D-homothetic deformation of an h-Einstein Para-Sasakian manifold

Indiwar Singh Chauhan

doi:10.17485/ijst/v13i13.198

Abstract

Objectives: The main purpose of this paper is to study the theory of Dhomothetic deformation of an h-Einstein Para-Sasakian manifold. Methods: A deformation technique is employed to solve the resulting problem. We provide its application in the general theory of relativity. Findings: Section 2 deals with recurrent and symmetric Para-Sasakian Manifolds. In section 3 we have defined and studied projectively symmetric Para-Sasakian manifold. The notion of ϕ -holomorphic sectional curvature in an h-Einstein Para-Sasakian manifold has been delineated in the section 4. In this very section, we shall study the relevant commutation formulae which give rise to the required curvature tensor. The obtained results were compared with the previous works(1--3) in this field and were found to be in good agreement. Novelty/Conclusion: The study concludes the following results: 1. If n-dimensional Para-Sasakian manifold is to be recurrent projective symmetric then the recurrent projective curvature tensor vanishes identically.2. If n-dimensional Para-Sasakian manifold is to be projective semi-symmetric then the semi-recurrent projective curvature tensor vanishes identically.3. If n-dimensional Para-Sasakian manifold is to be projective scalar-symmetric then the scalar projective curvature vanishes identically.4. If D-homothetic deformation of h-Einstein Para-Sasakian manifold is constant ϕ -holomorphic sectional curvature then we shall obtain the value of scalar fields. Keywords: ϕ -holomorphic sectional curvature; constant curvature; Para-Sasakian manifold; recurrent projective symmetric; D-homothetic deformation; h-Einstein manifold

Highlights

We consider an n-dimensional differentiable manifold Mn with a positive definite metric gα β which admits a unit covariant vector field ηα satisfying (1.1) ∇α ηβ − ∇β ηα = 0 and ( )(1.2) ∇α ∇β ηγ = − gαβ ηγ − gαγ ηβ + 2ηα ηβ ηγ, Singh Chauhan et al / Indian Journal of Science and Technology 2020;13(13):1435–1439 wherein ∇α denotes the operator of covariant differentiation with regard to gα β, such a space Mn is called Para-Sasakian manifold [4,5]
If D-homothetic deformation of η-Einstein Para-Sasakian manifold is constant φ -holomorphic sectional curvature we shall obtain the value of scalar fields
We have the following theorems: Theorem 2.1: The necessary and sufficient condition for the n-dimensional Para-Sasakian manifold to be recurrent symmetric manifold is that the recurrent curvature tensor vanishes

Summary

Introduction

We consider an n-dimensional differentiable manifold Mn with a positive definite metric gα β which admits a unit covariant vector field ηα satisfying (1.1) ∇α ηβ − ∇β ηα = 0 and (1.2) ∇α ∇β ηγ = − gαβ ηγ − gαγ ηβ + 2ηα ηβ ηγ , Singh Chauhan et al / Indian Journal of Science and Technology 2020;13(13):1435–1439 wherein ∇α denotes the operator of covariant differentiation with regard to gα β , such a space Mn is called Para-Sasakian manifold [4,5]. It is easy to verify that in a Para-Sasakian manifold, the following relations holds good [4,5]: (1.3) ηα = gαβ ξ β , (1.4) ηα ξ α = 1, (1.5) gαβ ηβ = ξ α ,.

Recurrent and Symmetric Para-Sasakian Manifolds

Projectively Symmetric Para-Sasakian Manifold

Φ-Holomorphic Sectional Curvature