Groupoïde d'homotopie d'un feuilletage Riemannien et réalisation symplectique de certaines variétés de Poisson. (Homotopy groupoid of a Riemannian foliation and symplectic realization of certain Poisson manifolds) (Q921677)

The Hamiltonian vector fields on a Poisson manifold generate a foliation in the sense of Sussmann and Stefan. If it is of dimension 2 and Riemannian, then under some further assumptions the Poisson manifold is shown to admit a symplectic realization in the following sense: there is a symplectic Lie groupoid (so that the graph of the multiplication is a Lagrange submanifold) such that the given Poisson manifold is the space of units and the source and target mappings are Poisson morphisms. On the way to this result it is shown that the homotopy groupoid of a Riemannian foliation is locally trivial.