TITLE: The Schwartz Space on a Manifold
QUESTION [13 upvotes]: I asked this question a couple of days ago on math.stackexchange, but have yet to receive a response, so I have decided to post this here.
This question is also vaguely related (both questions arose from the same thing I was working on) to this question I just asked last night.
The question is simple:  on a general manifold $M$, can one generalize the space of Schwartz functions on $\mathbb{R}^n$ to a space of smooth functions $\mathcal{S}(M)$ on $M$ that obeys similar properties?  I would like to be able to define the convolution of two Schwartz functions, so I guess I better require that $M$ at least be a (unimodular) Lie group.  Is it then possible to define $\mathcal{S}(M)$?  What about the Fourier transform?  Is there a natural definition of the Fourier Transform on $\mathcal{S}(M)$?

REPLY [8 votes]: To define a Schwartz space, you need a notion of decay at infinity, so you need a ``norm'', i.e. a distance to some origin. So the convenient framework is a complete Riemannian manifold. However, even on a Lie group, it is not enough to choose an invariant Riemannian structure to get a Schwartz space having the properties that you require (convolution algebra, good Fourier transformation...). See e.g. the subtlety in the definition of Harish-Chandra's Schwartz space on a semi-simple Lie group, where you have to throw in the $\Xi$-function.
For simply connected solvable Lie groups, the definition of the Schwartz algebra is (I believe) fairly recent: see a paper by Emilie David-Guillou: https://arxiv.org/pdf/1002.2185