A BEEMIAN SAMPLER: 1966-2002

15

a study of the Christoffel symbols and the geodesic ODEs) to a reparametrized null

geodesic'' with the same image as

/3,

but having

1'(1)

=

cr'(O) with c

1.

Hence

the new null geodesic

'Y

was future incomplete.

We will give Professor Beem the final word in this section and quote exactly a

result which he presented in a Special Session at a joint meeting of the American

and Canadian Mathematical Societies in Vancouver during the summer of 1993

[15]:

THEOREM

5.7.

Let (M,g) be a semi-Riemannian manifold. Assume that

(M,g) has an endless geodesic 'Y : (a, b)

~

M such that 'Y is incomplete in the

forward direction, (i.e., b

=f.

oo).

If

r

is not partially imprisoned in any compact

set as t

~

b, then there is a C

1

-neighborhood U(g) of g such that each g1 in U(g)

has at least one incomplete geodesic c. Furthermore, if 'Y is timelike (respectively,

null, spacelike}, then c may be taken correspondingly.

6. The Lorentzian Splitting Problem

During the academic year 1979-1980, a Special Year in Differential Geometry

was held at the Institute for Advanced Study in Princeton, New Jersey, with lead

organizer Professor Shing-Thng Yau. The first author was fortunate enough to have

been invited to participate in this program and elected to spend the second semester

at the Institute. In the waning days of this session, Yau delivered a series of lectures,

suggesting problems in differential geometry worthy of consideration. The list was

published a few years later in Yau

[94]

in the Annals of the Mathematics Studies

volume stemming from the Special Year in Differential Geometry at the Institute.

As the first author was attending these lectures, as a student of Professor Detlef

Gromoll at Stony Brook and also having come under the influence of Professor Jeff

Cheeger, he could not help but notice one of the problems Yau proposed:

CONJECTURE (YAu).

Show that a space-time (M,g) which is timelike geodesi-

cally complete, obeys the timelike convergence condition, and contains a complete

timelike line, splits as an isometric product

(IR x

V, -dt2 +h).

This problem was stated without motivation as a proposal to obtain the space-

time analogue of the celebrated Cheeger-Gromoll splitting theorem for Riemannian

manifolds (

cf.

[43]). As the first author thought about suggesting to Professor Beem

that we attack this problem with the aid of a visiting postdoctoral researcher from

Denmark, Dr. Steen Markvorsen, he was puzzled as to why Yau had formulated

the problem with the hypothesis of timelike geodesic completeness rather than

global hyperbolicity. For, recall from Section 1 that timelike geodesic completeness

does

not

guarantee the existence of maximal timelike geodesic segments between

chronologically related pairs of points, while global hyperbolicity

does

guarantee

that helpful property. But this question was to go unanswered for several months

until Professor G. Galloway, passing through Columbia for a short visit on the way

back to Miami from a sabbatical in San Diego, could enlighten us himself as to

what he had learned from S.-T. Yau.

At the time we began consideration of the problem, we were aware of various

results on related issues employing maximal hypersurface methods, (

cf.

Bartnik [8],

Gerhardt [58], and Galloway [55], among others). As decided nonexperts in maxi-

mal hypersurfaces, it seemed to us that it might be a wiser course to try to begin the

study of the Busemann function of a timelike geodesic ray, since that tool had been