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I have a basic notation related doubt as follows:

Let $M\subset \mathbb{R}^N$ be a manifold. What does $C^\infty(M)$ denote in $f \in C^\infty(M)$?

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notation tag added – qlinck Aug 18 '12 at 19:27
up vote 4 down vote accepted

Each $m \in M$ lies in some coordinate chart $(U, \phi)$, where $\phi: U \to U' \subset \mathbb{R}^k$ is a homeomorphism ($k$ may be less than $N$)

Then $f \in C^\infty(M)$ means that $f \circ \phi^{-1}: U' \to \mathbb{R}$ is smooth in the usual sense (ie, infinitely differentiable) for all charts containing $m$, for all $m \in M$.

(There must be some compatibility condition on the charts as well, but this is not what you are asking.)

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$C^\infty(M)$ denotes the space of smooth (infinitely differentiable) real-valued functions on the manifold $M$.

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