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 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 adₐ 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 H₍ Γ \ H n, where H₍ H n is n -dimensional Heisenberg Lie group.