Distributions on manifolds

Solution 1:

The case of a circle or products of circles is much nicer than the general case of manifolds, since there is a canonical invariant ("Haar") measure! Further, circles are abelian, compact Lie groups. And connected.

Thus, smooth functions are identifiable as Fourier expansions with rapidly decreasing coefficients, and distributions have Fourier expansions with at-most-polynomially-growing coefficients.

There is a useful gradation in between, by Levi-Sobolev spaces, etc.

One version of this is in my course notes http://www.math.umn.edu/~garrett/m/fun/notes_2012-13/04_blevi_sobolev.pdf

Solution 2:

There is some material distributed on the four volumes of "The Analysis of Linear Partial Differential Operators" by Lars Hörmander, where the very basic definitions can be found in section 6.3 "Distributions on a Manifold" of the first volume. (I am not entirely happy as there might be literature dealing specifically with this subject.)