Riemann–Poisson manifolds and Ka¨hler–Riemann foliations

Riemann–Poisson manifolds were introduced by the author in C. R. Acad. Sci. Paris, Ser. I 333 (2001) 763–768, and studied in detail in preprint math.DG/0206102. Ka¨hler–Riemann foliations form an interesting subset of the Riemannian foliations with remarkable properties (see Ann. Global Anal. Geom. 21 (2002) 377–399). In this Note we will show that to give a regular Riemann–Poisson structure on a manifold <f>P</f> is equivalent to give a Ka¨hler–Riemann foliation on <f>P</f> such that the leafwise symplectic form is invariant with respect to all local foliation-preserving perpendicular vector fields. Finally, we give some examples of such manifolds. To cite this article: M. Boucetta, C. R. Acad. Sci. Paris, Ser. I 336 (2003). [Copyright &y& Elsevier]