Stability of Completeness and Incompleteness

This chapter gives examples of Williams to show that both geodesic completeness and geodesic incompleteness may fail to be stable. It shows that nonimprisonment is a sufficient condition for the C stability of geodesic incompleteness. Sufficient conditions for the stability of geodesic completeness involve pseudoconvexity as well as nonimprisonment. An inextendible closed geodesic c is one which repeatedly retraces the same image. For spacelike and timelike geodesics this implies the geodesic is complete since these geodesics have constant nonzero speed and thus increase in affine parameter by the same amount for each circuit of the image. Geodesic completeness and geodesic incompleteness are stable properties in the Whitney C⁰ topology for both positive definite and negative definite semi-Riemannian manifolds.