The existence of light-like homogeneous geodesics in homogeneous Lorentzian manifolds

TL;DR

This paper refines an affine method to prove that all even-dimensional homogeneous Lorentzian manifolds have light-like homogeneous geodesics, highlighting differences with specific examples.

Contribution

It adapts a fundamental affine method to the pseudo-Riemannian case, proving the existence of light-like homogeneous geodesics in even-dimensional Lorentzian manifolds.

Findings

Any even-dimensional homogeneous Lorentzian manifold admits a light-like homogeneous geodesic.

An example of a 3-dimensional Lie group with an invariant metric lacking such geodesics is provided.

Abstract

In previous papers, a fundamental affine method for studying homogeneous geodesics was developed. Using this method and elementary differential topology it was proved that any homogeneous affine manifold and in particular any homogeneous pseudo-Riemannian manifold admits a homogeneous geodesic through arbitrary point. In the present paper this affine method is refined and adapted to the pseudo-Riemannian case. Using this method and elementary topology it is proved that any homogeneous Lorentzian manifold of even dimension admits a light-like homogeneous geodesic. The method is illustrated in detail with an example of the Lie group of dimension 3 with an invariant metric, which does not admit any light-like homogeneous geodesic.