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I asked this question in https://mathoverflow.net/, but was advised to ask it here. So here it is.
I just started a self-study of differential geometry and topology. And in several text I came accross the question, asking to show that the global coordinates cannot be defined on a circle $S_{1}$. It seems like quite easy question, but I cannot work the proof. I have a hunch that it has something to do with a Jacobian being zero at some points.
Suppose $S_{1}= [(x,y)\in R^{2} | x^{2}+y^{2}=1] $ and there exist a global coordinate system $u=f(x,y)$, $x=g(x,y)$. The Jacobian then is as follows: $ J=\begin{vmatrix} \frac{\partial f}{\partial x} & \frac{\partial f}{\partial y} \\\ \frac{\partial g}{\partial x} & \frac{\partial g}{\partial y} \end{vmatrix}$ Since $y=\pm \sqrt{1-x^{2}}$ $\frac{\partial f}{\partial y}=0$ $\frac{\partial g}{\partial y}=0$ So I have that Jacobian is equal to zero. But this also feels not quite right. I'm I missing something?

Comment: the global coordinates means that the mapping is smooth bijective and has a non-zero Jacobian everywhere.

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    @Tomas I'm asking the definition of your book.2012-12-12

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If there exists global coordinates on $S^1$, $S^1$ is homeomorphic to $R^n$ for some $n \ge 1$. But $S^1$ is compact while $R^n$ is not.

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    @Tomas Perhaps I misunderstood your problem.2012-12-11
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The notion of "global coordinate system" is somewhat soft. As $S^1$ is a one-dimensional manifold, any coordinate system, local or global, on $S^1$ has one coordinate variable, and not two, as you suggest using $(x,y)$, or $(u,v)$.

I'm interpreting your question in the following way: Prove that there is no map $f:\ ]a,b[\ \to S^1$ which is a diffeomorphism. As $S^1$ is locally diffeomorphically parametrized by the polar angle $\phi$, such a map, if it existed, would have the form $f:\quad ]a,b[\ \to S^1,\quad t\mapsto e^{i\phi(t)}\ ,$ and would install on $S^1$ a global coordinate system, namely the coordinate $t$ ranging in the interval $\ ]a,b[\ $ of ${\mathbb R}$.

Assume that $f$ has the required properties. In particular, the function $\phi(\cdot)$ is continuous and injective, therefore monotone, say, monotonically increasing. It follows that $\alpha:=\lim_{t\to a+}\phi(t)=\inf_{t\in\ ]a,b[}\phi(t)$ and $\beta:=\lim_{t\to b-} \phi(t)=\sup_{t\in\ ]a,b[}\phi(t)$ exist, and $\alpha<\beta\leq\alpha+2\pi$; otherwise $f$ would not be injective. But from $\beta\leq \alpha+2\pi$ and the fact that $\ ]a,b[\ $ has neither a minimal nor a maximal element it follows that the value $\alpha+2k\pi$ is not taken by $\phi$ for any $k\in{\mathbb Z}$, whence $e^{i\alpha}\in S^1$ is not taken by $f$. Therefore $f$ would not be surjective.

It follows that such an $f$ cannot exist.

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    Could you please explain, what does the function $\phi (t)$ means in $t\mapsto e^{i\phi(t)}\$ and where it came from? since I do not see the connections.2012-12-11