„¦ = „¦0 + s d• ’ • § ds.

(Here, we are using s as the coordinate on the ¬rst factor (’1, 1) and, as usual, we write

— —

„¦0 and • instead of π2 („¦0 ) and π2 (•) where π2 : (’1, 1) — M ’ M is the projection on

the second factor.)

The reader can check that „¦ is closed on (’1, 1) — M. Moreover, since „¦ pulls back

to each slice {s0 } — M to be the non-degenerate form „¦s0 it follows that „¦ has half-rank

n everywhere. Thus, the kernel N„¦ is 1-dimensional and is transverse to each of the slices

{t} — M. Hence there is a unique vector ¬eld X which spans N„¦ and satis¬es ds(X) = 1.

Now because M is compact, it is not di¬cult to see that each integral curve of X

projects by s = π1 di¬eomorphically onto (’1, 1). Moreover, it follows that there is a

smooth map φ: (’1, 1) — M ’ M so that, for each m, the curve t ’ φ(t, m) is the integral

curve of X which passes through (0, m).

It follows that the map ¦: (’1, 1) — M ’ (’1, 1) — M de¬ned by

¦(t, m) = t, φ(t, m)

carries the vector ¬eld ‚/‚s to the vector ¬eld X. Moreover, since „¦0 and „¦ have the

same value when pulled back to the slice {0} — M and since

L‚/‚s „¦0 = 0 LX „¦ = 0

and

X „¦ = 0,

‚/‚s „¦0 = 0

it follows easily that ¦— („¦) = „¦0 . In particular, φ— („¦t ) = „¦0 where φt is the di¬eomor-

t

phism of M given by φt (m) = φ(t, m).

Now let us turn to the general case. If „¦t for 0 ¤ t ¤ 1 is any continuous family of

smooth closed 2-forms for which the cohomology classes [„¦t ] are all equal to [„¦0 ], then for

any two values t1 and t2 in the unit interval, consider the 1-parameter family of 2-forms

Υs = (1 ’ s)„¦t1 + s„¦t2 .

Using the compactness of M, it is not di¬cult to show that for t2 su¬ciently close to t1 ,

the family Υs is a 1-parameter family of symplectic forms on M for s in some open interval

containing [0, 1]. Moreover, by hypothesis, [„¦t2 ’ „¦t1 ] = 0, so there exists a 1-form • on

M so that d• = „¦t2 ’ „¦t1 . Thus,

Υs = „¦t1 + s d•.

L.6.5 104

By the special case already treated, there exists a di¬eomorphism φt2 ,t1 of M so that

φ—2 ,t1 („¦t2 ) = „¦t1 .

t

Finally, using the compactness of the interval [0, t] for any t ∈ [0, 1], we can subdivide

this interval into a ¬nite number of intervals [t1 , t2 ] on which the above argument works.

Then, by composing di¬eomorphisms, we can construct a di¬eomorphism φt of M so that

φ— („¦t ) = „¦0 .

t

The reader may have wanted the family of di¬eomorphisms φt to depend continuously

on t and smoothly on t if the family „¦t is smooth in t. This can, in fact, be arranged.

However, it involves showing that there is a smooth family of 1-forms •t on M so that

d

dt „¦t = d•t , i.e., smoothly solving the d-equation. This can be done, but requires some

delicacy or use of elliptic machinery (e.g., Hodge-deRham theory).

Theorem 1 does not hold without the hypothesis of compactness. For example, if „¦

is the restriction of the standard structure on R2n to the unit ball B 2n , then for the family

„¦t = et „¦ there cannot be any family of di¬eomorphisms of the ball φt so that φ— („¦t ) = „¦

t

n nt n

since the integrals over B of the volume forms („¦t ) = e „¦ are all di¬erent.

Intuitively, Theorem 1 says that the “connected components” of the space of symplec-

tic structures on a manifold are orbits of the group Diff0 (M) of di¬eomorphisms isotopic

to the identity. (The reason this is only intuitive is that we have not actually de¬ned a

topology on the space of symplectic structures on M.)

It is an interesting question as to how many “connected components” the space of

symplectic structures on M has. The work of Gromov has yielded methods to attack this

problem and I will have more to say about this in Lecture 9.

Submanifolds of Symplectic Manifolds

We will now pass on to the study of the geometry of submanifolds of a symplectic

manifold. The following result describes the behaviour of symplectic structures near closed

submanifolds. This theorem, due to Weinstein (see [Weinstein]), can be regarded as a

generalization of Darboux™ Theorem. The reader will note that the proof is quite similar

to the proof of Theorem 1.

Theorem 2: Let P ‚ M be a closed submanifold and let „¦0 and „¦1 be symplectic

structures on M which have the property that „¦0 (p) = „¦1 (p) for all p ∈ P . Then there

exist open neighborhoods U0 and U1 of P and a di¬eomorphism φ: U0 ’ U1 satisfying

φ— („¦1 ) = „¦0 and which moreover ¬xes P pointwise and satis¬es φ (p) = idp : Tp M ’ Tp M

for all p ∈ P .

Proof: Consider the linear family of 2-forms

„¦t = (1 ’ t)„¦0 + t„¦1

L.6.6 105

which “interpolates” between the forms „¦0 and „¦1 . Since [0, 1] is compact and since, by

hypothesis, „¦0 (p) = „¦1 (p) for all p ∈ P , it easily follows that there is an open neighborhood

U of P in M so that „¦t is a symplectic structure on U for all t in some open interval

I = (’µ, 1 + µ) containing [0, 1].

We may even suppose that U is a “tubular neighborhood” of P which has a smooth

retraction R: [0, 1] — U ’ U into P . Since ¦ = „¦1 ’ „¦0 vanishes on P , it follows without

too much di¬cultly (see the Exercises) that there is a 1-form • on U which vanishes on P

and which satis¬es d• = ¦.

Now, on I — U, consider the 2-form

„¦ = „¦0 + s d• ’ • § ds.

This is a closed 2-form of half-rank n on I — U. Just as in the previous theorem, it follows

that there exists a unique vector ¬eld X on I — U so that ds(X) = 1 and X „¦ = 0.

Since • and d• vanish on P , the vector ¬eld X has the property that X(s, p) = ‚/‚s

for all p ∈ P and s ∈ I. In particular, the set {0} — P lies in the domain of the time 1

¬‚ow of X. Since this domain is an open set, it follows that there is an open neighborhood

U0 of P in U so that {0} — U0 lies in the domain of the time 1 ¬‚ow of X. The image of

{0} — U0 under the time 1 ¬‚ow of X is of the form {1} — U1 where U1 is another open

neighborhood of P in U.

Thus, the time 1 ¬‚ow of X generates a di¬eomorphism φ: U0 ’ U1 . By the arguments

of the previous theorem, it follows that φ— („¦1 ) = „¦0 . I leave it to the reader to check that

φ ¬xes P in the desired fashion.

Theorem 2 has a useful corollary:

Corollary : Let „¦ be a symplectic structure on M and let f0 and f1 be smooth embeddings

— —

of a manifold P into M so that f0 („¦) = f1 („¦) and so that there exists a smooth bundle

— —

isomorphism „ : f0 (T M) ’ f1 (T M) which extends the identity map on the subbundle

T P ‚ fi— (T M) and which identi¬es the symplectic structures on fi— (T M). Then there

exist open neighborhoods Ui of fi (P ) in M and a di¬eomorphism φ: U0 ’ U1 which

satis¬es φ— („¦) = „¦ and, moreover, φ —¦ f0 = f1 .

Proof: It is an elementary result in di¬erential topology that, under the hypotheses of the

Corollary, there exists an open neighborhood W0 of f0 (P ) in M and a smooth di¬eomorphic

embedding ψ: W0 ’ M so that ψ —¦ f0 = f1 and ψ f0 (p) : Tf0 (p)(M) ’ Tf1 (p)(M) is equal

to „ (p). It follows that ψ — („¦) is a symplectic form on W0 which agrees with „¦ along f0 (P ).

By Theorem 2, it follows that there is a neighborhood U0 of f0 (P ) which lies in W0 and a

smooth map ν: U0 ’ W0 which is a di¬eomorphism onto its image, ¬xes f0 (P ) pointwise,

satis¬es ν f0 (p) = idf0 (p) for all p ∈ P , and also satis¬es ν — ψ — („¦) = „¦. Now just take

φ = ψ —¦ ν.

We will now give two particularly important applications of this result:

If P ‚ M is a symplectic submanifold, then by using „¦, we can de¬ne a normal bundle

for P as follows:

ν(P ) = {(p, v) ∈ P — T M | v ∈ Tp M, „¦(v, w) = 0 for all w ∈ Tp P }.

L.6.7 106

The bundle ν(P ) has a natural symplectic structure on each of its ¬bers (see the Exer-

cises), and hence is a symplectic vector bundle. The following proposition shows that, up

to local di¬eomorphism, this normal bundle determines the symplectic structure „¦ on a

neighborhood of P .

Let (P, Υ) be a symplectic manifold and let f0 , f1 : P ’ M be two

Proposition 3:

symplectic embeddings of P as submanifolds of M so that the normal bundles ν0 (P ) and

ν1 (P ) are isomorphic as symplectic vector bundles. Then there are open neighborhoods

Ui of fi (P ) in M and a symplectic di¬eomorphism φ: U0 ’ U1 which satis¬es f1 = φ —¦ f0 .

Proof: It su¬ces to construct the map „ required by the hypotheses of Theorem 2.

Now, we have a symplectic bundle decomposition fi— (T M) = T P • νi (P ) for i = 1, 2. If

±: ν0 (P ) ’ ν1 (P ) is a symplectic bundle isomorphism, we then de¬ne „ = id • ± in the

obvious way and we are done.

At the other extreme, we want to consider submanifolds of M to which the form „¦

pulls back to be as degenerate as possible.

De¬nition 2: If „¦ is a symplectic structure on M 2n , an immersion f: P ’ M is said to be

isotropic if f — („¦) = 0. If the dimension of P is n, we say that f is a Lagrangian immersion.

If in addition, f is one-to-one, then we say that f(P ) is a Lagrangian submanifold of M.

Note that the dimension of an isotropic submanifold of M 2n is at most n, so the

Lagrangian submanifolds of M have maximal dimension among all isotropic submanifolds.

Example: Graphs of Symplectic Mappings. If f: M ’ N is a symplectic mapping

where „¦ and Υ are the symplectic forms on M and N respectively, then the graph of f

in M — N is an isotropic submanifold of M — N endowed with the symplectic structure

— —

(’„¦) • Υ = π1 (’„¦) + π2 (Υ). If M and N have the same dimension, then the graph of f

in M — N is a Lagrangian submanifold.

Example: Closed 1-forms. If ± is a 1-form on M, then the graph of ± in T — M is a

Lagrangian submanifold of T — M if and only if d± = 0. This follows because „¦ on T — M

has the “reproducing property” that ±— („¦) = d± for any 1-form on M.

Proposition 4: Let „¦ be a symplectic structure on M and let P be a closed Lagrangian

submanifold of M. Then there exists an open neighborhood U of the zero section in T — P

and a smooth map φ: U ’ M satisfying φ(0p ) = p which is a di¬eomorphism onto an open

neighborhood of P in M, and which pulls back „¦ to be the standard symplectic structure

on U.

Proof: From the earlier proofs, the reader probably can guess what we will do. Let

ι: P ’ M be the inclusion mapping and let ζ: P ’ T — P be the zero section of T — P . I

leave as an exercise for the reader to show that ζ — T (T — P ) = T P • T — P , and that the

induced symplectic structure Υ on this sum is simply the natural one on the sum of a

bundle and its dual:

Υ (v1 , ξ1 ), (v2 , ξ2 ) = ξ1 (v2 ) ’ ξ2 (v1 )

L.6.8 107

I will show that there is a bundle isomorphism „ : T P • T — P ’ ι— (T M) which restricts

to the subbundle T P to be ι : T P ’ ι— (T M).

First, select an n-dimensional subbundle L ‚ ι— (T M) which is complementary to

ι (T P ) ‚ ι— (T M). It is not di¬cult to show (and it is left as an exercise for the reader)

that it is possible to choose L so that it is a Lagrangian subbundle of ι— (T M) so that there

is an isomorphism ±: T — P ’ L so that „ : T P • T — P ’ ι (T P ) • L de¬ned by „ = ι • ±

is a symplectic bundle isomorphism.

Now apply the Corollary to Theorem 2.

Proposition 4 shows that the symplectic structure on a manifold M in a neighborhood

of a closed Lagrangian submanifold P is completely determined by the di¬eomorphism type

of P . This fact has several interesting applications. We will only give one of them here.

Proposition 5: Let (M, „¦) be a compact symplectic manifold with HdR (M, R) = 0.

1

Then in Diff(M) endowed with the C 1 topology, there exists an open neighborhood U of

the identity map so that any symplectomorphism φ: M ’ M which lies in U has at least

two ¬xed points.

Proof: Consider the manifold M — M endowed with the symplectic structure „¦ • (’„¦).

The diagonal ∆ ‚ M — M is a Lagrangian submanifold. Proposition 4 implies that

there exists an open neighborhood U of the zero section in T — M and a symplectic map

ψ: U ’ M — M which is a di¬eomorphism onto its image so that ψ(0p ) = (p, p).

Now, there is an open neighborhood U0 of the identity map on M in Diff(M) endowed

with the C 0 topology which is characterized by the condition that φ belongs to U0 if and

only if the graph of φ in M — M, namely id — φ lies in the open set ψ(U) ‚ M — M.

Moreover, there is an open neighborhood U ‚ U0 of the identity map on M in Diff(M)

endowed with the C 1 topology which is characterized by the condition that φ belongs to

U if and only if ψ ’1 —¦ (id — φ): M ’ T — M is the graph of a 1-form ±φ .

Now suppose that φ ∈ U is a symplectomorphism. By our previous discussion, it

follows that the graph of φ in M — M is Lagrangian. This implies that the graph of ±φ

is Lagrangian in T — M which, by our second example, implies that ±φ is closed. Since

HdR (M, R) = 0, this, in turn, implies that ±φ = dfφ for some smooth function f on M.

1

Since M is compact, it follows that fφ must have at least two critical points. However,

these critical points are zeros of the 1-form dfφ = ±φ . It is a consequence of our construction

that these points must then be places where the graph of φ intersects the diagonal ∆. In

other words, they are ¬xed points of φ.

This theorem can be generalized considerably. According to a theorem of Hamilton

[Ha], if M is compact, then there is an open neighborhood U of the identity map id in

Sp(„¦) (with the C 1 topology) so that every φ ∈ U is the time-one ¬‚ow of a symplectic

vector ¬eld Xφ ∈ sp(„¦). If Xφ is actually Hamiltonian (which would, of course, follow if

HdR (M, R) = 0), then ’Xφ „¦ = dfφ , so Xφ will vanish at the critical points of fφ and

1

these will be ¬xed points of φ.

L.6.9 108

Appendix: Lie™s Transformation Groups, II

The reader who is learning symplectic geometry for the ¬rst time may be astonished by

the richness of the subject and, at the same time, be wondering “Are there other geometries

like symplectic geometry which remain to be explored?” The point of this appendix is to

give one possible answer to this very vague question.

When Lie began his study of transformation groups in n variables, he modeled his

attack on the known study of the ¬nite groups. Thus, his idea was that he would ¬nd all

of the “simple groups” ¬rst and then assemble them (by solving the extension problem)

to classify the general group. Thus, if one “group” G had a homomorphism onto another

“group” H

1 ’’ K ’’ G ’’ H ’’ 1

then one could regard G as a semi-direct product of H with the kernel subgroup K.

Guided by this idea, Lie decided that the ¬rst task was to classify the transitive

transformation groups G, i.e., the ones which acted transitively on Rn (at least locally).

The reason for this was that, if G had an orbit S of dimension 0 < k < n, then the

restriction of the action of G to S would give a non-trivial homomorphism of G into a

transformation group in fewer variables.

Second, Lie decided that he needed to classify ¬rst the “groups” which, in his language,

“did not preserve any subset of the variables.” The example he had in mind was the group

of di¬eomorphisms of R2 of the form

φ(x, y) = f(x), g(x, y) .

Clearly the assignment φ ’ f provides a homomorphism of this group into the group of

di¬eomorphisms in one variable. Lie called groups which “did not preserve any subset of

the variables” primitive. In modern language, primitive is taken to mean that G does not

preserve any foliation on Rn (coordinates on the leaf space would furnish a “proper subset