# Questions on coordinate representation of a map between smooth manifolds (sphere with stereographic projections)

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:

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

Therefore,

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

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?