (dd), which shows that T is also semibasic. Proposition 4.7.2 We have
t*+ To= 0, where
5 is the associated semispray to IT.
(4.33)
212
Chapter 4. Connections in tangent bundles
Proof In fact,
by Proposition 4 . 6 . 1 . 0 With respect to the adapted frame, we obtain
t ( D i ) V j )= t(v;.,Vj) = 0 ,
(4.34)
T ( D i ) = (d(arf/dW')- r,")vk,T(K)= 0.
(4.35)
and
Proposition 4.7.3 I f T = 0, then t = 0. Proof If T = 0, then from (4.35)) we have
r,"= Or(ar;/aui). Hence we obtain
t(Di, D j ) = ( ( d r ; / / a d )- ( a r f / a U i ) ) v k
= t ( D j )0;).
Since t ( D j , D i ) = - t ( D ; , D j ) , we deduce that t(D;,D,) = 0. Hence t = 0, since it is semi basic.^ Let
be a linear connection. Then (4.32) and (4.33) become
4.8. Decomposition theorem
T(X) = o'xj(rfi- r:,)(a/auk).
213
(4.37)
Now, let U,V be vector fields on M . We define the torsion tensor T of r by
T(U,V)z = 4 , ( t ( U H , V H ) y )Y, E TZM, x E M .
(4.38)
We note that (4.38) does not depends on the choice of y. In fact,
t(uH,vH) = uivj(rfj- r$,)(a/auk),
(4.39)
where U = U'(d/ax'),V = Vi(a/ax'), and the right-side of (4.39) depends only on x. If we develop (4.38) in local coordinates, we obtain
=
vuv - vvu - [U,V],
where V is the covariant derivative defined by I?. Then T is, precisely, the torsion tensor defined by V (see Section 1.18).
4.8
Decomposition theorem
In this section, we shall prove that the strong torsion and the associated semispray characterize a connection in T M . We shall need the following result. and r' be two connections i n T M with the same strong torsion and the same associated semispray. Then r = r'.
Proposition 4.8.1 Let
r
Proof: Let J , t , T and H (resp. J',t',T' and H') be the associated semispray, weak torsion, strong torsion and tension of r (resp. I"). If we put B = I" - r, we have J B = 0 and B J = 0. Therefore B is a semibasic vector l-form on T M . Moreover
Chapter 4. Connections in tangent bundles
214
since (' = E. Now, let h (resp. h') be horizontal projector corresponding to I' (resp. I"). Then t'=
+
[J,h']= [J,h] (1/2)[J,B] = t + (1/2)[J,B],
T' = (t')' since h' = h
- H'
=T
+ (1/2)([J, B]' - [C,B]),
+ ( 1 / 2 ) B . But T' = T . Hence [ J , B ] ' = [C,B].
Therefore we deduce
BJ[t,XI - J [ E , BXI - B[E,JXI = 0 , (4.40) for every vector field X on T M .Since B X E I m J and B is semibasic, we obtain
J [ t ,B X ]= - B X by Proposition 4.3.7. On the other hand, we have
= -B(LtJ)X= -B(LtJ)(JIX), since B is semibasic (here
h is the horizontal
projector of I? = - L t J ) . But
( L ( J ) h = -6 Then we deduce
B J [ ( ,XI
-
B [ [ ,JX]= B(LX)= BX.
Thus (4.40) becomes
BX
+ B X = 2(BX)= 0.
So B X = 0, and then B vanishes. This ends the proof.0
215
4.8. Decomposition theorem
Theorem 4.8.2 (Decomposition theorem). Let ( be a semispray and T a semibasic 1-vector form on TM such that T o + (* = 0 . Then there ezists a unique connection J? i n T M such that its associated semispray i s ( and its strong torsion i s T . The connection r i s given by
r = -L[J+T. Proof: EXISTENCE. Let I' = - L t J + T . Then J r = J and r J = - J , since T is semibasic. So r is a connection in T M . Now, if h is the horizontal projector of I', we have
by Proposition 4.6.4. Furthermore, the weak torsion of t =
where
r is
[ J , h ]= f+ ( 1 / 2 ) [ J , T ]
f is the weak torsion of
r = -Lt J . Then t =
1/2[J,T]
(see Exercise 4.12.5). If we compute the tension H of
H = ( 1 / 2 ) ( L c F )= H
r, we obtain
+ (1/2)(LcT)
= ( 1 / 2 ) ( L c T- L p T ) ,
r
where H is the tension of = - Lt J (see Exercise 4.12.5). Therefore the strong torsion of I' is
T' = t o - H = 1 / 2 ( [ J , T I o+ [ C , T ]But an easy computation shows that
Consequently, we have
[ r J]). ,
2 16
Chapter 4. Connections in tangent bundles
+
T ' X = T ' ( K X ) = -(1/2){T([e1 J ] ( L X ) J [ e , T ( L X ) ] }
= -(1/2){-T(hX) - T ( K X ) } = T ( A X )= TX, since T and T' are semibasic and where h is the horizontal projector of Then T' = T . UNIQUENESS. It is a direct consequence of Proposition 4.8.1.0
r.
Corollary 4.8.3 Let I' be a connection in T M . Then the strong torsion o j 'I vanishes if and only if its weak torsion and tension vanish. Proof: If T = 0 then I' = - L t J and therefore t = 0 (see Exercise 4.12.5). On the other hand, since
T = to - H, we obtain H = 0. The converse is trivial. 0
Remark 4.8.4 From Corollary 4.8.3 we deduce that there are no nonhomogeneous connections with zero strong torsion.
4.9
Curvature
In this section, we introduce the curvature form of a connection in T M . We shall show that the curvature is the obstruction to the integrability of the horizontal distribution determined by the connection.
Definition 4.9.1 Let I' be a connection i n T M . The curvature form of I' is the vector &-form (or tensor field of type [1,2)) on T M given b y
where h is the horizontal projector of 'I
4.9. Curvature
217
Since h2 = h, we have
R ( X , Y ) = - [ h X , hY] + h [ h X , Y ]+ h [ X , h Y ]- h [ X , Y ] .
(4.41)
From (4.41), we obtain
R ( X , Y )= xiyj((arf/ad) - (ar:/azj) -
+ ri(ar:/aJ)
ri (arf/aul))(a/auk),
(4.42)
+
+
where X = X ' ( a / d z ' ) X i ( a / a u i ) , Y= Y i ( d / a z i ) Y i ( a / a u i ) . From (4.42), we see that R is a skew-symmetric semibasic form. If a linear connection, then (4.42) becomes
R ( X , Y ) = xiYjuk((arf,/az') -
-
is
(arik,)+ r;.&
rirrjl)(a/auk). l k
(4.43)
With respect to the adapted frame, we have
R(Di,~
((arf/azi)- (ar;/azj) + rf(ar:/aJ))
j =)
- rf(arf/aJ))vk,
(4.44)
R(Di,Vj) = R(K,Vj) = 0. Proposition 4.9.2 Let be a connection in T M . Then the horizontal distribution H is integrable if and only if R = 0.
Proof: Suppose that R = 0. Then, from (4.41), we obtain
[ h X ,h Y ] = h [ h X ,Y ]+ h [ X ,hY] - h [ X ,Y ] = h( [ h X ,Y ]
+ [X,h Y ] - [ X ,Y ] ) .
Hence H is integrable. Conversely, suppose that H is integrable. We have
R ( h X , hY) = - [ h X , h Y ] + h [ h X ,h Y ] (since h2 = h) = -u(hX,hY]
218
Chapter 4. Connections in tangent bundles = 0 (since H is integrable)
But, since R is semimbasic, then R = 0. Now, let l? be a linear connection in T M . We define the curvature tensor R of r to be the tensor field of type (1,3) on M given by
R z ( U , V , W )= 4 u ( R u ( V H , W H ) ) ,
(4.45)
where U,V ,W E T,M, x E M . If we develop (4.45) in local coordinates we easily obtain
R ( U , V , W )= VuVvW
-
VvVuW - V[u,vlW,
for any vector fields U,V,W E ( ( M ) . Then R is the curvature tensor of the linear connection V on M defined by I'.
4.10 Almost complex structures on tangent bundles (11) In this section, we generalize for connections in T M the results obtained in Section 3.5,for linear connections. Let I' be a connection in T M with horizontal projector h. Then we define a tensor field F of type (1,l) on T M by
F ( h X ) = -JX,F ( J X ) = h X )
(4.46)
for any vector field X on T M . It follows from (4.46) that F is an almost complex structure on T M . With respect to the adapted frame, we have
FDj = -K, FV, = Dj.
(4.47)
From a straightforward computation from (4.47), we easily see that
4.10. Almost complex structures on tangent bundles (11)
219
- ((ari/ad)- (arf/ad))Dk.
(4.48)
From (4.48)) we obtain the following. Theorem 4.10.1 F is integrable i f and only i f R = 0 and t = 0. Corollary 4.10.2 Let r be a linear connection in T M with couariant deriuatiue V . T h e n the almost complez structure F associated t o I' coincide with the corresponding o n e defined by V. Therefore, F i s integrable i f and only i f r i s flat, that is, R = 0 and T = 0 .
Now, let g be a metric in the vertical bundle V and TM.
a connection in
Definition 4.10.3 T h e Riemannian prolongation of g along R i e m a n n i a n metric g r o n T M defined by
i s the
g r ( X , Y ) = g ( J X , J Y )+ g ( u X , v Y ) , where
u
i s the vertical projector of
r.
It is easy t o see that, in fact, gr defines a Riemannian metric on T M . The next proposition (which proof is omitted) characterizes the Riemannian prolongations of metrics in V along connections in T M . Proposition 4.10.4 A R i e m a n n i a n metric g o n T M is the R i e m a n n i a n prolongation of a metric in V along r i f and only i f (1) 9 ( M = 0, (2) g ( h X , h Y ) = g ( J X , J Y ) = g ( J X ,J Y ) , f o r a n y vector fields X and Y o n T M .
w
Next, let F be the almost complex structure determined by
I?. We have
Proposition 4.10.5 ( I ) gr i s a Hermitian metric. (2) T h e Kiihler f o r m Kr of the almost Hermitian structure ( T M ,F , g r ) i s given by
Chapter 4. Connections in tangent bundles
220
Proof: (1) In fact, we easily see that g r ( F X , FYI = g r ( X , Y ) . (2) The Kahler form Kr of ( T M ,F , g r ) is defined by
since Fu = hF and Fh = -J . On the other hand
g r ( h X ,h F Y ) = g r ( J X , J F Y ) =gr(JX,uY)
(by Proposition 4.10.4)
(since J F = u).
Hence
gr (X, F Y ) = gr (JX, u y ) - gr ( v x ,J Y ) =gr(JX,Y)-gr(X,JY).u Now, let us suppose that ( M , g ) is a Riemannian manifold. We can define a metric # in V as follows:
SZ(X",Y")= 9 2 ( X ) Y ) , where X , Y E T,M, L E T,M. Let I' be the Riemannian connection defined by g and 5 the Sasaki metric on T M determined by g. It is easy to see that is the Riemannian prolongation of B along I?. Then we reobtain the results of section 3.5.3.
Remark 4.10.6 The results of 4.10 has been extended for connections in tangent bundles of higher order in de Le6n [29], de Le6n et al. [32].
4 . 1 1 . Connection in principal bundles
4.11
221
Connection in principal bundles
Let P ( M , G ) be a principal bundle over A4 with structure group G and projection r : P -+ M .
Definition 4.11.1 A connection I' in P i s a connection in the fibred manifold r : P -+ M such that the horizontal distribution H is invariant b y GI a.e.,
for a l l p E P, u E G . Then, given a connection I' in P, we have a decomposition
T,P = H p @ V,, for all p E P, where H p (resp. V,,) is the horizontal (resp. vertical) subspace at p . Let g be the Lie algebra of G . Then a connection I' in P determines a l-form w on P with values in g as follows. For each X E T,P, we define w ( X ) t o be the unique A E g such that
( X A ) , = uX, where u X is the vertical component of X . The form w is called the connection form of I'. Obviously, w ( X ) = 0 if and only if X is horizontal.
Proposition 4.11.2 The connection form w satisfies the following conditions: ( 1 ) w ( X A ) = A, for all A E g . (2) R:w = Ad(u-')w, i.e.,
w ( ( T R a ) X )= A ~ ( ~ - ' ) ( w ( x )f )o,r
x E T P and u E G.
Conversely, given a g-valued 1-form w on P satisfying (1) and (2), there is a unique connection I' in P whose connection form is w .
Proof If w is the connection form of a connection I?, then (1) and (2) follows directly from the definition of w and Proposition 1.20.6. Conversely, given a l-form w satisfying (1) and (2), we define
H p = { X E T , P / w ( X ) = O}.n
Chapter 4. Connections in tangent bundles
222
-
Now, let {$,a} be the transition functions corresponding t o an open covering {U,} of M (see Section 1.20). Let Q, : U , P be the section over U , defined by a,(z) = $;'(z,e), z E U,, where, ) t : r-'(U,) -+ U , x G is the local trivialization of P over U,. If 0 is the canonical g-valued l-form on G (see Exercise 1.22.21), then we define on U, n Up a g-valued l-form flaL3 by
For each a,we define a g-valued l-form w, on U, by
It is not hard to prove that wB = Ad($;;)w,
+ dab
on U, n Up.
(see Kobayashi and Nomizu [81] for a detailed proof). Next we shall define the curvature form of a connection
(4.49)
r in P.
Definition 4.11.3 Let w be the connection form of r. Then the curvature form o f r i s defined to be the g-valued 2-form R on P given by
R(X,Y) = dw(hX, h Y ) , for all X,Y E TP. It is easy to prove that: 0
fl is horizontal, i.e., a(X, Y) = 0 whenever at least one of the tangent vectors X,Y are vertical.
0
= Ad(a-')R, for all a E G.
Proposition 4.11.4 (Structure equation) We have 1 2
d U ( X , Y )= - - [ w ( X ) , w ( Y ) ] for all
+ R(X,Y),
X,Y E TP.
The proof is left to the reader as an exercise. The structure equation is sometimes written, for the sake of simplicity, as
4.11. Connection in principal bundles
-
1
&J= --[ w , w ]
223
+ n.
Now, let 7r : E M be a vector bundle of rank n associated with P (see Exercise 1.22.27). Then a connection I? in P induces a connection f' in E as follows. For each e E E , choose (p, I ) E P x R" such that e = [ p ,(1. Consider the map F
P-E defined by F ( q ) = [q,El. Then the horizontal subspace f i e at e is defined to be the image of Hp by F , i.e.,
gr,= dF(p)H,. One can easily prove that ?I defines a connection in E . Next, we consider the frame bundle FM of an m-dimensional manifold
M. Let I' be a connection in FM with connection form w . With respect to the canonical basis { of gl(m, R), m = dim M , we set
q.}
w = w'E3 j i , where w;. are l-forms on F M . Let (U,z') be a coordinate neighborhood in M and u : U section of FM given by
-
F M the
.(x) = { ( a / a d ) , , . . . ,( a / a q , } . We set U*Wj
= r:hdZJ,
where the m3 functions I?$k are called the Christoffel components of I?. Let (U,z'), ( U ,5') be two coordinate neighborhoods with U n # 8. If r ! k ) F $ k are the corresponding Christoffel components, a direct computation from (4.49) shows that (4.50)
Chapter 4. Connections in tangent bundles
224
The components I'ik permit us to reconstruct the connection I'. In fact, with respect to the induced coordinate system ( z i , X i ) on F M we have
wj = Y , ! ( ~+ x r;$ x ! d z a ) , where
(4.51)
(7') is the inverse matrix of ( X f ) .Hence, if for each coordinate system
(U,z') on M , we have a set of functions ,;?I satisfying (4.50)) then we can define wi.by (4.51) and w by w = wjE!. Now, let I' be a connection in F M . Since T M is a vector bundle associated with F M , then 'I induces a connection f in T M . It is a straigthforward computation to prove that f' is in fact a linear connection on M . Conversely, let f' be a linear connection on M . Then, proceeding as above, we can construct a connection in FM from the Christoffel components I'ik. Thus, a linear connection on M is a connection in F M .
4.12
Exercises
4.12.1 Prove that the vertical lift X u to T M of a vector field X on M is homogeneous of degree - 1. 4.12.2 Prove that for any semispray ( on T M and any vector field X on M [ X " ,€1 is a vertical vector field and [ X u ,(1 projects onto X . 4.12.3 Prove that if 2 is a vector field on T M and ( a semispray such that [?,(1 = 0 and J? = Xu,then 2 is the complete lift of X, i.e., 2 = X c . 4.12.4 Let ( be a semispray on T M . Prove that
4.12.5 Let I' = -LtJ be the connection in T M defined by an arbitrary semispray ( on T M . We denote by t , H and T the weak torsion, tension and strong torsion of I', respectively. Prove that: (i) t vanishes; (ii) H = - ( 1 / 2 ) [ r , J ] ; (iii) T = (1/2)[(*,J], where (* is the deviation of (. 4.12.6 Let 'I be a connection in a principal bundle P ( M , G ) .
(i) Prove that the horizontal lift X H to P of a vector field X on M is invariant by G , i.e., ( T R , ) X H = X H for all a E G. ] [X,YIH. (ii) Prove that the horizontal component of [ X H , Y H is (iii) Let A be an element of the Lie algebra of G and 2 a horizontal vector field on P. Prove that [ 2 , A A ]is horizontal. If, in particular, 2 is the horizontal lift X H of a vector field X on M , prove that [ X H , X A ]= 0.
4.12. Exercises
225
4.12.7 Let & ( M ) be a G-structure on M. Consider a connection r in B G ( M ) .Then we can define a connection f' in FM (i.e., a linear connection on M) as follows. For each p E FM, let be q E & ( M ) and a E G such that p = qa. Thus we define the horizontal subspace kpby k,= T R a ( H , ) , where Hq is the horizontal subspace at q defined by .'I Prove that f is in fact a connection in FM (I' is called a G-connection). 4.12.8 (i) Let P ( M , G ) be a principal bundle. Prove that there always exists a connection in P ( H i n t : use partitions of unity). (ii) Prove that, if & ( M ) is a G-structure on M , then there always exists a G-connection on M. (iii) Let J be an almost complex structure on a 2n-dimensional manifold M. Prove that a linear connection r is a Gl(n,C)-connection if and only if V J = 0, where V is the covariant derivative associated with I?. (iv) Extend (iii) for other polynomial structures (almost tangent structures, almost product structures, etc ...).ae
This Page Intentionally Left Blank
227
Chapter 5
Symplectic manifolds and cotangent bundles 5.1
Symplectic vector spaces
-
Let V be a real vector space of dimension m and w a skew-symmetric bilinear form (2-form) on V , i.e., w E A2V. Consider the linear mapping S, : V V*, where V * is the dual space of V , defined by u
-
S,(U) = iuw.
(5.1)
Here iuw is the interior product of the vector u by the form w , i.e., (iuw)(u) = w ( u , u), u E
v.
Let ImS, be the image of S, and ker S, = {u E V/i,w = 0) the kernel The rank of w , denoted by rank w , is the dimension of ImS,. The dimension of ker S, is called corank of w , denoted by corank w . If in particular corank w = 0 then dim V = rank w . In such a case we say that w is non-degenerate, of maximal rank or even regular. It is possible to show, without major difficulties the following assertion: “A necessary and sufficient condition for w E A2V be non-degenerate is that S, : V + V * defined by (5.1) be an isomorphism”. of S,.
Proposition 5.1.1 Let w E A2V. Then there i s a basis {u; I 1 5 i 5 m = dim V } for V and an integer 2s 5 m such that ( I ) w ( u ; , u # + ; )= -w(tl#+i,tL;) = I, for d l i 5 S , (2) for all other values, w(u;,uj) = 0.
Chapter 5. Syrnplectic manifolds and cotangent bundles
228
Proof: The proof is obtained by induction on dimension of V. The result is evident if dimV = 2. Let u1 E V be an arbitrary vector. Then there is a vector 112 such that w(u1, u2) # 0, since w is non zero. We may choose u2 in such a way that w ( u 1 , q ) = -w(uz,u1) = 1 (because, if necessary, we may change up by a numerical factor). Suppose that m 2 2. We may choose two vecton u l and u8+1 such that w(u1,u8+1) = - w ( u , + ~ , u l ) = 1. Let F be the space spanned by these vectors. Let W defined by
w = { v E V / w ( v , u1) = w ( v , U8+1) = o}. Then V = F @ W . Hence, by induction, there is an integer 2 s 5 m and a basis {ui,ud+;,uj/2 5 i 5 s, 2 s + 1 5 j 5 m } for W such that (1) and (2) hold. Then taking {u;, u,+;, u j / l 5 i 5 s, 2 s 1 5 j 5 m } as a basis for V one has the desired result.^ Let {up/l 5 i 5 m} be the corresponding dual basis for V * of {u;/l 5 i I m}. Then a direct computation obtained from the action of the forms ut A u:+~,1 2 i 5 s, on the pair of vectors ( u j , ul), shows that
+
w
= u;
A u:+~
* + . . . + u,* A u2,.
(54
Thus, rank w = 2s. Also, if we take the s-exterior product of w
=w
A
.. . A w = -(s)!
U;
A
U;
A
. .. A tl;,.
we see that w 8 # 0 and we+' = 0. In fact, if these two last properties hold for w E A 2 V then w has rank 2s. On the other hand, we have
From these comments we have the following.
Theorem 5.1.2 The rank of every &-form w on V i s an even number. The following assertions are equivalent: (1) rank w = 2 s 5 m = dim V , (2) the power w' # 0 and w 8 + l = 0, (3) there are 2 s linear independent forms (I-forms) on V, denoted by ui, . . ,u;,, such that
.
Aho, under these conditions, {ui, . . . ,uia} i s a basis for Im Sw.
5.1. Symplectic vector spaces
229
Definition 5.1.3 A symplectic structure on a finite (real) vector space V is given by a 2-form w on V such that w i s non-degenerate. The form w is called symplectic and the pair ( V , w ) a symplectic vector space. From the preceding considerations, we easily see that if ( V , w ) is a symplectic vector space then dim V is an even number, say 2n. The following proposition is a consequence of the results given above: Proposition 5.1.4 Let V be a vector space of even dimension 2n, and w a 2-form on V. Then the following assertions are equivalent: ( 1 ) (V, w ) i s a symplectic vector spaceJ (2) S, : V V * is an isomorphism, (9) w" i s a volume form on V.
-
From Theorem 5.1.2 and equality (2) of Proposition 5.1.4 we see that if w is a symplectic form on V there is a basis {ui 1 5 i 5 2n) for V such
1
that .-
n
w =~ u ~ A u ~ + , . i=l
We call such basis symplectic. The matrix of w with respect to a symplectic basis is
-
Definition 5.1.5 Let U and V be vector spaces and h : U V a linear mapping. Suppose that a and w are symplectic forms on U a n d V, respectively. We say that h is symplectic if the linear adjoint mapping h* : V * U* i s such that h*w = a, i.e.,
-
w ( h u , h v ) = a ( u , v ) , for all u , u E U. A symplectic linear mapping is injective. If dim U = dim V then h is called symplectic isomorphism. In particular, if U = V then h is said symplectic automorphism, and we say that h preserves the symplectic form: h*w = w .
Definition 5.1.6 The group of the symplectic automorphisms of a symplectic vector space ( V , w ) , endowed with the rule of composition of maps, is called symplectic group and it i s denoted by S p ( V ) . .
Chapter 5. Symplectic manifolds and cotangent bundles
230
Proposition 5.1.7 If (V,w) i s a symplectic vector space and h E Sp(V) then det h = 1. Proof: We remark that if h E Sp(V) then h preserves the volume form Since wn = h*w" = (deth)w", then det h = 1 (with respect to the volume form w"). 0 The reader is invited t o show the following:
w", i.e., h*(w") = w".
Proposition 5.1.8 If h i s an automorphism of V and H i s its matrix with respect to a symplectic basis then h is a syrnplectic automorphism i f and only if
From Proposition 5.1.7 one has that Sp(V) is isomorphic to the group of matrices Sp(n) = { H E G1(2n, R) I H t M H = M} which is a closed subgroup of G1(2n,R). Thus Sp(V) admits a Lie group structure.
Remark 5.1.9 Let V = R2" and
where (z', . . . ,z2") are the canonical coordinates on R2". Then (R2",w0)is a symplectic vector space. We can easily check that
where
H : R2"
-
R'". Then the Lie algebra of Sp(n) is
Definition 5.1.10 Let V be a vector space of finite dimension, w E A2V and K a subspace of V. Then the subspace
K'
= {u E V/w(u, v ) = 0, f o r all v E K }
i s called the orthocomplement of K in V with respect to w . ( I n particular, if u E V, then we may define v' = {u E V/w(u, v ) = 0)). One has:
5.1. Syrnplectic vector spaces
231
Ker S, = ' V implies corank w = dim V'; dim V + dim (V'n K ) = dim K+ dim K'; in particular, if w is symplectic then dim V = dim K+ dim K'.
Definition 5.1.11 Let ( V , w ) be a syrnplectic vector space and K a subspace of V. W e say that ( I ) K is isotropic if K c K'; (2) K is coisotropic if K' c K ; (3) K is Lagrangian if K i s a rnazirnal isotropic subspuce of ( V , w ) ; (4) K is symplectic if K n K' = 0. Proposition 5.1.12 Let (V, w) be a syrnplectic vector space. A necessary and suficient condition for a vector subspace K c V to be Lagrangian is K = K'. Proof If K is a proper subspace of K', K # K', then K is not maximal. In fact, we may choose u E K' such that u ,&K and so
+
for all 01, u2 E K and a l , a2 E R. Then K < u > 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/aql,.. . ,d/aqr >,
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> =<
=<
a/aqr+l,.. . ,a/aqrn,a/apr+l,.. . ,a/aprn> .
5.6. Darboux Theorem
249
Complete lifts of almost complex structures Let F be an almost complex structure on M . From Propositions 5.5.2 and 5.5.3 we have
Proposition 5.5.6 ( I ) F" is an almost complex structure on T*M if and only if F is integrable, i.e., F is a complez structure on M . (2) If F i s a complex structure on M , then F" is a complex structure on T * M .
5.6
Darboux Theorem
Regarding (5.4) we see that O M has a local expression similar t o the expression for symplectic forms on vector spaces. We show in this section a fundamental result due to Darboux which stablish the non-linear analogue of symplectic vector spaces. In order to prove the Darboux Theorem, we first introduce the notion of time-dependent vector field.
-
Definition 5.6.1 A time-dependent vector field on a manifold A4 is a C" map X : R x M T M such that X ( t , x ) E T,M. We remark that all the results obtained in Chapter 1 still hold for timedependent vector fields. Thus, we define (Pt,s(x) t o be the integral curve of Xt through time t = s, i.e.,
and
where Xt is the vector field on M given by Xt(x) = X ( t , z ) . In fact, 4 t , a is the (time-dependent) local l-parameter group generated by Xt. We have
The proof of the following result is left to the reader as an exercise.
Chapter 5. Symplectic manifolds and cotangent bundles
250
Proposition 5.6.2 Let X be
a time-dependent vector field on
M. Then we
have
for every p-form a on M.
Theorem 5.6.3 (Darbow Theorem). Let w be an almost symplectic form on a 2n-dimensional manifold S. Then dw = 0 if and only if for each x E S there exists a coordinate neighborhood U with local coordinates ( d ,. . . ,z2n) such that n
w = C d x ' A dznti i= 1
on U.
Proof: We will use an idea of Moser [98] adapted by Weinstein known as the "path method". (An alternative proof using the notion of class of a form may be founded in Godbillon [63]; we also remit to Arnold [4] for a beautiful and geometrical proof). First, we note that if w = dx' A dx"+i, then w is closed. In order to prove the converse, we can suppose that S = R2" and x = 0 E R2n.Let w1 be the constant 2-form on R2" defined by
xi
w l ( y ) = wo
= C d z ' Adznti, y
E
R2n,
i
where (XI,
. . . ,z2n)are the canonical coordinates in R2". We put Wt
=w
+ t(w1 - w ) , t E [0,1].
For each t E [0,1], w t ( 0 ) = W O . Thus w t ( 0 ) is non-degenerate for each t . Hence there exists a neighborhood U.of 0 such that wt is non-degenerate for all t E [0,1],since G1(2n, R) is an open set of g1(2n, R). We can suppose that U is an open ball centered at 0 E R2". Then, from the Poincard lemma, w1 - w = d a , for some 1-form a on U , with a(0) = 0 (since d(w1 - w ) = 0). Now, let Xt be a vector field defined by 1X,Wt
= -a.
So, Xt is a time-dependent vector field such that Xt(0)= 0, since
25 1
5.6. Darboux Theorem
iXt(O)W0 = -a(O)
= 0.
Then there exists an open ball V c U centered at 0 E R2" such that the (time-dependent) l-parameter group +t,o is defined for all t E [-1,1]. By Proposition 5.6.2, we have
= - $ q o ( d a - w1+ w ) = 0 . Thus 4;,0
w 1 = 4*00 w
= wo = w .
Therefore q51,o gives the required change of coordinates which transforms w to w 1 . 0
The coordinate neighborhoods given by Darboux Theorem are called symplectic and its coordinate functions (zl, . . . ,z2") are called symplectic (or canonical) coordinates. (If we set zi = q', z " + ~= pi, 1 5 i 5 n , then w is locally given by (5.4)). In terms of the theory of G-structures, the Darboux theorem can be rewritten as follows.
Corollary 5.6.4 Let Bsp(,,) be a Sp(n)- structure on S with almost s y m plectic f o r m w . Then Bsp(,,)is integrable if and only if w i s symplectic.
+
Definition 5.6.5 Let S be a manifold of dimension 2n r and w a closed ,?-form on S of constant rank 2n. Then w is said to be a presymplectic form ( o r structure) on S and the pair ( S , w ) i s called a presymplectic manifold. Let (S,w ) be a presymplectic manifold. Suppose that dim S = 2n+r and rank w = 2n. Let z be a point of S and (V,cp) a coordinate neighborhood at z such that p(z) = 0 E R2n+r. If we shrink U ,if necessary, we may suppose that there are neighborhoods V c R2", W c R' at the origin
Chapter 5. Syrnplectic manifolds and cotangent bundles
252
0 E R2", 0 E R', respectively, such that p(U) = V x W. Consider the pull-back (p-')*w on p(U) and denote by wy , resp. ww , the 2-forms on V, resp. W , defined by
wv = ((v-l)*w)/v, ww = ((p-')*w)/w. Since dwv = 0 and wv is of maximal rank 2n, from Darboux theorem, there is a coordinate system ( x l , . . . x2n) such that n
i= 1
Therefore we have the following.
Theorem 5.6.6 (Generalized Darboux Theorem).- Suppose that S is a (2n r)-dimensional manifold and w a &-form on S of constant rank 2n. Then w is closed (i.e., w i s a presymplectic form on S) if and only if for each point x E S there exists a coordinate neighborhood U with local coordinates (xl,. . . ,x2", y l , . . . ,y') such that
+
n
w =x d x i A
d@
i= 1
on U. Remark 5.6.7 We may prove directly the above generalized theorem using the path method and then obtain Darboux Theorem as a Corollary. Pulling back the form w to V x W and applying Frobenius theorem, we may prove that the form on V x W has a local expression of type n
i=l
Now, along V , defined by the local equations y2n+1 = ... - y2n+r = 0, we may also suppose that at the origin the matrix (a;,n+;) is of type
(L
a ) :
As R2" is a symplectic vector space, we consider a symplectic form on V with canonical expression and then we may apply the same procedure as in Darboux Theorem.
5.7. Almost cotangent structures
253
Remark 5.6.8 Another point of view for proving the generalized theorem consists in adopting induction on r. If r = 0, the result is just Darboux Theorem. Suppose that the assertion is true for r - 1. Then there is a (2n + r - l)-dimensional subspace of T,S, z E S , on which ~ ( z has ) rank 2n. If we apply Frobenius theorem for vector fields one obtain a submanifold N of S of dimension 2n r - 1. By induction we have a coordinate system on N such that w has the expression dyj A dy"+j on this coordinate system. We finish the proof by considering the distribution K e r S and choosing a vector field Y E K e r S, such that Y ( z )E T,N, z E N . Using the flow of such Y one obtains a coordinate system ( d , . . ,z2n,zl,. . . , z') such that w assumes the above expression (for further details see Robinson [ 1061)for example).
+
5.7
cy=l
Almost cotangent structures
The concept of an almost cotangent structure was introduced by Bruckheimer [lo] and interpreted by Clark and Goel [18] as a certain type of G-structure. Let T * M be the cotangent bundle of an m-dimensional manifold M and K M : T * M --+ M the canonical projection. Let (U, q i ) , ( 0 ,d ) be coordinate neigborhoods with U n 0 # 0 and (T*U,q',pi), ( T * U , $ , p i ) the induced coordinate neighborhoods on T * M . Then ( q i , p i ) , ($,pi) are related by a change of coordinates whose Jacobian matrix has the form
[
(A'l)'
]'
(5.16)
where A = ( a q j / d $ ) , B = [ ( a 2 ~ k / a q J a q ' ) ( a q ' / a d ) p k ] . This suggests the following definition.
Definition 5.7.1 Let N be a 2m-dimensional manifold carrying a G-structure whose group G consists of all 2m x 2m matrices of the form (5.16), where A E Gl(m, R ) and A t B = B'A. Such a Structure is called an almost cotangent structure, and such a manifold N is called an almost cotangent manifold. Now, let B be an almost cotangent structure on a 2m-dimensional manifold N . We define a 2-form w on N by specifying its components to be
254
Chapter 5. Symplectic manifolds and cotangent bundles
. . ,Xrn,Xrn+l,. . . ,Xzm} at x. Then w is relative to any adapted frame {XI,. well-defined and determines an almost symplectic structure on N. In fact, if { e l , . . . ,Om, O m + l , . . . 02m} is the dual coframe, then we have )
i=l
Furthermore let V be an m-dimensional distribution on
N defined by
Thus V is a Lagrangian distribution with respect to w , i.e.,
w ( X ,Y )= 0 for all X,Y E V. Conversely, suppose that N is a 2m- dimensional manifold endowed with an almost symplectic form w , together with a Lagrangian distribution V. Let x be a point of N and { Y l , .. . ,Yrn,Yrn+l,. . . ,Yzm} a frame at z which is adapted for V, i.e.,
Then the matrix of w relative to {Y,,1 5 a 5 2m} is
where Pt = - P and det Q as follows:
# 0.
We now construct a new frame
{X,}at x
and B = ;(Q-')'PQ-'. Then where A = Q-', C = Im
and the matrix of w relative to {X,} is wo. Now, let {X,},{Xa}be two frames at x as above. Since Vz =< X m + l , . . . , X 2 m >=< X m + l , .. . , X 2 m >, we deduce that they are related by a matrix of the form
255
5.7. Almost cotangent structures
[; :]
(5.17)
Moreover, since they are adapted to w , we deduce that C = (A-')' and A'B = B'A. Hence the matrix (5.17)belongs to G . So the set of such a frames {Xa} at all points of N defines a G-structure on M. Summing up, we have Proposition 5.7.2 Giving an almost cotangent structure on as giving an almost symplectic form tion V .
N i s the same
w , together with a Lagrangian distribu-
Example We have seen that the cotangent bundle T*M of an m-dimensional manifold M carries a canonical almost cotangent structure. Actually, we easily see that the corresponding almost symplectic form is precisely W M and the corresponding Lagrangian distribution is V ( T * M ) . Next, we stablish the integrability conditions of an almost cotangent structure.
Theorem 5.7.3 manifold
N is
An almost cotangent structure ( w , V ) on a 2m-dimensional integrable if and only if w i s symplectic and V i s involutive.
Proof Obviously, if ( w , V ) is integrable then V is involutive and dw = 0. We prove that this is also sufficient. In fact, let z be a point of N. Since V is involutive, then, from the Frobenius theorem, local coordinates ( q i ) y i ) may be introduced such that {a/ayi} span V . Choose a coframe field {t,!?). . . ,$ J ~at ~z adapted } to the almost cotangent structure ( w , V ) . Then we have $Ji
=
Aids, d e t A # 0.
We now define a new coframe field
Then
{el,.
{el,
. , . ,t92m}
at z by
. . ,02m} is adapted to ( w , V ) . Suppose that
Chapter 5. Syrnplectic manifolds and cotangent bundles
256
om+'
= p!dqj 1
+Qfdy,.
Since dw = 0, then we have
dw = d(6'
h 6"'")
= 0.
Thus,
and therefore we obtain
aQi/ayk = aQi/ay, It follows that the equations
a F ' / d y j = Q: admit differentiable solutions F' = F'(q, y ) on a neighborhood of x. Then we may construct a new local coordinates (q', zi) at x by setting
= Pjdq'
+dzj,
where = Pi - d F ' / a q j . The condition dw = 0 implies that (5.18) (5.19)
5.7. Almost cotangent structures
257
Now, consider the equations
aHipqj
-
a H j p q ' = p;
-
(5.20)
(t),
From (5.19) we deduce that the right-side of (5.20) depends only on and from (5.18) we deduce that there exist differentiable solutions H'(q) of (5.20) on a neighborhood of z. Next, we define functions p; by
Therefore we have 0' = dq',
dpi = em+;
+ (aH'/aqj
-
pi)ej.
Then (4,~;)is a local coordinate system at z. Furthermore, a simple computation shows that {dq',dpi} is a coframe field adapted to ( w , V ) since
aH'laqj is symmetric in i , j . Hence
v =< a i d p l , . . . , a p p , and
i= 1
This ends the proof.0
Remark 5.7.4 Notice that the example ( w M , V ( T * M )is) an integrable almost cot angent structure . Let ( w , V ) be an integrable almost cotangent structure on a 2mdimensional manifold N. Then w is a symplectic form and V an involutive distribution. Since Exercises 3.8.2 and 5.9.1, it is easy t o prove the following.
Proposition 5.7.5 If ( w , V ) i s an integrable almost cotangent structure on
M , then there ezists a symmetric linear connection V on M such that Vw = 0 and V x Y E V for all Y E V , i.e., V i s an almost cotangent connection.
Chapter 5. Symplectic manifolds and cotangent bundles
258
5.8
Integrable almost cotangent structures which define fibrations
As we have seen in Section 5.7, the integrability of an almost cotangent structure implies that it is locally equivalent to the cotangent bundle T*M of a differentiable manifold M. In this section we shall prove that an integrable almost cotangent structure verifying some global hypotheses is (globally) diffeomorphic to some cotangent bundle. This result is due to Thompson (1151.
Definition 5.8.1 Let
(0,V )
be a n integrable almost cotangent structure on a 2m-dimensional manifold N. Since V is involutive then V determines a foliation (also denoted by V). Let M be the space of leaves a n d A : N M the canonical projection. We say that ( w , V ) defines a fibration if M has a diflerentiable structure of manifold such that A is a surjective submersion (then M is a quotient manifold of N).
-
Suppose that ( w , V ) is a integrable almost cotangent structure on N which defines a fibration t : N + M. We now show that there is a construction which generalizes the vertical lift construction on a cotangent bundle. Let a be a l-form on M. We define the vertical lift ' a of a to N by ' a = -(SJl(t*(Y).
In adapted coordinates ( q i , p i ) , we have (5.21)
where a = ai (q)dgi Proposition 5.8.2 For all l-forms
a,p
on M, we have
[au,pu]= 0. Proof Directly from (5.21). Since ( w , V ) is integrable, there exists a symmetric almost cotangent connection V. We have the following. Proposition 5.8.3 V induces, by restriction, a flat connection on each leaf of v.
5.8. Integrable almost cotangent structures which define fibrations
259
We omit the proof which is similar to these of Proposition 2.10.3.
Theorem 5.8.4 Suppose that the flat connection induced from V on each leaf of V is geodesically complete and that each leaf is connected and simply connected. Then N is diffeomorphic to T * M . Moreover the diffeomorphism, F say, can be chosen such that w = F*(wM r&$), where I$ is a closed 2-form on M.
+
-
Proof: By similar arguments to those used in the proof of Theorem 2.10.4 we prove that r : N M is an affine bundle modeled on T * M . Now we choose a global section s of N over M. Then N may be identified with T*M, with s playing the role of zero section. We call the resulting diffeomorphism F : N T*M and consider the 2-form
-
R =w
-
F*WM
Then R is closed and verify ixfl = 0 for any vertical vector field X E V . Then there exists a closed 2-form 4 on N such that
Since X M o F = r , we have
w = F*(WM
+ rbI$).o
Corollary 5.8.5 Suppose that ( w , V ) verifies all the hypotheses of the Theorem 5.8.4 except that the leaves of V are simply connected. Then, if the leaves of V are assumed to be mutually diffeomorphic, T*M is a covering space of N and the leaves of V are of the form T kx Rm-k,0 5 k 5 m. Moreover, if the leaves of V are compact then T*M is a covering space of N and this leaves are diffeomorphic to T". The following definition was also introduced by Thompson [115].
Definition 5.8.6 We say that an almost cotangent structure ( N , w , V ) is regular if it verifies all the hypotheses of the Theorem 5.8.4. In such a case, (N, T , M, w , V ) is called a regular almost cotangent structure. If (N, T , M , w , V ) , ( N ,R , M , f i , V ) are two regular almost cotangent structures, they will be said to be equivalent if there ezists a bundle morphism F : N N over the identity of M such that
-
F*o - w = r * ( d a ) , for some l-form cr on M, i.e., F*fi
-w
is cohomologous to zero.
Chapter 5. Symplectic manifolds and cotangent bundles
260
Proposition 5.8.7 There ie a one-to-one correspondence between the set of equivalence classes of regular almost cotangent structures and elements of
P ( M ,R ) . Proof: Let (N, A , M , w , V ) be a regular almost cotangent structure. From Theorem 5.8.4, there exists a diffeomorphism Fa : N + T*M such that
is a closed 2-form on M and s is the section of N which is used t o where define Fa. Then Fa o s = SO (zero-section of T * M ) and we have
Let I be another section of N and Fa the corresponding diffeomorphism such that
Then there exists a section u of T*M (i.e., a l-form on M) such that ii = s u , i.e., if(%) = s(x) u ( x ) , for each point x E M. A simple computation shows that
+
+
Fa o ii = u. Then
= +a
+ du.
Thus #a and #., define the same cohomology class of H 2 ( M ,R). Now, we prove that the mapping defined above is surjective. Let [4] E H 2 ( M ,R). Then the corresponding regular almost cotangent structure is (T*M, AM, M , W M Ah4,v(T*M)). Finally, we prove that the mapping is injective. Suppose that (N,A, M , w , V ) , ( N , ii,M,o,P) are regular almost cotangent structures and that
+
5.9. Exercises
261
F and F are the respective diffeomorphism with T*M corresponding to the sections s and 3. Then we have
for some closed l-form
Q
on M . Now, a direct computation shows that
((F)-'
o
F)*Q = w
- x*da
Hence (p)-' o F is an equivalence of regular almost cotangent structures. This ends the proof.0 The last result of this section shows that the vanishing of the element of H 2 ( M ,R) characterizes T*M as a regular almost cotangent structure up to equivalence. Proposition 5.8.8 Suppose that (N,T, M , w , V ) is a regular almost cotangent structure. Then ( N , T , M , w , V ) is equivalent to (T*M,T M ,M , W M , V ( T * M ) )if and only if the element of H 2 ( M , R ) it determines is zero. In such a case w is exact, say w = -dA, and the equivalence F verifies F*AM = A .
The proof is left to the reader as an exercise.
Remark 5.8.9 In de Lebn et al. [35],[36]we introduce the concept of p almost cotangent structures which generalizes almost cotangent structures and prove some results similar to those proved in this section.
5.9
Exercises
5.9.1 (i) Let ( S , w ) be an almost symplectic manifold. Define a linear con-
nection V on S by
2w(VxY,2) = Xw(Y,2)
+ Y w ( X ,2) - Z w ( X ,Y )
+ W ( [ Z ,XI,Y)+ w ( K [Z,YI)>
+ w ( [ X ,Y ] ,2)
for all X , Y , 2 E x(S).Prove that V is an almost symplectic connection, i.e., Vw = 0. (ii) Prove that, if w is symplectic, then V is a locally Euclidean connection, i.e., T = 0 and R = 0, where T and R are the torsion and curvature
262
Chapter 5. Symplectic manifolds and cotangent bundles
tensors of V, respectively (Hint: compute the Christoffel components of V in canonical coordinates). 5.9.2 Show that if (V,w ) is a symplectic vector space and K is a subspace of V then dim V = dim K dim K'. 5.9.3 Show that (K')' = K . If K1 and K2 are subspaces of V then Kf n K k = ( K 1 + K 2 ) l and (K1 n K2)* = K ; Ki. 5.9.4 Let K be a coisotropic subspace of V . Then the symplectic form w induces a symplectic form on K / K * . 5.9.5 Is it true that every symplectic vector space admits a Lagrangian subspace? 5.9.6 Let (V1,wl) and (V2,wz) be symplectic vector spaces and 7r1 : Vl x V2 +V I , 7r2 : V1 x V2 +V2 the canonical projections. Show that T ; W ~9riw2 is a symplectic form on V1 x V2. 5.9.7 Show that h E Sp(V) if and only if the graph of h
+
+
is a Lagrangian subspace of V x V . 5.9.8 Show that every closed 2-form w on a manifold S of dimension 2s is symplectic if and only if w' is a volume form.=
263
Chapter 6
Hamiltonian systems 6.1
Harniltonian vector fields
-
Suppose that T * M is the cotangent bundle of a manifold M and H : T*M R a function on T * M . If W M is the canonical symplectic structure on T * M , then there exists a unique vector field X H on T*M such that i x H w M = d H ; X H is called in the literature Hamiltonian vector field of energy H . In this section we extend this definition to arbitrary symplectic manifolds.
-
-
Definition 6.1.1 Let ( S , w ) be a symplectic manifold and H : S R a function on S. Since the map S, : x(S) A'S is an isomorphism, there exists a unique vector field X H on S such that
W e call X H a Hamiltonian vector field with energy (or Hamiltonian energy) H . The triple ( S , W , X H )(or ( S ,w , H ) ) is called a Hamiltonian system. From Proposition 5.2.10, we deduce that a Hamiltonian vector field on ( S , w ) is symplectic. Conversely, a vector field X on ( S , w ) is said t o be locally Hamiltonian if for every point z E S there is a neighborhood U of x and a function H on U such that X = X H on U. From Proposition 5.2.10, we easily deduce that a vector field X on S is locally Hamiltonian if and only if X is a symplectic vector field.
Chapter 6. Harniltonian systems
264
Example A classical example of a local Hamiltonian vector field which cannot be Hamiltonian is the following. Consider the 2-torus T 2with local coordinates ( z , y ) . Then w = dz A dy
is a well-defined symplectic form on
T2.Let
x = alas + a / a y Then
which is closed in T 2(but not exact!). Then X is locally Hamiltonian. But X cannot be Hamiltonian. In fact, if X = X H for some function H defined on T2,then, since T 2is compact, H has a critical point and at this point dH vanishes. Hence X would correspondingly have a zero.
-
Let (S,w) be a symplectic manifold of dimension 2n and X H a Hamiltonian vector field with energy H . Let (q',p;) be canonical coordinates in S. Suppose that u : I = (--c,c) S is an integral curve of X H , i.e.,
X H ( u ( t ) )= b ( t ) , t E I In local coordinates we have
dq' a dp; a b ( t ) = -- + --. dt aq' dt api Consider the isomorphism S, : X E x ( S )
-
sw(x)= ixw E AIS
A simple computation shows that
s,(a/aqi)= dpi, 8,(a/api)= -dqi. Hence we obtain
265
6.1. Harniltonian vector fields
s,-l(dq') = -a/apj, s,-l(dPj)
i
= a/aq .
(6.3)
From (6.2) and (6.3),we deduce that if X is a vector field on S with local expression
x = r a / a q i + xia/api, then S,(Z)
= -X'dq'
+ X'dpj.
Also, if a is a 1-form on S locally given by
then
Then, since
we obtain
xH = s ; l ( d H )
= (aH/ap,)
a/aqi - (aH/aqi)a/apj.
(6.4)
From (6.1) and (6.4), we have
(.6.5) which are called the Hamilton (or canonical) equations. The equation
i x H w= d H is called the symplectic or intrinsical form of Hamilton equations.
Example In Classical Mechanics the phase space of momenta is the cotangent bundle of the configuration manifold M . Consider a function H on T*M given by
Chapter 6. Hamiltonian systems
266
where g is a metric in T*M and U : M in local coordinates ( q ' , p i ) we have
-
R is a C" function on M . Then
where g'J = g ( dp; ,dp,) . Now, suppose that M = R3 and consider the metric g on T*@ defined by .
N
R3 x @
3
where m > 0. Thus the Hamiltonian H is given by
and the Hamilton equations (6.5) are dq'ldt = p ; / m , d p i / d t = -aU/aq', 1
5 i 5 3,
Hence we have
which is Newton's second law for a particle of mass rn moving in a potential ~ ( q lq 2, , q S ) in P.
Proposition 6.1.2 Let (S,w) and (W,a)be symplectic manifolds of same dimension and let h be a symplectic transformation from (S,w) to (W,a). Then
for any function F on W .
6.2. Poisson brackets
267
Proof: Recall that for all k-form 0 on W we have h*(dO)= d(h*O) and
Hence, if 0 = w and Z = XF,where F : W have
-
R is a function on W , we
i(Th-l)XFw= h*(ixFw)= h*(dF) = d ( h * F ) = d ( F o h) since h*a = w . But, as w is non- degenerate, the map Z + i z w is an isomorphism. Since ixFohw= d ( F o h ) we deduce that ( T h - ' ) x ~= XFoh.0
6.2
Poisson brackets
Poisson brackets is the most important operation given by the symplectic structure.
Definition 6.2.1 Let (S,w ) be a symplectic manifold of dimension 2n. Let F and G be C" functions on S. Then the Poisson bracket of F and G is defined by
From (6.6) we deduce
Now, we observe that ( i x , w ) ( Y ) = ( d F ) Y implies w ( X F , Y ) = Y F . Thus, if we suppose Y = Xc for some function G on S the following equality holds:
Lx,F = X c ( F ) = w ( X F ,Xc)= { F , G } . The reader is invited t o prove the following:
Chapter 6. Harniltonian systems
268
Proposition 6.2.2 For all CM functions F, G and H on a symplectic manifold ( S ,w ) one has: (-1) { F , G } = - { G , F } ; (2) { F , G H } = { F , G } H + G { F , H } ; (3) { F, { G , H } } {G{ H , F } } { H , { F, G } } = 0 (Jacob; identity) (4) { a F , G } = a { F , G } , for all a E R; (5) { F G , H } = { F , H } { G , H } .
+
+
+
+
We recall that a real vector space endowed with an internal operation g such that g(.,b) = -s(b,a) and g(a,g(b,c)) g(b,g(c,a)) + g ( c , g ( a , b ) ) = 0 is called Lie algebra. Therefore, if we consider the real vector space CM(S) of all Coofunctions on the symplectic manifold ( S , w ) then Coo(S)is a Lie algebra with the Poisson bracket being the product (see Proposition 6.2.2). Next, we define Poisson brackets for l-forms.
+
Definition 6.2.3 Let ( S , w ) be a syrnplectic manifold and C Poisson bracket { c Y , ~i }s the 1-form given by
Y , E ~
A'S.
The
where ix,w = CY and i x B w = P.
Hence the following diagram
sw x sw A'S
I
X A'S
{J
-
I
sw
A'S
is commutative. Therefore A'S endowed with the Poisson bracket {,} of l-forms is a Lie algebra.
Proposition 6.2.4 We have
Proof: In fact, we have
269
6.2. Poisson brackets
++(Xa,
X,))(Z)*
Therefore we deduce
{(.,P} = -Lx,P Corollary 6.2.5 I f a and
P
+ L x p + d(ix,ix,w).o
are closed then {a,/?} is ezact.
Chapter 6. Hamiltonian systems
270
Proof: If
a!
and /3 are closed we have
L x J = ix,dP
+dixJ
L x , a = ix,da!
+ d i x p = dix,a!.
= dix,P,
Then, from Proposition 6.2.4, we obtain {a,P}=
-dix,P
+dix,a + d(ixaixaw)
The following result relates the Poisson brackets of functions and 1forms.
Corollary 6.2.6 We have d ( F , G } = (dF, d G }
Proof: From Proposition 6.2.4 we obtain
= -d{G, F }
+d{F,G} - d{F,G}
= d{F,G}.O
Corollary 6.2.7 We have x { F , G } = - [XF 9 XC]
6.2. Poisson brackets
271
Proof: In fact, we have
i x { F , G=l ~d{ F, G} = {dF, dG}
= -S,([XF,XG]) = -i[xF,xG]w*u Let us now recall that if ( S , w ) is a symplectic manifold of dimension 2n then w" is a volume form on S.
Theorem 6.2.8 (Liouville Theorem). Let (S,w) be a symplectic manifold of dimension 2n and pt the flow of a Hamiltonian vector field. Then p;' preserves the volume form w" for all t, i.e., pt w" = w", for all t . Proof: Since p f w = w it follows that
= (p;w)" = wn.n
To end this section we develop some computations in local coordinates. Let (S,w ) be a symplectic manifold and (q', , . . ,q",pl,. . . ,pn) canonical coordinates in S. From (6.4), we deduce that the Poisson bracket of two functions F and G is given by { F , G } = Lx,F = ( a F / d q ' ) ( a G / d p i )
-
(aF/dp;)(aG/dq')
(6.7)
k o m (6.7) we obtain the Poisson brackets of the canonical coordinates:
{F,qi} = -{q',F} = -dF/dpj, {F,pj} = -{pi, F }
1
aF/dq'.
Using (6.8), Hamilton equations may be written as dqi dt
-=
( 4 )H } ,
dpi dt
- = -{pj, H } .
272
Chapter 6. Hamiltonian systems
-
(S,w) Let us now show that a canonical transformation h : ( S , w ) preserves Hamilton equations. To see this it is sufficient to show that the Poisson brackets are invariant under the action of h. Invariant here means
h*{F,G} = {h*F,h*G}, i.e.,
{ F, G } o h = { F o h , G o h } . In fact, we have
h*{F,G}={F,G}oh= (X,F)oh = (((Th)XGoh)F) o h (by Proposition 6.1.2)
-
= (X,,,h)(F o h ) ) = { F o h,G o h } = {h*F,h*G}.
In particular, if h : ( q , p ) cal coordinates, we have
( q . ~ ) where , ( $ , p i ) and
h*{q', H } = {q' o h , H
o
h} =
(*,p;) are canoni-
d$ { d ,K } = -, dt
4%
h*{p;,H } = { p i o h , H o h } = {F;, K } = --,
dt
where K = h*H = Hoh. This shows that canonical transformations preserve the form of Hamilton equations.
6.3
First integrals
We say that a l-form a on a manifold M is a first integral of a vector field X if ixa = a ( X ) = 0. Also, a function F on M such that X F = 0 is called a first integral of X . Obviously, if F is a first integral of X then d F ( X ) = X F = 0 and thus dF is a first integral of X. Now, let ( S , w ) be a symplectic manifold. Then for every l-form a on S there exists a unique vector field X , such that a = ix,w. In such as case, as
6.3. First integrals
273
a is a first integral of X,. In particular, if LY = dH then H is a first integral of XH = X,. If follows that the Hamiltonian function H is constant along the integral curves of X H ;in fact, if a ( t ) is an integral curve of X H ,we have
This gives the well-known principle of ’energy conservation law”. A first integral F of XH is usually called a c o n s t a n t of motion. Then H is a constant of motion. More generally, let F and G be Coofunctions on ( S , w ) . Then there is a vector field XF,resp. XG such that
iXFw = dF, resp. i x c w = dG Hence
i.e.,
Thus we have Proposition 6.3.1 If F and G are functions on a symplectic manifold ( S , w ) such that X F ( G )= 0 then & ( F ) = 0.
Definition 6.3.2 If a and p are 1-forms on a symplectic manifold ( S , w ) such that w ( X , , X p ) = 0 then they are said i n involution. Two functions F and G on S are in involution i f d F and dG are in involution. Let us recall that if a and
p
are closed then
{a,@}= d ( - i x , P
(see Corollary 6 . 2 . 5 ) . Hence, if a and Also we have:
+ ixBa+ ix,p)
p are in involution we have { q p } = 0.
Chapter 6. Hamiltonian systems
274
Proposition 6.3.3 Two functions F and G on S are i n involution if and only if { F, G } = 0. Proof: Since X u = SF, XdG = X G , we obtain
Hence { F , G } = 0 if and only if w(Xd=,XdG) = 0, or, equivalently, F and G are in invo1ution.O We have seen before that
X G ( F ) = w ( X F , X G )= { F , G }
-XF(G).
1
So, if F and G are in involution, then X G ( F ) = X F ( G ) = 0, that is, F (resp. G ) is a first integral of XG (resp. XF)and conversely. As ix,w = a and ixaw = p we have
Thus a similar result holds for l-forms.
Proposition 6.3.4 Let X F be a Hamiltonian vector field on a symplectic manifold (S,W). Then i x F w i s a first integral of the Hamiltonian vector field X H if and only if [xF,xH]= 0. Proof: As we know
d { F , H } = { d F , d H } = -i[xF,xH1w. Therefore ( i x , w ) ( X ~ = ) 0 implies { F , H } = 0 and then [ X F ,X H ]= 0. Conversely [ X F , X H ]= 0 implies { d F , d H } = 0 and so F and H are in involution (see Proposition 6 . 3 . 3 ) . Hence (iX,W)(XH) = 0 . 0 Suppose that ,B is a closed l-form on a symplectic manifold ( S , w ) such that for all closed l-form a on S, a and p are in involution. Then p = 0. In particular, if /3 = d F such that for all function G , F and G are in involution then F is constant ( d F = 0) (in fact, F is locally constant if S is not supposed a connected manifold). This is called a regular condition for the Poisson brackets.
275
6.4. Lagrangian submanifolds
6.4
Lagrangian submanifolds
A set of CM functions fl, . .. , fk on a symplectic manifold (S,w) is said to be independent if the corresponding Hamiltonian vector fields X,,,. . . ,X f k are linearly independent (this is equivalent to say that the l-forms dfl,. . . ,dfk are linearly independent). Let K be a submanifold of codimension k of (S,w ) locally defined by the independent functions fl = . . . = fk = 0 , k 5 n .
Lemma 6.4.1 K is coisottopic if and only if {f;,fj} = 0 on fk = 0 , 1 5 i , j 5 k.
fl =
. .. =
Proof If z E K then the tangent space T,K is the orthonormal complement of the vector space spanned by Xfl (z),. . . ,X,, (z) with respect to ~ ( z ) In . fact, if Y E T,S, then
< X,,(~),-.,Xf&) > 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
which are the Hamilton equations for a non-autonomous Hamiltonian H.
Proposition 6.7.1 We have
aH L x , H = -. at
Now, let us define a 2-form WH
WH
on R x S by
= W'
+ dH A dt.
Chapter 6. Hamiltonian systems
294
Proposition 6.7.2 (1) WH is a closed &-form of rank 2n; (2) X H i s the unique vector field o n R x S such that ~X,WH
= 0, i x , d t = 1.
Proof (1) Obviously, W H is closed. Furthermore, d t # 0, since w' is of rank 2n. Then WH has rank 2n. (2) We have
(6.15) A (wH)" = dt A
(w')"
d t ( x H ( tz , ))= d t ( a / a t ) + xt(z))= d t ( a / a t ) = 1. Moreover, let Y be a tangent vector at ( t , z). We obtain
aH i x , d H = Lx,H = -, at
- d H ( t , z ) ( Y ) = 0, since
6.8. Hamiltonian systems with constraints
+
f3H
295
z ) d t ( Y ) = dH(t,z ) Y .
Finally, since Awnis of dimension I, then X H is unique.^ Now, suppose that ( X , w ) is a symplectic manifold of dimension 271 such that w = -do (this is the case when S is the cotangent bundle of a manifold). We set
Proposition 6.7.3 If H i s a non-vanishing function
on
R x S, then t l i s~
a contact form.
Proof: In fact, we have
that is,
Then 8~ does not vanish on A w H .Therefore, one easilly deduces that ( d 8 ~ #) 0~. 0
A
6.8
Hamiltonian systems with constraints
In this section we examine some topics in the geometric theory of constraints, inspired in Weber [124]. This study will be continued in Section 7.13. Let (S,w) be a symplectic manifold of dimension 2n.
Definition 6.8.1 A non-zero 1-form a on S i s called a constraint on S. A set C = ( ( ~ 1 , .. . ,ar}of r linearly independent l-forms on S is called a system of constraints on S. We say that a curve u i n S satisfies the constraints if
296
Chapter 6 . Hamiltonian systems
Now we consider a Hamiltonian system (S,w,H ) together with a system C of constraints on S. We call (S,w, H , C ) a Hamiltonian system with constraints. It is clear that a curve u satisfying the Hamilton equations for H will not satisfy the constraints in general. For a curve u to satisfy also the constraints it is necessary that some additional forces (called canonical constraint forces) act on the system besides the 'force" d H . Then the equation of motion becomes
ixw = d H
+ a,
(6.16)
where a is a l-form on S which is the resultant of the canonical constraint forces. We must require that if a,(Y) = 0, for all a , l 5 a 5 k, then a ( Y )= 0. But this condition holds if and only if a = Aaa,
Hence the equation (6.16) becomes ixw = dH
+ A'cx,,
a,(X) = 0.
(6.17)
(Here the A' are Lagrange multipliers). If we denote by X, the vector fields on S given by
ix,w = a,, 1 5 a <_ r and by XH the Hamiltonian vector field with energy H , i.e.,
then we easily deduce from (6.17) that
Now, suppose that a, is locally given by
where (qi ,pi) are canonical coordinates for (S,w). ) ( q ' ( t ) , p ; ( t ) ) in S is an integral It is easy to prove that a curve ~ ( t = curve of X if and only if it satisfy
297
6.9. Exercises
dq'/dt = a H / a p i dp;/dt = - a H / a q ' ( A a ) i ( d B / d t ) ( B a ) i ( d p i / d t ) = 0,
+ Aa(Ba)i,
+
(6.18)
- Aa(Aa)i,
l s i s n , 1saSr. Next, we shall explain the geometrical meaning of constraints. Let C be a system of constraints on S. Then we define a distribution D on S as follows. For each z E S, D ( z ) is given by
Thus D is a ( 2 n - r)-dimensional distribution on S
Definition 6.8.2 A system of Constraints C is called holonomic i f D is integrable; otherwise we call C anholonomic. Hence C is holonomic if and only if the ideal of A S generated by C is a differential ideal, i.e., d y c Obviously (6.18) holds for holonomic as well as anholonomic constraints. For a system of holonomic constraints the motion lies on a specific leaf of the foliation defined by D.
y.
Remark 6.8.3 Note that the energy H for a solution a ( t ) of (6.18) is conserved. In fact, from (6.17) we have 0 = ( i p ) ( X )= d H ( X ) = X(H).
6.9
Exercises
6.9.1 Show that a submanifold W of codimension r of a coisotropic manifold K of a symplectic manifold (S,w) is Lagrangian if and only if it is transverse to the integral manifolds of the distribution z (T,K)*. 6.9.2 Show that a transformation f : ( S , w ) ( S , w ) is canonical if the l-form a = pdq - pdq is exact, where
--
Show that the converse is also true locally. 6.9.3 Let ( S , w ) be a symplectic manifold with local coordinates S --+R a CM map with Hessian matrix regular. Set
(5,
y),
F :
298
Chapter 6. Hamiltonian systems
z = 2, z = ( a F / a g )
-
Then the map p : S + S defined by (z,y) (3,s)is a canonical transformation. The map F is called “generating f u n c t i o n of a canonical t r a n s f o r m a t i o n ” . (Hint: compute d(gz - F)). 6.9.4 Let { P 1 , n l } and {Pz,Q2} be Poisson manifolds and h : PI P2 a Coomap. Let { , } I , { , } 2 be the corresponding Poisson brackets. We say that h is a Poisson map if
-
-
R be a Cm function. We say for all Cm functions f , g on P2. Let f : PZ that the vector fields n l ( d ( f o h ) ) on PI and h22(df) on P2 are h-related if
Show that (*) holds if and only if (**) holds. 6.9.5 Let { P l , n , } , {P2,n2} and {Ps,Q3} be Poisson manifolds and
Coomappings. Show that if h12 and h23 are Poisson maps then h23 o h12 is also a Poisson map. 6.9.6 If h12 and h23 o hl2 are Poisson maps and hl2 is surjective then h23 is
a Poisson map? 6.9.7 Suppose that X is not a Liouville vector field with respect to a given
volume form 8 . Is it possible to find another volume form 8 such that X is Liouville with respect t o 8? (Note that the set of n-forms on a n-dimensional manifold N is of dimension one and therefore there exists a function f on N such that 8 = f0, f # 0). 6.9.8 Show that in local coordinates ( z A )the components X A of a Liouville form are given by
6.9. Exercises
299
where X is such that i X 0 = dX. 6.9.9 Let M be a configuration manifold, WM the canonical symplectic form on T * M and a a 2-form on M . Consider the form 0 = w &a and suppose that X and Y are Hamiltonian vector fields with respect to a and w , respectively. Show that X , Y can be Liouville with respect to the same volume f o r m a
+
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30 1
Chapter 7
Lagrangian systems 7.1
Lagrangian systems and almost tangent geometry
In this section, we shall study a Lagrangian formalism for Classical Mechanics due to J. Klein [77], [78], [79]. Let M be an m-dimensional manifold and T M its tangent bundle with canonical projection TM : T M M . T M is called the phase space of R be a velocities of the configuration manifold M . Let L : T M differentiable function on T M called the Lagrangian function. We consider the following closed 2-form on T M :
-
WL
-
-ddjL.
(7.1)
The function EL = C
L - L (or EL = L c L
-
L)
on T M is called the energy function associated to L .
Proposition 7.1.1 We have iJWL = 0
Proof In fact, ijwL
= - i j d d j L = i j d j d L (since d d j = - d j d )
Chapter 7. Lagrangian systems
302
= d J i J d L (since i j d j = d j i j ) = d;L
=0.0
Consider the equation
ixwL = dEL
(74
Then we shall see that (7.2) under a certain condition on X is the intrinsical expression of the Euler-Lagrange equations of motion. Let us first show the following.
Proposition 7.1.2 The f o r m WL is a symplectic f o r m o n T M if and only if, for any coordinate s y s t e m ( q i , v i ) , the Hessian matrix
(&) i s of maximal rank (i.e., an inversible matrix).
Proof: A straightforward computation shows that
and so the m-times exterior product w r = W L A . . . A W L gives w r = cdet
(
d!,z:vj)
~
dq'
A
. . . A dqm A dv'
A
. . . A dvm,
where c is a non-zero constant function. So W L is symplectic if and only if w r is a volume form if and only if det (/,2'a", # 0.0 If wz, is symplectic we say that L : TM -+ R is regular or non-
i>
degenerate. Otherwise L is said to be singular, irregular or degenerate. Suppose that L is regular. Then the equation (7.2) admits a unique solution E , i.e., itwL = dEL, because W L is symplectic. Moreover, we have
7.1. Lagrangian systems and almost tangent geometry
303
Proposition 7.1.3 The vector field ( given by (7.2) i s a semispray (or second-order differential equation).
Proof: It is sufficient to show that J f = C. One has
i c w ~ = - i c d d J L = i c d J d L (since ddJ = - d J d ) = i J d L - d J i c d L (since i c d J d J i c = i ~ ) = dJL- dJ(CL) = -dJ(CL- L) = -dJEL.
+
On the other hand,
and
Therefore, one deduces
and then
J ( = C, since W L is symplectic. Now, if ( is the semispray given by (7.2), we locally have
Therefore we obtain
On the other hand, we have
Chapter 7. Lagrangian systems
304
.aL EL = d-
aVJ
-
L
and thus
Since i e w ~= dEL, we deduce
-
-
Let u : R M be a p a t h of (, i.e., b : R T M is a n integral curve of (. If u ( t ) = ( q ' ( t ) ) , then b(t) = ( q ' ( t ) , ( d q ' l d t ) ) . Thus, u is a p a t h of E if and only if u verifies
(here the dots mean derivatives with respect to the time). T h a t is,
Hence, we have.
Theorem 7.1.4 The paths of equations (7.3).
E
are the solutions of the Euler-Lagrange
This suggests the following.
-
Definition 7.1.5 The unique semispray ( such that i f w ~= dEL, where L :TM R is a regular Lagrangian, is called Euler-Lagrange vector field for L and it i s sometimes represented b y FL.
of
We have LeLEL = ELEL = 0 and EL is constant along t h e integral curves (energy conservation law).
EL
Example Let L : T R 2 -+
R be defined by
7.1. Lagrangian systems and almost tangent geometry
305
L ( x , y , u , v ) = (1/2)[(u2 - v 2 ) - x2 - Y 2 ] . Then
and
The Hessian matrix of L is
and hence, L is regular and W L is a symplectic form on T R 2 . A simple computation shows that the Euler-Lagrange vector field for L is
and so
The Euler-Lagrange equations for L are
Definition 7.1.6 A mechanical system M is a triple ( M ,F, p ) , where M is an n-dimensional manifold, F is a diflerentiable function on T M and p is a semibasic f o r m on TM, called the force field. Proposition 7.1.7 Let M = ( M ,F , p ) be a mechanical system. Suppose that the 2-form W F = - d d J F is symplectic. Then there is a unique semispray satisfying the equation
+
dEF p , (7.4) where EF = C F - F i s the energy of F. The paths of ( are the solutions of the system of equations i p F =
- -pi, 1 5 i 5 where p = pidq'.
m,
Chapter 7. Lagrangian systems
306
Proof: By a similar procedure used above we deduce that JE = C. Furthermore, from (7.4), one easily obtains
Then if a ( t ) = ( q i ( t ) ) is a path of p (7.5) becomes
Definition 7.1.8 A mechanical system M = ( M ,F , p ) i s said to be conservative if the force field p i s a closed semibasic f o r m . Let M = (M, F, p ) be a conservative mechanical system. Then L ~ W = F di(WF = d(dEF p ) = 0, since p is closed. Also we have (at least locally) p = dV and thus L.c(EF + V) = 0. So we deduce the energy conservation law. Suppose that M = (M, F , p ) is a conservative mechanical system such that there exists a differentiable function U : M R with p = .&(dU) = d(U o T M ) . Then M is said to be a Lagrangian s y s t e m . If we set L = F U o TM then is not hard to see that (7.4) and (7.6) become
+
-
+
So, conservative mechanical systems are Lagrangian systems. Nonconservative mechanical systems are mechanical systems for which the force field p is not a closed semibasic form.
7.2
Homogeneous Lagrangians
In this section, we develop the Klein formalism for homogeneous Lagrangians.
-
Definition 7.2.1 A Lagrangian L : T M R is said t o be homogeneous if the function L is homogeneous of degree 2, i.e., C L = 2 L .
7.2. Homogeneous Lagrangians
307
Consequently, if L is an homogeneous Lagrangian, we have
.a L
us-
= 2L(q,v).
awl
Thus, the energy function EL associated to L coincides with L , i.e., EL = L .
Proposition 7.2.2 Let L be an homogeneous Lagrangian. Then: (1) W L is homogeneous of degree 1; and (2) the Euler-Lagrange vector field ( L is a spray. Proof: (1) In fact, if L is homogeneous then
WL
= - d d j L verifies
since Proposition 4.2.9. (2) We have
= d ( L c L ) - d L = d ( C L - L ) = dEL. Since itLwL = dEL, and W L is symplectic, it follows that [C,( L ] = For mechanical systems we have the following result.
e~.
Proposition 7.2.3 Let M = ( M ,F , p ) be a mechanical system such that F and p are homogeneous of degree k. Then the semispray ( satisfying the equation
Proof: We have
Chapter 7. Lagrangian systems
308
EF = C F - F = k F
-
= k(k - 1)dF
=
F =(k- l)F
+ kp,
(k - 1)dF + p = dEF
+p =~(WF. (1
Since W F is symplectic it follows that [C,
7.3
=
E.
Connection and Lagrangian systems
Let be an arbitrary semispray on T M . Then the endomorphism I' = - Lt J defined by
r ( y )=
- [ E l
JYI
+ J [ E , YI,
for all vector field Y on T M , is a connection in T M . Let
be the horizontal and vertical projectors associated to
r. Then we have
T ( T M )= I m h e I m u (we remark that dim Imh = i d i m ( T M ) = m, dim M = m). So, at every point z E T M , the tangent vectors are decomposed in vertical and horizontal components belonging to the vertical and horizontal subspaces of the tangent space of T M a t z . An interesting result about the above decomposition is given by the following theorem due to Crampin [21].
7.3. Connection and Lagrangian systems
309
Theorem 7.3.1 Let L be a regular Lagrangian o n T M . T h e n there i s a connection r L in T M such that the vertical and horizontal subspaces of the tangent space of each point of T M , determined by the associated projectors of I’ are both Lagrangian subspaces f o r W L . We first prove the following lemma.
Lemma 7.3.2 Let R be a 2-form and F a tensor field of type (1,l) o n a manifold N. T h e n
is a tensor field of type (0,Z) o n N . Moreover,
if X is a n arbitrary vector
field o n N, t h e n
Lx(FJR)= (LxF)JR+ F J ( L x R ) . Proof In fact,
for any functions f and g on N . Hence F I R is a tensor field of type ( 0 , 2 ) on N . Moreover, we have
= X ( n ( F y . 2 ) ) - R([X,FY],Z)+fl(FY,[X,Zl)
This ends the pro0f.O
Chapter 7. Lagrangian systems
310
Proof of Theorem 7.3.1: Consider the symplectic 2-form W L = -ddJ L and the Euler-Lagrange vector field (L for L . By Lemma 7.3.1,we have
since L€,wL = 0. We set I'L = - L t L J. Then
Now, since i j w L = 0, we deduce that J J W Lis symmetric, and, hence, I'LlwL is symmetric, or equivalently, i r , W L = 0. Then, for all vector fields X and Y on T M we obtain
L t , ( J l w ~ )is also symmetric. Thus
+
So, if h = ; ( I d Lc, J) and v = ; ( I d - L t LJ ) are the horizontal and vertical projectors associated to I'L, we have
WL(hX,Y ) - WL(X,v Y ) = 0. Therefore
W L ( h X , h Y ) = W L ( U X , O Y ) = 0.0 Next, we shall prove that there is an almost Hermitian structure on T M such that its Kahler form is precisely W L .
Proposition 7.3.3 Let L be a regular Lagrangian on TM. Then g ( J X , J Y ) = W L ( J X ,Y ) defines a metric on the vertical bundle V ( T M ) .
7.3. Connection and Lagrangian systems
311
Proof (1) In fact g is well-defined. If Y and Y' are vector fields on T M such that J Y = JY',then J ( Y - Y ' ) = 0 implies that Y - Y' is vertical. Thus W L ( J X , Y )= W L ( J X , Y ' ) . ( 2 ) ij is symmetric. In fact, g ( J Y , J X )= -WL(JY,x) = WL(X,JY) = -WL(Jx, Y ) = i j ( J X , J Y ) .
(3) g is non-degenerate, since if g ( J X , J Y ) = 0 for any vector field J Y on T M , it follows that W L ( J X , Y )= 0 for any vector field Y on T M . Since W L is non-degenerate, we deduce that J X = 0. In local coordinates (q', u') we have
-
d2L aviavj
-
Hence, if I' is an arbitrary connection in T M , we can prolongate J to a Riemannian metric gr on T M . Consider the almost Hermitian structure ( T M ,F, g r ) . We have Proposition 7.3.4 Let Kr be the Kahler form of ( T M ,F , g r ) . Then
Proof In fact,
Corollary 7.3.5 Let r L = - L c , J . ( T M ,F,gr,) coincides with W L .
Then the Kihler f o r m K r , of
Chapter 7. Lagrangian systems
312
Proof In fact,
by Theorem 7 . 3 . 1 . 0 Next we prove that there is a canonical connection in T M whose paths are precisely the solutions of the Euler-Lagrange equations. We have seen in Chapter 4 that the connection I'L = - L t LJ verifies:
EL + $ E ,
0
its associated semispray is
0
its tension is H = -:[(;,
0
its weak torsion vanishes,
0
its strong torsion is T = ; [ E ; , J ] .
J],
Therefore, if L is an homogeneous Lagrangian,
EL
is a spray and hence
Thus we have the following result due to Grifone [73].
Theorem 7.3.6 If L is an homogeneous Lagrangian then there is a one and only one homogeneous connection I'L in T M whose geodesics are the solutions of the Euler-Lagrange equations.
Proof: It follows directly from the Decomposition theorem (see Section 4.3). 0
Next we consider non-homogeneous Lagrangians. Our purpose is to find a connection in T M such that its associated semispray is precisely EL. From the Decomposition theorem, we need a semibasic vector l-form T on T M such that T o+(; = 0. Let be the metric on V ( T M )defined in Proposition 7.3.3, i.e.,
g(J X , J Y ) = W L ( J X ,Y ) . If we set
313
7.3. Connection and Lagrangian systems then one has: (1) e ( J x , y )= ~ ( T J X , J Y= ) 0, (2)
e(x,J Y )= ~ ( T xJY) , = 0.
(We remark that 8 is semibasic). The problem now is to find a scalar semibasic 2-form €3 verifying (1) and (2) and, for such fixed 8,we define T by (7.7). We have
i.e.,
8"= i T O W L . Now T must be such T o= -<;.
(74
Thus, (7.8) becomes
8"= -it;wL.
(7.9)
Let us consider the 2-form 8 = ( i c w ~0) 7, where 7 is a l-form on T M such that 7 ( J X ) = 0 and 0 denotes the symmetric product. Then 8 verifies (1) and (2). If we take the potential of €3, then
+
8" = ( ~ c w L ) " ~ " ( ~ c w L ) ,
i.e., - it;w
+
=(~cwL)"~ "(~cwL).
(7.10)
Therefore, we deduce that
and so
(7.11) F'rom (7.10) and (7.11) we have
Chapter 7. Lagrangian systems
314
from which one obtains 8 = i c w ~0 7 and a fortiori T . Hence we have proved the following.
Theorem 7.3.7 (Grifone /79]). Let L be a regular Lagrangian on TM. Then there is a canonical connection i n TM such that its paths are the solutions of the Euler-Lagrange equations for L. This connection is given by
FL=I'L+T=-L~~J+T, where T is defined by (7.7) and 7 by (7.12). Now, let be the metric in V defined in Proposition 7.3.3, I' an arbitrary connection in T M and g r the Riemannian prolongation of along I'.
Definition 7.3.8
r
is said to be simple if i r w L = 0.
Proposition 7.3.9 I' i s simple if and only if the Kihler form Kr coincides with W L . Proof From Proposition 7.3.4 we have
where v is the vertical projector of I?. Hence
1. 2
= W L - -aywL. Thus Kr = WL if and only if i r w L = 0 . 0 Let EL = C L - L be the energy function associated to L.
Definition 7.3.10 'I is said to be conservative i f d h E L = 0, where h is the horizontal projector of I?.
315
7.3. Connection and Lagrangian systems
Notice that dhEL = 0 is equivalent to the fact that EL is constant along the horizontal curves with respect to I?. In fact,
Proposition 7.3.11 Suppose that r i s simple. Then I' is conservative if and only if it associated semispray c is precisely ( L .
Proof Since
r
is simple we have X r = W L . Hence we have
= g r ( J E , X ) - g r ( € , J x )= g r ( C , X ) , since
is horizontal. Thus
( i . p ~ ) ( X= ) gr(C, O X )= gr(C,J F X ) (since u = J F ) = g(C, J F X ) = -w&,
F X ) = -(iCWL)(FX)
= (-i~icwL)(X).
Therefore we obtain
i
p =~- i ~ i ~ w ~ .
Since i c w ~= -dJEL (by Proposition 7.1.3), we have
+
But dEL = dhEL d,EL (since Id = h + v ) . Then i only if dhEL = 0, or, equivalently [ = € L if and only if dhEL = 0 . 0
p =~dEL if and
Chapter 7. Lagrangian systems
316
Theorem 7.3.12 Let L be an homogeneous Lagrangian. Then there ezists a unique conservative connection with zero strong torsion.
Proof: Since L is homogeneous then EL = L, CL is a spray and I'L = -Lt,J is a connection which associated semispray is precisely ( L . Moreover, I't is simple (see Theorem 7.3.1). Hence I'L is conservative. Now, let I' be a conservative connection with zero strong torsion. We have
=WL.
Thus 1. 2
-8rwL
1. = -;2h-IdWL 2
= i h w L - W L = 0,
which implies that 'I is simple. Hence, from Proposition 7.3.4, we see that the associated semispray to I' is ( L . Then, from the Decomposition theorem one has I' = F L . 0
Remark 7.3.13 If L is the kinetic energy defined by a Riemannian metric g on M, then Theorem 7.3.12 is the fundamental theorem of Riemannian geometry (see Theorem 1.18.10). Remark 7.3.14 In [27], (381 we have extended these results for higher order Lagrangians.
7.4. Semisprays and Lagrangian systems
7.4
317
Semisprays and Lagrangian systems
From the above results we have seen that the Lagrangian formulation of Classical Mechanics may also be developed intrinsically through a symplectic structure. However this formulation depends directly on the choice of the Lagrangian function. This shows a particular difference from the Hamiltonian situation where an intrinsical symplectic structure is canonically defined on the cotangent bundle of a given configuration manifold, independent of the choice of the Hamiltonian function. First-order differential equations (vector fields) are related with Hamiltonian functions via the symplectic structure. Let us now examine some results relating second-order differential equations (semisprays) with Lagrangian systems. Let E be an arbitrary semispray on T M . We denote by r = - L t J the connection defined by Then we have
c.
where ( = w'a/aq' Hence
+ cd/du'.
X H = X'(a/aq')-
1 ,x'(a(j/aq') (alawj),
(7.13)
where X = X' ( d / d q ' ) is a vector field on M . By using (7.13) we have
+ ('(a/aw'), X J a / a w J ]
[ < , X U= ] [.'(a/aq') =
-x'(a/aq')+ d ( a x ' / a q j ) ( a / a v ' )- xj(a~'/auJ)(a/a.') = -2X'(a/dq') -
xj(a~'/auj)(a/a.')
.
+xi(a/ a q' ) + j(ax'/aqj) (a/ aU i) = -2XH
+ xc.
Chapter 7. Lagrangian systems
318 Therefore
[(,XU] = -XH - ( X H
-
XC),
and, then, X H - X c is the vertical component of [(,Xu].
Proposition 7.4.1 (Crampin [ZO]). Suppose that w is a 2- f o r m on TM such that (1) Ltw = 0, (2) for every point z E T M the corresponding vertical subspace is Lagrangian with respect to w and iydw, where Y is an arbitrary horizontal vector field. Then w is closed. Proof: Let us first recall the following expression: d w ( X , Y , Z ) = X w ( Y , Z ) - Y w ( X , Z )+ Z W ( X , Y )
since w vanishes on pairs of vertical lifts (by (2)). Furthermore, for every vector field W on TM
dw(W,Y',Zu) = dw(hW
+ uW,YU,Zu)
= d w ( h W , Y U , Z " )= (ihWdw)(YU,Z")= 0.
Now,
7.4. Semisprays and Lagrangian systems
319
+ d w ( X H , Y " , [ E ) Z " ] )= 0. The first term vanishes because we have two vertical arguments in the expression. Now
[E,Y"]= - Y H - ( Y H - YC), [ ( , Z " ]= -ZH - ( Z H - ZC) and so only the horizontal components of [ f , Y " ]and [E,Z"]contribute to the remaining terms. Thus
d w ( X H , Y H , Z " ) = dw(XH,ZH,Y"). But then
d w ( X H , Y H , Z " ) = dw(XH,ZH,Y") = -dw(ZH,XH,Y") = -dw(ZH,YH,X") = dw(YH,ZH,X")
= d w ( Y H , X H , Z u= ) -dw(XH,YH,Zu) and so
d w ( X H , Y H , Z " )= 0. We conclude that dw vanishes except when all its arguments are horizontal vector fields. As ZH - Zc is vertical, d w ( X H , Y H , [&Z"]) = -dw(XH,YH,ZH). Using again (1) for d w ( X H , Y H , Z Uwe ) obtain
dw(XH,YH,[&Z"]= ) 0.0
Chapter 7. Lagrangian systems
320
Proposition 7.4.2 (Crampin (201). Let w be as in Proposition 7.4.1. Then there i s a CM function K defined on an open subset of T M such that
where J ia the canonical almost tangent structure on TM.
Proof: From Proposition 7.4.1 we have that locally w is exact, say w = d a , for some l-form a defined on some open subset U of T M . The restriction of d a to vertical subspaces is zero (they are Lagrangians for w ) . Thus the restriction of a to each fibre is exact and there is a function F on an open set V c U such that
a ( X u )= X " ( F ) = d F ( X " ) for any vector field X on I S M ( Vc) M . Set /3 =
- dF.
Then
d p = d a = w and @(Xu) = 0. From (1) and (2) of Proposition 7.4.1, we have
w " W U I , Y U )+ w ( X U ,[<,Y"l) = 0 and as only the horizontal components of the brackets signify,
W(XH,YU)= W ( Y H , XU). Thus, for vector fields X and Y on T M ( V )we , have
w(XH,YU) = XH(P(YU))- Y " ( P ( X H ) -) P ( [ X H , Y U ] ) = - Y " ( p ( X H ) ) = w ( Y H ,XU) = - x " ( p ( Y H ) ) , since [ X H , Y U is ] vertical. Put 7 = S*p, where S is the almost complex structure defined by r = - L t J . Let us recall that S is defined by
7.4. Semisprays and Lagrangian systems
32 1
Hence
X " ( q ( Y " ) )= X " ( @ ( Y H ) = ) Y " ( @ ( X H )= ) Y"(r)(X")). As before, the restriction of dr) to vertical subspaces is zero, and as before, there is a function K : W c V --+ R such that r)(X") = - d K ( X " ) = - X " ( K ) . Thus @(Xu) = 0 and p ( X H )= r)(X")= - d K ( X " ) , i.e.,
Therefore
and, locally, up to a constant function along the fibres of TM, one has
The following result is due to Crampin [20].
Theorem 7.4.3 A necessary and sufficient condition for a semispray ( on T M t o be a (regular) Euler-Lagrange vector field i s the existence of a 2-form w on TM satisfying (1) and (2) of Proposition 7.4.1. Proof: If ( is a Euler-Lagrange vector field then ( is the unique solution of
for some regular Lagrangian L : T M + R . The 2-form W L is symplectic and satisfies LEWL= 0. Also dwr, = 0 and then iydwr, = 0. From Theorem
7.3.1one has that the vertical subspaces are Lagrangian for W L . Conversely, by using Propositions 7.4.1 and 7.4.2, one obtains a 2- form (3 of maximal rank such that cj = - d d J K . The equation
determines a semispray (K. As L p = d ( i p ) = 0 there is at least locally some function E such that i t 0 = dE. Since (K and ( are both semisprays, then (K - ( is vertical and since
Chapter 7. Lagrangian systems
322
we deduce that EK - E is constant along the fibres vertical vector fields). Thus
(avanishes on pairs of
EK - E = G , where G is a function obtained by the pull-back of some function by TM : T M + M , defined on some open set of M . If we set L = K G then
+
C L - L = CK
-
K - G = EK - G = E
and so ( is (locally) a Euler-Lagrange vector field for such L . 0
Remark 7.4.4 (Cantrijn et al. [ll]). Let us suppose that X is a vector field on TM such that
( L x J )o J = J, J
o
( L x J ) = -J.
(7.14)
(i,e., - ( L x J ) is a connection in TM). Then in a local coordinate system (q', u') X has the following expression
x = (u' + A ' ( q )aq)a7 + X'(q,u)aayi, i.e.,
X is a semispray modulo a rM-projectable vector field
A'(q)(a/aq').
Now, suppose that W is a 2m-dimensional manifold endowed with an integrable almost tangent structure J and let 7 be a vector field on W such that L,J verifies (7.14).
Definition 7.4.5 We say that 7 Is a regular Lagrangian dynamical system if there ezists a symplectic form w on W such that (1) the subspaces ( I m J ) x of TxW are Lagrangian with respect to w, for all zEW; (2) L,w = 0 . Using (1) and (2) it can be shown that the symplectic form w and the almost tangent structure J are related by
7.5. A geometrical version of the inverse problem
323
for all vector fields X on W . We prove this result by using the fact that 7 defines “horizontal” and “vertical” distributions via the “connection” -L, J and we proceed along the same lines as the ones given above in the case where W = T M . We also remark that the above equality implies that all subspaces ( I m J)= are Lagrangian with respect to w . Now, using the same procedure as for the case T M and semisprays, we may show that if 7 is a regular Lagrangian dynamical system on an integrable almost tangent manifold ( W , J ) then there exists (locally) a function L on W such that
i,w = dEL, EL = ( J 7 ) L - L .
So, the Lagrangian formalism may be reobtained for the more general situation of integrable almost tangent manifolds.
7.5
A geometrical version of the inverse problem of Lagrangian dynamics
In this section, we will show that Theorem 7.4.2 gives a geometrical version of an old problem in Mechanics, attributed to Helmholtz (see for example known as the inverse problem of Lagrangian dynamics, which Santilli [107]), may be stated in the following form: given a semispray ( on T M when is it possible to find a Lagrangian L such that ( is precisely a Euler-Lagrange vector field for L? The answer involves some conditions, called Helmholtz conditions, concerning a matrix (cri,(q, q ) ) . If we develop the Euler-Lagrange equations in local coordinates ( q , q ) then we obtain an expression of the form gijQ’
+ hi = 0,
where g,, and h, are functions of q k ,qk and
A system of second-order differential equations is derivable from a Lagrangian if it is self-adjoint. Let us see the conditions for self-adjointness. We will consider only systems of second-order differential equations of type
Chapter 7. Lagrangian systems
324
Then the necessary and sufficient conditions for a system of differential equations in this form to be derivable from a Lagrangian is that there exists a regular matrix (gij) depending on ( q )q ) such that the equations g;jqj - gij f j = 0
are self-adjoint. In terms of the functions f i these conditions are: there should exist functions g;, such that
gij = gji,
(7.16)
(7.17)
(7.18)
(7.19)
Now, from the preceding geometrical study it is possible to show that (7.15)-(7.19) may be reobtained. In fact (we follow Crampin et al. [23])let us first set the semispray ( locally as
Let I' = -LcJ be the connection determined by (. Then the horizontal distribution (with respect to )'I is locally spanned by
7.5. A geometrical version of the inverse problem
The dual basis of
325
{ Q , K = a/au'} is { O ' , q g } , where
. . 8' = dq', q' = -(1/2)(af'/aui)dq'
+dug.
are
The Lie derivatives of these l-forms by
Lcei = (1/2)(afg/aUj)ej + q i ,
7;. = r(af'/auj) - (1/2)(af'/auj)
-
(7.20)
2(af'/aqj).
Suppose that the form w in Theorem 7.4.2 is locally expressed by
w = a,j#
A
ej + gijeg A qj,
where a;j+aj; = 0 (we may eliminate the terms in q'AqJ from the expression of w ) . If we use (7.20) and (7.21) we find that
L ~ = w (((a;,)
+ a;k(afk/aui)
-
(l/2)g;k7f)eg A 0'
+gijq' A qj.
As Ltw = 0, one finds that
((7.16)1
Sij = Sji.
Therefore a;, = 0,
Chapter 7. Lagrangian systems
326 Thus w = regular, i.e.,
gijqi A
qJ and w is of maximal rank if and only if
det(gij)
(gij)
is
((7.15)1
#0
Finally, (7.17) is obtained from the condition dw = 0. As the only terms of interest in dw are those involving e' A q j A q k and as the coefficients of such a term is ( d g i j / a v k ) one has that
and so
agjilavk= agjk/avi= a g i j p vk . The self-adjointness conditions are thus equivalent t o the conditions on the form w stated in Theorem 7.4.2.
7.6
-
The Legendre transformation
Let L : TM
Then
(YL is
R be a Lagrangian function and consider the l-form
locally expressed by CYL=
aL -dq',
,
av*
where ( q i , v i ) are the induced coordinates in TM. Hence (YL is a semibasic l-form on TM. If we take into account Theorem 4.2.16 one has the following
Theorem 7.6.1 There ezists a mapping Leg : T M following diagram
is commutative
(TM
and
AM
-
T*M such that the
are the canonical projections).
7.6. The Legendre transformation
327
We locally have
where pi = a L / d u ' . Moreover, if AM denotes the Liouville form on T * M , then
(see Corollary 4.2.17). Then if w~ = -dAM is the canonical symplectic form on T * M , we have
Leg*wM = W L , where
WL
= -ddJL.
Theorem 7.6.2 T h e following assertions are equivalent: ( I ) L i s a regular Lagrangian. (2) W L i s a symplectic f o r m o n TM. (8) Leg : T M + T*M i s a local diffeomorphism. I n such a case, t h e mapping Leg i s called the Legendre t r a n s f o r m a t i o n determined by L .
-
Proof: The equivalence between (1) and (2) was proved in Section 7.1. On the other hand from Leg*wM = W L it follows that W L is symplectic if and only if Leg : T M T * M is a symplectic m a p . 0 Now, let u : R M be a curve in M and u : R T M its natural prolongation to TM. Then along b, we have
-
pi = a L / a i ' ,
since u' = d q ' / d t , b ( t )= ( $ ( t ) , $ ( t ) ) . Hence
along 6 . We call aL/agt the momentum p,, 1 5 i 5 m. In general the Legendre transformation determined by a regular Lagrangian is not globally a diffeomorphism. If Leg is a global diffeomorphism then L is said to be hyperregular. As AM o Leg = TM we see that Leg is a global diffeomorphism if and only if its restriction to each fibre of TM : T M M is one-to-one.
-
Chapter 7. Lagrangian systems
328
Suppose that L is regular. Let EL = C L consider the equation
-
L be the energy of L and
i t L w= ~ dEL. Then the vector field ( = T Leg o CL o Leg-' on T * M is given by
Indeed,
Hence,
Proposition 7.6.3 If 7 is an integral curve of
ou
is an
The energy function EL of a regular Lagrangian L (and the function ELO LegvL on T * M if L is hyperregular) will be called Hamiltonian energy, corresponding to L and will be denoted by H . It is now clear that Leg is the map which permits to pass from a regular Lagrangian formalism t o a Hamiltonian formalism. Thus the above vector field f is the Hamiltonian vector field of energy H = EL o Leg-'. We have seen above the relation between the Lagrangian and Hamiltonian formulation of Classical Mechanics which are equivalent in the hyperregular case. Our exposition was presented in terms of the tangent bundle geometry on T M . Let us examine this theory without using the almost tangent procedure (see Sternberg [114], Mac Lane [92], Abraham and Marsden
PI). vE
-
R be a Lagrangian function. For each tangent vector Let L : T M T,M, z E M , we consider the natural identifications
and
7.6. The Legendre transformation
-
'pc
329
: T,*M -+ T,*(T,M).
Let L , : T,M R be the restriction of L to T , M . We define a mapping LegL : T M -+ T*M by h L ( U )
= ('p;)-l(dLz(U))*
Let (q',u') and ( q i , p i ) be the induced coordinates in T M and T * M , respectively. Then the isomorphism 'puis given by 'pu(a/dq'), = (apq'),.
(7.22)
M d 9 ' ) Z ) = (d9')u
(7.23)
Hence
From (7.22) and (7.23)) one has
i.e., LegL coincides with Leg. Sometimes it is found in the litterature the name of fiber derivative of L for LegL. Example.- Let g be a Riemannian metric on M and L the kinetic energy defined by 9 , i.e.,
L(u) = (1/2)g2(u)u), u E T,M, x E M . In local coordinates (q', u') we have
L(q', u') = (1/2)g'juivj. Hence EL = L and L is regular:
d2L auiauj
- Sij.
Thus the corresponding Legendre transformation is given by
Leg(q',u') = (q')gijuj), i.e., Leg(u) E T,*M is the cotangent vector of M at x defined by
Chapter 7. Lagrangian systems
330
-
< u , L e g ( v ) >= g z ( u , v ) , for all u E T,M. Hence Leg : T M T*M is a global diffeomorphism. In fact, Leg is injective since Leg(v) = Leg(ij) implies that g z ( u , u ) = g , ( u , i j ) , for all u E T,M and so v = ij. Moreover, Leg is surjective, since, if a E T,*M, then the tangent vector v E T,M such that
< u , a >= g z ( u , u ) , for all u E T z M , verifies Leg(v) = a. The same is true for a Lagrangian
: T M -+
R defined by
z= L + V O T M , where L is the kinetic energy of a Riemann metric g on M and V : M -+ is the potential energy.
7.7
R
Non-autonomous Lagrangians
In this section and in the followings we will use jet manifolds. More details about jets are given in Appendix B. Let ( E = M x R, p, R ) be a trivial bundle, where M is a m- dimensional manifold. As ( E ,p, R) is trivial we may identify mappings from R t o M with sections of ( E ,p, R) as well as their k-jets. The velocity space associated to M is the manifold J,'(R, M ) of all l-jets of mappings u : R + M with source a t the origin 0 E R. Thus we may identify J,'(R, M ) with the tangent bundle T M of M . The evolution space of M is the fibred manifold J ' ( R , M ) of all l-jets of mappings from R to M . A Coofunction L : J ' ( R , M ) R defined on the evolution space of M is said to be non-autonomous (or time-dependent). It is our purpose to stablish an intrinsical description of non-autonomous Lagrangian systems. For this, let us first notice that J 1 ( R ,M ) can be canonically identified with R x T M by the map
-
where j / u denotes the l-jet with source a t t of a curve u : R b ( t ) is the tangent vector of u at the point u ( t ) .
-
M and
7.7. Non-autonomous Lagrangians
331
If (q') are local coordinates for M then (t, q', u') are the induced coordinates for J 1 ( R ,M ) . Therefore we transport all geometric structures defined on T M to J 1 ( R , M ) , via this identification. Indeed J and C may be considered on J ' ( R , M ) (and J ( a / a t ) = 0). Furthermore, we define a new tensor field J" of type (1,l)on J ' ( R , M ) by
J"= J - C B d t From (7.24), we deduce that J" is locally characterized by .@/az)
=
-c, .@/aq')
= a/avi, J"(a/ava)= 0.
(7.24)
(7.25)
Hence J" has rank m and satisfies (J")2 = 0. We define the adjoint of J", denoted by J"*, as the endomorphism of the exterior algebra A ( J ' ( R , M ) ) of J ' ( R , M ) locally given by J"*(dt) = 0 ) J"*(dq') = 0 ) J"*(dv') = dq' - v'dt.
(7.26)
-
Definition 7.7.1 A p-form a on J ' ( R , M ) is said to be ~ 2 - s e m i b a s i c (resp. T-aemibasic) if a belongs to I m J* (resp. I m p ) . (Here 7r2 : J1(R,M) M and T : J ' ( R , M ) R x M are the canonical projections).
-
From (7.26) we easily deduce that a l-form a on J ' ( R , M ) is ~2-semibasic (resp. T-semibasic) if and only if (Y is locally expressed by a = a&, q , u)dq'
(7.27)
(resp. a = a i ( t ,q , v)dq' - f i i ( t ,q, ~ ) v ' d t )
(7.28)
Let a be a ~2-semibasic l-form on J 1 ( R ,M ) . Then we may define a differentiable mapping D : R x T M + R x T*M as follows:
where v E T,M and p E T,*M is given by
Chapter 7. Lagrangian systems
332
with X E T(t,,,(R x T M ) projecting onto X ,i.e., (.z),X locally given by (7.27), we have
= X. Hence if a. is
q t , q', v') = ( t , q ' , a;) Like in the autonomous situation we associate to on the algebra A(J1(R,M ) ) by
J" operators i j
and d j
P
( i j ( A J ) ( X l , .)XP) * . = C W ( X l , . ..,JX;, ...,Xp), i=l
d j = [ i j ,d] = i j d
-
dij,
and so from (7.25) we have ij (df) =
.P(df ) ,
i j ( d t ) = i j ( d q ' ) = 0, i j ( d v ' ) = dq' djf
- v'dt,
= ( d f / a v ' ) ( d g ' - v'dt),
(7.29)
d j ( d t ) = dj(dq') = 0, d j ( d v ' ) = -d(dp'
-
v'dt) = dv' A d t ,
for all f E CM(Jf ( R ,M ) ).
Remark 7.7.2 According to the theory of Frolicher-Nijenhuis i j (resp. d j ) is the derivation of type i , (resp. d,) associated to the tensor field J. Remark 7.7.3 In the following we will put 8' = dqi
-
d d t , 1 5 i 5 m,
(7.30)
which are l-forms on J1( R ,M ) .
-
Let us characterize a semispray on J ' ( R , M ) by means of the tensor fields J and J". We denote by : J 1 ( R ,M ) = R x T M R the canonical projection defined by xl(j/o) = t . Then the l-jet prolongation of a curve u : R M in M is a section denoted by jla of the fibred manifold (J'( R, M ) ,T I , R ) defined by
-
7.7. Non-au tonomous Lagrangians
j1c7: t E R
-
333
( j ' a ) ( t ) = j;. E J ' ( R , M ) .
Alternatively, j'a may also be regarded as a curve in J ' ( R , M ) and for simplicity we will call it the canonical prolongation of c7 t o J ' ( R , M ) . However it is clear that not every section of ( J ' ( R ,M ) , n l ,R) has this parR will be the canonical ticular form. A (local) section p of n1 : J ' ( R , M ) prolongation of a curve in M if and only if
-
p*ei
= 0, 1
5i5m
(7.31)
In such a case we say that p is holonomic.
Definition 7.7.4 A vector field E on J 1 ( R ,M ) whose integral curves are all holonomics is called a semispray (or second-order differential equation). It follows that a vector field
d t ( c ) = 1,
c on J 1 ( R ,M ) is a semispray if and only if e'(e)
= 0, 1 5 i
5 m.
Then a semispray ( on J ' ( R , M ) is locally given by
c = a p t + u'apq' + e'd/dv', where
= p ( t , q , u ) is a Coofunction on J ' ( R , M ) , 1
Proposition 7.7.5 A vector field i f J f = C andJ"<=O.
(7.32)
5 m.
c on J ' ( R , M ) is a semispray if and only
Proof In fact, if f is a semispray we easily deduce from (7.25) and (7.32) that JF = C and . f f = 0. Conversely, suppose that ( is locally expressed by f =r a p t
+ X'd/dq' + Y ' a p v ' .
As J f = C we obtain that X' = v'. Thus
J"(
= (1 - r)u'd/du' = 0
and so (1 - r)ui = 0, 1 5 i 5 m, where r = r ( t , q, v ) . If some vi # o , we deduce that r = 1 and by continuity T = 1 on all J ' ( R , M ) , showing that f is a semispray.0
Chapter 7. Lagrangian systems
334
Definition 7.7.6 Let ( be a semispray o n J ' ( R , M ) . A curve u in M i s called a path or solution of ( i f its canonical prolongation i s a n integral curve of (. Let CT be a curve in M ,locally given by ( q i ( t ) ) . Then the l-jet (j'o)(t)= ( t , q i ( t ) , ( d q i / d t ) ( t ) ) and so is a path of ( if and only if it satisfies the following non-autonomous system of second-order differential equations
where ( is given by (7.32). Suppose that a non-autonomous regular Lagrangian L is given on M, i.e., L is a function on J ' ( R , M) such that the matrix
.
.
(a2~ / a ~ ~ a ~ f ) is non-singular. Then we define a 2-form
RL on J ' ( R, M) by
RL = d d j L + d L A d t ;
+
RL is called the PoincarC-Cartan 2-form and d j L L d t is called the Poincard-cart an l-form. A straightforward computation in local coordinates shows that
+ ( a 2 L / a v i a v j ) d v i A d&
(7.33)
Therefore
nr = fd e t ( a 2 L / a v i a d ) d q 1 A . . . A dqm A dv'
A
. . . A durn.
Since L is regular we have
flr
A
d t # 0.
Then (RL,d t ) determines an almost cosymplectic structure on J ' ( R , M). Moreover, RL is exact, since
7.7. Non-au tonornous Lagrangians
RL = d ( d j L
335
+ Ldt).
Let
Rn, = {X E T ( J 1 ( R , M ) ) / i x f Z =~0 ) be the characteristic bundle of Q L . Then Rn, is an orientable line bundle and there exists a unique vector field (L on J ' ( R , M ) such that
i C L R ~ 0 , i,,dt = 1.
(7.34)
Proposition 7.7.7 Let L be a non-autonomous regular Lagrangian on J ' ( R , M ) and [ L the vector field given b y (7.34). Then (L is a semispray on J'( R, M ) whose paths are the solutions of the Euler-Lagrange equations
Proof Since it,dt = 1, then ( L is locally given by EL =
Furthermore,
(L
a p t + x'apqi + <'a/av'
satisfies i t , Q ~= 0. Therefore, from (7.33), we have
+ <'vj(a2L/avjav') = 0
(7.35)
uj(a2L/dvjdv') = x j ( a 2 L / a u j a v ' ) .
(7.37)
As L is regular, from (7.37), we have X' = d , 1 5 i 5 m. Hence (7.35) and (7.36) become
Chapter 7. Lagrangian systems
336
- (aL/dq'))= 0 )
(7.38)
( a 2 ~ / a u i a+t )uj(a2L/auiaqj) + ( j ( a 2 ~ / a U i a u-j a) L / a q i = 0. Now, let o be a path of
EL. Then from
(7.39)
(7.39) we have
- (aL/aq') = 0
(7.40)
along j'o. But (7.40) are the Euler-Lagrange equations for L. 0 We call [ L the Euler-Lagrange vector field for L.
7.8
Dynamical connections
-
The tensor fields J and J" on J ' ( R , M ) permit us to give a characterization of a kind of connections in the fibred bundle r : J ' ( R , M ) R x M.
Definition 7.8.1 By a dynamical connection on J ' ( R , M ) we mean a tensor field r of type (I,I) on J ' ( R , M ) satisfying
-
I
-
*
J r = J r = J , r J = - J , r J = -J.
(7.41)
From a straightforward computation from (7.41) we deduce that the local expressions of I' are
r(a/at) r(a/aq;)
r(a/avi)
= = =
- d ( a / a q i ) + ri(a/aui),
a p q i + r;(a/ad), -a/au*
1
(7.42)
The functions I" = I " ( t , q , u ) and I'i = r i ( t , q , u ) will be called the components of I?. From (7.42) we easily deduce that
r3- r = 0 and rank I' = 2m.
7.8. Dynamical connections
337
Then I' defines an f ( 3 , -1)-structure of rank 2rn on J ' ( R , M ) . Now, we can associate to r two canonical operators I! and rn given by
t = r2, m = -r 2
+ Id.
Then we have
and t and rn are complementary projectors. From (7.43) we deduce that t and rn are locally given by
!(apt)
=
- d ( a / a q i ) - (r' + v'r$(a/ad),
t ( a / a q ' ) = a/aq', t ( a / a v ' ) = a / a v ' , m(a/at) =
rn(a/aq')
a/at + v ' ( a / a q ' )
. .
+ (ri+ dr;)(a/ad), (7.44)
= rn(a/av') = 0.
If we set L = I m I ! , M = Irnrn then we have that L and mentary distributions on J 1 ( R ,M ) , i.e.,
M
are comple-
T ( J 1 ( RM , ) ) = M CB L. From (7.44) we deduce that L is 2rn-dimensional and is locally spanned by { d / d q ' , a / a v ' } . M is 1-dimensional and globally spanned by the vector field = rn(a/at). Taking into account the local expression of 6, we deduce that is a semispray which will called the canonical semispray associated to the dynamical connection r. Furthermore, we have r2t = t and rrn = 0. Thus I' acts on L as an is an almost product structure and trivially on M. Since M = Ker I?, f ( 3 , -1)-structure of rank 2m and parallelizable kernel. Moreover, r / L has eigenvalues +1 and -1. From (7.42)) the eigenspaces corresponding t o the eigenvalue -1 are the Ir-vertical subspaces V,, z E J ' ( R , M ) . Recall that for each z E J ' ( R , M ) , V, is the set of all tangent vectors t o J ' ( R , M ) a t z which are projected to 0 by T T . Thus V is a distribution given by z V,. The eigenspace at z E J 1 ( R ,M ) corresponding t o the eigenvalue +1 will be denoted by H , and called the strong horizontal subspace at z . We have a canonical decomposition
e
e
-
Chapter 7. Lagrangian systems
338
and then
-
T ( J ' ( R ,M))= M @ H CB v, where H is the distribution z
(7.45)
H,.
Remark 7.8.2 We note that a dynamical connection I' in J'(R, M) induces an almost product structure on J ' ( R , M) given by three complementary distributions for the three eigenvalues 0, +1 and -1. However, for a connection in TM,the corresponding almost product structure on TM was given by two complementary distributions corresponding to the eigenvalues $1 and -1 of r. Let us put HL = M, @ H,; H i will be called the weak horizontal s u b s p a c e at z . Then we have the following decompositions:
and
-
T ( J 1 ( R M)) , = H'CB V,
(7.46)
where z H i is the corresponding distribution. We notice that L , M , H and H' may be considered as vector bundles over J 1 ( R , M ) ; the bundles H and H' will be called strong and weak horizontal bundles, respectively. Thus, from (7.46)) I' defines a connection in the fibred manifold K : J ' ( R , M ) R x M , with horizontal bundle H'. A vector field X on J 1 ( R ,M) which belongs to H (resp. H') will be called a strong (resp. weak) horizontal vector field. From (7.46)) we have that the canonical projection K : J ' ( R , M ) R x M induces an isomorphism K* : HL T,c,, ( R x M),z E J' ( R ,M ). Then, if X is a vector field on R x M , there exists a unique vector field X H ' on J ' ( R , M ) which is weak horizontal and projects to X; XH'will be called the weak horizontal lift of X to J 1 ( R ,M). The projection of X H ' to H will be denoted by X H and called the strong horizontal lift of X to
-
-
-
J ' ( R ,M ) . From (7.42) and by a straightforward computation, we obtain
7.8. Dynarnical connections
339
Then, if we put D; = (a/c3q')H' and 5 = a / d v ' , one deduces that { t , D ; , V , } is a local basis of vector fields on J ' ( R , M ) . In fact, M =< ( >, H =< D; >, and V =< >. Then {(,D;,&} is an adapted basis to the f ( 3 , -1)-structure I?. In terms of { ( , D ; , K } . (7.47) becomes ( a / a t ) H ' = ( - dD;,( a / a q ; ) H ' = D;.
Therefore we obtain (a/at)H =
If X = r ( a / a t )
-viD;) ( a / a q ' ) H = D;.
+ X'(a/aq') is a vector field on R x A4 we have XH = (X'- r u ' p ; .
(7.48)
We notice that the dual local basis of l-forms of the adapted basis { t , D ; , K } is given by { d t , O i , q i } , where
Remark 7.8.3 If we set
then we have
H = I m h, V = Imu,
u( = 0 ,
UD;
= 0,
uv; = v;.
h and u will be called the strong horizontal and vertical projectors associated to r. If we set h' = m + h, then I m h' = H' and h' is called the weak horizontal projector.
Chapter 7. Lagrangian systems
340
-
Definition 7.8.4 Let r be a dynamical connection in J ' ( R , M ) . A curve a :R M is said t o be a path o f r i f and only i f j'a is a weak horizontal curwe in J'(R, M ) , i.e., the tangent vector j ' a ( t ) belong t o H$o(r)Jf o r every t E R. F'rom (7.44) we easily deduce that a is a path of r if and only if (T satisfies the following system of non-autonomous second-order differential equations: (7.49)
From (7.32)) (7.44) and (7.49) we obtain the following.
Proposition 7.8.5 A dynamical connection I' and i t s associated semispray have the s a m e paths.
<
Now, we introduce the curvature and torsion of a dynamical connection. Before proceeding further, we stablish the following lemma (the proof is obtained from a long by straightforward computation in local coordinates).
Lemma 7.8.6 Let I' be a dynamical connection in J ' ( R , M ) and {<,D ; , K } a n adapted basis t o
r.
T h e n we have
[K,Vj] = 0. where
(7.50)
7.8. Dynarnical connections
34 1
+(1/2) va r ;(arr,/ a d ) ,
- (i/4)rprr/av3).
Definition 7.8.7 The tensor fields R and R' of type (1,Z) o n J ' ( R , M ) given by 1
1
R = - [ h , h ] , R' = -[h',h']. 2 2 will be called, respectively the strong and weak curvatures of
(7.51)
r.
From (7.50) and (7.51) we easily deduce the following.
Proposition 7.8.8 W e have R ( < ,<) = R(<, 0,)= R ( < , K ) = R(D;,V,)= R(V,,Vj) R( H; , Hj) = C&V,, p((,() = R'(Di,Vj) = R'(V,,Vj) = 0 , R'(<,0 ; )= ArVr , R'(J,K) = Dl, P ( D ; , Dj) = qjvr, !*OR'= R . Definition 7.8.9 T h e tensor fields t,t',t" and given by t = [ J , h ] ,t' =
2 of type
0,
(1,Z) o n J'(R, M )
[J,h'],t"= [ . J , h ] ,2 = [J",h']
(7.52)
will be called, respectively, the strong J-torsion, weak J-torsion, strong J"-torsion and weak J"-torsion of I'.
Chapter 7. Lagrangian systems
342
Next, we associate to each dynamical connection I' in J'(R, M ) an almost contact structure on J'(R, M ) . Let I' be a dynamical connection in J 1 ( R ,M ) with canonical projectors m and !A and let h and u be the strong horizontal and vertical projectors of I', respectively. Let us now define a tensor field 4 of type ( 1 , l ) on J 1 ( R ,M ) by
Then, if {(, D ; , K } is an adapted basis to I?, we have
4% = Di,
dDi = -V,,
c$(
= 0.
(7.53)
Hence we deduce that
d2 = - I d
+ f @ q,
where q = dt and ( is the associated semispray to r. Then (4, <,q ) is an almost contact structure on J ' ( R, M ) which will called the canonical almost contact structure associated to r. Next, we characterize the normality of (4, (, q ) in terms of the connection r. As we have seen in Section 3.6, an almost contact structure (4, q ) is normal if and only if
c,
7.8. Dynamical connections
343
where N4 = (1/2)[4,4] is the Nijenhuis tensor of 4. In our case, r] = dt implies dq = 0. Hence the canonical almost contact structure (4, <,q ) associated to I' is normal if and only if N+ = 0. Now, from Lemma 7.8.6 and (7.53), we have the following. Proposition 7.8.11 The Nijenhuis tensor N4 of N4(€,€)
= 0,
N + ( t ,0;)
4
i s given b y
+
= (B' - (1/2)I'i)Dr (6: - A:)Vr, = (6; - A:)Dr ((1/2)I'i - Br)Vr, Nb(Di, D j ) = (1/2)((aI'i/avJ) - (dI'y/av'))Hr - C['Vr, N,j,(D;,Vj) = -C[jDr - (1/2)((dI'i/avJ) - (dI';/av'))Vr, N#(V;.,V,) = (C,Yj- (1/2)((aI':/dvJ) - (aI';/av'))Dr.
+
N#(t,K)
Moreover, the Lie derivative of
(Lt#)(V;.)= ((1/2)I':
4 with respect to
-
B:)Dr
is given b y
+ (A: - 6:)Vr.
From Propositions 7.8.8, 7.8.10 and 7.8.11, we easily deduce the following.
Theorem 7.8.12 e*ot=o.
(4, t, q ) is normal if and only if L t 4 = 0 , R = 0 and
Next, we shall construct an adapted metric to (4, t, q ) . Let g be a metric on the vertical bundle V over J 1 ( R ,M ) . Then we define a Riemannian metric gr on J ' ( R , M ) by
gr(X, Y ) = @X, JY)+ g ( v X v Y ) + q ( X ) r l ( Y ) * A simple computation from (7.54) shows that
(7.54)
344
Chapter 7. Lagrangian systems
g r ( € , C ) = 1, g r ( € , o ; ) = g r ( € , V I . ) = g r ( D i , V j ) = O .
(7.55)
From (7.55) one easily deduces that M,H and V are orthogonal distributions, q = itgr and gr is an adapted metric to (4, ( , q ) . Let us now consider the fundamental 2-form Kr associated t o the almost contact metric structure (4, (, q , gr) defined by
K}, we have In terms of an adapted basis { (, D;,
7.9
Dynamical connections and non-autonomous Lagrangian s
In this section we construct a dynamical connection canonically associated to a non-autonomous Lagrangian whose paths are the solutions of the EulerLagrange equations. First, we examine the relation between semisprays and dynamical connections. Let ( be an arbitrary semispray on J 1 ( R , M ) and suppose that ( is locally given by
Then a simple computation shows that
[(,a/at]
= -(a(j/at)(a/auj),
[(, a/aq;]
= -(a(j/aq’)(a/auj),
[E, a / a u i ] = Now, let
r = -L,J”.
-a/aq; - ( a p / a u ‘ ) ( a / a u j )
Then from (7.57) we have
1
(7.57)
7.9. Dynamical connections
r(a/at)
=
r(d/aqa) =
r(a/aUi)=
345
-d(a/aqi) - (d(ae/auJ)- c)(a/aUi), a/aqi + (ap/ad)(a/ad), -a/avi
(7.58)
From (7.58) we easily deduce the following.
Proposition 7.9.1 Let ( be a semispray on J ' ( R , M ) . Then r = -LtJ" i s a dynarnical connection i n J 1 ( R ,M ) whose associated semispray i s (. Now, let L be a non-autonomous regular Lagrangian on J ' ( R , M ) . We set
where
.
.
nL(D;,Vj) = -( d2L/dv'du'),
(7.59)
where { &, D,, K} is an adapted basis for L . Next we shall construct an adapted metric on J 1 ( R ,M) to the canonical almost contact structure ( q 5 ~ , ( ~ , qassociated ) t o I'L. First, we define a metric grL on the vertical bundle V by
gr,(JX,J"Y) = R L ( J X , Y ) - q ( Y ) .
(7.60)
In terms of an adapted basis { ( L , D,,V,} for r L we have
jr,(Vi,Vj) = ~
.
L ( K ~,
j
.
=) ( d 2 ~ / t W d u J ) .
Hence we can construct a Riemannian metric gr, defined by (7.54), i.e.,
Chapter 7. Lagrangian systems
346
~ ) ( 4 ~ , J ~ , q , g r ,is ) an Thus gr, is an adapted metric t o ( ~ L , J L , and almost contact metric structure on J ' ( R , M ) . Now, let Kr, be the fundamental 2-form associated t o ( t $ ~ , (L,q , gr,). Then we have
K r L ( D i , Q )= -grL(V(,Vj) = - ( d 2 ~ / a ~ ' a d ) .
(7.61)
From (7.59) and (7.61) we easily deduce the following.
Theorem 7.9.2 Let L be a non-autonomous regular Lagrangian on J 1 ( R ,M ) and JL the Euler-Lagrange vector field for L. Let ( 4 ~JL, ) q , g r L ) be the almost contact metric structure canonically associated to the dynarnical connection r L = - Lt,J". Then the PoincartGCartan form RL is the fundamen, v,grL), i.e., tal 2-form Kr, defined from ( 4 ~JL,
RL = Kr,. To end this section, we reobtain some results of Crampin et al. [23]. In fact, we have an adapted basis {JL, D;,&}to I'L, where
D,= a/a,+
+(1/2)(a~j/a~~)(a/a~j).
Thus the corresponding dual basis is {dt,O', q ' } , where qi =
-(c
-
(l/2)d(dJi/duJ))dt
-(1/2)(~3J'/auJ)dqj
+ du'.
The significance of this dual basis is that the form RL can be re-written as follows:
and so the semispray
CL is uniquely determined by the equations i t L @= itLq' = 0, it,dt = 1.
Remark 7.9.3 In [39]we have extended the results of this section for higher order Lagrangians.
7.10. The variational approach
7.10
347
The variational approach
Let M be a manifold of dimension m. Consider the evolution space J' ( R ,M ) which may be canonically identified with R x T M (see Section 7.7). If we consider the trivial fibred manifold ( R x M , p , R) then it is easy to see that we may identify the corresponding jet manifold of l-jets of sections of ( R x M , p , R ) with J ' ( R , M ) . In the following we set N = R x M , J'N = J ' ( R , M ) = R x T M . We have the following canonical projections: T
-
: J'N
N , p : J'N
-
R.
which are locally defined by T ( t , Q',
-
"') = ( t , q'), p ( t , q', v ' ) = t
Let t E J I N , y = T ( Z ) E N and s : R N a section such that s ( t ) = y, where t = p ( z ) . We define the following linear difference from T , ( J ' N ) to T U N :
T T - (2's)
o
(Tp) : T,(J'N)
using the tangent prolongations of trates this definition:
T ,s
TUN
(7.62)
and p. The following diagram illus-
TN
TR
c
c
N
R
Suppose that X E T , ( J ' N ) is locally given by
x = d / a t + xia/aqi+ Y i a / a v i Then
T+)(x)
-
p s ) ( t ) T + ) ) ( x )=
+ xia/aqi
Chapter 7. Lagrangian systems
348
= ra/at
-
+ xia/aqi- r a / a t - vira/aqi = (xi-
d)a/aqi,
(7.63)
i.e., (Tr - (7's) o ( T p ) ) ( X )is a vertical tangent vector with respect t o the fibration p : R x M R.
Definition 7.10.1 The TN-valued l-form given b y (7.62) is called structure (or canonical) 1 - f o r m and it is represented b y 8 .
-
Now, let s : R N be a section of p : N + R and consider the l-jet T ( J ' N ) gives prolongation j ' s . Then T ( j ' s ) : TR
T ( j l s ) ( r a / a t )= r a / a t
+ via/aqi+ zia/avi,
where s ( t ) = ( t , q i ( t ) ) ,v'(t) = ( d q ' / d t ) ( t ) and z ' ( t ) = ( d 2 q i / d t 2 ) ( t ) . Therefore
( z h ) ( T ( j l s ) ( r a / a t )= ) 7a/at
+ via/aqi,
and so
-
e ( z y j l s ) ( r a / a t ) )= 0. Also it is easy to verify that if 2 : N
(7.64)
T ( J ' N ) is vertical over N then
e(2) = T T ( 2 ) .
(7.65)
From (7.63) we may define l-forms O', i.e., setting ixei = X i have
8' = dq'
-
v'dt.
-
Proposition 7.10.2 For every section s : R the unique l-form verifying (7.64) and (7.65).
- m i , we
(7.66)
N the structure form 0 is
7.10. The variational approach
349
Proof Suppose that w is a V(N)-valued 1-form verifying (7.64) and (7.65). Then w ( z ) : T,(J'N)
-
-
Vu(N) c T U N ,
J ' N , y = n ( z ) . Suppose that I = j ' s ( t ) , where s is a section of p :N R and t E R. If X E T , ( J ' N ) then z E
where the last equality come from (7.64).0
Definition 7.10.3 Let W be a submanifold of N. We say that W is a crosssection submanifold of N i f p/W is a bijection of W onto a submanifold of R and f o r every y E W , T p : TuW Tp(,)Ris an isomorphism.
-
-
If we consider a section s : R N then s ( R ) defines a cross-section submanifold of N . Sometimes we set s( R) = R,. Also, the 1-jet prolongation j's : R J ' N defines a cross-section submanifold of J ' N and we set ( j ' s ) ( R )= RjlS.In the following we identify j ' s with Rji8 and we will say that j's is a cross-section submanifold of J ' N .
-
Proposition 7.10.4 The only cross-section submanifolds of J'N f o r which 0 vanishes ( 0 = 0 along such submanifolds) are the 1-jet prolongations of sections of ( N ,p , R ) .
-
This proposition is a consequence of the following: P r o p o s i t i o n 7.10.5 Let u : R J ' N be a section and R, the corresponding cross-section submanifold. Suppose that Nu c n-'(y), y E N . Then there is a section s : R N such that u = j's (i.e., u is a 1-jet prolongation) i f and only i f
Chapter 7. Lagrangian systems
350
Proof Suppose that there is a section s : R Then from Proposition 7.10.2
and so 8' vanishes alons
-
N such that
u = j's.
Nu,since
-
J'N such that Conversely, let be u : R are the local coordinates of y E N. We have
Nu c (T-')(t,q'), where ( t , q i )
u(t) = (t,qi,v'). But
8' = dq'
-
v'dt/Nu
E0
and so
. dq' . v' = - = Q'. dt Therefore taking s ( t ) = ( t , q i ( t ) )= (n o u ) ( t ) one has u = j ' s . 0 Now, let ( E , p ,Q)be a fibred manifold and a a k-form on E. We suppose that S is a compact k-dimensional submanifold with smooth boundary dS. We denote by S e c ( E ) the sections of (E,p, 9).
Definition 7.10.6 A variational problem with respect to a on a domain 5' is given b y a mapping J : S e c ( E ) + R defined b y J(s)=
s*a
-
(7.67)
called Hamiltonian functional of a for the sections s : S E of Sec( E ) . A variation of (7.67) is given b y a Coa 1-parameter family of sections st E S e c ( E ) , with so = s such that J(st) =
sla.
7.10. The variational approach
351
Let X be the vector field tangent t o the family
st
at t = 0 defined by
X ( z ) = -(Z)/t=o dst dt The Cartan formula for the Lie derivative is
where
= so and
CY
is a k-form on E.
D e f i n i t i o n 7.10.7 (Hamilton’s principle) We say that s E Sec(E) i s an extremal (or a critical section) of (7.67) if
In the following we restrict our study t o the case when (7.68)vanishes for all variations st of s such that st = s along as and X = 0 along as. In such a case we have from Stoke’s theorem that s is an extremal of (7.67)if and only if s*(iXda) = 0.
(7.69)
In the following we will assume that the vector fields X in (7.69) are vertical (with respect t o p ) since it can be shown that the ‘horizontal part” gives no contribution to the final results. So, as (7.69)is supposed t o hold for all vertical vector fields X on S such that X / a s = 0 one obtains s * ( i x d a ) = 0.
(7.70)
along S.We will say that s E Sec(E ) is an extremal of (7.67)if (7.70)holds. D e f i n i t i o n 7.10.8 A vector field Y o n E i s said to be projectable along S (or p-projectable) if there ezists a vector field X on S such that for every point z E E such that p ( z ) E S we have
Let us return to the previous situation
N
=R x M.
Chapter 7. Lagrangian systems
352
Definition 7.10.9 The l-jet prolongation of a vector field X on N is the projectable vector field 2' on J'N along N such that 2' is an infinitesimal contact transformation (i.c.t.), i.e.,
LX-' gi = r j e j
(7.71)
(the i.c.t. i s such that the ezterior diferential system generated by {O'} preserved under the action of the Lie derivative). The formula (7.71)allows us to compute explicitly nates. If we set
is
2' in local coordi-
and
a direct computation shows that
xi= ( a x ' p t ) - v j ( a r / a t ) + v j ( ( a x ' / a q j )- ( a r / a q j ) v ' ) . In case where X is a vertical vector field one has
f;
= ax'/aqj,
xi
=
axi/&+ d(axi/aqj)
(7.72)
Let us consider the set of S e c ( N ) but now with domain being a closed interval of R, say [ O , r ] (which will be represented by 5). Thus a variational problem is given by
-
(7.73)
-
where L : J'N R is a Lagrangian function. The form dt in (7.73) is identified with the pull-back p*(dt) where p : J'N R is the canonical projection. From the preceding considerations we shall say that a section s of N along 5 is an extremal of (7.73) if for all vertical vector fields X on N we have
7.10. The variational approach
353
p s ) * L * l (Ld t ) = 0, where 2' are such that rewritten 89
k1/(j~8)(ac) = 0. As
the above integral may be
LR1(Ld t ) = 0 (7.74) (r) and taking into account Stoke's theorem and the Cartan formula, (7.74) takes the form L 1 8 )
1
L*l (Ld t ) =
/
(7.75)
A dt) = 0
along ( j ' s ) ( ~ ) . Let us examine (7.75) in local coordinates. For X vertical we have
z1= X'(a/aq*) + ( ( a x ' / a t ) + d(ax*/a*j))(a/au*) (see (7.72)). Thus
L*1 ( L d t ) = X'(aL/aq')dt
+ ((ax'/at) + d (aX*/aqJ))(aL/du*)dt.
Taking into account that
(aL/au*)d(X') = d( (aL/aui)xi) - X*d(aL/au') and
(ax*/aqj)(aL/a 2)d dt = (ax*
.*
(aL/a ) (d+
-
ej)
we see that
L X I ( L d t )= X'(aL/aq')dt
= X'(aL/aq')dt
+ (aXi/at)(aL/au*)dt
+ (aL/aua)((aX*/at)dt+ (ax*/aqj)d+)
Chapter 7. Lagrangian systems
354
Now, the second term in the right side vanishes when we apply Stoke's theorem and as OJ = 0 along (j's)(c) (see Proposition 7.10.4)one has
and thus (as X i
# 0) (7.76)
Then if s : [ O , E ] + N is a section, s is an extremal of J ( s ) if and only if s ( t ) = ( t , q i ( t ) , $ ( t ) )is a solution of (7.76),i.e., one has
(7.77) Let us now examine the modified Hamilton's principle given by a functional I defined on Sec ( J ' N ) . For this we consider the l-form Oh
=djL
+ Ldt
(7.78)
where J" is the tensor field of type (1,l) on R x TM given by (7.24).In local coordinates
aL au* + Ldt.
OL = -0'
Note that if j's
is the l-jet prolongation of a secction s, then
355
7.10. The variational approach
= (Ldt)/ji,.
BL/jis
The modified Hamilton's principle is defined by (7.79) where now 1.1 : [O,c] + J'N is a section (not necessarily a 1-jet prolongation) and 5' = p ( [ O , c ] ) . The section p is an extremal of (7.79) if for all vertical vector field Y on J'N over R one has
which leads to the equation (7.80)
iydOL = 0 along 5' = p ( [ O , c ] ) . In local coordinates OL is written as OL = fidq'
where, for simplicity, we have set
fi
+ (L- fiv')dt, = (aL/dv'). Thus
For a vertical vector field Y on J'N along R we have
and (7.80) takes the form
iyddL =
(-2+
aL) Y'dt at
afi . +-Yldq' a91
-
. - -Yidq' afi a93
( 'a,:;)
Y'dt
Chapter 7. Lagrangian systems
356
afi +-Y
- j
av3
av3
-
dq' - -afi Y'dd
("'2) vjdt.
(7.81)
Now, (7.80) is considered along T1, thus the coordinates are taken along the curve p ( t ) , i.e., p ( t ) = ( t , q'(t), v ' ( t ) ) . Then
. dq' dq' = -dt, dt
. dv' d d = -dt, dt
and so (7.81) is
iydOL =
(-%
afi dq' + ( -a d dt
i.e.,
)
Y ' d t - (d-) .afi Y i d t
a 9'
at
Y'dt-
'
afi dvj
)
--
( a v j dt
Y'dt
7.11. Special symplec tic manifolds
357
which leads t o the following set of equations (from the coefficients of Y' and
Pi): (7.82)
(7.83)
k
-
In general the set of solutions of such system is different from the set of solutions of (7.76). However, if we suppose that L : J I N R is a regular function then one has
'
dq' _ ..1 _ -q - v .
dt
(7.84)
In other words, if L is regular, the expression p*(iydOL) = 0 is equivalent to (7.77). Thus the regularity of L gives the 1-jet prolongation condition (7.84) and the extremals are the same for both variational problems. We have shown the following:
Theorem 7.10.10 Let L : J'N -+ R be a regular Lagrangian function. Then the Hamilton's principle and the modified Hamilton's principle are equivalent in the sense that we may stablish an injective-surjective mapping between the set of eztremals of both problem.
7.11
Special symplectic manifolds
In Hamiltonian Classical Mechanics we have considered the cotangent bundle of the configuration manifold. However, sometimes it is convenient to consider symplectic manifolds diffeomorphic to cotangent bundles. In this section we introduce the notion of special symplectic manifolds due t o Tulczyjew [116], [117].
Definition 7.11.1 A special symplectic manifold is a quintuple ( X ,M,?r,O,A) where T : X -+ M is a fibred manifold, O is a I - f o r m on X and A : X T*M i s a difeomorphism such that ?r = ~ T Mo A and 0 = A*XM.
-
Chapter 7. Lagrangian systems
358
Now, let K be a submanifold of M and F : K + R a function. Set
N = {p E X/.lr(p) E K, < u,8 >=< T.lr(u),d L >, for any u E T X such that
TX(U)
= p and T s ( u )E T K
c TM}.
(7.85)
Suppose that dim K = k 5 dim M = m. Then we can choose local coordinates (q4,q'), 1 I a 5 k, k 1 5 r 5 m for M such that K is locally given by qk+' = . . . = qm = 0. Let (q',p;), 1 5 i 5 m be the induced coordinates for T*M. Since A is a diffeomorphism then (q',p;) may be taken as local coordinates for X in such a way that A is locally written as the identity map. Hence
+
8 = p;dqi and, from (7.85) we deduce that a point ( $ , p i ) of X is in q' = 0, pa = d F / a q 4 , 1
N if and only if
5 a 5 k, k + 1 5 r 5 m.
Thus (q4,pr) are local coordinates for N and dim N = rn. One can easily check that N is a Lagrangian submanifold in the symplectic manifold ( X , do); we call it the Lagrangian submanifold generated by L. Let Q be an arbitrary m-dimensional manifold and T*Q its cotangent bundle. Let BQ : TT*Q --+ T*T*Q be the vector bundle isomorphism defined by the canonical symplectic structure WQ on T*Q, i.e.,
Then we have ~ T * Q= =T*Q
0
PQ
If (&,pi, q',p;) are local coordinates for TT*Q one has PQ
(8 P i , 4' 9
ii)
= (Q i ,Pi ,- P i ,
d')
We can easily check that (TT*Q,T*Q, ~T*Q,xQ,PQ) is a special symplectic manifold, where X Q = ~ ; X T * Q . Hence XQ = -p;dq'
+ q'dp;.
(7.86)
Let H : T*Q + R be a Hamiltonian function. Hence d H = (aH/aq')dq'+ ( a H / a p ; ) d p ; . Choose u = ( t , p ; , $ , p ; ; 6qi,6p;,6$,6p;) E TTT*Q. Then
7.11. Special syrnplectic manifolds
359
and < U,XQ >=< T T T * Q ( u ) , ~>His equivalent to - Pi(6qi)
+ qi(SPi) = ( a H / a q i ) ( S q i )+ ( a H / a p i ) ( S P i )
(7.87)
From (7.64) we have
qi = a H / d p i , pi = - ( d H / d q i ) , 1 5 i 5 m, which are the Hamilton equations for H . Moreover, if XH is the Hamiltonian vector field on T*Q corresponding to H then N is the image of X, an XH = (&)-l o ( d H ) . The following diagram illustrates the above situation:
PQ
TT*Q
* T,*T*Q
-
Now, we can define a canonical diffeomorphism AQ : TT*Q T*TQ. Indeed, given u E TT*Q we must define AQ(u) E T*TQ by means of its pairing on any element w E T T Q , where T T Q ( W ) = T T Q ( u ) . Given two curves 7 : R --+ T Q and q : R T*Q such that +(o) = U Q ( W ) , i ( o ) = u and rQ 07 = T Q o q , we define
-
-
< W , AQ(u)>= d / d t < 7 ,q > / t = o
(here UQ : TTQ TTQ is the canonical involution of TTQ defined in Exercise 7.16.3). A simple computation shows that AQ is locally given by AQ(qi Set
)
Pi)
4') P i ) = ( q i )i' ,Pi, P i )
= A;)(XTQ).Then we locally have
CYQ
CYQ
=
+ p;dQi
(7.88)
Proposition 7.11.2 The quintuple (TT*Q,T Q , T T Q ,CYQ,A Q ) i s a special symplectic manifold.
360
Chapter 7. Lagrangian systems The proof is left t o the reader as an exercise.
-
Remark 7.11.3 We notice that XQ # CYQbut their sum is an exact l-form. : TT*Q R as follows: To show this, we define a canonical function
A simple computation in local coordinates shows that
F'rom (7.86) and (7.88) we obtain
Hence XQ and -CYQ define the same symplectic structure on TT*Q (see de Le6n and Lacomba [311). Now, we assume that L : TQ + R is a Lagrangian function. Consider the above construction for K = M = TQ, X = TT*Q. Hence
dL = ( d L / d q i ) d q i + ( d L / d q ' ) d q ' . Choose
T T x Q ( u )= ( q i , q', Sq', Sq') E TTQ and
is equivalent to
which implies (7.89)
7.11 . Special symplec tic manifolds
361
ji = a L p q i ,
(7.90)
1 I i I m. Thus (7.66) give the momentum and (7.67) re the EulerLagrange equations for L. Moreover we can check that the Lagrangian submanifold N given by (7.85) is
N = (&l
o dL)(TQ).
Then we have the following diagram:
Remark 7.11.4 We notice that the result holds even if L is a degenerate Lagrangian (see Chapter 8 for further information about degenerate Lagrangians). Also the Lagrangian L may be defined on a submanifold K of
TQ . In order to connect the Lagrangian with the Hamiltonian diagrams, which exist independently, we need the Lagrangian L to be regular. In such a case there exists a unique vector field (L on TQ such that
i t L w= ~ dEL is the Lagrange vector field for L ) . Let Leg : TQ + T*Q be the Legendre transformation defined by L . Hence the Lagrange vector field ( L : TQ TTQ satisfies the relation ((L
-
AQ oT Leg0 (L = d L
-
and we can construct the following diagram:
TTQ
T Leg
TT*Q
AQ
*
T*TQ
Chapter 7. Lagrangian systems
362
-
Since Leg is at least locally inversible, then we can define the Hamiltonian function H : T * Q R as H = EL o Leg-’. Also the equality N = X H ( T * Q )holds at least locally. Then we can construct the following diagram:
T Leg
-+ -
7 T*TQ
“0
TT*Q
PQ
T*T*Q
Remark 7.11.5 In de Le6n and Lacomba [30]this construction has been extended for higher order mechanical systems (see de Le6n and Rodrigues 13811.
7.12
Noether’s theorem. Symmetries
In this section we first discuss Noether’s theorem for mechanical Lagrangian systems and then symmetries of Euler-Lagrange vector fields. We follow Arnold [4] and Crampin [21].
-
-
Definition 7.12.1 (1) Let L : T M R be a regular Lagrangian function and F : M M a map. We say that L admits the map F if L o T F = L, i.e., the following diagram
7.12. Noether's theorem. Symmetries
363
is commutative. (2) Let X be a vector field on M and dt the local 1-parameter group of local transformations generated b y X . We say that L admits X i f L admits i j t , f o r all t .
Proposition 7.12.2 L admits X i f and only i f X c L = 0, where X c is the complete lift of X to T M . Proof In fact, X c is the infinitesimal generator of the local 1-parameter group of local transformations Tq&. Then L x c L = X c L = 0 if and only if L o Tdt = L , for all t. Example: Let L : T P R be a Lagrangian function defined by
-
L ( z ' , u') = 1/2
3 C(U')2 -
V ( 21 ,22 ),
1=1
where (z1,z2,z3) are the canonical coordinates on vector field X = a/&' on R3.
P.Then
L admits the
Theorem 7.12.3 (Noether'a Theorem) If L admits a vector field X on M , then X " L is a first integral of L, i.e., ( L ( X " L ) = 0, where [ L is the Euler-Lagrange vector field for L . Proof In fact,
(L
is given by the equation
Hence we have
= -(r.((JX", L)
+ X C ( C L )+ J [ € L , X C ] L
Chapter 7. Lagrangian systems
364
= -€L(xvL)
+XC(CL),
since J X c = X " , J ( L = C and J [ ( L ,X c ] = 0. On the other hand, we have
dEL(XC)= XCEL = XC(CL - L) = X C ( C L )- X C L . Thus, we obtain
€L(X"L) =XCL and, consequently, X c L = 0 implies ( L ( X " L )= 0.0 Example. Consider the Lagrangian
L=T$-R given by 3
~ ( x ' x, 2 , x3, tr1,tr2,
tr3)
=
C mi(~,?/2)- u(s',x2,x 3 ) i= 1
-
where the mi's are positive real numbers and U : R3 R. The Lagrangian L corresponds to a system of point masses with masses mi on R 3 . Suppose that L admits translations along the 2'-axis. This is equivalent t o say that L admits the vector field X = d/dz' on R3. (In fact, the 1-parameter group of transformations of R3 generated by X is r$t(xl,x 2 ,x 3 ) = (x' t , x 2 ,x 3 )). Then X c L = 0 and, according to Noether's theorem X " L is a first integral of (L (i.e., a constant of motion). But
+
is exactly the momentum p1. Thus p1 is conserved. Next we shall discuss symmetries.
Definition 7.12.4 Let ( be a semispray on T M . A Lie symmetry of 6 is a vector field X on M such that [ X c ,€1 = 0. From Exercise 1.22.4, we deduce that X is a Lie symmetry of ( if and only if every Tr$t commutes with every $,, where dt (resp. $,) is the local 1-parameter group of local transformations of M (resp. T M ) generated by X (resp. 0 .
7.12. Noether's theorem. Symmetries
365
Proposition 7.12.5 Let L be a regular Lagrangian on T M and ( L the corresponding Euler-Lagrange vector field. If X is a vector field on M such that L x c d J L is closed and d ( X " E L )= 0 , then X is a Lie symmetry of CL. Proof: In fact,
+
= d ( X C E L ) i[,d(LxcdJL)
=0.0
If, in addition, L x c d J L is exact, i.e., L X c d J L = d f , and X ' E L = 0 , then such a symmetry X is called a Noether symmetry. Clearly, a Noether symmetry of ( L is also a Lie symmetry, since Proposition 7.12.5.
Proposition 7.12.6 Suppose that X is f - X " L is a first integral of ( L . Proof: In fact, we have
a
Noether symmetry of
EL.
Then
Chapter 7. Lagrangian systems
366
Remark 7.12.7 Proposition 7.12.6 is a generalization of Noether's theorem. In fact, let us suppose that f is a constant function (for instance f =O). Then
L x c d j L = 0. Therefore we have
0 = (LXcdJL)((L)= X C ( ( d J L ) ( ( L )) d J L [ X " ,(Lit] = X " ( C L )- J[XC, (L]L = X C ( C L ) .
c
Definition 7.12.8 (1) A dynamical symmetry of a semispray on T M i s a vector field 2 on T M such that (1 = 0. (2) A Cartan symmetry of an Euler-Lagrange vector field EL i s a vector field 3 on T M such that L*(djL) i s ezact, namely, L*(dJL) = d f , and -%EL = 0.
[z,
Remark 7.12.9 The argument used in Proposition 7.12.5 applies to show that a Cartan symmetry of (L is also a dynamical symmetry. Proposition 7.12.10 I f 2 is a Cartan symmetry of [ L then f - ( J 2 ) L i s a first integral of EL. Proof In fact, we have
= -(LEL = 0.0 Thus, Cartan symmetries give rise t o constants of motion.
Remark 7.12.11 We remit t o Prince [lo51 for a discussion of symmetries when L is a time-dependent Lagrangian.
7.13. Lagrangian and Hamiltonian mechanical
7.13
367
Lagrangian and Hamiltonian mechanical systems with constraints
In this section we reformulate some results of Weber [124] for Lagrangian and Hamiltonian mechanical systems with constraints. Let L : T M R be a regular Lagrangian. Then ( T M , w L )is a symplectic manifold.
-
Definition 7.13.1 Let C = { e l , . . . ,er} be a system of constraints on T M . We call ( T M , w L , E L , C )a regular Lagrangian system with constraints. The constraints C are said to be classical constraints if the l-forms 8, are basic. Then holonomic classical constraints define foliations on the configuration manifold M , but holonomic constraints also admit foliations on the phase space of velocities T M . Next we seek conditions such as to make a Lagrangian system with constraints defines a mechanical system. Consider the equation (7.91)
Definition 7.13.2 We say that ( T M , w L ,E L , C ) defines a mechanical system with constraints if the vector field ( given b y (7.91) is a semisPray.
Theorem 7.13.3 ( T M ,W L , E c , C ) defines a mechanical system with constraints if and only if the l-forms e, are semibasic. Proof Let X, be the vector fields on T M given by
Then we have
where ( L is given by i t L q = dEL. Thus ( is a semispray if and only if JX, = 0 for all a, 1 5 a 5 r. Since ~ J W L= 0, we have
Chapter 7. Lagrangian systems
368
e
But i J 8 , = J*0,. Hence is a semispray if and only if J*8, = 0. Thus, since a l-form 0 on T M is semibasic if and only if J*0 = 0, we obtain the required result.
Remark 7.13.4 Let ( T M , w L , E L , C) be a mechanical system with constraints. From Theorem 7.13.3 the l-forms 8, are semibasic. Since m 1 semibasic l-forms on T M , m = dim M , are always linearly dependent, then we deduce that the number of constraints is r 5 m.
+
Let ( T M ,W L , EL,C ) be a mechanical system with constraints. Then the vector field 5 given by (7.91) is a semispray. One can easily check that the paths of satisfy the Euler-Lagrange equations with constraints:
e
where 8, = (8,)i(q,q)dqi. As we have seen, in Classical Mechanics, the phase spacce of momenta is the cotangent bundle T * M of the configuration manifold M . The bundle structure of T * M allows to define distinguished l-forms on T * M . Definition 7.13.5 A 1-form 0 on T * M i s said t o be semibasic if O ( X ) = 0 for all nM-vertical vector field X on T * M . A l - f o r m 8 on T * M is called basic if 0 = &q, where 9 i s a l - f o r m on M . Hence a l-form 8 on T*M is semibasic (resp. basic) if and only if it is locally expressed by
(resp.
0 =k(q)dd) where ( q ' , p ; ) are the induced coordinates for T * M . Next, we consider a Hamiltonian system ( T * M ,W M , H , C ) with constraints C ,where C = {Oa; 1 5 a 5 r } is a set of r linearly independent
7.13. Lagrangian and Hamiltonian mechanical
369
l-forms on T * M . As above, we may distinguish classical and non-classical constraints if the forms 0, are basic or semibasic. A first question is to seek conditions such that ( T * M ,W M , H, C) defines a mechanical system with constraints. For this we re-examine the relationship between Lagrangian and Hamiltonian formalisms. Let H : T * M R be a Hamiltonian and suppose that H is regular, i.e., the Hessian matrix
-
-
is non-singular. Then we define a map H a m : T*M --t T M as follows. Let x be a point of M and denote by H , : T,*M R the restriction of H to the fibre T,*M. If 7 E T Z M , then we have
dH,(7) : T7(T:M)
--+
R.
Since T7(T:M) may be canonically identified with T Z M , then dH,(7) is a linear map from TZM into R, i.e., an element of the dual space (TZM)* which may be canonically identified with T,M. We define
In local coordinates, we have
Since H is regular then H a m is a local diffeomorphism. Thus we can define a (local) function L on T M by
L(q',v') = (dH/ap')p'
-
H,
where ui = a H / a p ; holds. It is clear that since H is regular, then L is regular also. Actually, the Legendre transformation corresponding to L is the (local) inverse of H a m . Now, we can transport the regular Hamiltonian system with constraints ( T * M ,W M , H , C ) into the Lagrangian system with constraints ( T M ,W L , EL, C " ) ,where WL
= Leg*wM,
EL = C L - L , and C* = {O: = Leg*O,}.
Chapter 7. Lagrangian systems
370
Definition 7.13.6 We say that (T*M,W M , H , C ) defines a mechanical system wz'th constraints if ( T M ,W L , EL,C*) defines a mechanical system with constraints (as defined in Definition 7.13.2).
From Theorem 7.13.3 the Lagrangian system ( T M ,W L , E L ,C*) defines a mechanical system with constraints if and only if the l-forms 8: are semibasic. Since Leg : T M T * M is fibre preserving then 8; is semibasic on T M if and only if 8, is semibasic on T * M . Thus ( T * M , w M H , , C ) defines a mechanical system with constraints if and only if the l-forms 8, are semibasic. This shows that the essential condition to obtain mechanical systems for constrained (Lagrangian or Hamiltonian) systems is the semibasic character of the l-forms defining C . In both cases the number of constraints must be r 5 m.
-
Remark 7.13.7 Let C = { O , ; 1 5 a 5 r} be a system of constraints on the symplectic manifold ( T * M , ~ M )For . the holonomic case it is possible to show that there exists a local canonical transformation F : U c T*M T * M such that the original l-forms 8, can be transformated via F i n semibasic forms if and only if the l-forms ( 8 , ) are in involution (for a
-
proof see Jacobi's theorem in Duistermaat [50],p. 100 and the Lie's corollary in Abraham and Marsden [l],p. 419). This result is valid, under certain circumstances, for the non-holonomic case (see Weber [124]).
7.14 Euler-Lagrange equations on T * M @ T M In this section we will adopt the following concise notation. Suppose that M is an m-dimensional manifold and ( q ) = (q'), 1 5 i 5 m, are local coordinates for M ; then: ( q , u) = (q', u')
are local coordinates for T M ;
(q, p ) = ( q ' , p i ) are local coordinates for T * M ; (q, u, 6 q , 6u) = (q', u', 6qi, 6u') are local coordinates for T T M ;
( q , p , q , p )= ( q ' , p i , $ , p i ) are local coordinates for T T * M .
-
In the following W represents the Whitney sum of T * M and T M ; W = T * M @T M . A point of W is locally represented by ( q , u , p ) and :W
7.14. Euler-Lagrange equations on T * M @ T M
371
-
T * M , 7r2 : W T M are the obvious projections on the first and second factors of W . The tangent prolongations of T I and 7r2 are locally given by
Let Xz be a semispray (or second-order differential equation) on M .
Definition 7.14.1 W e will say that X2 is related to a vector field X I on T * M if for every solution (or path) u ( t ) of X2 there is a unique solution (or integral curve) r ( t ) ofX1 such that
-
In other words, X2 is related to a vector field X1 on T*M if and only if there is a Coomap p : T M T * M such that T p o X2 = X1 o p and T M o p = rM (p is fiber-preserving). The following diagram illustrates this situation :
TTM
x2
TT*M
Tp
1
'p
TM
*
1
x1
T*M
M In such a case X1 verifies the following equality:
T T M0 xi
0
p = IdTM
(7.92)
= rM then T ~ oM T p = TrM and so T T Mo T p o X2 = TrM o X2 = I d T M , since X2 is a semispray. But Tp o X2 = X1 0 p. Suppose that X2 is related to X1 and consider the graph of p, Graph p c T*M x T M . Since the diagram
In fact, as
'I~M o cp
Chapter 7. Lagrangian systems
372
-
W
I
A
M
T*MxTM
*
I
TM
x rM
M x M
-
-
where A is the diagonal mapping, is commutative, we shall consider Graph 'p c W . Then the related vector fields X2 : T M + T T M and X1 : T*M TT*M determine a vector field X : Graph 'p c W T W such that X projects by 7r1 and x2 to X1 and X2, respectively, i.e.,
Thus we have the following diagram:
T*M
X1
* TT*M
\W -TWX
x1
TM
* TTM
-
We set Graph 'p = W1 and we shall suppose that X = W1 TW1, i.e., X does not generate solutions curves leading off the (constraint) submanifold W1 (see Section 8.1.1). Suppose now that X : W + TW is an arbitrary vector field such that X projects by 7r1 to X1 and by 7r2 to a vector field on T M , i.e.,
Proposition 7.14.2 If the above X verifies
7.14. Euler-Lagrange equations on T*M @ T M
373
x = r2,
(7.95)
TITMo T r i o then [ in (7.94) is a semispray.
Proof: In fact, we have
which implies
Hence is a semispray.0 Equation (7.95) gives a sufficient condition for a vector field X on the Whitney sum W be projected by ~2 to a semispray. If X is locally given by ( q , p ,v , Sq, tip, S v ) and verifies (7.95) then a simple computation shows that 6 q = u and so T7r2 o X = ( q , u , u , b u ) is a semispray. Let AM be the Liouville form on T * M and set A@ = 'K:(XM).
Then A@ is locally given by
A@ = pidq'. Consider the function < 7 r 2 , ?rl > on W , i.e.,
< =2,a1 > (q,p,.) =< ( q , . ) , ( q , P ) '= P". A direct computation shows that
< X,A@>=<
7r2,7r1>
.
(7.96)
Let 2 E K e r T7i2 and consider the action of the Lie derivative L z on both sides of (7.96):
Chapter 7. Lagrangian systems
374
(7.97)
since A,( [Z, XI) = 0. Thus
L z < X,X@>=< for all
x,LZX@ >,
Z E K e r T7r2, and (7.97) becomes < x,LZX@ >= L z < A2,7r1 >,
from which one obtains
ixizdXe =i z d < ~
2 ~1,
since izX@= 0 and i z < T ~ , T >= I 0. Setting W@
we have
i.e.,
= -dX@,
>,
7.14. Euler-Lagrange equations on T*M @ T M
375
-
Theorem 7.14.3 (Skinner [IIO]) If we suppose that X = -d(lr$L) = - d ( L o T Z ) , where L : T M R i s a Lagrangian function, t h e n ixwe =d < ~
2 T,I
> -d(r;L)
= d ( < ~ 2 T,I >
-T;L >,
and the projections o n M of the integral curves of X are t h e Euler-Lagrange equations f o r L.
Proof It remains t o prove the second assertion. As X verifies (7.95), then X is locally given by
x = ( q ,P ,
0,
v , 6P, 6
4
Hence
i x w e = i x ( d q A dp) = ( i x d q ) d p- (iXdp)dq = -(6p)dq As <
7r2,nl
+ vdp.
>= p v , we have d ( < n2, T I > = pdv
-
-
= d ( p v ) - T,*(dL)
aL + v d p - -dq aq
dL -dq dq
aL
-dv
-
av
dL + v d p + ( p - -)dv. aV
so, 6p=
dL
-,
aq
dL p = -.
av
(7.98)
Now, if ( q ( t ) , p ( t ) , v ( t ) ) is an integral curve of X, then 6q = v = q = ( d / d ) ( q ( t ) )and 6 p = p = ( d / d t ) ( p ( t ) ) . Thus (7.98) implies
which are the Euler-Lagrange equations for L. 0 It is clear that the converse is also true, i.e., the solutions of the EulerLagrange equations for L are the solutions of the vector field X. So the intrinsical expression i x z w = ~ d E L of the Euler-Lagrange equations is equivalent t o the above i x w e = d D , where D : W R is the function defined by D =< ~ 2 T,I > -n;L. We finish this section by remarking that no hypothesis on the regularity of L was made, i.e., the above procedure may be adopted for regular (or hyperregular) Lagrangians as well as degenerate Lagrangians. We will return to this topic in Section 8.1.1.
-
Chapter 7. Lagrangian systems
3 76
7.15
More about semisprays
One of the most interesting results about the Hamiltonian formulation of Classical Mechanics is that vector fields and one-forms on the phase space are related by an isomorphism induced by the canonical symplectic form on the cotangent bundle of the configuration manifold. For the Lagrangian approach however this correspondence depends directly on the Lagrangian function, that is, for each regular Lagrangian we may define a symplectic form and so the induced isomorphism depends on this choice. Sarlet et al. [lo91 proposed a geometric study relating semisprays and one-forms for which it is possible to generate Lagrangian systems. The method is inspired on the definition of gradient vector fields and associates to every semispray on the tangent bundle a set of one-forms defined on such bundle. We start Sarlet et al. study by considering the conservative case. We consider a special set of one-forms A; defined by A: = {CZE
r \ ' ( T M ) / ( L [ o J * ) a = a},
where ( is a semispray on T M , A ' ( T M ) is the space of all l-forms on T M and J* is the adjoint endomorphism on A ' ( T M ) induced by the canonical almost tangent structure J on T M . We will see that for an appropiate choice of CY E A: we may obtain the Euler-Lagrange equations of motion in its intrinsical form. Let X be a vector field on TM. Then Ex is a R-linear operator on l-forms on T M defined by
Ex
= Id - LX 0 J*.
Thus A;
= Ker
Ex.
This set is in fact a vector space over R , by the linearity of E x . Moreover, we have
Therefore A; is not a module over the ring of real functions but it is a module over the ring of real functions satisfying X f = 0 (the constants of motion).
7.15. More about semisprays
377
Next we shall compute the local expressions of E x and A k . Suppose that X is locally expressed by
x = xia/aqi+ xia/avi, where
( 8 ,v ' )
are induced coordinates for TM. If we set
one easily deduces
ax] + (Yj(8! - -)dv'
axj
E x a = (a' - X ( 6 ' ) - &.-)dQ' dq'
3V'
(6: = Kronecker's delta). Hence, the elements of
A;
are locally of the form (7.99)
Proposition 7.15.1 Let R be a tensor field of type (1,l) on T M such that RJ = J R and J ( L x R ) = 0 . Then
R*o Ex = Ex
0
R*
Proof: Let a E A'(TM) and Y E x ( T M ) . Then
( R * ( E x a ) ) ( Y= ) ( E x a ) ( R Y )= a ( R Y ) - ( L x ( J * a ) ) ( R Y )
+
= a ( R Y ) - X ( a ( J R Y ) ) a ( J [ X ,R Y ] )
+a(RJ[X,Y]),
= a ( R Y )- X ( a ( R J Y ) )
since RJ = J R and J ( L x I 2 ) = 0. Thus
(R*( E x a ) )( Y ) = ( E x (R * a ) )( Y ). n Corollary 7.15.2 In order that a tensor field R of type (1,l) on T M preserves A; it is sufficient that R commutes with J and satisfies J ( L x R ) = 0 .
Chapter 7. Lagrangian systems
378
An example of a tensor field of type ( 1 , l ) on T M satisfying the hypothesis of Proposition 7.15.1 is obtained by using the theory of lifts of tensor as the complete lift of a fields studied in Chapter 2. If we take R = tensor field & on M then we have
where
4 is the vertical lift of &. Also, for a semispray < one has
= J [ € ,(&Y)7 - R : [ < , y c l= 0,
since [ ( , Y c ]is a vertical vector field for all Y E x ( M ) .
Proposition 7.15.3 Let Y be a vector field on T M . Then the tensor field
verifies the hypothesis of Proposition 7.13.1. The proof is obtained from a straightforward computation. We note that Ryw = RYC= 0 and Rt = I d for any semispray on T M . Now, suppose that X = is a semispray on T M . We call EE by Lagrange operator. Using the local expression of <,
<
<
we see that
E t a = (a;- <(&;))dq'. Therefore the elements of A i are locally characterized by
Moreover, from the above expression for Et we deduce that E p is a semibasic form, that is, J * ( E e a )= 0.
7.15. More about semisprays Definition 7.15.4 Let a E
379 T h e n a is said t o be regular i f the &-form
A:.
i s symplectic. A semispray i s called a regular Lagrangian vector field if there i s a regular f o r m a E A; (called the associated f o r m of 5) which is exact, i.e., a = d L f o r s o m e CM f u n c t i o n L : T M R.
-
Proposition 7.15.5 Let ( be a regular Lagrangian vector field. T h e n E&) = 0 i s the intrinsical f o r m of the Euler-Lagrange equations of motions, where a i s the associated regular f o r m of 5.
Proof E p = 0 implies a!
= Lt(J*a),
that is,
d L = L,c(J*dL)
LfdjL
Thus
i t d d J L = d ( L - i e d J L ) = -dEL, where EL = i t d J L - L = C L - L (since J E = C). Then the intrinsical form of the Euler-Lagrange equation is
where W L is defined by W L = - d d J L . O Since ( is supposed to be regular Lagrangian the form W L is symplectic and so L is a regular Lagrangian (of course the converse is also true). Now, we shall introduce a kind of dual of A;. We define a subset of X ( T M ) by
Chapter 7. Lagrangian systems
3 80
+
Then xt is a real vector space. As J [ ( , f Y ] = ((f) JY fJ[(Y , ] we have that xt is not a Cm(TM)-module. However it is a module over the ring of constants of motion. Locally the elements of xt have the form
Y = yia/aqi+ ( ( Y i ) a / a v i . Let us give a geometrical interpretation of xt. If Y E xe then J[Y,(1 = 0 , i.e., the Lie derivative L y ( of ( in the “direction” Y is a vertical vector field. Now, for every point x one has
( L Y € ) z = lim(l/t)[€(x) - (TPt(€))(41> where pt is the 1-parameter group generated by Y . Thus we may say that, at least to first order in t , (Tpt)( differs from ( by a vertical vector field. As ( is a semispray, J ( = C ,then, again at least t o first order, (Tpt)( is a semispray.
Definition 7.15.6 xt is called the set of all variation vector fields of the semispray (. This definition is suggested by the fact that the 1-parameter group pt is a variation from a point z to a point pt(x). A complete lift of any vector field X on M belongs t o x~ for any semispray ( on TM. In fact, there is a unique Y such that JY = X u which is Xc.As [Xc, (1 is vertical for any (, J [ X c ,(1 = 0 . Moreover
Proposition 7.15.7 (Crampin 1.211). Let ( be a semispray o n T M . If V is a vertical vector field on T M then there is a unique vector field i n xt, denoted by Vt, such that JVt = V .
-
Proof: It is sufficient to show that the linear mapping J : Y E X E JY E V ( T M )is bijective. It is injective. Suppose that JY = 0. Then Y is vertical. Consider the connection r = - L t J defined by (. Then, if h is the horizontal projector of I’, we have 1
O = hY = -(Y 2
-
1
( L ( J ) Y )= s ( Y - [(,J Y ]
which implies
Y = [(, JY] - J [ ( , Y ]= 0.
+J[(,Y])
7.15. More about semisprays
381
It is surjective. Let Y be a vertical vector field and define
and so Yc E xe.0 Let us consider the pairing < Y ,J * a >. Then
Le < Y , J * a >=< L e y , J*CU>
+ < Y ,L e ( J * a ) >
=< J ( L [ Y ) , a > + < Y , L e ( J * a ) > Thus, if Y E
xe and a E A:
we have
L , < Y ,J*CY>=< Y , a >
.O
The following proposition shows a kind of duality between
(7.100)
xc and A:.
Proposition 7.15.8 A vector field Y (resp. 1-form a) o n T M belongs to xf (resp. A' <) af a n d only if (7.100) holds for all a E A: (resp. Y E xe). Proof In fact, if Y E x~ and a E A:, then (7.100) holds. Now, let Y E x ( T M ) (resp. a E A1(TM)) such that (7.100) holds for all a E A: (resp. for all Y E xc). Then we have
< Y,(Y>=< J ( L [ Y ) , a > (resp.
+ < Y,a>
Chapter 7. Lagrangian systems
382
< Y,a >=< Y,L t ( J * a ) >) which implies J ( L t Y ) = J [ ( , Y ]= 0 (resp.
L t ( J * a ) = a).o Proposition 7.15.9 Let R be a tensor field of type ( I l l ) satisfying the hypothesis of Proposition 7.15.1. Then for all a E Af and Y E x t we have
< R Y , a >= LE(< RY, J * a >). Also,
R y = ( L y J ) o (LEJ).
Proof:
< RY,a >=< Y ,R*a >= Lt < Y ,J*R*a > (since R*a E A:) =
Lt < Y , R * J * a >= LE < R Y , J * a >
I f Y E xt then J [ c , Y ]= 0 and
.
so
Proposition 7.15.10 (“Conservation of energy”). Let semispray on T M . For each a E Af we have i t ( a - d < C , a >) = 0. Therefore, if a = d L , we have i t d E L = 0 .
be an arbitrary
383
7.15. More about semisprays
Proof: L t ( J * a ) = CY
a i e d J * a + d < (, J * a >= cx
uiedJ*a = a - d < J ( , a > i t d J * a = (Y - d < C,CU > Thus
- d < C , a >) = 0. If a = d L then a - d < C,Q >= - d E L , since
EL = C L - L = d L ( C ) - L =< C,Q> -L.O Proposition 7.15.11 (agauge freedom"). Let f be a Coofunction on M and F a semispray on TM.If a E A: then d =a
+ d f e E A;,
where f c i s the complete lift o f f to TM.
Proof In fact, J * ( d f e ) = df". Therefore
Et(a
+ df ') = Eta + E t ( d f c )
= E t ( d f c ) = dfc - L t ( J * ( d f c ) )
= dfe - L t d f v = 0)
since L t ( d f " ) = d ( L 6 f " ) = d f C . o If Q = d L in Proposition 7.15.11, then one has the well-known gaugefreedom in Lagrangian mechanics (see Exercise 7.17.1). Now, suppose that Y is a Lie symmetry of i.e., [Y, (1 = 0 and Y = Z c , for some vector field 2 on M. Then
c,
LyJ=O, L y o J = J o L y .
Chapter 7. Lagrangian systems
384
Proposition 7.15.12 Let Y be a Lie symmetry o f f . Then
Ly
0
Et = E ~ Loy
Proof: L y ( E t c ~= ) L y a - LyLt(J*a)
(1
= L y a - L e L y ( J * a ) (since [Y, = 0 )
= L y c ~- L e J * ( L y a )
Corollary 7.15.13 The Lie derivative with respect to a Lie symmetry Y of f preserves A:, i.e., L ~ ( A c~ A:.)
Now, let Y be a dynamical symmetry of Proposition 7.15.14 If isfies
Y is
a
(, i.e., [Y, €1 = 0.
dynamical symmetry of
J*R$cr = J*df, for some function f o n T M , then a' E
A;,
where
a' = LycY
-
d(
Proof: We have
J*o' = J * ( L y a ) - J * d ( € f ) = L Y ( J * c Y) (LyJ)*a- J*d((f),
since
and a E
A:
sat-
7.15. More about semisprays
385
(LyJ)* = L y o J* - J* o L y . Therefore
L t ( J * f f ' )= L&(J*a)
-
Le(LyJ)*a - L ( J * ( d ( ( f ) )
(7.101)
R*ya= Lt(J*R*ya)= L t ( J * d f ) As Ry = ( L y J ) o ( L t J ) we deduce
( L e J ) * ( L y J ) * a= ( L ( J ) * d f
+ J*L&f).
Operating on both sides with (LeJ)* we see that
( L y J ) * a = df
-
J*d(Ef).
Therefore, taking again the Lie derivative of the above expression with respect to E , we obtain
L e ( ( L y J ) * a )= 4Cf) - Le(J*d(Cf)). F'rom this we see that (7.101) reduces t o
L((J*cY') = L y a
-
d({f) = aI .
Thus a' E A i . 0
Remark 7.15.15 We have a converse of Proposition 7.15.14. In fact, if Y E x t and if there is a regular element a E A: such that J*R$a = J * d f , for some f : TM R with a' E A: then Y is a dynamical symmetry of C.
-
Chapter 7. Lagrangian systems
386
Finally, the following theorem gives a generalization of Noether’s theorem.
Theorem 7.15.16 Let a be a regular element of A:. If a vector field Y o n
TM satisfies (1) L y ( J * a ) = d f , f o r s o m e f : T M R (2) i y ( a - d < C,CY >) = 0, then g = f - < Y ,J * a > i s a first integral of (. Conversely, t o each first integral g of ( there corresponds a vector field Y o n T M satisfying (1) and (2). I n particular, i f a = d L then [Y,c] = 0 and so we have Noether’s theorem f o r Lagrangian mechanics. -----)
Proof: From (1) we obtain i y d ( J * a ) = df
-
d i y ( J * a ) = d( f
-
< Y,J*CY>) = dg.
But a E A: and so
ifd(J*a)=
-
d < C,a >
.
Consequently
i.e.,
Conversely, as d ( J * a ) is symplectic, each first integral g of unique vector field Y on T M satisfying
i y d ( J * a ) = dg. If we put f = g+ < Y,J*a >, then
Ly(J*a) = iyd(J*a) = dg
+diy(J*a)
+ d < Y ,J * a >= d f .
c defines a
7.15. More about semisprays
387
Furthermore, since Leg = 0, we obtain
0 = L4g = i t d g = i c i y d ( J * a )
Finally,
i [ y , e ] d ( J * a= ) Lyitd(J*a) = iyda. If a = d L , then
i [ y , ( ] d J * ( d L= ) i [ , [ ] d d J L= 0 and, as d d J L is symplectic, we obtain [Y, <] = 0. Then, if Y = X c for some vector field X on M , we deduce that X is a Noether symmetry of and we reobtain Proposition 7.12.6. We examine now the non-conservative case, i.e., mechanical systems ( M ,F, p ) where p is not a closed semibasic form. As the results obtained are similar to the conservative case we will only examine the main modifications. Let E be a semispray on T M . We associate to a R-bilinear operator
<
<
defined by
E [ , J ( Q , P )= E p + J*P. Now, let us put A t ,J =
Ker E ( J ,
i.e., A i , J = { ( c Y , ~E ) A ' ( T M ) X A'(TM)/EfQ=
The elements of
are locally of the form Q
= (<(&;)
+ P;)d*' + did??,
-J*p}.
Chapter 7. Lagrangian systems
388
We note that
E t , J ( f Q ,fp) =
(cf)(J*Q)+ f E t , J ( Q , P ) .
Hence is a real vector space but is not a module over the ring of the constants of motion. Next we shall relate A: and A : , J . Let j : A1(TM) A'(TM) x A 1 ( T M ) ,j ( a ) = (a,O)be the canonical injection. Then Et,J o j = Et and so j ( A i ) c A ; , J .
-
Definition 7.15.17 A pair of I-forms (a, p) is said to be regular if (a,,B) E and the 2-form wtr = - d ( J * a ) is symplectic. The f o r m p i s called the force field. A semispray c is called a non-conservative regular Lagrangian vector field if there is a regular pair ( a , p ) E such that Q i s ezact, say a = d F . The condition E t ( d F ) = - J * p which characterizes p may be rewritten in the form i(WF
def
= d E F ip (WF =
WdF),
where p = - J * p is a semibasic form on T M . We deduce
dF
aF
(z-) = -p;,
1 5 i s m = dim M ,
which yields a system of Euler-Lagrange equations with non-conservative forces. Similar results presented above may be re-obtained. For instance, it is easy to see that "If R is a tensor field of type (1,l) on T M satisfying J R = R J and J ( L t R ) = 0, then
R* o E ~ , = J E ~ , oJ (R* x R*) and so R* x R* preserves A:, J n . We may also introduce a kind of dual of A : , J . Let xt be the subset of x ( T M ) defined as above. If we define < Y,(a,p)>= (< 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(
Proof From the above definition of ar,p' we have
J*ar = J * ( L y a ) - J*d((f) = L y ( J * a ) - ( L y J ) * a - J*d(
L&*a')
= Lya
= L & y ( J * a ) ) - L t ( ( L y J ) * a )- L((J*d(€f))
+ L y ( J * p ) - L c ( ( L y J ) * a )- L&*d((f)).
On the other hand, (Fya,Fyp)E
(7.102)
then we have
RFa = L,t(J*R$a)- J*R;/3 = L.t(J*df) - J*(dg). As the vector field Y is a dynamical symmetry of (, then R y = ( L y J )o ( L t J ) . Thus
RGa = (LtJ)*df
+ J * ( L t ( d f ) )- J * ( d g ) .
Operating on both sides with ( L t J ) * ,we see that
(LyJ)*cY = df
-
J*d((f)
+ J*(dg).
Now, taking the Lie derivative of (7.103) with respecto to
(7.103)
6, we obtain
Chapter 7. Lagrangian systems
390
L[((LYJ)*cr)= d ( U )
+ Lt(J*d(€f))+ L t J * ( d g )
(7.104)
With (7.104), the relation (7.102) reduces to
L&*cr') = Lycr
+ L y ( J * / ? )- d ( E f ) + L t ( J * d g )
Thus a / -Lt (J*a') = - L y ( J * P )
+ Lt(J*dg)
( L t J ) * ( L y J ) * P= dg. Now, operating again on both sides with ( L t J ) * ,we deduce
(LYJ)*P= (LtJ)*dg. Hence ar -
L((J*cr') = - J * ( L y / ? - d ( € g ) )
or, equivalently,
Eta' = -J*p', and so ( a r , / ?E' )A i , J . U
Proposition 7.15.19 (Noether's Theorem). Let (a, p) be a regular element of A ; , ~ . IfY E ~ ( T Msatisfies ) ( I ) L y ( J * a ) = df, for some f E C m ( T M ) , (2) iy(a - d < C,CX > +J*P) = 0, then F = f - < Y,J*a > is a first integral of <. Conversely: to each first integral F of ( there corresponds a vector field Y satisfying (1) and (2). Proof The same as for the conservative case.U
7.16. Generalized Caplygin systems
7.16
Generalized Caplygin systems
-
Let P ( M ,G) be a principal bundle with projection a : P R is a G-invariant Lagrangian, i.e., that L : T P
391
-
M and suppose
L o T R , = L for all a E G. We impose as constraints on T P the horizontal subspaces determined by a connection I' in P . This means that only horizontal paths are allowed. We call ( P ( M , G ) ,L , r ) a generalized Caplygin system (see Koiller [82]). Given a generalized Caplygin system ( P ( M ,G), L , I?) we can define a Lagrangian L* : T M -+ R as follows:
where (X,)," is the horizontal lift of X , at p , ~ ( p=) q , with respect to the connection r. We notice that L* is well-defined since L is G-invariant and T R 4 ( X H= ) X H ,for any vector field X on M and for all a E G. Next, we shall obtain the Euler-Lagrange equations for L*. First, we choose local coordinates for P. For each point 2 of M there exist a coordinate neighborhood (U,q'), 1 5 i 5 m = dim M , of z and a diffeomorphism t+h : a-'(V) U x G. We need to introduce local coordiG be the exponential map from the Lie algebra nates for G. Let exp : g g of G into G. As we know exp gives a diffeomorphism of a neighborhood @ of 0 in g onto a neighborhood W of e in G. Thus, if we choose a basis { e , ; 1 5 r 5 n = dim G} of g we define local coordinates in W as follows: the point exp(C:,l a r e , ) has local coordinates (rr).For an arbitrary point a E G , we consider the coordinate neighborhood !,(W). These coordinates (rr)are called normal coordinates. So we obtain local coordinates ( q a , )'a in P . We denote by a : U P the local section over U given by a(q) = t+h-'(q,e). In local coordinates we have
--
Now, let w be the connection form of w . Then w = w'e,, where w' are l-forms on P . We have u*w = ( u * W r ) e r ,
where u*w' are l-forms on U . Thus
Chapter 7. Lagrangian systems
392
u*w r -- wrdq', 1 5 r
5
n.
(7.105)
Let X be a vector field on M and suppose that X is locally given by X = ba/aq' on U .Then the horizontal lift X H of X to P is given by
(xH)u(q) = d ' ( a / a ' l i ) u ( q ) + A'(a/arr)u(q)) since T r ( X H )= X . Now, from (7.105), we obtain
= [q'w;
+ Ar]er, (since (a/arr)u(q) = (Aer)u(q))
which implies
= qi(a/aqi)u(q)a - q'w:(TRa)(a/ar')u(q).
Now, in local coordinates we have R,(q', r') = ( Q i )q , where, if b = exp(r'e,), then ?i' = ~ ' ( a b )Hence . we obtain
or, for sake of simplicity
xH = 4'- a
a Q'
-
Qiw;(aiia/arr)a/aa8.
(7.107)
7.16. Generalized Caplygin systems
393
From the Campbell-Haussdorff formula (see Exercise 1.17.8) we have ii' = A'
+ r' + -21C ~ u ~ ' r+u terms of order
2 3,
where a = exp(r'e,.) and C& are the structure constants of G with respect to the basis {e,.}. Hence
and then
Thus we have
= L(q', q', -qj(w,"
1 + -w;C;"r")), 2
since L does not depends on 7'. An easy computation shows that
Thus d dt
-
(-)dL* dq'
-
(-)dqi
dL* d dL -= dq'
dt
dL
d dt
- -- -
dqi
(-)&isd L
(w;
1 + -w;C;"r") 2
(7.108)
Chapter 7. Lagrangian systems
394
If we evaluate the right-hand of (7.108) at the point a(q), i.e., = 0, i v= - q . j wvj ,
fV
then we obtain d dt
-
(-)a8 8~ *
-
awp
aL* dq’ =
dq’
1 + -c;,w;w; 2
aw; aqi
1
,
(7.109)
since the Euler-Lagrange equations for L are
If a solution q ( t ) is known, then we recover the dynamics in P by considering the horizontal lift of the curve q ( t ) in M t o P . We remark that (7.109) generalizes Caplygin’s formula by the appearance of terms involving the structure constants of g. In fact, if G is Abelian then (7.109) becomes
(see Neimark and Fufaev [99], and Koiller [82]). Next we examine the geometrical meaning of the right-hand of (7.109). Let us recall the structure equation for I?: dw
1 --[w,w]
2
+ $2,
(7.110)
where f l is the curvature form of I?. If
a*n = u*(nrer)=
A dq’,
then (7.110) becomes
aw;
awa1 --
aq’
aqi
1 --c;vw;wy 2
+ qj
and then ’
dL
(7.111)
7.17. Exercises
395
A simple computation shows that the right-hand of (7.111) does not depends on the choice of the coordinate neighborhood (U, q') and the section U . Thus there exists a l-form a on T M locally given by
Then (7.1 11) are the Euler-Lagrange equations for a mechanical system ( M ,L*, a ) . We notice that this result was obtained by Koiller [82] by using Hamel's formalism of '(quasi-coordinates" . We remit to the paper of Koiler for more details and examples.
7.17 Exercises 7.17.1 Let L I and Lz be two regular Lagrangians on T M and
e ~ EL^~ the,
corresponding Euler-Lagrange vector fields. Prove that the following two assertions are equivalent: (1) L1 = L2+&+ constant, where a is a closed l-form on M and & : T M R is the map defined by &(X,)=< X , , a ( z ) >, X, E T,M. (2) hl = €Lz and WLI = W L Z . 7.17.2 Let L : TM R be a regular Lagrangian. Prove that I m J is a Lagrangian subbundle of the symplectic vector bundle ( T M ,u ~ ) . 7.17.3 Let Q be an m-dimensional differentiable manifold and TQ its tangent bundle. If (q', u', Sq', So') are induced coordinates for TTQ then we define a diffeomorphism UQ : TTQ -+ TTQ locally given by
-
-
-
(i) Prove that UQ is, in fact, a global diffeomorphism such that 0: = I d . (ii) Prove that UQ : (TTQ,TTQ, T Q ) (TTQ,T T QTQ) , is a vector bundle isomorphism. UQ is called the canonical involution of TTQ.
7.17.4 The Helmholtz conditions for the non-autonomous case (Crampin et al. [23]). The Helmholtz conditions are the conditions that must be satisfied by a regular matrix g ; i ( t , q , q ) in order that a gives second- order ordinary differential equation = f'(t, q , q ) when written in the form q;,q'
+ h; = 0
(h; = - g ; , f J )
Chapter 7. Lagrangian systems
396
becomes the Euler-Lagrange equations for the Lagrangian L = L ( t ,q , q ) . Putting
where 7 is a semispray given by
7 = a / a t + $(a/&$)
+ fi(a/a,ji).
The necessary and sufficient conditions for the existence of a Lagrangian for the equations g ; j q J hi = 0 are that the functions g;, should satisfy the following Helmholtz conditions:
+
gjkd?
= gjktpf;
agij/aqk
= agik/a,jj.
Proceeding along the same lines as the autonomous case show that the geometrical approach of Section 7.5 for the time-dependent case may be applied for the Helmholtz conditions, that is, that there is a 2-form with suitable properties which are exactly equivalently to the Helmholtz conditions. 7.17.5 Show that A&,, = A\3uc = A& = 0, where Xu, X c and C are, respectively, the vertical, complete lift of X and the Liouville vector field. 7.17.6 Show that the local expression of a tensor field R which preserves A: is
+E(af)(a/au') 8 d q j ,
where a; = a ; ( q , u ) . 7.17.7 Prove that xc contains the complete lifts of arbitrary vector fields on
M. 7.17.8 Let G be a Lie group with Lie algebra g and { e l , . . . ,e,} a basis for g . Consider normal coordinates ( r i ) on G, i.e., if a = exp(Cin,l.lr'ei),
7.17. Exercises
397
then a has coordinates (x'). Suppose that a = expaie,, b = expbie;, i.e., . . .'(a) = a', x'(b) = b'. Then prove that Ir'(ab) = ai
1 + b' + -cjkaibk + terms of degree 2
2 3, (*)
where c j k are the structure constants with respect to the basis {e;} ((*) is called the Campbell-Hauadorff formula) (Hint: use Exercise 1.22.24).=
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399
Chapter 8
Presymplectic mechanical systems 8.1
The first-order problem and the Hamiltonian formalism
Supose that ( S , w ) is an arbitrary symplectic manifold. Then the linear mapping S, : x(S) -+ A ' s is an isomorphism. Hence, for every l-form field X , on S such that IX,W
-
Q
on S there is a unique vector
= Q.
For a presymplectic manifold (S,w ) we may also consider the linear mapping s, : x(S) A'S, and a vector bundle homomorphism, also denoted by S,, from T S to T*S. But now S, is not necessarily an isomorphism. So, if Q is a l-form on S , a condition for solving ixw = Q is that Q E Irn S,. We call this condition by the range condition. As Ker S, is not trivial (w is presymplectic), the range condition says that a solution is obtained modulo Ker S,. In fact, Ker S, is an obstruction to solve uniquely the equation. In the following we will resume some results of the presymplectic formulation. For further details the reader is invited to see two different viewpoints on the subject. One is due t o Gotay et al. [69] where the rank of the presymplectic form w is constant everywhere. The other is an approach where the
Chapter 8. Presymplectic mechanical systems
400
rank of w has a continous variation (Pnevmatikos [104]). The presymplectic situation occurs also when a manifold K is embedded into a symplectic manifold (S,w). If p : K S is the embedding, then in general the pair ( K , U ) ,where = p*w, is not a symplectic manifold. Supposing that rank of U is constant along K then ( K , U ) is a presymplectic manifold which we will call constraint manifold. We may have this situation in the Hamiltonian case where a canonical symplectic form is defined on the cotangent bundle of a given configuration manifold. In the Lagrangian case we may have the presymplectic or symplectic situation if the Lagrangian is degenerate or not. Throughout this chapter all submanifolds will be considered
-
as embedded submanifolds. 8.1.1
The presymplectic constraint algorithm
Suppose that (P1,wl)is a presymplectic manifold and a1 is a given 1-form on PI. The first-order problem is the question of finding a vector field X on PI such that i x w l = a1. More clearly, if a E Im S,, then for every point z E PI there is a tangent vector X a t z such that i x , w = a1 a t z. But in general there are only some points z E PI for which a ~ ( z ) is in Irn S,, (otherwise the problem is insolvable). So if we assume that the set of such points form a manifold P2, embedded into PI by a mapping p2 : P2 PI (and identifying P2 with p2(P2),TP2 with ( T p z ) ( T P ? )then ) we may consider the equation on P2 given by
-
As such equation possesses solutions we must demand that the integral curves be constrained to lie in P2, i.e., the vector field X must be tangent to P2 at every point z E P2 (otherwise the system will try to evolve off P2). This requirement will not necessarily be satisfied and so we consider a third submanifold P3 of P2 defined by
It is clear that the repetition of such arguments generates a sequence of manifolds
...Pi+l where
-- --P;
Pi-1
...
P2
8.1. The first-order problem and the Hamiltonian formalism
40 1
where, for simplicity, we have put 0 1 = 0... o pO;(w1). We call this process geometric algorithm of presymplectic systems, due to Gotay et al. [69].We call P, the r-ary constraint manifold. The process gives the following possibilities: 1. The algorithm produces an integer r such that
Pr
2. The algorithm produces an integer r such that P,
= 4,
# 4 but dim P, = 0.
3. There is an integer r such that Pr = Pt for all t 2 r and dim Pr
# 0.
4. The process is not finite.
The first case means that the dynamical equations have no solutions. The second case says that the constraint set consists of points and the unique solution is X = 0 (there is no dynamics). The third case gives a final constraint manifold in the sense that if N is any other submanifold along which the dynamical equations are verified then N c P,.. The last case says that the system has an infinite degree of freedom. In such a case we may stipulate that if P, is the intersection of all P, then we may reproduce one of the other possibilities.
Definition 8.1.1 Let (W,w) be a symplectic manifold, N c W a submanifold of W.W e say that N i s a manifold of regular consistent constraints if for every 1-form a o n W the equation i x w = cr when restricted t o N admits a solution X tangent to N . Thus the major interest in the study of the presymplectic systems is t o obtain a final submanifold defined by regular consistent constraints. So, a t least one solution X to the original equation restricted to the final constraint manifold P, exists and this solution is tangent t o P, at every point. This solution X is not unique for we can add to it any element of Ker Swl.
Remark 8.1.2 We may also re-characterize the definition of the constraint manifold. From the initial equation of motion we have
Chapter 8. Presymplectic mechanical systems
402
for all Y E x(P1). Thus iycq = 0 if and only if iyixwl = 0. For the above sequence of manifolds if we consider the orthocomplement with respect to W1
T&,
= {Y E T,Pl/iyiXwl(z) = 0 for all X E TZPi-1)
then we define
(where we have omitted the use of the embeddings p i , p ; - 1 , etc ...). The process is then constructed as follows: if X ,solution of the original equation, is tangent to P2, then for every vector field Y on PI
= -ix(cpi(iywi))- ( P ; ( ~ Y Q I ) ,
where 2 is such that T p ~ ( 2=) X . Thus, if Y is such that &(iywl) = 0, that is, Y E TP;, then p$(iycwl)= 0. This may not always be the case and so we must restrict our attention to these points of P2 where i y a l = 0, i.e., P3
= { z E P2/(p2 0 (03)*(iy(Yi)= 0, for all Y E T p 2 ( T z P i ) } ,
and so on. To conclude this section let us summarize the approach proposed by Skinner and Rusk [ 11 11, [ 1121 for degenerate Lagrangian systems. We have seen before in Section 7.14, that the Euler-Lagrange equation
ixwL = dEL is equivalent to a first-order equation
iyw@= dD (8.2) on the Whitney sum W = T*M $ T M (Skinner’s theorem). This theorem is valid even if the Lagrangian considered L : T M R is degenerate. In such
-
8.1. The first-order problem and the Harniltonian formalism
403
a case Eq.(8.2) is a presymplectic equation and we cannot say that there is Y globally defined on W . Therefore we may reproduce the procedure developed before for presymplectic systems. Let us first recall that the semibasic form CYL on T M defines a CM map Leg : T M T * M (which is not a diffeomorphism as L is degenerate) locally given by a (unique) solution
-
--
If we take p = Leg as it was considered in Section 7.14 and as Graph p C W = T * M @ T M ,we have a diffeomorphism 8 : T M Graph p c W . Now, let V"' be the vertical bundle with respect to ~1 : W T * M , i.e., V"' = Ker Tlrl. Then, if 2 E V"', izw@ = 0 (locally, 2 = (6v)a/au and we = dq A dp). But i z d D does not necessarily vanishes. Therefore we may define a manifold W1 by
W 1 = { z E W / i Z d D = O for all Z E
-
For the points z E W1, the first-order equation can be solved. Supposing that W1 is embedded into W by a mapping iw, : W1 W and setting tw,("1) = W1, one has W1 = Graph p c W and
-
To find a consistent solution of (8.1)) i.e., a final constraint submanifold T S ) we apply the above geometric algorithm. For instance, W , c W is the submanifold obtained at the r-th step, i.e., from S for which the solution X is a vector field on S (X : S
T W L , = { X E T W / w e ( X , Y )= 0 for all Y E TWr-1) we define
and we set W = Wo. Then, if the algorithm works, we find a final constraint manifold S on which is defined a solution of (8.1). For further details see Skinner and Rusk [lll],[112].
404
8.1.2
Chapter 8. Presymplectic mechanical systems
Relation to the Dirac-Bergmann theory of constraints
Around 1950 Dirac [47], [48] proposed a Hamiltonian theory for Lagrangian systems which cannot be directly put into this form (see also Bergmann [ S ] ) . This means that the condition on the non-singularity of the Lagrangian function with respect to the velocities breaks down. Taking into account the Implicit (or Inverse) function theorem we may say that in order to obtain an unconstrained Hamiltonian formalism in Classical Mechanics we need all momentum variables to be independent of the velocity variables. If we drop the non-singularity condition then there are a certain number of momenta which are not entirely independent of the velocities (of course we may develop a Hamiltonian formalism including these momenta but this is against to the usual procedure, see the Appendix A for further details in local coordinates). Therefore we need to restrict our attention to a certain subset K of the phase space manifold for which the momenta are, as in the standard theory, independent of the velocities (we may think that this corresponds to the regular part of the theory). This subset is taken as a C" manifold and is called the constraint manifold. Geometrically, Dirac-Bergmann constraint theory may be viewed as follows. Suppose that (W,w ) is an arbitrary symplectic manifold and K is a manifold embedded in W by a mapping 'p : K -+ W . Set 6 = 'p*w. Let h be a C" function on K . We want t o know: what are the conditions for h to be extended into a Coofunction H on W (at least locally) such that the corresponding Hamiltonian vector field XH is tangent t o K? We may also ask, if a Hamiltonian H is initially given on W , under what conditions the motion equation i x w = d H restricted to K (i.e., i x w = d('p*H)) admits a solution X tangent to K at every point? We will examine here only the first case: the extension of h to H . Recall that if H is an extension of h to all W (or on an open neighborhood) then from the isomorphism s,,,one obtains a unique solution XH solving i x w = d H , but such XH may not be tangent to K (and so XH is not an ordinary differential equation along K ) . The restriction of the dynamical equations to K does not assure us that there is a solution of the problem. We must have the above range condition verified: d('p*H) E S a ( T K ) . In such a case we cannot assure the unicity of XH. Let us suppose that a symplectic manifold ( W , w ) is initially given with a submanifold P such that ( P , 6 ) is presymplectic. Then after applying the algorithm method we assume that a final constraint manifold K in (W,w) is obtained (of the above third type) and a C" function h is defined on it. In
8.1. The first-order problem and the Hamiltonian formalism
405
fact, t o simplify things, we suppose that K is a primary constraint manifold Let ( z , y ) be a coordinate system defined on a neighborhood V c W of , with ~ vdim some point z E K such that y = y b = (y'+', . . . ,~ ~ " ) / o ,0 ~ K = r and dim W = 2n. We extend h to V by
=
H = h + u b yb, with
Ub
(8.3)
being arbitrary multipliers to be determined. We have the equation
extension of
ixu = dh.
(8.5)
Equation (8.4) has of course a solution X H defined for all points in V and in particular for those in U = K n V . But X H may not be tangent t o U . As we have supposed that dh satisfies the range condition ( d h E S a ( T U ) ) our main problem is the tangency question for this first-order problem (i.e., the consistency of the constraints). We also suppose that the extension H is Coo on V. Let z be a point in K and consider the orthocomplement T,K' of T,K in (T,W, w,):
T,K'
= { X E T,W/ for all Y E T,K, ( i x w ) ( z ) ( Y )= 0).
Locally the symplectic complement T,K' is generated by the family { X y b }of Hamiltonian vector fields corresponding t o { d y b } (see Lemma 6.4.1). In the following we represent xyb by Xb, r 1 5 b 5 2n. w e consider the following set
+
K(W,w) = { F E C " ( W ) / X F ( Z )E T,K for all z E K } , where X F is such that i x F w = d F . Proposition 8.1.3 We have
K(W,w) = { F E C " ( W ) / d F ( X ) = X F = 0 , for all
IE
K and X E T,K*}.
Proof It is sufficient t o show the proposition for the family {xb}. If F is a Coofunction on W then XF is tangent to K in a neighborhood of a
Chapter 8. Presymplectic mechanical systems
406
point z if and only if X F ( ~=~0,) since for any integral curve f ( t ) of X F passing through z we have
X F ( f ( t ) )= ( d / W ( f ( t ) ) / t = o . The local characterization of K by the functions yb tell us that all y's are constant along the trajectories of X F . So
But
which gives the above characterization of K ( W , w ) .0 Definition 8.1.4 The function F E K ( W , w ) is called offirst class. Functions which are not of first class are called of second class.
The LichRemark 8.1.5 The above definition is due t o Lichnerowicz [89]. nerowicz's method may be summarized as follows: in the place of searching a final constraint manifold K solving the problem we may study the class of functions for which the vector fields are first-order differential equations along K fixed. First class functions on constraint manifolds were characterized by Dirac as functions which are in involution with respect t o the Poisson brackets. So, if ( z , y ) is a local coordinate system in an open neighborhood V of W such that y = ( y b ) = 0 along U = V n K , then F is a first class function if and only if for all b one has { F , y b } = 0. In fact
We will say that ( y b ) are first class constraint functions if yb E K(W,w) for all b. So a first class constraint manifold of (W,W) is a submanifold K of W locally characterized by the vanishing of functions which are in K ( W ,w ) with respect to the Poisson brackets. Consider again a local coordinate system (2, y) = (z',yb) = (z', . . . ,z ' , yr+', . . . ,y2") on V such that U = V n K is defined by y = 0. If {a/&', c3/ayb} is the local basis induced by such coordinates then the restriction to U of (c3/ayb) generates a subbundle S of TW/* supplementary of T U in T W / u . The Hamiltonian vector fields Xb of d y b with respect to the
8.1. The first-order problem and the Hamiltonian formalism
symplectic form w generates a subbundle TU' may put
407
of T W / u and therefore we
for the direct decomposition T W / u = TU (I3 S , where Tb and Sb are sections of TU and S , respectively. A direct computation shows that
and & ( H ) = 0 if and only if Tb(h) = - & ( H ) . (8.3) we see that
Taking the differential of
Proposition 8.1.6 Let ( s , y ) be a coordinate s y s t e m as above. A necessary t o be extended along and s u f i c i e n t condition for a Coofunction h o n (K,w) a neighborhood V,
with the property that the vector field X H is tangent t o K at every point i n U is that the s y s t e m b
ua{Y , Y
a
= -Tb(h)
admits a solution u = (ua)for such coordinates.
Proof If (8.9) is verified then from (8.7) we have that for every vector field 2 E TU', Z ( H ) = 0 and from Proposition 8.1.2, XH is tangent to U . If H is in K ( W , w ) then Xb(H) = 0 and so $ ( H ) = -Tb(h). we develop S, along the basis { a / a y b ) / uand we obtain
408
Chapter 8. Presymplectic mechanical systems
-Tb(h) = { Yb, Y"} (aH/ay")/CJ. Therefore the functions { ( d H / a y a ) / ~ }are the desired solutions. 0 Thus, from Proposition 8.1.5 we see that the geometrical algorithm says that (8.9) admits a solution at the final constraint manifold. But Proposition 8.1.5 gives us another local method for searching such final constraint manifold (if there is one). The main question is in fact: when are there such coordinate systems (z,y)? If for a given system the matrix A = ({y",yb}) is of maximal rank then of course the problem is solvable. But in general this will not be so. We must discard the case where the system (8.9) is unsolvable for every coordinate system. In such a case Dirac [48] says: "it would mean that the Lagrangian equations of motion are inconsistent and we are excluding such a case". Suppose that for a given coordinate system the rank of A is not maximal. Since Tb(h) in (8.9) is arbitrary it may occurs that all components of such a vector field are not in the image of the regular part of 7r. Our system is independent of the u's and X(H)= 0 if Tb(h) = 0. This equation will not be satisfied except in an open subset U1 of U. This subset is characterized by a coordinate system (za, where now 1 5 a 5 k, ; k 1 5 A 5 t . Starting again with
sA),
+
for such situation, we analyze the system
where
is obtained from
If the above system is independent of the u's then we continue the process into another subset of U1 and so on. This algorithm originates a final local constraint submanifold U, for which a t least some of the components of the corresponding T ( h ) are in the image of the regular part of the respective matrix constructed for such domain.
8.2. The second-order problem and the Lagrangian formalism
409
If we suppose, for the original situation, that for our system ( z i , y b ) ,s elements of Tb(h) are in the image of the regular part of T , for simplicity (s is the rank of x ) then from Linear Algebra we know that the system (8.9) has a solution of type Ua
= iia
+
V j.' j " j a ,
where iia is fixed and v j i i j a is a linear combination of all solutions of the homogeneous equations associated to (8.9). If we put
f f = iijaya then
is the final expression of the extended Hamiltonian. We may verify that H is of first class. Finally, from the above algorithm we see that a t the end of the process the Hamiltonian will be of type
H = h + first class (primary/secondary) constraints +second class (primary/secondary) constraints. This finish our geometrical interpretation of Dirac-Bergmann formulation of the constraint theory.
Remark 8.1.7 Let (W,w) be a symplectic manifold, K a submanifold of W . We recall that K is respectively isotropic, coisotropic, Lagrangian if its tangent space at every point I of K is respectively of the corresponding type it9 a subspace of T,W. A coisotropic submanifold is a first class constraint manifold. A Lagrangian submanifold is a second class constraint manifold. Isotropic submanifolds are trivial constraint manifolds.
8.2
The second-order problem and the Lagrangian formalism
8.2.1
The constraint algorithm and the Legendre transformat ion
Let J be the canonical almost tangent structure on the tangent bundle of an m-dimensional manifold M and w = - d d ~L the (pre)symplectic form on
410
Chapter 8. Presymplectic mechanical systems
W induced by the Lagrangian L : T M
+R.
Then we may also consider the problem of finding a solution X for the problem ixwL = a!, where a! is a 1-form on T M . Naturally, if L is a regular function (i.e., W L is symplectic) then X exists and it is unique. So, let us suppose that L is degenerate (i.e., W L is presymplectic). Repeating the above procedure of the preceding section, we wish t o know if there exists a submanifold K of T M and a vector field X on T M solving the equation along K , with X being tangent to K at every point of K. As the points of T M for which the dynamical equation is inconsistent are those for which i y a # 0 for all Y E T(TM)', we define
K1 = { z E T M / ( i y a ) ( z )= 0, for all Y E T ( T M ) ' } . We repeat the same procedure to find a final constraint manifold K , (non-empty) defined by
Kr = { Z E Kr-l/ for all Y E TK,I_,, ( ; y e ) ( . ) = O } . The constrained algorithm for the Lagrangian situation is, however, insufficient since the solution X must be a semispray (second-order differential equation) on the final constraint manifold. Therefore we must incorporate t o the original problem the equation J X = C ,the geometrical condition for a dynamical system to be a semispray. This new problem is sometimes known as the second-order problem. Thus this new situation is now characterized by the set of equations
on some submanifold of T M (in fact, on some submanifold of the final constraint manifold obtained from the constraint algorithm for the first-order problem). We will examine this problem in the forthcoming paragraphs but let us first analyse the relation be tween Lagrangian and Hamiltonian systems for the degenerate Lagrangian case. Thus, we wish to know if there is a Hamiltonian counterpart of degenerate Lagrangian systems. A study in this direction was proposed by Gotay et al. [68], which we will see now. We first observe that the form W L is related to the canonical symplectic form WM on T*M by
Leg*wM = W L ,
8.2. The second-order problem and the Lagrangian formalism
--
411
where Leg : T M T * M is the Legendre transformation (see Chapter 7). If we suppose L : T M R regular (i.e., Leg is a diffeomorphism) then E L o Leg-' is the Hamiltonian on T*M ( E L is the energy for L) and the Hamiltonian counterpart of the Lagrangian L is defined. However, if Leg is not a diffeomorphism of T M onto T*M we cannot define the Hamiltonian to be EL o Leg-'. Defining a function hl on the image Leg ( T M ) implicitly by
hl
o
Leg = EL
then this definition will be well-defined on L e g ( T M ) if and only if for any two points x, y E T M one has: Leg (z) = Leg ( y ) implies E L ( z ) = E L ( Y ) . Thus we restrict our attention to a special class of Lagrangian functions verifying the following property:
-
Definition 8.2.1 A Lagrangian L : T M R is said to be almost regular if the Legendre mapping Leg : T M T*M is a submersion onto its image and the fibers Leg-'(Leg (x)) are connected for all x E T M . We now prove that every almost regular Lagrangian system has a special Hamiltonian counterpart. From the above implicit definition it is sufficient to show that hl o Leg = E L is a first integral (or a constant of motion) for all Y E K e r ( T Leg). A reason for this is that for almost regular Lagrangian systems the image Leg ( T M )can be canonically identified with the leaf space of the foliation of T M generated by the involutive distribution K e r (T Leg). The distribution K e r (T Leg) defines a local foliation on T M in the sense that for each t E T M there is a (local) submanifold Q of T M passing through z such that T Q = K e r (T L e g ) / g (in fact, as K e r ( T Leg) is integrable, from Frobenius theorem one has that K e r (T Leg) defines globally a foliation). Considering the family of almost regular Lagrangians it is possible to show that Leg (TM) may be identified via a diffeomorphism to the leaf space of the foliation generated by the distribution K e r (T Leg). Let Y E K e r (T Leg). As Y is a vertical vector field there is (locally) a vector field Z such that J Z = Y . Thus
(8.10) The Liouville vector field C is a vertical vector field and so there is (locally) a vector field X such that J X = C . Therefore W L ( Z , C= ) W L ( Z ,J X ) = - w L ( J Z , X ) = w L ( X , Y )
Chapter 8. Presymplectic mechanical systems
412
= (Leg*wM)(X,Y)= W M ( ( TLeg)X, (T L e g , Y ) ) = 0 ,
(8.11)
as Y E Ker(T Leg). But
L y EL = i y d E ~ = ~ E L ( Y=) Y (E L ) and so Y ( E L ) = 0 , showing that EL is a first integral for all Y E Ker(T Leg). Let us study now the relation between the motion equations for both formalisms.
-
-
Definition 8.2.2 Consider a Lagrangian system represented b y a triple ( T M , w L ,L ) , where L : T M R is the Lagrangian and W L = -ddJL. Let Leg : T M T * M be the Legendre transformation and set M I = L e g ( T M ) , w1 = w/M1 and hl o Leg = E L , where WM i s the canonical symplectic form on T * M and EL i s the energy associated to L . We say that the Hamiltonian system ( M l , w l ,hl) i s first-order equivalent to ( T M , ~ LL ,) i f (1) for every solution X L of the equation i x w ~= dEL the vector field (2' Leg)(XL) satisfies the Hamilton equations i x w l = dhl. (2) If xhl i s a solution of the Hamilton equations on MI then every X L E (TLeg)-'(Xh,) i s a solution of the Euler- Lagrange equations. The following theorem is due to Gotay and Nester (681.
Theorem 8.2.3 (The equivalence theorem). Let ( T M , L J LL,) be an almost regular Lagrangian system. Then ( T M , L J LL, ) admits a Hamiltonian counterpart ( M I ,w1, h l ) such that both systems are equivalent. Proof First we apply the constraint algorithm to find a final constraint manifold where a solution X L for the original Euler-Lagrange equations i x w ~ = EL, tangent to this manifold is obtained. For the sake of simplicity let us suppose that such X L is obtained at the first step, i.e., at PZ PI = T M , where PZ is embedded in PI via ' p 2 : PZ P I . Let M I = Leg(P1) and $1 : M I T * M the corresponding embedding of M I into T * M . We set Legl : PI --t M I , i.e., Legl o $1 = Leg. Also we define Leg2 : PZ M2 by Legl o ' p 2 = $2 o Leg2:
-
-
-
8.2. The second-order problem and the Lagrangian formalism
413
Let us recall that
P2 = {y E PI = T M / i z d E ~ ( y=) 0, for all 2 E Ker
WL
= (TPl)*}
is a secondary constraint manifold. As hl o Leg = EL and ( T P 1 ) l = (TM1)' @ K e r ( T L e g ) , using a local basis of vector fields on TP1 which locally span ( T P I ) l such that their prolongations by Leg1 exist and (locally) span (TMI)', we see that there is a secondary Hamiltonian constraint manifold M2 defined by
M2 = { Z E M1/iTLeg,Zdhl(Z)= 0). As Leg1 is a submersion, Leg2 is also a submersion. Suppose that XL satisfies the Euler-Lagrange equations. Then, if (T Legl) ( X , ) exists, we have
= Leg;*(iTLeglX'W1- dhl)/M2. As Leg1 is a submersion, T LeglXL solves the Hamilton equations
(iTLcgIXLW1 = d h l ) / M , . Conversely, requirement (2) of Definition 8.2.2 is satisfied by running the computations backwards. We remark that the above procedure may be adapted directly from the r-th step in the Lagrangian version transforming it to the Hamiltonian description via Leg, ( M , would be then the primary constraint manifold). 8.2.2
Almost tangent geometry and degenerate Lagrangians
Let us consider again the Euler-Lagrange equations of motion ~ X W L= dEL.
(8.12)
When L is a regular function there is a unique solution X which is automatically a semispray. Indeed, as we have seen before, from (8.12) one obtains
Chapter 8. Presymplectic mechanical systems
414
and the non-degeneracy of WL implies JX = C. If L : T M + R is not a regular function, i.e., W L is presymplectic then even if there is a vector field X solving (8.12) we cannot assure the second-order condition JX = C, i.e., that X is a semispray. In this section we develop a geometric formalism in order t o study degenerate Lagrangians. In the following, for simplicity, we will put Ker S,,
WL,
= Ker
WL.
Suppose that (P,Q) is an almost product structure on T M adapted to i.e., we have Ker P = Ker W L
Then I m P and I m Q are wL-orthogonal, i.e., WL
( P X ,Q Y ) = 0 , for all X,Y E x( M ) .
Also, if WL has rank 2r 5 2m, where dim M = m, then Ker P = I m Q is a subbundle of T T M with rank = co-rank W L = 2rn - 2r = 2(m - r ) . As S,
: I m P + Im P*
is an isomorphism, given a l-form a on T M , there exists a unique vector field X E I m P such that
Thus, there is a unique vector field ~ ( , w L=
IL E
I m P such that
P*(dEL), EL = C L - L.
(8.13)
is called P-Euler-Lagrange vector field. Obviously, the solution of (8.13) is parametrized by the kernel of S,,, i.e., is a solution on T M modulo Ker W L . Suppose that the adapted almost product structure ( P ,Q ) commutes with the canonical almost tangent structure J on T M , i.e., (L
e~
8.2. The second-order problem and the Lagrangian formalism
415
J P = P J (and hence JQ = Q J ) . Then it is easy to see that P is locally given by
1 5 i , j 5 m. Thus, P has associated matrix
Also,
Definition 8.2.4 We say that a vector field [ E I m P i s a P-semispray (or P-Pd order differential equation) if J [ = PC. Proposition 8.2.5 The solution
EL
of (8.18) is a P-semispray.
Proof: First, we compute the action of i~ on both sides of (8.13): ~ J ~ ( = ~ iJP*(dEL) w L = - i j t L w ~=
(J*P*)(~EL),
because
. .
[iJ,ieL] = r j t t , Now, recalling that
we have
-
icLij = - i j t L and ~ J W L= 0.
(8.14)
Chapter 8. Presymplectic mechanical systems
416
= d j L - d j ( C L ) = d j ( L - C L ) = -djEL.
Therefore ~ C W L=
-J*(dEL),
and we have
i p c w ~= -(P*J*)(dEL) = - ( J * P * ) ( d E L ) .
(8.15)
In fact, for every vector field Y
= - J*(dEL)(PC)= - ( P * J * ) ( d E L ) ( Y ) .
From (8.14) and (8.15) one obtains
i j t L w= ~ ipcw~.
(8.16)
, E I m P and so (8.16) implies J < L = PC. 0 But J ~ LPC Let us suppose that the almost product structure (P,Q) is integrable. Then I m P is an integrable distribution. Let S be an integrable manifold of I m P. Then S is a submanifold of T M , 4 : S T M (for sake of simplicity, we always suppose that all submanifolds are embedded). Also, the couple (S,PWL) is symplectic and the following equality holds:
-
+*P*(dEL) = d(EL o 4). To show this, let z be a point of S and X E T,S. Then
d E L ( T 4 ( X ) )= d(EL 0 + ) ( X I . The vector field ( L (as well as PC) is tangent to the submanifold S and so may be restricted to a vector field on S. Thus, for every point z E S
8.2. The second-order problem and the Lagrangian formalism
417
However we cannot say that ( L / S is a semispray. In fact, EL/S is only a Hamiltonian vector field associated to the function EL o 4 on the symplectic manifold (S,e w ~ )solution , of equation
To go on, we must study equation (8.13) in local coordinates and obtain some local relations useful for our main result. Locally one has
E~ = u i ( a L / a u i )- L . Suppose that ( L E Im P . Then there is a vector field X on TM such that ( L = P X , with
x
=
xi(a/aqi) +xi(a/aUi).
Proposition 8.2.6 Consider eq.(8.13). Then i n a local coordinate s y s t e m ( q i , 0 ' ) the following relations hold:
Proof The equalities (8.17), (8.18) and (8.19) are obtained from a straightforward commputation. We show (8.17) and we indicate how (8.18) and (8.19) are obtained. As ( L = P X , then
87P
Chapter 8. Presymplectic mechanical systems
418
But
As J ( L = PC, eq.(8.17) is verified. The other equalities (8.18) and (8.19) are obtained in the following manner: we compute i t L w which ~ gives
~ have used the relations where in the development of i t L w we
If we take the differential dEL and we apply P* to
EL we have
As i t L w = ~ P * ( ~ E Lwe ) obtain (8.18) and (8.19) using (8.17).0 Next, we shall prove the main result of this section.
8.2. The second-order problem and the Lagrangian formalism
-
419
R be a degenerate Lagrangian a n d ( P , Q ) a n adapted almost product structure to the presymplectic form W L . If (P,Q ) is the complete lift of a n integrable almost product structure (P0,Qo) on M then for every integral manifold SO of Im Po the restriction of L to TSo is regular. Therefore the equation
Theorem 8.2.7 Let L : T M
admits a unique solution vector field order differential equation.
SL
: TSo +TTSo which is a second-
Proof Suppose that W L has rank 2 r and let (PO,Qo ) be an integrable almost product structure on M such that the complete lift (P,C,Qg)is adapted to W L . Then POhas rank r and we have
Q i J = J Q ; = QE, (where Po" (resp. Qg) is the vertical lift of Po (resp. Qo) to T M ) . As (Po,Qo) is integrable, we may choose a coordinate system in M , say ( q i ) , such that Po and Qo are represented by the matrices
[ :],
1 5 i , j < r,
and
respectively, where 6: = Kronecker's delta. For the induced coordinates ( q i , v i ) on T M , the lift Pi is represented by the matrix
In such situation, equalities (8.18) and (8.19) of Proposition 8.2.6 become
Chapter 8. Presymplectic mechanical systems
420
m
(a2L/auiaqj)oi - (a2L/adaqi)d - C(a2 L/auiavj)c L ( v i ) i= 1
= (azL/aUiaqj)d- (aL/aqj), 1 5 j 5
(8.20)
r,
m
(a2L/auiaqj)d- (a2L/adaqi)d- C(a2L/auiad)cL(d)= 0 ,
(8.21)
i= 1
r
r
i=l
i=l
C(a2L/aviawj)vi= C(a2L/aoiauj)ui,1 5 j 5 r ,
(8.22)
r
C(a2L/auiauj)ui= 0, r + 1 5 j
5
(8.23)
m.
i=l
The equations (8.20)-(8.23) may be re-written as follows:
i=l
i=l
-
(aL/aqj) = 0, 1 5 j 5
i= 1
(8.24)
r,
i= 1
m
-
C(a2L/adauj)~L(ui) = 0, r + 1 5 j 5 m,
(8.25)
i=l r
m
i=l
i=l
C(a2L/auiawj)v'- C(a2L/auiaUj)d= 0 ,
15 j
5
r,
(8.26)
r
C(a2L/auiaUj)d= 0, r + 1 5 j i=l
5 m.
(8.27)
8.2. The second-order problem and the Lagrangian formalism
42 1
From (8.24)-(8.27), one deduces
Thus, for 1 5 j 5 r, one has
i=l
i=l
m
(8.29)
and, for r
+ 1 5 j 5 m,
i=l
i=l
i=l
(8.30) r
C(d2L/au'auj)u' = 0.
(8.31)
i=l
The local expression of f~ in such a case is
As X'P! = v'?',
then, when 1
5 j 5 r , we have
X' = u i , for 1 5 i 5 r. As there is no conditions for r + 1 5 j 5 m, we deduce EL
=
r
r
i=l
i=l
C u'(a/aq') + C x i ( a / a u i ) ,
= X'(qJ,uj), 1 5 j 5 m, i.e., Xi depends on all coordinates q's where and u's. Let So be an integral manifold of ImPo. Then So is locally determined by equations of type
422
Chapter 8. Presymplectic mechanical systems
qr+l = const., . . . ,qm = const.,
since ImPo is spanned by locally characterized by (f+l
{ a / a q l , .. . ,a/aqr}. The submanifold
of T M
= const., . . . ,qm = const., u'+l = 0 , . . . ,um = o
is an integral manifold of I m P i , since I m Pi is spanned by
and it is clear that such submanifold is TSo. Therefore P,CC restricted to TSo is precisely the Liouville vector field on TSo, say CO, and now
-
(8.32)
For this, we consider a path a : R the induced curve
--+
SO,a ( t ) = ( $ ( t ) ) ,of
( L I T S ~ Then .
in TSo is an integral curve of (LITS,,. From (8.28) we have
i=l
i=l
along b(t). But from eq.(8.29) one has
c m
(a2L/a$aqJ)q' = 0.
i=r+l
Since ( ~ ( 2=) 0 when r (on TSo) one deduces that
+ 1 5 i I m and ( L ( $ )
=
3 when
15 i
5r
8.2. The second-order problem and the Lagrangian formalism
U
423
+
C{(a2L/aqjaq')q' ( a Z L / a q J a q ' ) ~ ( a L / d q j= ) } 0, i=l
or, equivalently
d dt
- ( a ~ / a i j )- ( a L / a q j ) = 0, 1 5 j 5 r, which are the Euler-Lagrange equations on TSo. This shows that (LITSOis a semispray on TSo, solution of the corresponding Euler-Lagrange equations. Let us now show how we obtain the corresponding canonical formalism for degenerate Lagrangians from the present point of view. Suppose that (P0,Qo) is an integrable almost product structure on M . We may choose local coordinates ( q ' ) on M such that
Thus the complete lift ( P t ,Qi) of (PO, Qo) to T * M is an integrable almost product structure on T*M and we have
(see Section 5.5). (Here we put F E to denote the complete lift of a tensor field F on M to
T*M). As we have suppposing that W L has rank 2r and (PO,Qo) is an integrable almost product on M such that its complete lift (P6,Q;) t o T M is adapted to
WL,
we have Kerwr, = I m Q &
Therefore, when r
+ 1 5 k 5 m, we have
Chapter 8. Presymplectic mechanical systems
424
o = iB/a,,kWL
= -(a2L/avjavk)dd.
Therefore (8.33)
<
1 < j m, r + l < k 5 m. Thus, the Hessian matrix of L is
where det(d2L/duiduJ)# 0, 1 <_ i , j 5 r. Let a~ = d J L . Then t u is ~ semibasic and defines a map
Leg : T M
-
T*M
such that TM o
Leg = TM and Leg*wM = WL.
In local coordinates, we have
One can easily see that
(TL e g ) ( a / a u i )= ( a 2 L / a v i a v j ) a / a p j . Therefore, from (8.33), we see that the Jacobian matrix of Leg is
r
I,
I
(a2L/aqjaui)
0
(a2L/auiauj) o
1 .
8.2. The second-order problem and the Lagrangian formalism
425
Now, since Pt is locally given by
then we have r
(P:)*wM=
C dq'
A
dp;
i=l
and
-
Leg*(P:)*wM = ( ~ t ) * u L . H
Let us suppose now that there is a CO" map H : T*M R such that Leg = E L . Then there is a unique vector field ( H E I m Pl such that
0
i g WM = (P:) *( d H ) . This comes from the fact that w/P: is symplectic, i.e., (Im Po", w / P ' , ) is a symplectic subbundle of T ( T * M ) (see Definition 5.1.14). If So is an integral manifold of ImPo, then TSo is an integral manifold of I m Pi and the cotangent bundle T*So is an integral manifold of I m Po". If we set cp : TSo
----+
T M , 1c, : T*So + T * M
for the corresponding embeddings, then ~ symplectic forms on TSo and T*So. Also: 0
0
0
* W and L ~
W
are, M respectively,
( L / T S Ois the Euler-Lagrange vector field associated t o the regular Lagrangian L o cp on TSo; &/T*So is the Hamiltonian vector field associated to the Hamiltonian H o $ on T*So; and
-
Leg : T M T*M may be restricted to a map, denoted also by Leg from TSo to T*So. This map is precisely the Legendre transformation determined by the Lagrangian L o c p .
Chapter 8. Presymplectic mechanical systems
426
From these considerations it is now easy to obtain the corresponding Hamiltonian form of the Euler-lagrange equations (8.32). Example Let us consider M = R 2 , TM = T R 2 and the Lagrangian
L ( x , y , u , u ) = (1/2)(u2
-
x2)
Then
d j L = tLdx, W L = - d d j L = dx A du. Therefore
WL
is a presymplectic form on T R 2 of rank 2. If we set
A direct computation shows that EL = C L - L = (1/2)(u2 + x 2 ) , dEL = xdx From the equality
~ ( W L= dEL
(1
+ tLdiL.
we deduce
=U)
(1
= -x.
Then f assumes the general expression
f = U(a/az)
+ €2(a/ay) - x ( a / a u ) + ( " a / a u ) .
Now,
Kerwr, = { Y / Y = Y2(d/dy)
+Y2((a/au)}.
So, let (P,Q) be an almost product structure on T R 2 adapted to
i.e.,
KerwL = I m Q = Ker P. We have
+
I m p = {Y/Y= ~ ' ( a / d z ) ~ ~ ( a / a u ) }
WL,
8.2. The second-order problem and the Lagrangian formalism
427
Therefore P is given by
P(a/az) = a / a z , P(a/ay) = 0) P(a/au) = a/au, P(a/au) = 0, and then it is represented by the matrix
;
;I1
1 0 0 0
p = [ 0; ;0 0 0
This shows that P J = J P . In fact, P is the complete lift of the almost product structure (P0,Qo) on R2 represented by the matrix
Since P * ( ~ E L=) z dz
EL
+ u du, then the P-Euler-Lagrange
vector field
E I m P such that
is given by (L
= u(a/az)
-
z(a/au).
Now, if we consider the straight line a ( t ) = (z(t),O), we have on T o ( t ) = ( z ( t ) , O , i ( t ) , O )that (L =
i(a/az) - z(a/au).
As the image of the Liouville vector field C under P is precisely the Liouville vector field restricted to T a ( t ) ,i.e.,
c = i(a/au), we have the second-order condition J ( L = C along T a ( t ) . We observe that there exist degenerate Lagrangian for which this method don't works. For instance, consider the Lagrangian
Chapter 8. Presymplectic mechanicaJ systems
428
L = (l/2)(yu - xw
-
x 2 - y2)
Then
K e r w L = { Y / Y = Y'(a/au)
+ y2(a/aw)}.
Suppose that there exists an adapted almost product structure (P,Q). Then I m Q = K e r P = K e r w L . But J P # P J . In fact
JX = J ( X ' ( a / a x ) + X2(a/ay) + X'(a/au) = X'(a/au)
+ X2((a/aw))
+ X2(a/aw) E I m Q
and so P ( J X ) = 0. As
ImP = {Y/Y = Y'(d/dX)
+ Y2(a/ay)},
we see that
J(PX) = x'(a/au)
8.2.3
+ X2(a/aw)# P(JX).
Other approaches
In this section, we summarize two different approaches for the problem of finding simultaneously solutions of equations ~ X W L=
EL, JX = C
for degenerate Lagrangians. The first study is due t o Gotay and Nester [68] and consists in finding, under certain hypotheses, a constraint manifold S and a vector field X on S such that (iXWL
- d E L ) / S = 0, (JX - C ) / S = 0.
(8.34)
Definition 8.2.8 A Lagrangian system ( T M , w L L , ) is said to be admissible if the leaf space T^M of the foliation 7 on T M defined b y the integrable distribution D = K e r W L n V ( T M ) admits a manifold structure such that the canonical projection p : T M -+ T M is a submersion.
8.2. The second-order problem and the Lagrangian formalism
429
Suppose that P is the final constraint manifold associated to the Lagrangian system ( T M ,w ~L ), by the algorithm. If the Lagrangian system is admissible then so is the constrained system in the following sense:
-
Proposition 8.2.9 D restricts to an involutive distribution on TP such that ?p = 3 / P foliates P. Furthermore, the leaf space fi = P/?p is a manifold embedded in T% and the induced projection r : P 'f is a submersion.
Proof Let Y cz D be a vertical vector field and Z a vector field such that locally JZ = Y . From (8.10), we have
and as C is vertical there is X such that locally J X = C. Thus
LY (dEL) = d(WL(X,J Z ) ) = d(WL(X,Y ) ) .
(8.35)
Now, the final constraint manifold P is obtained from a sequence of su bmanifolds .
a
.
- -p,+1
'Pi+ 1
P;
...
and will show that D / p c T P by induction on the constraint manifolds pi. Of course D c T ( T M ) = T P l . Suppose that D / p i c TP;. The constraint manifold P;+l is characterized by the vector fields Y E D such that
for all W E (TP,)' (and so Y is tangent t o pi+l). Now
+
Y ( i w d E ~=) i [ y , W ] d E ~ i w L y d E L
(8.36)
and from (8.35), as Y E D c Ker W L , the second term in (8.36) vanishes. Letting Z E TP; be arbitrary, one has along P;
-W(W1[Y, 21) = 0 by the assumptions on Y , Z and W . Consequently [Y,W]/piE (Tpi)* and so ( i [ y , W l d E ~ ) / p ~=+0, which implies Y ( i w d E ~= ) 0, i.e.,
Chapter 8. Presymplectic mechanical systems
430
D / p gives rise to a foliation 3 p of P. The leaf space space P of 7 p can be identified with the image of P under the map T = P o p, where p : P T M is the embedding. Consequently, 7 p inherits a manifold structure from 7 such that T : P fi is a submersion and ’p : 7 p 7 is an embedding. The main theorem of Gotay and Nester [68] is the following.
-
-
-
Theorem 8.2.10 Let us consider an admisible Lagrangian system with f i nal constraint manifold P embedded in T M . Then there exists at least one submanifold S of P and a unique (for fixed S) vector field X on S which simultaneously satisfies (i*WL
- dEL)/S = 0 , ( J X
- C ) / S = 0.
Every such manifold S i s difleomorphic to
7p.
The proof of the theorem will be broken into two steps: the first consists in a local argument, in the sense that, there exists a unique point at each leaf of 3p at which every solution X satisfies J X = C. In the second step it is shown that these points define a submanifold of P diffeomorphic t o 7 p and that there exists a unique X solution of the Euler-Lagrange equations tangent to this submanifold First step: Local existence and uniqueness We start by supposing that X E T P is a solution of ( i x w ~ d E ~ ) /= p 0 of the admissible system ( T M ,W L , L ) .
Definition 8.2.11 The deviation vector field W y of a vector field Y is the vector field Wy = JY - C. It is not hard to see that i ~ ( i y w - ~EL) = i w y w ~and if Y solves the Euler-Lagrange equations then Wy E D / p . A vector field X is prolongable if it projects to T^M, the leaf space of the foliation 3 on T M defined by the involutive distribution D = K e r W L n V ( T M ) .This means that T p ( X ) = 2 is well-defined. Now, if one replaces “almost regular” by “admissible” and ( L e g ( T M ) ,w1, d h l ) by (T^M,G ,d z ~ ) where W L = p*$~, d J 3 ~= p * ( d E ~then ) a generalization of the equivalence Theorem suffices to show that prolongable solutions of ( i x w ~- d E ~ ) / = p 0 always exist when ( T M , w L ,L ) is an admissible system. More clearly, we have
8.2. The second-order problem and the Lagrangian formalism
43 1
Lemma 8.2.12 There ezists at least one solution X of the Euler-Lagrange equations (i*GL
- dhL)/p = 0.
Any vector field X E T ; ' { k } will then solve the Euler-Lagrange equations. Any vector field X is semi-prolongable if it is prolongable modulo
V(TM). Proposition 8.2.13 Let ( T M , ~ LL ), be admissible and X a vector field on the final constraint manifold P. Suppose that X is a semi-prolongable solution of ( i x w L - d E ~ ) / p= 0 . Then there ezists a unique point in each leaf of ?p = ? / p at which X is a semispray. Proof: For each point z E P we denote by ?p(z) the leaf of 7 p through z . Our assertion is: n x = T T M ( X ( Z is ) )the required point. To show this it is necessary to prove that: (i) tax is independent of the choice of z ; (ii) q x is in 3p(z); (iii) X is a semispray at q x . Let us put X = Y 2, as X is semi-prolongable (Y is the prolongable part of X and 2 is vertical). As D is vertical there is a map p : T(T^M) + T M such that the following diagram
+
is commutative. Consequently,
TTM(X(Z))= TTM(Y(Z))= ( P O T P ) ( Y ( Z ) ) . But T P ( Y ( z ) )is independent t o the choice of z E ?p(z) and so q x does not depend on z . To show (ii) we consider the vertical integral curves of ( - 1 ) W x and as (-1)Wx E D / p if Y is a solution of ( i x w L - d E ~ ) / = p 0, these trajectories are contained in the leaves of 7p. Locally C(q,u ) = u'd/au' and ( J X ) ( q ,U) = q&(a/au')
(8.37)
Chapter 8. Presymplectic mechanical systems
432
and the equation determining the integrals of (- 1)Wx is du'(t)/dt = u ' ( t ) - qk.
rlk
But, by (i) the functions are constant on 7 p ( z )and the integral curve starting at z = (q0,vO) is thus
-
-
As t -00, d(t) q$ so that rlx is a limit point of u ( t ) . But u ( t ) E 3 p ( z ) for all t and asp is continous one has 3 p ( z ) closed. Consequently
rlx
E
5+).
Finally, X is a semispray a t qx because Wx(qx)= 0 and from (8.37) we see that X is a semispray only at qx. III Second step: Global existence and uniqueness Two semi-prolongable solutions X , Y of (ixw~, -dE~)/p = 0 are said to be J-equivalent if J ( X - Y) = 0. We have, of course, an equivalence relation and J-equivalence classes of vector fields X are denoted by [ X I . If follows that if X, Y E [XI, then W x = W y , so that qx depends only upon [XI, denoted by qxl. Now, let S p ] be the union of such qx],for each leaf of 3 p . This set depends only upon [XI. The map
defined by
is CM and injective and it is independent of the choice of z E P - l ( i ) . Also, a computation in local coordinates suffices t o show that TalX]is regular and, as S[XIis the image of B under the action of a[x]one has that S [ x ] is a submanifold of P diffeomorphic to P . On the other hand, if [XI # [Y] then Six] # S [ y ~i.e., , each J-equivalence class [XI of solutions of the EulerLagrange equations defines a unique Six].Now, the solutions of ( i x w ~d E ~ ) / p= 0 are unique modulo vector fields in K e r W L n T P ; thus the set of all submanifolds S p ]is parametrized by (ker W L n T P ) / V ( T M ) . Let X be a semi-prolongable solution of the Euler-Lagrange equations. Then X verifies
8.2. The second-order problem and the Lagrangian formalism
433
(ixw - d&)/s,,, = 0 , (JX- C)/S[,] = 0 (8.38) on a submanifold SIX]of P but X/S[X] is not necessarily tangent to SIX]. Thus we search a X E [XIverifying (8.38) and X E TSlx]. If we fix a prolongable solution X and if we set satisfies ( i @ ~- d , ? ? ~ ) /= ~ 0. Now
T.[X](X) and since p o
XI
2 E
= T P ( X ) by Lemma 8.2.12,
2
q x ]
= I d / B we have
and hence it is vertical. Therefore
showing the following
Proposition 8.2.14 The vector field Tcu[x](X) is tangent to SIX] and solves (8.98) simultaneously. Proposition 8.2.15 There i s a unique vector field Y E TS[x]satisfying (8.95) simultaneously.
Chapter 8. Presyrnplectic mechanical systems
434
Proof If Y and 2 are two solutions then W y = W z = 0 and so J(Y 2 ) = 0. From the first equation of (8.38) we have Y - 2 E K e r W L . Thus Y -2 E D / p which gives a contradiction since TSix]nD= 0 and T : P P is a diffeomorphism. Now, let us examine a second approach due to Cantrijn et al. [ll]. Let L : TM R be a Lagrangian function. Then it is easy t o see that L is regular if and only if K e r WL n V ( T M ) = 0. Thus, if L is degenerate K e r W L n V ( T M )is non-trivial. Furthermore, K e r W L n V ( T M )defines an integrable distribution on T M and if we consider the restriction J / K e r w L then K e r ( J / K ~ w, ,. ) = K e r W L n V ( T M ) and J ( K e r W L ) c K e r W L n V ( T M ) .Let EL be the energy of L and take Y E S;l(dEL), i.e.,
-
-
i y w= ~ dEL. From the equality
we see that X = JY - C E K e r W L and as X E V ( T M ) we have X E K e r W L n V ( T M ) . The following result is due to Carifiena et al. (1985).
Proposition 8.2.16 Suppose dim K e r W L = 2 dim ( K e r Then there is a vector field 2 E K e r W L such that J Z = X .
WL
nV ( T M ) ) .
Proof The following diagram is commutative:
K e r WL
JIKerwL t
where r$ is an injective homomorphism and Now, in general dim ( K e r WL)
T
K e r WL nV ( T M )
is the canonical projection.
5 2dim ( K e r W L n V ( T M ) )
and as we are supposing the equality, 4 is also surjective. Thus the assertion is proved. The vector field Y - 2 verifies
8.2. The second-order problem and the Lagrangian formalism
J(Y
-
435
2) = J Y - J Z = JY - X = C
Thus, if we assume that the degenerate Lagrangian L admits a global dynamics, i.e., a globally defined vector field Y such that i y w ~= EL, then there is a semispray = Y - Z on T M which verifies icwL = d E L . In such a case, a general solution is any particular solution plus a vector field belonging to Ker W L . Now, from the proof of Proposition 8.2.16, we have the following types of Lagrangians: Type I: if d i m ( K e r W L ) = d i m ( K e r W L n V ( T M ) )= 0 , Type 11: if d i m ( K e r W L ) = 2dim ( K e r W L n V ( T M ) )# 0. Type 111: if d i m (Ker W L ) < 2dim ( K e r wr, n V ( T M ) ) . Clearly, type I Lagrangians are regular Lagrangians. We consider in the rest of this section only type I1 Lagrangians. In such a case, if the Lagrangian L admits a global dynamics - which will be of type A B , A being any particular solution and B E Ker W L - then there exists a semispray which is a global solution. We also assume that all degenerate Lagrangians of type I1 admits a global dynamics. The third assumption will concern Ker W L . Consider the quotient space
<
+
-
If we assume that the foliation defined by Ker W L is a fibration with projection .li : T M T M then there is a unique symplectic form LJL on T ? such that W L = A*&, (for a proof see Guillemin and Sternberg [74], Theorem 2 . 5 . 2 ) . Suppose that Ker W L verifies such condition. Then it is possible to show that J projects onto a tensor field of type ( 1 , l ) j on ?M if Im ( L z J ) c Ker W L , for all 2 E Ker W L (we call this condition by the kernel condition). Moreover, j is an integrable almost tangent structure on T^M, the subspaces Im jzare Lagrangian with respect to GL(z), for each is the above semispray then the projected vector field 2 E T^M and if verifies
i
Therefore (see Remark 7.4.4) we have
-
R be a degenTheorem 8.2.17 (Cantrijn et al. 1111). Let L : T M erate Lagrangian of type 11, with a global dynamics, such that the foliation
Chapter 8. Presymplectic mechanical systems
436
defined by K e r W L is a fibration. Suppose that K e r W L verifies the kernel j , i )i s a regular Lagrangian system (called “reduced condition. Then ( T Z , Lagrangian system ”1. Now, let be a (local Lagrangian) for (. Putting L1 = t o 2, we can easily prove that the Lagrangian L1 is gauge (locally) equivalent to the original L in the sense that they determine the same dynamics.
Remark 8.2.18 Actually, we can relate this approach with the one given in Section 8.2.2. From the assumptions in Section 8.2.2, one has a global solution for the Euler-Lagrange equations. Moreover, the assumption that J P = P J (and hence J Q = Q J ) is equivalent t o say that L is of type 11. In fact ImQ = Ker
WL
implies that
= I m Q n V ( T M )= K e r
WL
nV ( T M ) .
Furthermore, the assumption that (P,Q ) is integrable is equivalent to the assumption that K e r WL defines a fibration.
8.3
Exercises
c
In Section 7.15 we have examined the set of 1-forms A;, where is a semispray. If we associate to each an appropriate 1-form (I! € A: and if a = d L , for some regular function L then we obtain the Euler-Lagrange equations. Now, let ( P , Q ) be an almost product structure on T M which commutes with J and E a P-semispray (see Definition 8.2.4). We set Ee=Id-LeoJ*; A; =
A form (I! E
Ker Ef;
5 ; is called P-regular
P is adapted to w, and
B=P*oEe.
5; = K e r if wa = - d ( J * a ) is presymplectic and Euler-Lagrange vector field
c is said to be P-regular
437
8.3. Exercises
if there is a P-regular a E such that a = d L , (for what follows we refer sections 7.15 and 8.2.2). 8.3.1 Show that such L is degenerate and that the motion equation associated to L is P*(Lf(J*(dL))) = P*(dL). Show that this last equation is i p L = P * ( d E t ) . Thus, if P is as in Theorem 8.2.7 one obtains the regular equations of motion for L. 8.3.2 Compute the local expressions of E t a and &a ( H i n t : as J ( = PC, take ( = PX, where X is a vector field such that J X = C. Use the local expression of P given in p. 415 to show that XiFf = wiP,!, where X i is the (d/dpi)-component of X and ui the (d/dwi)-component of C. ~8.3.3 Suppose that R is a tensor field of type ( 1 , l ) on T M which commutes with J , P and satisfies J(LcR) = 0. Show that R*oEe = EcoR*, R*oEc = o R* and that R preserves A: and A:. -~ 8.3.4 Let P be the complete lift t o T M of an integrable almost product structure POon M . Let A0 be a tensor field of type ( 1 , l ) on M which commutes with Po. Show that A; commutes with J and Po" and satisfies J ( L f A 6 )= 0. Hence (Ah)* preserves A: and A:.
-
8.3.5 Let a be a P-regular element of
A:.
Prove that
i ( P * ( a - d < PC, Q >) = 0. Show that this equation represents a generalization of the usual law of conservation of energy ( H i n t : EL = i p c d L - L -dEL = dLd < PC, dL >, if is a P-regular Euler-Lagrange vector field and cr = dL. Show that EL is constant along the integrals of 6 ) . 8.3.6 Let P as in Exercise 8.3.4. Let f : M R be a smooth function. Prove that, if cr E A: then a' = a d(
<
-
+
8.3.8 Let a be a P-regular element of that (1)
(2)
A:
and Y a vector field on T M such
P * L y ( J * a ) = P*(df), f : T M
-
R
iy(P*(a- d < PC,a >)) = 0.
438
Chapter 8. Presymplectic mechanical systems
Show that F = f - < Y,J*a > is a first integral of (. Conversely, prove that to each first integral F of ( there corresponds a vector field satisfying (1) and ( 2 ) . 8.3.9 Show that, if L is degenerate then Ker W L n V ( T M ) defines an integrable distribution on TM. Show that dim (Ker W L ) 2 dim (Ker wL n V ( T M ) ) . 8.3.10 Suppose that the foliation defined by Ker W L is a fibration with projection 6 : TM + TM/Ker W L . Show that J projects onto a tensor field of type ( 1 , l ) j on TM/Ker W L if I m ( L z J ) c Ker W L for all 2 E Ker W L. Prove that if j is integrable almost tangent structure then I m j , are Lagrangian with respect to c j ~ ( z ) z, E TM/Ker W L . 8.3.11 Let g be a degenerate metric on a differentiable m-dimensional manifold M. Let N be the distribution defined by
<
N , = { X E T,M; g , ( X , Y ) = 0, for all Y E T,M} and L defined by L ( v ) = ( 1 / 2 ) g 2 ( vw, ) , v E T,M, 2 E M. Suppose that the elements of N are Killing vector fields for g . Show that Ker W L = N C
439
Appendix A
A brief summary of particle mechanics in local coordinates A.l A.l.1
Newtonian Mechanics Elementary principles
Let us consider a single particle and let us denote by ?the position with re7 7 spect to the canonical referencial (1 ,3 , k), from the origin to the particle. Suposing that ?is a function depending on the time t , Newton’s second law of motion says that
2 = m(d2/dt2)?=
(d/dt)g,
(A4
where F’ is the total force on the particle, rn is a constant (the mass of the particle) and p = m(d/dt)?is the (linear) momentum. The forces are supposed dependent only on the time, position and velocity. Eq.(A.l) may be extended to a mechanical system involving a great number of particles, say N. If { P I , .. . , PN} is a family of particles, then for each a, 1 5 a 5 N , Eq.(A.l) takes the following vectorial form:
P = ma(d2/dt2)P = ( d / d t ) j ~ ,
(A4
where ?is the position vector of Pa,with mass ma and p” = ma(d/dt)< We can express the above equations in its Cartesian form. If
440Appendix A
( z ? ( t ) , m ) , z w )=
denotes the position of the form
(6) % 43
(A.3)
Pa at the instant t , then the motion equations take
= ( d / d t ) p ; = p;; 1 5
A!
5 3, 1 5 a! 5 N
(A.4)
where the dots mean the time derivatives and rny,mt,mg represents the mass of Pa under the condition my = m; = mg. The mechanical system { P I ,. . . ,PN} of N-particles in the Euclidean space can be considered as being a single particle P in the Euclidean space RSN (in fact, this is not correct: we must take into account that two or more particles cannot occupy the same position at the same time; thus the position of the system is given by R3N - W , where W is defined as the set of coordinates for which the particles have same position. Also some constraint relations like the system is restricted to a sphere may be considered. Having in mind these restrictions we now assume that P is in all R 3 N ) .For this, we represent the coordinates (A.3) of Pa by (A-5) that is, the initial position of the system in R3 is represented in pN by (23a-2,23a-l, Z3a)
Let us put k = 3 N . Then the fundamental equations (A.4) are expressed in Rk by
F.* -- rn.x. $ 0 I l i l k ,
(A4 where F l , . . . ,Fk are forces acting on the particle. More clearly, the force Fi is acting on some particle Pa in the sense that for each i there is a! E { 1,. . . ,N} such that i E (3a - 2,3a - 1,3a!}. As the particle P represents the initial system of N particles, the fundamental equation in Rk,equivalent to eq.(A.l), is k
C ( F i - mi(d2/dt2)zi) = 0, i= 1
A. 1. Newtonian Mechanics
44 1
that is,
Theorem A . l . l (Momentum conservation law) If the sum of all forces acting on the mechanical system {PI,. . . ,P N } (or P) vanishes, then the total momentum of the system i s conserved.
In fact, p ; = m;i; = m ; ( d / d t ) x ; and as we are assuming that m; is invariant with respect to the time t , p; = ( d / d t ) ( m ; x ; ) . So, as C F; = 0, one has Cp; = constant.
A. 1.2
Energies
-
k-forces F1, . . . , F k acting on a particle P E Rk are said to be conservatives if there is a diferentiable mapping f : Rk R, at least of class C2,such that F; = ( a f / a x ; ) , 1 5 i 5 k. The potential energy V of P is defined by V = - f + constant. Therefore, a force is conservative if F; = -(aV/z;). The potential energy V : Rk -+ R is a function depending explicitly on the position coordinates and implicity on the time. So
and
The integral
vanishes. Reciprocally, the existence of a function V (at least of class C2) verifying (A.8) is determined by condition (A.9). This is a consequence of Green’s Theorem and the value of V at a point z is the value of the integral
442Appendix A along an arbitrary path from xo t o x. The kinetic energy of P E Rk,at time t , is defined by the expression
From this definition we have k
dT -= dt i= 1
k
Cm;i;X;= C Fix;,
(A.lO)
i= 1
called the la* form of the energy equation.
Theorem A.1.2 (Energy Conservation Law) Let us suppose that on P, act only conservative forces. Then there i s a constant function E, called the 'total energy" so that
+
T ( t ) V ( t ) = E, for all t E R
(A.ll)
In fact, the time derivative of T and V gives
and so 0=
d C(m;X; - F ; ) i ; = -dtd( T + V ) = -E. dt
In general, we will say that a mechanical system is conservative if equality ( A . l l ) holds. This expression is called the 2ndform of the energy equation. If the system is not conservative, then we may have the 1" form if all forces aren't conservatives. However, if there are some conservative forces, then we put
Fi = F/
+ F:
where F/ = conservative force and Ff = non-conservative force. The 1'' form takes the expression
A. 2. Classical Mechanics: Lagrangian and Hamiltonian formalisms
443
and so,
d
-(T dt
+ V ) = C F:&, i
called the 3‘d form of the energy equation. Up to now we have supposed no restrictions on the motion of the particle. However, in general, there are some restrictions, like, for example, a particle moving on a plane, sphere, etc. We call these restrictions by constraints. We will be concerned with only a type of constraints, called holonomic, which are characterized by equations of type
depending only on the position.
A.2 A.2.1
Classical Mechanics: Lagrangian and Hamilt onian formalisms Generalized coordinates
Suppose that P is a mechanical system on P Nand the number of holonomic constraints is r . Then the family of functions fj(z1,...,%3N)=O,1 S . i L r
which characterize the constraint relations define a subspace C of p N of dimension m = 3 N - r. We call C the configuration space (it is assumed to be an embedded manifold in PN). Locally C is characterized by a family of functions
where V is an open neighborhood of R3N such that the Jacobian of q A ( q , . . . , z 3 ~ with ) respect to the z’s is non-vanishing at every point of V n C. In the following we will identify C with the Euclidean space Rm. We call the functions q A , 1 5 A 5 m, by generalized coordinates of the system P . Thus the introduction of the generalized coordinates allows us consider a new set of independent coordinates in terms of which the old coordinates are expressed by the relations
444Appendix A
Z i = Z i ( q 1) . . . ) q m ) )
1<;<3N
as we are supposing the regularity of the Jacobian of q A with respect to the 2's. Thus, we have
and so (A.12)
The derivation of the kinetic energy T with respect to qB (resp. q B ) gives (A.13)
(A.14)
where the last expression in (A.14) is obtained from
Now, if we derive (A.14) with respect to the time t one has
d dt
(-)aaTp
+xm.i.(-) ' 3N
= x3 Nm i z diX vi i=l
i= 1
d 'dt
aqB
Taking into account (A.14) one obtains (A.15)
A.2. Classical Mechanics: Lagrangian and Hamiltonian formalisms
445
since
If the mechanical system is conservative we have
and so (A.15) takes the form
A.2.2 Euler-Lagrange and Hamilton equations As the potential energy is supposed to be a function depending only on position (and time) we have (aV/dqB)= 0 and the above expression is
d (-(Ta Setting L = T
a
- V ) ) - -(T aqB
dt
ap
-V
, we have
- V ) = 0.
(A.16) known by the name of Euler-Lagrange equations for a conservative meA rn, is called chanical system. The function L = L ( t , @ , q A ) , 1 the Lagrangian of the system. In the non-conservative situation (A.16) assumes the form
<
d
-
dt
dL (-)aqB
-
aL
-=
dqB
<
KB
with K B = Q~ - (av/aqB). In the following we suppose that L is a function depending implicitly in t , i.e., L : Rm x Rm R. Let u s set
-
uA = iA;LA = ( a L / a v A ) ( qB , U B ) and assume that
446Appendix A
is non-singular at a point (qB,ijB). Consider the Euclidean space Rm x R"' x Rm with coordinates ($,uA,pA). For ( F , i j B )let , -B - B
LA(q
,U
- -A
)-P , 1 L B I : m , l l A < m
(A.17)
and define the mappings
Setting
we have from (A.17) B - B -B
G(Q , v ,P
= (o,**.,o)*
On the other hand the matrix (dG/avA)($,uB,pB) is
which is non-singular at ( p ,c B , p B ) . Thus, from the Implicit function theorem there is locally a unique function ( $ 1 , . . . ,$ M ) such that
So, there are m-cordinates (p', . . . ,p") such that the velocities d ,. . . , urn are locally dependent functions of (q', . . . , q m , p l , . . . ,p"). In the following we suppose that such result is valid at least on an open subset U of Rm x Rm. Furthermore, we will set (see(A.17))
aL
pA==,
I ~ A F ~ .
(A.18)
These coordinates are called generalized momenta. Hence, the introduction of such coordinates is a consequence of the non-degeneracy of
A.2. Classical Mechanics: Lagrangian and Hamiltonian formalisms
447
the Hessian matrix of L with respect to the velocities, (at least on an open neighborhood of Rm x IF).In such a case we say that L is regular (if the non-degeneracy property is valid on all R"' x Rm then we say that L is hyperregular). Let us consider again the (conservative) Euler-Lagrange equations:
If we develop this expression we obtain
i.e., the above equations are of type m
(A.19) B= 1
where
Now, let us set qA = t A . Then we may replace in an equivalent way the m Euler-Lagrange equations (A.19) by the 2m differential equations of first-order Q* A --- A ,
~ A = ~ A B( ,Q rB )
(A.20)
This is a well-known method to search solutions of (A.19). However we will develop a different view point which consists in the utilization'of new coordinates due to Hamilton. For this, consider the transformation
which is at least a local diffeomorphism and (A.18) permits to express the velocities 1's iii terms of the momenta p's. Conversely, the inverse of the above transformation gives the momenta in terms of the positions and velocities. To give such inverse consider the function .._
H =
C pAqA A=l
-
L
(A.21)
448Appendix A This function depends on ( q A , q A , p A ) but as we are supposing L regular
Thus
-
gives the regularity condition for the p's be functions of the q's and q's. The function H : R"' x Rm R defined by (A.21) is called Hamiltonian function of the mechanical system. The space of the variables ( f l , p A ) is called phase space of the mechanical system. Now, consider the differential of (A.21): m
dH = C ( p A d q A A= 1
dL + 4A dpA ) - -d$ dqA
-
m
= C q .A d p A - - da$L A= 1
(A.22)
&IA
On the other hand
dH =
dH dH C -dqA + -dPA A=l dqA dpA
(A.23)
Thus, (A.22) and (A.23) give (A.24) but from the Euler-Lagrange equations we have
-d=L- ( - )d= - (dL p dqA dt dqA
A ) - - . pA
dt
and so (A.24) is (A.25) These equations are called Hamilton equations, canonical form of Euler-Lagrange equations or simply canonical equations.
A.2. Classical Mechanics: Lagrangian and Hamiltonian formalisms
449
We have used above an important mapping called Legendre transformation. It can be considered as a transformation which associates a regular function L defined on Rm x R"' a function H from R"' x R m to R defined by (A.21). We represent by L e g L : Rm x Rm --Ri " x Rm the map ( $ , q A ) ( $ , p A ) , where p A is given by (A.18) and the inverse A A ( m L ) - ' : ( q ,P ) ($AA) where
--
with L = CpAqA- H , i.e., ( L e g L ) - ' is L e g H . Let us consider a holonomic conservative mechanical system and suppose H = H ( t , q A ( t ) ,p A ( t ) ) . Then
aH d - H = - + E dt at
dH
=a+ U L
dH -d'A
aH
+
A= 1 a q A
-dpA A= 1
c(-p + c
aPA
m
+
A
)q
A
qAjA
(from (A.25)
A= 1
A=l
and so
Thus, if H does not depend explicitly on t , H assumes the same value along the motion of the system, i.e., H = constant. If we consider the kinetic energy
as
A
we have
A
45OAppendix A
H = 2T
-
L = 2T
-T
+ V = T + V = constant = E ,
the total energy of the system. Thus a sufficient condition for the Hamiltonian be the total energy of the mechanical system is: (a) the system is conservative; (b) the Lagrangian (resp. Hamiltonian) is only implicitly dependent on the time. The converse is not generally true (see Goldstein for example). Finally a first integral of a system of ordinary differential equations is a function which is constant along every integral curve (solution) of the system. Let p = p(qA,pA)be a function. For every integral curve (fl(t),pA(t))we have
ap
ap aPA ap
ap
aH
d p -- -d$ aH + -- - -- - -dt a q A dt apA dt aqAapA apAaqA
(+% if p depend explicitly on t ) . Thus, p is a first integral if (A.26) The expression (A.26) is represented by {p,H}, called Poisson brackets. Thus, for autonomous mechanical systems we have {H, H} = 0, i.e., the Hamiltonian is a first integral.
45 1
Appendix B
Higher order tangent bundles. Generalities The theory of Jet manifolds introduced by Ch. Ehresmann around 1950 is an important topic in modern differential geometry. We shall give a brief outline of this theory and we shall examine a particular case concerning higher order tangent bundles the place where the geometric formulation of higher order Lagrangian particle mechanics is developed. For a more general description involving Lagrangians depending in many independent variables with higher order derivatives see our book in this series (de Le6n and Rodrigues [38]). Our article [41] gives also another approach for this subject.
B.l
Jets of mappings (in one independent variable)
Let M be a manifold of dimension m and R the Euclidean space with coordinate t . In what follows we shall assume that all the mappings are Co3-class. M and g : R M two mapping such that f ( t ) = g ( t ) , for a Let f : R fixed t E R. We say that f and g are Nk-related (or that they are tangent to the k-th order) at t E R if for all function h : M R the function
-
-
-
is “flat” of order k at t , i.e., this function and all their derivatives up to order k, included, vanishes at t . The equivalence classes determinedd by -k are called jets of order k, or simply, k-jets, with source t and same target.
Appendix €3. Higher order tangent bundles. Generalities
452
-
M at t E R is denoted by j!f or f k ( t ) . The k-jet of a mapping f : R The set of all k- jets at t is denoted by $ ( R , M ) and we set J k ( R ,M) for the union J k ( R ,M ) =
u J / ( R ,M ) tER
(we remark that we can define also in a similar way k-jets of local m a p pings). It can be shown that J k ( R ,M ) has a (k l ) m dimensional manifold structure. J k ( R ,M ) can be fibered in different ways: 0
a Bource projection ' a : J k ( R ,M )
0
a target projection p k : J k ( R ,M )
-
--
+
R;
(Y k
(f"k ( t ) )= t
M ; p k (f"k ( t ) )= f ( t )
and the projection p,k : J k ( R ,M ) J ' ( R , M ) ; p , k ( f k ( t ) ) = f;(t), where r 5 k. The mapping which associates to each point t E R the k-jet of f : R -----t M at t is called the k-jet prolongation of f and is represented by j k f or f k . hence f k : R J k ( R ,M ) is defined by t f k ( t ) and f k is a section of the fibred manifold ( J k M , a k R). ,
-
-
B.2 Higher order tangent bundles Let us now consider the particular situation t = 0 E R. Then the submanifold J t ( R ,M ) is denoted by T k M and called the tangent bundle of order k of M . One has, of course, the same type of fibrations as above. In fact, if r 5 k, we have the canonical projection p: : T k M T'M given k - k (0)) = G'(0) and the target projection p k : T k M M given by by p,(a p k (Gk (0)) = o(O), where D : R + M is a function. Obviously p i = p k and
--
TOM is identified with M if k = 0 and T ' M with T M if k = 1. We shall now describe the local coordinates for T k M . Let U be a chart of M with local coordinates yA, 1 5 A 5 m, u : R M a curve in M such that ~ ( 0E)U and set u A = # 0 6 ,1 5 A 5 m. Then the k-jet Gk(0) is uniquely represented in ( p k ) - l ( U )= T k U by
-
where
B.2. Higher order tangent bundles
453
(we will set yA = z;). Then we have a chart ( p k ) - l ( U ) in T k M with local coordinates (zoA ,z1A , . . . ,zf). The factor (l/i!) appears only for technical reasons. We may consider the following coordinate system in ( p k ) - ' ( U ) : A
A
A
(9 ,q1 , - * * , q k ) , 1 5 A 5 m,
where qA = " A ( 0 ) and = i! z t , 0 5 i 5 k, 1 5 A 5 m. Now, let u be a curve in M . We shall denote by G k the canonical prolongation of u t o T k M defined as follows: 6 k ( t ) = 6!(0),
+
where at(.) = a ( s t ) . If k = 1 we put 6' = 6.Along the prolongation G k to T k M of a curve a in M we have q t = ( d ' a A / d t ' ) , 0 5 i 5 k. Let h : M -+ R be a function. The lift of h to T k M is defined as follows: for each i E { 1 , 2 , . . . ,k} the < i >-lift h<'> of h is
h<'>(Gk(0))= (l/i!)(d/dt')(h
o o)(O),
for every curve a in M . Clearly zf = ( t A ) < ' > Let. X be a vector field on M . Then the < i >-lift of X t o T k M , denoted by X<'> is the unique vector field on T k M such that
for every function f : M X A ( d / a z A )then
-
R and j E { 1 , 2 , . . . ,k}. Locally, if X =
(we set"'f = 0 if l > k). In particular ( d / 8 z A ) < J >= ( a / d z f ) . We are now able to define the < j >-lift of a tensor field F of type ( 1 , l ) on M to a tensor field F<j' of type ( 1 , l ) on T k M . The < j >-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<'> = ( F X )
for every vector field X on M and i E { 1 , 2 , . . . ,k}. We call F the complete lift of F to T k M .
Appendix B. Higher order tangent bundles. Generalities
454
B.3
The canonical almost tangent structure of order k
The use of the above lifting procedure on the identity operator IM of M permits us to define a unique tensor field J1 which endows T k M with an almost tangent structure of order k. It is the tensor field of type ( 1 , l ) giving by J1 = I&’’ with rank J1 = km and J: = J1 o . . . o J1 (k-times) # 0 and Jf+l = 0. The composition of J1 r-times gives more (k - 1) tensor fields which are locally expressed by
(in particular k- 1 i=O
An exterior calculus generated by these tensor fields may be constructed: an interior product iJrdefined by P
( ; J , w ) ( x ~* ,*
xp)
=
C ~(x1,. Jrxi,.. . , *.
>
xp)
(B-1)
i=l
where w is a p f o r m and Xi are vector fields on T k M , and an exterior differentiation d J , defined by
where d is the usual exterior differentiation.
B.4
The higher-order Poincar6-cart an form
-
-
Let ( o k : T k M --+ T ( T k - ’ M ) be the mapping given by j$u j i r , where r :R Tk-’M is defined by t --t ~ ( t =) $-‘at with u t ( s ) = u(s t). Then locally
+
B.4. The higher-order Poincar6-Cartan form
455
We use this map to construct the Tulczyjew differential operator, represented by d ~which , maps each function f on T k M into a function dTf on Tk+'M defined by dT
f ($+ 4 = df bo".
(P h + 1 (2'
1.
Locally, we have k
dTf(P,Z?) =
C(i+ l)z&(af/az?). i=O
In particular, dT(z$) = ( i
+ l)e+l and (FT(zA)= dT
0..
.o
= r!zt.
The operator dT extends in a very natural way to an operator which maps p-forms on T k M into p f o r m s on Tk++'M.Also, we have ddT = dTd. We may now construct the canonical vector fields on T k M , generalizing the Liouville vector field on the tangent bundle T M . In order to do this let us recall that the vertical lift of a vector field Y on Tk-'M to T k M with respect to the projecction p k : T k M M is the unique vector field Y " k on T kM given by
-
) Y'k = for every function f on M . Locally, if Y = C f z , l T A ( ~ / ~ z ?then Cfzt Y,A(a/a~&l). Now we construct the canonical vector field C1 on T k M as follows:
"'
c 1(j,"0) = ('Pk (jo"0))
and locally we have k-1 c1 =
C(i+ l)ziA+l(a/a%$l).
i=O
One obtains a family of vector fields C, on T k M , 2 5 r 5 k, defined by
C, = JICr-l and locally k-r
Cr =
C(i+ I)Z&l(a/az,A+i).
i=O
456
Appendix B. Higher order tangent bundles. Generalities
We may transport the geometric structures defined on T k M to R x T k M (which may identified with J k ( R ,M)): we set
and we may define in a similar way to ( B . l ) and (B.2) the operators iJ, and d J r . One has, for instance
d J , f = iJrdf = (Jr)*(df) = ( J r ) * ( d f )- ( C r f ) d t .
-
With such structures we define the Poincard-Cartan form for a higherR by order time-dependent Lagrangian L : R x T k M
If we develop this definition in local coordinates q t = i ! @ , 0 5 i 5 k and if we set 0s = d q t - q$,dt, then one obtains the expression k- 1
i=O where k-i-1
~ 2=’ C
(-l)Jd’/dtJ(aL/aqiA,j+l), 0 5 i
I k - 1.
j=O
It is possible to show that this approach maintains the main ideas of tangent bundle geometry and we suggest the reader the references quoted in the introduction of this Appendix.
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471
Index Abelian Lie algebra, 83 action of a group, on the left, 92 on the right, 91 arc length of a curve, 80 adapted coframe, t o a linear connection, 131 field, 102 adapted frame, 101 field, 101 t o an almost complex structure, 152 t o an almost Hermitian structure, 162 to an almost contact structure, 171 t o an almost product structure, 148 t o an almost tangent structure, 112 to an f-structure, 176 t o a linear connection, 131 adapted projector, t o a presymplectic form on a vector space, 233 adjoint, operator J * , 181 representation, of a Lie algebra, 90 of a Lie group, 89 admissible Lagrangian system, 428
affine, bundle, 138 morphism, 138 space, 138 affine space modeled on a vector space, 138 almost complex, connection, 156 manifold, 15 1 structure, 151 integrable almost complex structure, 155 on the tangent bundle, 165, 166, 218 on the cotangent bundle, 249 almost contact, metric structure, 170 structure, 169, 172 associated t o a dynamical connection, 342 compatible metric for an almost contact structure, 170 almost cosymplectic structure, 172 almost cotangent, manifolds, 253 structures, 253 integrable, 255 regular, 259 equivalent, 259 almost product, connection, 150
472 integrable almost product structure, 149 manifold, 147 structure, 147 on the cotangent bundle, 248 almost symplectic, connection, 261 form, 234 manifold, 234 structure, 234 almost tangent, canonical almost tangent structure on T N , 115 connection, 119 integrable almost tangent structure, 116 manifold, 111 structure, 111 alternating mapping, 38 automorphism preserving the symplectic form, 229 basic form on T * M , 368 +basis, 171 Bianchi's identities, 79 boundary, of H m , 66 manifold with boundary, 67 manifold without boundary, 67 bracket product, 82 of derivations, 45 of skew-derivations, 45 of a derivation and a skew-derivation, 45 bump function, 10 bundle, 22 base space of a bundle, 22 isomorphism, 23 morphism, 23
INDEX of linear frames, 95 projection of a bundle, 22 total space of a bundle, 22 trivial bundle, 22
Canonical, coordinates, 251 equations, 265, 448 l-form, 348 transformation, 236 Campbell-Haussdorff formula, 393, 397 Caplygin systems, generalized Caplygin systems, 391 Cart an, formula, 54 symmetry, 366 Theorem, 86 Cartesian product of vector bundles, 25 C'-atlas, 6 change of variables rule, 65 characteristic vector field, 291 Christoffel components, 76,201,223 coisotropic, submanifold, 236 subspace, 231 vector subbundle, 232 compact support, 10, 44 CM-compatible coordinate neighborhoods, 6 complete vector field, 20 complex, coordinates on a complex manifold, 154 frame, 152 manifold, 154 structure, 154
INDEX components, of a contravariant tensor, 34 of a covariant tensor, 34 of a tensor field, 36 of a tensor of type (r, s), 34 configuration, manifold, 265, 301 space, 443 connection, form, 221 in a fibred manifold, 197 in a principal bundle, 221 in a tangent bundle, 200 path of a connection, 204 symmetric connection, 78 conservative, connection, 314 force, 441 contact, form, 169 manifold, 169 metric structure, 172 structure, 169 constants of motion, 273 constraint, 295 anholonomic constraint, 297 canonical constraint forces, 296 classical constraint, 367 holonomic constraint, 297 manifold, 400 system of constraints, 295 contraction operator, 35 contravariant, tensor space of degree r , 33 tensor of degree r, 33 convex neighborhood, 81 coordinate, functions, 5 neighborhood, 5, 67, 154
473 corank, of a 2-form, 227, 234 of a Poisson tensor field, 285 cotangent, bundle, 23, 30 vector space, 28 covariant derivative, of a curve, 204 of a vector field, 76, 203 of a tensor field, 78 covariant tensor, space of degree s, 33 of degree s, 33 critical section, 351 cross-section submanifold, 349 cup product, 71 curvature form, 216, 222 curvature tensor, 78, 218 Riemannian curvature tensor, 107 curve, 15 Darboux Theorem, 250 Generalized Darboux Theorem, 252 de Rham, algebra, 71 cohomology group, 71 theorem, 72 Decomposition Theorem, 215 derivation, of C*"(Z), 21 of AM, 45, 46 derivation of A M of degree k, 45 determinant, of a linear map, 107 of a differentiable mapping, 107 deviation vector field, 430
474 diffeomorphic manifold, 8 diffeomorphism, 4,8 preserving orientations, 62 reversing orientations, 62 preserving volume forms, 62 differentiable, function, 3 invariant, 72 manifold, 6 mapping, 4,8,66 structure, 6 differential, form, 43 of a mapping, 17 differential equation, first order differential equation, 18 second order differential equation, 193,333 distance function, 81 distinguished coordinates, 60 distribution, 55 completely integrable distribution, 55 involutive distribution, 55 local basis of a distribution, 55 dynamical connection, 336 components of a dynamical connection, 336 path of a dynamical connection, 340 dynamical symmetry, 366 embedded submanifold, 12 embedding, 12 of principal bundles, 97 energy, conservation law, 273,382,442
INDEX first form of the energy equation, 442 function, 263 second form of the energy equation, 442 third form of the energy equation, 443 Equivalence Theorem, 412 (e)-structures, 102 Euler-Lagrange, equations, 304,335, 445 with constraints, 368 vector field, 304,336 evolution space, 330 exponential map, 84 exterior, algebra, 40 derivative, 47 product, 39 extremal, 351 f-basis, 176 fibration, defined by an almost cotangent structure, 258 defined by an almost tangent structure, 140 fiber derivative, 329 fibre, of a bundle, 22 of a submersion, 14 fibred manifold, 22 final constraint manifold, 401 first class, constraint function, 406 constrained manifold, 275 function of first class, 406 first integral, of a vector field, 272
INDEX of a system of differential equations, 450 first-order, problem, 400 equivalent Hamiltonian systems, 412 foliation, 60 leaves of a foliation, 60 force field, 305 l-forms, 28, 29 p f o r m , 43 closed p f o r m , 71 exact p f o r m , 71 frame bundle, 95 Frobenius Theorem, 56 f-structure, 176 integrable f-structure, 178 metric f-structure, 178 partially integrable f-structure, 178 Riemannian metric adapted to an f-structure, 178 f(3,l)-structure, 176 f(3, -1)-structure, 179 fundamental form, of an almost contact metric structure, 172 of an almost Hermitian structure, 163 gauge equivalent Lagrangians, 436 gauge freedom, 383 G-connection, 225 general linear group, 7, 82 complex general linear group, 85 generalized, coordinates, 443 momenta, 446
475 generating function, of a canonical transformation, 298 of a Lagrangian submanifold, 279 geodesic, of a linear connection, 76, 204 minimal geodesic, 81 geodesically complete, 77 geometric algorithm of presymplectic systems, 401 germ of a function, 21 global dynamics, 435 Grassman algebra, 40 Green’s Theorem, 70 G-structures, 100 automorphism of G-structures, 103 integrable G-structures, 105 isomorphism of G-structures, 103 local automorphism of G-structures, 104 local isomorphism of G-structures, 104 locally isomorphic G-structures, 103 standard G-structures, 104 Hamilton equations, 265, 293, 448 intrinsical form of Hamilton equations, 265 symplectic form of Hamilton equations, 265 Hamilton’s principle, 351 modified Hamilton’s principle, 354 Hamilton-Jacobi method, 279 Hamiltonian,
476 energy, 263, 328 functional, 350 regular Hamiltonian, 369 system, 263 with constraints, 296 vector field, 263, 282 locally Hamiltonian vector field, 263 Helmholtz conditions, 323, 395 Hermi tian, almost Hermitian manifold, 161 almost Hermitian structure, 161 manifold, 161 metric, 161 holonomic section, 333 homogeneous, connection, 202 form, 189 function, 187 vector field, 188 vector form, 190 Hopf-Rinow Theorem, 81 horizontal, bundle, 197 component of a vector field, 197 distribution, 130, 197 projector, 197 subspace, 197 tangent vector, 197 vector field, 197 Ideal, differential ideal, 60 of AM,60 image of a vector bundle homomorphism, 25 immersed submanifold, 11 immersion, 11 independent functions, 275
INDEX induced coordinates, in TM, 27 in T*M, 30 infinitesimal con tact transformation, 352 inner automorphism of a Lie group, 89 integral, curve, 18 lattice, 115 manifold, 55 of a form, 66 interior product, 51, 52 Inverse Function Theorem, 4 involution, canonical involution of TTQ, 359,395 l-forms in involution, 273 functions in involution, 273 isometry, 104 isomorphic, bundles, 23 vector bundles, 23 isotropic, submanifold, 236 subspace, 231 vector subbundle, 232 Jacobi, field, 129 identity, 22, 82 Theorem, 276 Jacobian matrix, 4 jets, k-jets, 451 of order k, 451 Kahler, almost Kahler manifold, 165 form, 163
INDEX manifold, 165 kernel, of a vector bundle homomorphism, 25 condition, 435 Killing vector field, 110 kinetic energy, 329 Lagrange operator, 378 Lagr angian, almost regular Lagrangian, 411 degenerate Lagrangian, 302 function, 301, 445 admitting a map, 362 admitting a vector field, 363 homogeneous Lagrangian, 306 hyperregular Lagrangian, 327, 447 irregular Lagrangian, 302 non-autonomous Lagrangian, 330 non-degenerate Lagrangian, 302 regular Lagrangian, 302, 447 singular Lagrangian, 302 system, 306 with constraints, 367 submanifold, 236 generated by a Lagrangian, 279, 358 subspace, 231 time-dependent Lagrangian, 330 vector subbundle, 232 left, invariant vector field, 83 invariant form, 83 translation, 83 Legendre transformation, 327,449 Leibniz rule, 21 Levi-Civita connection, 79
477 Lie, algebra, 82 automorphism, 88 homomorphism, 88 isomorphism, 88 of a Lie group, 84 bracket, 22 derivative of a form, 52 derivative of a tensor field, 55 derivative of a vector field, 106 group, 81 acting effectively on the left, 92 on the right, 91 acting freely on the left, 92 on the right, 91 automorphism, 88 homomorphism, 88 isomorphism, 88 subalgebra, 85 subgroup, 85 symmetry, 364 transformation group, 91 lift, complete lift of a distribution to T N , 125 complete lift of a function to T N , 121 complete lift of a linear connection to T N ,126 complete lift of a tensor field of type (1,r) to T N , 124 complete lift of a tensor field of type (0,r) to T N , 124 complete lift of a tensor field of type (1,l)to T * M , 244 complete lift of a tensor field of type (1,2) to T * M , 245 complete lift of a vector field
478
INDEX
to T N , 122 complete lift of a vector field to T * M , 243 horizontal lift of a distribution to T N , 134 horizontal lift of a of a l-form to T N , 131 horizontal lift of a linear connection to T N , 134 horizontal lift of a tensor field of type ( 1 , l ) to T N , 132 horizontal lift of a tensor field of type (1,r) to T N , 133 horizontal lift of a of a tensor field of type (0,2) to T N , 133 horizontal lift of a vector field to T N , 130,201 natural lift, 103 vertical lift of afunction to T N , 121 vertical lift of a tangent vector to T N , 114 vertical lift of a tensor field of type (1,r) to T N , 120 vertical lift of a tensor field of type (0,r) to T N , 121 vertical lift of a vector field to T N , 114 vertical lift of afunction to T * M , 240 vertical lift of a a l-form to T * M , 241 lifts to the tangent bundle of order
k, of functions, 453 of tensor fields, 453 of vector fields, 453 linear connection, 75, 205, 224
in a tangent bundle, 202 symmetric linear connection, 78 flat linear connection, 79 linear frame, 94 Liouville, form, 238 property, 287 Theorem, 271 vector field, 181, 287 locally Liouville, 287 local, coordinate system, 5 frame field, 96 locally finite covering, 9 Maurer-Cartan equation, 91 maximal, Cco-atlas, 6 integral manifold, 60 mechanical system, 305 conservative mechanical system, 306,442 non-conservative mechanical system, 306 with constraints, 367 metric in a vector bundle, 106 mixed tensor space of type (r, s), 34 momentum, 327 Momentum conservation law, 441 Newlander-Niremberg Theorem, 156 Newton’s second law of motion, 439 Nijenhuis tensor, of an almost complex structure, 156 of a1 almost tangent structure, 116 Nijenhuis torsion, 37 Noether symmetry, 365
INDEX Noether’s Theorem, 363, 390 non-degenerate, 2-form, 227, 234 Poisson tensor field, 285 normal, almost contact metric structure, 174 coordinates, 391 n-submanifold property, 12 O(m)-structures, 102 open submanifold, 7 orientation, induced orientation, 68 on vector spaces, 61 orientable manifold, 62 oriented vector space, 61 orthocomplement respect t o a symplectic form, 230 orthogonal group, 87, 102 automorphism, 109 complex orthogonal group, 87 special orthogonal group, 88 orthonormal frame, 103 bundle, 103 paracompact topological space, 9 parallel, curve, 204 vector field, 76 tensor field, 78 parallelizable manifold, 102 l-parameter group of transformations, 19 time-dependent, 249 infinitesimal generator of a 1parameter group, 19 local l-parameter group of local transformations, 19
479 l-parameter subgroup of a Lie group, 84 partial derivative, 3 partition of unity subordinate to an atlas, 10 P-Euler-Lagrange vector field, 414 phase space, of momenta, 265 of velocities, 301 Poincar6 Lemma, 72 Poincar6-Cart an, l-form, 334 2-form, 334 form of higher order, 456 Poisson, bracket of l-forms, 268 bracket of functions, 267, 282, 450 manifold, 282, 284 map, 298 structure, 282 tensor field, 283 polar decomposition, 109 positive definite automorphism, 109 positively oriented, basis, 62, 68 coordinate neighborhood, 63 potential energy, 441 presymplectic, forms on manifolds, 251 forms on vector spaces, 233 manifold, 25 1 vector space, 233 structure, 251 principal bundle, 93 base space of a principal bundle, 94 bundle space of a principal bundle, 94
480 homomorphism, 96 projection of a principal bundle, 93 section of a principal bundle, 96 total space of a principal bundle, 94 transition functions of a principal bundle, 97 trivial principal bundle, 94 vertical subspace of a principal bundle, 94 vertical tangent vector to a principal bundle, 94 principal subbundle, 97 product manifold, 7 projectable vector field, 351 prolongable vector field, 430 prolongation, canonical prolongation, 333,453 l-jet prolongation, 332, 352 k-jet prolongation, 452 P-second order differential equation, 415 P-semispray, 415 quotient manifold, 14 r-ary constraint manifold, 401 range condition, 399 rank, 2-form of maximal rank, 227, 234 of a function, 4 of a mapping, 8 of a 2-form, 227, 234 of a Poisson tensor field, 285 theorem, 5 theorem for manifolds, 9 real projective space, 14
INDEX reduced bundle, 97 reduced Lagrangian system, 436 Reeb vector field, 169, 290 refinement, 9 regular, condition for Poisson brackets, 274 consistent constraints, 401 2-form, 227 Lagrangian dynamical system, 322 Lagrangian vector field, 379 non-conservative regular Lagrangi vector field, 388 representation, of a Lie algebra, 88 of a Lie group, 88 Ricci tensor, 108 Riemannian, connection, 79 induced Riemannian metric, 107 metric, 37 prolongation, 219 pseudo-Riemannian metric, 38 structure, 102 right translation, 83 Sasaki metric, 135 Sasakian almost contact metric structure, 175 scalar curvature, 108 Schouten bracket, 283 second class, function of second class, 406 second-order problem, 410 section of a bundle, 22 sectional curvature, 108 semibasic forms, 190 algebra of semibasic forms, 191
INDEX on T ” M , 368 potential of semibasic forms, 195 semibasic vector form, 193 potential of a semibasic vector form, 195 7r-semibasic forms, 331 7r2-semibasic forms, 331 semi-prolongable vector fields, 431 J-equivalen t semi-prolongable vector fields, 432 semispray, 193, 333 associated semispray, 206, 337 deviation of a semispray, 194 path of a semispray, 194 solution of a semispray, 194 signature, of a symmetric tensor of type (0,2), 145 of a pseudo-Riemannian metric, 145 skew-derivation of A M , 45 skew-derivation of degree k, 45 skew-Hermitian matrix, 87 skewsymmetric matrix, 87 shuffles permutations, 39 simple connection, 314 source projection, 452 special, linear group, 87 real special linear group, 87 symplectic manifold, 357 unitary group, 86 spray, 195 quadratic spray, 195 standard orientation of R”, 62 Stoke’s Theorem, 68 strong, curvature, 341
48 1 J-torsion, 341 J-torsion, 341 strong horizontal, bundle, 338 lift, 338 projector, 339 subspace, 338 vector field, 338 structure, constants, 91 equation, 222 l-form, 348 group of a principal bundle, 94 reducible group of a principal bundle, 97 reduction of the group of a principal bundle, 97 submersion, 13 support, of a function, 10 of a form, 44 symmetric, automorphism, 109 product, 39 symmetrizing mapping, 38 symplectic, automorphism, 229 basis, 229 canonical symplectic form on T * M , 239 coordinate neighborhood, 251 coordinates, 251 diffeomorphism, 236 form on manifolds, 236 form on a vector space, 229 frame, 234 group, 229 isomorphism, 229 linear mapping, 229
482 manifold, 236 vector bundle, 232 vector field, 237 vector space, 229 structure on manifolds, 236 structure on a vector space, 229 subspace, 231 submanifold, 236 transformation, 236 infinitesimal symplectic transformation, 237 vector subbundle, 232 symplectomorphism, 104, 236 tangent, bundle, 23, 27 bundle of order k, 452 covector, 28 curves at a point, 15 curves at the order k, 451 space, 16 vector, of a curve, 15 target projection, 452 tension of a connection, 202 tensor, algebra, 35 field of type ( r , s ) on a manifold, 36 of type ( r , s), 34 product, 31 space of type ( r , s), 34 skew-symmetric tensor, 36 symmetric tensor, 36 topological, invariant, 72 manifold, 5 torsion, strong torsion, 211
INDEX tensor, 78, 213 weak torsion, 211 total force, 439 transformation of a manifold, 8 Tulczyjew differential operator, 455 unitary, frame, 162 group, 86 universal factorization property, 31 variation, 350 variation vector field, 380 variational problem, 350 vector bundle, 23 homomorphism, 23 induced vector bundle, 24 M-vector bundle homomorphism, 23 real vector bundle, 23 isomorphism, 23 M-vector bundle isomorphism, 23 local triviality for a vector bundle, 23 quotient vector bundle, 25 restricting a vector bundle, 24 transition functions of a vector bundle, 110 trivial vector bundle, 24 vector field, 17 fundamental vector field, 95 time-dependent vector field, 249 vector fields related by a mapping, 106, 298, 371 vector form, 190 vector subbundle, 25 vertical, bundle of T N ,114 bundle of T * M , 240
INDEX component of a vector field, 197 derivation on T M , 183 distribution, 197 differentiation on T M , 184 projector, 197, 339 vector field on T N , 114 vector field on T * M , 241 subspace to T N ,197 tangent vector to T N , 114 tangent vector to T * M , 240 volume form, on a manifold, 62 natural volume form, 63 on a vector space, 61 weak, curvature, 341 J-torsion, 341 1-torsion, 341 weak horizontal, bundle, 338 lift, 338 projector, 339 subspace, 338 vector field, 338 Whitney sum of vector bundles, 25
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