$\eta-$Ricci solitons on contact pseudo-metric manifolds

TL;DR

This paper investigates $ ext{eta}$-Ricci solitons on contact pseudo-metric manifolds, establishing conditions under which these manifolds are $ ext{eta}$-Einstein and analyzing the properties of the potential vector fields.

Contribution

It provides new results linking $ ext{eta}$-Ricci solitons to $ ext{eta}$-Einstein structures and explores the behavior of such solitons on various contact pseudo-metric manifolds.

Findings

Sasakian pseudo-metric manifolds with $ ext{eta}$-Ricci solitons are $ ext{eta}$-Einstein.

Gradient $ ext{eta}$-Ricci solitons on $K$-contact pseudo-metric manifolds are $ ext{eta}$-Einstein.

Conditions for $ ext{eta}$-Ricci solitons with potential vector fields colinear with the Reeb vector field.

Abstract

In this paper, we prove that a Sasakian pseudo-metric manifold which admits an $\eta-$Ricci soliton is an $\eta-$Einstein manifold, and if the potential vector field of the $\eta-$Ricci soliton is not a Killing vector field then the manifold is $\mathcal{D}-$homothetically fixed, and the vector field leaves the structure tensor field invariant. Next, we prove that a $K-$contact pseudo-metric manifold with a gradient $\eta-$Ricci soliton metric is $\eta-$Einstein. Moreover, we study contact pseudo-metric manifolds admitting an $\eta-$Ricci soliton with a potential vector field point-wise colinear with the Reeb vector field. Finally, we study gradient $\eta-$Ricci solitons on $(\kappa, \mu)$-contact pseudo-metric manifolds.