symplectic manifold

Definition

A symplectic manifold is a pair where is a smooth manifold and is a differential 2-form () such that

1. is non-degenerate (pointwise). i.e. for every , is non-degenerate as a bilinear form on .
2. is closed (i.e. ).

Isomorphism

An isomorphism in the category of symplectic manifolds is a symplectomorphism.

Examples

• Let with coordinates . Then the standard symplectic form

is symplectic. (Note for my sanity that ).

• Let with (linear) coordinates . The form

is symplectic. If we think of this as a 2n-dimensional real manifold with the identification that (i.e. ) then this form is the same as the standard symplectic form above.

• Let . We may take the volume form

This is symplectic. It is closed since it is top degree, and non-degenerate by a volume argument.

• The cotangent bundle of a smooth manifold is always canonically a symplectic manifold using the tautological 1-form.

Cohomology of symplectic manifolds

Given a 2n-dimensional symplectic manifold , is always canonically orientable using the symplectic volume form which is nonvanishing.

We can also see that if is compact then the de Rham cohomology class using Stokes theorem.

Proof: Assume is exact, that is i.e. for some . Stokes theorem says that

This is a contradiction as is nonvanishing so the integral cannot be 0.

Thus, so . So by the de Rham theorem, .

This means that the only sphere with a symplectic structure is , all other spheres () have .