On warped product manifolds satisfying some pseudosymmetric type conditions

TL;DR

This paper characterizes warped product manifolds satisfying specific pseudosymmetric conditions related to the projective curvature tensor, providing necessary and sufficient conditions and examples for various pseudosymmetry types.

Contribution

It introduces a unified framework for characterizing warped product manifolds under pseudosymmetric conditions involving the projective curvature tensor, including new theorems and examples.

Findings

Characterization theorems for warped product manifolds under pseudosymmetric conditions

Necessary and sufficient conditions for various pseudosymmetry types

Examples illustrating the theoretical results

Abstract

The object of the present paper is to study the characterization of warped product manifolds satisfying some pseudosymmetric type conditions, especially, due to projective curvature tensor. For this purpose we consider a warped product manifold satisfying the pseudosymmetric type condition $R\cdot R = L_1 Q(g,R) + L_2 Q(S,R)$ and evaluate its characterization theorem. As special cases of $L_1$ and $L_2$ we find out the necessary and sufficient condition for a warped product manifold to satisfy various pseudosymmetric type, such as pseudosymmetry, Ricci generalized pseudosymmetry, semisymmetry due to projective curvature tensor ($P\cdot R = 0$), pseudosymmetry due to projective curvature tensor ($P\cdot R = L Q(g,R)$) etc. Finally we present some suitable examples of warped product manifolds satisfying such pseudosymmetric type conditions.