symplectic volume

Definition

For a symplectic manifold , the form is called the symplectic volume form (or the Liouville volume) of .

Non-vanishing

Given a -dimensional vector space with a symplectic form , the n’th exterior product is non-zero.

Proof

Note that there is a basis for such that the symplectic form can be written as

where are the dual basis vectors for the above basis.

Then we see that

Then we can continue this, wedging more, but each time multiplying by the choices to get

Importance

This gives that is a volume form on , and thus that every symplectic manifold is orientable.