The Whitney embedding theorem states that any smooth manifold can be embedded in Euclidean space.
In the Riemannian setting this naturally leads to the question whether this can be done in such a way that the Riemannian metric is preserved, that is, whether we can find an embedding such that the inherited metric as a submanifold of Euclidean space is the original metric?
I guess there are partial results on this topic (e.g. spaces of constant curvature), but is there anything like the Whitney embedding theorem?
Any answers, links, book recommendations would be very much appreciated!
Thanks in advance for your always great responses,
Edit: Since there was a very short definite answer to my original question in form of a Wikipedia link, I would like to ask for recommendations for books on Riemannian geometry (where this fact is proved).
I would really like to pursue this topic a little further.