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arXiv · 2404.17090

Killing Fields on Compact m-Quasi-Einstein Manifolds

Eric Cochran
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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.

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Eric Cochran. 2024-04-26. Killing Fields on Compact m-Quasi-Einstein Manifolds. https://arxiv.org/abs/2404.17090

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