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„¦ = „¦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

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