On Nilpotent and Solvable Quasi-Einstein Manifolds
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866916973981466624 |
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| author | Valiyakath, Nazia |
| author_facet | Valiyakath, Nazia |
| contents | 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 Heisenberg. 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 $\operatorname{ad}_a$ of $S$ is a normal derivation, we obtain a full classification: these groups are standard and their nilradical must be Heisenberg Lie algebra. As an application, we prove that the only near-horizon geometries on a nilmanifold are $Γ\backslash H_{n}$, where $ H_{n}$ is $n$-dimensional Heisenberg Lie group. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_19674 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On Nilpotent and Solvable Quasi-Einstein Manifolds Valiyakath, Nazia Differential Geometry Mathematical Physics 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 Heisenberg. 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 $\operatorname{ad}_a$ of $S$ is a normal derivation, we obtain a full classification: these groups are standard and their nilradical must be Heisenberg Lie algebra. As an application, we prove that the only near-horizon geometries on a nilmanifold are $Γ\backslash H_{n}$, where $ H_{n}$ is $n$-dimensional Heisenberg Lie group. |
| title | On Nilpotent and Solvable Quasi-Einstein Manifolds |
| topic | Differential Geometry Mathematical Physics |
| url | https://arxiv.org/abs/2507.19674 |