16 PAUL E. EHRLICH AND KEVIN L. EASLEY a key ingredient rediscovered by Cheeger and Gromoll for use in their proof of the Riemannian splitting theorem. The first author had studied Busemann functions during his stay in Bonn in connection with manifolds of negative curvature, and of course Beem was a student of Busemann himself, familiar with the text Busemann [38]. Since we had decided to take the course of action of exploring the Busemann function of a timelike geodesic ray, the first author returned once again to Cheeger and Gromoll [43], which he had always found a difficult paper to understand. With the great emphasis on ellipticity of the Laplacian, while the d'Alembertian of General Relativity is hyperbolic, it also seemed like a daunting task to make any of this transform to the space-time case. Fortunately, just before we began our studies, we received an unexpected preprint in the mail which was published later as Eschenburg and Heintze [52]. At a first glance, it looked like an approach to the splitting problem that had a good chance of adapting to the space-time setting, and so with Markvorsen we set to work. Many standard elementary methods in basic Riemannian geometry, in the con- text of constructing asymptotic geodesics, rely on the compactness of the set of unit vectors based at a given point in the manifold. For space-times, however, the set of future unit timelike vectors based at a point (while closed) is noncompact. A further difficulty is that a sequence of unit timelike tangent vectors can never converge to a null vector n, even though the set of nonspacelike directions in TpM is itself compact, so that direction( Vn) ---+ direction( n) is indeed possible. Here is the basic type of construction that is involved in both the Riemannian and Lorentzian setting. Let 1: [O,+oo)---+(M,g) be a unit timelike geodesic ray, e.g., suppose that (6.1) L(ll[o,tJ) = d(1(0),1(t)) for all t 2::0. Take any p in M with p in the chronological past of 1 and any sequence Sn 0 with sn ---+ +oo. Assuming that (M, g) is globally hyperbolic, construct unit speed maximal timelike geodesic segments Cn from p to qn = l(sn)· Then in the Riemannian case, one could turn to the sequence of unit vectors {en' (0)} and extract a convergent subsequence to define an asymptotic geodesic c to 1 starting at p. But as we have indicated above, in the space-time case such a convergence is not guaranteed. However, already at hand is the limit curve machinery for strongly causal space- times mentioned in Section 4. So instead, one may let c be a nonspacelike limit curve of the timelike geodesic segments {en}· Then two issues which must be dealt with are: (i) why is c timelike rather than null, and (ii) why is c future complete? Inspired by the somewhat more general approach taken to the asymptotic ge- odesic construction in Busemann [38] ( cf. Busemann [37] for apparently the first appearance of what would later be termed by others the Busemann function), in Beem, Ehrlich, Markvorsen, and Galloway [25] the following definition was adopted
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