Classification of generalized Einstein metrics on 3-dimensional Lie groups

TL;DR

This paper develops a theory for left-invariant generalized pseudo-Riemannian metrics on 3D Lie groups, computes their Ricci tensors, and classifies all generalized Einstein metrics in this setting.

Contribution

It introduces a framework for analyzing Einstein metrics within generalized geometry on Lie groups and provides a complete classification for three-dimensional cases.

Findings

Explicit formulas for Ricci tensors in terms of Lie algebra data

Polynomial homogeneous expressions for Einstein equations

Complete classification of generalized Einstein metrics on 3D Lie groups

Abstract

We 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 two 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.