Locally Homogeneous Non-gradient Quasi-Einstein 3-Manifolds

TL;DR

This paper classifies compact locally homogeneous non-gradient m-quasi Einstein 3-manifolds, revealing the invariance of potential vector fields and characterizing specific product and circle manifolds with such metrics.

Contribution

It provides a complete classification of these manifolds, showing potential vector fields are left invariant and identifying the unique role of $S^1$ in admitting nontrivial m-quasi Einstein Einstein metrics.

Findings

Potential vector field $X$ is left invariant and Killing.

Classification of nontrivial m-quasi Einstein metrics as products of Einstein metrics.

$S^1$ is the only compact manifold with nontrivially m-quasi Einstein and Einstein metrics.

Abstract

In this paper, we classify the compact locally homogeneous non-gradient $m$-quasi Einstein 3-manifolds. Along the way, we prove that given a compact quotient of a Lie group of any dimension that is $m$-quasi Einstein, the potential vector field $X$ must be left invariant and Killing. We also classify the nontrivial $m$-quasi Einstein metrics that are a compact quotient of be the product of two Einstein metrics. We also show that $S^1$ is the only compact manifold of any dimension which admits a metric which is nontrivially $m$-quasi Einstein and Einstein.