Geodesically Complete Lorentzian Metrics on Some Homogeneous 3 Manifolds

TL;DR

This paper investigates conditions for geodesic completeness of left invariant pseudo-Riemannian metrics on 3D Lie groups, establishing sufficiency for unimodular groups and identifying complete metrics on certain Lorentzian 3-manifolds.

Contribution

It provides a characterization of geodesic completeness for these metrics, highlighting differences between unimodular and non-unimodular Lie groups in three dimensions.

Findings

Necessary condition for completeness is also sufficient for unimodular groups.

Necessary condition is not sufficient for non-unimodular groups.

Complete Lorentzian 3-manifolds identified among compact locally homogeneous cases.

Abstract

In this work it is shown that a necessary condition for the completeness of the geodesics of left invariant pseudo-Riemannian metrics on Lie groups is also sufficient in the case of 3-dimensional unimodular Lie groups, and not sufficient for 3-dimensional non unimodular Lie groups. As a consequence it is possible to identify, amongst the compact locally homogeneous Lorentzian 3-manifolds with non compact (local) isotropy group, those that are geodesically complete.