A BEEMIAN SAMPLER: 1966-2002
21
for the evaluation of curvature, geodesic equations, etc., must be altered accordingly.
It should also be noted that in Beem, Ehrlich, and Easley
[24],
p. 95, as in Beem's
early work, a Lorentzian warped product is defined as the product of a base ( M, g)
of signature ( -,
+, ... , +)
with a Riemannian fiber
(F,
gp ).
The class of Riemannian warped product manifolds was first defined and stud-
ied by R. Bishop and B. O'Neill in 1969 in
[36],
where it was used to construct
a variety of complete Riemannian manifolds of everywhere negative sectional cur-
vature. The introduction of the warped product concept into semi-Riemannian
geometry and General Relativity was first made by Beem, Ehrlich, and Powell ( cf.
[21], [26]),
who observed that many of the well-known exact solutions to Einstein's
field equations are semi-Riemannian warped products.
With the publication of their respective books ( cf. Table
1)
in 1981 and 1983,
Beem and O'Neill introduced the power and utility of warped product techniques-
especially in relativistic studies-to a much wider audience. In the First Edition of
the research monograph
Global Lorentzian Geometry [21],
Beem and Ehrlich
devoted fully 25 pages (Section 2.6) to the study of Lorentzian warped products, ex-
ploring issues of causality, completeness, chronology, etc.-all of the general themes
of the monograph. With the publication of O'Neill's
Semi-Riemannian Ge-
ometry [78]
in 1983-a text suitable as an introduction for first-year graduate
students-warped products were made accessible to an even wider audience. More
recently, warped product manifolds have begun to figure prominently in many stan-
dard references, both in Lorentzian and Riemannian geometry ( cf. Besse
[35],
Dug-
gal and Bejancu
[47],
Duggal and Sharma
[48],
Kriele
[70],
and Petersen
[80]).
Turning aside from historical matters, we now return to the question of exact
solutions to Einstein's equations. There is a rather commonly held misconception
that warped product techniques are of limited use in General Relativity, since the
frequent appearance of warped products as exact solutions to Einstein's equations
is merely indicative of the many unrealistic, idealized physical assumptions that
must be made in solving these equations. For example, the strong assumptions
of spatial homogeneity and isotropy, together with the idealized treatment of the
stress-energy content of the universe as a "perfect fluid," lead inexorably to the
warped product structures of the Robertson-Walker cosmological models, with their
constant curvature Riemannian fibers.
The above objection somewhat misses the point, however, as Professor Beem
was fond of observing. After one has used the advantages afforded by the warped
product formalism to derive a result of physical interest, one must then address the
issue of the result's stability under perturbations of the metric in the appropriate
topology on Lor(M). If stability can be established, the "realist's" qualms about
using warped products are, to some extent at least, undermined ( cf. p. 268 of
[24]).
We must remark also that, for a given smooth manifold M, the set of metrics in
Lor(M) which factor as warped products is a "sparse" set under the Whitney
cr
topologies for r ;:::: 1. This raises the interesting question of why the above strategy
should work as often as it does, an issue we unfortunately cannot pursue further in
this short summary.
The basic causal properties of Lorentzian warped products were first explored
by Beem and Ehrlich in the First Edition of
Global Lorentzian Geometry [21].
We now consider a small selection from these early results.
In [36],
as previously noted, Bishop and O'Neill proved that the Riemannian
warped product M
=
B
Xf
F is complete iff both Riemannian factors (B,gB) and
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