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.