Three-dimensional Ricci-degenerate Riemannian manifolds satisfying geometric equations

TL;DR

This paper classifies three-dimensional Ricci-degenerate Riemannian manifolds satisfying a specific geometric equation, refining existing models and explicitly describing metrics and potential functions for key subclasses.

Contribution

It develops a general approach for solutions to the geometric equation and classifies Ricci-degenerate spaces in three dimensions, including gradient Ricci solitons, V-static spaces, and critical point metrics.

Findings

Explicit classifications of metrics and potential functions for key subclasses.

Refined analysis of previously studied classes like quasi Einstein and static spaces.

Development of a unified approach for solutions to the geometric equation.

Abstract

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 $\psi,\phi$ 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 $\nabla f$ 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 $f$.