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On weakly Einstein Lie groups (2411.12311v1)
Published 19 Nov 2024 in math.DG
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.