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If Y is a lagrangian submanifold of (M1 Г— M2 , Пү), then its вҖңtwistвҖқ Y Пғ := Пғ(Y )
is a lagrangian submanifold of (M1 Г— M2 , Пү).
Recipe for producing symplectomorphisms M1 = T вҲ— X1 вҶ’ M2 = T вҲ— X2 :
1. Start with a lagrangian submanifold Y of (M1 Г— M2 , Пү).
2. Twist it to obtain a lagrangian submanifold Y Пғ of (M1 Г— M2 , Пү).
3. Check whether Y Пғ is the graph of some diп¬Җeomorphism П• : M1 вҶ’ M2 .
4. If it is, then П• is a symplectomorphism.
Let i : Y Пғ вҶ’ M1 Г— M2 be the inclusion map
YПғ
В  d
pr1 в—¦ i В  d pr2 в—¦ i
В  d
В  d
В
В‚
П•? E M2
M1
Step 3 amounts to checking whether pr1 в—¦ i and pr2 в—¦ i are diп¬Җeomorphisms. If
yes, then П• := (pr2 в—¦ i) в—¦ (pr1 в—¦ i)вҲ’1 is a diп¬Җeomorphism.
T вҲ— (X1 Г— X2 ), we
In order to obtain lagrangian submanifolds of M1 Г— M2
can use the method of generating functions.

22
23
4.2 Method of Generating Functions

4.2 Method of Generating Functions

For any f вҲҲ C вҲһ (X1 Г— X2 ), df is a closed 1-form on X1 Г— X2 . The lagrangian
submanifold generated by f is

Yf := {((x, y), (df )(x,y) ) | (x, y) вҲҲ X1 Г— X2 } .

We adopt the notation
вҲ—
:= (df )(x,y) projected to Tx X1 Г— {0},
dx f
вҲ—
:= (df )(x,y) projected to {0} Г— Ty X2 ,
dy f

which enables us to write

Yf = {(x, y, dx f, dy f ) | (x, y) вҲҲ X1 Г— X2 }

and
YfПғ = {(x, y, dx f, вҲ’dy f ) | (x, y) вҲҲ X1 Г— X2 } .
When YfПғ is in fact the graph of a diп¬Җeomorphism П• : M1 вҶ’ M2 , we call П• the
symplectomorphism generated by f , and call f the generating function,
of П• : M1 вҶ’ M2 .
So when is YfПғ the graph of a diп¬Җeomorphism П• : M1 вҶ’ M2 ?
Let (U1 , x1 , . . . , xn ), (U2 , y1 , . . . , yn ) be coordinate charts for X1 , X2 , with
associated charts (T вҲ— U1 , x1 , . . . , xn , Оҫ1 , . . . , Оҫn ), (T вҲ— U2 , y1 , . . . , yn , О·1 , . . . , О·n ) for
M1 , M2 . The set

YfПғ = {(x, y, dx f, вҲ’dy f ) | (x, y) вҲҲ X1 Г— X2 }

is the graph of П• : M1 вҶ’ M2 if and only if, for any (x, Оҫ) вҲҲ M1 and (y, О·) вҲҲ M2 ,
we have

П•(x, Оҫ) = (y, О·) вҮҗвҮ’ Оҫ = dx f and О· = вҲ’dy f .

Therefore, given a point (x, Оҫ) вҲҲ M1 , to п¬Ғnd its image (y, О·) = П•(x, Оҫ) we must
solve the вҖңHamiltonвҖқ equations
пЈ±
вҲ‚f
пЈҙ Оҫi =
пЈІ (x, y) ()
вҲ‚xi
пЈҙ О· = вҲ’ вҲ‚f (x, y)
пЈіi ()
вҲ‚yi

If there is a solution y = П•1 (x, Оҫ) of ( ), we may feed it to ( ) thus obtaining
О· = П•2 (x, Оҫ), so that П•(x, Оҫ) = (П•1 (x, Оҫ), П•2 (x, Оҫ)). Now by the implicit function
theorem, in order to solve ( ) locally for y in terms of x and Оҫ, we need the
condition n
вҲ‚ вҲ‚f
det =0.
вҲ‚yj вҲ‚xi i,j=1
24 4 GENERATING FUNCTIONS

This is a necessary local condition for f to generate a symplectomorphism П•.
Locally this is also suп¬ғcient, but globally there is the usual bijectivity issue.
2
Rn , and f (x, y) = вҲ’ |xвҲ’y| , the
R n , X2 = U 2
Example. Let X1 = U1 2
square of euclidean distance up to a constant.
The вҖңHamiltonвҖқ equations are
пЈ± пЈ±
вҲ‚f
пЈҙ Оҫi = пЈІ yi = xi + Оҫ i
пЈІ = y i вҲ’ xi
вҲ‚xi вҮҗвҮ’
пЈҙ О· = вҲ’ вҲ‚f = y вҲ’ x пЈі
пЈіi О·i = Оҫ i
i i
вҲ‚yi
The symplectomorphism generated by f is

П•(x, Оҫ) = (x + Оҫ, Оҫ) .

If we use the euclidean inner product to identify T вҲ— Rn with T Rn , and hence
regard П• as П• : T Rn вҶ’ T Rn and interpret Оҫ as the velocity vector, then the sym-
plectomorphism П• corresponds to free translational motion in euclidean space.

B
ВЁ
ВЁВЁ
rВЁ
Оҫ BВЁ
ВЁ
ВЁ x+Оҫ
ВЁВЁ
ВЁ
r
x

в™¦

4.3 Application to Geodesic Flow

Let V be an n-dimensional vector space. A positive inner product G on V
is a bilinear map G : V Г— V вҶ’ R which is

symmetric : G(v, w) = G(w, v) , and
positive-deп¬Ғnite : G(v, v) > 0 when v=0.

Deп¬Ғnition 4.1 A riemannian metric on a manifold X is a function g which
assigns to each point x вҲҲ X a positive inner product gx on Tx X.
A riemannian metric g is smooth if for every smooth vector п¬Ғeld v : X вҶ’
T X the real-valued function x вҶ’ gx (vx , vx ) is a smooth function on X.

Deп¬Ғnition 4.2 A riemannian manifold (X, g) is a manifold X equipped with
a smooth riemannian metric g.
25
4.3 Application to Geodesic Flow

The arc-length of a piecewise smooth curve Оі : [a, b] вҶ’ X on a riemannian
manifold (X, g) is
b
dОі dОі
gОі(t) , dt .
dt dt
a

y = Оі(b)
r

Оі
E

r
x = Оі(a)

Deп¬Ғnition 4.3 The riemannian distance between two points x and y of a
connected riemannian manifold (X, g) is the inп¬Ғmum d(x, y) of the set of all
arc-lengths for piecewise smooth curves joining x to y.
A smooth curve joining x to y is a minimizing geodesic7 if its arc-length
is the riemannian distance d(x, y).
A riemannian manifold (X, g) is geodesically convex if every point x is
joined to every other point y by a unique minimizing geodesic.

Example. On X = Rn with T X Rn Г— Rn , let gx (v, w) = v, w , gx (v, v) =
|v|2 , where В·, В· is the euclidean inner product, and | В· | is the euclidean norm.
Then (Rn , В·, В· ) is a geodesically convex riemannian manifold, and the rieman-
nian distance is the usual euclidean distance d(x, y) = |x вҲ’ y|. в™¦

Suppose that (X, g) is a geodesically convex riemannian manifold. Consider
the function
d(x, y)2
f : X Г— X вҲ’вҶ’ R , f (x, y) = вҲ’ .
2
What is the symplectomorphism П• : T вҲ— X вҶ’ T вҲ— X generated by f ?
The metric gx : Tx X Г— Tx X вҶ’ R induces an identiп¬Ғcation

вҲ—
вҲ’вҶ’ Tx X
g x : Tx X
вҲ’вҶ’ gx (v, В·)
v

Use g to translate П• into a map П• : T X вҶ’ T X.
7 Inriemannian geometry, a geodesic is a curve which locally minimizes distance and
whose velocity is constant.
26 4 GENERATING FUNCTIONS

We need to solve
gx (v) = Оҫi = dx f (x, y)
= вҲ’dy f (x, y)
gy (w) = О·i

for (y, О·) in terms of (x, Оҫ) in order to п¬Ғnd П•, or, equivalently, for (y, w) in terms
(x, v) in order to п¬Ғnd П•.
Let Оі be the geodesic with initial conditions Оі(0) = x and dОі (0) = v.
dt


Вў
Вў Оі
vВў E
Вў
Вў
Вў
r
x

Then the symplectomorphism П• corresponds to the map

вҲ’вҶ’ T X
П•: TX
вҲ’вҶ’ (Оі(1), dОі (1)) .
(x, v) dt

This is called the geodesic п¬‚ow on X (see Homework 4).
Homework 4: Geodesic Flow
This set of problems is adapted from .
Let (X, g) be a riemannian manifold. The arc-length of a smooth curve
Оі : [a, b] вҶ’ X is

b
dОі dОі dОі dОі
arc-length of Оі := dt , where := gОі(t) , .
dt dt dt dt
a

1. Show that the arc-length of Оі is independent of the parametrization of
Оі, i.e., show that, if we reparametrize Оі by П„ : [a , b ] вҶ’ [a, b], the new
curve Оі = Оі в—¦ П„ : [a , b ] вҶ’ X has the same arc-length. A curve Оі is
called a curve of constant velocity when dОі is independent of t. Show
dt
that, given any curve Оі : [a, b] вҶ’ X (with dОі never vanishing), there is
dt
a reparametrization П„ : [a, b] вҶ’ [a, b] such that Оі в—¦ П„ : [a, b] вҶ’ X is of
constant velocity.
2
b
dОі
2. Given a smooth curve Оі : [a, b] вҶ’ X, the action of Оі is A(Оі) := dt.
dt
a
Show that, among all curves joining x to y, Оі minimizes the action if and
only if Оі is of constant velocity and Оі minimizes arc-length.
Hint: Suppose that Оі is of constant velocity, and let П„ : [a, b] вҶ’ [a, b] be
a reparametrization. Show that A(Оі в—¦ П„ ) вүҘ A(Оі), with equality only when
П„ = identity.

3. Assume that (X, g) is geodesically convex, that is, any two points x, y вҲҲ X
are joined by a unique (up to reparametrization) minimizing geodesic; its
arc-length d(x, y) is called the riemannian distance between x and y.
Assume also that (X, g) is geodesically complete, that is, every minimizing
geodesic can be extended indeп¬Ғnitely. Given (x, v) вҲҲ T X, let exp(x, v) :
R вҶ’ X be the unique minimizing geodesic of constant velocity with initial
conditions exp(x, v)(0) = x and d exp(x,v) (0) = v.
dt
1
Consider the function П• : X Г—X вҶ’ R given by П•(x, y) = вҲ’ 2 В·d(x, y)2 . Let
вҲ—
dП•x and dП•y be the components of dП•(x,y) with respect to T(x,y) (X Г—X)
вҲ— вҲ—
Tx X Г— Ty X. Recall that, if

О“Пғ = {(x, y, dП•x , вҲ’dП•y ) | (x, y) вҲҲ X Г— X}
П•

is the graph of a diп¬Җeomorphism f : T вҲ— X вҶ’ T вҲ— X, then f is the symplec-
tomorphism generated by П•. In this case, f (x, Оҫ) = (y, О·) if and only if
Оҫ = dП•x and О· = вҲ’dП•y .
Show that, under the identiп¬Ғcation of T X with T вҲ— X by g, the sym-
plectomorphism generated by П• coincides with the map T X вҶ’ T X,
(x, v) вҶ’ exp(x, v)(1).

27
28 HOMEWORK 4

Hint: The metric g provides the identiп¬Ғcations Tx Xv Оҫ(В·) = gx (v, В·) вҲҲ
вҲ— X. We need to show that, given (x, v) вҲҲ T X, the unique solution of
Tx
gx (v, В·) = dП•x (В·)
is (y, w) = (exp(x, v)(1), d exp(x,v) (1)).
()
gy (w, В·) = вҲ’dП•y (В·) dt
Look up the Gauss lemma in a book on riemannian geometry. It asserts that
geodesics are orthogonal to the level sets of the distance function.
d exp(x,v)
To solve the п¬Ғrst line in ( ) for y, evaluate both sides at v = (0).
dt
Conclude that y = exp(x, v)(1). Check that dП•x (v ) = 0 for vectors v вҲҲ Tx X
orthogonal to v (that is, gx (v, v ) = 0); this is a consequence of П•(x, y) being
the arc-length of a minimizing geodesic, and it suп¬ғces to check locally.
The vector w is obtained from the second line of ( ). Compute
d exp(x,v)
вҲ’dП•y ( (1)). Then evaluate вҲ’dП•y at vectors w вҲҲ Ty X orthogo-
dt
nal to d exp(x,v) (1); this pairing is again 0 because П•(x, y) is the arc-length
dt
of a minimizing geodesic. Conclude, using the nondegeneracy of g, that
w = d exp(x,v) (1).
dt
For both steps, it might be useful to recall that, given a function f : X вҶ’ R
d
and a tangent vector v вҲҲ Tx X, we have dfx (v) = [f (exp(x, v)(u))] u=0 .
du
5 Recurrence

5.1 Periodic Points
Let X be an n-dimensional manifold. Let M = T вҲ— X be its cotangent bundle
with canonical symplectic form Пү.
Suppose that we are given a smooth function f : X Г—X вҶ’ R which generates
a symplectomorphism П• : M вҶ’ M , П•(x, dx f ) = (y, вҲ’dy f ), by the recipe of the
previous lecture.
What are the п¬Ғxed points of П•?
Deп¬Ғne ПҲ : X вҶ’ R by ПҲ(x) = f (x, x).

Proposition 5.1 There is a one-to-one correspondence between the п¬Ғxed points
of П• and the critical points of ПҲ.

Proof. At x0 вҲҲ X, dx0 ПҲ = (dx f + dy f )|(x,y)=(x0 ,x0 ) . Let Оҫ = dx f |(x,y)=(x0 ,x0 ) .

x0 is a critical point of ПҲ вҮҗвҮ’ dx0 ПҲ = 0 вҮҗвҮ’ dy f |(x,y)=(x0 ,x0 ) = вҲ’Оҫ .

Hence, the point in О“Пғ corresponding to (x, y) = (x0 , x0 ) is (x0 , x0 , Оҫ, Оҫ). But
f
О“Пғ is the graph of П•, so П•(x0 , Оҫ) = (x0 , Оҫ) is a п¬Ғxed point. This argument also
f
works backwards.
Consider the iterates of П•,

П•(N ) = П• в—¦ П• в—¦ . . . в—¦ П• : M вҶ’ M , N = 1, 2, . . . ,
N

each of which is a symplectomorphism of M . According to the previous propo-
sition, if П•(N ) : M вҶ’ M is generated by f (N ) , then there is a correspondence

critical points of
1вҲ’1
п¬Ғxed points of П•(N ) вҶҗвҶ’ (N )
: X вҶ’ R , ПҲ (N ) (x) = f (N ) (x, x)
ПҲ

Knowing that П• is generated by f , does П•(2) have a generating function?
The answer is a partial yes:
Fix x, y вҲҲ X. Deп¬Ғne a map

вҲ’вҶ’ R
X
вҲ’вҶ’ f (x, z) + f (z, y)
z

Suppose that this map has a unique critical point z0 , and that z0 is nondegen-
erate. Let
f (2) (x, y) := f (x, z0 ) + f (z0 , y) .

Theorem 5.2 The function f (2) : X Г— X вҶ’ R is smooth and is a generating
function for П•(2) .

29
30 5 RECURRENCE

Proof. The point z0 is given implicitly by dy f (x, z0 ) + dx f (z0 , y) = 0. The
nondegeneracy condition is

вҲ‚ вҲ‚f вҲ‚f
det (x, z) + (z, y) =0.
вҲ‚zi вҲ‚yj вҲ‚xj

By the implicit function theorem, z0 = z0 (x, y) is smooth.
As for the second assertion, f (2) (x, y) is a generating function for П•(2) if and
only if

П•(2) (x, dx f (2) ) = (y, вҲ’dy f (2) )

(assuming that, for each Оҫ вҲҲ Tx X, there is a unique y вҲҲ X for which dx f (2) = Оҫ).
вҲ—

Since П• is generated by f , and z0 is critical, we obtain,

= П•(П•(x, dx f (2) (x, y))
П•(2) (x, dx f (2) (x, y)) = П•(z0 , вҲ’dy f (x, z0 ))
=dx f (x,z0 )
= (y, вҲ’dy f (z0 , y) ) .
= П•(z0 , dx f (z0 , y))
=вҲ’dy f (2) (x,y)

Exercise. What is a generating function for П•(3) ?
Hint: Suppose that the function

X Г— X вҲ’вҶ’ R
(z, u) вҲ’вҶ’ f (x, z) + f (z, u) + f (u, y)

has a unique critical point (z0 , u0 ), and that it is a nondegenerate critical point.
Let ПҲ (3) (x, y) = f (x, z0 ) + f (z0 , u0 ) + f (u0 , y). в™¦

5.2 Billiards

Let ПҮ : R вҶ’ R2 be a smooth plane curve which is 1-periodic, i.e., ПҮ(s+1) = ПҮ(s),
dПҮ
and parametrized by arc-length, i.e., = 1. Assume that the region Y
ds
enclosed by ПҮ is convex, i.e., for any s вҲҲ R, the tangent line {ПҮ(s) + t dПҮ | t вҲҲ R}
ds
intersects X := вҲ‚Y (= the image of ПҮ) at only the point ПҮ(s).
31
5.2 Billiards

'

X = вҲ‚Y
r ПҮ(s)

Suppose that we throw a ball into Y rolling with constant velocity and
bouncing oп¬Җ the boundary with the usual law of reп¬‚ection. This determines a
map
П• : R/Z Г— (вҲ’1, 1) вҲ’вҶ’ R/Z Г— (вҲ’1, 1)
(x, v) вҲ’вҶ’ (y, w)

by the rule

when the ball bounces oп¬Җ x with angle Оё = arccos v, it will next collide with y
and bounce oп¬Җ with angle ОҪ = arccos w.

4
4
%
4
4
4
xr
В˜
В˜
В˜
В˜
В˜
j В˜
В˜ 4
В˜ 4
B
В˜ 4
В˜4
r
y

Let f : R/Z Г— R/Z вҶ’ R be deп¬Ғned by f (x, y) = вҲ’|x вҲ’ y|; f is smooth oп¬Җ
the diagonal. Use ПҮ to identify R/Z with the image curve X.
Suppose that П•(x, v) = (y, w), i.e., (x, v) and (y, w) are successive points on
32 5 RECURRENCE

the orbit described by the ball. Then
пЈ± df xвҲ’y
пЈҙ =вҲ’ projected onto Tx X = v
пЈҙ
пЈҙ dx
пЈІ |x вҲ’ y|
пЈҙ df
пЈҙ yвҲ’x
пЈҙ
пЈі =вҲ’ = вҲ’w
projected onto Ty X
|x вҲ’ y|
dy

or, equivalently,
пЈ±d y вҲ’ x dПҮ
пЈҙ В·
пЈҙ ds f (ПҮ(s), y) = = cos Оё = v
пЈҙ
пЈІ |x вҲ’ y| ds
пЈҙd
пЈҙ x вҲ’ y dПҮ
пЈҙ
пЈі В· = вҲ’ cos ОҪ = вҲ’w
f (x, ПҮ(s)) =
|x вҲ’ y| ds
ds

We conclude that f is a generating function for П•. Similar approaches work
for higher dimensional billiards problems.
Periodic points are obtained by п¬Ғnding critical points of

X Г—... Г—X вҲ’вҶ’ R , N >1
N
(x1 , . . . , xN ) вҲ’вҶ’ f (x1 , x2 ) + f (x2 , x3 ) + . . . + f (xN вҲ’1 , xN ) + f (xN , x1 )
= |x1 вҲ’ x2 | + . . . + |xN вҲ’1 вҲ’ xN | + |xN вҲ’ x1 |

that is, by п¬Ғnding the N -sided (generalized) polygons inscribed in X of critical
perimeter.
Notice that

R/Z Г— (вҲ’1, 1) {(x, v) | x вҲҲ X, v вҲҲ Tx X, |v| < 1} A

is the open unit tangent ball bundle of a circle X, that is, an open annulus A.
The map П• : A вҶ’ A is area-preserving.

5.3 PoincarВҙ Recurrence
e

Theorem 5.3 (PoincarВҙ Recurrence Theorem) Suppose that П• : A вҶ’ A
e
is an area-preserving diп¬Җeomorphism of a п¬Ғnite-area manifold A. Let p вҲҲ A,
and let U be a neighborhood of p. Then there is q вҲҲ U and a positive integer N
such that П•(N ) (q) вҲҲ U.

Proof. Let U0 = U, U1 = П•(U), U2 = П•(2) (U), . . .. If all of these sets were
disjoint, then, since Area (Ui ) = Area (U) > 0 for all i, we would have

Area A вүҘ Area (U0 вҲӘ U1 вҲӘ U2 вҲӘ . . .) = Area (Ui ) = вҲһ .
i
33
5.3 PoincarВҙ Recurrence
e

To avoid this contradiction we must have П•(k) (U) вҲ© П•(l) (U) = вҲ… for some k > l,
which implies П•(kвҲ’l) (U) вҲ© U = вҲ….
Hence, eternal return applies to billiards...
Remark. Theorem 5.3 clearly generalizes to volume-preserving diп¬Җeomor-
в™¦
phisms in higher dimensions.

Theorem 5.4 (PoincarВҙвҖ™s Last Geometric Theorem) Suppose П• : A вҶ’
e
A is an area-preserving diп¬Җeomorphism of the closed annulus A = R/Z Г— [вҲ’1, 1]
which preserves the two components of the boundary, and twists them in opposite
directions. Then П• has at least two п¬Ғxed points.

This theorem was proved in 1925 by Birkhoп¬Җ, and hence is also called the
PoincarВҙ-Birkhoп¬Җ theorem. It has important applications to dynamical sys-
e
tems and celestial mechanics. The Arnold conjecture (1966) on the existence of
п¬Ғxed points for symplectomorphisms of compact manifolds (see Lecture 9) may
be regarded as a generalization of the PoincarВҙ-Birkhoп¬Җ theorem. This con-
e
jecture has motivated a signiп¬Ғcant amount of recent research involving a more
general notion of generating function; see, for instance, [34, 45].
Part III
Local Forms
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 Dar-
boux theorem) and of a lagrangian submanifold (the Weinstein theorems), inside
a symplectic manifold. The main tool is the Moser trick, explained in Lecture 7,
which leads to the crucial Moser theorems and which is at the heart of many
arguments in symplectic geometry.
In order to prove the normal forms, we need the (non-symplectic) ingredients
discussed in Lecture 6; for more on these topics, see, for instance, [18, 54, 94].

6 Preparation for the Local Theory

6.1 Isotopies and Vector Fields

Let M be a manifold, and ПҒ : M Г— R вҶ’ M a map, where we set ПҒt (p) := ПҒ(p, t).

Deп¬Ғnition 6.1 The map ПҒ is an isotopy if each ПҒt : M вҶ’ M is a diп¬Җeomor-
phism, and ПҒ0 = idM .

Given an isotopy ПҒ, we obtain a time-dependent vector п¬Ғeld, that is, a
family of vector п¬Ғelds vt , t вҲҲ R, which at p вҲҲ M satisfy

d
q = ПҒвҲ’1 (p) ,
vt (p) = ПҒs (q) where t
ds s=t

i.e.,
dПҒt
= vt в—¦ ПҒt .
dt
Conversely, given a time-dependent vector п¬Ғeld vt , if M is compact or if the
vt вҖ™s are compactly supported, there exists an isotopy ПҒ satisfying the previous
ordinary diп¬Җerential equation.
Suppose that M is compact. Then we have a one-to-one correspondence
1вҲ’1
{isotopies of M } вҶҗвҶ’ {time-dependent vector п¬Ғelds on M }
ПҒt , t вҲҲ R вҶҗвҶ’ vt , t вҲҲ R

Deп¬Ғnition 6.2 When vt = v is independent of t, the associated isotopy is called
the exponential map or the п¬‚ow of v and is denoted exp tv; i.e., {exp tv :
M вҶ’ M | t вҲҲ R} is the unique smooth family of diп¬Җeomorphisms satisfying

d
exp tv|t=0 = idM and (exp tv)(p) = v(exp tv(p)) .
dt

35
36 6 PREPARATION FOR THE LOCAL THEORY

Deп¬Ғnition 6.3 The Lie derivative is the operator
d
(exp tv)вҲ— Пү|t=0 .
Lv : в„¦k (M ) вҲ’вҶ’ в„¦k (M ) Lv Пү :=
deп¬Ғned by
dt
When a vector п¬Ғeld vt is time-dependent, its п¬‚ow, that is, the corresponding
isotopy ПҒ, still locally exists by PicardвҖ™s theorem. More precisely, in the neigh-
borhood of any point p and for suп¬ғciently small time t, there is a one-parameter
family of local diп¬Җeomorphisms ПҒt satisfying
dПҒt
= vt в—¦ ПҒt and ПҒ0 = id .
dt
Hence, we say that the Lie derivative by vt is
d
(ПҒt )вҲ— Пү|t=0 .
Lvt : в„¦k (M ) вҲ’вҶ’ в„¦k (M ) Lvt Пү :=
deп¬Ғned by
dt

Exercise. Prove the Cartan magic formula,

Lv Пү = Д±v dПү + dД±v Пү ,

and the formula
dвҲ—
ПҒ t Пү = ПҒ вҲ— Lvt Пү , ()
t
dt
where ПҒ is the (local) isotopy generated by vt . A good strategy for each formula
is to follow the steps:
1. Check the formula for 0-forms Пү вҲҲ в„¦0 (M ) = C вҲһ (M ).
2. Check that both sides commute with d.
3. Check that both sides are derivations of the algebra (в„¦вҲ— (M ), вҲ§). For
instance, check that

Lv (Пү вҲ§ О±) = (Lv Пү) вҲ§ О± + Пү вҲ§ (Lv О±) .

4. Notice that, if U is the domain of a coordinate system, then в„¦вҖў (U) is gen-
erated as an algebra by в„¦0 (U) and dв„¦0 (U), i.e., every element in в„¦вҖў (U) is
a linear combination of wedge products of elements in в„¦0 (U) and elements
in dв„¦0 (U).
в™¦
We will need the following improved version of formula ( ).

Theorem 6.4 For a smooth family Пүt , t вҲҲ R, of d-forms, we have

dвҲ— dПүt
ПҒ t Пүt = ПҒ вҲ— L v t Пүt + .
t
dt dt
37
6.2 Tubular Neighborhood Theorem

Proof. If f (x, y) is a real function of two variables, by the chain rule we have
d d d
f (t, t) = f (x, t) + f (t, y) .
dt dx dy
x=t y=t

Therefore,
dвҲ— dвҲ— dвҲ—
ПҒ Пүt = ПҒ Пүt + ПҒ Пүy
dt t dx x dy t
x=t y=t

dПүy
ПҒвҲ— L v x Пү t by ( ) ПҒвҲ—
x t dy
x=t y=t

dПүt
= ПҒ вҲ— L v t Пүt + .
t
dt

6.2 Tubular Neighborhood Theorem
Let M be an n-dimensional manifold, and let X be a k-dimensional submanifold
where k < n and with inclusion map
i:X вҶ’M .
At each x вҲҲ X, the tangent space to X is viewed as a subspace of the tangent
space to M via the linear inclusion dix : Tx X вҶ’ Tx M , where we denote x =
i(x). The quotient Nx X := Tx M/Tx X is an (n вҲ’ k)-dimensional vector space,
known as the normal space to X at x. The normal bundle of X is
N X = {(x, v) | x вҲҲ X , v вҲҲ Nx X} .
The set N X has the structure of a vector bundle over X of rank n вҲ’ k under the
natural projection, hence as a manifold N X is n-dimensional. The zero section
of N X,
i0 : X вҶ’ N X , x вҶ’ (x, 0) ,
embeds X as a closed submanifold of N X. A neighborhood U0 of the zero
section X in N X is called convex if the intersection U0 вҲ© Nx X with each п¬Ғber
is convex.
Theorem 6.5 (Tubular Neighborhood Theorem) There exist a convex
neighborhood U0 of X in N X, a neighborhood U of X in M , and a diп¬Җeomor-
phism П• : U0 вҶ’ U such that
П• E U вҠҶM
N X вҠҮ U0
d
s В

d В
d В  commutes.
i0 d В i
d В
X
38 6 PREPARATION FOR THE LOCAL THEORY

Outline of the proof.
вҖў Case of M = Rn , and X is a compact submanifold of Rn .

Theorem 6.6 (Оµ-Neighborhood Theorem)
Let U Оµ = {p вҲҲ Rn : |p вҲ’ q| < Оµ for some q вҲҲ X} be the set of points at a
distance less than Оµ from X. Then, for Оµ suп¬ғciently small, each p вҲҲ U Оµ
has a unique nearest point q вҲҲ X (i.e., a unique q вҲҲ X minimizing |q вҲ’x|).
ПҖ
Moreover, setting q = ПҖ(p), the map U Оµ вҶ’ X is a (smooth) submersion
with the property that, for all p вҲҲ U Оµ , the line segment (1 вҲ’ t)p + tq,
0 вү¤ t вү¤ 1, is in U Оµ .

The proof is part of Homework 5. Here are some hints.
At any x вҲҲ X, the normal space Nx X may be regarded as an (n вҲ’ k)-
dimensional subspace of Rn , namely the orthogonal complement in Rn of
the tangent space to X at x:

{v вҲҲ Rn : v вҠҘ w , for all w вҲҲ Tx X} .
Nx X

We deп¬Ғne the following open neighborhood of X in N X:

N X Оµ = {(x, v) вҲҲ N X : |v| < Оµ} .

Let
вҲ’вҶ’ Rn
exp : NX
вҲ’вҶ’ x + v .
(x, v)
Restricted to the zero section, exp is the identity map on X.
Prove that, for Оµ suп¬ғciently small, exp maps N X Оµ diп¬Җeomorphically onto
U Оµ , and show also that the diagram

exp E UОµ
NXОµ
d В
d В
d В  commutes.
ПҖ0 d В ПҖ
В‚
d В
X

вҖў Case where X is a compact submanifold of an arbitrary manifold M .
Put a riemannian metric g on M , and let d(p, q) be the riemannian distance
between p, q вҲҲ M . The Оµ-neighborhood of a compact submanifold X is

U Оµ = {p вҲҲ M | d(p, q) < Оµ for some q вҲҲ X} .

Prove the Оµ-neighborhood theorem in this setting: for Оµ small enough, the
following assertions hold.
39
6.3 Homotopy Formula

вҖ“ Any p вҲҲ U Оµ has a unique point q вҲҲ X with minimal d(p, q). Set q = ПҖ(p).
ПҖ
вҖ“ The map U Оµ вҶ’ X is a submersion and, for all p вҲҲ U Оµ , there is a unique
geodesic curve Оі joining p to q = ПҖ(p).
вҖ“ The normal space to X at x вҲҲ X is naturally identiп¬Ғed with a subspace
of Tx M :

{v вҲҲ Tx M | gx (v, w) = 0 , for any w вҲҲ Tx X} .
Nx X

Let N X Оµ = {(x, v) вҲҲ N X | gx (v, v) < Оµ}.
вҖ“ Deп¬Ғne exp : N X Оµ вҶ’ M by exp(x, v) = Оі(1), where Оі : [0, 1] вҶ’ M
is the geodesic with Оі(0) = x and dОі (0) = v. Then exp maps N X Оµ
dt
diп¬Җeomorphically to U Оµ .

вҖў General case.
When X is not compact, adapt the previous argument by replacing Оµ by
an appropriate continuous function Оµ : X вҶ’ R+ which tends to zero fast
enough as x tends to inп¬Ғnity.

Restricting to the subset U 0 вҠҶ N X from the tubular neighborhood theorem,
ПҖ0 вҲ’1
we obtain a submersion U0 вҲ’вҶ’ X with all п¬Ғbers ПҖ0 (x) convex. We can carry
this п¬Ғbration to U by setting ПҖ = ПҖ0 в—¦ П•вҲ’1 :

U0 вҠҶ NX U вҠҶM
is a п¬Ғbration =вҮ’ is a п¬Ғbration
ПҖ0 вҶ“ ПҖвҶ“
X X

This is called the tubular neighborhood п¬Ғbration.

6.3 Homotopy Formula

Let U be a tubular neighborhood of a submanifold X in M . The restriction iвҲ— :
d d
HdeRham(U) вҶ’ HdeRham (X) by the inclusion map is surjective. As a corollary
of the tubular neighborhood п¬Ғbration, iвҲ— is also injective: this follows from the
homotopy-invariance of de Rham cohomology.

Corollary 6.7 For any degree , HdeRham (U) HdeRham (X).

At the level of forms, this means that, if Пү is a closed -form on U and iвҲ— Пү
is exact on X, then Пү is exact. We will need the following related result.

Theorem 6.8 If a closed -form Пү on U has restriction iвҲ— Пү = 0, then Пү is
exact, i.e., Пү = dВµ for some Вµ вҲҲ в„¦dвҲ’1 (U). Moreover, we can choose Вµ such that
Вµx = 0 at all x вҲҲ X.
40 6 PREPARATION FOR THE LOCAL THEORY

Proof. Via П• : U0 вҲ’вҶ’ U, it is equivalent to work over U0 . Deп¬Ғne for every
0 вү¤ t вү¤ 1 a map
U0 вҲ’вҶ’ U0
ПҒt :
(x, v) вҲ’вҶ’ (x, tv) .
This is well-deп¬Ғned since U0 is convex. The map ПҒ1 is the identity, ПҒ0 = i0 в—¦
ПҖ0 , and each ПҒt п¬Ғxes X, that is, ПҒt в—¦ i0 = i0 . We hence say that the family
{ПҒt | 0 вү¤ t вү¤ 1} is a homotopy from i0 в—¦ ПҖ0 to the identity п¬Ғxing X. The
map ПҖ0 : U0 вҶ’ X is called a retraction because ПҖ0 в—¦ i0 is the identity. The
submanifold X is then called a deformation retract of U.
A (de Rham) homotopy operator between ПҒ0 = i0 в—¦ ПҖ0 and ПҒ1 = id is a
linear map
Q : в„¦d (U0 ) вҲ’вҶ’ в„¦dвҲ’1 (U0 )
satisfying the homotopy formula

Id вҲ’ (i0 в—¦ ПҖ0 )вҲ— = dQ + Qd .

When dПү = 0 and iвҲ— Пү = 0, the operator Q gives Пү = dQПү, so that we can take
0
Вµ = QПү. A concrete operator Q is given by the formula:
1
ПҒвҲ— (Д±vt Пү) dt ,
QПү = t
0

where vt , at the point q = ПҒt (p), is the vector tangent to the curve ПҒs (p) at
s = t. The proof that Q satisп¬Ғes the homotopy formula is below.
In our case, for x вҲҲ X, ПҒt (x) = x (all t) is the constant curve, so vt vanishes
at all x for all t, hence Вµx = 0.
To check that Q above satisп¬Ғes the homotopy formula, we compute
1 1
ПҒвҲ— (Д±vt dПү)dt ПҒвҲ— (Д±vt Пү)dt
QdПү + dQПү = +d
t t
0 0

1
ПҒвҲ— (Д±vt dПү + dД±vt Пү)dt ,
= t
0
Lvt Пү

where Lv denotes the Lie derivative along v (reviewed in the next section), and
we used the Cartan magic formula: Lv Пү = Д±v dПү + dД±v Пү. The result now follows
from
dвҲ—
ПҒ t Пү = ПҒ вҲ— Lvt Пү
t
dt
and from the fundamental theorem of calculus:
1
dвҲ—
ПҒ Пү dt = ПҒвҲ— Пү вҲ’ ПҒвҲ— Пү .
QdПү + dQПү = 1 0
dt t
0
Homework 5: Tubular Neighborhoods in Rn

1. Let X be a k-dimensional submanifold of an n-dimensional manifold M .
Let x be a point in X. The normal space to X at x is the quotient space

Nx X = Tx M/Tx X ,

and the normal bundle of X in M is the vector bundle N X over X
whose п¬Ғber at x is Nx X.

(a) Prove that N X is indeed a vector bundle.
(b) If M is Rn , show that Nx X can be identiп¬Ғed with the usual вҖңnormal
spaceвҖқ to X in Rn , that is, the orthogonal complement in Rn of the
tangent space to X at x.

2. Let X be a k-dimensional compact submanifold of Rn . Prove the tubular
neighborhood theorem in the following form.

(a) Given Оµ > 0 let UОµ be the set of all points in Rn which are at a distance
less than Оµ from X. Show that, for Оµ suп¬ғciently small, every point
p вҲҲ UОµ has a unique nearest point ПҖ(p) вҲҲ X.
(b) Let ПҖ : UОµ вҶ’ X be the map deп¬Ғned in (a) for Оµ suп¬ғciently small.
Show that, if p вҲҲ UОµ , then the line segment (1 вҲ’ t) В· p + t В· ПҖ(p),
0 вү¤ t вү¤ 1, joining p to ПҖ(p) lies in UОµ .
(c) Let N XОµ = {(x, v) вҲҲ N X such that |v| < Оµ}. Let exp : N X вҶ’ Rn
be the map (x, v) вҶ’ x + v, and let ОҪ : N XОµ вҶ’ X be the map
(x, v) вҶ’ x. Show that, for Оµ suп¬ғciently small, exp maps N XОµ dif-
feomorphically onto UОµ , and show also that the following diagram
commutes:
exp E UОµ
N XОµ
d В
d В
d В
ОҪd В ПҖ
В‚
d В
X

3. Suppose that the manifold X in the previous exercise is not compact.
Prove that the assertion about exp is still true provided we replace Оµ by a
continuous function
Оµ : X вҶ’ R+
which tends to zero fast enough as x tends to inп¬Ғnity.

41
7 Moser Theorems

7.1 Notions of Equivalence for Symplectic Structures

Let M be a 2n-dimensional manifold with two symplectic forms Пү0 and Пү1 , so
that (M, Пү0 ) and (M, Пү1 ) are two symplectic manifolds.

Deп¬Ғnition 7.1 We say that
вҖў (M, Пү0 ) and (M, Пү1 ) are symplectomorphic if there is a diп¬Җeomorphism
П• : M вҶ’ M with П•вҲ— Пү1 = Пү0 ;
вҖў (M, Пү0 ) and (M, Пү1 ) are strongly isotopic if there is an isotopy ПҒt :
M вҶ’ M such that ПҒвҲ— Пү1 = Пү0 ;
1

вҖў (M, Пү0 ) and (M, Пү1 ) are deformation-equivalent if there is a smooth
family Пүt of symplectic forms joining Пү0 to Пү1 ;
вҖў (M, Пү0 ) and (M, Пү1 ) are isotopic if they are deformation-equivalent with
[Пүt ] independent of t.

Clearly, we have

strongly isotopic =вҮ’ symplectomorphic , and

isotopic =вҮ’ deformation-equivalent .
We also have
strongly isotopic =вҮ’ isotopic
because, if ПҒt : M вҶ’ M is an isotopy such that ПҒвҲ— Пү1 = Пү0 , then the set Пүt :=
1
вҲ—
ПҒt Пү1 is a smooth family of symplectic forms joining Пү1 to Пү0 and [Пүt ] = [Пү1 ],
вҲҖt, by the homotopy invariance of de Rham cohomology. As we will see below,
the Moser theorem states that, on a compact manifold,

isotopic =вҮ’ strongly isotopic .

7.2 Moser Trick

Problem. Given a 2n-dimensional manifold M , a k-dimensional submanifold
X, neighborhoods U0 , U1 of X, and symplectic forms Пү0 , Пү1 on U0 , U1 , does
there exist a symplectomorphism preserving X? More precisely, does there
exist a diп¬Җeomorphism П• : U0 вҶ’ U1 with П•вҲ— Пү1 = Пү0 and П•(X) = X?
At the two extremes, we have:
Case X = point: Darboux theorem вҖ“ see Lecture 8.
Case X = M : Moser theorem вҖ“ discussed here:

Let M be a compact manifold with symplectic forms Пү0 and Пү1 .

42
43
7.2 Moser Trick

вҖ“ Are (M, Пү0 ) and (M, Пү1 ) symplectomorphic?
I.e., does there exist a diп¬Җeomorphism П• : M вҶ’ M such that П•вҲ— Пү0 = Пү1 ?
1

Moser asked whether we can п¬Ғnd such an П• which is homotopic to id M . A
necessary condition is [Пү0 ] = [Пү1 ] вҲҲ H 2 (M ; R) because: if П• вҲј idM , then, by
the homotopy formula, there exists a homotopy operator Q such that

idвҲ— Пү1 вҲ’ П•вҲ— Пү1 = dQПү1 + Q dПү1
M
0
вҲ—
=вҮ’ Пү1 = П• Пү1 + d(QПү1 )
[Пү1 ] = [П•вҲ— Пү1 ] = [Пү0 ] .
=вҮ’

вҖ“ If [Пү0 ] = [Пү1 ], does there exist a diп¬Җeomorphism П• homotopic to idM such
that П•вҲ— Пү1 = Пү0 ?

Moser  proved that the answer is yes, with a further hypothesis as in
Theorem 7.2. McDuп¬Җ showed that, in general, the answer is no; for a coun-
terexample, see Example 7.23 in .

Theorem 7.2 (Moser Theorem вҖ“ Version I) Suppose that [Пү0 ] = [Пү1 ] and
that the 2-form Пүt = (1 вҲ’ t)Пү0 + tПү1 is symplectic for each t вҲҲ [0, 1]. Then there
exists an isotopy ПҒ : M Г— R вҶ’ M such that ПҒвҲ— Пүt = Пү0 for all t вҲҲ [0, 1].
t

In particular, П• = ПҒ1 : M вҲ’вҶ’ M , satisп¬Ғes П•вҲ— Пү1 = Пү0 .
The following argument, due to Moser, is extremely useful; it is known as
the Moser trick.
Proof. Suppose that there exists an isotopy ПҒ : M Г— R вҶ’ M such that ПҒвҲ— Пүt =
t
Пү0 , 0 вү¤ t вү¤ 1. Let
dПҒt
в—¦ ПҒвҲ’1 , tвҲҲR.
vt = t
dt
Then
d dПүt
0 = (ПҒвҲ— Пүt ) = ПҒвҲ— Lvt Пүt +
t t
dt dt
dПүt
вҮҗвҮ’ L v t Пүt + =0. ()
dt
Suppose conversely that we can п¬Ғnd a smooth time-dependent vector п¬Ғeld
vt , t вҲҲ R, such that ( ) holds for 0 вү¤ t вү¤ 1. Since M is compact, we can
integrate vt to an isotopy ПҒ : M Г— R вҶ’ M with
dвҲ—
ПҒ вҲ— Пүt = ПҒ вҲ— Пү0 = Пү 0 .
(ПҒ Пүt ) = 0 =вҮ’ 0
dt t t

So everything boils down to solving ( ) for vt .
First, from Пүt = (1 вҲ’ t)Пү0 + tПү1 , we conclude that
dПүt
= Пү1 вҲ’ Пү0 .
dt
44 7 MOSER THEOREMS

Second, since [Пү0 ] = [Пү1 ], there exists a 1-form Вµ such that

Пү1 вҲ’ Пү0 = dВµ .

Third, by the Cartan magic formula, we have

Lvt Пүt = dД±vt Пүt + Д±vt dПүt .
0

Putting everything together, we must п¬Ғnd vt such that

dД±vt Пүt + dВµ = 0 .

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

Theorem 7.3 (Moser Theorem вҖ“ Version II) Let M be a compact man-
ifold with symplectic forms Пү0 and Пү1 . Suppose that Пүt , 0 вү¤ t вү¤ 1, is a smooth
family of closed 2-forms joining Пү0 to Пү1 and satisfying:
d d
(1) cohomology assumption: [Пүt ] is independent of t, i.e., dt [Пүt ] = dt Пүt = 0,
(2) nondegeneracy assumption: Пүt is nondegenerate for 0 вү¤ t вү¤ 1.
Then there exists an isotopy ПҒ : M Г— R вҶ’ M such that ПҒвҲ— Пүt = Пү0 , 0 вү¤ t вү¤ 1.
t

Proof. (Moser trick) We have the following implications from the hypotheses:
(1) =вҮ’ вҲғ family of 1-forms Вµt such that
dПүt
0вү¤tвү¤1.
= dВµt ,
dt
We can indeed п¬Ғnd a smooth family of 1-forms Вµt such that dПүt = dВµt . The
dt
argument involves the PoincarВҙ lemma for compactly-supported forms,
e
together with the Mayer-Vietoris sequence in order to use induction on
the number of charts in a good cover of M . For a sketch of the argument,
see page 95 in .
(2) =вҮ’ вҲғ unique family of vector п¬Ғelds vt such that

Д±v t Пү t + Вµ t = 0 (Moser equation) .

Extend vt to all t вҲҲ R. Let ПҒ be the isotopy generated by vt (ПҒ exists by
compactness of M ). Then we indeed have
dвҲ— dПүt
(ПҒt Пүt ) = ПҒвҲ— (Lvt Пүt + ) = ПҒвҲ— (dД±vt Пүt + dВµt ) = 0 .
t t
dt dt
45
7.3 Moser Local Theorem

The compactness of M was used to be able to integrate vt for all t вҲҲ R.
If M is not compact, we need to check the existence of a solution ПҒt for the
diп¬Җerential equation dПҒt = vt в—¦ ПҒt for 0 вү¤ t вү¤ 1.
dt

Picture. Fix c вҲҲ H 2 (M ). Deп¬Ғne Sc = {symplectic forms Пү in M with [Пү] = c}.
The Moser theorem implies that, on a compact manifold, all symplectic forms
on the same path-connected component of Sc are symplectomorphic.

7.3 Moser Local Theorem

Theorem 7.4 (Moser Theorem вҖ“ Local Version) Let M be a manifold,
X a submanifold of M , i : X вҶ’ M the inclusion map, Пү0 and Пү1 symplectic
forms in M .

Hypothesis: Пү0 |p = Пү1 |p , вҲҖp вҲҲ X .
Conclusion: There exist neighborhoods U0 , U1 of X in M ,
and a diп¬Җeomorphism П• : U0 вҶ’ U1 such that
П• E U1
U0
d
s В

d В
d В  commutes
id В i
d В
X
and П•вҲ— Пү1 = Пү0 .

Proof.
1. Pick a tubular neighborhood U0 of X. The 2-form Пү1 вҲ’ Пү0 is closed on U0 ,
and (Пү1 вҲ’ Пү0 )p = 0 at all p вҲҲ X. By the homotopy formula on the tubular
neighborhood, there exists a 1-form Вµ on U0 such that Пү1 вҲ’ Пү0 = dВµ and
Вµp = 0 at all p вҲҲ X.
2. Consider the family Пүt = (1 вҲ’ t)Пү0 + tПү1 = Пү0 + tdВµ of closed 2-forms
on U0 . Shrinking U0 if necessary, we can assume that Пүt is symplectic for
0 вү¤ t вү¤ 1.
3. Solve the Moser equation: Д±vt Пүt = вҲ’Вµ. Notice that vt = 0 on X.
4. Integrate vt . Shrinking U0 again if necessary, there exists an isotopy ПҒ :
U0 Г— [0, 1] вҶ’ M with ПҒвҲ— Пүt = Пү0 , for all t вҲҲ [0, 1]. Since vt |X = 0, we have
t
ПҒt |X = idX .
Set П• = ПҒ1 , U1 = ПҒ1 (U0 ).

Exercise. Prove the Darboux theorem. (Hint: apply the local version of the
Moser theorem to X = {p}, as in the next lecture.) в™¦
8 Darboux-Moser-Weinstein Theory

8.1 Classical Darboux Theorem

Theorem 8.1 (Darboux) Let (M, Пү) be a symplectic manifold, and let p be
any point in M . Then we can п¬Ғnd a coordinate system (U, x1 , . . . , xn , y1 , . . . yn )
centered at p such that on U
n
dxi вҲ§ dyi .
Пү=
i=1

As a consequence of Theorem 8.1, if we prove for (R2n , dxi вҲ§ dyi ) a lo-
cal assertion which is invariant under symplectomorphisms, then that assertion
holds for any symplectic manifold.
Proof. Apply the Moser local theorem (Theorem 7.4) to X = {p}:
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 вҲ§ dyi
Пүp = .
p

There are two symplectic forms on U : the given Пү0 = Пү and Пү1 = dxi вҲ§
dyi . By the Moser theorem, there are neighborhoods U0 and U1 of p, and a
diп¬Җeomorphism П• : U0 вҶ’ U1 such that

П•вҲ— ( dxi вҲ§ dyi ) = Пү .
П•(p) = p and

Since П•вҲ— ( dxi вҲ§ dyi ) = d(xi в—¦ П•) вҲ§ d(yi в—¦ П•), we only need to set new
coordinates xi = xi в—¦ П• and yi = yi в—¦ П•.
If in the Moser local theorem (Theorem 7.4) we assume instead

Hypothesis: X is an n-dimensional submanifold with
iвҲ— Пү0 = iвҲ— Пү1 = 0 where i : X вҶ’ M is inclusion, i.e.,
X is a submanifold lagrangian for Пү0 and Пү1 ,

then Weinstein  proved that the conclusion still holds. We need some
algebra for the Weinstein theorem.

8.2 Lagrangian Subspaces

Suppose that U, W are n-dimensional vector spaces, and в„¦ : U Г— W вҶ’ R
is a bilinear pairing; the map в„¦ gives rise to a linear map в„¦ : U вҶ’ W вҲ— ,
в„¦(u) = в„¦(u, В·). Then в„¦ is nondegenerate if and only if в„¦ is bijective.

46
47
8.2 Lagrangian Subspaces

Proposition 8.2 Suppose that V is a 2n-dimensional vector space and в„¦ :
V Г— V вҶ’ R is a nondegenerate skew-symmetric bilinear pairing. Let U be a
lagrangian subspace of (V, в„¦) (i.e., в„¦|U Г—U = 0 and U is n-dimensional). Let W
be any vector space complement to U , not necessarily lagrangian.
Then from W we can canonically build a lagrangian complement to U .

в„¦
Proof. The pairing в„¦ gives a nondegenerate pairing U Г— W вҶ’ R. Therefore,
в„¦ : U вҶ’ W вҲ— is bijective. We look for a lagrangian complement to U of the
form
W = {w + Aw | w вҲҲ W } ,
A : W вҶ’ U being a linear map. For W to be lagrangian we need

вҲҖ w1 , w2 вҲҲ W , в„¦(w1 + Aw1 , w2 + Aw2 ) = 0

=вҮ’ в„¦(w1 , w2 ) + в„¦(w1 , Aw2 ) + в„¦(Aw1 , w2 ) + в„¦(Aw1 , Aw2 ) = 0
вҲҲU

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

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

вҲҖ w1 , w2 вҲҲ W , в„¦(w1 , w2 ) = A (w2 )(w1 ) вҲ’ A (w1 )(w2 ) .

The canonical choice is A (w) = вҲ’ 1 в„¦(w, В·). Then set A = (в„¦ )вҲ’1 в—¦ A .
2

Proposition 8.3 Let V be a 2n-dimensional vector space, let в„¦0 and в„¦1 be
symplectic forms in V , let U be a subspace of V lagrangian for в„¦0 and в„¦1 , and
let W be any complement to U in V . Then from W we can canonically construct
a linear isomorphism L : V вҶ’ V such that L|U = IdU and LвҲ— в„¦1 = в„¦0 .

Proof. From W we canonically obtain complements W0 and W1 to U in V such
that W0 is lagrangian for в„¦0 and W1 is lagrangian for в„¦1 . The nondegenerate
bilinear pairings
в„¦
в„¦0 : W0 вҲ’вҶ’ U вҲ—
0
W0 Г— U вҲ’вҶ’ R
give isomorphisms
в„¦
в„¦1 : W1 вҲ’вҶ’ U вҲ— .
1
W1 Г— U вҲ’вҶ’ R

Consider the diagram
в„¦
UвҲ—
0
вҲ’вҶ’
W0
BвҶ“ вҶ“ id
в„¦
UвҲ—
1
вҲ’вҶ’
W1
48 8 DARBOUX-MOSER-WEINSTEIN THEORY

where the linear map B satisп¬Ғes в„¦1 в—¦ B = в„¦0 , i.e., в„¦0 (Пү0 , u) = в„¦1 (BПү0 , u),
вҲҖПү0 вҲҲ W0 , вҲҖu вҲҲ U . Extend B to the rest of V by setting it to be the identity
on U :
L := IdU вҠ• B : U вҠ• W0 вҲ’вҶ’ U вҠ• W1 .
Finally, we check that LвҲ— в„¦1 = в„¦0 .

(LвҲ— в„¦1 )(u вҠ• w0 , u вҠ• w0 ) = в„¦1 (u вҠ• BПү0 , u вҠ• BПү0 )
= в„¦1 (u, BПү0 ) + в„¦1 (BПү0 , u )
= в„¦0 (u, Пү0 ) + в„¦0 (Пү0 , u )
в„¦0 (u вҠ• w0 , u вҠ• w0 ) .
=

8.3 Weinstein Lagrangian Neighborhood Theorem

Theorem 8.4 (Weinstein Lagrangian Neighborhood Theorem )
Let M be a 2n-dimensional manifold, X an n-dimensional submanifold, i : X вҶ’
M the inclusion map, and Пү0 and Пү1 symplectic forms on M such that iвҲ— Пү0 =
iвҲ— Пү1 = 0, i.e., X is a lagrangian submanifold of both (M, Пү0 ) and (M, Пү1 ).
Then there exist neighborhoods U0 and U1 of X in M and a diп¬Җeomorphism
П• : U0 вҶ’ U1 such that

П• E U1
U0
d
s 
В
d В
П• вҲ— Пү1 = Пү 0 .
d В  commutes and
id В i
d В
X
The proof of the Weinstein theorem uses the Whitney extension theorem.

Theorem 8.5 (Whitney Extension Theorem) Let M be an n-dimensional
manifold and X a k-dimensional submanifold with k < n. Suppose that at
each p вҲҲ X we are given a linear isomorphism Lp : Tp M вҶ’ Tp M such that
Lp |Tp X = IdTp X and Lp depends smoothly on p. Then there exists an embedding
h : N вҶ’ M of some neighborhood N of X in M such that h|X = idX and
dhp = Lp for all p вҲҲ X.

The linear maps L serve as вҖңgermsвҖқ for the embedding.
Proof of the Weinstein theorem. Put a riemannian metric g on M ; at each
p вҲҲ M , gp (В·, В·) is a positive-deп¬Ғnite inner product. Fix p вҲҲ X, and let V = Tp M ,
U = Tp X and W = U вҠҘ = orthocomplement of U in V relative to gp (В·, В·).
Since iвҲ— Пү0 = iвҲ— Пү1 = 0, U is a lagrangian subspace of both (V, Пү0 |p ) and
(V, Пү1 |p ). By symplectic linear algebra, we canonically get from U вҠҘ a linear
49
8.3 Weinstein Lagrangian Neighborhood Theorem

isomorphism Lp : Tp M вҶ’ Tp M , such that Lp |Tp X = IdTp X and LвҲ— Пү1 |p = Пү0 |p .
p
Lp varies smoothly with respect to p since our recipe is canonical!
By the Whitney theorem, there are a neighborhood N of X and an embed-
ding h : N вҶ’ M with h|X = idX and dhp = Lp for p вҲҲ X. Hence, at any
p вҲҲ X,
(hвҲ— Пү1 )p = (dhp )вҲ— Пү1 |p = LвҲ— Пү1 |p = Пү0 |p .
p

Applying the Moser local theorem (Theorem 7.4) to Пү0 and hвҲ— Пү1 , we п¬Ғnd a
neighborhood U0 of X and an embedding f : U0 вҶ’ N such that f |X = idX and
f вҲ— (hвҲ— Пү1 ) = Пү0 on Uo . Set П• = h в—¦ f .
Sketch of proof for the Whitney theorem.
Case M = Rn :
For a compact k-dimensional submanifold X, take a neighborhood of the
form
U Оµ = {p вҲҲ M | distance (p, X) вү¤ Оµ}
For Оµ suп¬ғciently small so that any p вҲҲ U Оµ has a unique nearest point in X,
deп¬Ғne a projection ПҖ : U Оµ вҶ’ X, p вҶ’ point on X closest to p. If ПҖ(p) = q, then
p = q + v for some v вҲҲ Nq X where Nq X = (Tq X)вҠҘ is the normal space at q;
see Homework 5. Let
h : U Оµ вҲ’вҶ’ Rn
p вҲ’вҶ’ q + Lq v
where q = ПҖ(p) and v = p вҲ’ ПҖ(p) вҲҲ Nq X. Then hX = idX and dhp = Lp for
p вҲҲ X. If X is not compact, replace Оµ by a continuous function Оµ : X вҶ’ R+
which tends to zero fast enough as x tends to inп¬Ғnity.
General case:
Choose a riemannian metric on M . Replace distance by riemannian distance,
replace straight lines q + tv by geodesics exp(q, v)(t) and replace q + Lq v by the
value at t = 1 of the geodesic with initial value q and initial velocity Lq v.

In Lecture 30 we will need the following generalization of Theorem 8.4. For
a proof see, for instance, either of [47, 57, 105].

Theorem 8.6 (Coisotropic Embedding Theorem) Let M be a manifold
of dimension 2n, X a submanifold of dimension k вүҘ n, i : X вҶ’ M the inclusion
map, and Пү0 and Пү1 symplectic forms on M , such that iвҲ— Пү0 = iвҲ— Пү1 and X is
coisotropic for both (M, Пү0 ) and (M, Пү1 ). Then there exist neighborhoods U0 and
U1 of X in M and a diп¬Җeomorphism П• : U0 вҶ’ U1 such that

П• E U1
U0
d
s В

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