# Riemann surface

**Riemann surfacescompact Riemann surfaceconformally invariantconformal invarianceRiemann sheetsanalytic structurecompact Riemann surfacescomplex curveconformal invarianthyperbolic surface**

In mathematics, particularly in complex analysis, a Riemann surface is a one-dimensional complex manifold.wikipedia

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### Bernhard Riemann

**RiemannGeorg Friedrich Bernhard RiemannRiemann, Bernhard**

These surfaces were first studied by and are named after Bernhard Riemann.

His contributions to complex analysis include most notably the introduction of Riemann surfaces, breaking new ground in a natural, geometric treatment of complex analysis.

### Multivalued function

**multi-valued functionmulti-valuedset-valued analysis**

Riemann surfaces are nowadays considered the natural setting for studying the global behavior of these functions, especially multi-valued functions such as the square root and other algebraic functions, or the logarithm.

These problems are resolved in the theory of Riemann surfaces: to consider a multivalued function f(z) as an ordinary function without discarding any values, one multiplies the domain into a many-layered covering space, a manifold which is the Riemann surface associated to f(z).

### Riemann–Roch theorem

**Riemann-Roch theoremRiemann–Roch formulaRiemann–Roch theorem for algebraic curves**

The Riemann–Roch theorem is a prime example of this influence.

It relates the complex analysis of a connected compact Riemann surface with the surface's purely topological genus g, in a way that can be carried over into purely algebraic settings.

### Complex manifold

**complex structurecomplex manifoldscomplex structures**

In mathematics, particularly in complex analysis, a Riemann surface is a one-dimensional complex manifold. Every Riemann surface is a two-dimensional real analytic manifold (i.e., a surface), but it contains more structure (specifically a complex structure) which is needed for the unambiguous definition of holomorphic functions.

Riemann surfaces, two dimensional manifolds equipped with a complex structure, which are topologically classified by the genus, are an important example of this phenomenon.

### Mathematics

**mathematicalmathmathematician**

In mathematics, particularly in complex analysis, a Riemann surface is a one-dimensional complex manifold.

While some areas might seem unrelated, the Langlands program has found connections between areas previously thought unconnected, such as Galois groups, Riemann surfaces and number theory.

### Surface (topology)

**surfaceclosed surfacesurfaces**

Every Riemann surface is a two-dimensional real analytic manifold (i.e., a surface), but it contains more structure (specifically a complex structure) which is needed for the unambiguous definition of holomorphic functions.

This added structures can be a smoothness structure (making it possible to define differentiable maps to and from the surface), a Riemannian metric (making it possible to define length and angles on the surface), a complex structure (making it possible to define holomorphic maps to and from the surface—in which case the surface is called a Riemann surface), or an algebraic structure (making it possible to detect singularities, such as self-intersections and cusps, that cannot be described solely in terms of the underlying topology).

### Conformal geometry

**conformal structureconformal manifoldconformal**

In a real two dimensional space, conformal geometry is precisely the geometry of Riemann surfaces.

### Manifold

**manifoldsboundarymanifold with boundary**

Every Riemann surface is a two-dimensional real analytic manifold (i.e., a surface), but it contains more structure (specifically a complex structure) which is needed for the unambiguous definition of holomorphic functions.

Riemannian manifolds and Riemann surfaces are named after Riemann.

### Unit disk

**unit discopen unit diskopen unit disc**

Considered as a Riemann surface, the open unit disk is therefore different from the complex plane.

### Complex analysis

**complex variablecomplex functioncomplex functions**

In mathematics, particularly in complex analysis, a Riemann surface is a one-dimensional complex manifold.

Sometimes, as in the case of the natural logarithm, it is impossible to analytically continue a holomorphic function to a non-simply connected domain in the complex plane but it is possible to extend it to a holomorphic function on a closely related surface known as a Riemann surface.

### Stein manifold

**Levi problemSteinStein space**

In fact, every non-compact Riemann surface is a Stein manifold.

Let X be a connected, non-compact Riemann surface.

### Uniformization theorem

**uniformizationuniformisationUniformisation Theorem**

In mathematics, the uniformization theorem says that every simply connected Riemann surface is conformally equivalent to one of three Riemann surfaces: the open unit disk, the complex plane, or the Riemann sphere.

### Klein bottle

**Klein bottlesbottleKlein**

So the sphere and torus admit complex structures, but the Möbius strip, Klein bottle and real projective plane do not.

A Klein surface is, as for Riemann surfaces, a surface with an atlas allowing the transition maps to be composed using complex conjugation.

### Teichmüller space

**Teichmüller theoryBers compactificationTeichmüller metric**

Each point in T(S) may be regarded as an isomorphism class of "marked" Riemann surfaces, where a "marking" is an isotopy class of homeomorphisms from S to itself.

### Modular curve

**Hauptmodulmodular curvesExtended complex upper-half plane**

In number theory and algebraic geometry, a modular curve Y is a Riemann surface, or the corresponding algebraic curve, constructed as a quotient of the complex upper half-plane H by the action of a congruence subgroup Γ of the modular group of integral 2×2 matrices SL(2, Z).

### Riemann sphere

**complex projective lineextended complex planecomplex sphere**

However, there always exist non-constant meromorphic functions (holomorphic functions with values in the Riemann sphere C ∪ {∞}).

In geometry, the Riemann sphere is the prototypical example of a Riemann surface, and is one of the simplest complex manifolds.

### Fuchsian model

In the remaining cases X is a hyperbolic Riemann surface, that is isomorphic to a quotient of the upper half-plane by a Fuchsian group (this is sometimes called a Fuchsian model for the surface).

In mathematics, a Fuchsian model is a representation of a hyperbolic Riemann surface R as a quotient of the upper half-plane H by a Fuchsian group.

### Algebraic geometry and analytic geometry

**GAGAChow's theoremRiemann's existence theorem**

On the other hand, every projective complex manifold is necessarily algebraic, see Chow's theorem.

Riemann surface theory shows that a compact Riemann surface has enough meromorphic functions on it, making it an algebraic curve.

### Field extension

**extensionextension fieldsubfield**

More precisely, the function field of X is a finite extension of C(t), the function field in one variable, i.e. any two meromorphic functions are algebraically dependent.

Given a Riemann surface M, the set of all meromorphic functions defined on M is a field, denoted by \C(M).

### Riemann mapping theorem

**Riemann mapRiemann mappingconformal mappings**

His proof used Riemann surfaces and it was simplified by Paul Koebe two years later in a way which did not require them.

### Fuchsian group

**Fuchsian symmetry groups**

In the remaining cases X is a hyperbolic Riemann surface, that is isomorphic to a quotient of the upper half-plane by a Fuchsian group (this is sometimes called a Fuchsian model for the surface).

Fuchsian groups are used to create Fuchsian models of Riemann surfaces.

### Ramification (mathematics)

**ramificationunramifiedramified**

This is because holomorphic and meromorphic maps behave locally like so non-constant maps are ramified covering maps, and for compact Riemann surfaces these are constrained by the Riemann–Hurwitz formula in algebraic topology, which relates the Euler characteristic of a space and a ramified cover.

This is the standard local picture in Riemann surface theory, of ramification of order n.

### Bolza surface

**Bolza curve**

In mathematics, the Bolza surface, alternatively, complex algebraic Bolza curve (introduced by ), is a compact Riemann surface of genus 2 with the highest possible order of the conformal automorphism group in this genus, namely GL 2 (3) of order 48.

### Klein quartic

**Klein quartic curveKlein**

In hyperbolic geometry, the Klein quartic, named after Felix Klein, is a compact Riemann surface of genus

### Hurwitz's automorphisms theorem

**Hurwitz groupHurwitz automorphism theoremHurwitz automorphisms theorem**

In mathematics, Hurwitz's automorphisms theorem bounds the order of the group of automorphisms, via orientation-preserving conformal mappings, of a compact Riemann surface of genus g > 1, stating that the number of such automorphisms cannot exceed 84(g − 1).