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I am a little rusty on my Riemannian geometry. In addressing a problem in PDE's I came across a situation that I cannot reconcile with the Hopf-Rinow Theorem. If $\Omega \subset \mathbb{R}^n$ is a bounded, open set with smooth boundary, then $\mathbb{R}^n - \Omega$ is a Riemannian manifold with smooth boundary. Since $\mathbb{R}^n - \Omega$ is closed in $\mathbb{R}^n$, it follows that $\mathbb{R}^n - \Omega$ is a complete metric space. However, the Hopf-Rinow Theorem seems to indicate that $\mathbb{R}^n - \Omega$ (endowed with the usual Euclidean metric) is not a complete metric space since not all geodesics $\gamma$ are defined for all time. Am I missing something here? Do the hypotheses of the Hopf-Rinow theorem have to be altered to accommodate manifolds with boundary?

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Hopf-Rinow concerns, indeed, Riemannian manifolds with no boundary.

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Thanks! I thought that was the case, but I wanted to check. Just out of curiosity, if one allows for say broken geodesics (allowing geodesics to be reflected off the boundary) using the metric, do you know if you can extend the Hopf-Rinow theorem to manifolds with boundary? – John May 27 '13 at 21:33
Hmm, I don't know about this. It starts to sound like various billiard-ball problems ... – Ted Shifrin May 27 '13 at 21:35

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