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

Locally Homogeneous Non-gradient Quasi-Einstein 3-Manifolds