# Lorentzian left invariant metrics on three dimensional unimodular Lie groups and their curvatures

## TL;DR

This paper classifies all Lorentzian left invariant metrics on five three-dimensional unimodular Lie groups, detailing their curvature properties, and identifying special cases like constant curvature and Ricci solitons.

## Contribution

It explicitly lists all Lorentzian left invariant metrics on these groups and analyzes their curvature characteristics, including Ricci and scalar curvatures.

## Key findings

- Classification of all Lorentzian left invariant metrics on five Lie groups
- Identification of metrics with constant curvature and Ricci solitons
- Analysis of curvature signatures and special features

## Abstract

There are five unimodular simply connected three dimensional unimodular non abelian Lie groups: the nilpotent Lie group $\mathrm{Nil}$, the special unitary group $\mathrm{SU}(2)$, the universal covering group $\widetilde{\mathrm{PSL}}(2,\mathbb{R})$ of the special linear group, the solvable Lie group $\mathrm{Sol}$ and the universal covering group $\widetilde{\mathrm{E}_0}(2)$ of the connected component of the Euclidean group. For each $G$ among these Lie groups, we give explicitly the list of all Lorentzian left invariant metrics on $G$, up to un automorphism of $G$. Moreover, for any Lorentzian left invariant metric in this list we give its Ricci curvature, scalar curvature, the signature of the Ricci curvature and we exhibit some special features of these curvatures. Namely, we give all the metrics with constant curvature, semi-symmetric non locally symmetric metrics and the Ricci solitons.