Can every Riemmanian Manifold be completed?

Question

I had two trails of though.. is either of them fruitful?

1. I know every metric space can be completed, my question is: can a Riemmanian manifold $M$ be embedded smoothly and isometrically into it's metric completion $\hat{M}$? If so I could conclude using the Hopf–Rinow theorem that $M$ can be embedded into a geodesically complete manifold $\hat{M}$...
2. Can we adjoin the boundary to M and then it becomes compact since Riemmanian manifolds satisfy the HB property?

Answer 1

The metric completion of $M$ might not be a manifold. For an example, take the Alexander horned sphere $A \subset S^3$. There are two complementary components of $A$; let $M$ be one of them. Then the metric completion of $M$ is $M \cup A$ which is not a manifold-with-boundary.

This example has a coincidental side-effect that $M$ can be isometrically embedded into a geodesically complete manifold, namely under its inclusion into $S^3$. But I think it won't be too hard to construct an $M$ which does not have this coincidental property either.