A classical result states that a smooth, connected one-dimensional manifold is diffeomorphic to an interval or to $S^1$.
I need to explicitly show the diffeomorphism in the second case. It seems to be a calculus exercise, but I'm having problems on it!
Let $f : (0,1) \rightarrow \mathbb{R}$ and $g: (0,1) \rightarrow \mathbb{R}$ two charts of $M$.
Considering the previos notation, I already proved that
$\bullet$ if $f(0,1) \cap g(0,1)$ has only one connected component, there exist a parametrization (I explicity showed) $\varphi:(0,1) \rightarrow M$ such that $\varphi(0,1) = f(0,1) \cup g(0,1)$.
Now I need to do the same in the case in which $f(0,1) \cap g(0,1)$ has two connected componentes. To do that, I tried to find explicitly a parametrization such that $\varphi (0,2\pi) = f(0,1) \cup g(0,1)$ and compose it with another diffeomorphism from $S^1$ to $(0, 2\pi]$. Of course, the compositions must be smooth.
I'm having problems on proving the differentiability of the candidates I find.
Thanks a lot!