1Symplectic manifolds

III Symplectic Geometry



1.2 Symplectic manifolds
We are now ready to define symplectic manifolds.
Definition
(Symplectic manifold)
.
A symplectic manifold is a manifold
M
of dimension 2
n
equipped with a 2-form
ω
that is closed (i.e. d
ω
= 0) and
non-degenerate (i.e. ω
n
6= 0). We call ω the symplectic form.
The closedness condition is somewhat mysterious. There are some motivations
from classical mechanics, for examples, but they are not very compelling, and
the best explanation for the condition is “these manifolds happen to be quite
interesting”.
Observe that by assumption, the form
ω
n
is nowhere-vanishing, and so it is
a volume form. If our manifold is compact, then pairing the volume form with
the fundamental class (i.e. integrating it over the manifold) gives the volume,
and in particular is non-zero. Thus,
ω
n
6
= 0
H
2n
dR
(
M
). Since the wedge is
well-defined on cohomology, an immediate consequence of this is that
Proposition.
If a compact manifold
M
2n
is such that
H
2k
dR
(
M
) = 0 for some
k < n, then M does not admit a symplectic structure.
A more down-to-earth proof of this result would be that if
ω
2k
= d
α
for some
α, then
Z
ω
n
=
Z
ω
2k
ω
2(nk)
=
Z
d(α ω
2(nk)
) = 0
by Stokes’ theorem.
Example. S
n
does not admit a symplectic structure unless
n
= 2 (or
n
= 0, if
one insists).
On the other hand, S
2
is a symplectic manifold.
Example.
Take
M
=
S
2
. Take the standard embedding in
R
3
, and for
p S
2
,
the tangent space consists of the vectors perpendicular to
p
. We can take the
symplectic form to be
ω
p
(u, v) = p ·(u × v).
Anti-symmetry and non-degeneracy is IA Vectors and Matrices.
Definition
(Symplectomorphism)
.
Let (
X
1
, ω
1
) and (
X
2
, ω
2
) be symplectic
manifolds. A symplectomorphism is a diffeomorphism
f
:
X
1
X
2
such that
f
ω
2
= ω
1
.
If we have a single fixed manifold, and two symplectic structures on it, there
are other notions of “sameness” we might consider:
Definition
(Strongly isotopic)
.
Two symplectic structures on
M
are strongly
isotopic if there is an isotopy taking one to the other.
Definition
(Deformation equivalent)
.
Two symplectic structures
ω
0
, ω
1
on
M
are deformation equivalent if there is a family of symplectic forms
ω
t
that start
and end at ω
0
and ω
1
respectively.
Definition
(Isotopic)
.
Two symplectic structures
ω
0
, ω
1
on
M
are isotopic
if there is a family of symplectic forms
ω
t
that start and end at
ω
0
and
ω
1
respectively, and further the cohomology class [ω
t
] is independent of t.
A priori, it is clear that we have the following implications:
symplectomorphic strongly isotopic isotopic deformation equivalent.
It turns out when M is compact, isotopic implies strongly isotopic:
Theorem
(Moser)
.
If
M
is compact with a family
ω
t
of symplectic forms with
[ω
t
] constant, then there is an isotopy ρ
t
: M M with ρ
t
ω
t
= ω
0
.
The key idea is to express the condition
ρ
t
ω
t
=
ω
0
as a differential equation
for
v
t
=
d
dt
ρ
t
, and ODE theory guarantees the existence of a
v
t
satisfying the
equation. Then compactness allows us to integrate it up to get a genuine ρ
t
.
Proof.
We set
ρ
0
to be the identity. Then the equation
ρ
t
ω
t
=
ω
0
is equivalent
to ρ
t
ω
t
being constant. If v
t
is the associated vector field to ρ
t
, then we need
0 =
d
dt
(ρ
t
ω
t
) = ρ
t
L
v
t
ω
t
+
dω
t
dt
.
So we want to solve
L
v
t
ω
t
+
dω
t
dt
= 0.
To solve for this, since [
dω
t
dt
] = 0, it follows that there is a family
µ
t
of 1-forms
such that
dω
t
dt
= dµ
t
. Then our equation becomes
L
v
t
ω
t
+ dµ
t
= 0.
By assumption, dω
t
= 0. So by Cartan’s magic formula, we get
dι
v
t
ω
t
+ dµ
t
= 0.
To solve this equation, it suffices to solve Moser’s equation,
ι
v
t
ω
t
+ µ
t
= 0,
which can be solved since ω
t
is non-degenerate.
There is also a relative version of this.
Theorem
(Relative Moser)
.
Let
X M
be a compact manifold of a manifold
M
,
and
ω
0
, ω
1
symplectic forms on
M
agreeing on
X
. Then there are neighbourhoods
U
0
, U
1
of X and a diffeomorphism ϕ : U
0
U
1
fixing X such that ϕ
ω
1
= ω
0
.
Proof. We set
ω
t
= (1 t)ω
0
+
1
.
Then this is certainly closed, and since this is constantly
ω
0
on
X
, it is non-
degenerate on
X
, hence, by compactness, non-degenerate on a small tubular
neighbourhood U
0
of X (by compactness). Now
d
dt
ω
t
= ω
1
ω
0
,
and we know this vanishes on
X
. Since the inclusion of
X
into a tubular
neighbourhood is a homotopy equivalence, we know [
ω
1
ω
0
] = 0
H
1
dR
(
U
0
).
Thus, we can find some
µ
such that
ω
1
ω
0
= d
µ
, and by translation by a
constant, we may suppose
µ
vanishes on
X
. We then solve Moser’s equation,
and the resulting ρ will be constant on X since µ vanishes.
We previously had the standard form theorem for symplectic vector spaces,
which is not surprising — we have the same for Riemannian metrics, for example.
Perhaps more surprisingly, give a symplectic manifold, we can always pick
coordinate charts where the symplectic form looks like the standard symplectic
form throughout:
Theorem
(Darboux theorem)
.
If (
M, ω
) is a symplectic manifold, and
p M
,
then there is a chart (U, x
1
, . . . , x
n
, y
1
, . . . , y
n
) about p on which
ω =
X
dx
i
dy
i
.
Proof. ω
can certainly be written in this form at
p
. Then relative Moser with
X = {p} promotes this to hold in a neighbourhood of p.
Thus, symplectic geometry is a global business, unlike Riemannian geometry,
where it is interesting to talk about local properties such as curvature.
A canonical example of a symplectic manifold is the cotangent bundle of a
manifold
X
. In physics,
X
is called the configuration space, and
M
=
T
X
is
called the phase space.
On a coordinate chart (
U, x
1
, . . . , x
n
) for
X
, a generic 1-form can be written
as
ξ =
n
X
i=1
ξ
i
dx
i
.
Thus, we have a coordinate chart on
T
X
given by (
T
U, x
1
, . . . , x
n
, ξ
1
, . . . , ξ
n
).
On T
U, there is the tautological 1-form
α =
n
X
i=1
ξ
i
dx
i
.
Observe that this is independent of the chart chosen, since it can be characterized
in the following coordinate-independent way:
Proposition.
Let
π
:
M
=
T
X X
be the projection map, and
π
:
T
X
T
M the pullback map. Then for ξ M, we have α
ξ
= π
ξ.
Proof. On a chart, we have
π
ξ
x
j
= ξ
x
j
= ξ
j
= α
ξ
x
j
,
and similarly both vanish on
ξ
j
.
This tautological 1-form is constructed so that the following holds:
Proposition.
Let
µ
be a one-form of
X
, i.e. a section
s
µ
:
X T
X
. Then
s
µ
α = µ.
Proof. By definition,
α
ξ
= ξ dπ.
So for x X, we have
s
µ
α
µ(x)
= µ(x) dπ ds
µ
= µ(x) d(π s
µ
) = µ(x) d(id
X
) = µ(x).
Given this, we can set
ω = dα =
n
X
i=1
dx
i
dξ
i
,
the canonical symplectic form on
T
X
. It is clear that it is anti-symmetric,
closed, and non-degenerate, because it looks just like the canonical symplectic
form on R
2n
.
Example.
Take
X
=
S
1
, parametrized by
θ
, and
T
X
=
S
1
×R
with coordinates
(θ, ξ
θ
). Then
ω = dθ dξ
θ
.
We can see explicitly how this is independent of coordinate. Suppose we
parametrized the circle by
τ
= 2
θ
. Then d
τ
= 2d
θ
. However, since we de-
fined ξ
θ
by
ξ = ξ
θ
(ξ) dθ
for all ξ T
S
1
, we know that ξ
τ
=
1
2
ξ
θ
, and hence
dθ dξ
θ
= dτ dξ
τ
.
Of course, if we have a diffeomorphism
f
:
X Y
, then the pullback map
induces a symplectomorphism
f
:
T
Y T
X
. However, not all symplecto-
morphisms arise this way.
Example.
If
X
=
S
1
and
T
X
=
S
1
× R
is given the canonical symplectic
structure, then the vertical translation
g
(
θ, ξ
) = (
θ, ξ
+
c
) is a symplectomorphism
that is not a lift of a diffeomorphism S
1
S
1
.
So when does a symplectomorphism come from a diffeomorphism?
Exercise.
A symplectomorphism
y
:
T
X T
X
is a lift of a diffeomorphism
X X iff g
α = α.