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If you have a Riemannian manifold $(M,g)$ (maybe with other assumptions as need), is there a natural way to extend it to a smooth manifold with boundary? For example, the Lobachevsky space viewed as an open disk has a natural extension to a closed disk. Are there any references about this?

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  • $\begingroup$ For the hyperbolic space you mentioned, these are called points at infinity. And there are different ways of defining them using intrinsic structure of the space. But I think this whole thing relies on the specific geometry/topology of this space. For sphere, for instance, all geodesics are periodic, and the manifold is complete. SO, there does not seem to be any missing at the infinity (towards the margins) that you could think of as the boundary. $\endgroup$ – Behnam Esmayli Jun 4 '16 at 15:47
  • $\begingroup$ @Duohead, have you found an answer to your question? The one that you have accepted shows only how to construct compact (sometimes metric) spaces out of Riemannian manifolds, but these are no longer smooth manifolds and don't remember almost anything about the Riemannian or metric structures on the original manifold. $\endgroup$ – Alex M. Jun 13 '18 at 15:48
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You can search for "compactifications of Riemannian manifolds". For example, this article is a survey discussing several possible different compactifications under some assumptions and provides further references.

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