18 PAUL E. EHRLICH AND KEVIN L. EASLEY We now briefly summarize how the proof of the splitting theorem was extended from the sectional curvature hypothesis to the desired timelike convergence con- dition that Ric( v, v) 2 0 for all timelike (hence all nonspacelike) tangent vectors. J.-H. Eschenburg [51] obtained the first important breakthrough in realizing that instead of trying for global control of the timelike co-rays on !("!) as in Beem, Ehrlich, Markvorsen, and Galloway [25], it was sufficient to obtain a splitting in a tubular neighborhood of the given timelike geodesic line and then extend the split- ting to all of ( M, g) through a procedure similar to the one used in making analytic continuation type arguments in complex analysis. Hence, in [51], the splitting the- orem was obtained under the assumption of both timelike geodesic completeness and global hyperbolicity in the Ricci curvature case. Working with this new idea, Galloway was able to remove the hypothesis of ge- odesic completeness shortly thereafter ( cf. Galloway [56]). Then Newman returned to the original question of Yau and obtained the splitting for timelike geodesic completeness rather than global hyperbolicity ( cf. Newman [76]). Here, Newman had to confront the issue that maximal nonspacelike geodesic segments could not be constructed without global hyperbolicity, so he had to work with almost maxi- mizers instead of geodesics, introducing a higher level of complexity. A philosophy which emerged is that the existence of a maximal geodesic segment implies that things work out better in a tubular neighborhood of this maximal segment, in terms of the behavior of almost maximizers, the Busemann function, etc. In Galloway and Horta [57], these ideas were given a much simplified exposition and especially the original timelike co-ray condition of the earlier work [25] morphed into the generalized timelike co-ray condition. Out of all of these results, the Lorentzian Splitting Theorem emerged. THEOREM 6.3. Let (M,g) be a space-time of dimension n 2 3 which satisfies each of the following conditions: 1. ( M, g) is either globally hyperbolic or timelike geodesically complete. 2. ( M, g) satisfies the timelike convergence condition. 3. (M, g) contains a complete timelike line. Then (M, g) splits isometrically as a product (JR x V, -dt2 +h), where (H, h) is a complete Riemannian manifold. As mentioned above, Beem and the first author had no inkling as to why Yau had proposed the problem of proving a Lorentzian splitting theorem (which was not mentioned by Yau in his lectures at the Institute or the Problem List written from those lectures), but that was explained to us by Galloway when he passed through Columbia en route to Miami from San Diego and spoke about what was published as [55]. Yau's motivation had been the idea that timelike geodesic completeness should interface with the concept of "curvature rigidity" which had been formulated during the 1960s and 1970s in global Riemannian geometry, cf. especially the exposition in the introduction to the text Cheeger and Ebin [42], where it was first widely publicized. Recall our earlier statement of the Sphere Theorem of global Riemannian ge- ometry if a complete, simply connected Riemannian manifold has sectional curva- tures strictly 1/ 4-pinched, then it is homeomorphic to the n-sphere of the same dimension. In the statement of this result, there is a curvature condition of strict inequality. For curvature rigidity, the condition of strict inequality is relaxed to include the possibility of equality as well, and one tries to show that either the

Purchased from American Mathematical Society for the exclusive use of nofirst nolast (email unknown) Copyright 2004 American Mathematical Society. Duplication prohibited. Please report unauthorized use to cust-serv@ams.org. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community.