Geometry of $\mathcal{P}\mathcal{R}$-semi-invariant warped product submanifolds in paracosymplectic manifold

TL;DR

This paper investigates the geometric structure of $\

Contribution

It establishes integrability of distributions and characterizes $\

Findings

Distributions in $\

Shape operator conditions for warped product structure

Abstract

The purpose of this paper is to study $\mathcal{P}\mathcal{R}$-semi-invariant warped product submanifolds of a paracosymplectic manifold $\widetilde{M}$. We prove that the distributions associated with the definition of $\mathcal{P}\mathcal{R}$-semi-invariant warped product submanifold $M$ are always integrable. A necessary and sufficient condition for an isometrically immersed $\mathcal{P}\mathcal{R}$-semi-invariant submanifold of $\widetilde{M}$ to be a $\mathcal{P}\mathcal{R}$-semi-invariant warped product submanifold is obtained in terms of the shape operator.