A BEEMIAN SAMPLER: 1966-2002
5
metrics, this particular warped product (with f(t)
=
et
in O'Neill's convention) is
geodesically incomplete; indeed,
sO
provides an example of an incomplete null geodesic. Here one checks that the
geodesic equations for the warped product are satisfied and also
II'Y'(s)ll2
=-
(s12)
+
e(2lns) (- s12) 2
-1
s
2
=
-2
+4 =O.
s s
Hence, arbitrary warped products (even of definite factors) in the semi-Riemannian
case may fail to be geodesically complete. So that is what became of (2.1) and (2.3)
in O'Neill
[78]!
After this initial collaboration, Beem wrote his dissertation, which appeared
as a Memoir of the American Mathematical Society, along with work of a second
Busemann student
[34].
During his early years at Missouri, Beem produced seven papers in the area of
Finsler geometry and indefinite metrics and had one Ph.D. student in this field, M.
A. Kishta. As one example of this work, Beem and Kishta published a paper in
the Indiana Mathematics Journal with the title "On generalized indefinite Finsler
spaces," cf.
[29].
During this time, the first author was in his two-year postdoctoral
appointment at Bonn University and working with H.-C. ImHof on the geometry
of the Dirichlet fundamental region for Riemannian manifolds without conjugate
points. In this theory, the sides of the Dirichlet region are formed by bisectors
B
={mE
M:
d(p,m)
=
d(q,m)}
for two distinct points p, q E M. The first author recalls browsing in the mathemat-
ics reading room at Bonn and seeing J. Beem
[10]
"Pseudo-Riemannian manifolds
with totally geodesic bisectors" appear in the new journals shelf. As a young job
applicant sending applications back to the U.S., this title certainly caught the au-
thor's eye, but Beem and the first author would never employ bisectors in any of
their joint work.
3. Studies of Nonspacelike
Geodesics
If any point p in a space-time
(M,
g) is selected, then emanating from p we
have the three families of timelike, spacelike, and null geodesics. Since the null
geodesics are, in a naive, imprecise fashion, limits of the spacelike and timelike
geodesics, it was hoped at one time that, possibly, continuity results could be
obtained for the different types of geodesic completeness. For example, perhaps
timelike and spacelike geodesic completeness (or incompleteness) might force null
geodesic completeness (or incompleteness). However, in a series of examples, these
earlier hopes were found to be too optimistic. Kundt
[71]
gave an example which
was timelike and null geodesically complete, but not spacelike complete. Geroch
[59]
gave a globally hyperbolic example, conformal to Minkowski 2-space, which
was null and spacelike complete, but not timelike complete.
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