Statement
Let be a Poisson manifold, and let . There exists coordinates centered at such that
where and are smooth functions of such that .
Proof
Notes
This means that we can shrink the open set small enough so that the local structure of the Poisson manifold under the Weinstein splitting chart becomes
Thus, the charts give local Poisson isomorphisms
Where is the canonical Poisson bracket on with coordinates which is given by
and
The important thing to notice is that this Poisson structure vanishes at the point .
The first chart on gives a result that is very close to the standard symplectic form that comes from Darboux charts on a symplectic manifold.
The second structure that is defined on shows that we can “squish” the manifold to a place where the “rest” of the bivector doesn’t do anything at the point where it is centered at. Thus, we can safely make a submanifold, since the Poisson structure vanishes, leaving only the canonical structure.
Reference
@weinstein1983 @crainic2021 - Chapter 3