ABSTRACT
This chapter considers the Lorentzian analogue of the Bonnet-Myers Theorem and in so doing, study the timelike diameter of space-times. Classes of space-times with finite timelike diameter, including the “Wheeler universes,” have been studied in general relativity. It provides Lorentzian versions of two well-known comparison theorems in Riemannian geometry, the index comparison theorem and the Rauch Comparison Theorem. The chapter also considers analogues of the Hadamard-Cartan Theorem. The Hopf-Rinow Theorem guarantees the geodesic connectedness of any complete Riemannian manifold. However, complete Lorentzian manifolds may fail to be geodesically connected. Of course, global hyperbolicity yields a geodesic segment joining any two causally related points. Lorentzian version of the Hadamard-Cartan Theorem is actually valid for any manifold with an affine connection. The usual completeness assumption used in the Riemannian version is replaced by disprisonment and pseudoconvexity assumptions.
