obtained first with the assumption of constant curvature in Kamishima
and in
full generality in Romero and Sanchez [82].
To help the reader recall the time frame in which Beem and his co-authors'
earlier work was set, the following table shows the dates of publication for various
influential texts in differential geometry and General Relativity.
R. Penrose, Techniques of Topology in General Relativity 1972
S. Hawking and G. Ellis, The Large Scale Structure of Space-time 1973
C. Misner, K. Thorne, and J. Wheeler, Gravitation 1973
R. Sachs and H. Wu, General Relativity for Mathematicians 1977
J. Beem and P. Ehrlich, Global Lorentzian Geometry 1981
B. O'Neill, Semi-Riemannian Geometry 1983
Publication Dates for Selected Standard References in
Differential Geometry and General Relativity
2. Early Beginnings
A half-decade before any of the standard reference works mentioned in Table
1 had appeared, Professor Herbert Busemann of the Department of Mathematics
at the University of Southern California, Beem's future advisor, decided to develop
a geometric theory for indefinite metrics analogous to the theory of metric G-
spaces which he had developed for the positive definite case,
Busemann [39].
Hence, Busemann formulated the concept of timelike spaces, general Hausdorff
spaces having a partial ordering with properties similar to those of the chronological
partial ordering
p q
of a space-time. Also, Busemann supposed that these
timelike spaces were equipped with a function d( , ) which behaves just like the
Lorentzian distance function of a chronological space-time restricted to the set
{(x,y) EM x M : x:::;; y}. Thus the function d(x,y) on l! was assumed to
satisfy the following three conditions:
(a) d(x,x)
0 for all x,
(b) d(x,y) 0 if x y, and
(c) d(x, z)
d(x, y)
d(y, z)
x y z.
(Interestingly, several theoretical physicists rediscovered similar axiom systems and
lattice theoretic concepts in the 1990s, apparently unaware of Busemann's seminal
work of 1967.) For this class of nondifferentiable spaces, Busemann observed that
the length of continuous curves could be defined and that the length functional is
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