Generalised Einstein metrics on Lie groups
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911965036675072 |
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| author | Cortés, Vicente Freibert, Marco Galdeano, Mateo |
| author_facet | Cortés, Vicente Freibert, Marco Galdeano, Mateo |
| contents | We continue the systematic study of left-invariant generalised Einstein metrics on Lie groups initiated in arXiv:2206.01157. Our approach is based on a new reformulation of the corresponding algebraic system. For a fixed Lie algebra $\mathfrak{g}$, the unknowns of the system consist of a scalar product $g$ and a $3$-form $H$ on $\mathfrak{g}$ as well as a linear form $δ$ on $\mathfrak{g}\oplus\mathfrak{g}^*$. As in arXiv:2206.01157, the Lie bracket of $\mathfrak{g}$ is considered part of the unknowns. In the Riemannian case, we show that the generalised Einstein condition always reduces to the commutator ideal and we provide a full classification of solvable generalised Einstein Lie groups. In the Lorentzian case, under the additional assumption $δ=0$, we classify -- up to one case -- all almost Abelian generalised Einstein Lie groups. We then particularize to four dimensions and provide a full classification of generalised Einstein Riemannian Lie groups as well as generalised Einstein Lorentzian Lie groups with $δ=0$ and non-degenerate commutator ideal. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_16562 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Generalised Einstein metrics on Lie groups Cortés, Vicente Freibert, Marco Galdeano, Mateo Differential Geometry 53D18 (Primary) 53C25, 53C30 (Secondary) We continue the systematic study of left-invariant generalised Einstein metrics on Lie groups initiated in arXiv:2206.01157. Our approach is based on a new reformulation of the corresponding algebraic system. For a fixed Lie algebra $\mathfrak{g}$, the unknowns of the system consist of a scalar product $g$ and a $3$-form $H$ on $\mathfrak{g}$ as well as a linear form $δ$ on $\mathfrak{g}\oplus\mathfrak{g}^*$. As in arXiv:2206.01157, the Lie bracket of $\mathfrak{g}$ is considered part of the unknowns. In the Riemannian case, we show that the generalised Einstein condition always reduces to the commutator ideal and we provide a full classification of solvable generalised Einstein Lie groups. In the Lorentzian case, under the additional assumption $δ=0$, we classify -- up to one case -- all almost Abelian generalised Einstein Lie groups. We then particularize to four dimensions and provide a full classification of generalised Einstein Riemannian Lie groups as well as generalised Einstein Lorentzian Lie groups with $δ=0$ and non-degenerate commutator ideal. |
| title | Generalised Einstein metrics on Lie groups |
| topic | Differential Geometry 53D18 (Primary) 53C25, 53C30 (Secondary) |
| url | https://arxiv.org/abs/2407.16562 |