A BEEMIAN SAMPLER: 1966-2002 19 old possibility still obtains, or if it fails to be true, it fails in an isometric (hence "rigid") way. Thus in the Riemannian example, if one relaxes the pinching on the sectional curvature to ~ ::: : K::: : 1, then either the Riemannian manifold remains homeomor- phic to then-sphere (the old alternative), or if not, it is isometric to a symmetric space of rank one. Already in Geroch [60], the idea had been presented that most space-times should be nonspacelike geodesically incomplete and also that a space-time should fail to be nonspacelike geodesically incomplete only under special circumstances (in the paragraph below, a white dot represents a geodesically complete space-time, a black dot a nonspacelike geodesically incomplete space-time): "Thus we expect that the diagrams for closed universes will be almost entirely black. There are, however, at least a few white points. There exist closed, geodesically complete, flat space-times. . . . Perhaps there are a few other nonsingular closed universes, but these may be expected to appear either as isolated points or at least regions of lower dimensionality in an otherwise black diagram." To see how the idea of curvature rigidity could apply in the context of timelike geodesic incompleteness, first let us state a simple prototype singularity theorem: THEOREM 6.4. Let (M, g) be a space-time of dimension n ~ 3 which satisfies each of the following conditions: 1. (M,g) contains a compact Cauchy surface. 2. (M,g) satisfies Ric(v,v) ~ 0 on all nonspacelike tangent vectors v. 3. Every inextendible nonspacelike geodesic satisfies the generic condition. Then (M, g) contains an incomplete nonspacelike geodesic. In this result, condition (2) already allows for equality, so that cannot be weak- ened. Thus, here curvature rigidity would call for dropping the requirement (3) of the generic condition that some curvature quantity is nonzero at some point of the geodesic. Hence, that is how the conjectured rigidity of timelike geodesic complete- ness arises. This was apparently first published by one of Yau's Ph.D. students in Bartnik [9] as follows: CONJECTURE 6.5. Let (M,g) be a space-time of dimension n ~ 3 which 1. contains a compact Cauchy surface, and 2. satisfies the timelike convergence condition Ric(v,v) ~ 0 for all timelike v. Then either (M,g) is timelike geodesically incomplete, or (M,g) splits isometrically as a product (JR x V, -dt2 +h), where (H, h) is a compact Riemannian manifold. The isometric splitting in the second alternative is precisely the manifestation of curvature rigidity here. The idea to solve this conjecture is a proof by contra- diction. Suppose the space-time is not timelike geodesically incomplete. From the hypotheses, produce a nonspacelike line (recall Theorem 4.2 above), and prove that the line is timelike rather than null. Then under the assumption of timelike geo- desic completeness the line is complete, so the Lorentzian Splitting Theorem may be applied to give the second alternative. Indeed, a result of this sort was obtained in [25] under the stronger sectional curvature hypothesis. A survey of later progress on this conjecture may be found in Beem, Ehrlich and Easley [24], Section 14.5.
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