Sobolev spaces and geometry This is a very naive question, is there a way to geometrically understand Sobolev spaces without going through analysis and PDE's? To my knowledge, Sobolev spaces where created precisely to study PDE's so it might be a bit nonsensical to not want to go through that path, but perhaps one can understand these spaces via some construction on manifolds or something analogous? 
In case the answer to the above is "No." then I would ask what you would consider to be the nicest use of Sobolev spaces to geometry (including solving a particular PDE and stuff like that, of course)?
Also, it would be nice to know a bit of the history behind the modern usage of Sobolev spaces...
Thanks!
 A: No time to give a complete answer but just a hint to a possible direction. Sobolev spaces in $R^n$ arise as the largest possible spaces on which some functional ('energy') can be defined. So they are the natural domain of some important functionals, the basic example being the Dirichlet functional $\int|\nabla u|^2dx$. This is the most synthetic point of view I can think of, and I wouldn't say it has a geometric nature. However, you have natural generalizations of this kind of functionals on manifolds, so if that is your background, this might give you a better hold on the nature of these spaces.
A: In my view, in decreasing order of importance, Sobolev spaces have a tremendous impact on geometry because:


*

*On $\mathbb R^n$ they have an extremely nice behavior with respect to Fourier transform.
This leads to pseudo differential operators and fourier integral operators and give very powerful tools for solving linear PDE's. See [Shubin: Pseudodifferential operators and spectral theory, Springer] for a compact treatment.

*A priori estimates for elliptic equations, the Sobolev inequality, and Rellich's lemma 
have uses in linear theory and nonlinear theory. In particular, they lead to module properties of Sobolev spaces (that they are Banach algebras, etc.)

*Using 1 and 2 one can view certain nonlinear PDE's is smooth (Lipschitz) vector fields on suitable Sobolev spaces. This can be used to do geometry on manifolds of mapping, diffeomorphisms, shapes, etc. 
Now one can argue, that juggling flows of vector fields is a good part part of differential geometry (essentially this part which goes beyond algebraic geometry for smooth spaces). Just an indication: horizontal lifts of flows and their commutation properties leads directly to curvature 
I hope that this is helpful. 
EDIT: added a reference to 1.
A: Unlike their Holder cousins, most Sobolev spaces are reflexive Banach spaces. Reflexivity is a highly  desirable feature  for  variational problems because it gives a little bit of compactness enough so you can prove  the existence of  minimizers (or more general critical points) of various energy   functionals.   Such critical points  satisfy the Euler-Lagrange equations   and thus you obtain existence  of (weak) solutions  of many important equations in geometry or physics.  Given that their norms have an integral description, the Sobolev spaces are tailor made for energy functionals    described by integrals. The  famous and for a while controversial  Dirichlet  principle states that any function $u$ defined on a compact smooth domain $\newcommand{\bR}{\mathbb{R}}$ $\Omega\subset \bR^N$ which  is zero on the boundary of $\Omega$ and minimizes  the energy functional
$$E(u)=\int_\Omega\left(\frac{1}{2}|\nabla u(x)|^2 -f(x) u(x)\right) dx $$
must  be a solution of the Poisson problem $\Delta u=f$ in $\Omega$ satisfying the boundary conditions $u=0$ on $\partial \Omega$.
Weierstrass pointed out a major flaw in the classical understanding of this principle by constructing, in a special case,   a minimizer  of this energy functional  which is not twice differentiable so the Laplacian does not make sense.
That stopped  things in their tracks for a while until Hilbert, in his famous 1900 Paris  Conference talk  included this in the list of his famous 27 problems. In particular, he hinted to a way out by stating that any variational problem has a solution provided that, if need be, we suitably define the concept of solution.
You can read more about this and see many applications of Sobolev spaces to geometry in these lectures. 
