14

PAUL E. EHRLICH AND KEVIN L. EASLEY

It is interesting that while neither causal pseudoconvexity nor causal dispris-

onment is a stable property by itself, together causal pseudoconvexity and causal

disprisonment are

C

1-stable in Lor(

M).

Indeed, the combination of these two prop-

erties may be regarded as a generalization of global hyperbolicity, for which Geroch

[61]

observed the

C

0-stability in Lor(M).

In place of the assumption of Riemannian completeness, Beem and Parker

[31]

proved the following working tool for a manifold M with linear connection \7.

LEMMA

5.3. Let (M, \7) be both pseudoconvex and disprisoning. Assume that

Pn

-+ p and

Qn

-+ q for distinct p, q in M. If each pair

Pn, Qn

can be joined by a

geodesic segment, then there exists a geodesic segment from p to q.

Using this tool, Beem and Parker obtained a result akin to the type of thing

classically obtained in Riemannian geometry for Cartan-Hadamard manifolds.

THEOREM

5.4. Let (M,

\7)

be both pseudoconvex and disprisoning. If (M,

\7)

has no conjugate points, then (M, \7) is geodesically connected. Thus for each p in

M the exponential map expP : TpM -+ M is a diffeomorphism of M with JRn.

Here is an example of how these two conditions under consideration rescue the

stability of geodesic completeness.

THEOREM

5.5. Let (M, g) be causally pseudoconvex and causally disprisoning.

If ( M, g) is nonspacelike geodesically complete, then there is a fine C

1

-neighborhood

U(g) of gin Lor(M) such that all g' in U(g) are nonspacelike geodesically complete.

In Beem and Ehrlich [23], a more conceptual study of certain of the construc-

tions in Williams

[92]

was made. First, in a partial return to the roots of Ehrlich

[50],

a study was made of how conformal changes interfaced with null geodesic

completeness. It was found that "small" conformal changes will destroy neither

null completeness nor null incompleteness for pseudo-Riemannian manifolds (note

that (5.2) and (5.3) are not conformal changes of metric). As a consequence, it

follows that for a compact manifold M, all metrics in the conformal class C(M,g)

are either null geodesically complete or null geodesically incomplete. In the 1990s,

Romero and Sanchez at Granada and others conducted studies of the geodesic be-

havior on compact space-times, obtaining a detailed and rich understanding ( cf.

[82] and [84] for two examples out of many).

Secondly, in the spirit of the Williams' examples which contained closed null

geodesics, the following general result was obtained:

THEOREM

5.6.

Let (M, g) be a pseudo-Riemannian manifold with a closed null

geodesic (3: [0, 1]-+ M satisfying (3'(0)

=

(3'(1). Then each C

00

-fine neighborhood

U(g) of g in Pseudo(M) contains a metric g1 which contains an incomplete closed

null geodesic (and thus is null incomplete).

The proof relies on a very non-Riemannian phenomenon important in certain

aspects of General Relativity: given a smooth closed null geodesic 'Y : [0, 1] -+

( M, g), it is not automatically the case, as it is for Riemannian ( timelike or space-

like) geodesics, that "Y'(O)

=

1'(1),

forcing the geodesic to be complete. Instead,

all that can be guaranteed in the null case is that "Y'(O) and "'(1 (1) are proportional.

If these two vectors are unequal, then the given null geodesic is either future in-

complete or past incomplete ( cf. [24], pages 243-244 for a proof). So in [23], the

given complete closed null geodesic

f3

was perturbed in a tubular neighborhood (by