ABSTRACT
A common assumption made in studying Riemannian manifolds is that the spaces under consideration are Cauchy complete or, equivalently, geodesically complete. This assumption seems reasonable since a large number of important Riemannian manifolds are complete. A large number of the more important Lorentzian manifolds used as models in general relativity fail to be geodesically complete. This chapter is concerned with establishing theorems which guarantee the nonspacelike geodesic incompleteness of a large class of space-times. These space-times contain at least one nonspacelike geodesic which is both inextendible and incomplete. The chapter shows that if (M, g) is a space-time of dimension at least three which satisfies the generic and timelike convergence conditions, then every complete nonspacelike geodesic contains a pair of conjugate points. The basic technique in proving nonspacelike incompleteness is to first use physical or geometric assumptions on (M, g) to construct an inextendible nonspacelike geodesic which is maximal and hence contains no conjugate points.
