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