On the sphere $S^n$, consider the atlas given by the stereographic projection: $\{(U_1, \phi_1), (U_2, \phi_2)\}$, where $U_1 = S^n \backslash \{N\}$, $U_2 = S^n \backslash \{S\}$, $N$ and $S$ are respectively the north and the south poles of the sphere, $\phi_{1,2}: U_{1,2} \mapsto \mathbb{R}^n, (x_1, \dots, x_{n+1}) \mapsto \frac{1}{1 \mp x_{n+1}} (x_1, \dots, x_n)$. Determine the representation of the following maps in such coordinates:

$p_n: S^1 \mapsto S^1, z \mapsto z^n$ and $\alpha: S^n \mapsto S^n, x \mapsto -x$. Verify they are both $C^\infty$.

My attempt:

A map $F:M\mapsto N$ between differentiable manifolds $M$ and $N$ (with $dim(M)=m$ and $dim(N)=n$) is differentiable if and only if for any $(U, \phi)$ chart on $M$ and for any $(V, \psi)$ chart on $N$, $\psi \circ F \circ \phi^{-1}$ is differentiable as a map between open subsets of $\mathbb{R}^m$ and $\mathbb{R}^n$ respectively. This composition, $\psi \circ F \circ \phi^{-1}$, is by definition the coordinate representation of the map.

On $S^1 = \{ z \in \mathbb{C}: |z|=1 \}$, we can write

$\phi_{1,2}: U_{1,2} \mapsto \mathbb{R}, (x_1, x_2) \mapsto \frac{1}{1 \mp x_{2}} x_1$.

The inverse is easily computable:

$\phi_{1,2}^{-1}: \mathbb{R} \mapsto U_{1,2}, u \mapsto \frac{1}{1+|u|^2} (2u, \pm(|u|^2-1))$.


$\phi_1 \circ p_n \circ \phi_1^{-1}(u) = \phi_n \circ p_n\left(\frac{1}{1+|u|^2} (2u, |u|^2-1)\right) =...$

My questions:

Is this procedure correct? To continue the computations, do I have to see $p_n$ as a complex map and use something like the tangent half-angle substitution? Does this computation have to be repeated for all charts, that is, $\phi_{1,2} \circ p_n \circ \phi_{1,2}^{-1}$? Similar reasoning applies to $\alpha$, doesn't it?


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