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Lectures on Symplectic Geometry




Ana Cannas da Silva1




1
E-mail: acannas@math.ist.utl.pt or acannas@math.berkeley.edu
Foreword
These notes approximately transcribe a 15-week course on symplectic geometry
I taught at UC Berkeley in the Fall of 1997.
The course at Berkeley was greatly inspired in content and style by Victor
Guillemin, whose masterly teaching of beautiful courses on topics related to
symplectic geometry at MIT, I was lucky enough to experience as a graduate
student. I am very thankful to him!
That course also borrowed from the 1997 Park City summer courses on
symplectic geometry and topology, and from many talks and discussions of the
symplectic geometry group at MIT. Among the regular participants in the MIT
informal symplectic seminar 93-96, I would like to acknowledge the contributions
of Allen Knutson, Chris Woodward, David Metzler, Eckhard Meinrenken, Elisa
Prato, Eugene Lerman, Jonathan Weitsman, Lisa Je¬rey, Reyer Sjamaar, Shaun
Martin, Stephanie Singer, Sue Tolman and, last but not least, Yael Karshon.
Thanks to everyone sitting in Math 242 in the Fall of 1997 for all the com-
ments they made, and especially to those who wrote notes on the basis of which
I was better able to reconstruct what went on: Alexandru Scorpan, Ben Davis,
David Martinez, Don Barkauskas, Ezra Miller, Henrique Bursztyn, John-Peter
Lund, Laura De Marco, Olga Radko, Peter Pˇib´ Pieter Collins, Sarah Pack-
r ±k,
man, Stephen Bigelow, Susan Harrington, Tolga Etg¨ and Yi Ma.
u
I am indebted to Chris Tu¬„ey, Megumi Harada and Saul Schleimer who
read the ¬rst draft of these notes and spotted many mistakes, and to Fernando
Louro, Grisha Mikhalkin and, particularly, Jo˜o Baptista who suggested several
a
improvements and careful corrections. Of course I am fully responsible for the
remaining errors and imprecisions.
The interest of Alan Weinstein, Allen Knutson, Chris Woodward, Eugene
Lerman, Jiang-Hua Lu, Kai Cieliebak, Rahul Pandharipande, Viktor Ginzburg
and Yael Karshon was crucial at the last stages of the preparation of this
manuscript. I am grateful to them, and to Mich`le Audin for her inspiring
e
texts and lectures.
Finally, many thanks to Faye Yeager and Debbie Craig who typed pages of
messy notes into neat L TEX, to Jo˜o Palhoto Matos for his technical support,
a
A

and to Catriona Byrne, Ina Lindemann, Ingrid M¨rz and the rest of the Springer-
a
Verlag mathematics editorial team for their expert advice.


Ana Cannas da Silva

Berkeley, November 1998
and Lisbon, September 2000




v
vii
CONTENTS


Contents


Foreword v

Introduction 1


I Symplectic Manifolds 3

1 Symplectic Forms 3
1.1 Skew-Symmetric Bilinear Maps . . . . . . . . . . . . . . . . . . . 3
1.2 Symplectic Vector Spaces . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Symplectic Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4 Symplectomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . 7

Homework 1: Symplectic Linear Algebra 8

2 Symplectic Form on the Cotangent Bundle 9
2.1 Cotangent Bundle . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Tautological and Canonical Forms in Coordinates . . . . . . . . . 9
2.3 Coordinate-Free De¬nitions . . . . . . . . . . . . . . . . . . . . . 10
2.4 Naturality of the Tautological and Canonical Forms . . . . . . . 11

Homework 2: Symplectic Volume 13


II Symplectomorphisms 15

3 Lagrangian Submanifolds 15
3.1 Submanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.2 Lagrangian Submanifolds of T — X . . . . . . . . . . . . . . . . . . 16
3.3 Conormal Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.4 Application to Symplectomorphisms . . . . . . . . . . . . . . . . 19

Homework 3: Tautological Form and Symplectomorphisms 20

4 Generating Functions 22
4.1 Constructing Symplectomorphisms . . . . . . . . . . . . . . . . . 22
4.2 Method of Generating Functions . . . . . . . . . . . . . . . . . . 23
4.3 Application to Geodesic Flow . . . . . . . . . . . . . . . . . . . . 24

Homework 4: Geodesic Flow 27
viii CONTENTS


5 Recurrence 29
5.1 Periodic Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
5.2 Billiards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
5.3 Poincar´ Recurrence . . . . . . . . . . . . . . . . . . . . . . . . . 32
e


III Local Forms 35
6 Preparation for the Local Theory 35
6.1 Isotopies and Vector Fields . . . . . . . . . . . . . . . . . . . . . 35
6.2 Tubular Neighborhood Theorem . . . . . . . . . . . . . . . . . . 37
6.3 Homotopy Formula . . . . . . . . . . . . . . . . . . . . . . . . . . 39

Homework 5: Tubular Neighborhoods in Rn 41

7 Moser Theorems 42
7.1 Notions of Equivalence for Symplectic Structures . . . . . . . . . 42
7.2 Moser Trick . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
7.3 Moser Local Theorem . . . . . . . . . . . . . . . . . . . . . . . . 45

8 Darboux-Moser-Weinstein Theory 46
8.1 Classical Darboux Theorem . . . . . . . . . . . . . . . . . . . . . 46
8.2 Lagrangian Subspaces . . . . . . . . . . . . . . . . . . . . . . . . 46
8.3 Weinstein Lagrangian Neighborhood Theorem . . . . . . . . . . . 48

Homework 6: Oriented Surfaces 50

9 Weinstein Tubular Neighborhood Theorem 51
9.1 Observation from Linear Algebra . . . . . . . . . . . . . . . . . . 51
9.2 Tubular Neighborhoods . . . . . . . . . . . . . . . . . . . . . . . 51
9.3 Application 1:
Tangent Space to the Group of Symplectomorphisms ....... 53
9.4 Application 2:
Fixed Points of Symplectomorphisms . . . . . . . . . ....... 55


IV Contact Manifolds 57
10 Contact Forms 57
10.1 Contact Structures . . . . . . . . . . . . . . . . . . . . . . . . . . 57
10.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
10.3 First Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

Homework 7: Manifolds of Contact Elements 61
ix
CONTENTS


11 Contact Dynamics 63
11.1 Reeb Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . . 63
11.2 Symplectization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
11.3 Conjectures of Seifert and Weinstein . . . . . . . . . . . . . . . . 65


V Compatible Almost Complex Structures 67
12 Almost Complex Structures 67
12.1 Three Geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
12.2 Complex Structures on Vector Spaces . . . . . . . . . . . . . . . 68
12.3 Compatible Structures . . . . . . . . . . . . . . . . . . . . . . . . 70

Homework 8: Compatible Linear Structures 72

13 Compatible Triples 74
13.1 Compatibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
13.2 Triple of Structures . . . . . . . . . . . . . . . . . . . . . . . . . . 75
13.3 First Consequences . . . . . . . . . . . . . . . . . . . . . . . . . . 75

Homework 9: Contractibility 77

14 Dolbeault Theory 78
14.1 Splittings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
14.2 Forms of Type ( , m) . . . . . . . . . . . . . . . . . . . . . . . . . 79
14.3 J-Holomorphic Functions . . . . . . . . . . . . . . . . . . . . . . 80
14.4 Dolbeault Cohomology . . . . . . . . . . . . . . . . . . . . . . . . 81

Homework 10: Integrability 82


VI K¨hler Manifolds
a 83
15 Complex Manifolds 83
15.1 Complex Charts . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
15.2 Forms on Complex Manifolds . . . . . . . . . . . . . . . . . . . . 85
15.3 Di¬erentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

Homework 11: Complex Projective Space 89

16 K¨hler Forms
a 90
16.1 K¨hler Forms . . . . . . . . . . . . . . .
a . . . . . . . . . . . . . . 90
16.2 An Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
16.3 Recipe to Obtain K¨hler Forms . . . . .
a . . . . . . . . . . . . . . 92
16.4 Local Canonical Form for K¨hler Forms
a . . . . . . . . . . . . . . 94

Homework 12: The Fubini-Study Structure 96
x CONTENTS


17 Compact K¨hler Manifolds
a 98
17.1 Hodge Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
17.2 Immediate Topological Consequences . . . . . . . . . . . . . . . . 100
17.3 Compact Examples and Counterexamples . . . . . . . . . . . . . 101
17.4 Main K¨hler Manifolds . . . . . . . . . . .
a . . . . . . . . . . . . . 103


VII Hamiltonian Mechanics 105
18 Hamiltonian Vector Fields 105
18.1 Hamiltonian and Symplectic Vector Fields . . . . . . . . . . . . . 105
18.2 Classical Mechanics . . . . . ........ . . . . . . . . . . . . . 107
18.3 Brackets . . . . . . . . . . . ........ . . . . . . . . . . . . . 108
18.4 Integrable Systems . . . . . ........ . . . . . . . . . . . . . 109

Homework 13: Simple Pendulum 112

19 Variational Principles 113
19.1 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . 113
19.2 Principle of Least Action . . . . . . . . . . . . . . . . . . . . . . 114
19.3 Variational Problems . . . . . . . . . . . . . . . . . . . . . . . . . 114
19.4 Solving the Euler-Lagrange Equations . . . . . . . . . . . . . . . 116
19.5 Minimizing Properties . . . . . . . . . . . . . . . . . . . . . . . . 117

Homework 14: Minimizing Geodesics 119

20 Legendre Transform 121
20.1 Strict Convexity . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
20.2 Legendre Transform . . . . . . . . . . . . . . . . . . . . . . . . . 121
20.3 Application to Variational Problems . . . . . . . . . . . . . . . . 122

Homework 15: Legendre Transform 125


VIII Moment Maps 127
21 Actions 127
21.1 One-Parameter Groups of Di¬eomorphisms . . . . . . . . . . . . 127
21.2 Lie Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
21.3 Smooth Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
21.4 Symplectic and Hamiltonian Actions . . . . . . . . . . . . . . . . 129
21.5 Adjoint and Coadjoint Representations . . . . . . . . . . . . . . . 130

Homework 16: Hermitian Matrices 132
xi
CONTENTS


22 Hamiltonian Actions 133
22.1 Moment and Comoment Maps . . . . . . . . . . . . . . . . . . . 133
22.2 Orbit Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
22.3 Preview of Reduction . . . . . . . . . . . . . . . . . . . . . . . . . 136
22.4 Classical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 137

Homework 17: Coadjoint Orbits 139


IX Symplectic Reduction 141
23 The Marsden-Weinstein-Meyer Theorem 141
23.1 Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
23.2 Ingredients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
23.3 Proof of the Marsden-Weinstein-Meyer Theorem . . . . . . . . . 145

24 Reduction 147
24.1 Noether Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
24.2 Elementary Theory of Reduction . . . . . . . . . . . . . . . . . . 147
24.3 Reduction for Product Groups . . . . . . . . . . . . . . . . . . . 149
24.4 Reduction at Other Levels . . . . . . . . . . . . . . . . . . . . . . 149
24.5 Orbifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

Homework 18: Spherical Pendulum 152


X Moment Maps Revisited 155
25 Moment Map in Gauge Theory 155
25.1 Connections on a Principal Bundle . ........ . . . . . . . . 155
25.2 Connection and Curvature Forms . . ........ . . . . . . . . 156
25.3 Symplectic Structure on the Space of Connections . . . . . . . . 158
25.4 Action of the Gauge Group . . . . . ........ . . . . . . . . 158
25.5 Case of Circle Bundles . . . . . . . . ........ . . . . . . . . 159

Homework 19: Examples of Moment Maps 162

26 Existence and Uniqueness of Moment Maps 164
26.1 Lie Algebras of Vector Fields . . . . . . . . . . . . . . . . . . . . 164
26.2 Lie Algebra Cohomology . . . . . . . . . . . . . . . . . . . . . . . 165
26.3 Existence of Moment Maps . . . . . . . . . . . . . . . . . . . . . 166
26.4 Uniqueness of Moment Maps . . . . . . . . . . . . . . . . . . . . 167

Homework 20: Examples of Reduction 169
xii CONTENTS


27 Convexity 170
27.1 Convexity Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 170
27.2 E¬ective Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
27.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

Homework 21: Connectedness 175


XI Symplectic Toric Manifolds 177
28 Classi¬cation of Symplectic Toric Manifolds 177
28.1 Delzant Polytopes . . . . . . . . . . . . . . . . . . . . . . . . . . 177
28.2 Delzant Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
28.3 Sketch of Delzant Construction . . . . . . . . . . . . . . . . . . . 180

29 Delzant Construction 183
29.1 Algebraic Set-Up . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
29.2 The Zero-Level . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
29.3 Conclusion of the Delzant Construction . . . . . . . . . . . . . . 185
29.4 Idea Behind the Delzant Construction . . . . . . . . . . . . . . . 186

Homework 22: Delzant Theorem 189

30 Duistermaat-Heckman Theorems 191
30.1 Duistermaat-Heckman Polynomial . . . . . . . . . . . . . . . . . 191
30.2 Local Form for Reduced Spaces . . . . . . . . . . . . . . . . . . . 192
30.3 Variation of the Symplectic Volume . . . . . . . . . . . . . . . . . 195

Homework 23: S 1 -Equivariant Cohomology 197


References 199

Index 207
Introduction

The goal of these notes is to provide a fast introduction to symplectic geometry.
A symplectic form is a closed nondegenerate 2-form. A symplectic mani-
fold is a manifold equipped with a symplectic form. Symplectic geometry is
the geometry of symplectic manifolds. Symplectic manifolds are necessarily
even-dimensional and orientable, since nondegeneracy says that the top exte-
rior power of a symplectic form is a volume form. The closedness condition is
a natural di¬erential equation, which forces all symplectic manifolds to being
locally indistinguishable. (These assertions will be explained in Lecture 1 and
Homework 2.)
The list of questions on symplectic forms begins with those of existence
and uniqueness on a given manifold. For speci¬c symplectic manifolds, one
would like to understand the geometry and the topology of special submanifolds,
the dynamics of certain vector ¬elds or systems of di¬erential equations, the
symmetries and extra structure, etc.
Two centuries ago, symplectic geometry provided a language for classical
mechanics. Through its recent huge development, it conquered an independent
and rich territory, as a central branch of di¬erential geometry and topology.
To mention just a few key landmarks, one may say that symplectic geometry
began to take its modern shape with the formulation of the Arnold conjec-
tures in the 60™s and with the foundational work of Weinstein in the 70™s. A
paper of Gromov [49] in the 80™s gave the subject a whole new set of tools:
pseudo-holomorphic curves. Gromov also ¬rst showed that important results
from complex K¨hler geometry remain true in the more general symplectic cat-
a
egory, and this direction was continued rather dramatically in the 90™s in the
work of Donaldson on the topology of symplectic manifolds and their symplectic
submanifolds, and in the work of Taubes in the context of the Seiberg-Witten
invariants. Symplectic geometry is signi¬cantly stimulated by important inter-
actions with global analysis, mathematical physics, low-dimensional topology,
dynamical systems, algebraic geometry, integrable systems, microlocal analysis,
partial di¬erential equations, representation theory, quantization, equivariant
cohomology, geometric combinatorics, etc.
As a curiosity, note that two centuries ago the name symplectic geometry
did not exist. If you consult a major English dictionary, you are likely to ¬nd
that symplectic is the name for a bone in a ¬sh™s head. However, as clari¬ed
in [103], the word symplectic in mathematics was coined by Weyl [108, p.165]
who substituted the Greek root in complex by the corresponding Latin root, in
order to label the symplectic group. Weyl thus avoided that this group connote
the complex numbers, and also spared us from much confusion that would have
arisen, had the name remained the former one in honor of Abel: abelian linear
group.
This text is essentially the set of notes of a 15-week course on symplectic
geometry with 2 hour-and-a-half lectures per week. The course targeted second-
year graduate students in mathematics, though the audience was more diverse,

1
2 INTRODUCTION


including advanced undergraduates, post-docs and graduate students from other
departments. The present text should hence still be appropriate for a second-
year graduate course or for an independent study project.
There are scattered short exercises throughout the text. At the end of most
lectures, some longer guided problems, called homework, were designed to com-
plement the exposition or extend the reader™s understanding.
Geometry of manifolds was the basic prerequisite for the original course, so
the same holds now for the notes. In particular, some familiarity with de Rham
theory and classical Lie groups is expected.
As for conventions: unless otherwise indicated, all vector spaces are real and
¬nite-dimensional, all maps are smooth (i.e., C ∞ ) and all manifolds are smooth,
Hausdor¬ and second countable.
Here is a brief summary of the contents of this book. Parts I-III explain
classical topics, including cotangent bundles, symplectomorphisms, lagrangian
submanifolds and local forms. Parts IV-VI concentrate on important related
areas, such as contact geometry and K¨hler geometry. Classical hamiltonian
a
theory enters in Parts VII-VIII, starting the second half of this book, which
is devoted to a selection of themes from hamiltonian dynamical systems and
symmetry. Parts IX-XI discuss the moment map whose preponderance has
been growing steadily for the past twenty years.
There are by now excellent references on symplectic geometry, a subset of
which is in the bibliography. However, the most e¬cient introduction to a sub-
ject is often a short elementary treatment, and these notes attempt to serve that
purpose. The author hopes that these notes provide a taste of areas of current
research, and will prepare the reader to explore recent papers and extensive
books in symplectic geometry, where the pace is much faster.
Part I
Symplectic Manifolds
A symplectic form is a 2-form satisfying an algebraic condition “ nondegeneracy
“ and an analytical condition “ closedness. In Lectures 1 and 2 we de¬ne
symplectic forms, describe some of their basic properties, introduce the ¬rst
examples, namely even-dimensional euclidean spaces and cotangent bundles.


1 Symplectic Forms

1.1 Skew-Symmetric Bilinear Maps

Let V be an m-dimensional vector space over R, and let „¦ : V — V ’ R be
a bilinear map. The map „¦ is skew-symmetric if „¦(u, v) = ’„¦(v, u), for all
u, v ∈ V .

Theorem 1.1 (Standard Form for Skew-symmetric Bilinear Maps)
Let „¦ be a skew-symmetric bilinear map on V . Then there is a basis
u1 , . . . , uk , e1 , . . . , en , f1 , . . . , fn of V such that

for all i and all v ∈ V ,
„¦(ui , v) = 0 ,
„¦(ei , ej ) = 0 = „¦(fi , fj ) , for all i, j, and
„¦(ei , fj ) = δij , for all i, j.

Remarks.

1. The basis in Theorem 1.1 is not unique, though it is traditionally also
called a “canonical” basis.

2. In matrix notation with respect to such basis, we have
® ® 
|
0 0 0
„¦(u, v) = [ u ] ° 0 Id » ° v » .
0
0 ’Id 0 |

™¦

Proof. This induction proof is a skew-symmetric version of the Gram-Schmidt
process.
Let U := {u ∈ V | „¦(u, v) = 0 for all v ∈ V }. Choose a basis u1 , . . . , uk of
U , and choose a complementary space W to U in V ,

V =U •W .

3
4 1 SYMPLECTIC FORMS


Take any nonzero e1 ∈ W . Then there is f1 ∈ W such that „¦(e1 , f1 ) = 0.
Assume that „¦(e1 , f1 ) = 1. Let

W1 = span of e1 , f1
„¦
= {w ∈ W | „¦(w, v) = 0 for all v ∈ W1 } .
W1

„¦
Claim. W1 © W1 = {0}.
„¦
Suppose that v = ae1 + bf1 ∈ W1 © W1 .

0 = „¦(v, e1 ) = ’b
=’ v=0.
0 = „¦(v, f1 ) = a

„¦
Claim. W = W1 • W1 .
Suppose that v ∈ W has „¦(v, e1 ) = c and „¦(v, f1 ) = d. Then

v = (’cf1 + de1 ) + (v + cf1 ’ de1 ) .
∈W1 „¦
∈W1

„¦ „¦
Go on: let e2 ∈ W1 , e2 = 0. There is f2 ∈ W1 such that „¦(e2 , f2 ) = 0.
Assume that „¦(e2 , f2 ) = 1. Let W2 = span of e2 , f2 . Etc.
This process eventually stops because dim V < ∞. We hence obtain

V = U • W 1 • W2 • . . . • W n

where all summands are orthogonal with respect to „¦, and where Wi has basis
ei , fi with „¦(ei , fi ) = 1.
The dimension of the subspace U = {u ∈ V | „¦(u, v) = 0, for all v ∈ V }
does not depend on the choice of basis.
=’ k := dim U is an invariant of (V, „¦) .
Since k + 2n = m = dim V ,
=’ n is an invariant of (V, „¦); 2n is called the rank of „¦.

1.2 Symplectic Vector Spaces
Let V be an m-dimensional vector space over R, and let „¦ : V — V ’ R be a
bilinear map.

De¬nition 1.2 The map „¦ : V ’ V — is the linear map de¬ned by „¦(v)(u) =
„¦(v, u).

The kernel of „¦ is the subspace U above.

De¬nition 1.3 A skew-symmetric bilinear map „¦ is symplectic (or nonde-
generate) if „¦ is bijective, i.e., U = {0}. The map „¦ is then called a linear
symplectic structure on V , and (V, „¦) is called a symplectic vector space.
5
1.2 Symplectic Vector Spaces


The following are immediate properties of a symplectic map „¦:

1. Duality: the map „¦ : V ’ V — is a bijection.
2. By the standard form theorem, k = dim U = 0, so dim V = 2n is even.
3. By Theorem 1.1, a symplectic vector space (V, „¦) has a basis
e1 , . . . , en , f1 , . . . , fn satisfying
„¦(ei , fj ) = δij and „¦(ei , ej ) = 0 = „¦(fi , fj ) .
Such a basis is called a symplectic basis of (V, „¦). With respect to a
symplectic basis, we have
® 
|
0 Id °
v ».
„¦(u, v) = [ u ]
’Id 0
|

Not all subspaces W of a symplectic vector space (V, „¦) look the same:
• A subspace W is called symplectic if „¦|W is nondegenerate. For instance,
the span of e1 , f1 is symplectic.
• A subspace W is called isotropic if „¦|W ≡ 0. For instance, the span of
e1 , e2 is isotropic.
Homework 1 describes subspaces W of (V, „¦) in terms of the relation between
W and W „¦ .
The prototype of a symplectic vector space is (R2n , „¦0 ) with „¦0 such
that the basis
n

e1 = (1, 0, . . . , 0), . . . , en = (0, . . . , 0, 1 , 0, . . . , 0),
f1 = (0, . . . , 0, 1 , 0, . . . , 0), . . . , fn = (0, . . . , 0, 1) ,
n+1

is a symplectic basis. The map „¦0 on other vectors is determined by its values
on a basis and bilinearity.

De¬nition 1.4 A symplectomorphism • between symplectic vector spaces
(V, „¦) and (V , „¦ ) is a linear isomorphism • : V ’ V such that •— „¦ = „¦.
(By de¬nition, (•— „¦ )(u, v) = „¦ (•(u), •(v)).) If a symplectomorphism exists,
(V, „¦) and (V , „¦ ) are said to be symplectomorphic.
The relation of being symplectomorphic is clearly an equivalence relation in
the set of all even-dimensional vector spaces. Furthermore, by Theorem 1.1,
every 2n-dimensional symplectic vector space (V, „¦) is symplectomorphic to the
prototype (R2n , „¦0 ); a choice of a symplectic basis for (V, „¦) yields a symplecto-
morphism to (R2n , „¦0 ). Hence, positive even integers classify equivalence classes
for the relation of being symplectomorphic.
6 1 SYMPLECTIC FORMS


1.3 Symplectic Manifolds

Let ω be a de Rham 2-form on a manifold M , that is, for each p ∈ M , the map
ωp : Tp M — Tp M ’ R is skew-symmetric bilinear on the tangent space to M
at p, and ωp varies smoothly in p. We say that ω is closed if it satis¬es the
di¬erential equation dω = 0, where d is the de Rham di¬erential (i.e., exterior
derivative).

De¬nition 1.5 The 2-form ω is symplectic if ω is closed and ωp is symplectic
for all p ∈ M .

If ω is symplectic, then dim Tp M = dim M must be even.

De¬nition 1.6 A symplectic manifold is a pair (M, ω) where M is a man-
ifold and ω is a symplectic form.


Example. Let M = R2n with linear coordinates x1 , . . . , xn , y1 , . . . , yn . The
form
n
dxi § dyi
ω0 =
i=1

is symplectic as can be easily checked, and the set

‚ ‚ ‚ ‚
,..., , ,...,
‚x1 ‚xn ‚y1 ‚yn
p p p p


™¦
is a symplectic basis of Tp M .

Example. Let M = Cn with linear coordinates z1 , . . . , zn . The form
n
i
dzk § d¯k
ω0 = z
2
k=1

is symplectic. In fact, this form equals that of the previous example under the
identi¬cation Cn R2n , zk = xk + iyk . ™¦

Example. Let M = S 2 regarded as the set of unit vectors in R3 . Tangent
vectors to S 2 at p may then be identi¬ed with vectors orthogonal to p. The
standard symplectic form on S 2 is induced by the inner and exterior products:

for u, v ∈ Tp S 2 = {p}⊥ .
ωp (u, v) := p, u — v ,

This form is closed because it is of top degree; it is nondegenerate because
p, u — v = 0 when u = 0 and we take, for instance, v = u — p. ™¦
7
1.4 Symplectomorphisms


1.4 Symplectomorphisms

De¬nition 1.7 Let (M1 , ω1 ) and (M2 , ω2 ) be 2n-dimensional symplectic man-
ifolds, and let g : M1 ’ M2 be a di¬eomorphism. Then g is a symplectomor-
phism if g — ω2 = ω1 .1

We would like to classify symplectic manifolds up to symplectomorphism.
The Darboux theorem (proved in Lecture 8 and stated below) takes care of
this classi¬cation locally: the dimension is the only local invariant of symplec-
tic manifolds up to symplectomorphisms. Just as any n-dimensional manifold
looks locally like Rn , any 2n-dimensional symplectic manifold looks locally like
(R2n , ω0 ). More precisely, any symplectic manifold (M 2n , ω) is locally symplec-
tomorphic to (R2n , ω0 ).
Theorem 8.1 (Darboux) Let (M, ω) be a 2m-dimensional symplectic man-
ifold, and let p be any point in M .
Then there is a coordinate chart (U, x1 , . . . , xn , y1 , . . . , yn ) centered at p such
that on U
n
dxi § dyi .
ω=
i=1


A chart (U, x1 , . . . , xn , y1 , . . . , yn ) as in Theorem 8.1 is called a Darboux
chart.
By Theorem 8.1, the prototype of a local piece of a 2n-dimensional
symplectic manifold is M = R2n , with linear coordinates (x1 , . . . , xn , y1 , . . . , yn ),
and with symplectic form
n
dxi § dyi .
ω0 =
i=1




1 Recall that, by de¬nition of pullback, at tangent vectors u, v ∈ Tp M1 , we have
(g — ω 2 )p (u, v)= (ω2 )g(p) (dgp (u), dgp (v)).
Homework 1: Symplectic Linear Algebra
Given a linear subspace Y of a symplectic vector space (V, „¦), its symplectic
orthogonal Y „¦ is the linear subspace de¬ned by
Y „¦ := {v ∈ V | „¦(v, u) = 0 for all u ∈ Y } .
dim Y + dim Y „¦ = dim V .
1. Show that
What is the kernel and image of the map
Hint:
Y — = Hom(Y, R)
V ’’ ?
v ’’ „¦(v, ·)|Y

2. Show that (Y „¦ )„¦ = Y .
3. Show that, if Y and W are subspaces, then
Y ⊆ W ⇐’ W „¦ ⊆ Y „¦ .

4. Show that:
Y is symplectic (i.e., „¦|Y —Y is nondegenerate) ⇐’ Y © Y „¦ = {0}
⇐’ V = Y • Y „¦ .
5. We call Y isotropic when Y ⊆ Y „¦ (i.e., „¦|Y —Y ≡ 0).
1
Show that, if Y is isotropic, then dim Y ¤ dim V .
2

6. An isotropic subspace Y of (V, „¦) is called lagrangian when dim Y =
1
2 dim V .
Check that:
Y is lagrangian ⇐’ Y is isotropic and coisotropic ⇐’ Y = Y „¦ .

7. Show that, if Y is a lagrangian subspace of (V, „¦), then any basis e1 , . . . , en
of Y can be extended to a symplectic basis e1 , . . . , en , f1 , . . . , fn of (V, „¦).
Choose f1 in W „¦ , where W is the linear span of {e2 , . . . , en }.
Hint:

8. Show that, if Y is a lagrangian subspace, (V, „¦) is symplectomorphic to
the space (Y • Y — , „¦0 ), where „¦0 is determined by the formula
„¦0 (u • ±, v • β) = β(u) ’ ±(v) .

In fact, for any vector space E, the direct sum V = E •E — has a canonical
symplectic structure determined by the formula above. If e1 , . . . , en is a
basis of E, and f1 , . . . , fn is the dual basis, then e1 • 0, . . . , en • 0, 0 •
f1 , . . . , 0 • fn is a symplectic basis for V .
9. We call Y coisotropic when Y „¦ ⊆ Y .
Check that every codimension 1 subspace Y is coisotropic.


8
2 Symplectic Form on the Cotangent Bundle

2.1 Cotangent Bundle

Let X be any n-dimensional manifold and M = T — X its cotangent bundle. If
the manifold structure on X is described by coordinate charts (U, x1 , . . . , xn )
with xi : U ’ R, then at any x ∈ U, the di¬erentials (dx1 )x , . . . (dxn )x form
n
— —
a basis of Tx X. Namely, if ξ ∈ Tx X, then ξ = i=1 ξi (dxi )x for some real
coe¬cients ξ1 , . . . , ξn . This induces a map

T —U ’’ R2n
’’ (x1 , . . . , xn , ξ1 , . . . , ξn ) .
(x, ξ)

The chart (T — U, x1 , . . . , xn , ξ1 , . . . , ξn ) is a coordinate chart for T — X; the co-
ordinates x1 , . . . , xn , ξ1 , . . . , ξn are the cotangent coordinates associated to
the coordinates x1 , . . . , xn on U. The transition functions on the overlaps are
smooth: given two charts (U, x1 , . . . , xn ), (U , x1 , . . . , xn ), and x ∈ U © U , if

ξ ∈ Tx X, then
n n
‚xi
ξ= ξi (dxi )x = ξi (dxj )x = ξj (dxj )x
‚xj
i=1 i,j j=1


‚xi
is smooth. Hence, T — X is a 2n-dimensional manifold.
where ξj = i ξi ‚xj

We will now construct a major class of examples of symplectic forms. The
canonical forms on cotangent bundles are relevant for several branches, including
analysis of di¬erential operators, dynamical systems and classical mechanics.

2.2 Tautological and Canonical Forms in Coordinates

Let (U, x1 , . . . , xn ) be a coordinate chart for X, with associated cotangent co-
ordinates (T — U, x1 , . . . , xn , ξ1 , . . . , ξn ). De¬ne a 2-form ω on T — U by
n
dxi § dξi .
ω=
i=1

In order to check that this de¬nition is coordinate-independent, consider the
1-form on T — U
n
±= ξi dxi .
i=1

Clearly, ω = ’d±.
Claim. The form ± is intrinsically de¬ned (and hence the form ω is also intrin-
sically de¬ned) .

9
10 2 SYMPLECTIC FORM ON THE COTANGENT BUNDLE


Proof. Let (U, x1 , . . . , xn , ξ1 , . . . , ξn ) and (U , x1 , . . . , xn , ξ1 , . . . , ξn ) be two
cotangent coordinate charts. On U © U , the two sets of coordinates are re-
‚x
‚x
lated by ξj = i ξi ‚x i . Since dxj = i ‚xj dxi , we have i
j



±= ξi dxi = ξj dxj = ± .
i j




The 1-form ± is the tautological form and 2-form ω is the canonical
symplectic form. The following section provides an alternative proof of the
intrinsic character of these forms.


2.3 Coordinate-Free De¬nitions

Let
M = T —X —
ξ ∈ Tx X
p = (x, ξ)
“π “
X x
be the natural projection. The tautological 1-form ± may be de¬ned point-
wise by
±p = (dπp )— ξ ∈ Tp M ,



where (dπp )— is the transpose of dπp , that is, (dπp )— ξ = ξ —¦ dπp :

p = (x, ξ) Tp M Tp M
‘ (dπp )—
“π “ dπp

x Tx X Tx X

Equivalently,
for v ∈ Tp M ,
±p (v) = ξ (dπp )v ,

Exercise. Let (U, x1 , . . . , xn ) be a chart on X with associated cotangent coor-
n
dinates x1 , . . . , xn , ξ1 , . . . , ξn . Show that on T — U, ± = ™¦
ξi dxi .
i=1


The canonical symplectic 2-form ω on T — X is de¬ned as

ω = ’d± .
n
dxi § dξi .
Locally, ω = i=1

Exercise. Show that the tautological 1-form ± is uniquely characterized by the
property that, for every 1-form µ : X ’ T — X, µ— ± = µ. (See Lecture 3.) ™¦
11
2.4 Naturality of the Tautological and Canonical Forms


2.4 Naturality of the Tautological and Canonical Forms
Let X1 and X2 be n-dimensional manifolds with cotangent bundles M1 = T — X1
and M2 = T — X2 , and tautological 1-forms ±1 and ±2 . Suppose that f : X1 ’
X2 is a di¬eomorphism. Then there is a natural di¬eomorphism

f : M1 ’ M 2

which lifts f ; namely, if p1 = (x1 , ξ1 ) ∈ M1 for x1 ∈ X1 and ξ1 ∈ Tx1 X1 , then
we de¬ne
x2 = f (x1 ) ∈ X2 and
f (p1 ) = p2 = (x2 , ξ2 ) , with
ξ1 = (dfx1 )— ξ2 ,

where (dfx1 )— : Tx2 X2 ’ Tx1 X1 , so f |Tx1 is the inverse map of (dfx1 )— .
— — —


Exercise. Check that f is a di¬eomorphism. Here are some hints:
f
’’
M1 M2
π1 “ “ π2
1. commutes.
f
’’
X1 X2
2. f : M1 ’ M2 is bijective.
3. f and f ’1 are smooth.
™¦

Theorem 2.1 The lift f of a di¬eomorphism f : X1 ’ X2 pulls the tautolog-
ical form on T — X2 back to the tautological form on T — X1 , i.e.,

(f )— ±2 = ±1 .

Proof. At p1 = (x1 , ξ1 ) ∈ M1 , this identity says

(df )p1 (±2 )p2 = (±1 )p1 ()

where p2 = f (p1 ).
Using the following facts,
• De¬nition of f :
p2 = f (p1 ) ⇐’ p2 = (x2 , ξ2 ) where x2 = f (x1 ) and (dfx1 )— ξ2 = ξ1 .
• De¬nition of tautological 1-form:
(±1 )p1 = (dπ1 )—1 ξ1 (±2 )p2 = (dπ2 )—2 ξ2 .
and
p p

f
’’
M1 M2
• The diagram π1 “ “ π2 commutes.
f
’’
X1 X2
12 2 SYMPLECTIC FORM ON THE COTANGENT BUNDLE


the proof of ( ) is:

(df )—1 (±2 )p2 = (df )—1 (dπ2 )—2 ξ2 = (d(π2 —¦ f ))p1 ξ2
p p p

= (dπ1 )—1 (df )— 1 ξ2
= (d(f —¦ π1 ))p1 ξ2 p x
= (dπ1 )—1 ξ1 = (±1 )p1
p




Corollary 2.2 The lift f of a di¬eomorphism f : X1 ’ X2 is a symplecto-
morphism, i.e.,
(f )— ω2 = ω1 ,
where ω1 , ω2 are the canonical symplectic forms.

In summary, a di¬eomorphism of manifolds induces a canonical symplecto-
morphism of cotangent bundles:

f : T — X1 ’’ T — X2

’’
f: X1 X2

Example. Let X1 = X2 = S 1 . Then T — S 1 is an in¬nite cylinder S 1 — R.
The canonical 2-form ω is the area form ω = dθ § dξ. If f : S 1 ’ S 1 is any
di¬eomorphism, then f : S 1 — R ’ S 1 — R is a symplectomorphism, i.e., is an
™¦
area-preserving di¬eomorphism of the cylinder.
If f : X1 ’ X2 and g : X2 ’ X3 are di¬eomorphisms, then (g —¦ f ) =
g —¦ f . In terms of the group Di¬(X) of di¬eomorphisms of X and the group
Sympl(M, ω) of symplectomorphisms of (M, ω), we say that the map

Di¬(X) ’’ Sympl(M, ω)
f ’’ f

is a group homomorphism. This map is clearly injective. Is it surjective? Do all
symplectomorphisms T — X ’ T — X come from di¬eomorphisms X ’ X? No:
for instance, translation along cotangent ¬bers is not induced by a di¬eomor-
phism of the base manifold. A criterion for which symplectomorphisms arise as
lifts of di¬eomorphisms is discussed in Homework 3.
Homework 2: Symplectic Volume

1. Given a vector space V , the exterior algebra of its dual space is
dim V
— —
§k (V — ) ,
§ (V ) =
k=0

k

where §k (V — ) is the set of maps ± : V — · · · — V ’ R which are linear
in each entry, and for any permutation π, ±(vπ1 , . . . , vπk ) = (sign π) ·
±(v1 , . . . , vk ). The elements of §k (V — ) are known as skew-symmetric
k-linear maps or k-forms on V .
(a) Show that any „¦ ∈ §2 (V — ) is of the form „¦ = e— § f1 + . . . + e— § fn ,
— —
1 n
where u— , . . . , u— , e— , . . . , e— , f1 , . . . , fn is a basis of V — dual to the
— —
1 k1 n
standard basis (k + 2n = dim V ).
(b) In this language, a symplectic map „¦ : V — V ’ R is just a nonde-
generate 2-form „¦ ∈ §2 (V — ), called a symplectic form on V .
Show that, if „¦ is any symplectic form on a vector space V of di-
mension 2n, then the nth exterior power „¦n = „¦ § . . . § „¦ does not
n
vanish.
(c) Deduce that the nth exterior power ω n of any symplectic form ω on
a 2n-dimensional manifold M is a volume form.2
Hence, any symplectic manifold (M, ω) is canonically oriented by the
n
symplectic structure. The form ω is called the symplectic volume
n!
or the Liouville form of (M, ω).
Does the M¨bius strip support a symplectic structure?
o
(d) Conversely, given a 2-form „¦ ∈ §2 (V — ), show that, if „¦n = 0, then
„¦ is symplectic.
Standard form.
Hint:

2. Let (M, ω) be a 2n-dimensional symplectic manifold, and let ω n be the
volume form obtained by wedging ω with itself n times.
(a) Show that, if M is compact, the de Rham cohomology class [ω n ] ∈
H 2n (M ; R) is non-zero.
Stokes™ theorem.
Hint:

(b) Conclude that [ω] itself is non-zero (in other words, that ω is not
exact).
(c) Show that if n > 1 there are no symplectic structures on the sphere
S 2n .

2A volume form is a nonvanishing form of top degree.


13
Part II
Symplectomorphisms
Equivalence between symplectic manifolds is expressed by a symplectomorphism.
By Weinstein™s lagrangian creed [103], everything is a lagrangian manifold! We
will study symplectomorphisms according to the creed.


3 Lagrangian Submanifolds

3.1 Submanifolds

Let M and X be manifolds with dim X < dim M .

De¬nition 3.1 A map i : X ’ M is an immersion if dip : Tp X ’ Ti(p) M is
injective for any point p ∈ X.
An embedding is an immersion which is a homeomorphism onto its image.3
A closed embedding is a proper4 injective immersion.


Exercise. Show that a map i : X ’ M is a closed embedding if and only if i
is an embedding and its image i(X) is closed in M .
Hints:

• If i is injective and proper, then for any neighborhood U of p ∈ X, there
is a neighborhood V of i(p) such that f ’1 (V) ⊆ U.

• On a Hausdor¬ space, any compact set is closed. On any topological
space, a closed subset of a compact set is compact.

• An embedding is proper if and only if its image is closed.

™¦


De¬nition 3.2 A submanifold of M is a manifold X with a closed embedding
i : X ’ M .5

Notation. Given a submanifold, we regard the embedding i : X ’ M as an
inclusion, in order to identify points and tangent vectors:

Tp X = dip (Tp X) ‚ Tp M .
p = i(p) and
3 Theimage has the topology induced by the target manifold.
4A map is proper if the preimage of any compact set is compact.
5 When X is an open subset of a manifold M , we refer to it as an open submanifold.




15
16 3 LAGRANGIAN SUBMANIFOLDS


Lagrangian Submanifolds of T — X
3.2

De¬nition 3.3 Let (M, ω) be a 2n-dimensional symplectic manifold. A sub-
manifold Y of M is a lagrangian submanifold if, at each p ∈ Y , Tp Y is
a lagrangian subspace of Tp M , i.e., ωp |Tp Y ≡ 0 and dim Tp Y = 1 dim Tp M .
2
Equivalently, if i : Y ’ M is the inclusion map, then Y is lagrangian if and
only if i— ω = 0 and dim Y = 1 dim M .
2


Let X be an n-dimensional manifold, with M = T — X its cotangent bundle.
If x1 , . . . , xn are coordinates on U ⊆ X, with associated cotangent coordinates
x1 , . . . , xn , ξ1 , . . . , ξn on T — U , then the tautological 1-form on T — X is

±= ξi dxi

and the canonical 2-form on T — X is

ω = ’d± = dxi § dξi .

The zero section of T — X

X0 := {(x, ξ) ∈ T — X | ξ = 0 in Tx X}




is an n-dimensional submanifold of T — X whose intersection with T — U is given
ξi dxi vanishes on X0 © T — U .
by the equations ξ1 = . . . = ξn = 0. Clearly ± =
In particular, if i0 : X0 ’ T — X is the inclusion map, we have i— ± = 0. Hence,
0
i— ω = i— d± = 0, and X0 is lagrangian.
0 0
What are all the lagrangian submanifolds of T — X which are “C 1 -close to
X0 ”?
Let Xµ be (the image of) another section, that is, an n-dimensional sub-
manifold of T — X of the form

Xµ = {(x, µx ) | x ∈ X, µx ∈ Tx X} ()

where the covector µx depends smoothly on x, and µ : X ’ T — X is a de Rham
1-form. Relative to the inclusion i : Xµ ’ T — X and the cotangent projection
π : T — X ’ X, Xµ is of the form ( ) if and only if π —¦ i : Xµ ’ X is a
di¬eomorphism.
When is such a Xµ lagrangian?

Proposition 3.4 Let Xµ be of the form ( ), and let µ be the associated de
Rham 1-form. Denote by sµ : X ’ T — X, x ’ (x, µx ), be the 1-form µ regarded
exclusively as a map. Notice that the image of sµ is Xµ . Let ± be the tautological
1-form on T — X. Then
s— ± = µ .
µ
3.2 Lagrangian Submanifolds of T — X 17


Proof. By de¬nition of ± (previous lecture), ±p = (dπp )— ξ at p = (x, ξ) ∈ M .
For p = sµ (x) = (x, µx ), we have ±p = (dπp )— µx . Then

(s— ±)x = (dsµ )— ±p
µ x
= (dsµ )— (dπp )— µx
x
= (d(π —¦ sµ ))— µx = µx .
x

idX




Suppose that Xµ is an n-dimensional submanifold of T — X of the form ( ),
with associated de Rham 1-form µ. Then sµ : X ’ T — X is an embedding with
image Xµ , and there is a di¬eomorphism „ : X ’ Xµ , „ (x) := (x, µx ), such
that the following diagram commutes.

sµ E T —X
X
d  

d  
d  
„d  i

d  


We want to express the condition of Xµ being lagrangian in terms of the form
µ:
Xµ is lagrangian ⇐’ i— d± = 0
⇐’ „ — i— d± = 0
⇐’ (i —¦ „ )— d± = 0
⇐’ s— d± = 0
µ
⇐’ ds— ± = 0
µ
⇐’ dµ = 0
⇐’ µ is closed .
Therefore, there is a one-to-one correspondence between the set of lagrangian
submanifolds of T — X of the form ( ) and the set of closed 1-forms on X.
1
When X is simply connected, HdeRham (X) = 0, so every closed 1-form
µ is equal to df for some f ∈ C ∞ (X). Any such primitive f is then called a
generating function for the lagrangian submanifold Xµ associated to µ. (Two
functions generate the same lagrangian submanifold if and only if they di¬er by
a locally constant function.) On arbitrary manifolds X, functions f ∈ C ∞ (X)
originate lagrangian submanifolds as images of df .
Exercise. Check that, if X is compact (and not just one point) and f ∈ C ∞ (X),
then #{Xdf © X0 } ≥ 2. ™¦

There are lots of lagrangian submanifolds of T — X not covered by the de-
scription in terms of closed 1-forms, starting with the cotangent ¬bers.
18 3 LAGRANGIAN SUBMANIFOLDS


3.3 Conormal Bundles

Let S be any k-dimensional submanifold of an n-dimensional manifold X.

De¬nition 3.5 The conormal space at x ∈ S is
— —
Nx S = {ξ ∈ Tx X | ξ(v) = 0 , for all v ∈ Tx S} .

The conormal bundle of S is

N — S = {(x, ξ) ∈ T — X | x ∈ S, ξ ∈ Nx S} .





Exercise. The conormal bundle N — S is an n-dimensional submanifold of T — X.
Hint: Use coordinates on X adapted6 to S. ™¦


Proposition 3.6 Let i : N — S ’ T — X be the inclusion, and let ± be the tauto-
logical 1-form on T — X. Then
i— ± = 0 .


Proof. Let (U, x1 , . . . , xn ) be a coordinate system on X centered at x ∈ S
and adapted to S, so that U © S is described by xk+1 = . . . = xn = 0. Let
(T — U, x1 , . . . , xn , ξ1 , . . . , ξn ) be the associated cotangent coordinate system. The
submanifold N — S © T — U is then described by

xk+1 = . . . = xn = 0 and ξ 1 = . . . = ξk = 0 .

ξi dxi on T — U, we conclude that, at p ∈ N — S,
Since ± =


(i— ±)p = ±p |Tp (N — S) = ξi dxi =0.
i>k ‚
span{ ‚x ,i¤k}
i




Corollary 3.7 For any submanifold S ‚ X, the conormal bundle N — S is a
lagrangian submanifold of T — X.

Taking S = {x} to be one point, the conormal bundle L = N — S = Tx X is a


cotangent ¬ber. Taking S = X, the conormal bundle L = X0 is the zero section
of T — X.
6A
coordinate chart (U , x1 , . . . , xn ) on X is adapted to a k-dimensional submanifold S if
S © U is described by xk+1 = . . . = xn = 0.
19
3.4 Application to Symplectomorphisms


3.4 Application to Symplectomorphisms
Let (M1 , ω1 ) and (M2 , ω2 ) be two 2n-dimensional symplectic manifolds. Given
a di¬eomorphism • : M1 ’’ M2 , when is it a symplectomorphism? (I.e., when
is •— ω2 = ω1 ?)
Consider the diagram of projection maps
M1 — M 2
(p1 , p2 ) (p1 , p2 )
  d
  d pr2
pr1
  d
  d
 
© ‚
d
c c
p1 M1 M2 p2
Then ω = (pr1 )— ω1 + (pr2 )— ω2 is a 2-form on M1 — M2 which is closed,
dω = (pr1 )— dω1 + (pr2 )— dω2 = 0 ,
0 0

and symplectic,
n n
2n
ω 2n = (pr1 )— ω1 § (pr2 )— ω2 =0.
n
More generally, if »1 , »2 ∈ R\{0}, then »1 (pr1 )— ω1 + »2 (pr2 )— ω2 is also a sym-
plectic form on M1 —M2 . Take »1 = 1, »2 = ’1 to obtain the twisted product
form on M1 — M2 :
ω = (pr1 )— ω1 ’ (pr2 )— ω2 .
The graph of a di¬eomorphism • : M1 ’’ M2 is the 2n-dimensional sub-
manifold of M1 — M2 :
“• := Graph • = {(p, •(p)) | p ∈ M1 } .
The submanifold “• is an embedded image of M1 in M1 — M2 , the embedding
being the map
γ : M1 ’’ M1 — M2
p ’’ (p, •(p)) .
Theorem 3.8 A di¬eomorphism • is a symplectomorphism if and only if “ •
is a lagrangian submanifold of (M1 — M2 , ω).

Proof. The graph “• is lagrangian if and only if γ — ω = 0.
γ —ω = γ — pr— ω1 ’ γ — pr— ω2
1 2
= (pr1 —¦ γ)— ω1 ’ (pr2 —¦ γ)— ω2 .
But pr1 —¦ γ is the identity map on M1 and pr2 —¦ γ = •. Therefore,
γ—ω = 0 • — ω2 = ω 1 .
⇐’
Homework 3:
Tautological Form and Symplectomorphisms

This set of problems is from [52].


1. Let (M, ω) be a symplectic manifold, and let ± be a 1-form such that

ω = ’d± .

Show that there exists a unique vector ¬eld v such that its interior product
with ω is ±, i.e., ±v ω = ’±.
Prove that, if g is a symplectomorphism which preserves ± (that is, g — ± =
±), then g commutes with the one-parameter group of di¬eomorphisms
generated by v, i.e.,

(exp tv) —¦ g = g —¦ (exp tv) .
Hint: Recall that, for p ∈ M , (exp tv)(p) is the unique curve in M solving
the ordinary di¬erential equation
d
(exp tv(p)) = v(exp tv(p))
dt
(exp tv)(p)|t=0 = p

for t in some neighborhood of 0. Show that g —¦ (exp tv) —¦ g ’1 is the one-
parameter group of di¬eomorphisms generated by g— v. (The push-forward of v
by g is de¬ned by (g— v)g(p) = dgp (vp ).) Finally check that g preserves v (that
is, g— v = v).



2. Let X be an arbitrary n-dimensional manifold, and let M = T — X. Let
(U, x1 , . . . , xn ) be a coordinate system on X, and let x1 , . . . , xn , ξ1 , . . . , ξn
be the corresponding coordinates on T — U.
Show that, when ± is the tautological 1-form on M (which, in these coor-
dinates, is ξi dxi ), the vector ¬eld v in the previous exercise is just the

vector ¬eld ξi ‚ξi .
Let exp tv, ’∞ < t < ∞, be the one-parameter group of di¬eomorphisms
generated by v.
Show that, for every point p = (x, ξ) in M ,

pt = (x, et ξ) .
(exp tv)(p) = pt where




20
21
HOMEWORK 3




3. Let M be as in exercise 2.
Show that, if g is a symplectomorphism of M which preserves ±, then

g(x, ξ) = (y, ·) =’ g(x, »ξ) = (y, »·)

for all (x, ξ) ∈ M and » ∈ R.
Conclude that g has to preserve the cotangent ¬bration, i.e., show that
there exists a di¬eomorphism f : X ’ X such that π —¦ g = f —¦ π, where
π : M ’ X is the projection map π(x, ξ) = x.
Finally prove that g = f# , the map f# being the symplectomorphism of
M lifting f .
Hint: Suppose that g(p) = q where p = (x, ξ) and q = (y, ·).
Combine the identity
(dgp )— ±q = ±p
with the identity
dπq —¦ dgp = dfx —¦ dπp .
(The ¬rst identity expresses the fact that g — ± = ±, and the second identity is
obtained by di¬erentiating both sides of the equation π —¦ g = f —¦ π at p.)



4. Let M be as in exercise 2, and let h be a smooth function on X. De¬ne
„h : M ’ M by setting

„h (x, ξ) = (x, ξ + dhx ) .

Prove that
„h ± = ± + π — dh



where π is the projection map

M (x, ξ)
“π “
X x

Deduce that

„h ω = ω ,
i.e., that „h is a symplectomorphism.
4 Generating Functions
4.1 Constructing Symplectomorphisms
Let X1 , X2 be n-dimensional manifolds, with cotangent bundles M1 = T — X1 ,
M2 = T — X2 , tautological 1-forms ±1 , ±2 , and canonical 2-forms ω1 , ω2 .
Under the natural identi¬cation
M 1 — M 2 = T — X1 — T — X2 T — (X1 — X2 ) ,
the tautological 1-form on T — (X1 — X2 ) is
± = (pr1 )— ±1 + (pr2 )— ±2 ,
where pri : M1 — M2 ’ Mi , i = 1, 2 are the two projections. The canonical
2-form on T — (X1 — X2 ) is
ω = ’d± = ’dpr— ±1 ’ dpr— ±2 = pr— ω1 + pr— ω2 .
1 2 1 2

In order to describe the twisted form ω = pr— ω1 ’pr— ω2 , we de¬ne an involution
1 2
of M2 = T — X2 by
M2 ’’ M2
σ2 :
(x2 , ξ2 ) ’’ (x2 , ’ξ2 )

which yields σ2 ±2 = ’±2 . Let σ = idM1 — σ2 : M1 — M2 ’ M1 — M2 . Then
σ — ω = pr— ω1 + pr— ω2 = ω .

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