The Nash embedding theorem tells us that every smooth Riemannian m-manifold can be embedded in $R^n$ for, say, $n = m^2 + 5m + 3$. What can we say in the special case of 2-manifolds? For example, can we always embed a 2-manifold in $R^3$?



Nash embedding theorem for 2D manifolds
The Nash-Kuiper embedding theorem states that any orientable 2-manifold is isometrically ${\cal C}^1$-embeddable in $\mathbb{R}^3$.
A theorem of Thompkins [cited below] implies that as soon as one moves to ${\cal C}^2$, even
compact flat $n$-manifolds cannot be isometrically ${\cal C}^2$-immersed in $\mathbb{R}^{2n-1}$.
So the answer to your question for smooth embeddings is: *No*, as others have pointed out.
I believe Gromov reduced the dimension you quote of the space needed for any compact surface to 5,
but I don't have a precise reference for that.


Tompkins, C. "Isometric embedding of flat manifolds in Euclidean space," *Duke Math.J.* **5**(1): 1939, 58-61.


**Edit.** Both Deane Yang and Willie Wong were correct that the Gromov result is in
*Partial Differential Relations*. I believe this is it, on p.298: "We construct here an isometric $\cal{C}^\infty$
($\cal{C}^{\mathrm{an}}$)-imbedding of $(V,g) \rightarrow \mathbb{R}^5$ for all compact surfaces $V$."
$g$ is a Riemannian metric on $V$.


