The German mathematician

Bernhard Riemann Georg Friedrich Bernhard Riemann (; ; 17September 182620July 1866) was a German mathematician who made profound contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the f ...

(1826–1866) is the

eponym An eponym is a noun after which or for which someone or something is, or is believed to be, named. Adjectives derived from the word ''eponym'' include ''eponymous'' and ''eponymic''. Eponyms are commonly used for time periods, places, innovati ...

of many things.

"Riemann" (by field)

Riemann bilinear relations In mathematics, a Riemann form in the theory of abelian varieties and modular forms, is the following data: * A lattice Λ in a complex vector space Cg. * An alternating bilinear form α from Λ to the integers satisfying the following Riemann bil ...

* Riemann conditions * Riemann form * Riemann function *

Riemann–Hurwitz formula In mathematics, the Riemann–Hurwitz formula, named after Bernhard Riemann and Adolf Hurwitz, describes the relationship of the Euler characteristics of two surfaces when one is a ''ramified covering'' of the other. It therefore connects ramifica ...

* Riemann matrix * Riemann operator * Riemann singularity theorem ** Riemann-Kempf singularity theorem *

Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...

Compact Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...

** Planar Riemann surface * Cauchy–Riemann manifold ** The tangential Cauchy–Riemann complex *

Zariski–Riemann space In algebraic geometry, a Zariski–Riemann space or Zariski space of a subring ''k'' of a field ''K'' is a locally ringed space whose points are valuation rings containing ''k'' and contained in ''K''. They generalize the Riemann surface of a c ...

Analysis

Cauchy–Riemann equations In the field of complex analysis in mathematics, the Cauchy–Riemann equations, named after Augustin-Louis Cauchy, Augustin Cauchy and Bernhard Riemann, consist of a system of differential equations, system of two partial differential equatio ...

Riemann integral In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval. It was presented to the faculty at the University of Gö ...

** Generalized Riemann integral ** Riemann multiple integral * Riemann invariant *

Riemann mapping theorem In complex analysis, the Riemann mapping theorem states that if U is a non-empty simply connected open subset of the complex number plane \mathbb which is not all of \mathbb, then there exists a biholomorphic mapping f (i.e. a bijective hol ...

** Measurable Riemann mapping theorem *

Riemann problem A Riemann problem, named after Bernhard Riemann, is a specific initial value problem composed of a conservation equation together with piecewise constant initial data which has a single discontinuity in the domain of interest. The Riemann prob ...

Riemann solver A Riemann solver is a numerical method used to solve a Riemann problem A Riemann problem, named after Bernhard Riemann, is a specific initial value problem composed of a conservation equation together with piecewise constant initial data whi ...

Riemann sphere In mathematics, the Riemann sphere, named after Bernhard Riemann, is a Mathematical model, model of the extended complex plane (also called the closed complex plane): the complex plane plus one point at infinity. This extended plane represents ...

Riemann–Hilbert correspondence In mathematics, the term Riemann–Hilbert correspondence refers to the correspondence between regular singular flat connections on algebraic vector bundles and representations of the fundamental group, and more generally to one of several generali ...

* Riemann–Hilbert problem * Riemann–Lebesgue lemma *

Riemann–Liouville integral In mathematics, the Riemann–Liouville integral associates with a real function f: \mathbb \rightarrow \mathbb another function of the same kind for each value of the parameter . The integral is a manner of generalization of the repeated antid ...

Riemann–Roch theorem The Riemann–Roch theorem is an important theorem in mathematics, specifically in complex analysis and algebraic geometry, for the computation of the dimension of the space of meromorphic functions with prescribed zeros and allowed poles. It re ...

** Arithmetic Riemann–Roch theorem ** Riemann–Roch theorem for smooth manifolds **

Riemann–Roch theorem for surfaces In mathematics, the Riemann–Roch theorem for surfaces describes the dimension of linear systems on an algebraic surface. The classical form of it was first given by , after preliminary versions of it were found by and . The sheaf-theoretic ver ...

** Grothendieck–Hirzebruch–Riemann–Roch theorem **

Hirzebruch–Riemann–Roch theorem In mathematics, the Hirzebruch–Riemann–Roch theorem, named after Friedrich Hirzebruch, Bernhard Riemann, and Gustav Roch, is Hirzebruch's 1954 result generalizing the classical Riemann–Roch theorem on Riemann surfaces to all complex algeb ...

Riemann–Stieltjes integral In mathematics, the Riemann–Stieltjes integral is a generalization of the Riemann integral, named after Bernhard Riemann and Thomas Joannes Stieltjes. The definition of this integral was first published in 1894 by Stieltjes. It serves as an inst ...

Riemann series theorem In mathematics, the Riemann series theorem, also called the Riemann rearrangement theorem, named after 19th-century German mathematician Bernhard Riemann, says that if an infinite series of real numbers is conditionally convergent, then its terms ...

Riemann sum In mathematics, a Riemann sum is a certain kind of approximation of an integral by a finite sum. It is named after nineteenth century German mathematician Bernhard Riemann. One very common application is in numerical integration, i.e., approxima ...

Number theory

* Riemann–von Mangoldt formula *

Riemann hypothesis In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part . Many consider it to be the most important unsolved problem in pure ...

Generalized Riemann hypothesis The Riemann hypothesis is one of the most important conjectures in mathematics. It is a statement about the zeros of the Riemann zeta function. Various geometrical and arithmetical objects can be described by so-called global ''L''-functions, whi ...

Grand Riemann hypothesis In mathematics, the grand Riemann hypothesis is a generalisation of the Riemann hypothesis and generalized Riemann hypothesis. It states that the nontrivial zeros of all Automorphic L-function, automorphic ''L''-functions lie on the critical line \f ...

Riemann hypothesis for curves over finite fields In mathematics, the local zeta function (sometimes called the congruent zeta function or the Hasse–Weil zeta function) is defined as :Z(V, s) = \exp\left(\sum_^\infty \frac (q^)^k\right) where is a non-singular -dimensional projective algebr ...

* Riemann theta function * Riemann Xi function *

Riemann zeta function The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for and its analytic c ...

Riemann–Siegel formula In mathematics, the Riemann–Siegel formula is an asymptotic formula for the error of the approximate functional equation of the Riemann zeta function, an approximation of the zeta function by a sum of two finite Dirichlet series. It was found by ...

Riemann–Siegel theta function In mathematics, the Riemann–Siegel theta function is defined in terms of the gamma function as :\theta(t) = \arg \left( \Gamma\left(\frac+\frac\right) \right) - \frac t for real values of ''t''. Here the argument (complex analysis), argum ...

Physics

Free Riemann gas In mathematical physics, the primon gas or Riemann gas discovered by Bernard Julia is a model illustrating correspondences between number theory and methods in quantum field theory, statistical mechanics and dynamical systems such as the Lee-Yang ...

also called primon gas * Riemann invariant * Riemann–Cartan geometry *

Riemann–Silberstein vector In mathematical physics, in particular electromagnetism, the Riemann–Silberstein vector or Weber vector named after Bernhard Riemann, Heinrich Martin Weber and Ludwik Silberstein, (or sometimes ambiguously called the "electromagnetic field") is ...

* Riemann-Lebovitz formulation *

Riemann curvature tensor Georg Friedrich Bernhard Riemann (; ; 17September 182620July 1866) was a German mathematician who made profound contributions to mathematical analysis, analysis, number theory, and differential geometry. In the field of real analysis, he is mos ...

also called Riemann tensor * Riemann tensor (general relativity)

Riemannian

Pseudo-Riemannian manifold In mathematical physics, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the ...

Riemannian bundle metric In differential geometry, the notion of a metric tensor can be extended to an arbitrary vector bundle, and to some principal fiber bundles. This metric is often called a bundle metric, or fibre metric. Definition If ''M'' is a topological manifold ...

Riemannian circle In mathematics, a metric circle is the metric space of arc length on a circle, or equivalently on any rectifiable simple closed curve of bounded length. The metric spaces that can be embedded into metric circles can be characterized by a four-p ...

* Riemannian cobordism *

Riemannian connection In mathematics, a metric connection is a connection in a vector bundle ''E'' equipped with a bundle metric; that is, a metric for which the inner product of any two vectors will remain the same when those vectors are parallel transported along ...

Riemannian connection on a surface In mathematics, the Riemannian connection on a surface or Riemannian 2-manifold refers to several intrinsic geometric structures discovered by Tullio Levi-Civita, Élie Cartan and Hermann Weyl in the early part of the twentieth century: parallel t ...

* Riemannian cubic * Riemannian cubic polynomials * Riemannian foliation *

Riemannian geometry Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, defined as manifold, smooth manifolds with a ''Riemannian metric'' (an inner product on the tangent space at each point that varies smooth function, smo ...

** Fundamental theorem of Riemannian geometry * Riemannian graph * Riemannian group *

Riemannian holonomy In differential geometry, the holonomy of a connection on a smooth manifold is the extent to which parallel transport around closed loops fails to preserve the geometrical data being transported. Holonomy is a general geometrical consequence ...

Riemannian manifold In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...

also called Riemannian space * Riemannian metric tensor * Riemannian Penrose inequality * Riemannian polyhedron * Riemannian singular value decomposition * Riemannian submanifold *

Riemannian submersion In differential geometry, a branch of mathematics, a Riemannian submersion is a submersion from one Riemannian manifold to another that respects the metrics, meaning that it is an orthogonal projection on tangent spaces. Formal definition Let ( ...

Riemannian volume form In mathematics, a volume form or top-dimensional form is a differential form of degree equal to the differentiable manifold dimension. Thus on a manifold M of dimension n, a volume form is an n-form. It is an element of the space of sections of th ...

* Riemannian wavefield extrapolation *

Sub-Riemannian manifold In mathematics, a sub-Riemannian manifold is a certain type of generalization of a Riemannian manifold. Roughly speaking, to measure distances in a sub-Riemannian manifold, you are allowed to go only along curves tangent to so-called ''horizontal s ...

Riemannian symmetric space In mathematics, a symmetric space is a Riemannian manifold (or more generally, a pseudo-Riemannian manifold) whose group of isometries contains an inversion symmetry about every point. This can be studied with the tools of Riemannian geomet ...

Riemann's

* Riemann's differential equation * Riemann's existence theorem *

Riemann's explicit formula In mathematics, the explicit formulae for L-functions are relations between sums over the complex number zeroes of an L-function and sums over prime powers, introduced by for the Riemann zeta function. Such explicit formulae have been appli ...

* Riemann's minimal surface * Riemann's theorem on removable singularities

Non-mathematical

* Riemann (crater) * 4167 Riemann {{DEFAULTSORT:List Of Topics Named After Bernhard Riemann

Riemann Georg Friedrich Bernhard Riemann (; ; 17September 182620July 1866) was a German mathematician who made profound contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first ...

Bernhard Riemann