The Poisson saturation of coregular submanifolds

TL;DR

This paper investigates the local structure of coregular submanifolds in Poisson geometry, providing normal forms and uniqueness results that extend to Dirac geometry, enhancing understanding of Poisson and Dirac manifold embeddings.

Contribution

It introduces a normal form for the Poisson saturation of coregular submanifolds and extends results to Dirac geometry, offering new insights into their local structure and embeddings.

Findings

The Poisson saturation of coregular submanifolds is an embedded Poisson submanifold.

A normal form for the Poisson saturation around coregular submanifolds is established.

A uniqueness result for coisotropic embeddings of Dirac manifolds is proved.

Abstract

This paper is devoted to coregular submanifolds in Poisson geometry. We show that their local Poisson saturation is an embedded Poisson submanifold, and we give a normal form for this Poisson submanifold around the coregular submanifold. This result recovers the normal form around Poisson transversals, and it yields Poisson versions of some normal form/rigidity results around constant rank submanifolds in symplectic geometry. As an application, we prove a uniqueness result concerning coisotropic embeddings of Dirac manifolds in Poisson manifolds. We also show how our results generalize to the setting of coregular submanifolds in Dirac geometry.