Local sections of Hopf fibration $(S^3,\pi,S^2)$

Question

In the lecture we showed the local triviality of the Hopf fibration $(S^3,\pi,S^2)$ as a principal-$S^1$-bundle by constructing local sections $$s_1:S^2\setminus\{\infty\}\cong\mathbb{C}\to S^3,\qquad s_1(z):=\frac{1}{\sqrt{1+\lvert z\rvert^2}}\left(z,1\right)$$ and $$s_2:S^2\setminus\{0\}\to S^3,\qquad s_2(z):=\frac{1}{\sqrt{1+\lvert z\rvert^2}}(\lvert z\rvert,\lvert z\rvert).$$ At least that is what I have written in my notes, but in general $s_2(z)\notin S^3$. So, what is a correct section $s_2$?

Answer 1

We want $s_{1}$ and $s_{2}$ to be compatible with each other under "inversion," i.e. $z\mapsto 1/z$, so the right formula for $s_{2}(z)$ would be just $s_{1}(1/z)=\frac{1}{\sqrt{1+|z|^{2}}}(\frac{|z|}{z},|z|)$.