ABSTRACT
Global Riemannian geometry was first widely disseminated in 1975 in the first two paragraphs of the preface to the influential monograph by J. Cheeger and D. Ebin. In 1932 Busemann introduced an analytic method of studying generalizations of the classical horospheres of hyperbolic geometry to a very general class of metric geometries. In the proof of the Cheeger-Gromoll Splitting Theorem given in Cheeger and Gromoll, this particular function was independently rediscovered during the course of the proof as a key ingredient. A Riemannian proof contained in Eschenburg and Heintze for the Cheeger-Gromoll Riemannian Splitting Theorem, different than that originally given in Cheeger and Gromoll, provided a helpful model for obtaining results in the space-time context. Subsequent proofs of the Lorentzian Splitting Theorem in the globally hyperbolic case under the weaker Ricci curvature assumption dealt with showing that the asymptotic geodesic construction would be well behaved in some neighborhood of the given timelike ray.
