Classification of odd generalized Einstein metrics on 3-dimensional 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 and F. 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 Ricci^{\delta} of the pair (E_{-}, \delta ). The odd generalized metric E_{-} is called odd generalized Einstein with divergence \delta if Ricci^{\delta} =0. We describe all odd generalized Einstein metrics of arbitrary left-invariant divergence on all 3-dimensional Lie groups.