Three-dimensional Ricci-degenerate Riemannian manifolds satisfying geometric equations

Abstract

In this paper, we study a three-dimensional Ricci-degenerate Riemannian manifold \((M^3,g)\) that admits a smooth nontrivial solution f to the equation $$\begin{aligned} \nabla df=\psi Rc+\phi g, \end{aligned}$$ (1) 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 Jordan (Gen Relativ Gravit 41(9):2191–2280, 2009) and Kim and Shin (Math Nachr 292(8): 1727–1750, 2019), respectively. In this paper, we refine them and develop a general approach for the solutions of (1). We specify the shape of the metric g satisfying (1) 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.