On the existence of a Hofer type metric for Poisson manifolds

An analogue of the Hofer metric rho H on the Hamiltonian group Ham (M, Lambda) of a Poisson manifold (M, Lambda) can be defined, but there is the problem of its nondegeneracy. First, we observe that rho H is a genuine metric on Ham (M, Lambda), when the union of all proper leaves 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 rho H is a Hofer type metric for every Poisson manifold, which admits a Hausdorff integration.