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
Email: 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˜odegenerada. 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, estabelecendose 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 ±sicamatem´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¸ase 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˜
estudamse campos hamiltonianos, ac¸oes hamiltonianas e algumas das suas aplica
c˜
coes pr´ticas no ambito da mecˆnica e dos sistemas dinˆmicos. Ao longo do
¸˜ a ˆ a a
texto fornecemse exemplos simples e exerc´ ±cios relevantes. Pressup˜emse 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´sgradua¸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 2form. 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, lowdimensional 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 SkewSymmetric 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 MarsdenWeinsteinMeyer 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 OneParameter 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 EulerLagrange 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 2form 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
socalled Moser trick, which takes advantage of the main features of a symplectic
form in order to show the equivalence of symplectic structures.
1.1 SkewSymmetric Bilinear Maps
Let V be an mdimensional vector space over R, and let „¦ : V — V ’ R be
a bilinear map. The map „¦ is skewsymmetric if „¦(u, v) = ’„¦(v, u), for all
u, v ∈ V .
Theorem 1.1 (Standard Form for Skewsymmetric Bilinear Maps) Let „¦
be a skewsymmetric 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 skewsymmetric version of the GramSchmidt
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 mdimensional 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 skewsymmetric 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 evendimensional vector spaces. Furthermore, by Theorem 1.1,
every 2ndimensional 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 onedimensional 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 2form on a manifold M , that is, for each p ∈ M , the map
ωp : Tp M — Tp M ’ R is skewsymmetric 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 2form 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 skewsymmetric klinear maps or kforms
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
2form „¦ ∈ §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 2form „¦ ∈ §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
2ndimensional 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 2ndimensional 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 nonzero. Hence,
[ω] itself is nonzero (in other words, ω is not exact). This reveals a necessary
topological condition for a compact 2ndimensional 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 2ndimensional 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 ndimensional manifold looks locally like Rn ,
any 2ndimensional 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 deformationequivalent 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 deformationequivalent with
[ωt ] independent of t.
Clearly, we have
strongly isotopic =’ symplectomorphic , and
isotopic =’ deformationequivalent .
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 2ndimensional manifold M , a kdimensional 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 (nonsymplectic) 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 2form 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 timedependent 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 1form 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 2forms 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 1forms µt such that
dωt
0¤t¤1.
= dµt ,
dt
We can indeed ¬nd a smooth family of 1forms µt such that dωt = dµt .
dt
The argument involves the Poincar´ lemma for compactlysupported forms,
e
together with the MayerVietoris 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 pathconnected component of Sc are symplectomorphic.
11
1.8. MOSER LOCAL THEOREM
Exercises 4
Any oriented 2dimensional 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 2dimensional
manifold M representing the same de Rham cohomology class, i.e.,
2
[ω0 ] = [ω1 ] ∈ HdeRham (M ).
Prove that there is a 1parameter 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 1parameter 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 nonzero 2cohomology 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 2form 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 1form 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 2forms 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 2ndimensional 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 2ndimensional
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 2dimensional case, using the fact that every
nonvanishing 1form 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 skewsymmetric bilinear form ω ξ on u(n) =
T1 U(n) = iH (space of skewhermitian 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 2forms on the orbits H » .
Show that these 2forms 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 skewhermitian 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 2form ω on T — U by
n
dxi § dξi .
ω=
i=1
1 Let X be any ndimensional 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 2ndimensional manifold.
where ξj = i ξi ‚xj
15
16 LECTURE 2. COTANGENT BUNDLES
In order to check that this de¬nition is coordinateindependent, consider the 1form
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 1form ± is the tautological form and 2form 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 1form ± 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 2form ω on T — X is de¬ned as
ω = ’d± .
n
dxi § dξi .
Locally, ω = i=1
Exercise 8
Show that the tautological 1form ± is uniquely characterized by the property
that, for every 1form µ : X ’ T — X, µ— ± = µ. (See Section 2.4.)
2.2 Naturality of the Canonical Forms
Let X1 and X2 be ndimensional manifolds with cotangent bundles M1 = T — X1
and M2 = T — X2 , and tautological 1forms ±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 1form:
(±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 2form 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
™¦
areapreserving 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 1form 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 oneparameter 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 oneparameter group of di¬eomorphisms generated by g— v. (The pushforward
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 ndimensional 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 1form 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 oneparameter 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 1form ±.
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 2ndimensional 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 ndimensional 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 1form on T — X is
±= ξi dxi
and the canonical 2form 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 ndimensional 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 ndimensional 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
1form. 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
1form. Denote by sµ : X ’ T — X, x ’ (x, µx ), be the 1form µ regarded exclu
sively as a map. Notice that the image of sµ is Xµ . Let ± be the tautological 1form
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 ndimensional submanifold of T — X of the form ( ),
with associated de Rham 1form 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
‚
Xµ
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 onetoone correspondence between the set of lagrangian
submanifolds of T — X of the form ( ) and the set of closed 1forms on X.
1
When X is simply connected, HdeRham (X) = 0, so every closed 1form 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 1forms from the previous section, starting with the cotangent
¬bers.
Let S be any kdimensional submanifold of an ndimensional 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 ndimensional submanifold of T — X.
2Acoordinate chart (U , x1 , . . . , xn ) on X is adapted to a kdimensional 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 1form 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 ndimensional 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 ndimensional 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 2ndimensional vector space and „¦ : V —
V ’ R is a nondegenerate skewsymmetric bilinear pairing. Let U be a lagrangian
subspace of (V, „¦) (i.e., „¦U —U = 0 and U is ndimensional). 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 2ndimensional 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 LU = 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
„¦