Let's consider two maps: $H:\mathbb{S}^3\subset \mathbb{C}^2\to \mathbb{CP}^1$ with $H(z_0,z_1)=[z_0:z_1]$ and $h:\mathbb{S}^3\to \mathbb{S^2}$ with $h(x,y,z,t)=(x^2+y^2-z^2-t^2, 2(yz-xt), 2(xz+yt))$. I have to prove that
(1) $h$ and $H$ are smooth submersions.
(2) $\mathbb{CP}^1$ is diffeomorphic to $\mathbb{S}^2$.
I have some problems solving both points because if I take some charts of $\mathbb{S}^2$ or $\mathbb{S}^3$, then the calculations are extremely complicated.
My attempt:
(1) I can consider the natural extension of $h$ to $\mathbb{R}^4$, $\widetilde{h}:\mathbb{R}^4\to \mathbb{R}^3$, which has only one critical point 0. Then, since $h=\widetilde{h}\circ \iota\mid_{\mathbb{S}^3}$, by chain rule we get, $(dh)_p=(d\widetilde{h})_p\circ (d\iota\mid_{\mathbb{S}^3})_p$ where $(d\widetilde{h})_p$ is surjective and $(d\iota\mid_{\mathbb{S}^3})_p$. However, it's not trivial to conclude that $(df)_p$ is surjective. I don't know if it's the right way to do this.
(2) Let's consider the map $\Phi: \mathbb{CP}^1\to \mathbb{S}^2$ such that $h=\Phi\circ H$. $h$ is a surjective smooth submersion, then $\Phi$ is a surjective smooth submersion too. Moreover, since $\mathbb{CP}^1$ and $\mathbb{S}^2$ have the same dimension, i get that $\Phi$ is an immersion and therefore a local diffeomorphism. However, I still have to prove that $\Phi$ is injective. Any hint?