Now I’ll actually give the proof of the Darboux theorem that a symplectic manifold is locally symplectomorphic to \mathbb{R}^{2n} with the usual form.

Proof of the Darboux theorem

We will prove the equivalent:

Theorem 1 Let {M} be a manifold with closed symplectic forms {\omega_0, \omega_1}, and {p \in M} with {\omega_0(p) = \omega_1(p)}. Then there are neighborhoods {U,V} of {p} and a diffeomorphism {f: U \rightarrow V} with {f^*\omega_1 = \omega}.


The idea is to consider the continuously varying family of 2-forms

\displaystyle \omega_t = (1-t) \omega_0 + t \omega_1 = \omega_0 + t \alpha


\displaystyle \alpha = \omega_1 - \omega_0 .

We will consider a small neighborhood {U} of {p} and a smooth map { G: U \times [0,1] \rightarrow M } such that {G_t:=G(\cdot, t)} is a diffeomorphism, {G_0 = id}, and

\displaystyle G_t^*(\omega_t) = \omega_0 \ (*). (more…)