In my view, in decreasing order of importance, Sobolev spaces have a tremendous impact on geometry because:

1. 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.

2. 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.)

3. 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.

In my view, in decreasing order of importance, Sobolev spaces have a tremendous impact on geometry because:

1. 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.

2. 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.)

3. 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.

In my view, in decreasing order of importance, Sobolev spaces have a tremendous impact on geometry because:

1. 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.

2. 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.)

3. 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.

In my view, in decreasing order of importance, Sobolev spaces have a tremendous impact on geometry because:

1. 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.

2. 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.)

3. 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.