Remarks on almost $\eta$-Ricci solitons in $(\varepsilon)$-para Sasakian manifolds

TL;DR

This paper studies almost $ abla$-Ricci solitons within $( ext{ε})$-para Sasakian manifolds, providing curvature estimates and conditions for soliton expansion or shrinking based on the Ricci operator's properties.

Contribution

It introduces new results on the behavior of almost $ abla$-Ricci solitons in $( ext{ε})$-para Sasakian manifolds, including curvature bounds and classification under Codazzi Ricci operator conditions.

Findings

Estimation of Ricci curvature norm in gradient case

Expression of scalar curvature in terms of soliton functions

Classification of solitons as expanding or shrinking based on Ricci operator properties

Abstract

We consider almost $\eta$-Ricci solitons in $(\varepsilon)$-para Sasakian manifolds satisfying certain curvature conditions. In the gradient case we give an estimation of the Ricci curvature tensor's norm and express the scalar curvature of the manifold in terms of the functions that define the soliton. We also prove that if the Ricci operator is Codazzi, then the gradient $\eta$-Ricci soliton is expanding if $M$ is spacelike or shrinking if $M$ is timelike.