On the completeness of geodesics obtained as a limit

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Abstract
The process to obtain a geodesic as a limit from a sequence of geodesics in a Lorentzian manifold is analyzed in detail. It is found that the limit of incomplete geodesics in a compact manifold is not incomplete in general. A new class of incomplete compact Lorentzian manifolds, one of them showing this anomalous behavior, is constructed. Thus, a counterexample to the proof, given by U. Yurtsever [J. Math. Phys. 33, 1295 (1992)] to the result originally proven by Y. Carrière [Invent. Math. 95, 615 (1989)] (in a quite different way) is given. That proof should imply that an incomplete compact Lorentzian manifold must be null incomplete. It is not known if this assertion is true, and, hence, remains as an open question.

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