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Generalised Einstein metrics on Lie groups
- Publication Year :
- 2024
-
Abstract
- 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 $\delta$ 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 $\delta=0$, 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 $\delta =0$ and non-degenerate commutator ideal.<br />Comment: 58 pages
- Subjects :
- Mathematics - Differential Geometry
53D18 (Primary) 53C25, 53C30 (Secondary)
Subjects
Details
- Database :
- arXiv
- Publication Type :
- Report
- Accession number :
- edsarx.2407.16562
- Document Type :
- Working Paper