22
PAUL E. EHRLICH AND KEVIN L. EASLEY
(F,
gp)
are complete. One remarkable feature of this result is that completeness
holds for
any
warping function
f
on
B.
The Busemann-Beem example described in
detail in Section 2 shows that there is no exact Lorentzian analogue of this result,
even when both factors are
definite,
once again illustrating the striking differences
between Lorentzian and Riemannian geometry.
It
is natural to inquire what can
be salvaged in the Lorentzian case.
The decomposition of the warped product metric g
=
g
EB
fh
implies that
the basic projection maps
1r :
M x
f
H
---+
M and
a :
M x
f
H
---+
H have the
properties that
1r!Mxb
---+
M
is an
isometry
for each
b
E
H,
and
aimxH
---+
H
is a
homothety with factor 1/
f(m)
for each
mE
M.
Consequently, given any tangent
vector
v
E
T(M x H), one then has the useful inequality
g(1r*V,1r*v) ::; g(v,v),
yielding two immediate consequences: (i) the map
1r* :
Tp(M x H)
---+
T1r(p)M
maps nonspacelike vectors to nonspacelike vectors, and
(ii)
the map
1r
is length
nondecreasing on nonspacelike curves.
Within the context of the Robertson-Walker cosmological models, these fun-
damental relations were used to obtain the following satisfying Lorentzian analogue
for the Riemannian completeness theorem
(cf.
Beem and Ehrlich [21], p. 65; Beem,
Ehrlich, and Easley [24], p. 103) in the simpler product case with base manifold
(JR,
-dt2
).
THEOREM
7.2.
Suppose that
(H,
h) is a Riemannian manifold and that lR x
H
is given the product Lorentzian metric -dt2
EB
h. Then the following are equivalent:
1. ( H, h) is geodesically complete.
2.
(JR
x H, -dt2
EB
h) is geodesically complete.
3.
(JR
x H, -dt2
EB
h) is globally hyperbolic.
If
the dimension of the base manifold (
M,
g)
is two or larger, then the situation
naturally becomes more difficult. The following result (
cf.
[21], p. 66; [24], p. 104)
shows what can be recovered with an additional assumption on the base manifold
(M,g).
THEOREM
7.3.
Let
(M,
g) be a space-time, and let
(H,
h) be a Riemannian
manifold. Then the Lorentzian warped product (Mx fH, gEBfh) is globally hyperbolic
iff both of the following conditions are satisfied:
1. ( M, g) is globally hyperbolic.
2.
(H,
h) is a complete Riemannian manifold.
Along the way to establishing Theorems 7.2 and 7.3, Beem and Ehrlich ([21], p.
61; [24], p. 100) proved the following preliminary result, which is also independent
of the warping function.
PROPOSITION
7.4. Let
(M,
g) be a space-time, and let
(H,
h) be a Riemann-
ian manifold. Then the Lorentzian warped product ( M
x
f
H, g
EB
f h) is strongly
causal (respectively, chronological, causal) iff
(M,
g) is strongly causal (respectively,
chronological, causal).
Considered together, the above remarks produce a theme of sharp contrasts.
In the case of Riemannian warped products with complete base and fiber, geodesic
completeness is independent of the choice of warping function f; in the space-time
case, by contrast, only certain causal aspects of the warped product are independent
of the choice of warping function. There is also the additional complication of dis-
tinct dimensional cases, as illustrated in Theorems 7.2 and 7.3, which characterizes
so many results in Lorentzian warped product analysis.
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