Warped Poisson Brackets on Warped Products

TL;DR

This paper extends the concept of Poisson structures to warped product pseudo-Riemannian manifolds, introducing a new warped bivector field and analyzing its geometric properties including connections and curvature.

Contribution

It generalizes the product Poisson structure to warped products and constructs a new warped bivector field with associated geometric analysis.

Findings

Construction of three bivector fields on product manifolds.

Identification of conditions for these bivectors to be Poisson structures.

Calculation of contravariant Levi-Civita connection and curvature for warped bivector fields.

Abstract

In this paper, we generalize the geometry of the product pseudo-Riemannian manifold equipped with the product Poisson structure (\cite{Nas2}) to the geometry of a warped product of pseudo-Riemannian manifolds equipped with a warped Poisson structure. We construct three bivector fields on a product manifold and show that each of them lead under certain conditions to a Poisson structure. One of these bivector fields will be called the warped bivector field. For a warped product of pseudo-Riemannian manifolds equipped with a warped bivector field, we compute the corresponding contravariant Levi-Civita connection and the curvatures associated with.