Abstract
This article generalizes some geometric structures on warped product manifolds equipped with a Poisson structure to doubly warped products of pseudo-Riemannian manifolds equipped with a doubly warped Poisson structure. First, we introduce the notion of Poisson doubly warped product manifold (B-f x(b) F, Pi= mu(v)Pi(h)(B) + nu(h)Pi(v)(F), g) and express the Levi-Civita contravariant connection, curvature and metacurvature of ((f) B x(b) F, Pi, g) in terms of Levi-Civita connections, curvatures and metacurvatures of components (B, Pi(B), g(B)) and (F, Pi(F), g(F)). We also study compatibility conditions related to the Poisson structure Pi and the contravariant metric g on B-f x(b) F, so that the compatibility conditions on (B, Pi(B), g(B)) and (F, Pi(F), g(F)) remain consistent in the Poisson doubly warped product manifold (B-f x(b )F, Pi, g).