Completeness of compact Locally symmetric Lorentz manifolds

Abstract

The geodesic completeness of compact locally symmetric Lorentz manifolds has been established in several important cases, namely, the constant curvature, indecomposable, and Brinkmann settings. In this paper, we prove geodesic completeness in all remaining cases, thereby confirming the conjecture that all compact, locally symmetric Lorentz manifolds are geodesically complete. Along the way, we use the completeness result to give a comprehensive overview on four-dimensional compact locally symmetric Lorentz manifolds.

18 pages. In Appendix A, the action of the $\mathbb{R}$-factor on the Heisenberg group has been corrected. The statement of Theorem 5.3 now assumes that $G$ is semisimple rather than merely algebraic. The proof of Theorem 5.3 has been revised accordingly under this assumption on $G$. The necessary corrections are contained in Statements 5.6--5.9. The proof of Proposition 7.1 is shortened