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.