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One-dimensional complex manifold
Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces
Riemann_surface
Relation between genus, degree, and dimension of function spaces over surfaces
poles. 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
Riemann–Roch_theorem
Model of the extended complex plane plus a point at infinity
function whose codomain is the Riemann sphere. In geometry, the Riemann sphere is the prototypical example of a Riemann surface, and is one of the simplest
Riemann_sphere
Riemann surface (or schlichtartig Riemann surface) is a Riemann surface sharing the topological properties of a connected open subset of the Riemann sphere
Planar_Riemann_surface
German mathematician (1826–1866)
the Riemann integral, and his work on Fourier series. His contributions to complex analysis include most notably the introduction of Riemann surfaces, breaking
Bernhard_Riemann
Conformal structure admits a Hodge dual of 1-forms without even specifying a metric
In mathematics, differential forms on a Riemann surface are an important special case of the general theory of differential forms on smooth manifolds
Differential forms on a Riemann surface
Differential_forms_on_a_Riemann_surface
Class of mathematical function
of a meromorphic function can be defined for every Riemann surface. When D is the entire Riemann sphere, the field of meromorphic functions is simply
Meromorphic_function
Graph drawing used to study Riemann surfaces
mathematics, a dessin d'enfant is a type of graph embedding used to study Riemann surfaces and to provide combinatorial invariants for the action of the absolute
Dessin_d'enfant
Mathematical theorem
In mathematics, the Riemann–Roch theorem for surfaces describes the dimension of linear systems on an algebraic surface. The classical form of it was
Riemann–Roch theorem for surfaces
Riemann–Roch_theorem_for_surfaces
Study of space and shapes locally given by a convergent power series
include Riemann surfaces for algebraic functions and zeros for algebraic functions. A Riemann surface, first studied by and named after Bernhard Riemann, is
Geometric_function_theory
Logarithm of a complex number
^{*}} . Ways of dealing with this include branches, the associated Riemann surface, and partial inverses of the complex exponential function. The principal
Complex_logarithm
Conjecture on zeros of the zeta function
zeros of the Riemann zeta function have a real part equal to one half? More unsolved problems in mathematics In mathematics, the Riemann hypothesis is
Riemann_hypothesis
Two-dimensional manifold
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)
Surface_(topology)
differential geometry, Riemann's minimal surface is a one-parameter family of minimal surfaces described by Bernhard Riemann in a posthumous paper published
Riemann's_minimal_surface
Mathematical formula of two surfaces
the Riemann–Hurwitz formula, named after Bernhard Riemann and Adolf Hurwitz, describes the relationship of the Euler characteristics of two surfaces when
Riemann–Hurwitz_formula
Generalizations of codimension-1 subvarieties of algebraic varieties
divisors on a compact Riemann surface X is the free abelian group on the points of X. Equivalently, a divisor on a compact Riemann surface X is a finite linear
Divisor_(algebraic_geometry)
Theorem in complex analysis
every connected Riemann surface is second-countable (has a countable base for its topology). The Prüfer surface is an example of a surface with no countable
Radó's theorem (Riemann surfaces)
Radó's_theorem_(Riemann_surfaces)
On the Euler characteristic of a holomorphic vector bundle on a compact complex manifold
is Hirzebruch's 1954 result generalizing the classical Riemann–Roch theorem on Riemann surfaces to all complex algebraic varieties of higher dimensions
Hirzebruch–Riemann–Roch theorem
Hirzebruch–Riemann–Roch_theorem
Characteristic property of holomorphic functions
In mathematics, the Cauchy–Riemann equations are two partial differential equations that characterize differentiability of complex functions. The equations
Cauchy–Riemann_equations
Smooth closed surface with g holes
3 surface. The Klein quartic is a compact Riemann surface of genus 3 with the highest possible order automorphism group for compact Riemann surfaces of
Genus_g_surface
Mathematical theory
{\displaystyle {\text{Spec}}({\mathcal {O}}_{K})} such that it extends to a Riemann surface X ∞ = X ( C ) {\displaystyle X_{\infty }={\mathfrak {X}}(\mathbb {C}
Arakelov_theory
determine that the surfaces which have a Riemannian metric of constant curvature 0 {\displaystyle 0} − 1 {\displaystyle -1} i.e. Riemann surfaces with a complete
Non-positive_curvature
In mathematics, a quadratic differential on a Riemann surface is a section of the symmetric square of the holomorphic cotangent bundle. If the section
Quadratic_differential
Geometric representation of the complex numbers
finite everywhere on the Riemann surface, except when z = 0 (that is, f is holomorphic, except when z = 0). How can the Riemann surface for the function w =
Complex_plane
Natural number
groups. An icosagon is a polygon with 20 edges. Bring's curve is a Riemann surface, whose fundamental polygon is a regular hyperbolic icosagon. The largest
20_(number)
Parametrizes complex structures on a surface
{\displaystyle T(S)} may be regarded as an isomorphism class of "marked" Riemann surfaces, where a "marking" is an isotopy class of homeomorphisms from S {\displaystyle
Teichmüller_space
Simply connected Riemann surface is equivalent to an open disk, complex plane, or sphere
connected Riemann surface is conformally equivalent to one of three Riemann surfaces: the open unit disk, the complex plane, or the Riemann sphere. The
Uniformization_theorem
In mathematics, the identity theorem for Riemann surfaces is a theorem that states that a holomorphic function is completely determined by its values on
Identity theorem for Riemann surfaces
Identity_theorem_for_Riemann_surfaces
Property of algebraic varieties and complex manifolds
of complex varieties, (the complex loci of) non-singular curves are Riemann surfaces. The algebraic definition of genus agrees with the topological notion
Geometric_genus
Branch of mathematics
area of study in the work of Bernhard Riemann in his study of Riemann surfaces. Work in the spirit of Riemann was carried out by the Italian school of
Geometry
Algebraic surface
curve (also called Bring's surface and, by analogy with the Klein quartic, the Bring sextic) is a highly-symmetric Riemann surface. It can be defined as an
Bring's_curve
Analytic function on the upper half-plane with a certain behavior under the modular group
space G\H∗. What is more, it can be endowed with the structure of a Riemann surface, which allows one to speak of holo- and meromorphic functions. Important
Modular_form
Iranian mathematician (1977–2017)
awarded the Fields Medal for her work in "the dynamics and geometry of Riemann surfaces and their moduli spaces", becoming the first woman and the first Iranian
Maryam_Mirzakhani
Result in algebraic geometry
itself a generalisation of the classical Riemann–Roch theorem for line bundles on compact Riemann surfaces. Riemann–Roch type theorems relate Euler characteristics
Grothendieck–Riemann–Roch theorem
Grothendieck–Riemann–Roch_theorem
Theorem in complex analysis
function on a compact Riemann surface is necessarily constant. Let f ( z ) {\displaystyle f(z)} be holomorphic on a compact Riemann surface M {\displaystyle
Liouville's theorem (complex analysis)
Liouville's_theorem_(complex_analysis)
Term in mathematics
connected Riemann surface is a Stein manifold if and only if it is not compact. This can be proved using a version of the Runge theorem for Riemann surfaces, due
Stein_manifold
Topological space that locally resembles Euclidean space
manifolds, also known as a 2D surfaces embedded in our common 3D space, were considered by Riemann under the guise of Riemann surfaces, and rigorously classified
Manifold
Type of continuous map in topology
introduced by Riemann as domains on which naturally multivalued complex functions become single-valued. These spaces are now called Riemann surfaces. Covering
Covering_space
In mathematics, a Riemann surface
mathematics, the Bolza surface, alternatively, complex algebraic Bolza curve (introduced by Oskar Bolza (1887)), is a compact Riemann surface of genus 2 {\displaystyle
Bolza_surface
Correspondsnce between Higgs bundles and fundamental group representations
and unitary representations of the fundamental group of a compact Riemann surface. In fact the Narasimhan–Seshadri theorem may be obtained as a special
Nonabelian Hodge correspondence
Nonabelian_Hodge_correspondence
Point of interest for complex multi-valued functions
{\displaystyle n} values. Multi-valued functions are rigorously studied using Riemann surfaces, and the formal definition of branch points employs this concept. Branch
Branch_point
Kepler–Poinsot polyhedron
covering of the Riemann sphere by a Riemann surface of genus 4, with branch points at the center of each pentagram. This Riemann surface, called Bring's
Small_stellated_dodecahedron
surface is a surface obtained from identifying the sides of a polygon in the Euclidean plane by translations. An equivalent definition is a Riemann surface
Translation_surface
Mathematics theory
the usual Teichmüller theory that describes the uniformization of Riemann surfaces and their moduli. It was introduced and developed by Shinichi Mochizuki (1996
P-adic_Teichmüller_theory
Theorem in complex analysis
mathematics, specifically complex analysis, Riemann's existence theorem states that the category of compact Riemann surfaces is equivalent to the category of complex
Riemann's_existence_theorem
Diffeomorphism that has a hyperbolic structure on the tangent bundle
section develops the case of the Anosov flow on the tangent bundle of a Riemann surface of negative curvature. This flow can be understood in terms of the
Anosov_diffeomorphism
Mathematical theorem
In complex analysis, the Riemann mapping theorem states that if U {\displaystyle U} is a non-empty simply connected open subset of the complex number
Riemann_mapping_theorem
Concept in algebraic geometry
the Riemann surface of a complex curve. Zariski–Riemann spaces were introduced by Zariski (1940, 1944) who (rather confusingly) called them Riemann manifolds
Zariski–Riemann_space
System of partial differential equations used in Higgs field theory
over a Riemann surface, written down by Nigel Hitchin in 1987. Hitchin's equations are locally equivalent to the harmonic map equation for a surface into
Hitchin's_equations
Riemann matrix Riemann operator Riemann singularity theorem Riemann-Kempf singularity theorem Riemann surface Compact Riemann surface Planar Riemann surface
List of things named after Bernhard Riemann
List_of_things_named_after_Bernhard_Riemann
J-holomorphic curve) is a smooth map, from a Riemann surface into an almost complex manifold, that satisfies the Cauchy–Riemann equations. Introduced in 1985 by Mikhail
Pseudoholomorphic_curve
Theorem in algebraic geometry
automorphisms, via orientation-preserving conformal mappings, of a compact Riemann surface of genus g > 1, stating that the number of such automorphisms cannot
Hurwitz's automorphisms theorem
Hurwitz's_automorphisms_theorem
Two closely related mathematical subjects
advances are listed here in chronological order. Riemann surface theory shows that a compact Riemann surface has enough meromorphic functions on it, making
Algebraic geometry and analytic geometry
Algebraic_geometry_and_analytic_geometry
Non-Euclidean geometry
hyperbolic surfaces can also be understood according to the language of Riemann surfaces. According to the uniformization theorem, every Riemann surface is either
Hyperbolic_space
Study of complex manifolds and several complex variables
by Bernhard Riemann during his original work on Riemann surfaces. The classification theory is most well known for compact Riemann surfaces. By the classification
Complex_geometry
Curve defined as zeros of polynomials
curve Riemann–Roch theorem for algebraic curves Weber's theorem (Algebraic curves) Riemann–Hurwitz formula Riemann–Roch theorem for Riemann surfaces Riemann
Algebraic_curve
Mathematical metric for Riemann surfaces
Riemann surfaces with n marked points. It was introduced by André Weil (1958, 1979) using the Petersson inner product on forms on a Riemann surface (introduced
Weil–Petersson_metric
Polygon associated with a compact Riemann surface
defined for every compact Riemann surface of genus greater than 0. It encodes not only information about the topology of the surface through its fundamental
Fundamental_polygon
Mathematical space with two coordinates
non-uniform curvature are called Riemannian surfaces. (Not to be confused with Riemann surfaces.) Some surfaces are embedded in three-dimensional Euclidean
Two-dimensional_space
Mathematical theorem
Arthur–Selberg trace formula. When Γ is the fundamental group of a Riemann surface, the Selberg trace formula describes the spectrum of differential operators
Selberg_trace_formula
Tensor field in Riemannian geometry
field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the
Riemann_curvature_tensor
Connects non-singular algebraic curves with compact Riemann surfaces
algebraic number coefficients, represents a compact Riemann surface which is a ramified covering of the Riemann sphere, ramified at three points only. This is
Belyi's_theorem
Neighborhood of a singularity of cusp type
points near a cusp singularity. The cusp neighborhood for a hyperbolic Riemann surface can be defined in terms of its Fuchsian model. Suppose that the Fuchsian
Cusp_neighborhood
Branching out of a mathematical structure
the standard local picture in Riemann surface theory, of ramification of order n. It occurs for example in the Riemann–Hurwitz formula for the effect
Ramification_(mathematics)
In mathematics, a Prym differential of a Riemann surface is a differential form on the universal covering space that transforms according to some complex
Prym_differential
Compact Riemann surface of genus 3
geometry, the Klein quartic, named after Felix Klein, is a compact Riemann surface of genus 3 with the highest possible order automorphism group for this
Klein_quartic
In Riemann surface theory and hyperbolic geometry, the MacBeath surface, also called MacBeath curve or the Fricke–MacBeath surface curve, is the genus-7
MacBeath_surface
In Riemann surface theory and hyperbolic geometry, a Hurwitz surface, named after Adolf Hurwitz, is a compact Riemann surface with precisely 84(g − 1)
Hurwitz_surface
Mathematical behavior near singularities
f} . In the case of F = C {\displaystyle \mathbb {F} =\mathbb {C} } Riemann surface theory enters and allows for the geometric interpretation given above
Monodromy
Algebra describing 2D conformal symmetry
meromorphic vector fields with two poles on a genus 0 Riemann surface. On a higher-genus compact Riemann surface, the Lie algebra of meromorphic vector fields
Virasoro_algebra
Type of function in mathematics
analytic continuations of it leads in general to a Riemann surface that covers an open subset of the Riemann sphere. The global analytic function obtained
Analytic_function
Surface that locally minimizes its area
x_{3}):M\rightarrow \mathbb {R} ^{3}} is an isometric immersion of a Riemann surface into 3-space, then X {\displaystyle X} is said to be minimal whenever
Minimal_surface
Branch of mathematics studying functions of a complex variable
extend it to a holomorphic function on a closely related surface known as a Riemann surface. All this refers to complex analysis in one variable. There
Complex_analysis
Term used in the theories of Riemann surfaces and algebraic curves
differential of the first kind is a traditional term used in the theories of Riemann surfaces (more generally, complex manifolds) and algebraic curves (more generally
Differential of the first kind
Differential_of_the_first_kind
Dianalytic manifold of complex dimension 1
have a boundary and need not be orientable. Klein surfaces generalize Riemann surfaces. While the latter are used to study algebraic curves over the complex
Klein_surface
Natural number
of uniform polytopes in five-dimensional space. Bring's curve is a Riemann surface of genus four, with a domain that is a regular hyperbolic 20-sided
12_(number)
Group of isotopy classes of a topological automorphism group
Teichmüller space and the quotient is the moduli space of Riemann surfaces homeomorphic to the surface. These groups exhibit features similar both to hyperbolic
Mapping_class_group
Group representation of a Riemann surface
representation of a hyperbolic Riemann surface R as a quotient of the upper half-plane H by a Fuchsian group. Every hyperbolic Riemann surface admits such a representation
Fuchsian_model
Mathematical problems related to differential equations
In mathematics, Riemann–Hilbert problems, named after Bernhard Riemann and David Hilbert, are a class of problems that arise in the study of differential
Riemann–Hilbert_problem
Special functions of several complex variables
application of the Riemann theta function is that it allows one to give explicit formulas for meromorphic functions on compact Riemann surfaces, as well as other
Theta_function
Mathematical idealization of the trace left by a moving point
the topological point of view, is not a curve, but a surface, and is often called a Riemann surface. Although not being curves in the common sense, algebraic
Curve
Type of vector bundle
\wedge \varphi =0} (which is vacuous in Hitchin's original set-up on Riemann surfaces) was introduced later by Carlos Simpson. A Higgs bundle can be thought
Higgs_bundle
Index of lists with the same name
function is (usually) a function analogous to the original example, the Riemann zeta function ζ ( s ) = ∑ n = 1 ∞ 1 n s . {\displaystyle \zeta (s)=\sum
List_of_zeta_functions
Theorem in differential geometry
surface's Euler characteristic. Compactness of the surface is of crucial importance. Consider for instance the open unit disc, a non-compact Riemann surface
Gauss–Bonnet_theorem
Finnish mathematician (1907–1996)
analysis during the 20th century. He is remembered for his work on Riemann surfaces, quasiconformal mappings and Teichmüller spaces, and for his textbook
Lars_Ahlfors
Type of mathematical functions
compact (closed) Riemann surface, because since the Riemann-Roch theorem (Riemann's inequality) holds for compact Riemann surfaces (Therefore, the theory
Function of several complex variables
Function_of_several_complex_variables
Identity between theta functions of Riemann surfaces
Fay's trisecant identity is an identity between theta functions of Riemann surfaces introduced by John Fay. Fay's identity holds for theta functions of
Fay's_trisecant_identity
Indian mathematician (1932–2021)
which proved the necessary conditions for stable vector bundles on a Riemann surface. He was a recipient of the Padma Bhushan, India's third highest civilian
M._S._Narasimhan
Branch of mathematics
algebraic number theory. In the 1850s, Riemann introduced the fundamental concept of a Riemann surface. Riemann's methods relied on an assumption he called
Abstract_algebra
Supergeometric generalization of a manifold
_{z}} . More generally, for a split super Riemann surface, the reduced space is an ordinary Riemann surface together with a spin structure, that is, a
Supermanifold
Concept in complex analysis
The simplest examples of such curves are the complex plane and the Riemann surface. This extension is done by transferring structures and properties through
Zeros_and_poles
Theorem about the range of an analytic function
Theorem (meromorphic version): If M is a Riemann surface, w a point on M, P1(C) = C ∪ {∞} denotes the Riemann sphere and f : M\{w} → P1(C) is a holomorphic
Picard_theorem
German mathematician (1885–1955)
of a Riemann Surface), which gave a unified treatment of Riemann surfaces. In it Weyl utilized point set topology, in order to make Riemann surface theory
Hermann_Weyl
Branch of topology
simply connected Riemann surface is conformally equivalent to one of the three domains: the open unit disk, the complex plane, or the Riemann sphere. In particular
Low-dimensional_topology
Superconductivity theory
Ginzburg–Landau functional for the manifold M {\displaystyle M} being a Riemann surface, and taking n = 1 {\displaystyle n=1} ; i.e., a line bundle. The phenomenon
Ginzburg–Landau_theory
Mathematics of smooth surfaces
spaces by Riemann and led to what is known today as Riemannian geometry. The nineteenth century was the golden age for the theory of surfaces, from both
Differential geometry of surfaces
Differential_geometry_of_surfaces
Finitely generated extension field of positive transcendence degree
{\displaystyle M(X)} of meromorphic functions defined on a connected Riemann surface X {\displaystyle X} is a function field of one variable over the complex
Algebraic_function_field
bundle is a line bundle over the space of Cauchy–Riemann operators of a vector bundle over a Riemann surface, introduced by Quillen (1985). Quillen proved
Quillen determinant line bundle
Quillen_determinant_line_bundle
Mathematical group
Riemann surfaces and hyperbolic geometry, the triangle group (2,3,7) is particularly important for its connection to Hurwitz surfaces, namely Riemann
(2,3,7)_triangle_group
Bernhard Riemann. In 1857, Riemann introduced the concept of Riemann surfaces as part of a study of the process of analytic continuation; Riemann surfaces are
History of manifolds and varieties
History_of_manifolds_and_varieties
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