Riemann Poisson Manifolds and Kähler-riemann Foliations

1 Riemann Poisson manifolds were introduced by the author in [1] and studied in more details in [2]. Kähler-Riemann foliations form an interesting subset of the Riemannian foliations with remarkable properties ( see [3]). In this paper we will show that to give a regular Riemann Poisson structure on a manifold M is equivalent to to give a Kähler-Riemann foliation on M such that the leafwise symplectic form is invariant with respect to all local foliate perpendicular vector fields. We show also that the sum of the vector space of leafwise cohomology and the vector space of the basic forms is a subspace of the space of Poisson cohomology.