Einstein Poisson warped product space

Abstract

In this paper, we provide some results on Poisson manifold (M, Π) with contravariant Levi–Civita connection associated to pair (Π, g). We introduce the notion of Einstein Poisson warped product space (M = B × f F, Π, g f ) (where Π = Π1 + Π2). Moreover, we show that if M is an Einstein Poisson warped product space with nonpositive scalar curvature and compact base B, J 1 is a field endomorphism on T*B satisfies , then M is simply a Riemannian Poisson product. For a contravariant Lorentzian Poisson warped space (M = B × f F, g, Π) (where ) one can determine contravariant Einstein equations and the cosmological constant Λ corresponding to the contravariant Einstein equation G = −Λg. Moreover, it is shown that Einstein equation G = −Λg, induces the contravariant Einstein equation with cosmological constant Λ F on fiber space (F, g F , Π F ).