Classification of Odd Generalized Einstein Metrics on 3-Dimensional Non-unimodular Lie Groups

Vicente Cortés () and Liana David ()
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Vicente Cortés: University of Hamburg, Department of Mathematics and Center for Mathematical Physics
Liana David: Institute of Mathematics ‘Simion Stoilow’ of the Romanian Academy

A chapter in Real and Complex Geometry, 2025, pp 109-136 from Springer

Abstract: Abstract An odd generalized metric E − $$E_{-}$$ on a Lie group G of dimension n with Lie algebra 𝔤 $$\mathfrak {g}$$ is a left-invariant generalized metric defined on a Courant algebroid E H , F $$E_{H, F}$$ of type B n $$B_{n}$$ over G, with left-invariant twisting forms H and F. We say that E − $$E_{-}$$ is generalized Einstein with left-invariant divergence δ ∈ ( 𝔤 ⊕ 𝔤 ∗ ⊕ ℝ ) ∗ $$\delta \in (\mathfrak {g}\oplus \mathfrak {g}^{*}\oplus \mathbb {R})^{*}$$ if the Ricci tensor of the pair ( E − , δ ) $$(E_{-}, \delta )$$ vanishes. In this paper we describe all odd generalized Einstein metrics (together with their divergences and Courant algebroids E H , F $$E_{H, F}$$ on which they are defined) on 3-dimensional non-unimodular Lie groups.

Keywords: Courant algebroids; Lie groups; Invariant generalized metrics; Generalized Einstein equation; 53D18 (Generalized geometry a la Hitchin); 35Q76 (Einstein’s equations); 53C30; (Homogeneous; manifolds) (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/978-3-031-92297-8_5

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