note-math

Embedding ${\mathrm{𝕊}}^2\hookrightarrow {\mathrm{ℝ}}^3\hookrightarrow {\mathrm{ℂ}}^2$, in ${\mathrm{ℝ}}^3\simeq \mathrm{ℝ}\times \mathrm{ℂ}$ use $\frac{𝑤}{𝑧}\in \mathrm{ℂ}$ as #link(<stereographic-projection>)[] coordinates

$$\begin{array}{ccccc}{\mathrm{ℂ}}^2& ⟶& \mathrm{ℂ}& ⟶& \mathrm{ℝ}\times \mathrm{ℂ}\\ 𝑧,𝑤& \twoheadrightarrow & \frac{𝑤}{𝑧}& ⟶& \frac{1-{\left|\frac{𝑤}{𝑧}\right|}^2,2\frac{𝑤}{𝑧}}{1+{\left|\frac{𝑤}{𝑧}\right|}^2}\\ & & & =& \frac{|𝑧{|}^2-|𝑤{|}^2,2𝑤{𝑧}^{\ast }}{|𝑧{|}^2+|𝑤{|}^2}\\ & \twoheadrightarrow & \frac{{𝑧}^{\ast }}{{𝑤}^{\ast }}& ⟶& \frac{-\left(1-{\left|\frac{𝑧}{𝑤}\right|}^2\right),2\frac{{𝑧}^{\ast }}{{𝑤}^{\ast }}}{1+{\left|\frac{𝑧}{𝑤}\right|}^2}\end{array}$$

The transformation function of the two coordinates of stereographic projection $\frac{𝑤}{𝑧}⇝\frac{{𝑧}^{\ast }}{{𝑤}^{\ast }}$ or $𝜉⇝\frac{1}{{𝜉}^{\ast }}$

$𝜆\left(𝑧,𝑤\right),𝜆\in \text{GL}\left(1,\mathrm{ℂ}\right)$ does not change the projective result e.g. $\frac{𝜆𝑧}{𝜆𝑤}=\frac{𝑧}{𝑤}$

${\mathrm{ℂ}}^2\setminus 0$ is a $\text{GL}\left(1,\mathrm{ℂ}\right)$ bundle on ${\mathrm{𝕊}}^2={\mathrm{ℂ}\mathrm{\mathbb{P}}}^1$

Construct bundle coordinates with two stereographic projection coordinates

$\left(𝑧,𝑤\right)⇝\left(2\frac{𝑤}{𝑧},𝑧\right)$ and $\left(𝑧,𝑤\right)⇝\left(2\frac{𝑧}{𝑤},𝑤\right)$

And the transformation function $\left(2\frac{𝑤}{𝑧},𝑧\right)=\left(2\frac{{𝑧}^{\ast }}{{𝑤}^{\ast }},{𝑤}^{\ast }\right)$ or $\left(𝜉,𝜆\right)⇝\left(\frac{1}{{𝜉}^{\ast }},{\left(\frac{1}{2}𝜉𝜆\right)}^{\ast }\right)$

You can first quotient $\frac{{\mathrm{ℂ}}^2}{{\mathrm{ℝ}}_{>0}}\simeq {\mathrm{𝕊}}^3=\left\{|𝑧{|}^2+|𝑤{|}^2=1\right\}$

At this point, the ${\mathrm{ℝ}}^3\simeq \mathrm{ℝ}\times \mathrm{ℂ}$ representation of stereographic projection

$$\left(𝑧,𝑤\right)\twoheadrightarrow \left(|𝑧{|}^2-|𝑤{|}^2,2𝑤{𝑧}^{\ast }\right)$$

$𝜆\left(𝑧,𝑤\right),𝜆\in \text{ U }\left(1\right)$ does not change the projective result

${\mathrm{𝕊}}^3$ is a $\#𝑈\left(1\right)$ bundle on ${\mathrm{𝕊}}^2$

Construct bundle coordinates with two stereographic projection coordinates

$\left(𝑧,𝑤\right)⇝\left(2\frac{𝑤}{𝑧},\frac{𝑧}{\left|𝑧\right|}\right)$ and $\left(𝑧,𝑤\right)⇝\left(2\frac{𝑧}{𝑤},\frac{𝑤}{\left|𝑤\right|}\right)$

And the transformation function $\left(2\frac{𝑤}{𝑧},\frac{𝑧}{\left|𝑧\right|}\right)=\left(2\frac{{𝑧}^{\ast }}{{𝑤}^{\ast }},\frac{{𝑤}^{\ast }}{\left|{𝑤}^{\ast }\right|}\right)$ or $\left(𝜉,𝜆\right)⇝\left(\frac{1}{{𝜉}^{\ast }},{\left(\frac{𝜉}{\left|𝜉\right|}𝜆\right)}^{\ast }\right)$