symplectic manifold

A symplectic manifold is a pair $(M,\omega )$ where $M$ is a smooth manifold and $\omega$ is a differential 2-form ($\omega \in {\Omega }_M^2$) such that

1. $\omega$ is non-degenerate (pointwise). i.e. for every $p\in M$, $\omega (p)\in {\bigwedge }^2(T_p^{\ast }M)$ is non-degenerate as a bilinear form on $T_{p}M$.
2. $\omega$ is closed (i.e. $d\omega =0$).

Morphisms

A morphism in the category of symplectic manifolds is a symplectomorphism.