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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!

Cleto Pereira
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    Here are some possible reasons you're not getting responses. (1) "One-dimensional surface" is an oxymoron. Surfaces are two-dimensional. Presumably you mean "one-dimensional manifold." (2) your first sentence says that you want to show that two things are homeomorphic, but further down you talk about showing that something is a diffeomorphism. (3) You introduce two charts $f$ and $g$ without saying why, or what they have to do with the problem at hand. (4) you introduce a parametrization $\varphi$ whose domain is $(0,1)$, but then you talk about $\varphi(\mathbb S^1)$, which makes no sense. – Jack Lee Jan 10 '20 at 23:08
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    I addition, it's not clear to me what "the candidates" you found are, so it's not clear to me what a proof of their differentiability would look like. – Jason DeVito - on hiatus Jan 11 '20 at 15:45

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