Metrics on symplectic groupoids and Killing-Poisson structures.

The problem consist o in the study of compatibility conditions between the Riemannian metric of an integrable smooth manifold M endowed with a Killing¿Poisson structure and possible metrics on the symplectic groupoid that integrates, such that different geometric structures are preserved through the fibration. In particular, we intend to study the behavior of the symplectic form along the g-gradient flows of Casimir functions, the relation between the corresponding Levi¿Civita connections on the groupoid and the base manifold and their relevance to the choice of possible symplectic connections associated to the symplectic form.