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I am trying to prove that the unit disk $S=\{(x,y) \mid x^{2} + y^{2} \leq 1 \}$ is a 2-manifold with boundary. This seems to be a well-established fact, but I can't find a reference that explicitly gives the charts for this. Specifically, the boundary charts are what is confusing me. Obviously, $\partial S$, the topological boundary of $S$, will also be the boundary in the manifold sense, but for each such point $z \in \partial S$, we must find a local diffeomorphism

$$ \phi: U \cap S \rightarrow \mathbb{R} \times \{0\}$$

such that $z \in U$. Does anybody know what such a diffeomorphism would look like, or have a reference to the charts explicitly written out? Thanks in advance!

Mark
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  • You need a diffeomorphism to $\mathbb R \times [0,\infty)$, not to $\mathbb R\times {0}$. – Arctic Char Sep 30 '20 at 14:31
  • I want to write an answer but this is probably a duplicate of this one? Try to use Stereographic projection to construct boundary charts. Another way is by using regular value theorem. – Kelvin Lois Sep 30 '20 at 14:39
  • @SiKucing Thanks for the link. It seems, unless I am mistaken, that the sterographic projection maps from (in my example) $S \subseteq \mathbb{R}^{2}$ to $\mathbb{R}^{1}$, and not to $\mathbb{R} \times [0,\infty) \subseteq \mathbb{R}^{2}$. – Mark Sep 30 '20 at 15:29
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    In the answer in the given link, we consider $S \subseteq \mathbb{R}^2$ as a subspace of $\mathbb{R}^3$ and then use stereographic projection from $\mathbb{S}^2\smallsetminus {\text{north pole}} \to \mathbb{R}^2$. – Kelvin Lois Sep 30 '20 at 15:34
  • Thank you very much – Mark Oct 01 '20 at 13:35

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Well you can use the linear fractional transformation $f$ from $\bar{\mathbb{H}}^{2} \rightarrow S$ defined by $f(z) = \frac{z-i}{z+i}$ this maps the $\bar{\Bbb{H}}^{2}$ to $S \setminus \{ (1,0)\}$. Now define $g(z) = \frac{i-z}{z+i}$ then $g$ maps the upper half plane to the $S \setminus \{(-1,0)\}$. for $ z \in \bar{\Bbb H}^{2} \setminus \{(1,0) , (-1,0) \}$ then $g^{-1}\circ f(z) = -z$ which is clearly a smooth function. So these two charts define a smooth atlas on $\bar{\Bbb H }^{2} \setminus \{(1,0),(-1,0) \}$.

Amirhossein
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