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!
Best Answer
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.