We now briefly summarize how the proof of the splitting theorem was extended
from the sectional curvature hypothesis to the desired timelike convergence con-
dition that Ric(
v, v)
2 0 for all timelike (hence all nonspacelike) tangent vectors.
J.-H. Eschenburg
obtained the first important breakthrough in realizing that
instead of trying for global control of the timelike co-rays on
as in Beem,
Ehrlich, Markvorsen, and Galloway
it was sufficient to obtain a splitting in a
tubular neighborhood of the given timelike geodesic line and then extend the split-
ting to all of (
M, g)
through a procedure similar to the one used in making analytic
continuation type arguments in complex analysis. Hence, in
the splitting the-
orem was obtained under the assumption of both timelike geodesic completeness
and global hyperbolicity in the Ricci curvature case.
Working with this new idea, Galloway was able to remove the hypothesis of ge-
odesic completeness shortly thereafter ( cf. Galloway
Then Newman returned
to the original question of Yau and obtained the splitting for timelike geodesic
completeness rather than global hyperbolicity ( cf. Newman
Here, Newman
had to confront the issue that maximal nonspacelike geodesic segments could not
be constructed without global hyperbolicity, so he had to work with almost maxi-
mizers instead of geodesics, introducing a higher level of complexity. A philosophy
which emerged is that the existence of a maximal geodesic segment implies that
things work out better in a tubular neighborhood of this maximal segment, in terms
of the behavior of almost maximizers, the Busemann function, etc. In Galloway
and Horta
these ideas were given a much simplified exposition and especially
the original timelike co-ray condition of the earlier work
morphed into the
generalized timelike co-ray condition.
Out of all of these results, the Lorentzian Splitting Theorem emerged.
6.3. Let (M,g) be a space-time of dimension n
3 which satisfies
each of the following conditions:
1. ( M, g) is either globally hyperbolic or timelike geodesically complete.
2. ( M, g) satisfies the timelike convergence condition.
3. (M, g) contains a complete timelike line.
Then (M, g) splits isometrically as a product
(JR x
V, -dt2 +h), where (H, h) is a
complete Riemannian manifold.
As mentioned above, Beem and the first author had no inkling as to why Yau
had proposed the problem of proving a Lorentzian splitting theorem (which was not
mentioned by Yau in his lectures at the Institute or the Problem List written from
those lectures), but that was explained to us by Galloway when he passed through
Columbia en route to Miami from San Diego and spoke about what was published as
Yau's motivation had been the idea that timelike geodesic completeness should
interface with the concept of "curvature rigidity" which had been formulated during
the 1960s and 1970s in global Riemannian geometry, cf. especially the exposition
in the introduction to the text Cheeger and Ebin
where it was first widely
Recall our earlier statement of the Sphere Theorem of global Riemannian ge-
ometry; if a complete, simply connected Riemannian manifold has sectional curva-
tures strictly 1/ 4-pinched, then it is homeomorphic to the n-sphere of the same
dimension. In the statement of this result, there is a curvature condition of strict
inequality. For curvature rigidity, the condition of strict inequality is relaxed to
include the possibility of equality as well, and one tries to show that either the
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