стр. 1(всего 5)СОДЕРЖАНИЕ >>
Introduction to
Symplectic and Hamiltonian Geometry

Notes for a Short Course at IMPA
Rio de Janeiro, February 2002

Ana Cannas da Silva1

Revised Version вЂ“ March 19, 2002

1
E-mail: acannas@math.ist.utl.pt
PrefВґcio
a
A geometria simplВґctica Вґ a geometria de variedades equipadas com uma forma
e e
simplВґctica, ou seja, com uma forma de grau 2 fechada e nЛњo-degenerada. A geo-
e a
e e
cЛњ cЛњ
por simetrias.
HВґ cerca de dois sВґculos, a geometria simplВґctica fornecia a linguagem para
a e e
a mecЛ†nica clВґssica; pelo seu rВґpido crescimento recente, conquistou um rico
a a a
territВґrio, estabelecendo-se como um ramo central da geometria e da topologia
o
diferenciais. AlВґm da sua actividade como disciplina independente, a geometria
e
simplВґctica Вґ signiп¬Ѓcativamente estimulada por interacВёoes importantes com sis-
e e cЛњ
temas dinЛ†micos, anВґlise global, fВґ
a a Д±sica-matemВґtica, topologia em baixas dimensЛњes,
a o
teoria de representaВёoes, anВґlise microlocal, equaВёoes diferenciais parciais, geo-
cЛњ a cЛњ
metria algВґbrica, geometria riemanniana, anВґlise combinatВґrica geomВґtrica, co-
e a o e
homologia equivariante, etc.
Este texto cobre fundamentos da geometria simplВґctica numa linguagem
e
moderna. ComeВёa-se por descrever as variedades simplВґcticas e as suas trans-
c e
formaВёoes, e por explicar ligaВёoes a topologia e outras geometrias. Seguidamente
cЛњ cЛњ
estudam-se campos hamiltonianos, acВёoes hamiltonianas e algumas das suas aplica-
cЛњ
coes prВґticas no ambito da mecЛ†nica e dos sistemas dinЛ†micos. Ao longo do
ВёЛњ a Л† a a
texto fornecem-se exemplos simples e exercВґ Д±cios relevantes. PressupЛњem-se conheci-
o
mentos prВґvios de geometria de variedades diferenciВґveis, se bem que os principais
e a
factos requeridos estejam coleccionados em apЛ†ndices. e
Estas notas reproduzem aproximadamente o curso curto de geometria sim-
plВґctica, constituВґ por cinco liВёoes dirigidas a estudantes de pВґs-graduaВёao e
e Д±do cЛњ o cЛњ
a a
e Aplicada, no Rio de Janeiro, em Fevereiro de 2002. Alguns trechos deste texto
sЛњo rearranjos do Lectures on Symplectic Geometry (Springer LNM 1764).
a
Fico grata ao IMPA pelo acolhimento muito proveitoso, e em especial ao
Marcelo Viana por me ter gentilmente proporcionado a honra e o prazer desta
visita, e a Suely Torres de Melo pela sua inestimВґvel ajuda perita com os prepa-
` a
rativos locais.

Ana Cannas da Silva

Lisboa, Janeiro de 2002, e
Rio de Janeiro, Fevereiro de 2002

v
Foreword
Symplectic geometry is the geometry of manifolds equipped with a symplectic
form, that is, with a closed nondegenerate 2-form. Hamiltonian geometry is the
geometry of (symplectic) manifolds equipped with a moment map, that is, with a
collection of quantities conserved by symmetries.
About two centuries ago, symplectic geometry provided a language for clas-
sical mechanics; through its recent fast development, it conquered a rich territory,
asserting itself as a central branch of diп¬Ђerential geometry and topology. Besides its
activity as an independent subject, symplectic geometry is signiп¬Ѓcantly stimulated
by important interactions with dynamical systems, global analysis, mathemati-
cal physics, low-dimensional topology, representation theory, microlocal analysis,
partial diп¬Ђerential equations, algebraic geometry, riemannian geometry, geometric
combinatorics, equivariant cohomology, etc.
This text covers foundations of symplectic geometry in a modern language.
We start by describing symplectic manifolds and their transformations, and by ex-
plaining connections to topology and other geometries. Next we study hamiltonian
п¬Ѓelds, hamiltonian actions and some of their practical applications in the context
of mechanics and dynamical systems. Throughout the text we provide simple ex-
amples and relevant exercises. We assume previous knowledge of the geometry of
smooth manifolds, though the main required facts are collected in appendices.
These notes approximately transcribe the short course on symplectic geome-
try, delivered in п¬Ѓve lectures mostly for graduate students and researchers, held at
the summer program of Instituto de MatemВґtica Pura e Aplicada, Rio de Janeiro,
a
in February of 2002. Some chunks of this text are rearrangements from Lectures
on Symplectic Geometry (Springer LNM 1764).
I am grateful to IMPA for the very rewarding hospitality, and specially to
Marcelo Viana for kindly providing me the honour and the pleasure of this visit,
and to Suely Torres de Melo for her invaluable expert help with local arrangements.

Ana Cannas da Silva

Lisbon, January 2002, and
Rio de Janeiro, February 2002

vi
Contents

PrefВґcio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
a v
Foreword . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

1 Symplectic Forms 1
1.1 Skew-Symmetric Bilinear Maps . . . . . . . . . . . . . . . . . . . . 1
1.2 Symplectic Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Special Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 Symplectic Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.5 Symplectic Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.6 Equivalence for Symplectic Structures . . . . . . . . . . . . . . . . 7
1.7 Moser Trick . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.8 Moser Local Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.9 Darboux Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2 Cotangent Bundles 15
2.1 Tautological and Canonical Forms . . . . . . . . . . . . . . . . . . 15
2.2 Naturality of the Canonical Forms . . . . . . . . . . . . . . . . . . 17
2.3 Symplectomorphisms of T в€— X . . . . . . . . . . . . . . . . . . . . . 19
2.4 Lagrangian Submanifolds of T в€— X . . . . . . . . . . . . . . . . . . . 20
2.5 Conormal Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.6 Lagrangian Complements . . . . . . . . . . . . . . . . . . . . . . . 23
2.7 Lagrangian Neighborhood Theorem . . . . . . . . . . . . . . . . . . 25
2.8 Weinstein Tubular Neighborhood Theorem . . . . . . . . . . . . . 26
2.9 Symplectomorphisms as Lagrangians . . . . . . . . . . . . . . . . . 28

3 Generating Functions 31
3.1 Constructing Symplectomorphisms . . . . . . . . . . . . . . . . . . 31
3.2 Method of Generating Functions . . . . . . . . . . . . . . . . . . . 32
3.3 Riemannian Distance . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.4 Geodesic Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.5 Periodic Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.6 Billiards . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

vii
viii CONTENTS

3.7 PoincarВґ Recurrence . . . . . . . . . . . . . . . . . . . . . . . . . .
e 41
3.8 Group of Symplectomorphisms . . . . . . . . . . . . . . . . . . . . 42
3.9 Fixed Points of Symplectomorphisms . . . . . . . . . . . . . . . . . 44

4 Hamiltonian Fields 47
4.1 Hamiltonian and Symplectic Vector Fields . . . . . . . . . . . . . . 47
4.2 Hamilton Equations . . . . ......... . . . . . . . . . . . . . 49
4.3 Brackets . . . . . . . . . . . ......... . . . . . . . . . . . . . 50
4.4 Integrable Systems . . . . . ......... . . . . . . . . . . . . . 53
4.5 Pendulums . . . . . . . . . ......... . . . . . . . . . . . . . 55
4.6 Symplectic and Hamiltonian Actions . . . . . . . . . . . . . . . . . 57
4.7 Moment Maps . . . . . . . ......... . . . . . . . . . . . . . 58
4.8 Language for Mechanics . . ......... . . . . . . . . . . . . . 63
4.9 Existence and Uniqueness of Moment Maps . . . . . . . . . . . . . 65

5 Symplectic Reduction 69
5.1 Marsden-Weinstein-Meyer Theorem . . . . . . . . . . . . . . . . . . 69
5.2 Ingredients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.3 Proof of the Reduction Theorem . . . . . . . . . . . . . . . . . . . 75
5.4 Elementary Theory of Reduction . . . . . . . . . . . . . . . . . . . 76
5.5 Reduction for Product Groups . . . . . . . . . . . . . . . . . . . . 77
5.6 Reduction at Other Levels . . . . . . . . . . . . . . . . . . . . . . . 78
5.7 Orbifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.8 Symplectic Toric Manifolds . . . . . . . . . . . . . . . . . . . . . . 79
5.9 DelzantвЂ™s Construction . . . . . . . . . . . . . . . . . . . . . . . . . 83

A Prerequisites from Diп¬Ђerential Geometry 91
A.1 Isotopies and Vector Fields . . . . . . . . . . . . . . . . . . . . . . 91
A.2 Submanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
A.3 Tubular Neighborhood Theorem . . . . . . . . . . . . . . . . . . . 94
A.4 Homotopy Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
A.5 Whitney Extension Theorem . . . . . . . . . . . . . . . . . . . . . 98

B Prerequisites from Lie Group Actions 101
B.1 One-Parameter Groups of Diп¬Ђeomorphisms . . . . . . . . . . . . . 101
B.2 Lie Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
B.3 Smooth Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
B.4 Adjoint and Coadjoint Representations . . . . . . . . . . . . . . . . 103
B.5 Orbit Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

C Variational Principles 107
C.1 Principle of Least Action . . . . . . . . . . . . . . . . . . . . . . . 107
C.2 Variational Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 109
C.3 Solving the Euler-Lagrange Equations . . . . . . . . . . . . . . . . 111
ix
CONTENTS

C.4 Legendre Transform . . . . . . . . . . . . . . . . . . . . . . . . . . 113
C.5 Application to Variational Problems . . . . . . . . . . . . . . . . . 117

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
Lecture 1

Symplectic Forms
A symplectic form is a 2-form satisfying an algebraic condition вЂ“ nondegeneracy
вЂ“ and an analytical condition вЂ“ closedness. In this lecture we deп¬Ѓne symplectic
forms, describe some of their basic properties, and introduce the п¬Ѓrst examples.
We conclude by exhibiting a major technique in the symplectic trade, namely the
so-called Moser trick, which takes advantage of the main features of a symplectic
form in order to show the equivalence of symplectic structures.

1.1 Skew-Symmetric Bilinear Maps
Let V be an m-dimensional vector space over R, and let в„¦ : V Г— V в†’ R be
a bilinear map. The map в„¦ is skew-symmetric if в„¦(u, v) = в€’в„¦(v, u), for all
u, v в€€ V .
Theorem 1.1 (Standard Form for Skew-symmetric Bilinear Maps) Let в„¦
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 в€€ V ,
в„¦(ui , v) = 0 ,
в„¦(ei , ej ) = 0 = в„¦(fi , fj ) , for all i, j, and
в„¦(ei , fj ) = Оґij , for all i, j.

Remarks.
1. The basis in Theorem 1.1 is not unique, though it is traditionally also called
a вЂњcanonicalвЂќ basis.
2. In matrix notation with respect to such basis, we have
пЈ® пЈ№пЈ® пЈ№
|
0 0 0
в„¦(u, v) = [ u ] пЈ° 0 Id пЈ» пЈ° v пЈ» .
0
0 в€’Id 0 |
в™¦

1
2 LECTURE 1. SYMPLECTIC FORMS

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

V =U вЉ•W .

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

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

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

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

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

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

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

V = U вЉ• W 1 вЉ• W2 вЉ• . . . вЉ• W n

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

1.2 Symplectic Vector Spaces
Let V be an m-dimensional vector space over R, and let в„¦ : V Г— V в†’ R be a
bilinear map.
3
1.2. SYMPLECTIC VECTOR SPACES

Deп¬Ѓnition 1.2 The map в„¦ : V в†’ V в€— is the linear map deп¬Ѓned by в„¦(v)(u) =
в„¦(v, u).
The kernel of в„¦ is the subspace U in the previous section.
Deп¬Ѓnition 1.3 A skew-symmetric bilinear map в„¦ is symplectic (or nonde-
generate) if в„¦ is bijective, i.e., U = {0}. The map в„¦ is then called a linear
symplectic structure on V , and (V, в„¦) is called a symplectic vector space.
The following are immediate properties of a symplectic map в„¦:
вЂў Duality: the map в„¦ : V в†’ V в€— is a bijection.
вЂў By Theorem 1.1, we must have that k = dim U = 0, so dim V = 2n is even.
вЂў Also by Theorem 1.1, a symplectic vector space (V, в„¦) has a basis
e1 , . . . , en , f1 , . . . , fn satisfying
в„¦(ei , fj ) = Оґij and в„¦(ei , ej ) = 0 = в„¦(fi , fj ) .
Such a basis is called a symplectic basis of (V, в„¦). With respect to a
symplectic basis, we have
пЈ® пЈ№
|
0 Id пЈ°
v пЈ».
в„¦(u, v) = [ u ]
в€’Id 0
|

The prototype of a symplectic vector space is (R2n , в„¦0 ) with в„¦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 в„¦0 on other vectors is determined by its values on
a basis and bilinearity.

Deп¬Ѓnition 1.4 A symplectomorphism П• between symplectic vector spaces (V, в„¦)
and (V , в„¦ ) is a linear isomorphism П• : V в†’ V such that П•в€— в„¦ = в„¦. (By deп¬Ѓ-
nition, (П•в€— в„¦ )(u, v) = в„¦ (П•(u), П•(v)).) If a symplectomorphism exists, (V, в„¦) and
(V , в„¦ ) 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, в„¦) is symplectomorphic to the
prototype (R2n , в„¦0 ); a choice of a symplectic basis for (V, в„¦) yields a symplecto-
morphism to (R2n , в„¦0 ). Hence, positive even integers classify equivalence classes
for the relation of being symplectomorphic.
4 LECTURE 1. SYMPLECTIC FORMS

1.3 Special Subspaces

Given a linear subspace Y of a symplectic vector space (V, в„¦), its symplectic
orthogonal Y в„¦ is the linear subspace deп¬Ѓned by

Y в„¦ := {v в€€ V | в„¦(v, u) = 0 for all u в€€ Y } .

By considering the kernel and image of the map

в€’в†’ Y в€— = Hom(Y, R)
V
в€’в†’ в„¦(v, В·)|Y
v

we obtain that
dim Y + dim Y в„¦ = dim V .
By nondegeneracy of в„¦, we have that (Y в„¦ )в„¦ = Y . It is also easily checked that, if
Y and W are subspaces, then

Y вЉ† W в‡ђв‡’ W в„¦ вЉ† Y в„¦ .

Not all subspaces W of a symplectic vector space (V, в„¦) look the same:

вЂў A subspace Y is called symplectic if в„¦|Y Г—Y is nondegenerate. This is the
same as saying that Y в€© Y в„¦ = {0}, or, by counting dimensions, that V =
Y вЉ• Y в„¦.

вЂў A subspace Y is called isotropic if в„¦|Y Г—Y в‰Ў 0. If Y is isotropic, then
dim Y в‰¤ 1 dim V . Every one-dimensional subspace is isotropic.
2

вЂў A subspace is called coisotropic if its symplectic orthogonal is isotropic. If
1
Y is coisotropic, then dim Y в‰Ґ 2 dim V . Every codimension 1 subspace is
coisotropic.

For instance, if e1 , . . . , en , f1 , . . . , fn is a symplectic basis of (V, в„¦), then:

вЂў the span of e1 , f1 is symplectic,

вЂў the span of e1 , e2 is isotropic, and

вЂў the span of e1 , . . . , en , f1 , f2 is coisotropic.
1
An isotropic subspace Y of (V, в„¦) is called lagrangian when dim Y = dim V .
2
We have that

Y is lagrangian в‡ђв‡’ Y is isotropic and coisotropic в‡ђв‡’ Y = Y в„¦ .
5
1.4. SYMPLECTIC MANIFOLDS

Exercise 1
Show that, if Y is a lagrangian subspace of (V, в„¦), then any basis e1 , . . . , en of
Y can be extended to a symplectic basis e 1 , . . . , en , f1 , . . . , fn of (V, в„¦).

Hint: Choose f1 in W в„¦ , where W is the linear span of {e2 , . . . , en }.

If Y is a lagrangian subspace, then (V, в„¦) is symplectomorphic to the space
(Y вЉ• Y в€— , в„¦0 ), where в„¦0 is determined by the formula

в„¦0 (u вЉ• О±, v вЉ• ОІ) = ОІ(u) в€’ О±(v) .

Moreover, for any vector space E, the direct sum V = E вЉ• E в€— 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 вЉ• 0, . . . , en вЉ• 0, 0 вЉ• f1 , . . . , 0 вЉ• fn is a
symplectic basis for V .

1.4 Symplectic Manifolds
Let П‰ be a de Rham 2-form on a manifold M , that is, for each p в€€ M , the map
П‰p : Tp M Г— Tp M в†’ R is skew-symmetric bilinear on the tangent space to M at p,
and П‰p varies smoothly in p. We say that П‰ is closed if it satisп¬Ѓes the diп¬Ђerential
equation dП‰ = 0, where d is the de Rham diп¬Ђerential (i.e., exterior derivative).

Deп¬Ѓnition 1.5 The 2-form П‰ is symplectic if П‰ is closed and П‰ p is symplectic
for all p в€€ M .

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

Deп¬Ѓnition 1.6 A symplectic manifold is a pair (M, П‰) where M is a manifold
and П‰ is a symplectic form.

Examples.

1. Let M = R2n with linear coordinates x1 , . . . , xn , y1 , . . . , yn . The form
n
dxi в€§ dyi
П‰0 =
i=1

is symplectic as can be easily checked; the set

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

is a symplectic basis of Tp M .
6 LECTURE 1. SYMPLECTIC FORMS

2. Let M = Cn with linear coordinates z1 , . . . , zn . The form
n
i
dzk в€§ dВЇk
П‰0 = z
2
k=1

is symplectic. In fact, this form equals that of the previous example under
the identiп¬Ѓcation Cn R2n , zk = xk + iyk .

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

for u, v в€€ Tp S 2 = {p}вЉҐ .
П‰p (u, v) := p, u Г— v ,

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

в™¦

Exercise 2
Consider cylindrical polar coordinates (Оё, h) on S 2 away from its poles, where
0 в‰¤ Оё < 2ПЂ and в€’1 в‰¤ h в‰¤ 1. Show that, in these coordinates, the form of the
previous example is
П‰ = dОё в€§ dh .

1.5 Symplectic Volume

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

k

where в€§k (V в€— ) is the set of maps О± : V Г— В· В· В· Г— V в†’ R which are linear in each
entry, and for any permutation ПЂ, О±(vПЂ1 , . . . , vПЂk ) = (sign ПЂ) В· О±(v1 , . . . , vk ). The
elements of в€§k (V в€— ) are known as skew-symmetric k-linear maps or k-forms
on V .

Exercise 3
Show that any в„¦ в€€ в€§2 (V в€— ) is of the form в„¦ = eв€— в€§ f1 + . . . + eв€— в€§ fn , where
в€— в€—
n
1
в€— , . . . , uв€— , eв€— , . . . , eв€— , f в€— , . . . , f в€— is a basis of V в€— dual to the standard basis
u1 n1 n
k1
(k + 2n = dim V ).
7
1.6. EQUIVALENCE FOR SYMPLECTIC STRUCTURES

In this language, a symplectic map в„¦ : V Г— V в†’ R is just a nondegenerate
2-form в„¦ в€€ в€§2 (V в€— ), called a symplectic form on V . By the previous exercise, if в„¦
is any symplectic form on a vector space V of dimension 2n, then the nth exterior
power в„¦n = в„¦ в€§ . . . в€§ в„¦ does not vanish. Conversely, given a 2-form в„¦ в€€ в€§2 (V в€— ),
n
if в„¦n = 0, then в„¦ is symplectic.
We conclude that the nth exterior power П‰ n of any symplectic form П‰ on a
2n-dimensional manifold M is a volume form.1 Hence, any symplectic manifold
(M, П‰) is canonically oriented by the symplectic structure, and any nonorientable
n
manifold cannot be symplectic. The form П‰ is called the symplectic volume of
n!
(M, П‰).
Let (M, П‰) be a 2n-dimensional symplectic manifold, and let П‰ n be the volume
form obtained by wedging П‰ with itself n times. By StokesвЂ™ theorem., if M is
compact, the de Rham cohomology class [П‰ n ] в€€ H 2n (M ; R) is non-zero. Hence,
[П‰] itself is non-zero (in other words, П‰ is not exact). This reveals a necessary
topological condition for a compact 2n-dimensional manifold to be symplectic:
there must exist a degree 2 cohomology class whose nth power is a volume form.
In particular, for n > 1 there are no symplectic structures on the sphere S 2n .

1.6 Equivalence for Symplectic Structures
Let M be a 2n-dimensional manifold with two symplectic forms П‰0 and П‰1 , so that
(M, П‰0 ) and (M, П‰1 ) are two symplectic manifolds.

Deп¬Ѓnition 1.7 A symplectomorphism between (M 1 , П‰1 ) and (M2 , П‰2 ) is a dif-
feomorphism П• : M1 в†’ M2 such that П•в€— П‰2 = П‰1 .2

We would like to classify symplectic manifolds up to symplectomorphism. The
Darboux theorem (stated and proved in Section 1.9) takes care of this classiп¬Ѓcation
locally: the dimension is the only local invariant of symplectic manifolds up to
symplectomorphisms. Just as any n-dimensional manifold looks locally like Rn ,
any 2n-dimensional symplectic manifold looks locally like (R2n , П‰0 ). More precisely,
any symplectic manifold (M 2n , П‰) is locally symplectomorphic to (R2n , П‰0 ).

Deп¬Ѓnition 1.8 We say that

вЂў (M, П‰0 ) and (M, П‰1 ) are symplectomorphic if there is a diп¬Ђeomorphism
П• : M в†’ M with П•в€— П‰1 = П‰0 ;

вЂў (M, П‰0 ) and (M, П‰1 ) are strongly isotopic if there is an isotopy ПЃt : M в†’
M such that ПЃв€— П‰1 = П‰0 ;
1
1A volume form is a nonvanishing form of top degree.
2 Recall that, by deп¬Ѓnition of pullback, at tangent vectors u, v в€€ Tp M1 , we have
в€— П‰ ) (u, v) = (П‰ )
(П• 2 p 2 П•(p) (dП•p (u), dП•p (v)).
8 LECTURE 1. SYMPLECTIC FORMS

вЂў (M, П‰0 ) and (M, П‰1 ) are deformation-equivalent if there is a smooth fam-
ily П‰t of symplectic forms joining П‰0 to П‰1 ;

вЂў (M, П‰0 ) and (M, П‰1 ) are isotopic if they are deformation-equivalent with
[П‰t ] independent of t.

Clearly, we have

strongly isotopic =в‡’ symplectomorphic , and

isotopic =в‡’ deformation-equivalent .
We also have
strongly isotopic =в‡’ isotopic
because, if ПЃt : M в†’ M is an isotopy such that ПЃв€— П‰1 = П‰0 , then the set П‰t := ПЃв€— П‰1
1 t
is a smooth family of symplectic forms joining П‰1 to П‰0 and [П‰t ] = [П‰1 ], в€Ђt, by
the homotopy invariance of de Rham cohomology. As we will see below, the Moser
theorem states that, on a compact manifold,

isotopic =в‡’ strongly isotopic .

The remainder of this lecture concerns the following problem:
Problem. Given a 2n-dimensional manifold M , a k-dimensional submanifold X,
neighborhoods U0 , U1 of X, and symplectic forms П‰0 , П‰1 on U0 , U1 , does there exist
a symplectomorphism preserving X? More precisely, does there exist a diп¬Ђeomor-
phism П• : U0 в†’ U1 with П•в€— П‰1 = П‰0 and П•(X) = X?
At the two extremes, we have:
Case X = point: Darboux theorem вЂ“ see Section 1.9.
Case X = M : Moser theorem вЂ“ see Section 1.7.
Inspired by the elementary normal form in symplectic linear algebra (Theo-
rem 1.1), we will go on to describe normal neighborhoods of a point (the Darboux
theorem) and of a lagrangian submanifold (the Weinstein theorems), inside a sym-
plectic manifold. The main tool is the Moser trick, explained below, which leads
to the crucial Moser theorems and which is at the heart of many arguments in
symplectic geometry. We need some (non-symplectic) ingredients discussed in Ap-
pendix A; for more on these topics, see, for instance, [12, 25, 41].

1.7 Moser Trick

Let M be a compact manifold with symplectic forms П‰0 and П‰1 . Moser asked
whether we can п¬Ѓnd a symplectomorphism П• : M в†’ M which is homotopic to
9
1.7. MOSER TRICK

idM . A necessary condition is [П‰0 ] = [П‰1 ] в€€ H 2 (M ; R) because: if П• в€ј idM , then,
by the homotopy formula, there exists a homotopy operator Q such that
idв€— П‰1 в€’ П•в€— П‰1 = dQП‰1 + Q dП‰1
M
0
П‰1 = П•в€— П‰1 + d(QП‰1 )
=в‡’
[П‰1 ] = [П•в€— П‰1 ] = [П‰0 ] .
=в‡’

Suppose now that [П‰0 ] = [П‰1 ]. Moser  proved that the answer to the
question above is yes, with a further hypothesis as in Theorem 1.9. McDuп¬Ђ showed
that, in general, the answer is no; for a counterexample, see Example 7.23 in .

Theorem 1.9 (Moser Theorem вЂ“ Version I) Suppose that [П‰ 0 ] = [П‰1 ] and
that the 2-form П‰t = (1 в€’ t)П‰0 + tП‰1 is symplectic for each t в€€ [0, 1]. Then there
exists an isotopy ПЃ : M Г— R в†’ M such that ПЃв€— П‰t = П‰0 for all t в€€ [0, 1].
t

In particular, П• = ПЃ1 : M в€’в†’ M , satisп¬Ѓes П•в€— П‰1 = П‰0 . The following argu-
ment, due to Moser, is extremely useful; it is known as the Moser trick.
Proof. Suppose that there exists an isotopy ПЃ : M Г— R в†’ M such that ПЃв€— П‰t = П‰0 ,
t
0 в‰¤ t в‰¤ 1. Let
dПЃt
в—¦ ПЃв€’1 , tв€€R .
vt = t
dt
Then
d dП‰t
0 = (ПЃв€— П‰t ) = ПЃв€— Lvt П‰t +
t t
dt dt
dП‰t
в‡ђв‡’ L v t П‰t + =0. ()
dt
Suppose conversely that we can п¬Ѓnd a smooth time-dependent vector п¬Ѓeld
vt , t в€€ R, such that ( ) holds for 0 в‰¤ t в‰¤ 1. Since M is compact, we can integrate
vt to an isotopy ПЃ : M Г— R в†’ M with
dв€—
ПЃ в€— П‰t = ПЃ в€— П‰0 = П‰ 0 .
(ПЃ П‰t ) = 0 =в‡’ 0
dt t t

So everything boils down to solving ( ) for vt .
First, from П‰t = (1 в€’ t)П‰0 + tП‰1 , we conclude that
dП‰t
= П‰1 в€’ П‰0 .
dt
Second, since [П‰0 ] = [П‰1 ], there exists a 1-form Вµ such that

П‰1 в€’ П‰0 = dВµ .

Third, by the Cartan magic formula, we have

Lvt П‰t = dД±vt П‰t + Д±vt dП‰t .
0
10 LECTURE 1. SYMPLECTIC FORMS

Putting everything together, we must п¬Ѓnd vt such that
dД±vt П‰t + dВµ = 0 .

It is suп¬ѓcient to solve Д±vt П‰t + Вµ = 0. By the nondegeneracy of П‰t , we can solve
this pointwise, to obtain a unique (smooth) vt .

Theorem 1.10 (Moser Theorem вЂ“ Version II) Let M be a compact manifold
with symplectic forms П‰0 and П‰1 . Suppose that П‰t , 0 в‰¤ t в‰¤ 1, is a smooth family
of closed 2-forms joining П‰0 to П‰1 and satisfying:
d d
(1) cohomology assumption: [П‰t ] is independent of t, i.e., dt [П‰t ] = dt П‰t = 0,
(2) nondegeneracy assumption: П‰t is nondegenerate for 0 в‰¤ t в‰¤ 1.
Then there exists an isotopy ПЃ : M Г— R в†’ M such that ПЃв€— П‰t = П‰0 , 0 в‰¤ t в‰¤ 1.
t

Proof. (Moser trick) We have the following implications from the hypotheses:
(1) =в‡’ There is a family of 1-forms Вµt such that
dП‰t
0в‰¤tв‰¤1.
= dВµt ,
dt
We can indeed п¬Ѓnd a smooth family of 1-forms Вµt such that dП‰t = dВµt .
dt
The argument involves the PoincarВґ lemma for compactly-supported forms,
e
together with the Mayer-Vietoris sequence in order to use induction on the
number of charts in a good cover of M . For a sketch of the argument, see
page 95 in .
(2) =в‡’ There is a unique family of vector п¬Ѓelds vt such that
Д±v t П‰ t + Вµ t = 0 (Moser equation) .

Extend vt to all t в€€ R. Let ПЃ be the isotopy generated by vt (ПЃ exists by
compactness of M ). Then we indeed have
dв€— dП‰t
(ПЃt П‰t ) = ПЃв€— (Lvt П‰t + ) = ПЃв€— (dД±vt П‰t + dВµt ) = 0 .
t t
dt dt

The compactness of M was used to be able to integrate vt for all t в€€ R. If M
is not compact, we need to check the existence of a solution ПЃt for the diп¬Ђerential
equation dПЃt = vt в—¦ ПЃt for 0 в‰¤ t в‰¤ 1.
dt

Picture. Fix c в€€ H 2 (M ). Deп¬Ѓne Sc = {symplectic forms П‰ in M with [П‰] = c}.
The Moser theorem implies that, on a compact manifold, all symplectic forms on
the same path-connected component of Sc are symplectomorphic.
11
1.8. MOSER LOCAL THEOREM

Exercises 4
Any oriented 2-dimensional manifold with an area form is a symplectic mani-
fold.
(a) Show that convex combinations of two area forms П‰ 0 and П‰1 that induce
the same orientation are symplectic.
This is wrong in dimension 4: п¬Ѓnd two symplectic forms on the vector
space R4 that induce the same orientation, yet some convex combination
of which is degenerate. Find a path of symplectic forms that connect
them.
(b) Suppose that we have two area forms П‰ 0 , П‰1 on a compact 2-dimensional
manifold M representing the same de Rham cohomology class, i.e.,
2
[П‰0 ] = [П‰1 ] в€€ HdeRham (M ).
Prove that there is a 1-parameter family of diп¬Ђeomorphisms П• t : M в†’
M such that П•в€— П‰0 = П‰1 , П•0 = id, and П•в€— П‰0 is symplectic for all t в€€
1 t
[0, 1].
Such a 1-parameter family П• t is a strong isotopy between П‰0 and П‰1 .
In this language, this exercise shows that, up to strong isotopy, there is
a unique symplectic representative in each non-zero 2-cohomology class
of M .

1.8 Moser Local Theorem
Theorem 1.11 (Moser Theorem вЂ“ Local Version) Let M be a manifold, X
a submanifold of M , i : X в†’ M the inclusion map, П‰0 and П‰1 symplectic forms
in M .
Hypothesis: П‰0 |p = П‰1 |p , в€Ђp в€€ X .
Conclusion: There exist neighborhoods U0 , U1 of X in M ,
and a diп¬Ђeomorphism П• : U0 в†’ U1 such that
П• E U1
U0
d
s 
В
d В
d В  commutes
id В i
d В
X
and П•в€— П‰1 = П‰0 .

Proof.
1. Pick a tubular neighborhood U0 of X. The 2-form П‰1 в€’ П‰0 is closed on U0 ,
and (П‰1 в€’ П‰0 )p = 0 at all p в€€ X. By the homotopy formula on the tubular
neighborhood, there exists a 1-form Вµ on U0 such that П‰1 в€’ П‰0 = dВµ and
Вµp = 0 at all p в€€ X.
2. Consider the family П‰t = (1 в€’ t)П‰0 + tП‰1 = П‰0 + tdВµ of closed 2-forms on U0 .
Shrinking U0 if necessary, we can assume that П‰t is symplectic for 0 в‰¤ t в‰¤ 1.
12 LECTURE 1. SYMPLECTIC FORMS

3. Solve the Moser equation: Д±vt П‰t = в€’Вµ. Notice that vt = 0 on X.
4. Integrate vt . Shrinking U0 again if necessary, there exists an isotopy ПЃ :
U0 Г— [0, 1] в†’ M with ПЃв€— П‰t = П‰0 , for all t в€€ [0, 1]. Since vt |X = 0, we have
t
ПЃt |X = idX . Set П• = ПЃ1 , U1 = ПЃ1 (U0 ).

1.9 Darboux Theorem
We will apply the local version of the Moser theorem to X = {p} in order to prove:
Theorem 1.12 (Darboux) Let (M, П‰) be a 2n-dimensional symplectic mani-
fold, and let p be any point in M . Then there is a coordinate chart (U, x 1 , . . . , xn ,
y1 , . . . , yn ) centered at p such that on U
n
dxi в€§ dyi .
П‰=
i=1

As a consequence of Theorem 1.12, if we show for (R2n , dxi в€§ dyi ) a local
assertion which is invariant under symplectomorphisms, then that assertion holds
for any symplectic manifold.
Proof. Use any symplectic basis for Tp M to construct coordinates (x1 , . . . , xn ,
y1 , . . . yn ) centered at p and valid on some neighborhood U , so that

dxi в€§ dyi
П‰p = .
p

There are two symplectic forms on U : the given П‰0 = П‰ and П‰1 = dxi в€§ dyi . By
the Moser theorem (Theorem 1.11) applied to X = {p}, there are neighborhoods
U0 and U1 of p, and a diп¬Ђeomorphism П• : U0 в†’ U1 such that

П•в€— ( dxi в€§ dyi ) = П‰ .
П•(p) = p and

Since П•в€— ( dxi в€§dyi ) = d(xi в—¦П•)в€§d(yi в—¦П•), we only need to set new coordinates
xi = xi в—¦ П• and yi = yi в—¦ П•.
A chart (U, x1 , . . . , xn , y1 , . . . , yn ) as in Theorem 1.12 is called a Darboux
chart.
By Theorem 1.12, 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 в€§ dyi .
П‰0 =
i=1
13
1.9. DARBOUX THEOREM

Exercise 5
Prove the Darboux theorem in the 2-dimensional case, using the fact that every
nonvanishing 1-form on a surface can be written locally as f dg for suitable
functions f, g.
Hint: П‰ = df в€§ dg is nondegenerate в‡ђв‡’ (f, g) is a local diп¬Ђeomorphism.

Exercise 6
Let H be the vector space of n Г— n complex hermitian matrices. The unitary
group U(n) acts on H by conjugation: AВ·Оѕ = AОѕAв€’1 , for A в€€ U(n) , Оѕ в€€ H.
For each О» = (О»1 , . . . , О»n ) в€€ Rn , let HО» be the set of all nГ—n complex hermitian
matrices whose spectrum is О».
(a) Show that the orbits of the U(n)-action are the manifolds H О» .
For a п¬Ѓxed О» в€€ Rn , what is the stabilizer of a point in HО» ?

Hint: If О»1 , . . . , О»n are all distinct, the stabilizer of the diagonal matrix
is the torus T n of all diagonal unitary matrices.
(b) Show that the symmetric bilinear form on H, (X, Y ) в†’ trace (XY ) ,
is nondegenerate.
For Оѕ в€€ H, deп¬Ѓne a skew-symmetric bilinear form П‰ Оѕ on u(n) =
T1 U(n) = iH (space of skew-hermitian matrices) by
П‰Оѕ (X, Y ) = i trace ([X, Y ]Оѕ) , X, Y в€€ iH .
Check that П‰Оѕ (X, Y ) = i trace (X(Y Оѕ в€’ ОѕY )) and Y Оѕ в€’ ОѕY в€€ H.
Show that the kernel of П‰Оѕ is KОѕ := {Y в€€ u(n) | [Y, Оѕ] = 0}.
(c) Show that KОѕ is the Lie algebra of the stabilizer of Оѕ.

Hint: Diп¬Ђerentiate the relation AОѕAв€’1 = Оѕ.
Show that the П‰Оѕ вЂ™s induce nondegenerate 2-forms on the orbits H О» .
Show that these 2-forms are closed.
Conclude that all the orbits HО» are compact symplectic manifolds.
(d) Describe the manifolds HО» .
When all eigenvalues are equal, there is only one point in the orbit.
Suppose that О»1 = О»2 = . . . = О»n . Then the eigenspace associated
with О»1 is a line, and the one associated with О»2 is the orthogonal
C Pnв€’1 . We
hyperplane. Show that there is a diп¬Ђeomorphism H О»
nв€’1 , on for each
have thus exhibited a lot of symplectic forms on C P
pair of distinct real numbers.

Hint: When the eigenvalues О»1 < . . . < О»n are all distinct, any element
in HО» deп¬Ѓnes a family of pairwise orthogonal lines in Cn : its eigenspaces.
(e) Show that, for any skew-hermitian matrix X в€€ u(n), the vector п¬Ѓeld
on H generated by X в€€ u(n) for the U(n)-action by conjugation is
#
XОѕ = [X, Оѕ].
Lecture 2

Cotangent Bundles

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
diп¬Ђerential operators, dynamical systems and classical mechanics.

2.1 Tautological and Canonical Forms

Let (U, x1 , . . . , xn ) be a coordinate chart for X, with associated cotangent coordi-
nates1 (T в€— U, x1 , . . . , xn , Оѕ1 , . . . , Оѕn ). Deп¬Ѓne a 2-form П‰ on T в€— U by
n
dxi в€§ dОѕi .
П‰=
i=1

1 Let X be any n-dimensional manifold and M = T в€— X its cotangent bundle. If the manifold
structure on X is described by coordinate charts (U , x1 , . . . , xn ) with xi : U в†’ R, then at
в€— в€—
any x в€€ U , the diп¬Ђerentials (dx1 )x , . . . (dxn )x form a basis of Tx X. Namely, if Оѕ в€€ Tx X, then
n
Оѕ = i=1 Оѕi (dxi )x for some real coeп¬ѓcients Оѕ1 , . . . , Оѕn . This induces a map

R2n
T в€—U в€’в†’
(x, Оѕ) в€’в†’ (x1 , . . . , xn , Оѕ1 , . . . , Оѕn ) .

The chart (T в€— U , x1 , . . . , xn , Оѕ1 , . . . , Оѕn ) is a coordinate chart for T в€— X; the coordinates
x1 , . . . , xn , Оѕ1 , . . . , Оѕn are the cotangent coordinates associated to the coordinates x1 , . . . , xn
on U . The transition functions on the overlaps are smooth: given two charts (U , x 1 , . . . , xn ),
в€—
(U , x1 , . . . , xn ), and x в€€ U в€© U , if Оѕ в€€ Tx X, then
n n
в€‚xi
Оѕ= Оѕi (dxi )x = Оѕi (dxj )x = Оѕj (dxj )x
в€‚xj
i=1 i,j j=1

в€‚xi
is smooth. Hence, T в€— X is a 2n-dimensional manifold.
where Оѕj = i Оѕi в€‚xj

15
16 LECTURE 2. COTANGENT BUNDLES

In order to check that this deп¬Ѓnition is coordinate-independent, consider the 1-form
on T в€— U
n
О±= Оѕi dxi .
i=1

Clearly, П‰ = в€’dО±.
Claim. The form О± is intrinsically deп¬Ѓned (and hence the form П‰ is also intrinsi-
cally deп¬Ѓned) .

Proof. Let (U, x1 , . . . , xn , Оѕ1 , . . . , Оѕn ) and (U , x1 , . . . , xn , Оѕ1 , . . . , Оѕn ) be two cotan-
gent coordinate charts. On U в€© U , the two sets of coordinates are related by
в€‚x
в€‚x
Оѕj = i Оѕi в€‚x i . Since dxj = i в€‚xj dxi , we have i
j

О±= Оѕi dxi = Оѕj dxj = О± .
i j

The 1-form О± is the tautological form and 2-form П‰ is the canonical
symplectic form. Next we provide an alternative proof of the intrinsic character
of these forms. Let

M = T в€—X в€—
Оѕ в€€ Tx X
p = (x, Оѕ)
в†“ПЂ в†“
X x

be the natural projection. The tautological 1-form О± may be deп¬Ѓned pointwise
by
О±p = (dПЂp )в€— Оѕ в€€ Tp M ,
в€—

where (dПЂp )в€— is the transpose of dПЂp , that is, (dПЂp )в€— Оѕ = Оѕ в—¦ dПЂp :
в€—
p = (x, Оѕ) Tp M Tp M
в†‘ (dПЂp )в€—
в†“ПЂ в†“ dПЂp
в€—
x Tx X Tx X

Equivalently,
for v в€€ Tp M .
О±p (v) = Оѕ (dПЂp )v ,

Exercise 7
Let (U , x1 , . . . , xn ) be a chart on X with associated cotangent coordinates
n
x1 , . . . , xn , Оѕ1 , . . . , Оѕn . Show that on T в€— U , О± = Оѕi dxi .
i=1
17
2.2. NATURALITY OF THE CANONICAL FORMS

The canonical symplectic 2-form П‰ on T в€— X is deп¬Ѓned as
П‰ = в€’dО± .
n
dxi в€§ dОѕi .
Locally, П‰ = i=1

Exercise 8
Show that the tautological 1-form О± is uniquely characterized by the property
that, for every 1-form Вµ : X в†’ T в€— X, Вµв€— О± = Вµ. (See Section 2.4.)

2.2 Naturality of the Canonical Forms
Let X1 and X2 be n-dimensional manifolds with cotangent bundles M1 = T в€— X1
and M2 = T в€— X2 , and tautological 1-forms О±1 and О±2 . Suppose that f : X1 в†’ X2
is a diп¬Ђeomorphism. Then there is a natural diп¬Ђeomorphism
f : M1 в†’ M 2
в€—
which lifts f ; namely, if p1 = (x1 , Оѕ1 ) в€€ M1 for x1 в€€ X1 and Оѕ1 в€€ Tx1 X1 , then we
deп¬Ѓne
x2 = f (x1 ) в€€ X2 and
f (p1 ) = p2 = (x2 , Оѕ2 ) , with
Оѕ1 = (dfx1 )в€— Оѕ2 ,
where (dfx1 )в€— : Tx2 X2 в†’ Tx1 X1 , so f |Tx1 is the inverse map of (dfx1 )в€— .
в€— в€— в€—

Exercise 9
Check that f is a diп¬Ђeomorphism. Here are some hints:
f
M1 в€’в†’ M2
1. commutes;
ПЂ1 в†“ в†“ ПЂ2
f
X1 в€’в†’ X2
2. f : M1 в†’ M2 is bijective;
3. f and f в€’1 are smooth.

Proposition 2.1 The lift f of a diп¬Ђeomorphism f : X1 в†’ X2 pulls the tautolog-
ical form on T в€— X2 back to the tautological form on T в€— X1 , i.e.,
(f )в€— О±2 = О±1 .

Proof. At p1 = (x1 , Оѕ1 ) в€€ M1 , this identity says that
в€—
(df )p1 (О±2 )p2 = (О±1 )p1 ()

where p2 = f (p1 ).
Using the following facts,
18 LECTURE 2. COTANGENT BUNDLES

вЂў deп¬Ѓnition of f :
p2 = f (p1 ) в‡ђв‡’ p2 = (x2 , Оѕ2 ) where x2 = f (x1 ) and (dfx1 )в€— Оѕ2 = Оѕ1 ,
вЂў deп¬Ѓnition of tautological 1-form:
(О±1 )p1 = (dПЂ1 )в€—1 Оѕ1 (О±2 )p2 = (dПЂ2 )в€—2 Оѕ2 ,
and
p p

f
в€’в†’
M1 M2
вЂў the diagram ПЂ1 в†“ в†“ ПЂ2 commutes,
f
в€’в†’
X1 X2
the proof of ( ) is:
в€—
(df )в€—1 (О±2 )p2 = (df )в€—1 (dПЂ2 )в€—2 Оѕ2 = (d(ПЂ2 в—¦ f ))p1 Оѕ2
p p p
в€—
= (dПЂ1 )в€—1 (df )в€— 1 Оѕ2
= (d(f в—¦ ПЂ1 ))p1 Оѕ2 p x
= (dПЂ1 )в€—1 Оѕ1 = (О±1 )p1 .
p

Corollary 2.2 The lift f of a diп¬Ђeomorphism f : X1 в†’ X2 is a symplectomor-
phism, i.e.,
(f )в€— П‰2 = П‰1 ,
where П‰1 , П‰2 are the canonical symplectic forms.
In summary, a diп¬Ђeomorphism of manifolds induces a canonical symplecto-
morphism of cotangent bundles:
f : T в€— X1 в€’в†’ T в€— X2
в†‘
в€’в†’
f: X1 X2

Example. Let X1 = X2 = S 1 . Then T в€— S 1 is an inп¬Ѓnite cylinder S 1 Г— R. The
canonical 2-form П‰ is the area form П‰ = dОё в€§ dОѕ. If f : S 1 в†’ S 1 is any dif-
feomorphism, then f : S 1 Г— R в†’ S1 Г— R is a symplectomorphism, i.e., is an
в™¦
area-preserving diп¬Ђeomorphism of the cylinder.
If f : X1 в†’ X2 and g : X2 в†’ X3 are diп¬Ђeomorphisms, then (g в—¦ f ) = g в—¦ f .
In terms of the group Diп¬Ђ(X) of diп¬Ђeomorphisms of X and the group Sympl(M, П‰)
of symplectomorphisms of (M, П‰), we say that the map
Diп¬Ђ(X) в€’в†’ Sympl(M, П‰)
f в€’в†’ f
is a group homomorphism. This map is clearly injective. Is it surjective? Do all
symplectomorphisms T в€— X в†’ T в€— X come from diп¬Ђeomorphisms X в†’ X? No: for
instance, translation along cotangent п¬Ѓbers is not induced by a diп¬Ђeomorphism
of the base manifold. A criterion for which symplectomorphisms arise as lifts of
diп¬Ђeomorphisms is discussed in the next section.
2.3. SYMPLECTOMORPHISMS OF T в€— X 19

Symplectomorphisms of T в€— X
2.3

Let (M, П‰) be a symplectic manifold, and let О± be a 1-form such that

П‰ = в€’dО± .

There exists a unique vector п¬Ѓeld v such that its interior product with П‰ is О±, i.e.,
Д±v П‰ = в€’О±.

Proposition 2.3 If g is a symplectomorphism which preserves О± (that is, g в€— О± =
О±), then g commutes with the one-parameter group of diп¬Ђeomorphisms generated
by v, i.e.,
(exp tv) в—¦ g = g в—¦ (exp tv) .

Proof. Recall that, for p в€€ M , (exp tv)(p) is the unique curve in M solving the
ordinary diп¬Ђerential equation
d
dt (exp tv(p)) = v(exp tv(p))
(exp tv)(p)|t=0 = p

for t in some neighborhood of 0. From this is follows that g в—¦ (exp tv) в—¦ g в€’1 must be
the one-parameter group of diп¬Ђeomorphisms generated by gв€— v. (The push-forward
of v by g is deп¬Ѓned by (gв€— v)g(p) = dgp (vp ).) Finally we have that gв€— v = v, i.e.,
that g preserves v.
Let X be an arbitrary n-dimensional manifold, and let M = T в€— X. Let
(U, x1 , . . . , xn ) be a coordinate system on X, and let x1 , . . . , xn , Оѕ1 , . . . , Оѕn be
the corresponding coordinates on T в€— U. When О± is the tautological 1-form on M
(which, in these coordinates, is Оѕi dxi ), the vector п¬Ѓeld v above is just the vector
в€‚
Оѕi в€‚Оѕi . Let exp tv, в€’в€ћ < t < в€ћ, be the one-parameter group of diп¬Ђeomor-
п¬Ѓeld
phisms generated by v.

Exercise 10
Show that, for every point p = (x, Оѕ) in M ,
pt = (x, et Оѕ) .
(exp tv)(p) = pt where

If g is a symplectomorphism of M = T в€— X which preserves О±, then we must
have that
g(x, Оѕ) = (y, О·) =в‡’ g(x, О»Оѕ) = (y, О»О·)
for all (x, Оѕ) в€€ M and О» в€€ R. In fact, if g(p) = q where p = (x, Оѕ) and q = (y, О·),
this assertion follows from a combination of the identity

(dgp )в€— О±q = О±p
20 LECTURE 2. COTANGENT BUNDLES

with the identity
dПЂq в—¦ dgp = dfx в—¦ dПЂp .
(The п¬Ѓrst identity expresses the fact that g в€— О± = О±, and the second identity is
obtained by diп¬Ђerentiating both sides of the equation ПЂв—¦g = f в—¦ПЂ at p.) We conclude
that g has to preserve the cotangent п¬Ѓbration, i.e., there exists a diп¬Ђeomorphism
f : X в†’ X such that ПЂ в—¦ g = f в—¦ ПЂ, where ПЂ : M в†’ X is the projection map
ПЂ(x, Оѕ) = x. Moreover, g = f# , the map f# being the symplectomorphism of M
lifting f . Hence, the symplectomorphisms of T в€— X of the form f# are those which
preserve the tautological 1-form О±.
Here is a diп¬Ђerent class of symplectomorphisms of M = T в€— X. Let h be a
smooth function on X. Deп¬Ѓne П„h : M в†’ M by setting

П„h (x, Оѕ) = (x, Оѕ + dhx ) .

Then
П„h О± = О± + ПЂ в€— dh
в€—

where ПЂ is the projection map
M (x, Оѕ)
в†“ПЂ в†“
X x

Therefore,
в€—
П„h П‰ = П‰ ,
so all such П„h are symplectomorphisms.

Lagrangian Submanifolds of T в€—X
2.4
Let (M, П‰) be a 2n-dimensional symplectic manifold.

Deп¬Ѓnition 2.4 A submanifold Y of M is a lagrangian submanifold if, at each
p в€€ Y , Tp Y is a lagrangian subspace of Tp M , i.e., П‰p |Tp Y в‰Ў 0 and dim Tp Y =
1
2 dim Tp M . Equivalently, if i : Y в†’ M is the inclusion map, then Y is lagrangian
if and only if iв€— П‰ = 0 and dim Y = 1 dim M .
2

Let X be an n-dimensional manifold, with M = T в€— X its cotangent bundle.
If x1 , . . . , xn are coordinates on U вЉ† X, with associated cotangent coordinates
x1 , . . . , xn , Оѕ1 , . . . , Оѕn on T в€— U , then the tautological 1-form on T в€— X is

О±= Оѕi dxi

and the canonical 2-form on T в€— X is

П‰ = в€’dО± = dxi в€§ dОѕi .
2.4. LAGRANGIAN SUBMANIFOLDS OF T в€— X 21

The zero section of T в€— X,
X0 := {(x, Оѕ) в€€ T в€— X | Оѕ = 0 in Tx X} ,
в€—

is an n-dimensional submanifold of T в€— X whose intersection with T в€— U is given by
Оѕi dxi vanishes on X0 в€© T в€— U .
the equations Оѕ1 = . . . = Оѕn = 0. Clearly О± =
In particular, if i0 : X0 в†’ T в€— X is the inclusion map, we have iв€— О± = 0. Hence,
0
iв€— П‰ = iв€— dО± = 0, and X0 is lagrangian.
0 0
What are all the lagrangian submanifolds of T в€— X which are вЂњC 1 -close to
X0 вЂќ?
Let XВµ be (the image of) another section, that is, an n-dimensional subman-
ifold of T в€— X of the form
в€—
XВµ = {(x, Вµx ) | x в€€ X, Вµx в€€ Tx X} ( )
where the covector Вµx depends smoothly on x, and Вµ : X в†’ T в€— X is a de Rham
1-form. Relative to the inclusion i : XВµ в†’ T в€— X and the cotangent projection
ПЂ : T в€— X в†’ X, XВµ is of the form ( ) if and only if ПЂ в—¦ i : XВµ в†’ X is a
diп¬Ђeomorphism.
When is such a XВµ lagrangian?
Proposition 2.5 Let XВµ be of the form ( ), and let Вµ be the associated de Rham
1-form. Denote by sВµ : X в†’ T в€— X, x в†’ (x, Вµx ), be the 1-form Вµ regarded exclu-
sively as a map. Notice that the image of sВµ is XВµ . Let О± be the tautological 1-form
on T в€— X. Then
sв€— О± = Вµ .
Вµ

Proof. By deп¬Ѓnition of the tautological form О±, О±p = (dПЂp )в€— Оѕ at p = (x, Оѕ) в€€ M .
For p = sВµ (x) = (x, Вµx ), we have О±p = (dПЂp )в€— Вµx . Then
(sв€— О±)x = (dsВµ )в€— О±p
Вµ x
= (dsВµ )в€— (dПЂp )в€— Вµx
x
= (d(ПЂ в—¦ sВµ ))в€— Вµx = Вµx .
x

idX

Suppose that XВµ is an n-dimensional submanifold of T в€— X of the form ( ),
with associated de Rham 1-form Вµ. Then sВµ : X в†’ T в€— X is an embedding with
image XВµ , and there is a diп¬Ђeomorphism П„ : X в†’ XВµ , П„ (x) := (x, Вµx ), such that
the following diagram commutes.
sВµ E T в€—X
X
d 
В
d В
d В
П„d В i
d
В‚ В
XВµ
22 LECTURE 2. COTANGENT BUNDLES

We want to express the condition of XВµ being lagrangian in terms of the form Вµ:

iв€— dО± = 0
в‡ђв‡’
XВµ is lagrangian
П„ в€— iв€— dО± = 0
в‡ђв‡’
(i в—¦ П„ )в€— dО± = 0
в‡ђв‡’
sв€— dО± = 0
в‡ђв‡’ Вµ
dsв€— О± = 0
в‡ђв‡’ Вµ
в‡ђв‡’ dВµ = 0
в‡ђв‡’ Вµ is closed .

Therefore, there is a one-to-one correspondence between the set of lagrangian
submanifolds of T в€— 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 Вµ is
equal to df for some f в€€ C в€ћ (X). Any such primitive f is then called a generat-
ing function for the lagrangian submanifold XВµ associated to Вµ. (Two functions
generate the same lagrangian submanifold if and only if they diп¬Ђer by a locally
constant function.) On arbitrary manifolds X, functions f в€€ C в€ћ (X) originate
lagrangian submanifolds as images of df .

Exercise 11
Check that, if X is compact (and not just one point) and f в€€ C в€ћ (X), then
#{Xdf в€© X0 } в‰Ґ 2.

2.5 Conormal Bundles
There are lots of lagrangian submanifolds of T в€— X not covered by the description
in terms of closed 1-forms from the previous section, starting with the cotangent
п¬Ѓbers.
Let S be any k-dimensional submanifold of an n-dimensional manifold X.

Deп¬Ѓnition 2.6 The conormal space at x в€€ S is
в€— в€—
Nx S = {Оѕ в€€ Tx X | Оѕ(v) = 0 , for all v в€€ Tx S} .

The conormal bundle of S is

N в€— S = {(x, Оѕ) в€€ T в€— X | x в€€ S, Оѕ в€€ Nx S} .
в€—

By using coordinates on X adapted2 to S, one sees that the conormal bundle
N в€— S is an n-dimensional submanifold of T в€— X.
2Acoordinate chart (U , x1 , . . . , xn ) on X is adapted to a k-dimensional submanifold S if S в€©U
is described by xk+1 = . . . = xn = 0.
23
2.6. LAGRANGIAN COMPLEMENTS

Proposition 2.7 Let i : N в€— S в†’ T в€— X be the inclusion, and let О± be the tautolog-
ical 1-form on T в€— X. Then
iв€— О± = 0 .

Proof. Let (U, x1 , . . . , xn ) be a coordinate system on X centered at x в€€ S
and adapted to S, so that U в€© S is described by xk+1 = . . . = xn = 0. Let
(T в€— U, x1 , . . . , xn , Оѕ1 , . . . , Оѕn ) be the associated cotangent coordinate system. The
submanifold N в€— S в€© T в€— U is then described by
xk+1 = . . . = xn = 0 and Оѕ 1 = . . . = Оѕk = 0 .
Оѕi dxi on T в€— U, we conclude that, at p в€€ N в€— S,
Since О± =

(iв€— О±)p = О±p |Tp (N в€— S) = Оѕi dxi =0.
i>k в€‚
span{ в€‚x ,iв‰¤k}
i

Corollary 2.8 For any submanifold S вЉ‚ X, the conormal bundle N в€— S is a la-
grangian submanifold of T в€— X.
Taking S = {x} to be one point, the conormal bundle L = N в€— S = Tx X is a
в€—

cotangent п¬Ѓber. Taking S = X, the conormal bundle L = X0 is the zero section
of T в€— X.

2.6 Lagrangian Complements
Normal neighborhoods of lagrangian submanifolds are described by the theorems
in the following two sections. It was proved by Weinstein  that the conclusion
of the Moser local theorem (Theorem 1.11) still holds if we assume instead
Hypothesis: X is an n-dimensional submanifold with
iв€— П‰0 = iв€— П‰1 = 0 where i : X в†’ M is inclusion, i.e.,
X is a submanifold lagrangian for П‰0 and П‰1 .
We need some algebra for the Weinstein theorem.
Suppose that U and W are n-dimensional vector spaces, and в„¦ : U Г—W в†’ R is
a bilinear pairing; the map в„¦ gives rise to a linear map в„¦ : U в†’ W в€— , в„¦(u) = в„¦(u, В·).
Then в„¦ is nondegenerate if and only if в„¦ is bijective.
Proposition 2.9 Suppose that V is a 2n-dimensional vector space and в„¦ : V Г—
V в†’ R is a nondegenerate skew-symmetric bilinear pairing. Let U be a lagrangian
subspace of (V, в„¦) (i.e., в„¦|U Г—U = 0 and U is n-dimensional). Let W be any vector
space complement to U , not necessarily lagrangian.
Then from W we can canonically build a lagrangian complement to U .
24 LECTURE 2. COTANGENT BUNDLES

в„¦
Proof. The pairing в„¦ gives a nondegenerate pairing U Г— W в†’ R. Therefore,
в„¦ : U в†’ W в€— is bijective. We look for a lagrangian complement to U of the form

W = {w + Aw | w в€€ W } ,

A : W в†’ U being a linear map. For W to be lagrangian we need

в€Ђ w 1 , w2 в€€ W , в„¦(w1 + Aw1 , w2 + Aw2 ) = 0

=в‡’ в„¦(w1 , w2 ) + в„¦(w1 , Aw2 ) + в„¦(Aw1 , w2 ) + в„¦(Aw1 , Aw2 ) = 0
в€€U

0
= в„¦(Aw2 , w1 ) в€’ в„¦(Aw1 , w2 )
=в‡’ в„¦(w1 , w2 )
= в„¦ (Aw2 )(w1 ) в€’ в„¦ (Aw1 )(w2 ) .

Let A = в„¦ в—¦ A : W в†’ W в€— , and look for A such that

в€Ђ w 1 , w2 в€€ W , в„¦(w1 , w2 ) = A (w2 )(w1 ) в€’ A (w1 )(w2 ) .
1
The canonical choice is A (w) = в€’ 2 в„¦(w, В·). Then set A = (в„¦ )в€’1 в—¦ A .

Proposition 2.10 Let V be a 2n-dimensional vector space, let в„¦0 and в„¦1 be
symplectic forms in V , let U be a subspace of V lagrangian for в„¦0 and в„¦1 , and
let W be any complement to U in V . Then from W we can canonically construct
a linear isomorphism L : V в†’ V such that L|U = IdU and Lв€— в„¦1 = в„¦0 .

Proof. From W we canonically obtain complements W0 and W1 to U in V such
that W0 is lagrangian for в„¦0 and W1 is lagrangian for в„¦1 . The nondegenerate
bilinear pairings
в„¦
 стр. 1(всего 5)СОДЕРЖАНИЕ >>