Extending Riemannian Manifold to Boundary

Question

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?

Answer 1

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.

Answer 2

You need some additional assumptions. For example, if you take a cone, and remove its vertex, the resulting object is a Riemannian manifold, but I think there's no way to extend it to a manifold with boundary.