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?
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.