On the existence of a Hofer type metric for Poisson manifolds

TL;DR

This paper investigates the conditions under which a Hofer-type metric can be defined on the Hamiltonian group of a Poisson manifold, establishing non-degeneracy in certain cases and proving its existence for integrable Poisson manifolds with Hausdorff integrations.

Contribution

It introduces a Hofer-type metric for Poisson manifolds, proves its non-degeneracy under specific conditions, and shows its existence for integrable Poisson manifolds with Hausdorff symplectic groupoids.

Findings

The Hofer-type metric is genuine when the union of all closed leaves is dense.

The metric is well-defined and non-degenerate on certain Poisson manifolds.

Existence of the Hofer-type metric is proven for integrable Poisson manifolds with Hausdorff integrations.

Abstract

An analogue of the Hofer metric $\varrho_H$ on the Hamiltonian group $Ham(M,\Lambda)$ of a Poisson manifold $(M,\Lambda)$ can be defined but there is the problem of its non-degeneracy. First we observe that $\varrho_H$ is a genuine metric on $Ham(M,\Lambda)$ when the union of all closed leaves (as subsets of $M$) of the corresponding symplectic foliation is dense. Next we deal with the important class of integrable Poisson manifolds. Recall that a Poisson manifold is called integrable if it can be realized as the space of units of a symplectic groupoid. Our main result states that $\varrho_H$ is a Hofer type metric for every Poisson manifold which admits a Hausdorff integration.