Abstract
AbstractWe develop the theory of left-invariant generalized pseudo-Riemannian metrics on Lie groups. Such a metric accompanied by a choice of left-invariant divergence operator gives rise to a Ricci curvature tensor, and we study the corresponding Einstein equation. We compute the Ricci tensor in terms of the tensors (on the sum of the Lie algebra and its dual) encoding the Courant algebroid structure, the generalized metric, and the divergence operator. The resulting expression is polynomial and homogeneous of degree 2 in the coefficients of the Dorfman bracket and the divergence operator with respect to a left-invariant orthonormal basis for the generalized metric. We determine all generalized Einstein metrics on three-dimensional Lie groups.
Publisher
Canadian Mathematical Society
Reference15 articles.
1. Supergravity as generalised geometry I: type II theories;Coimbra;J. High Energy Phys.,2011
2. Contact metric three manifolds and Lorentzian geometry with torsion in six-dimensional supergravity
3. [CEHL] Cortés, V. , Ehlert, J. , Haupt, A. , and Lindemann, D. , Classification of left-invariant Einstein metrics on $\mathrm{SL}\left(2,\mathbb{R}\right)\times \mathrm{SL}\left(2,\mathbb{R}\right)$ that are bi-invariant under a one-parameter subgroup, Ann. Glob. Anal. Geom., accepted November 25, 2022. arXiv:2201.07343 [math.DG]
4. Generalized Connections, Spinors, and Integrability of Generalized Structures on Courant Algebroids
5. [ADG] Álvarez-Cónsul, L. , De Arriba de La Hera, A. , and Garcia-Fernandez, M. , (0,2) mirror symmetry on homogeneous Hopf surfaces. Preprint, 2021. arXiv:2012.01851 [math.DG]
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