Einstein Poisson warped product space

Abstract

Abstract In this paper, we provide some results on Poisson manifold (M, Π) with contravariant Levi–Civita connection D 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 J 1 2 = I , then M is simply a Riemannian Poisson product. For a contravariant Lorentzian Poisson warped space (M = B × f F, g, Π) (where B = I × R ) 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 G F i j = − Λ F g F i j with cosmological constant Λ F on fiber space (F, g F , Π F ).