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