∗-η-Ricci Soliton and Gradient Almost ∗-η-Ricci Soliton Within the Framework of Para-Kenmotsu Manifolds

Santu Dey,Nasser Bin Turki

doi:10.3389/fphy.2022.809405

Santu Dey, Nasser Bin Turki

Open Access

https://doi.org/10.3389/fphy.2022.809405

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Abstract

The goal of the present study is to study the ∗-η-Ricci soliton and gradient almost ∗-η-Ricci soliton within the framework of para-Kenmotsu manifolds as a characterization of Einstein metrics. We demonstrate that a para-Kenmotsu metric as a ∗-η-Ricci soliton is an Einstein metric if the soliton vector field is contact. Next, we discuss the nature of the soliton and discover the scalar curvature when the manifold admits a ∗-η-Ricci soliton on a para-Kenmotsu manifold. After that, we expand the characterization of the vector field when the manifold satisfies the ∗-η-Ricci soliton. Furthermore, we characterize the para-Kenmotsu manifold or the nature of the potential vector field when the manifold satisfies the gradient almost ∗-η-Ricci soliton.

Highlights

We can say that the metric g is a Ricci soliton if there exists a smooth vector field V and a constant λ such that
The term “gradient almost p-η-Ricci soliton” denotes a gradient p-η-Ricci soliton for which we considered λ as a smooth function
We considered a para-Kenmotsu metric as p-η-Ricci solitons and gradient p-η-Ricci solitons

Summary

INTRODUCTION

If the potential vector field V is the gradient of a smooth function f, denoted by Df the soliton equation reduces to. If we consider the potential vector field V as the gradient of a smooth function f, the p-η-Ricci soliton equation can be rewritten as follows: Hessf + Sp + λg + μη ⊗ η 0. Blaga studied certain aspects of η-Ricci solitons on para-Kenmotsu and Lorentzian para-Sasakian manifolds (see [32,33,34]) Motivated by these results, we considered a para-Kenmotsu metric as p-η-Ricci solitons and gradient p-η-Ricci solitons. We have explained that if the metric g represents a p-η-Ricci soliton and if the soliton vector field V is contact, V is a strictly infinitesimal contact transformation and the manifold is Einstein. Either M is Einstein or there exists an open set where the potential vector field V is pointwise collinear with the characteristic vector field ξ

SOME PRELIMINARIES ON PARA-KENMOTSU MANIFOLDS

GRADIENT ALMOST p-η-RICCI SOLITON ON PARA-KENMOTSU MANIFOLDS

DATA AVAILABILITY STATEMENT