Alphabeta Math
TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passaudited 2026-08-28
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Every biholomorphic self-map of the Riemann sphere is Möbius

Statement

Every biholomorphic self-map of the Riemann sphere is Möbius.

Facts & Assumptions

Given: A biholomorphic self-map F of C^.

[L1]

Meromorphic self-maps of the sphere are exactly rational maps (Meromorphic functions on the Riemann sphere are exactly the rational functions).

[L2]

A nonconstant rational map has every fibre of total multiplicity equal to its degree (A nonconstant rational map has total fibre multiplicity equal to its degree).

Proof

technique · direct
1.1

Because F is holomorphic on the sphere, [L1] makes it a rational map. Bijectivity means every sphere value has exactly one preimage, and that preimage has multiplicity 1.

L1given
2.1

Applying [L2] to any fibre forces the degree of F to be 1, and a degree-1 rational self-map is exactly a Möbius transformation.

L2givenalgebra∎

Depends on

Used by

Dependency tree · two levels

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Sources