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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.

salmonella
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    Do you already know that conformal self-maps of the Riemann sphere are Moebius? – Moishe Kohan Feb 06 '24 at 15:45
  • @MoisheKohan No, I do not. – salmonella Feb 06 '24 at 15:55
  • Then, my suggestion you prove this first (it is not hard, reduce to the case of confirmal maps fixing infinity) instead of doing uninformative brute-force calculations. Then prove that the stereographic projection is conformal. – Moishe Kohan Feb 06 '24 at 16:03
  • @MoisheKohan can I do this with basic knowledge of complex analysis? – salmonella Feb 06 '24 at 16:38
  • Absolutely! Can you prove that an entire function without an essential singularity at infinity is polynomial? – Moishe Kohan Feb 06 '24 at 17:01
  • @MoisheKohan is it the same as proving that if an entire function has a pole at infinity then it is a polynomial? – salmonella Feb 06 '24 at 17:23
  • Of course...... – Moishe Kohan Feb 06 '24 at 17:25
  • @MoisheKohan then I can – salmonella Feb 06 '24 at 17:52
  • @MoisheKohan, I’ve proven that conformal self-maps of the Riemann sphere are Möbius, but I don’t know how I can show that the given composition is conformal(since it’s not just a stereographic map). Am I missing something? – salmonella Feb 06 '24 at 21:14
  • Rotation of the sphere is clearly conformal. Now you compose three conformal maps. – Moishe Kohan Feb 06 '24 at 21:30
  • @MoisheKohan, makes sense. And can I prove the fact that the stereographic projection is conformal geometrically or should I do it using some statements equating conformity with holomorphy under certain conditions? (I know how to prove that stereographic projection from $S^2 \ {N}$ to $C$ is conformal geometrically, but it's not quite the same(?)) – salmonella Feb 06 '24 at 21:43
  • The proof that I know uses the fact that it is a restriction of a Moebius transformation of the 3-space. But you should be able to prove conformality by computing the derivative – Moishe Kohan Feb 06 '24 at 21:46
  • @MoisheKohan, and how are conformality and the derivative related? – salmonella Feb 06 '24 at 21:50
  • How do you prove that a smooth map preserves angles? – Moishe Kohan Feb 06 '24 at 21:58
  • @MoisheKohan, I’ve thought about it a bit. Is it true that to prove that the stereographic projection is conformal I need to calculate a Jacobian matrix and check that it is a scaled multiple of a rotation matrix? – salmonella Feb 06 '24 at 22:07
  • https://math.stackexchange.com/questions/4534283/stereographic-projections-preserve-angles – Moishe Kohan Feb 06 '24 at 22:25

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