# Riemannian Immersions into Euclidean Space?

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,

S.L.

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.

• Dear S.L., The answer is yes: en.wikipedia.org/wiki/Nash_embedding_theorem – Akhil Mathew Jan 8 '11 at 1:00
• As Akhil said: yes. In fact, this is called the Isometric Embedding Theorem. Anyway, I asked a more technical question on something similar on MO (about he required minimal dimension of the ambiant Euclidean space). I got very interesting comments here: mathoverflow.net/questions/37708/… – user2093 Jan 8 '11 at 1:11
• Thanks a lot! Interesting =) – Sam Jan 8 '11 at 1:27
• @S. L.: I believe the 3 volume set of Spivak's "A comprehensive introduction to differential geometry" covers this (a better question to ask is: what does it NOT cover?). But it may be a little dense for first reading. John Lee's "An introduction to differentiable manifolds" would be a good prelude to Spivak. – user2093 Jan 8 '11 at 6:22
• Technically, immersions are different than embeddings. The title of this question says immersions but the question seems to ask about embeddings. – timur Jan 9 '11 at 17:31