Two questions regarding the $n$-sphere:
- construct a parametrization
- what is the minimum number of charts needs for a smooth manifold of $\mathbb{S}^n$
With regards to the first question:
$$\varphi \in (0,2\pi)\quad \theta_n = (0,\pi)$$ $$\mathbb{S}^1(\theta,\varphi) = (\cos \varphi, \sin \varphi)$$ $$\mathbb{S^n}(\theta_1,\dots,\theta_n,\varphi) = (\sin\theta_n)\mathbb{S}^{n-1}\times (\cos\theta_n)$$ by which I mean the coordinates of the parametrization are: $$ \mathbb{S^n}_i = (\sin\theta_n)\mathbb{S^{n-1}}_i \quad i\in\{1,\dots,n-1\} $$$$ \mathbb{S^n}_n = \cos\theta_n $$ Is there a way to write the parametrization without the use of recursion?
With regards to the second question:
We need a minimum of $2$ charts to parametrize the circle into a smooth manifold.
We need a minimum of $4$ charts to parametrize the sphere into a smooth manifold.
So we need $2^n$ charts to parametrize $\mathbb{S}^n$, because for each incremental increase of the dimension we require two more charts to keep the charts injective. Is this reasoning correct?
EDIT: with regards to the second question, I found out that you can have a generalised stereographic projection in such a way that only two charts are required to construct a smooth manifold:
$$\mathcal{A}=\{ (\mathbb{R}^n, x_N), (\mathbb{R}^n, x_N) \}$$ where $x_N$ is the projection from the generalized North Pole and $x_S$ is respectively the projection from the South Pole.