Stereographic Projection Atlas for the n-Sphere

Definition (Stereographic Atlas for $S^n$).

Let $n\ge 1$. The $n$-sphere $S^n\subset {\mathbb{R}}^{n+1}$ is the set

$$S^n=\{(x,t)\in {\mathbb{R}}^n\times \mathbb{R}:\Vert x{\Vert }^2+t^2=1\}.$$

The north and south poles are $N=(0,\dots ,0,1)$ and $S=(0,\dots ,0,-1)$, respectively. Define the open sets $U_N=S^n\setminus \{N\}$ and $U_S=S^n\setminus \{S\}$. The charts ${\varphi }_N:U_N\to {\mathbb{R}}^n$ and ${\varphi }_S:U_S\to {\mathbb{R}}^n$ are given by stereographic projection:

$${\varphi }_N(x,t)=\frac{x}{1-t},{\varphi }_S(x,t)=\frac{x}{1+t}.$$

The inverse maps ${\varphi }_N^{-1}:{\mathbb{R}}^n\to U_N$ and ${\varphi }_S^{-1}:{\mathbb{R}}^n\to U_S$ are given by

$${\varphi }_N^{-1}(y)=\left(\frac{2y}{\Vert y{\Vert }^2+1},\frac{\Vert y{\Vert }^2-1}{\Vert y{\Vert }^2+1}\right),{\varphi }_S^{-1}(y)=\left(\frac{2y}{\Vert y{\Vert }^2+1},-\frac{\Vert y{\Vert }^2-1}{\Vert y{\Vert }^2+1}\right).$$

The atlas for $S^n$ is the set $\mathcal{A}=\{(U_N,{\varphi }_N),(U_S,{\varphi }_S)\}$.

Smoothness of the Atlas.

Proof. The atlas $\mathcal{A}$ is smooth if its transition maps are smooth. The domain of the transition maps is the overlap $U_N\cap U_S=S^n\setminus \{N,S\}$. The image of this overlap under the charts is ${\varphi }_N(U_N\cap U_S)={\mathbb{R}}^n\setminus \{0\}$ and ${\varphi }_S(U_N\cap U_S)={\mathbb{R}}^n\setminus \{0\}$.

For any $y\in {\mathbb{R}}^n\setminus \{0\}$, the transition map ${\varphi }_S\circ {\varphi }_N^{-1}:{\mathbb{R}}^n\setminus \{0\}\to {\mathbb{R}}^n\setminus \{0\}$ is computed as follows:

$$({\varphi }_S\circ {\varphi }_N^{-1})(y)={\varphi }_S\left(\frac{2y}{\Vert y{\Vert }^2+1},\frac{\Vert y{\Vert }^2-1}{\Vert y{\Vert }^2+1}\right)=\frac{\frac{2y}{\Vert y{\Vert }^2+1}}{1+\frac{\Vert y{\Vert }^2-1}{\Vert y{\Vert }^2+1}}=\frac{2y}{\Vert y{\Vert }^2+1}\cdot \frac{\Vert y{\Vert }^2+1}{2\Vert y{\Vert }^2}=\frac{y}{\Vert y{\Vert }^2}.$$

The map $y\mapsto y/\Vert y{\Vert }^2$ is the inversion map on ${\mathbb{R}}^n\setminus \{0\}$, which is $C^{\infty }$. This map is its own inverse, so the inverse transition map ${\varphi }_N\circ {\varphi }_S^{-1}$ is identical and also $C^{\infty }$. Therefore, the charts are smoothly compatible, and $\mathcal{A}$ is a smooth atlas on $S^n$. $\square$