20 PAUL E. EHRLICH AND KEVIN L. EASLEY The first author came to his collaboration with Professor Beem from a back- ground of graduate work in global Riemannian geometry and three semesters of the basic large lecture physics in college. Professor Beem studied much more physics in his undergraduate work at the University of Southern California and indeed, was for a time torn between majoring in mathematics or physics. Thus the first author always admired Professor Beem's deep love for the physics and astronomy behind the differential geometry on which we worked, as well as his keen sense for constructing examples and counterexamples. As a tribute to the influence of the physical motivation on Professor Beem's work, we conclude this section by quoting the first author's favorite passage from the seminal text by Hawking and Ellis ([64], p. 134): "However, we are not able to make cosmological models with- out some admixture of ideology. In the earlier cosmologies, man placed himself in a commanding position at the centre of the uni- verse. Since the time of Copernicus we have been steadily demoted to a medium sized planet going round a medium sized star on the outer edge of a fairly average galaxy, which is itself simply one of a local group of galaxies. Indeed we are now so democratic that we would not claim that our position in space is specially distin- guished in any way. We shall, following Bondi (1960), call this assumption the Copernican principle." 7. Warped Product Manifolds in General Relativity One particular class of manifolds, semi-Riemannian warped products, has been central to much of Professor Beem's work, appearing in his individual research, in many of his collaborative efforts, and especially in the work of his students. Indeed, Beem, his collaborators, and his students have collectively been responsible for much of the early, foundational work on applications of warped product manifolds to global Lorentzian geometry. Given the confinements of space in this paper, we shall content ourselves with providing a brief historical perspective and a few of the more interesting and representative results that have been achieved. Warped products are the natural first-order generalization of product mani- folds, including products as a special case: DEFINITION 7.1. Let (B,gB) and (F,gp) be semi-Riemannian manifolds, and let f : B -+ ( 0, +oo) be a smooth function on B. The warped product M = B x f F is the product manifold B x F equipped with the metric tensor g = gB EB f gp. With such a representation, the standard mathematical maneuver of decom- posing a complex object into (presumably) simpler parts is achieved the principal components in the geometry of (M, g) are the geometry of the base (B, gB) and fiber (F, gp) manifolds, and various derivatives of the warping function f. We use the convention on the warping function which was first adopted by Beem and Ehrlich in [21] because it is consistent with so much of Beem's early work in warped products readers who first learned their semi-Riemannian geometry from O'Neill's influential text [78] will be more accustomed to the definition g = gB EB P gp. The difference is relatively insignificant, but the fundamental formulae
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