finding two charts to cover the manifold $S^n$

Question

i have a problem in stereography of Manifold Sphere $S^n$ for finding two charts to cover it , still stereography is hard for me to understand but the problem is how can i find these 2 charts to cover $S^n$ and in how should i write the prove of locally euclidean property of this manifold (with math notation not explaining)

Answer 1

Let $\varepsilon\in\{-1,1\}$, $U_\varepsilon:=S^n\setminus\{(0,\ldots,0,\varepsilon)\}$ and $\varphi_\varepsilon\colon U_\varepsilon\rightarrow\mathbb{R}^n$ defined by: $$\varphi_\varepsilon(x_0,\ldots,x_n)=\left(\frac{x_0}{1-\varepsilon x_n},\ldots,\frac{x_{n-1}}{1-\varepsilon x_n}\right),$$ then $\varphi_\varepsilon$is an homeomorphism which inverse is given by: $${\varphi_\varepsilon}^{-1}(x)=\left(\frac{2x_1}{1+\|x\|^2},\ldots,\frac{2x_n}{1+\|x\|^2},\varepsilon\frac{\|x\|^2-1}{1+\|x\|^2}\right),$$ where $\|x\|^2={x_1}^2+\ldots+{x_n}^2$. Furtheremore, it is easily checked that $\varphi_1\circ\varphi_{-1}=\varphi_{-1}\circ\varphi_1$ is: $$x\mapsto\frac{x}{\|x\|^2},$$ which is a smooth map. Whence, $\{(U_\varepsilon,\varphi_\varepsilon)\}_{\varepsilon\in\{-1,1\}}$ is a smooth atlas of $S^n$ formed by $2$ charts.

Does this little discussion answer your question? I am bit confused since you seem to already know about stereographic projections...