Three dimensional m-quasi Einstein manifolds with degenerate Ricci tensor

TL;DR

This paper classifies three-dimensional m-quasi Einstein manifolds with degenerate Ricci tensor, providing explicit metrics, potential functions, and warped product Einstein manifolds, advancing understanding of their geometric structure.

Contribution

It offers a comprehensive classification of 3D m-quasi Einstein manifolds with degenerate Ricci tensor, including explicit descriptions and analysis of associated warped products.

Findings

Explicit local and complete metrics derived.

Characterization of potential functions and warped products.

Resolution of cases where the gradient of the potential is not a Ricci-eigen vector.

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 $\nabla f$ of the potential function is not a Ricci-eigen vector field.