On Nilpotent and Solvable Quasi-Einstein Manifolds

Abstract

In this paper, we investigate nilpotent and unimodular solvable Lie groups that admit quasi-Einstein metrics $(M,g,X)$ with $X$ a left-invariant vector field, which we call \emph{totally left-invariant quasi-Einstein metrics}. We give a complete classification of nilpotent Lie groups admitting such metrics, proving that this occurs if and only if the group is isomorphic to a Heisenberg Lie group. For unimodular solvable Lie groups $S$, we show that the existence of a non-flat totally left-invariant quasi-Einstein metric forces the center of $S$ to be one-dimensional. Furthermore, under the additional assumption that the adjoint action ${\mathrm{ad}}_a$ of $S$ is a normal derivation, we obtain a full classification: these groups are standard and their nilradical must be a Heisenberg Lie algebra. As an application, we prove that the only near-horizon geometries on a compact nilmanifold are $\Gamma \backslash H_n$, where $H_n$ is $n$-dimensional Heisenberg Lie group.

arXiv comment25 pages

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• On homogeneous warped product Einstein metrics