On Weakly Einstein Lie Groups

期刊论文
发表日期：

2025年5月24日

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Yunhee Euh,
Sinhwi Kim,
Yuri Nikolayevsky,
JeongHyeong Park

摘要

A Riemannian manifold is called weakly Einstein if the tensor $R_{iabc}R_{j}^{~~abc}$ $R_{iabc} R_{j}^{a b c}$ is a scalar multiple of the metric tensor $g_{ij}$ $g_{ij}$ . We consider weakly Einstein Lie groups with a left-invariant metric. 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.