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I have some issues understanding the notion of completeness of a Riemannian manifold with boundary. In the case of Riemannian manifolds without boundary, I found that completeness is usually defined via the existence of geodesics, i.e. every geodesic is defined on the whole real line.

With Hopf-Rinow, this is equivalent to various conditions, for example, the manifold is complete as a metric space.

In the context of a manifold with boundary clearly Hopf-Rinow makes no sense because there will usually be geodesics which cease to exist (reach the boundary) after finite time. How do we define completeness in this case? Just via completeness as a metric space? If yes, isn't it true that every compact manifold with boundary is complete?

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  • $\begingroup$ To the final question, note that any compact metric space is complete. $\endgroup$ – Harald Hanche-Olsen Apr 21 '14 at 9:41
  • $\begingroup$ yeah, that's why I thought that there is a different way to define completeness than just via completeness as a metric space $\endgroup$ – duke-bongu Apr 21 '14 at 9:51
  • $\begingroup$ Yes, for manifolds with boundary, Cauchy completeness is the only way to go. $\endgroup$ – Moishe Kohan Apr 22 '14 at 17:24

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