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
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