Symmetric Spaces, Geometric Manifolds and Geodesic Completeness

Abstract

This thesis explores topics related to the geodesic completeness of compact locally sym metric Lorentzian manifolds. In particular, it discusses some important results relating to locally and globally symmetric spaces as well as the theory of geometric manifolds. These results are used to present a proof of a key proposition in Klingler (1996), which proves the geodesic completeness of compact Lorentzian manifolds with constant curvature. We also prove a new result, that compact Lorentzian manifolds which are locally isometric to the product of Cahen-Wallach space and flat Riemannian space are geodesically complete by extending methods used in Leistner & Schliebner (2016). These results may be helpful in the study of geodesic completeness of compact locally symmetric Lorentzian manifolds more generally as they reduce the number of open cases.