Contemporary Mathematics Volume 1 TO, 1994 Stability of Geodesic Incompleteness JOHN K. BEEM ABSTRACT. Geodesic incompleteness is shown to be stable for many semi- Riemannian manifolds. If the semi-Riemannian manifold (M, g) has an incomplete geodesic which is not partially imprisoned in any compact set, then there is a C 1 -fine neighborhood U(g) of g in the space of semi- Riemannian metrics on M such that all metrics in U (g) are geodesically incomplete. Consequently, causal geodesic incompleteness is C 1 -stable for strongly causal spacetimes. This yields stability results for singularity theorems in general relativity. 1. Introduction It is well known that compact positive definite Riemannian manifolds are always complete. On the other hand, compact spacetimes need not be com- plete, compare Fierz and Jost [9]. Examples of Williams [15] show that both geodesic completeness and geodesic incompleteness are, in general, unstable for spacetimes. In fact, the examples show that both of these properties may fail to be stable for compact spacetimes as well as noncompact spacetimes. These instabilities are related to questions involving sprays and the Levi-Civita map, see Del Riego and Dodson [8]. Furthermore, it is natural to ask for reasonable sufficient conditions to guarantee the stability of each of these properties. The main result of this paper is that geodesic incompleteness is C 1 -fine stable if there is a geodesic with an incomplete end which is not partially imprisoned in any compact set, see Theorem 3.4. For positive definite Riemannian metrics both geodesic completeness and geo- desic incompleteness are stable in the Whitney cr -fine topologies for all r :::: 0. Because of the geometric and physical importance of completeness and incom- pleteness, it is of clear interest to establish sufficient conditions for these proper- ties to be stable for semi-Riemannian manifolds. The following stability theorem 1991 Mathematics Subject Classification. Primary 53C50, 53C80. This paper is in final form and no version of it will be submitted for publication elsewhere © 1994 American Mathematical Society http://dx.doi.org/10.1090/conm/170/01740
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