On generalized quasi-Einstein manifolds

TL;DR

This paper establishes rigidity results for compact and noncompact generalized quasi-Einstein manifolds with constant scalar curvature, characterizes those conformal to Euclidean space with non-constant scalar curvature, and provides explicit examples.

Contribution

It introduces new rigidity theorems for generalized quasi-Einstein manifolds and classifies those conformal to Euclidean space with explicit examples.

Findings

Rigidity results for compact manifolds with constant scalar curvature

Rigidity for noncompact manifolds under geometric assumptions

Classification of conformal generalized quasi-Einstein manifolds with explicit examples

Abstract

A rigidity result for a class of compact generalized quasi-Einstein manifolds with constant scalar curvature is obtained. Moreover, under some geometric assumptions, the rigidity for the noncompact case is also proved. Considering non constant scalar curvature, we characterize the generalized quasi-Einstein manifolds which are conformal to the Euclidean space and we show that there exist two classes of complete manifolds, which are obtained by considering potential functions and conformal factors either to be radial or invariant under the action of an (n-1)-dimensional translation group. Explicit examples are given.