On the existence of a Hofer type metric for Poisson manifolds

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.