Regular Foliations and Poisson Structures on Orientable Manifolds

TL;DR

This paper investigates conditions under which regular Poisson structures exist on orientable manifolds with a given foliation, exploring obstructions and special classes of Poisson structures.

Contribution

It provides necessary and sufficient conditions for the existence of regular and unimodular Poisson structures aligned with a specified foliation.

Findings

Characterization of when regular Poisson structures exist

Obstructions to unimodular Poisson structures identified

Conditions for transversally constant Poisson structures derived

Abstract

On an orientable manifold M, we consider a regular even dimensional foliation F which is globally defined by a set of k-independent 1-forms. We give necessary and sufficient conditions for the existence of a regular Poisson structure on M whose Characteristic foliation is precisely F. Moreover, introducing a special class of the foliated 1-cohomology we describe obstructions for the existence of unimodular Poisson structures with a given characteristic foliation. In the same lines, we also give conditions for the existence of transversally constant Poisson structures.