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
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-
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:
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
is the product manifold B x F equipped with the metric tensor
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
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
will be more accustomed to the definition g
gp. The difference is relatively insignificant, but the fundamental formulae
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