Killing Fields on Compact m-Quasi-Einstein Manifolds

TL;DR

This paper proves that on compact m-quasi Einstein manifolds, the potential vector field is Killing if and only if the scalar curvature is constant, extending previous results and providing new conditions for the existence of Killing fields.

Contribution

It extends known results by establishing the equivalence of Killing fields and constant scalar curvature on compact m-quasi Einstein manifolds, including new cases where m = -2.

Findings

X is Killing iff scalar curvature is constant

Provides a sufficient condition for non-gradient m-quasi Einstein metrics to admit Killing fields

Offers an alternative proof that works for m = -2

Abstract

We show that given a compact, connected $m$-quasi Einstein manifold $(M,g,X)$ without boundary, the potential vector field $X$ is Killing if and only if $(M, g)$ has constant scalar curvature. This extends a result of Bahuaud-Gunasekaran-Kunduri-Woolgar, where it is shown that $X$ is Killing if $X$ is incompressible. We also provide a sufficient condition for a compact, non-gradient $m$-quasi Einstein metric to admit a Killing field. We do this by following a technique of Dunajski and Lucietti, who prove that a Killing field always exists in this case when $m=2$. This condition provides an alternate proof of the aforementioned result of Bahuaud-Gunasekaran-Kunduri-Woolgar. This alternate proof works in the $m = -2$ case as well, which was not covered in the original proof.