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

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

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  • 发表日期：

    2025年

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  • Vicente Cortés,
  • Liana David

摘要

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