Math 2.0

Functions of a complex variable Definition

Holomorphic map between Riemann surfaces

Builds on Riemann surfaceContinuous functionHolomorphic function

Definition

Let XX and YY be Riemann surfaces. A map f ⁣:X→Yf \colon X \to Y is holomorphic if:

  1. ff is continuous, and
  2. for every chart φ ⁣:U→ℂ\varphi \colon U \to \C of XX and every chart ψ ⁣:V→ℂ\psi \colon V \to \C of YY, the map ψ∘f∘φ−1 ⁣:φ(U∩f−1(V))→ℂ\psi \circ f \circ \varphi^{-1} \colon \varphi(U \cap f^{-1}(V)) \to \C is holomorphic.

A holomorphic map is a biholomorphism if it is bijective; its inverse is then also holomorphic.

Remarks

Continuity makes U∩f−1(V)U \cap f^{-1}(V) open, so clause 2 makes sense. It suffices to check clause 2 for charts from one atlas of XX and one atlas of YY. When XX and YY are open subsets of ℂ\C with the identity chart, this is the usual notion of a holomorphic function.

Example: f ⁣:ℂ^→ℂ^f \colon \hat\C \to \hat\C with f(z)=z2f(z) = z^2 for z∈ℂz \in \C and f(∞)=∞f(\infty) = \infty. In the chart zz near finite points it is z↦z2z \mapsto z^2. Near ∞\infty, in the chart w=1/zw = 1/z on both sides, it reads w↦1/(1/w)2=w2w \mapsto 1/(1/w)^2 = w^2, which is holomorphic.

Non-example: complex conjugation z↦zˉz \mapsto \bar z on ℂ\C is continuous, but in the identity chart it is not holomorphic, so clause 2 fails.

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Complex numbers, object, no evidence levelComplex numbersobjectLimit point of a set, definition, no evidence levelLimit point of a setdefinitionReal numbers, object, no evidence levelReal numbersobjectRiemann surface, concept, no evidence levelRiemann surfaceconceptSet bounded above, definition, no evidence levelSet bounded abovedefinitionReal and imaginary parts, definition, no evidence levelReal and imaginary pa…definitionLimit of a function, definition, no evidence levelLimit of a functiondefinitionSupremum, definition, no evidence levelSupremumdefinitionContinuity at a point, definition, no evidence levelContinuity at a pointdefinitionCompleteness axiom, statement, no evidence levelCompleteness axiomstatementContinuous function, definition, no evidence levelContinuous functiondefinitionExistence of square roots of nonnegative reals, statement, no evidence levelExistence of square r…statementSquare root, definition, no evidence levelSquare rootdefinitionModulus of a complex number, definition, no evidence levelModulus of a complex …definitionComplex differentiability, definition, no evidence levelComplex differentiabi…definitionHolomorphic function, definition, no evidence levelHolomorphic functiondefinitionHolomorphic map between Riemann surfaces, definition, no evidence levelHolomorphic map betwe…definition
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  1. Complex numbers object
  2. Limit point of a set definition
  3. Real numbers object
  4. Riemann surface concept
  5. Set bounded above definition
  6. Real and imaginary parts definition
  7. Limit of a function definition
  8. Supremum definition
  9. Continuity at a point definition
  10. Completeness axiom statement
  11. Continuous function definition
  12. Existence of square roots of nonnegative reals statement
  13. Square root definition
  14. Modulus of a complex number definition
  15. Complex differentiability definition
  16. Holomorphic function definition
  17. Holomorphic map between Riemann surfaces definition

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Riemann_surface
About this page
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Kinddefinition
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MSC30F10, 30F99
DefinesHolomorphic map between Riemann surfaces
UsesRiemann surface, Continuous function, Holomorphic function
Changed2 November 2026

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