2
PAUL E. EHRLICH AND KEVIN L. EASLEY
Moreover, if any one of
(1)
through
(4)
holds, then (N,
go)
also satisfies
5. minimal geodesic connectability: given any p, q
E
N, there exists a
smooth geodesic segment
c: [0,
1]
---
N with
c(O)
=
p,
c(1)
=
q, and
L(c)
=
do(p,q).
Finally, the Heine-Borel property of basic topology implies (via (4)) that all
Riemannian metrics for a compact manifold are automatically complete. Also, the
basic examples that one introduces in a beginning Riemannian geometry course,
such as
sn, JRpn, IRn,
and
rn,
all carry complete Riemannian metrics as usually
described. From one viewpoint, one may regard the Hopf-Rinow Theorem as as-
serting that complete Riemannian metrics are the proper objects of study in the
global differential geometry of Riemannian manifolds.
In a somewhat related vein, thanks to (1) of the Hopf-Rinow Theorem, it is
well known that if the space of all Riemannian metrics Riem(N) for a fixed smooth
manifold N is considered, then
(1.1) both geodesic completeness and
geodesic incompleteness are
C0
-stable in Riem(N).
An elementary proof may be found in
[23].
Now if we leave the Riemannian world and enter the realm of General Relativ-
ity, then unlike the basic complete or compact examples explained in elementary
Riemannian geometry courses, we first find that several basic examples such as
Schwarzschild space-time and the big bang cosmological models are nonspacelike
geodesically incomplete. Moreover, in the 1970s attention was primarily focused on
noncompact manifolds, because any compact space-time contains a closed timelike
curve, thus violating the basic chronology condition of General Relativity.
The Hopf-Rinow Theorem fails to hold in general space-times. Indeed, we have
explicitly recalled the statement of this result because so much of Beem's research
has been set in the arena of what can be rescued for space-times and, later, semi-
Riemannian manifolds. (For example, Beem
[11],
which we will not discuss below,
concerned what could be done with finite compactness, especially in the globally
hyperbolic case.)
Also, nothing as simple as (1.1) holds for the space Lor(M) of all Lorentzian
metrics for a given smooth manifold M without imposing further conditions on the
background space-time
(M,
g) in question.
As one basic aspect of the failure of the Hopf-Rinow Theorem for space-times,
it should be emphasized at the outset that compactness of the underlying manifold
M by itself does not imply geodesic completeness of the space-time ( M, g). A
well-known result in basic Riemannian geometry is the proof that a homogeneous
Riemannian metric on an arbitrary smooth manifold is automatically geodesically
complete. This result fails to hold for indefinite metrics, but in Marsden
[73]
it
was noted that a compact space-time with a homogeneous metric is geodesically
complete (providing an early example of some less na'ive aspects of the Hopf-Rinow
Theorem being valid for space-times).
Much later, in Carriere
[41],
it was shown that a compact, flat space-time is
geodesically complete; thus, adding the requirement that the Riemannian curva-
ture tensor vanish rescues that aspect of the Hopf-Rinow Theorem. A second, more
recent example which may be cited is the result that a compact Lorentzian man-
ifold which admits a timelike Killing field is automatically geodesically complete,
Previous Page Next Page