Quasi-Einstein manifolds admitting conformal vector fields
Volume 174 / 2023
Colloquium Mathematicum 174 (2023), 81-87
MSC: Primary 53C21; Secondary 53C25.
DOI: 10.4064/cm8903-6-2023
Published online: 5 October 2023
Abstract
We study an $m$-quasi-Einstein manifold $(M,g,f,\lambda )$ with finite $m$, and a non-homothetic conformal vector field $U$ that leaves the potential vector field and the scalar curvature both invariant, and show that either $M$ is trivial, or $U$ is Killing on the set of regular points of $f$. In the case when $M$ is a gradient Ricci soliton, it is trivial. Finally, for an $m$-quasi-Einstein manifold with finite $m$, and a homothetic vector field $U$ leaving the potential vector field invariant, we show that either (i) $M$ is Ricci-flat and $f$ is constant, or (ii) $U$ is Killing.
