On Nilpotent and Solvable Quasi-Einstein Manifolds

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 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 Heisenberg. 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 Heisenberg Lie algebra. As an application, we prove that the only near-horizon geometries on a nilmanifold are $\mathrm{\Gamma }\mathrm{\backslash }{H}_n$, where $H_n$ is $n$-dimensional Heisenberg Lie group.