TITLE: Usefulness of Nash embedding theorem
QUESTION [37 upvotes]: Nash embedding theorem states that every smooth Riemannian manifold can be smoothly isometrically embedded into some Euclidean space $E^N$. This result is of fundamental importance, for it unifies the intrinsic and extrinsic points of view of Riemannian geometry, however, it is less clear that it is also useful. Most if not all results in differential geometry that I am aware of seem to have been obtained using the intrinsic point of view so avoiding any recurse to Nash embedding theorem.  
Can you mention results that used in their proof the Nash embedding in an essential way, or results whose proof was considerably simplified by the Nash embedding result? If not can you explain why Nash theorem is less useful and powerful than expected?

REPLY [14 votes]: Sobolev mappings between Riemannian manifolds. 
Let $N$ be closed, and $M$ is compact, possibly with boundary.
A natural definition of Sobolev mappings between Riemannian manifolds $W^{1,p}(M,N)$  requires the isometric embedding ot $N$ into an Euclidean space $N\subset\mathbb{R}^\nu$ which is the Nash theorem. Then the space is defined as
$$
W^{1,p}(M,N)=\{u\in W^{1,p}(M,\mathbb{R}^\nu):\, u(x)\in N \text{ a.e.}\}
$$
The space is equipped with the metric  inherited from $W^{1,p}(M,\mathbb{R}^\nu)$.
The space does not depend on the isometric embedding of $N$. If $\iota_1:N\to\mathbb{R}^{\nu_1}$ and $\iota_2:N\to\mathbb{R}^{\nu_2}$ are two isometric embeddings and we denote by
$$
W^{1,p}_{\iota_1}(M,N)
\quad
\text{and}
\quad
W^{1,p}_{\iota_2}(M,N)
$$
the spaces obtained with respect to these embeddings, then 
for a mapping $u:M\to N$ we have that 
$$
\iota_1\circ u\in W^{1,p}_{\iota_1}(M,N)
\quad
\text{if and only if}
\quad
\iota_2\circ u\in W^{1,p}_{\iota_2}(M,N).
$$
However, the metric in the space depends on the embedding, but the map
$$
W^{1,p}_{\iota_1}(M,N)\ni u\mapsto \iota_2\circ\iota_1^{-1}\circ u\in
W^{1,p}_{\iota_2}(M,N)
$$
is a homeomorphism of spaces.
This class of mappings appear in a natural way in the study of geometric variational problems for mappings between manifolds. Like for example, the theory of harmonic mappings. One of the early problems was the question whether smooth mappings are dense. That led to a very fruitful research showing deep connections to algebraic topology.
You can find some basic information about this space as well as references in the survey paper (available on my website):
P. Hajłasz, Sobolev mappings between manifolds and metric spaces. In: Sobolev Spaces in Mathematics I. Sobolev type Inequalities pp. 185-222. International Mathematical Series. Springer 2009.