On weakly Einstein Lie groups

TL;DR

This paper investigates the class of weakly Einstein Lie groups, establishing non-existence results for certain nilpotent groups and characterizing almost abelian groups that satisfy the weakly Einstein condition.

Contribution

It provides the first classification results for weakly Einstein Lie groups, including non-existence theorems and a characterization of almost abelian cases.

Findings

No weakly Einstein non-abelian 2-step nilpotent Lie groups.

No weakly Einstein non-abelian nilpotent Lie groups of dimension ≤ 5.

Almost abelian Lie groups are weakly Einstein iff their Lie algebra involves a normal operator with a square multiple of identity.

Abstract

A Riemannian manifold is called \emph{weakly Einstein} if the tensor $R_{iabc}R_{j}^{~~abc}$ is a scalar multiple of the metric tensor $g_{ij}$. We consider weakly Einstein Lie groups with a left-invariant metric which are weakly Einstein. We prove that there exist no weakly Einstein non-abelian $2$-step nilpotent Lie groups and no weakly Einstein non-abelian nilpotent Lie groups whose dimension is at most $5$. We also prove that an almost abelian Lie group is weakly Einstein if and only if at the Lie algebra level it is defined by a normal operator whose square is a multiple of the identity.