III Symplectic Geometry - Symplectic manifolds

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

of dimension 2

equipped with a 2-form

that is closed (i.e. d

= 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

∧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,

∧n

= 0

∈ H

(

). Since the wedge is

well-defined on cohomology, an immediate consequence of this is that

Proposition.

If a compact manifold

is such that

(

) = 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

= d

for some

α, then

∧ ω

2(n−k)

d(α ∧ω

2(n−k)

) = 0

by Stokes’ theorem.

Example. S

does not admit a symplectic structure unless

= 2 (or

= 0, if

one insists).

On the other hand, S

is a symplectic manifold.

Example.

Take

. Take the standard embedding in

, and for

p ∈ S

the tangent space consists of the vectors perpendicular to

. We can take the

symplectic form to be

(u, v) = p ·(u × v).

Anti-symmetry and non-degeneracy is IA Vectors and Matrices.

Definition

(Symplectomorphism)

Let (

, ω

) and (

, ω

) be symplectic

manifolds. A symplectomorphism is a diffeomorphism

→ X

such that

∗

= ω

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

are strongly

isotopic if there is an isotopy taking one to the other.

Definition

(Deformation equivalent)

Two symplectic structures

, ω

are deformation equivalent if there is a family of symplectic forms

that start

and end at ω

and ω

respectively.

Definition

(Isotopic)

Two symplectic structures

, ω

are isotopic

if there is a family of symplectic forms

that start and end at

and

respectively, and further the cohomology class [ω

] 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)

is compact with a family

of symplectic forms with

[ω

] constant, then there is an isotopy ρ

: M → M with ρ

∗

= ω

The key idea is to express the condition

∗

as a differential equation

for

, and ODE theory guarantees the existence of a

satisfying the

equation. Then compactness allows us to integrate it up to get a genuine ρ

Proof.

We set

to be the identity. Then the equation

∗

is equivalent

to ρ

∗

being constant. If v

is the associated vector field to ρ

, then we need

0 =

(ρ

∗

) = ρ

∗



dω



So we want to solve

dω

= 0.

To solve for this, since [

dω

] = 0, it follows that there is a family

of 1-forms

such that

dω

= dµ

. Then our equation becomes

+ dµ

= 0.

By assumption, dω

= 0. So by Cartan’s magic formula, we get

dι

+ dµ

= 0.

To solve this equation, it suffices to solve Moser’s equation,

+ µ

= 0,

which can be solved since ω

is non-degenerate.

There is also a relative version of this.

Theorem

(Relative Moser)

Let

X ⊆ M

be a compact manifold of a manifold

and

, ω

symplectic forms on

agreeing on

. Then there are neighbourhoods

, U

of X and a diffeomorphism ϕ : U

→ U

fixing X such that ϕ

∗

= ω

Proof. We set

= (1 − t)ω

+ tω

Then this is certainly closed, and since this is constantly

, it is non-

degenerate on

, hence, by compactness, non-degenerate on a small tubular

neighbourhood U

of X (by compactness). Now

= ω

− ω

and we know this vanishes on

. Since the inclusion of

into a tubular

neighbourhood is a homotopy equivalence, we know [

− ω

] = 0

∈ H

(

Thus, we can find some

such that

− ω

= d

, and by translation by a

constant, we may suppose

vanishes on

. 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

, . . . , x

, y

, . . . , y

) about p on which

ω =

∧ dy

Proof. ω

can certainly be written in this form at

. 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

. In physics,

is called the configuration space, and

∗

called the phase space.

On a coordinate chart (

U, x

, . . . , x

) for

, a generic 1-form can be written

ξ =

i=1

Thus, we have a coordinate chart on

∗

given by (

∗

U, x

, . . . , x

, ξ

, . . . , ξ

On T

∗

U, there is the tautological 1-form

α =

i=1

Observe that this is independent of the chart chosen, since it can be characterized

in the following coordinate-independent way:

Proposition.

Let

∗

X → X

be the projection map, and

∗

X →

∗

M the pullback map. Then for ξ ∈ M, we have α

= π

∗

ξ.

Proof. On a chart, we have

∗



∂

∂x



= ξ



∂

∂x



= ξ

= α



∂

∂x



and similarly both vanish on

∂

∂ξ

This tautological 1-form is constructed so that the following holds:

Proposition.

Let

be a one-form of

, i.e. a section

X → T

∗

. Then

∗

α = µ.

Proof. By definition,

= ξ ◦dπ.

So for x ∈ X, we have

∗

µ(x)

= µ(x) ◦ dπ ◦ ds

= µ(x) ◦ d(π ◦ s

) = µ(x) ◦ d(id

) = µ(x).

Given this, we can set

ω = −dα =

i=1

∧ dξ

the canonical symplectic form on

∗

. It is clear that it is anti-symmetric,

closed, and non-degenerate, because it looks just like the canonical symplectic

form on R

Example.

Take

, parametrized by

, and

∗

×R

with coordinates

(θ, ξ

). Then

ω = dθ ∧ dξ

We can see explicitly how this is independent of coordinate. Suppose we

parametrized the circle by

= 2

. Then d

= 2d

. However, since we de-

fined ξ

ξ = ξ

(ξ) dθ

for all ξ ∈ T

∗

, we know that ξ

, and hence

dθ ∧ dξ

= dτ ∧ dξ

Of course, if we have a diffeomorphism

X → Y

, then the pullback map

induces a symplectomorphism

∗

Y → T

∗

. However, not all symplecto-

morphisms arise this way.

Example.

and

∗

× R

is given the canonical symplectic

structure, then the vertical translation

(

θ, ξ

) = (

θ, ξ

) is a symplectomorphism

that is not a lift of a diffeomorphism S

→ S

So when does a symplectomorphism come from a diffeomorphism?

Exercise.

A symplectomorphism

∗

X → T

∗

is a lift of a diffeomorphism

X → X iff g

∗

α = α.