Suppose \(\mathbb{S}^n\) denotes a sphere . For \(i=1,\ldots ,n+1\), let
\[ U_i^+=\{(x^1,\ldots ,x^{n+1})\in \mathbb{R}^{n+1}\mid x^i>0\}, \]and, similarly, let \(U_i^-\) be the set where \(x^i< 0\). Let \(\varphi^\pm_i \colon U_i^\pm \cap \mathbb{S}^n \to \mathbb{B}^n\) with
\[ \varphi_i^\pm (x^1,\ldots ,x^{n+1})=(x^1,\ldots , \hat{x^i}, \ldots , x^{n+1}), \]where the hat indicates that \(x^i\) is omitted. Since,
\[ (\varphi_i^\pm )^{-1}(x^1, \ldots , x^n) = (x^1, \ldots , x^{i-1}, \pm \sqrt{1-\lvert x\rvert^2}, x^{i+1},\ldots ,x^n), \]the tuples \((U_i^\pm ,\varphi_i^\pm )\) are local charts forming an atlas on \(\mathbb{S}^n\). These, are the graph coordinates on \(\mathbb{S}^n\).
Remarks
- The atlas containing all graph coordinates on \(\mathbb{S}^n\) has \(2n+2\) charts.
- This atlas is smooth which can be easily verified.
- This atlas determines the standard smooth structure on \(\mathbb{S}^n\) .