Under what conditions can a Riemannian manifold be embedded isometrically as a submanifold of a complete one of the same dimension? 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)?



When is a Riemannian manifold an open subset of a complete one?
I take the opportunity to advertise the work of a colleague Charles Frances, which is somehow related.
There are counter-examples to a more flexible question: given a (pseudo-)riemannian manifold, is it always possible to *conformally*, non-trivially embed it into another?


A counter-example to this question gives a counter-example to yours since a conformal class on a (non-compact !) manifold contains a non-complete riemannian metric.


Details can be found at the following adress: <http://www.math.u-psud.fr/~frances/boundary-frances.pdf>


