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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
Branch of mathematics
curves and surfaces in the three-dimensional Euclidean space, and the study of these shapes formed the basis for development of modern differential geometry
Differential_geometry
In mathematics, a real differential one-form ω on a surface is called a harmonic differential if ω and its conjugate one-form, written as ω∗, are both
Harmonic_differential
Concept of vector calculus
and differential topology, a closed form is a differential form α whose exterior derivative is zero (dα = 0); and an exact form is a differential form, α
Closed and exact differential forms
Closed_and_exact_differential_forms
Expression that may be integrated over a region
In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, volumes, and higher-dimensional manifolds. The
Differential_form
Study of complex manifolds and several complex variables
Foundations of Differential Geometry (Wiley Classics Library) Volume 1, 2. Miranda, Rick (1997). Algebraic curves and Riemann surfaces. Graduate studies
Complex_geometry
Two-dimensional manifold
in topology and differential geometry, it may not. A surface is a two-dimensional space; this means that a moving point on a surface may move in two directions
Surface_(topology)
Relation between genus, degree, and dimension of function spaces over surfaces
globally defined (algebraic) one-forms (see Kähler differential). Finally, meromorphic functions on a Riemann surface are locally represented as fractions
Riemann–Roch_theorem
Mathematics of smooth surfaces
the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian
Differential geometry of surfaces
Differential_geometry_of_surfaces
Tensor field in Riemannian geometry
mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel)
Riemann_curvature_tensor
Surface that locally minimizes its area
immersion of a Riemann surface into 3-space, then X {\displaystyle X} is said to be minimal whenever x i {\displaystyle x_{i}} is a harmonic function on M {\displaystyle
Minimal_surface
Term used in the theories of Riemann surfaces and algebraic curves
In mathematics, differential of the first kind is a traditional term used in the theories of Riemann surfaces (more generally, complex manifolds) and
Differential of the first kind
Differential_of_the_first_kind
In mathematics, a planar Riemann surface (or schlichtartig Riemann surface) is a Riemann surface sharing the topological properties of a connected open
Planar_Riemann_surface
Branch of mathematics
first appeared as a distinct area of study in the work of Bernhard Riemann in his study of Riemann surfaces. Work in the spirit of Riemann was carried out
Geometry
Mathematical problems related to differential equations
mathematics, Riemann–Hilbert problems, named after Bernhard Riemann and David Hilbert, are a class of problems that arise in the study of differential equations
Riemann–Hilbert_problem
Analytic function on the upper half-plane with a certain behavior under the modular group
logarithmic differential, while cusp forms give holomorphic differentials. The dimensions of spaces of modular forms can be computed using the Riemann–Roch theorem
Modular_form
Mathematical manifold theory
on M, every cohomology class has a canonical representative, a differential form that vanishes under the Laplacian operator of the metric. Such forms
Hodge_theory
Concept in mathematics
(specifically differential equations). Classically, David Hilbert posed his twenty-first problem, referencing earlier work by Bernhard Riemann. The basic
Riemann–Hilbert correspondence
Riemann–Hilbert_correspondence
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
number Liouville one-form Liouville operator Liouville space Liouville surface Liouville–Neumann series Liouvillian function Riemann–Liouville integral
List of things named after Joseph Liouville
List_of_things_named_after_Joseph_Liouville
Belgian mathematician
Deligne's 1980 paper contains a much more general version of the Riemann hypothesis. From 1970 until 1984, Deligne was a permanent member of the IHÉS staff
Pierre_Deligne
Parametrizes complex structures on a surface
{\displaystyle T(S)} of a (real) topological (or differential) surface S {\displaystyle S} is a space that parametrizes complex structures on S {\displaystyle
Teichmüller_space
Operation in calculus
product and the calculus of differential forms makes sense in arbitrary dimension and on more general manifolds (curves, surfaces, and their higher-dimensional
Integral
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
Intrinsic geometric structures in mathematics
Gauss on the differential geometry of surfaces and the subsequent emergence of the concept of Riemannian manifold initiated by Bernhard Riemann in the
Riemannian connection on a surface
Riemannian_connection_on_a_surface
Mathematical equation
form: y 2 = 4 x 3 − g 2 x − g 3 . {\displaystyle y^{2}=4x^{3}-g_{2}x-g_{3}.\,} Note that the j-invariant is an isomorphism from the Riemann surface H
Picard–Fuchs_equation
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
This is a list of differential geometry topics. See also glossary of differential and metric geometry and list of Lie group topics. List of curves topics
List of differential geometry topics
List_of_differential_geometry_topics
Property of algebraic varieties and complex manifolds
of) non-singular curves are Riemann surfaces. The algebraic definition of genus agrees with the topological notion. On a nonsingular curve, the canonical
Geometric_genus
Study of rates of change
mathematics, differential calculus is a subfield of calculus that studies the rates at which quantities change. The primary objects of study in differential calculus
Differential_calculus
Topological space that locally resembles Euclidean space
the graph of a function. Hermann Weyl gave an intrinsic definition for differentiable manifolds in his lecture course on Riemann surfaces in 1911–1912
Manifold
Mathematical notion of infinitesimal difference
notions. Abelian differentials usually mean differential one-forms on an algebraic curve or Riemann surface. Quadratic differentials (which behave like
Differential_(mathematics)
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
American mathematician (1952–2002)
mathematician known for his work in spectral geometry, Riemann surfaces, circle packings, and differential geometry. Brooks was born in 1952 in Washington,
Robert_W._Brooks
translations. An equivalent definition is a Riemann surface together with a holomorphic 1-form. These surfaces arise in dynamical systems where they can
Translation_surface
Mathematical space with two coordinates
Topology of Surfaces. Springer. doi:10.1007/978-1-4612-0899-0. ISBN 0-387-94102-9. Needham, Tristan (2021). Visual Differential Geometry and Forms. Princeton
Two-dimensional_space
Branch of differential geometry
geometry is the branch of differential geometry that studies Riemannian manifolds. An example of a Riemannian manifold is a surface, on which distances are
Riemannian_geometry
Polygon associated with a compact Riemann surface
In mathematics, a fundamental polygon can be defined for every compact Riemann surface of genus greater than 0. It encodes not only information about the
Fundamental_polygon
Product of the principal curvatures of a surface
In differential geometry, the Gaussian curvature or Gauss curvature (symbol Κ, named after Carl Friedrich Gauss) of a smooth surface in three-dimensional
Gaussian_curvature
Integration over a non-flat region in 3D space
{d} x+\omega _{3}\mathrm {d} x\wedge \mathrm {d} y} be a differential 2-form defined on a surface S in R 3 {\displaystyle \mathbb {R} ^{3}} , and let r
Surface_integral
Mathematical metric for Riemann surfaces
inner product on forms on a Riemann surface (introduced by Hans Petersson). If a point of Teichmüller space is represented by a Riemann surface R, then the
Weil–Petersson_metric
Numerical method used to solve a Riemann problem
A Riemann solver is a numerical method used to solve a hyperbolic partial differential equation based on the solution of the corresponding Riemann problem
Riemann_solver
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
Construction in algebraic geometry
bundle on C. By definition, this is the space of globally defined holomorphic differential forms on C, so we can choose g linearly independent forms ω 1
Abel–Jacobi_map
Notion in calculus
In calculus, the differential represents the principal part of the change in a function y = f ( x ) {\displaystyle y=f(x)} with respect to changes in
Differential_of_a_function
Correspondsnce between Higgs bundles and fundamental group representations
a compact Riemann surface. Hitchin showed that a polystable Higgs bundle corresponds to a solution of Hitchin's equations, a system of differential equations
Nonabelian Hodge correspondence
Nonabelian_Hodge_correspondence
Seven mathematical problems with a US$1 million prize for each solution
problem, Riemann hypothesis, Yang–Mills existence and mass gap, and the Poincaré conjecture at the Millennium Meeting held on May 24, 2000. Thus, on the official
Millennium_Prize_Problems
Definite integral of a scalar or vector field along a path
scalar function on the curve (commonly arc length or, for a vector field, the scalar product of the vector field with a differential vector in the curve)
Line_integral
Notion in geometry
number at a given point. Riemann introduced an abstract and rigorous way to define curvature for these manifolds, now known as the Riemann curvature tensor
Curvature of Riemannian manifolds
Curvature_of_Riemannian_manifolds
Compact Riemann surface of genus 3
hyperbolic geometry, the Klein quartic, named after Felix Klein, is a compact Riemann surface of genus 3 with the highest possible order automorphism group
Klein_quartic
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
Generalization of elliptic integrals
are given a Riemann surface S {\displaystyle S} and on it a differential 1-form ω {\displaystyle \omega } that is everywhere holomorphic on S {\displaystyle
Abelian_integral
Circulation density in a vector field
context of differential forms, which involves a number of steps. In short, they correspond to the derivatives of 0-forms, 1-forms, and 2-forms, respectively
Curl_(mathematics)
Study of space and shapes locally given by a convergent power series
functions. A Riemann surface, first studied by and named after Bernhard Riemann, is a one-dimensional complex manifold. Riemann surfaces can be thought
Geometric_function_theory
Theorem in geometric topology
Betti numbers, which associate to any manifold a list of nonnegative integers. Riemann showed that a closed connected two-dimensional manifold is fully
Poincaré_conjecture
Branch of mathematics
of infinitesimals, it has two major branches, differential calculus and integral calculus. Differential calculus studies instantaneous rates of change
Calculus
Mathematical condition
electromagnetism and differential geometry, where it relates to the fact that the boundary of a boundary is always empty, i.e. if you have a surface (a 2-form) and you
Poincaré_lemma
Operator in fractional calculus
common forms are: The Riemann–Liouville differintegral This is the simplest and easiest to use, and consequently it is the most often used. It is a generalization
Differintegral
Differential form in commutative algebra
In mathematics, Kähler differentials provide an adaptation of differential forms to arbitrary commutative rings or schemes. The notion was introduced
Kähler_differential
manifolds and Riemann surfaces are named after Bernhard Riemann. In 1857, Riemann introduced the concept of Riemann surfaces as part of a study of the
History of manifolds and varieties
History_of_manifolds_and_varieties
Basic integral in elementary calculus
In real analysis, the Riemann integral is a rigorous definition of the integral of a function on an interval. It defines the integral by approximating
Riemann_integral
Mathematics of real numbers and real functions
and area. The Riemann integral formalizes the integral using approximations by finite sums over intervals. The Riemann integral is put on a firm foundation
Real_analysis
23 mathematical problems stated in 1900
solved, or the 8th problem (the Riemann hypothesis), which still remains unresolved, were presented precisely enough to enable a clear affirmative or negative
Hilbert's_problems
Set of points equidistant from a center
coordinates with r held constant. A sphere of any radius centered at zero is an integral surface of the following differential form: x d x + y d y + z d z = 0
Sphere
Class of differential equations expressible in differential algebra
a global theory of linear differential equations, and has been developed to include substantive results in the algebraic theory (including a Riemann-Hilbert
Algebraic differential equation
Algebraic_differential_equation
Mathematical formula
field of differential geometry, the Simons formula (also known as the Simons identity, and in some variants as the Simons inequality) is a fundamental
Simons'_formula
Space of complex matrices with positive definite imaginary part
compact Riemann surfaces and complex abelian varieties. Let S {\displaystyle S} be a compact Riemann surface of genus g {\displaystyle g} , and choose a symplectic
Siegel_upper_half-space
Theorem in differential geometry
In differential geometry, the Gauss–Bonnet theorem (or Gauss–Bonnet formula) is a fundamental formula which links the curvature of a surface to its underlying
Gauss–Bonnet_theorem
Ties Euler characteristic of a closed even-dimensional Riemannian manifold to curvature
geometry. The Riemann–Roch theorem and the Atiyah–Singer index theorem are other generalizations of the Gauss–Bonnet theorem. One useful form of the Chern
Chern–Gauss–Bonnet_theorem
Theorem in vector calculus
to the surface at the point Σ ( u , v ) {\displaystyle \mathbf {\Sigma } (u,v)} . The equality can be expressed in terms of differential forms, with ∧
Stokes'_theorem
Equations describing classical electromagnetism
Maxwell's equations are a set of coupled partial differential equations that describe how electric and magnetic fields are generated by electric charges
Maxwell's_equations
Second-order partial differential equation
In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its
Laplace's_equation
topology) Riemann–Roch theorem for smooth manifolds (differential topology) Rokhlin's theorem (geometric topology) S–cobordism theorem (differential topology)
List_of_theorems
1938 article on the theory of elliptic partial differential equations on two-dimensional domains, leading later to the measurable Riemann mapping theorem
Isothermal_coordinates
Map from multiple vectors to an underlying field of scalars, linear in each argument
a differential 0-form: f ∈ C 0 ( U ) = Ω 0 ( U ) {\displaystyle f\in C^{0}(U)=\Omega ^{0}(U)} . We first construct differential 1-forms from 0-forms and
Multilinear_form
Construct allowing differentiation of tangent vector fields of manifolds
In differential geometry, an affine connection is a geometric object on a smooth manifold which connects nearby tangent spaces, so it permits tangent vector
Affine_connection
Fundamental formulas linking the metric and curvature tensor of a manifold
is, in particular, a local embedding, the above formulas also hold for immersions. In classical differential geometry of surfaces, the Codazzi–Mainardi
Gauss–Codazzi_equations
Differentiable manifold
mathematics, a CR manifold, or Cauchy–Riemann manifold, is a differentiable manifold together with a geometric structure modeled on that of a real hypersurface
CR_manifold
canonical class. In terms of holomorphic line bundles L on a connected compact Riemann surface, it is therefore L such that L2 is the canonical bundle
Theta_characteristic
Manifold with Riemannian, complex and symplectic structure
2-manifold is a Riemann surface in a canonical way; this is known as the existence of isothermal coordinates. Conversely, every Riemann surface is Kähler
Kähler_manifold
Operation on differential forms
On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The
Exterior_derivative
Brill–Noether theory Genus (mathematics) Riemann surface Riemann–Hurwitz formula Riemann–Roch theorem Abelian integral Differential of the first kind Jacobian variety
List of algebraic geometry topics
List_of_algebraic_geometry_topics
Concept in algebraic geometry
formula d(1/t) = −dt/t2, for example, a meromorphic differential with double pole at the origin on the Riemann sphere. In particular, KC and its multiples
Canonical_bundle
Branch of mathematical analysis
introduce fractional differential operators with non-singular nonlocal kernel. Their fractional differential operators are given below in Riemann–Liouville sense
Fractional_calculus
Conjecture in algebraic geometry
Riemann surfaces of genus g with n distinct marked points x1,...,xn, and Mg,n is its Deligne–Mumford compactification. There are n line bundles Li on
Witten_conjecture
2D surface which extends indefinitely
a plane is a two-dimensional space or flat surface that extends indefinitely. A plane is the two-dimensional analogue of a point (zero dimensions), a
Plane_(mathematics)
Mathematical theorem
When Γ is the fundamental group of a Riemann surface, the Selberg trace formula describes the spectrum of differential operators such as the Laplacian in
Selberg_trace_formula
Metric on a determinant line bundle
to give a differential-geometric interpretation of the ample line bundle over the moduli space of vector bundles on a compact Riemann surface, known as
Quillen_metric
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
Divisor_(algebraic_geometry)
Manifold
orientable: a biholomorphic map to (a subset of) Cn gives an orientation, as biholomorphic maps are orientation-preserving). Riemann surfaces. Calabi–Yau
Complex_manifold
with a fixed point is rotation Euler's theorem (differential geometry) – Orthogonality of the directions of the principal curvatures of a surface Euler's
List of topics named after Leonhard Euler
List_of_topics_named_after_Leonhard_Euler
Technique in integral evaluation
the chain rule "backwards." This involves differential forms. Before stating the result rigorously, consider a simple case using indefinite integrals. Compute
Integration_by_substitution
Infinitesimal calculus on functions defined on a geometric algebra
including vector calculus, differential geometry, and differential forms. With a geometric algebra given, let a {\displaystyle a} and b {\displaystyle b}
Geometric_calculus
Surface with constant mean curvature
In differential geometry, constant-mean-curvature (CMC) surfaces are surfaces with constant mean curvature. This includes minimal surfaces as a subset
Constant-mean-curvature surface
Constant-mean-curvature_surface
Projective variety that is also an algebraic group
complex numbers, a polarised abelian variety can be defined as an abelian variety A together with a choice of a Riemann form H. Two Riemann forms H 1 {\displaystyle
Abelian_variety
Theorem in algebraic geometry
conformal mappings, of a compact Riemann surface of genus g > 1, stating that the number of such automorphisms cannot exceed 84(g − 1). A group for which the
Hurwitz's automorphisms theorem
Hurwitz's_automorphisms_theorem
Differential calculus on function spaces
to lie on a surface in space, then the solution is less obvious, and possibly many solutions may exist. Such solutions are known as geodesics. A related
Calculus_of_variations
curve can be embedded in a unique compact Riemann surface called its smooth completion. The projection of the Riemann surface to C P 1 {\displaystyle \mathbb
Smooth_completion
Concept in mathematical analysis
of the notion of a definite integral to cases that violate the usual assumptions for that kind of integral. In the context of Riemann integrals (or, equivalently
Improper_integral
Branch of differential geometry and differential topology
is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds equipped with a closed
Symplectic_geometry
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