Möbius transformations are holomorphic and bijective on the Riemann sphere, establishing them as automorphisms of the Riemann sphere as a complex manifold. The discussion raises the question of whether the converse holds true, specifically if all holomorphic and 1-to-1 mappings on the Riemann sphere are Möbius transformations. Participants express confidence in the existence of counter-examples, indicating a need for further exploration of this topic.
PREREQUISITESMathematicians, complex analysts, and students studying complex variables who are interested in the properties of Möbius transformations and their implications in the context of the Riemann sphere.
https://en.wikipedia.org/wiki/Möbius_transformationThe Möbius transformations are exactly the bijective conformal maps from the Riemann sphere to itself, i.e., the automorphisms of the Riemann sphere as a complex manifold