A BEEMIAN SAMPLER: 1966-2002 5 metrics, this particular warped product (with f(t) = et in O'Neill's convention) is geodesically incomplete indeed, sO provides an example of an incomplete null geodesic. Here one checks that the geodesic equations for the warped product are satisfied and also II'Y'(s)ll2 =- (s12) + e(2lns) (- s12) 2 -1 s 2 = -2 +4 =O. s s Hence, arbitrary warped products (even of definite factors) in the semi-Riemannian case may fail to be geodesically complete. So that is what became of (2.1) and (2.3) in O'Neill [78]! After this initial collaboration, Beem wrote his dissertation, which appeared as a Memoir of the American Mathematical Society, along with work of a second Busemann student [34]. During his early years at Missouri, Beem produced seven papers in the area of Finsler geometry and indefinite metrics and had one Ph.D. student in this field, M. A. Kishta. As one example of this work, Beem and Kishta published a paper in the Indiana Mathematics Journal with the title "On generalized indefinite Finsler spaces," cf. [29]. During this time, the first author was in his two-year postdoctoral appointment at Bonn University and working with H.-C. ImHof on the geometry of the Dirichlet fundamental region for Riemannian manifolds without conjugate points. In this theory, the sides of the Dirichlet region are formed by bisectors B ={mE M: d(p,m) = d(q,m)} for two distinct points p, q E M. The first author recalls browsing in the mathemat- ics reading room at Bonn and seeing J. Beem [10] "Pseudo-Riemannian manifolds with totally geodesic bisectors" appear in the new journals shelf. As a young job applicant sending applications back to the U.S., this title certainly caught the au- thor's eye, but Beem and the first author would never employ bisectors in any of their joint work. 3. Studies of Nonspacelike Geodesics If any point p in a space-time (M, g) is selected, then emanating from p we have the three families of timelike, spacelike, and null geodesics. Since the null geodesics are, in a naive, imprecise fashion, limits of the spacelike and timelike geodesics, it was hoped at one time that, possibly, continuity results could be obtained for the different types of geodesic completeness. For example, perhaps timelike and spacelike geodesic completeness (or incompleteness) might force null geodesic completeness (or incompleteness). However, in a series of examples, these earlier hopes were found to be too optimistic. Kundt [71] gave an example which was timelike and null geodesically complete, but not spacelike complete. Geroch [59] gave a globally hyperbolic example, conformal to Minkowski 2-space, which was null and spacelike complete, but not timelike complete.
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