A metric on Sobolev maps? Let $M$ and $N$ be Riemannian Manifolds, say, of same dimension $d$, and let $W^{1,p}(M,N)$ be the space of Sobolev maps with $p>d$ (for simplicity). What would be a "natural" metric for this space, which would generalize the standard metric when $N$ is a linear space? 
 A: One option is to use a (deep and complicated) result, the Nash isometric embedding theorem and choose an isometric embedding $N\rightarrow \mathbb{R}^k$ for some sufficiently large $k$, and the look at this as a supspace of $W^{1,p}(M,  \mathbb{R}^k)$
For this to work you should know that the restriction to the image makes sense, i.e. require continuity or possibly some weaker condition. The norm will then depend on the choice of embedding.
Depending on what exactly you want to do it might also be an option to work with different spaces altogether. For elliptic PDE, as an example, the space of differentiable functions with finite Hölder norm of the highest derivatives is a good space to work with. For these type of spaces the manifold versions are easy to define and are Banach manifolds. The difficulty with these comes from the fact that they are not separable.
A: Besides using Nash, you can use normal coordinates on $N$ as follows:
Start with a locally finite cover of $N$ by normal balls $B(a_i, r_i), i\in I,$ with respect to the Riemannian metric on $N$. For each ball you have the logarithmic map
$$
\log_{a_i}: B(a_i, r_i)\to B({\mathbf 0}, r_i)\subset T_{a_i}N.
$$
These maps are not isometric, but are isometric at the centers $a_i$ and you get better accuracy as your net $\{a_i\}_{i\in I}$ becomes denser in $N$. 
Extend these maps smoothly to the rest of $N$ using suitable bump-functions, to maps $\phi_i: N\to {\mathbb R}^n$, $n=dim(N)$. Thus, you obtain a collection of composition maps 
$$
\Phi_i: W^{1,p}(M,N)\to W^{1,p}(M, {\mathbb R}^n), i\in I.
$$ 
If $N$ is compact and, thus, $I$ is finite, you can take, say,
$$
d(f, g)= \sum_{i\in I} ||\Phi_i(f)- \Phi_i(g))||_{1,p}.
$$
If $N$ is noncompact and $I$ is infinite, you have many inequivalent choices, pick one depending on the problem you are solving. 
Again, depending on the problem, instead of normal coordinates, you can use harmonic coordinates. 
Of course, these constructions are noncanonical, but neither is the one using isometric embeddings.  
Lastly, using Nash (in conjunction with the tubular neighborhood theorem which is a standard way to eliminate the difficulties mentioned by Thomas) is quite standard, see e.g. P. Hajlasz, Sobolev mappings between manifolds and metric spaces. In: Sobolev Spaces in Mathematics I. Sobolev type Inequalities pp. 185-222. International Mathematical Series. Springer 2009. However, personally, I find it unsatisfactory, since it relies upon a very implicit and "big" hammer (Nash isometric embedding theorem) and violates the spirit of working with manifolds "locally". 
