Left invariant Lorentzian metrics and curvatures on non-unimodular Lie groups of dimension three

TL;DR

This paper classifies left invariant Lorentzian metrics on three-dimensional non-unimodular Lie groups, analyzing how curvature properties change with different metrics, and extends previous work on Riemannian and unimodular cases.

Contribution

It provides a comprehensive classification of Lorentzian metrics on non-unimodular Lie groups and explores curvature variations, extending prior studies on related Lie groups.

Findings

Explicit formulas for Ricci operator and scalar curvature.

Classification of metrics up to automorphism.

Analysis of curvature variation under metric changes.

Abstract

For each connected and simply connected three-dimensional non-unimodular Lie group, we classify the left invariant Lorentzian metrics up to automorphism, and study the extent to which curvature can be altered by a change of metric. Thereby we obtain the Ricci operator, the scalar curvature, and the sectional curvatures as functions of left invariant Lorentzian metrics on the three-dimensional non-unimodular Lie groups. Our study is a continuation and extension of the previous studies done in \cite{HL2009_MN} for Riemannian metrics on three-dimensional Lie groups and in \cite{BC} for Lorentzian metrics on three-dimensional unimodular Lie groups.