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

Abstract

We study some properties of the $3$-dimensional manifold $\mathbb R^3$ 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 obstructions 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.