A BEEMIAN SAMPLER:

1966-2002

7

THEOREM 3 .1. Timelike geodesic completeness, null geodesic completeness,

and spacelike geodesic completeness are logically inequivalent.

In a somewhat related area, Beem also studied another important issue in

space-time geodesic geometry at the time. A well-known result of Nomizu and

Ozeki

[77]

asserts that an arbitrary Riemannian metric for a smooth manifold

can be made geodesically complete by a conformal change of metric. We were all

exposed to at least a statement of this result in any basic differential geometry

course! On the other hand, the situation for space-times had been seen to be

more complicated. Misner

[7

4] gave a 2-dimensional null geodesically incomplete

example which could not be made complete by any conformal change (since the

incomplete null geodesics were future trapped in a compact set and would remain

future trapped null pre geodesics under any conformal change of metric). Seifert [86]

showed that if ( M,

g)

were stably causal, then a conformal change of metric could be

made which would produce future nonspacelike geodesic completeness. Clarke [46]

showed that a strongly causal space-time could be made null geodesically complete.

So that is the setting for Beem's paper

[12]

on "Conformal changes and geodesic

completeness."

In this paper, Beem formulated what he termed "Condition N" for nonimpris-

onment:

DEFINITION 3.2. The causal space-time (M,g) will satisfy Condition N if, for

each compact subset K of M, there is no future inextendible nonspacelike curve

x(t) which is totally future imprisoned in K.

Here it should be noted that while in Riemannian geometry attention is often

simply restricted to geodesics, typically in General Relativity one has to consider

all nonspacelike curves, not just geodesics. Here also the curve x(t) is said to be

totally future imprisoned in K if there exists

h

so that x ( t)

E

K for all

t

2:

h.

By making a sequence of conformal changes related to a compact exhaustion

and taking the infinite product of those functions for the final conformal factor,

Beem established

THEOREM 3.3. Let (M,g) be a causal space-time which satisfies Condition N.

There is some conformal factor

n

such that (M,

n2

g) is null and timelike geodesi-

cally complete.

As a corollary, it followed that if

(M, g)

were distinguishing, strongly causal,

stably causal or globally hyperbolic, then Condition N held, so that (

M, g)

could

be made nonspacelike geodesically complete by a conformal change of metric. In

Beem and Powell [33] an interesting study was made of Condition N for doubly

warped products.

Since conformal changes are being discussed, comments may also be made in

this section about the first collaboration of Beem and the first author in

[17],

"Con-

formal deformations, Ricci curvature and energy conditions on globally hyperbolic

space-times." The first author's thesis research was originally motivated by efforts

to better understand a result of T. Aubin

[7],

which considered the question of

deforming a Riemannian metric of nonnegative Ricci curvature and all Ricci cur-

vatures positive at some point, to a metric of everywhere positive Ricci curvature.

Looking at

[7]

led to the study of local convex deformations: conformal de-

formations of a given Riemannian metric expressed in terms of the distance to the

boundary of a convex metric ball. It was found that if the given Ricci curvature was