1Symplectic manifolds

III Symplectic Geometry



1.5 Lagrangian submanifolds and fixed points
Recall what we have got so far. Let (
M, ω
) be a symplectic manifold,
ϕ
:
M M
.
Then the graph of
ϕ
is a subset of
M ×M
. Again let
˜ω
be the twisted product form
on
M ×M
. Then we saw that a morphism
ϕ
:
M M
is a symplectomorphism
iff the graph of
ϕ
is Lagrangian in (
M ×M, ˜ω
). Moreover, the set of fixed points
is exactly graph ϕ, where ∆ = graph id
M
is the diagonal.
If
ϕ
is “close to” the identity, map, then its graph is close to ∆. Thus, we are
naturally led to understanding the neighbourhoods of the identity. An important
theorem is the following symplectic version of the tubular neighbourhood theorem:
Theorem
(Lagrangian neighbourhood theorem)
.
Let (
M, ω
) be a symplectic
manifold,
X
a compact Lagrangian submanifold, and
ω
0
the canonical symplectic
form on
T
X
. Then there exists neighbourhoods
U
0
of
X
in
T
X
and
U
of
X
in M and a symplectomorphism ϕ : U
0
U sending X to X.
An equivalent theorem is the following:
Theorem
(Weinstein)
.
Let
M
be a 2
n
-dimensional manifold,
X n
-dimensional
compact submanifold, and
i
:
X M
the inclusion, and symplectic forms
ω
0
, ω
1
on
M
such that
i
ω
0
=
i
ω
1
= 0, i.e.
X
is Lagrangian with respect to both
symplectic structures. Then there exists neighbourhoods
U
0
, U
1
of
X
in
M
such
that ρ|
X
= id
X
and ρ
ω
1
= ω
0
.
We first prove these are equivalent. This amounts to identifying the (dual of
the) cotangent bundle of X with the normal bundle of X.
Proof of equivalence.
If (
V,
Ω) is a symplectic vector space,
L
a Lagrangian
subspace, the bilinear form
Ω : V/L × L R
([v], u) Ω(v, u).
is non-degenerate and gives a natural isomorphism
V/L
=
L
. Taking
V
=
T
p
M
and L = T
p
X, we get an isomorphism
NX = T M|
X
/T X
=
T
X.
Thus, by the standard tubular neighbourhood theorem, there is a neighbourhood
N
0
of
X
in
NX
and a neighbourhood
N
of
X
in
M
, and a diffeomorphism
ψ
:
N
0
N
. We now have two symplectic forms on
N
0
the one from the
cotangent bundle and the pullback of that from
M
. Then applying the second
theorem gives the first.
Conversely, if we know the first theorem, then applying this twice gives us
symplectomorphisms between neighbourhoods of
X
in
M
under each symplectic
structure with a neighbourhood in the cotangent bundle.
It now remains to prove the second theorem. This is essentially an application
of the relative Moser theorem, which is where the magic happens. The bulk of
the proof is to get ourselves into a situation where relative Moser applies.
Proof of second theorem.
For
p X
, we define
V
=
T
p
M
and
U
=
T
p
X
, and
W
any complement of
U
. By assumption,
U
is a Lagrangian subspace of both
(
V, ω
0
|
p
=
0
) and (
V, ω
1
|
p
=
1
). We apply the following linear-algebraic
lemma:
Lemma.
Let
V
be a 2
n
-dimensional vector space,
0
,
1
symplectic structures
on
V
. Suppose
U
is a subspace of
V
Lagrangian with respect to both
0
and
1
, and
W
is any complement of
V
. Then we can construct canonically a linear
isomorphism H : V V such that H|
U
= id
U
and H
1
= Ω
2
.
Note that the statement of the theorem doesn’t mention
W
, but the con-
struction of
H
requires a complement of
V
, so it is canonical only after we pick
a W .
By this lemma, we get canonically an isomorphism
H
p
:
T
p
M T
p
M
such
that
H
p
|
T
p
X
=
id
T
p
X
and
H
p
ω
1
|
p
=
ω
0
|
p
. The canonicity implies
H
p
varies
smoothly with p. We now apply the Whitney extension theorem
Theorem
(Whitney extension theorem)
.
Let
X
be a submanifold of
M
,
H
p
:
T
p
M T
p
M
smooth family of isomorphisms such that
H
p
|
T
p
X
=
id
T
p
X
. Then
there exists an neighbourhood
N
of
X
in
M
and an embedding
h
:
N M
such that h|
X
= id
X
and for all p X, dh
p
= H
p
.
So at
p X
, we have
h
ω
1
|
p
= (d
h
p
)
ω
1
|
P
=
h
p
ω
1
|
p
=
ω
0
|
p
. So we are done
by relative Moser.
Example.
We can use this result to understand the neighbourhood of the iden-
tity in the group
Symp
(
M, ω
) of symplectomorphisms of a symplectic manifold
(M, ω).
Suppose
ϕ, id Symp
(
M, ω
). Then the graphs Γ
ϕ
and = Γ
id
are La-
grangian submanifolds of (
M × M, ˜ω
). Then by our theorem, there is a neigh-
bourhood
U
of in (
M × M, ˜ω
) that is symplectomorphic to a neighbourhood
U
0
of the zero section of (T
M, ω
0
).
Suppose
ϕ
is sufficiently
C
0
-close to
id
. Then Γ
ϕ
U
. If
ϕ
is sufficiently
C
1
-close to the identity, then the image of Γ
ϕ
in
U
0
is a smooth section
X
µ
for
some 1-form µ.
Now
ϕ
is a symplectomorphism iff Γ
ϕ
is Lagrangian iff
X
µ
is Lagrangian iff
µ
is closed. So a small
C
1
-neighbourhood of
id
in
Sym
(
M, ω
) is “the same as” a
small
C
1
-neighbourhood of the zero-section in the space of closed 1-forms on
X
.
We can also use this to understand fixed points of symplectomorphisms
(again!).
Theorem.
Let (
M, ω
) be a compact symplectic manifold such that
H
1
dR
(
M
) = 0.
Then any symplectomorphism
ϕ
:
M M
sufficiently close to the identity has
at least two fixed points.
Proof.
The graph of
ϕ
corresponds to a closed 1-form on
M
. Since
µ
is closed
and
H
1
dR
(
M
) = 0, we know
µ
= d
h
for some
h C
. Since
M
is compact,
h
has at least two critical points (the global maximum and minimum). Since the
fixed points corresponding to the points where
µ
vanish (so that Γ
ϕ
intersects
∆), we are done.
Counting fixed points of symplectomorphisms is a rather popular topic in
symplectic geometry, and Arnold made some conjectures about these. A version
of this is
Theorem
(Arnold conjecture)
.
Let (
M, ω
) be a compact symplectic manifold
of dimension 2
n
, and
ϕ
:
M M
a symplectomorphism. Suppose
ϕ
is exactly
homotopic to the identity and non-degenerate. Then the number of fixed points
of ϕ is at least
P
2n
i=0
dim H
i
(M, R).
We should think of the sum
P
2n
i=0
dim H
i
(
M, R
) as the minimal number of
critical points of a function, as Morse theory tells us.
We ought to define the words we used in the theorem:
Definition
(Exactly homotopic)
.
We say
ϕ
is exactly homotopic to the identity
if there is isotopy
ρ
t
:
M M
such that
ρ
0
=
id
and
ρ
1
=
ϕ
, and further there
is some 1-periodic family of functions
h
t
such that
ρ
t
is generated by the vector
field v
t
defined by ι
v
t
ω = dh
t
.
The condition
ι
v
t
ω
d
h
t
says
v
t
is a Hamiltonian vector field, which we will
discuss soon after this.
Definition
(Non-degenerate function)
.
A endomorphism
ϕ
:
M M
is non-
degenerate iff all its fixed points are non-degenerate, i.e. if
p
is a fixed point,
then det(id dϕ
p
) 6= 0.
Example.
In the original statement of the Arnold conjecture, which is the case
(T
2
, dθ
1
dθ
2
), any symplectomorphism has at least 4 fixed points.
In the case where
h
t
is actually not time dependent, Arnold’s conjecture is
easy to prove.