Distributions on manifolds Wikipedia entry on distributions contains a seemingly innocent sentence > With minor modifications, one can also define complex-valued distributions, and one can replace $\mathbb{R}^n$ by any (paracompact) smooth manifold. without any reference cited. I went through Vladimirov, Demidov, Gel'fand & Shilov but could not find a single mention of the latter concept. Of course, I have an intuitive feeling of how to go about this, but I would need to use generalized functions on $S^1$ in my work and I don't want to inefficiently re-discover the whole theory if it exists anywhere already. Could anyone point me at a reference where I could learn more about distributions on smooth manifolds? NB this is **not** the same question as distributions _supported_ , or _concentrated_ , on a manifold embedded in $\mathbb{R}^n$. My space of test functions would also be defined on the same manifold so transverse derivatives are not defined.

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 <