On weakly Einstein Lie groups
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866909395653230592 |
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| author | Euh, Yunhee Kim, Sinhwi Nikolayevsky, Yuri Park, JeongHyeong |
| author_facet | Euh, Yunhee Kim, Sinhwi Nikolayevsky, Yuri Park, JeongHyeong |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_12311 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On weakly Einstein Lie groups Euh, Yunhee Kim, Sinhwi Nikolayevsky, Yuri Park, JeongHyeong Differential Geometry 53C25, 53C30, 17B30 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. |
| title | On weakly Einstein Lie groups |
| topic | Differential Geometry 53C25, 53C30, 17B30 |
| url | https://arxiv.org/abs/2411.12311 |