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On $\mathbb{R}^n$, the Schwartz space is an incredibly nice space of functions, and in many ways is more natural than $C_c^\infty (\mathbb{R}^n)$. On a manifold $M$, it of course still makes sense to talk about $C_c^\infty(M)$, but what about $\mathcal{S}(M)$, the Schwartz space on $M$? Is there a way we can define this on a general smooth manifold $M$?

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It probably makes sense for a manifold which is flat outside of a compact set. –  Grumpy Parsnip Nov 2 '11 at 23:17
You should probably post a link to MO, in case someone stops by here later –  user16299 Nov 5 '11 at 2:05
@YemonChoi Good idea. I have asked this question on MO as well: mathoverflow.net/questions/80094/… –  Jonathan Gleason Nov 5 '11 at 2:50
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