On some $3$-dimensional almost $\eta$-Ricci solitons with diagonal metrics

TL;DR

This paper investigates properties of 3-dimensional almost η-Ricci solitons with diagonal metrics, focusing on potential vector fields, metric constraints, and conditions for flatness, supported by explicit examples.

Contribution

It provides explicit characterizations and conditions for 3D almost η-Ricci solitons with diagonal metrics, including potential vector fields and flatness criteria.

Findings

Determined potential vector fields under certain assumptions.

Derived metric constraints for specific potential vector fields.

Established conditions for the manifold to be flat.

Abstract

We study some properties of a $3$-dimensional manifold with a diagonal Riemannian metric as an almost $\eta$-Ricci soliton from the following points of view: under certain assumptions, we determine the potential vector field if $\eta$ is given; we get constraints on the metric when the potential vector field has a particular expression; we compute the defining functions of the soliton when both the potential vector field and the $1$-form are prescribed. Moreover, we find conditions for the manifold to be flat. Based on the theoretical results, we provide examples.