Poisson structure on manifolds with corners

TL;DR

This paper develops a framework for defining Poisson structures on manifolds with corners using sheaf theory, extending classical Poisson geometry to include boundary and corner cases.

Contribution

It introduces a sheaf-theoretic approach to Poisson structures on manifolds with corners, generalizing the classical smooth case.

Findings

Defines Poisson structures via sheaf morphisms on manifolds with corners.

Establishes conditions for Leibniz rule and Jacobi identity in this setting.

Provides a foundation for further study of Poisson geometry with boundary conditions.

Abstract

Since a Poisson Structure is a smooth bivector field, we use a ring-valued sheaf $\OO_{X}$ on a manifold with corners $X$, we can interpret $\OO_{X}(U)$ as the ring of admissible smooth functions where $U$ is an open subset on $X$, in this way, a poisson structure on $(X, \OO_{X})$ is a sheaf morphism $$ \{-, -\}: \OO_{X} \times \OO_{X} \longrightarrow \OO_{X} $$ which satisfies the Leibniz rule an also the Jacobi Identity.