Product of statistical manifolds with a non-diagonal metric

TL;DR

This paper extends dualistic structures to generalized warped product manifolds, deriving curvature expressions and establishing conditions under which these manifolds are statistical, thus broadening the understanding of their geometric properties.

Contribution

It introduces a generalization of dualistic structures on warped products and relates curvature and statistical properties to base and fiber manifolds.

Findings

Dualistic structures on base and fiber induce structures on the product

Curvature expressions relate to base, fiber, and warping functions

Conditions for the product manifold to be a statistical manifold

Abstract

In this paper, we generalize the dualistic structures on warped product manifolds to the dualistic structures on generalized warped product manifolds. we develop an expression of curvature for the connection of the generalized warped product in relation to those corresponding analogues of its base and fiber and warping functions. we show that the dualistic structures on the base $M_{_1}$ and the fiber $M_{_2}$ induces a dualistic structure on the generalized warped product $M_1\times M_2$ and conversely, moreover, $(M_{_1}\times M_{_1},G_{_{f_{_1}f_{_2}}})$ or $(M_{_1}\times M_{_1},\tilde{g}_{_{f_{_1}f_{_2}}})$ is statistical manifold if and only if $(M_{_1},g_{_1})$ and $(M_{_1},g_{_1})$ are. Finally, Some interesting consequences are also given.