Is there an analog of "completion" for Riemannian manifolds--more accurately, under

Under what conditions can a Riemannian manifold be embedded isometrically as a submanifold of a complete one , especially of the same dimension? (I guess "completion" may not be the right word if density doesn't necesarily occur.) There should some kinds of necessary conditions. For instance, any ball in $M$ (considered as a metric space) must be totally bounded. Is this sufficient?

I am curious because it seems that many theorems are stated and proved only for the complete case, and I was wondering how to what extent they could be generalized using a completion tool (if it existed).

Also, is there any kind of uniqueness (there is for $C^{\omega}$ manifolds--implied by the Myers-Rinow theorem)?

Is there an analog of "completion" for Riemannian manifolds--more accurately, under what conditions can a Riemannian manifold be embedded isometrically as a submanifold of a complete one, especially of the same dimension? (I guess "completion" may not be the right word if density doesn't necesarily occur.) There should some kinds of necessary conditions. For instance, any ball in $M$ (considered as a metric space) must be totally bounded. Is this sufficient?

I am curious because it seems that many theorems are stated and proved only for the complete case, and I was wondering how to what extent they could be generalized using a completion tool (if it existed).

Also, is there any kind of uniqueness (there is for $C^{\omega}$ manifolds--implied by the Myers-Rinow theorem)?