24
PAUL E. EHRLICH AND KEVIN L. EASLEY
product manifolds. The students are listed below in chronological order, along with
a brief summary of their thesis research.
M. A. Kishta
[69]
wrote a thesis on generalized indefinite and Einstein-Finsler
spaces; his work
[29]
with Beem was discussed earlier in Section 2.
T. G. Powell
[81], [33], [26]
investigated Lorentzian manifolds with non-smooth
metrics and, working both individually and jointly with Beem and Ehrlich, stud-
ied issues of nonspacelike completeness and stability of timelike incompleteness of
Lorentzian warped products, as well as issues relating to the cut and conjugate loci
of warped product manifolds.
D. E. Allison
[1],
inspired by the work of Kemp [
68],
investigated semi-Riemannian
warped products ofthe form
(a, b)
1 x
H,
where
(a, b)
is a real interval equipped with
the negative definite metric
-dt2
, (
H, h)
is a Riemannian manifold,
f
is a smooth,
positive, real-valued function on H, and the metric on the product is of the form
- P
dt
2
EB
h.
Allison investigated issues of geodesic completeness and curvature and
energy conditions in this class of standard static space-times
[2], [3]
and, in later
work, explored pseudo-convexity in doubly warped products
[4]. In [5],
he studied
Lorentzian submersions from a space-time onto a Riemannian base, generalizing
his earlier work on standard static space-times. See also Allison's joint work with
B. Unal cited below.
K.
L. Easley
[49], [24]
derived necessary and sufficient conditions for the local ex-
istence of warped product metrics on a given semi-Riemannian
(M, g),
defined and
studied the class of multiply warped products of the form
Box
11
B
1
x
h
x · · · x
!k
Bk
with metric
g = gBo
EB
I:i
fi 2gBi and, inspired by Beem and Parker
[32],
stud-
ied tidally destructive null directions in semi-Riemannian warped products, show-
ing in particular that under mild physical assumptions, all null directions in the
Robertson-Walker cosmological models are tidally destructive. Recently, multiply
warped products have been found useful in obtaining a more detailed understand-
ing of the issue of geodesic connectability, especially in certain standard space-time
models, cf. Flores and Sanchez
[54],
among others.
J. Choi
[44], [45]
studied multiply warped products with nonsmooth metrics and
warped product spaces with non-smooth warping functions. Choi also used the
multiply warped product formalism to study the Schwarzschild black hole.
B. Unal [88] explored the geometry of doubly and multiply warped products, specif-
ically studying global hyperbolicity and geodesic completeness on Riemannian and
Lorentzian multiply warped products
[89],
and geodesic completeness, Killing and
conformal fields on doubly warped products
[90].
In later joint work, Allison and
Unal
[6]
obtained conditions which guarantee nonreturning and pseudo-convex ge-
odesic systems on standard static space-times. Also, Unal, Kupeli, and others
studied sufficient conditions for a twisted product to be a warped product
[53].
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