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Introduction to
Symplectic and Hamiltonian Geometry



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




Ana Cannas da Silva1

Revised Version “ March 19, 2002




1
E-mail: acannas@math.ist.utl.pt
Pref´cio
a
A geometria simpl´ctica ´ a geometria de variedades equipadas com uma forma
e e
simpl´ctica, ou seja, com uma forma de grau 2 fechada e n˜o-degenerada. A geo-
e a
metria hamiltoniana ´ a geometria de variedades (simpl´cticas) equipadas com
e e
uma aplica¸ao momento, ou seja, com uma colec¸ao de quantidades conservadas
c˜ c˜
por simetrias.
H´ cerca de dois s´culos, a geometria simpl´ctica fornecia a linguagem para
a e e
a mecˆnica cl´ssica; pelo seu r´pido crescimento recente, conquistou um rico
a a a
territ´rio, estabelecendo-se como um ramo central da geometria e da topologia
o
diferenciais. Al´m da sua actividade como disciplina independente, a geometria
e
simpl´ctica ´ signi¬cativamente estimulada por interac¸oes importantes com sis-
e e c˜
temas dinˆmicos, an´lise global, f´
a a ±sica-matem´tica, topologia em baixas dimens˜es,
a o
teoria de representa¸oes, an´lise microlocal, equa¸oes diferenciais parciais, geo-
c˜ a c˜
metria alg´brica, geometria riemanniana, an´lise combinat´rica geom´trica, co-
e a o e
homologia equivariante, etc.
Este texto cobre fundamentos da geometria simpl´ctica numa linguagem
e
moderna. Come¸a-se por descrever as variedades simpl´cticas e as suas trans-
c e
forma¸oes, e por explicar liga¸oes a topologia e outras geometrias. Seguidamente
c˜ c˜
estudam-se campos hamiltonianos, ac¸oes hamiltonianas e algumas das suas aplica-

coes pr´ticas no ambito da mecˆnica e dos sistemas dinˆmicos. Ao longo do
¸˜ a ˆ a a
texto fornecem-se exemplos simples e exerc´ ±cios relevantes. Pressup˜em-se conheci-
o
mentos pr´vios de geometria de variedades diferenci´veis, se bem que os principais
e a
factos requeridos estejam coleccionados em apˆndices. e
Estas notas reproduzem aproximadamente o curso curto de geometria sim-
pl´ctica, constitu´ por cinco li¸oes dirigidas a estudantes de p´s-gradua¸ao e
e ±do c˜ o c˜
investigadores, integrado no programa de Ver˜o do Instituto de Matem´tica Pura
a a
e Aplicada, no Rio de Janeiro, em Fevereiro de 2002. Alguns trechos deste texto
s˜o 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 inestim´vel 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 di¬erential geometry and topology. Besides its
activity as an independent subject, symplectic geometry is signi¬cantly stimulated
by important interactions with dynamical systems, global analysis, mathemati-
cal physics, low-dimensional topology, representation theory, microlocal analysis,
partial di¬erential 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
¬elds, 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 ¬ve lectures mostly for graduate students and researchers, held at
the summer program of Instituto de Matem´tica 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

Pref´cio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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 — X . . . . . . . . . . . . . . . . . . . . . 19
2.4 Lagrangian Submanifolds of T — 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 Poincar´ 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 Delzant™s Construction . . . . . . . . . . . . . . . . . . . . . . . . . 83

A Prerequisites from Di¬erential 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 Di¬eomorphisms . . . . . . . . . . . . . 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 “ nondegeneracy
“ and an analytical condition “ closedness. In this lecture we de¬ne symplectic
forms, describe some of their basic properties, and introduce the ¬rst 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 „¦ : 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 |
™¦


1
2 LECTURE 1. SYMPLECTIC FORMS

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 .

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. Hence, k := dim U is an invariant of (V, „¦).
Since k + 2n = m = dim V , we have that 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.
3
1.2. SYMPLECTIC VECTOR SPACES

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 in the previous section.
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.
The following are immediate properties of a symplectic map „¦:
• Duality: the map „¦ : V ’ V — is a bijection.
• By Theorem 1.1, we must have that k = dim U = 0, so dim V = 2n is even.
• Also 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
|

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.
4 LECTURE 1. SYMPLECTIC FORMS

1.3 Special Subspaces

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 } .

By considering the kernel and image of the map

’’ Y — = Hom(Y, R)
V
’’ „¦(v, ·)|Y
v

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

Y ⊆ W ⇐’ W „¦ ⊆ Y „¦ .

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

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

• A subspace Y is called isotropic if „¦|Y —Y ≡ 0. If Y is isotropic, then
dim Y ¤ 1 dim V . Every one-dimensional subspace is isotropic.
2

• A subspace is called coisotropic if its symplectic orthogonal is isotropic. If
1
Y is coisotropic, then dim Y ≥ 2 dim V . Every codimension 1 subspace is
coisotropic.

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

• the span of e1 , f1 is symplectic,

• the span of e1 , e2 is isotropic, and

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

Y is lagrangian ⇐’ Y is isotropic and coisotropic ⇐’ Y = Y „¦ .
5
1.4. SYMPLECTIC MANIFOLDS

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

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


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

„¦0 (u • ±, v • β) = β(u) ’ ±(v) .

Moreover, 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 .


1.4 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 ω p 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 manifold
and ω is a symplectic form.

Examples.

1. 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; the set

‚ ‚ ‚ ‚
,..., , ,...,
‚x1 ‚xn ‚y1 ‚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 § 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 .

3. 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.

™¦


Exercise 2
Consider cylindrical polar coordinates (θ, h) on S 2 away from its poles, where
0 ¤ θ < 2π and ’1 ¤ h ¤ 1. Show that, in these coordinates, the form of the
previous example is
ω = dθ § dh .




1.5 Symplectic Volume

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 .

Exercise 3
Show that any „¦ ∈ §2 (V — ) is of the form „¦ = e— § f1 + . . . + e— § fn , where
— —
n
1
— , . . . , u— , e— , . . . , e— , f — , . . . , f — is a basis of V — 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 „¦ : V — V ’ R is just a nondegenerate
2-form „¦ ∈ §2 (V — ), called a symplectic form on V . By the previous exercise, if „¦
is any symplectic form on a vector space V of dimension 2n, then the nth exterior
power „¦n = „¦ § . . . § „¦ does not vanish. Conversely, given a 2-form „¦ ∈ §2 (V — ),
n
if „¦n = 0, then „¦ is symplectic.
We conclude that the nth exterior power ω n of any symplectic form ω on a
2n-dimensional manifold M is a volume form.1 Hence, any symplectic manifold
(M, ω) is canonically oriented by the symplectic structure, and any nonorientable
n
manifold cannot be symplectic. The form ω is called the symplectic volume of
n!
(M, ω).
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. By Stokes™ theorem., if M is
compact, the de Rham cohomology class [ω n ] ∈ H 2n (M ; R) is non-zero. Hence,
[ω] itself is non-zero (in other words, ω 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 ω0 and ω1 , so that
(M, ω0 ) and (M, ω1 ) are two symplectic manifolds.

De¬nition 1.7 A symplectomorphism between (M 1 , ω1 ) and (M2 , ω2 ) is a dif-
feomorphism • : M1 ’ M2 such that •— ω2 = ω1 .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 classi¬cation
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 , ω0 ). More precisely,
any symplectic manifold (M 2n , ω) is locally symplectomorphic to (R2n , ω0 ).

De¬nition 1.8 We say that

• (M, ω0 ) and (M, ω1 ) are symplectomorphic if there is a di¬eomorphism
• : M ’ M with •— ω1 = ω0 ;

• (M, ω0 ) and (M, ω1 ) are strongly isotopic if there is an isotopy ρt : M ’
M such that ρ— ω1 = ω0 ;
1
1A volume form is a nonvanishing form of top degree.
2 Recall that, by de¬nition of pullback, at tangent vectors u, v ∈ Tp M1 , we have
— ω ) (u, v) = (ω )
(• 2 p 2 •(p) (d•p (u), d•p (v)).
8 LECTURE 1. SYMPLECTIC FORMS

• (M, ω0 ) and (M, ω1 ) are deformation-equivalent if there is a smooth fam-
ily ωt of symplectic forms joining ω0 to ω1 ;

• (M, ω0 ) and (M, ω1 ) are isotopic if they are deformation-equivalent with
[ωt ] independent of t.

Clearly, we have

strongly isotopic =’ symplectomorphic , and

isotopic =’ deformation-equivalent .
We also have
strongly isotopic =’ isotopic
because, if ρt : M ’ M is an isotopy such that ρ— ω1 = ω0 , then the set ωt := ρ— ω1
1 t
is a smooth family of symplectic forms joining ω1 to ω0 and [ωt ] = [ω1 ], ∀t, by
the homotopy invariance of de Rham cohomology. As we will see below, the Moser
theorem states that, on a compact manifold,

isotopic =’ 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 ω0 , ω1 on U0 , U1 , does there exist
a symplectomorphism preserving X? More precisely, does there exist a di¬eomor-
phism • : U0 ’ U1 with •— ω1 = ω0 and •(X) = X?
At the two extremes, we have:
Case X = point: Darboux theorem “ see Section 1.9.
Case X = M : Moser theorem “ 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 ω0 and ω1 . Moser asked
whether we can ¬nd a symplectomorphism • : M ’ M which is homotopic to
9
1.7. MOSER TRICK

idM . A necessary condition is [ω0 ] = [ω1 ] ∈ H 2 (M ; R) because: if • ∼ idM , then,
by the homotopy formula, there exists a homotopy operator Q such that
id— ω1 ’ •— ω1 = dQω1 + Q dω1
M
0
ω1 = •— ω1 + d(Qω1 )
=’
[ω1 ] = [•— ω1 ] = [ω0 ] .
=’

Suppose now that [ω0 ] = [ω1 ]. Moser [37] proved that the answer to the
question above is yes, with a further hypothesis as in Theorem 1.9. McDu¬ showed
that, in general, the answer is no; for a counterexample, see Example 7.23 in [35].

Theorem 1.9 (Moser Theorem “ Version I) Suppose that [ω 0 ] = [ω1 ] and
that the 2-form ωt = (1 ’ t)ω0 + tω1 is symplectic for each t ∈ [0, 1]. Then there
exists an isotopy ρ : M — R ’ M such that ρ— ωt = ω0 for all t ∈ [0, 1].
t

In particular, • = ρ1 : M ’’ M , satis¬es •— ω1 = ω0 . The following argu-
ment, due to Moser, is extremely useful; it is known as the Moser trick.
Proof. Suppose that there exists an isotopy ρ : M — R ’ M such that ρ— ωt = ω0 ,
t
0 ¤ t ¤ 1. Let
dρt
—¦ ρ’1 , t∈R .
vt = t
dt
Then
d dωt
0 = (ρ— ωt ) = ρ— Lvt ωt +
t t
dt dt
dωt
⇐’ L v t ωt + =0. ()
dt
Suppose conversely that we can ¬nd a smooth time-dependent vector ¬eld
vt , t ∈ R, such that ( ) holds for 0 ¤ t ¤ 1. Since M is compact, we can integrate
vt to an isotopy ρ : M — R ’ M with
d—
ρ — ωt = ρ — ω0 = ω 0 .
(ρ ωt ) = 0 =’ 0
dt t t


So everything boils down to solving ( ) for vt .
First, from ωt = (1 ’ t)ω0 + tω1 , we conclude that
dωt
= ω1 ’ ω0 .
dt
Second, since [ω0 ] = [ω1 ], there exists a 1-form µ such that

ω1 ’ ω0 = dµ .

Third, by the Cartan magic formula, we have

Lvt ωt = d±vt ωt + ±vt dωt .
0
10 LECTURE 1. SYMPLECTIC FORMS

Putting everything together, we must ¬nd vt such that
d±vt ωt + dµ = 0 .

It is su¬cient to solve ±vt ωt + µ = 0. By the nondegeneracy of ωt , we can solve
this pointwise, to obtain a unique (smooth) vt .

Theorem 1.10 (Moser Theorem “ Version II) Let M be a compact manifold
with symplectic forms ω0 and ω1 . Suppose that ωt , 0 ¤ t ¤ 1, is a smooth family
of closed 2-forms joining ω0 to ω1 and satisfying:
d d
(1) cohomology assumption: [ωt ] is independent of t, i.e., dt [ωt ] = dt ωt = 0,
(2) nondegeneracy assumption: ωt is nondegenerate for 0 ¤ t ¤ 1.
Then there exists an isotopy ρ : M — R ’ M such that ρ— ωt = ω0 , 0 ¤ t ¤ 1.
t


Proof. (Moser trick) We have the following implications from the hypotheses:
(1) =’ There is a family of 1-forms µt such that
dωt
0¤t¤1.
= dµt ,
dt
We can indeed ¬nd a smooth family of 1-forms µt such that dωt = dµt .
dt
The argument involves the Poincar´ 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 [35].
(2) =’ There is a unique family of vector ¬elds vt such that
±v t ω t + µ t = 0 (Moser equation) .

Extend vt to all t ∈ R. Let ρ be the isotopy generated by vt (ρ exists by
compactness of M ). Then we indeed have
d— dωt
(ρt ωt ) = ρ— (Lvt ωt + ) = ρ— (d±vt ωt + dµt ) = 0 .
t t
dt dt


The compactness of M was used to be able to integrate vt for all t ∈ R. If M
is not compact, we need to check the existence of a solution ρt for the di¬erential
equation dρt = vt —¦ ρt for 0 ¤ t ¤ 1.
dt

Picture. Fix c ∈ H 2 (M ). De¬ne Sc = {symplectic forms ω in M with [ω] = 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 ω 0 and ω1 that induce
the same orientation are symplectic.
This is wrong in dimension 4: ¬nd 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 ω 0 , ω1 on a compact 2-dimensional
manifold M representing the same de Rham cohomology class, i.e.,
2
[ω0 ] = [ω1 ] ∈ HdeRham (M ).
Prove that there is a 1-parameter family of di¬eomorphisms • t : M ’
M such that •— ω0 = ω1 , •0 = id, and •— ω0 is symplectic for all t ∈
1 t
[0, 1].
Such a 1-parameter family • t is a strong isotopy between ω0 and ω1 .
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 “ Local Version) Let M be a manifold, X
a submanifold of M , i : X ’ M the inclusion map, ω0 and ω1 symplectic forms
in M .
Hypothesis: ω0 |p = ω1 |p , ∀p ∈ X .
Conclusion: There exist neighborhoods U0 , U1 of X in M ,
and a di¬eomorphism • : U0 ’ U1 such that
• E U1
U0
d
s 
 
d  
d   commutes
id  i
d  
X
and •— ω1 = ω0 .

Proof.
1. Pick a tubular neighborhood U0 of X. The 2-form ω1 ’ ω0 is closed on U0 ,
and (ω1 ’ ω0 )p = 0 at all p ∈ X. By the homotopy formula on the tubular
neighborhood, there exists a 1-form µ on U0 such that ω1 ’ ω0 = dµ and
µp = 0 at all p ∈ X.
2. Consider the family ωt = (1 ’ t)ω0 + tω1 = ω0 + tdµ of closed 2-forms on U0 .
Shrinking U0 if necessary, we can assume that ωt is symplectic for 0 ¤ t ¤ 1.
12 LECTURE 1. SYMPLECTIC FORMS

3. Solve the Moser equation: ±vt ωt = ’µ. Notice that vt = 0 on X.
4. Integrate vt . Shrinking U0 again if necessary, there exists an isotopy ρ :
U0 — [0, 1] ’ M with ρ— ωt = ω0 , for all t ∈ [0, 1]. Since vt |X = 0, we have
t
ρt |X = idX . Set • = ρ1 , U1 = ρ1 (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, ω) 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 § dyi .
ω=
i=1

As a consequence of Theorem 1.12, if we show for (R2n , dxi § 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 § dyi
ωp = .
p

There are two symplectic forms on U : the given ω0 = ω and ω1 = dxi § dyi . By
the Moser theorem (Theorem 1.11) applied to X = {p}, there are neighborhoods
U0 and U1 of p, and a di¬eomorphism • : U0 ’ U1 such that

•— ( dxi § dyi ) = ω .
•(p) = p and

Since •— ( dxi §dyi ) = d(xi —¦•)§d(yi —¦•), we only need to set new coordinates
xi = xi —¦ • and yi = yi —¦ •.
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 § dyi .
ω0 =
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: ω = df § dg is nondegenerate ⇐’ (f, g) is a local di¬eomorphism.




Exercise 6
Let H be the vector space of n — n complex hermitian matrices. The unitary
group U(n) acts on H by conjugation: A·ξ = AξA’1 , for A ∈ U(n) , ξ ∈ H.
For each » = (»1 , . . . , »n ) ∈ Rn , let H» be the set of all n—n complex hermitian
matrices whose spectrum is ».
(a) Show that the orbits of the U(n)-action are the manifolds H » .
For a ¬xed » ∈ Rn , what is the stabilizer of a point in H» ?

Hint: If »1 , . . . , »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 ) ’ trace (XY ) ,
is nondegenerate.
For ξ ∈ H, de¬ne a skew-symmetric bilinear form ω ξ on u(n) =
T1 U(n) = iH (space of skew-hermitian matrices) by
ωξ (X, Y ) = i trace ([X, Y ]ξ) , X, Y ∈ iH .
Check that ωξ (X, Y ) = i trace (X(Y ξ ’ ξY )) and Y ξ ’ ξY ∈ H.
Show that the kernel of ωξ is Kξ := {Y ∈ u(n) | [Y, ξ] = 0}.
(c) Show that Kξ is the Lie algebra of the stabilizer of ξ.

Hint: Di¬erentiate the relation AξA’1 = ξ.
Show that the ωξ ™s induce nondegenerate 2-forms on the orbits H » .
Show that these 2-forms are closed.
Conclude that all the orbits H» are compact symplectic manifolds.
(d) Describe the manifolds H» .
When all eigenvalues are equal, there is only one point in the orbit.
Suppose that »1 = »2 = . . . = »n . Then the eigenspace associated
with »1 is a line, and the one associated with »2 is the orthogonal
C Pn’1 . We
hyperplane. Show that there is a di¬eomorphism H »
n’1 , on for each
have thus exhibited a lot of symplectic forms on C P
pair of distinct real numbers.
What about the other cases?

Hint: When the eigenvalues »1 < . . . < »n are all distinct, any element
in H» de¬nes a family of pairwise orthogonal lines in Cn : its eigenspaces.
(e) Show that, for any skew-hermitian matrix X ∈ u(n), the vector ¬eld
on H generated by X ∈ u(n) for the U(n)-action by conjugation is
#
Xξ = [X, ξ].
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
di¬erential 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 — U, x1 , . . . , xn , ξ1 , . . . , ξn ). De¬ne a 2-form ω on T — U by
n
dxi § dξi .
ω=
i=1

1 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 a basis of Tx X. Namely, if ξ ∈ Tx X, then
n
ξ = i=1 ξi (dxi )x for some real coe¬cients ξ1 , . . . , ξn . This induces a map

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

The chart (T — U , x1 , . . . , xn , ξ1 , . . . , ξn ) is a coordinate chart for T — X; the coordinates
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 , x 1 , . . . , 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



15
16 LECTURE 2. COTANGENT BUNDLES

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 intrinsi-
cally de¬ned) .

Proof. Let (U, x1 , . . . , xn , ξ1 , . . . , ξn ) and (U , x1 , . . . , xn , ξ1 , . . . , ξn ) be two cotan-
gent coordinate charts. On U © U , the two sets of coordinates are related by
‚x
‚x
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. Next we provide an alternative proof of the intrinsic character
of these forms. 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 pointwise
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 7
Let (U , x1 , . . . , xn ) be a chart on X with associated cotangent coordinates
n
x1 , . . . , xn , ξ1 , . . . , ξn . Show that on T — U , ± = ξi dxi .
i=1
17
2.2. NATURALITY OF THE CANONICAL FORMS

The canonical symplectic 2-form ω on T — X is de¬ned as
ω = ’d± .
n
dxi § dξi .
Locally, ω = i=1

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




2.2 Naturality of the 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 9
Check that f is a di¬eomorphism. Here are some hints:
f
M1 ’’ M2
1. commutes;
π1 “ “ π2
f
X1 ’’ X2
2. f : M1 ’ M2 is bijective;
3. f and f ’1 are smooth.



Proposition 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 that

(df )p1 (±2 )p2 = (±1 )p1 ()

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

• 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
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 symplectomor-
phism, 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 dif-
feomorphism, then f : S 1 — R ’ S1 — 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¬eomorphism
of the base manifold. A criterion for which symplectomorphisms arise as lifts of
di¬eomorphisms is discussed in the next section.
2.3. SYMPLECTOMORPHISMS OF T — X 19

Symplectomorphisms of T — X
2.3

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

ω = ’d± .

There exists a unique vector ¬eld v such that its interior product with ω is ±, i.e.,
±v ω = ’±.

Proposition 2.3 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) .

Proof. Recall that, for p ∈ M , (exp tv)(p) is the unique curve in M solving the
ordinary di¬erential 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 —¦ (exp tv) —¦ g ’1 must be
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 we have that g— v = v, i.e.,
that g preserves v.
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. When ± is the tautological 1-form on M
(which, in these coordinates, is ξi dxi ), the vector ¬eld v above is just the vector

ξi ‚ξi . Let exp tv, ’∞ < t < ∞, be the one-parameter group of di¬eomor-
¬eld
phisms generated by v.

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


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

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

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.) We conclude
that g has to preserve the cotangent ¬bration, i.e., 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. Moreover, g = f# , the map f# being the symplectomorphism of M
lifting f . Hence, the symplectomorphisms of T — X of the form f# are those which
preserve the tautological 1-form ±.
Here is a di¬erent class of symplectomorphisms of M = T — X. Let h be a
smooth function on X. De¬ne „h : M ’ M by setting

„h (x, ξ) = (x, ξ + dhx ) .

Then
„h ± = ± + π — dh



where π is the projection map
M (x, ξ)
“π “
X x

Therefore,

„h ω = ω ,
so all such „h are symplectomorphisms.


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

De¬nition 2.4 A submanifold 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
2 dim Tp M . 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 .
2.4. LAGRANGIAN SUBMANIFOLDS OF T — X 21

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 by
ξi dxi vanishes on X0 © T — U .
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 subman-
ifold 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 2.5 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 exclu-
sively as a map. Notice that the image of sµ is Xµ . Let ± be the tautological 1-form
on T — X. Then
s— ± = µ .
µ

Proof. By de¬nition of the tautological form ±, ±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
‚  

22 LECTURE 2. COTANGENT BUNDLES

We want to express the condition of Xµ being lagrangian in terms of the form µ:

i— d± = 0
⇐’
Xµ is lagrangian
„ — 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 generat-
ing 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 11
Check that, if X is compact (and not just one point) and f ∈ C ∞ (X), then
#{Xdf © X0 } ≥ 2.




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

De¬nition 2.6 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} .



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

Proposition 2.7 Let i : N — S ’ T — X be the inclusion, and let ± be the tautolog-
ical 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 2.8 For any submanifold S ‚ X, the conormal bundle N — S is a la-
grangian 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.


2.6 Lagrangian Complements
Normal neighborhoods of lagrangian submanifolds are described by the theorems
in the following two sections. It was proved by Weinstein [44] 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
i— ω0 = i— ω1 = 0 where i : X ’ M is inclusion, i.e.,
X is a submanifold lagrangian for ω0 and ω1 .
We need some algebra for the Weinstein theorem.
Suppose that U and W are n-dimensional vector spaces, and „¦ : U —W ’ R is
a bilinear pairing; the map „¦ gives rise to a linear map „¦ : U ’ W — , „¦(u) = „¦(u, ·).
Then „¦ is nondegenerate if and only if „¦ is bijective.
Proposition 2.9 Suppose that V is a 2n-dimensional vector space and „¦ : V —
V ’ R is a nondegenerate skew-symmetric bilinear pairing. Let U be a lagrangian
subspace of (V, „¦) (i.e., „¦|U —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

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

W = {w + Aw | w ∈ W } ,

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

∀ w 1 , w2 ∈ W , „¦(w1 + Aw1 , w2 + Aw2 ) = 0

=’ „¦(w1 , w2 ) + „¦(w1 , Aw2 ) + „¦(Aw1 , w2 ) + „¦(Aw1 , Aw2 ) = 0
∈U

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

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

∀ w 1 , w2 ∈ W , „¦(w1 , w2 ) = A (w2 )(w1 ) ’ A (w1 )(w2 ) .
1
The canonical choice is A (w) = ’ 2 „¦(w, ·). Then set A = („¦ )’1 —¦ A .


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

Proof. From W we canonically obtain complements W0 and W1 to U in V such
that W0 is lagrangian for „¦0 and W1 is lagrangian for „¦1 . The nondegenerate
bilinear pairings
„¦

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