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On complete Riemannian manifolds, there is a characterization of the time $t_0$ when a geodesic $c$ stops being minimizing: either $c(t_0)$ is conjugate to $c(0)$ along $c$, or there exists a geodesic $\gamma$ from $c(0)$ to $c(t_0)$ with $\gamma \neq c$ and $L(\gamma) = L(c)$. (See Cheeger--Ebin, Comparison Theorems in Riemannian Geometry, Lemma 5.2.)

Does anyone know if there is a similar (at least partial) characterization for incomplete Riemannian manifolds? Any references would be much appreciated.

Thanks in advance.

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When I need to generalize something, I first list the properties which I want and then use it as a definition. For example, you might want your number to be smaller (or bigger) than any number which comes from an embedding into complete Riemannian manifold. – Anton Petrunin Jun 13 '11 at 19:01
Thanks for that perspective. The case I'm interested in doesn't admit an embedding into a complete Riemannian manifold, but I guess I was hoping for something similar, like: either one of the two cases for complete manifolds hold, or there is an equally short path in the (metric) completion that passes through (at least) one of the completion points. – Brian Clarke Jun 13 '11 at 19:56
Brian, in general, the trouble with metric completion is lack of local compactness which prevents you from generalizing the argument from the complete case. Instead, you should consider the "constant ultralimit" of your Riemannian manifold (regarded as a metric space), I think, it would do the job. See e.g. my paper for arguments of this type. The point is that a sequence of geodesics in the original metric space yields a geodesic in the ultralimit. – Misha Mar 11 '12 at 22:32

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