4
PAUL E. EHRLICH AND KEVIN L. EASLEY
upper semicontinuous in a topology of uniform convergence. As part of this work,
Busemann studied finite compactness and metric completeness in the spirit of (1)
and (
4)
of the Hopf-Rinow Theorem, as recalled in Section 1.
In Beem's first co-authored work, Busemann and Beem
[40],
undertaken before
Beem started work on his thesis, aspects of generalized indefinite metric spaces were
studied. In particular, the authors offered two basic examples of such spaces-
Minkowski space on the first page of their paper, and what has become known as
the Lorentz-Poincare model, discussed in the last two pages of their paper. Recall
that in basic complex variables or differential geometry, if
(2.1)
U
=
{(x,y)
E
1R2
:
y 0}
=
lR
X
(O,+oo)
then the Riemannian metric
(2.2)
furnishes
U
with a geodesically complete, constant curvature metric with
K
=
-1.
If desired, by a change of variables this could be recast as a warped product metric
on JR2
.
In
[40],
the set
U
of (2.1) was retained, but for this second class of examples,
the Lorentzian metric
(2.3)
was considered. It was noted that this Lorentz analogue to the Poincare upper
half-plane, while of constant curvature, fails to be geodesically complete. Later, in
Beem [14], a more detailed study would be made of this type of Lorentz space.
In O'Neill's text (Example 41, p. 209 of [78]), this model is given as a fun-
damental example, but in a form which does not on the face of it resemble (2.1)
and (2.3). O'Neill has been discussing the geodesic equations for a general semi-
Riemannian warped product metric and obtains the helpful result that for any pair
of complete Riemannian factors and any warping function, a Riemannian warped
product is automatically geodesically complete. To fit the Busemann-Beem exam-
ple into this framework, first make a change of variables
t
=
ln y : (0,
+oo)
----
JR, so
that
Now make a change of variables
t
~---+
-t,
and one has the Lorentzian metric
on the manifold
M
=
JRx 1R In Example 41, O'Neill observes that this is an impor-
tant metric because while the factors
(JR,
-dt2
)
and
(JR,
dx
2
)
have complete definite
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