Characterizations of $3$-manifolds with diagonal metrics as almost η-Ricci solitons

We characterize the $3$-dimensional manifold R³ with a diagonal Riemannian metric as an almost η-Ricci soliton from the following points of view: under certain assumptions, we determine the potential vector field if η is given; we get obstructions on the metric when the potential vector field has a particular expression; and 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 or Ricci-flat. Based on the theoretical results we provide many examples.