Odd generalized Einstein metrics on Lie groups

Abstract

An odd generalized metric \(E_{-}\) on a Lie group G of dimension n is a left-invariant generalized metric on a Courant algebroid \(E_{H, F}\) of type \(B_{n}\) over G with left-invariant twisting forms \(H\in \Omega ^{3}(G)\) and \(F\in \Omega ^{2}(G)\). Given an odd generalized metric \(E_{-}\) on G we determine the affine space of left-invariant Levi-Civita generalized connections of \(E_{-}\). Given in addition a left-invariant divergence operator \(\delta \) we show that there is a left-invariant Levi-Civita generalized connection of \(E_{-}\) with divergence \(\delta \) and we compute the corresponding Ricci tensor \(\textrm{Ric}^{\delta }\) of the pair \((E_{-}, \delta )\). The odd generalized metric \(E_{-}\) is called odd generalized Einstein with divergence \(\delta \) if \(\textrm{Ric}^{\delta }=0\). As an application of our theory, we describe all odd generalized Einstein metrics of arbitrary left-invariant divergence on all 3-dimensional unimodular Lie groups.