18

PAUL E. EHRLICH AND KEVIN L. EASLEY

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

[51]

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

[25],

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

[51],

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

[56]).

Then Newman returned

to the original question of Yau and obtained the splitting for timelike geodesic

completeness rather than global hyperbolicity ( cf. Newman

[76]).

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

[57],

these ideas were given a much simplified exposition and especially

the original timelike co-ray condition of the earlier work

[25]

morphed into the

generalized timelike co-ray condition.

Out of all of these results, the Lorentzian Splitting Theorem emerged.

THEOREM

6.3. Let (M,g) be a space-time of dimension n

2

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

[55].

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

[42],

where it was first widely

publicized.

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