A Study of Generalized Projective  P − Curvature Tensor on Warped Product Manifolds

Uday Chand De,Suliman Alsaeed,Abdallah Abdelhameed Syied,Nasser Bin Turki,Antonio Masiello

doi:10.1155/2021/7882356

Abstract

The main aim of this study is to investigate the effects of the P − curvature flatness, P − divergence-free characteristic, and P − symmetry of a warped product manifold on its base and fiber (factor) manifolds. It is proved that the base and the fiber manifolds of the P − curvature flat warped manifold are Einstein manifold. Besides that, the forms of the P − curvature tensor on the base and the fiber manifolds are obtained. The warped product manifold with P − divergence-free characteristic is investigated, and amongst many results, it is proved that the factor manifolds are of constant scalar curvature. Finally, P − symmetric warped product manifold is considered.

Highlights

Curvature tensors play a significant role in mathematics and physics. is is why many researchers have introduced and studied many curvature tensors in various ways, as well as they have shown the importance of these curvature tensors
It is proved that pseudo-Riemannian manifolds M will be Einstein manifold if M admits a traceless P−curvature tensor and will be of constant scalar curvature if M is of P−curvature flat
We prove that the base and the fiber manifolds of the P−curvature flat warped product manifold are Einstein manifold; in this case, the forms of the P−curvature tensor on the base and the fiber manifolds are obtained

Summary

Research Article

A Study of Generalized Projective P − Curvature Tensor on Warped Product Manifolds. Uday Chand De, Abdallah Abdelhameed Syied ,2 Nasser Bin Turki, and Suliman Alsaeed. E main aim of this study is to investigate the effects of the P−curvature flatness, P−divergence-free characteristic, and P−symmetry of a warped product manifold on its base and fiber (factor) manifolds. It is proved that the base and the fiber manifolds of the P−curvature flat warped manifold are Einstein manifold. The forms of the P−curvature tensor on the base and the fiber manifolds are obtained. E warped product manifold with P−divergence-free characteristic is investigated, and amongst many results, it is proved that the factor manifolds are of constant scalar curvature.

Introduction

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