Codimension one symplectic foliations and regular Poisson structures

TL;DR

This paper characterizes a class of compact corank one Poisson manifolds with specific symplectic foliations, showing they are mapping tori if they have a compact leaf, and discusses their extension as Poisson b-manifolds.

Contribution

It provides a complete characterization of certain compact corank one Poisson manifolds with symplectic foliations and explores their extension as Poisson b-manifolds.

Findings

If such a manifold has a compact leaf, then all leaves are compact.

The manifold is a mapping torus of a compact leaf.

These structures can be extended as the critical hypersurface of a Poisson b-manifold.

Abstract

In this short note we give a complete characterization of a certain class of compact corank one Poisson manifolds, those equipped with a closed one-form defining the symplectic foliation and a closed two-form extending the symplectic form on each leaf. If such a manifold has a compact leaf, then all the leaves are compact, and furthermore the manifold is a mapping torus of a compact leaf. These manifolds and their regular Poisson structures admit an extension as the critical hypersurface of a Poisson b-manifold as we consider in a later paper.