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 nonzero solution f to the equationwhere ψ, φ 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 [26,24], 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 ∇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 .