John Beem

Cut points, conjugate points and Lorentzian comparison theorems

1. Introduction . The purpose of this paper is to study the global geometry of a space–time ( M, g ), which is related to the Lorentzian distance function induced on the manifold M by the Lorentzian structure. We will use the signature convention (−, +, …, +) for g and assume that ( M, g ) is time orientated. The first part of this paper deals with cut points and maximal geodesies, both of which were defined in (1) using the Lorentzian distance function in analogy to the standard concepts in Rie-mannian geometry. In (1), sections 2 and 3, some elementary properties of maximal geodesies were established. In particular, the principle that, for strongly causal space-times, a limit curve of a sequence of future-directed nonspacelike ‘almost maximal’ curves is a maximal geodesic was used to prove nonspacelike incompleteness ((1), theorem 6·3). Also null cut points were used to obtain results on null incompleteness ((1), section 5). In (2) we studied deeper properties of maximal geodesies and cut points using the technical tools developed in (1), sections 2 and 3. The first part of the present paper continues these investigations.

Constructing maximal geodesics in strongly causal space-times

Let ( M, g ) be an arbitrary space-time of dimension ≥ 2 and let d = d ( g ): M × M → ℝ ∪ {∞} (where d ( p, q ) = 0 for q ∉ J + ( p )) denote the Lorentzian distance function of ( M, g ). Also let C ( M, g ) denote the space of Lorentzian metrics for M globally con-formal to g . Here g 1 is said to be globally conformal to g if there exists a smooth function Ω: M → (0, ∞) such that g 1 = Ω g .