I’m reading O.Foster’s «Lectures on Riemann surfaces» and trying to solve the edited version of exercise 1.3. O. Foster’s «Lectures on Riemann surfaces» Instead of proving that the mapping is biholomorphic I need to prove that it is a Möbius transformation.
As far as I understand, I need to manually check if it’s true for generators of $SO_{3}$.
I’ve written the stereographic projection $\sigma : S^2 \rightarrow \mathbb{C}P^1$ as $(s, u, t) \mapsto [s + it: 1 - u]$ and the reverse one as $\sigma^{-1}: \mathbb{C}P^1 \mapsto S^2$, $[x : y] \mapsto (2Re(x\bar{y})/(|x|^2 + |y|^2), 2Im(x\bar{y})/(|x|^2 + |y|^2), (|x|^2 - |y|^2)/(|x|^2 + |y|^2))$.
But when I use this mapping with, for example,$$ \begin{bmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{bmatrix}$$ and get, that $[x : y] \mapsto (2Im(x\bar{y})/(|x|^2 + |y|^2) + i*(|x|^2 - |y|^2)/(|x|^2 + |y|^2), 1 - 2Re(x\bar{y})/(|x|^2 + |y|^2))$.
As I see, I’ve chosen inefficient way to prove that this is a möbius transformation.