Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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 $f \in C^\infty(M)$?

share|improve this question
    
notation tag added –  qlinck Aug 18 '12 at 19:27

2 Answers 2

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.)

share|improve this answer

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

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.