A simple proof of geodesical completeness for compact space-times of zero curvature

Publisher Website
Google Scholar
Abstract
Any Riemannian metric on a compact manifold is geodesically complete. By contrast, it is widely known that incomplete compact Lorentzian manifolds exist. In this article it will be proven that a compact manifold with a smooth Lorentz metric must be complete if the metric is flat everywhere. This question of completeness; which has significant implications for the classification of (compact) Lorentzian space forms, has remained unresolved until quite recently, when it was finally established by Carrière as a special case of an even more general result. Here, a simple, alternative proof of this special case of Carrière’s theorem is given; the proof requires minimal mathematical machinery but still involves the use of some intriguing connections between the topology and global geometry of a compact flat space‐time.

This publication has 8 references indexed in Scilit: