The Normal Form Theorem around Poisson Transversals

TL;DR

This paper establishes a canonical normal form theorem for Poisson structures around Poisson transversals, generalizing key symplectic and Poisson splitting theorems and including an equivariant version.

Contribution

It introduces a unified normal form theorem for Poisson structures around transversals, extending classical results and providing an essentially canonical approach.

Findings

Proves a normal form theorem for Poisson structures around transversals.

Generalizes Weinstein's symplectic neighborhood and splitting theorems.

Provides an equivariant version of the splitting theorem.

Abstract

We prove a normal form theorem for Poisson structures around Poisson transversals (also called cosymplectic submanifolds), which simultaneously generalizes Weinstein's symplectic neighborhood theorem from symplectic geometry and Weinstein's splitting theorem. Our approach turns out to be essentially canonical, and as a byproduct, we obtain an equivariant version of the latter theorem.