A BEEMIAN SAMPLER: 1966-2002 9 as explained in Hawking and Ellis [64] (indeed, nonspacelike cut points had not yet been formulated in the literature). An intrinsic Morse index theorem for null geodesic segments in arbitrary space-times had not yet been published, despite several works such as Uhlenbeck [87] and Woodhouse [93], among others. Working some of these issues out was accomplished in Beem and Ehrlich [18], [19], [20], and in greater detail in the First Edition of Global Lorentzian Geom- etry [21]. In place of the complete metric of Riemannian geometry, what emerged was an interesting interplay between the causal properties of the given space-time and the continuity (and other properties) of the Lorentzian distance function. Philo- sophically, this aspect emerges since d(p, q) 0 iff q E J+(p). For example, at the one extreme of totally vicious space-times, the Lorentzian distance always takes on the value +oo ( cf. p. 137 in [24]). Less drastically, if ( M, g) contains a closed time- like curve passing through p, then d(p, q) = +oo for all q E J+(p). In an allied vein, a space-time (M, g) is chronological iff its distance function vanishes identically on the diagonal !:!.(M) = {(p,p) : p EM}. In general, the Lorentzian distance function is only lower semi-continuous. The strength of the additional property of upper semi-continuity is seen in the result that if a distinguishing space-time has a continuous distance function, then it is causally continuous. At the other extreme from totally vicious space-times in the hierarchy of causality are globally hyperbolic space-times. In some sense, these space-times share many of the properties of complete Riemannian manifolds. For instance, the Lorentzian distance function of a globally hyperbolic space-time is both continu- ous and finite-valued. (Indeed, it may be shown that a strongly causal space-time ( M, g) is globally hyperbolic iff all Lorentz metrics g' in the conformal class C ( M, g) also have finite-valued distance functions d(g').) Second, for globally hyperbolic space-times, Seifert [85] and others had established the important working tool of maximal nonspacelike geodesic connectability: given any p, q E M with p ~ q, there exists a nonspacelike geodesic segment c : [0, 1]---+ (M,g) with c(O) = p, c(1) = q, and L(c) = d(p, q), in exact analogy with property (5) of the Hopf-Rinow Theorem stated in Section 1. In view of the last several properties of the distance function, the theories of the timelike and null cut loci were studied for globally hyperbolic space-times in Beem and Ehrlich [18]. Now let us turn to the main aim of this section, the discussion of causally disconnected space-times. Here is a general pattern which is common in global Riemannian geometry: (4.3) complete Riemannian metric and curvature inequality implies topological or geometric conclusion.
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