Wednesday 16 September 2015 from 11:00–12:00 in Carslaw 535A

Geodesic completeness of compact Lorentzian manifolds

Thomas Leistner (Adelaide)

Abstract

A semi-Riemannian manifold is geodesically complete (or for short, complete) if its maximal geodesics are defined for all times. For Riemannian metrics the compactness of the manifold implies completeness. In contrast, there are very simple compact Lorentzian manifolds that are not complete. Nevertheless, completeness plays an important role for fundamental geometric questions in Lorentzian geometry such as the classification of compact Lorentzian manifolds of constant curvature and in particular for a Lorentzian version of Bieberbach's theorem. We will study the completeness for compact manifolds that arise from the classification of Lorentzian holonomy groups, which we will briefly review in the talk. These manifolds carry a parallel null vector field that can be used to study their completeness. They include the co-called plane fronted waves for which we determine the universal covering and show that they are complete. In the talk we will explain this result and further work in progress, both being joint work with A. Schliebner (Humboldt-University Berlin).