Three-dimensional Ricci-degenerate Riemannian manifolds satisfying geometric equations

In this paper, we study a three-dimensional Ricci-degenerate Riemannian manifold $(M^3,g)$ that admits a smooth nonzero solution $f$ to the equation \begin{align} \label{a1a} \nabla df=\psi Rc+\phi g, \end{align} where $\psi ,\varphi$ are given smooth functions of $f$, $Rc$ is the Ricci tensor of $g$. Spaces of this type include various interesting classes, namely gradient Ricci solitons, $m$-quasi Einstein metrics, (vacuum) static spaces, $V$-static spaces, and critical point metrics. The $m$-quasi Einstein metrics and vacuum static spaces were previously studied in \cite{JJ,JEK}, respectively. In this paper, we refine them and develop a general approach for the solutions of (\ref{a1a}); we specify the shape of the metric $g$ satisfying (\ref{a1a}) when $\mathrm{\nabla }f$ is not a Ricci-eigen vector. Then we focus on the remaining three classes, namely gradient Ricci solitons, $V$-static spaces, and critical point metrics. Furthermore, we present classifications of local three-dimensional Ricci-degenerate spaces of these three classes by explicitly describing the metric $g$ and the potential function $f$.