Homogeneous geodesics of non-unimodular Lorentzian Lie groups and naturally reductive Lorentzian spaces in dimension three

TL;DR

This paper classifies all homogeneous geodesics in three-dimensional non-unimodular Lorentzian Lie groups and completes the classification of three-dimensional Lorentzian g.o. and naturally reductive spaces.

Contribution

It provides a complete characterization of homogeneous geodesics in non-unimodular Lorentzian Lie groups and advances the classification of Lorentzian g.o. and naturally reductive spaces.

Findings

Identified all homogeneous geodesics in three-dimensional non-unimodular Lorentzian Lie groups.

Completed the classification of three-dimensional Lorentzian g.o. spaces.

Classified naturally reductive Lorentzian spaces in dimension three.

Abstract

We determine, for all three-dimensional non-unimodular Lie groups equipped with a Lorentzian metric, the set of homogeneous geodesics through a point. Together with the results of [C] and [CM2], this leads to the full classification of three-dimensional Lorentzian g.o. spaces and naturally reductive spaces.