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!