Completeness of certain compact Lorentzian locally symmetric spaces

TL;DR

This paper establishes conditions under which certain compact Lorentzian locally symmetric spaces are geodesically complete, focusing on the structure of their Lorentzian and flat factors in the de Rham-Wu decomposition.

Contribution

It identifies specific geometric conditions involving Cahen-Wallach type factors and one-dimensional time-like flat factors that guarantee geodesic completeness.

Findings

Spaces with Cahen-Wallach type Lorentzian factors are complete.

Spaces with one-dimensional time-like flat factors are complete.

The proof leverages recent and classical results in Lorentzian geometry.

Abstract

We show that a compact Lorentzian locally symmetric space is geodesically complete if the Lorentzian factor in the local de Rham-Wu decomposition is of Cahen-Wallach type or if the maximal flat factor is one-dimensional and time-like. Our proof uses a recent result by Mehidi and Zeghib and an earlier result by Romero and S\'{a}nchez.