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).