Three dimensional m‐quasi Einstein manifolds with degenerate Ricci tensor

Abstract

In this article we give a classification of three dimensional m‐quasi Einstein manifolds with two distinct Ricci‐eigen values. Our study provides explicit description of local and complete metrics and potential functions. We also describe the associated warped product Einstein manifolds in detail. For the proof we present a Codazzi tensor on any three dimensional m‐quasi Einstein manifold and use geometric properties of the tensor which help to analyze the m‐quasi Einstein equation effectively. A technical advance over preceding studies is made by resolving the case when the gradient ∇f of the potential function is not a Ricci‐eigen vector field.

$m$-quasi Einstein manifolds with convex potential

“…Einstein manifolds with parallel Ricci tensor, and also in [13], they classified m-quasi Einstein manifolds with all eigenvalues of the Ricci tensor as constant. Kim and Shin [14] also classified 3-dimensional m-quasi Einstein manifolds. Barros and Gomes [2] studied compact m-quasi Einstein manifolds and also showed that if such a manifold is Einstein, then its potential vector field vanishes.…”

Three-dimensional Ricci-degenerate Riemannian manifolds satisfying geometric equations

“…In this paper, we study three-dimensional Riemannian manifolds with two distinct Ricci eigenvalues having a nonzero solution to (2). This paper is the result of efforts devoted toward refining and generalizing [26], which concerns three-dimensional m-quasi Einstein manifolds. We preferentially study the common properties of spaces for general ψ(f ) and φ(f ).…”

“…

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.

…”

“…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.…”