On the geometry of compatible Poisson and Riemannian structures

We consider compatibility conditions between Poisson and Riemannian structures on smooth manifolds by means of a contravariant Levi-Civita connection. These include Riemann–Poisson structures (as defined by M. Boucetta), and the class of almost Kähler–Poisson manifolds, introduced with the aid of a contravariant f -structure, that will be called partially co-complex structure , in analogy with complex ones on Kähler manifolds. Additionally, we study the geometry of the symplectic foliation, and the behavior of these compatibilities under structure preserving maps and symmetries.