I asked [this question](https://math.stackexchange.com/questions/78358/schwartz-space-on-a-manifold) 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](https://mathoverflow.net/questions/80007/topology-on-the-space-of-schwartz-distributions) 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 $\mathcal{S}(M)$?



The Schwartz Space on a Manifold
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](https://en.wikipedia.org/wiki/Harish-Chandra's_Schwartz_space), 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>


