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Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and
differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
. In the field of real analysis, he is mostly known for the first rigorous formulation of the integral, the Riemann integral, and his work on
Fourier series A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or ''p ...
. His contributions to
complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
include most notably the introduction of Riemann surfaces, breaking new ground in a natural, geometric treatment of complex analysis. His 1859 paper on the
prime-counting function In mathematics, the prime-counting function is the function counting the number of prime numbers less than or equal to some real number ''x''. It is denoted by (''x'') (unrelated to the number ). History Of great interest in number theory is t ...
, containing the original statement of the
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 ...
, is regarded as a foundational paper of
analytic number theory In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Peter Gustav Lejeune Dirichlet's 1837 introduction of Diric ...
. Through his pioneering contributions to differential geometry, Riemann laid the foundations of the mathematics of general relativity. He is considered by many to be one of the greatest mathematicians of all time.


Biography


Early years

Riemann was born on 17 September 1826 in Breselenz, a village near Dannenberg in the Kingdom of Hanover. His father, Friedrich Bernhard Riemann, was a poor Lutheran pastor in Breselenz who fought in the Napoleonic Wars. His mother, Charlotte Ebell, died before her children had reached adulthood. Riemann was the second of six children, shy and suffering from numerous nervous breakdowns. Riemann exhibited exceptional mathematical talent, such as calculation abilities, from an early age but suffered from timidity and a fear of speaking in public.


Education

During 1840, Riemann went to Hanover to live with his grandmother and attend lyceum (middle school years). After the death of his grandmother in 1842, he attended high school at the
Johanneum Lüneburg Johanneum may refer to: * Johanneum (Dresden) The Johanneum is a 16th-century Renaissance building, originally named ''Stallgebäude'' because it was constructed as the royal mews. It is located at the Neumarkt in Dresden. Today the Johanneum is ...
. In high school, Riemann studied the Bible intensively, but he was often distracted by mathematics. His teachers were amazed by his ability to perform complicated mathematical operations, in which he often outstripped his instructor's knowledge. In 1846, at the age of 19, he started studying philology and
Christian theology Christian theology is the theology of Christianity, Christian belief and practice. Such study concentrates primarily upon the texts of the Old Testament and of the New Testament, as well as on Christian tradition. Christian theology, theologian ...
in order to become a pastor and help with his family's finances. During the spring of 1846, his father, after gathering enough money, sent Riemann to the University of Göttingen, where he planned to study towards a degree in theology. However, once there, he began studying
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
under Carl Friedrich Gauss (specifically his lectures on the method of least squares). Gauss recommended that Riemann give up his theological work and enter the mathematical field; after getting his father's approval, Riemann transferred to the University of Berlin in 1847. During his time of study, Carl Gustav Jacob Jacobi, Peter Gustav Lejeune Dirichlet, Jakob Steiner, and Gotthold Eisenstein were teaching. He stayed in Berlin for two years and returned to Göttingen in 1849.


Academia

Riemann held his first lectures in 1854, which founded the field of Riemannian geometry and thereby set the stage for Albert Einstein's general theory of relativity. In 1857, there was an attempt to promote Riemann to extraordinary professor status at the University of Göttingen. Although this attempt failed, it did result in Riemann finally being granted a regular salary. In 1859, following the death of Dirichlet (who held Gauss's chair at the University of Göttingen), he was promoted to head the mathematics department at the University of Göttingen. He was also the first to suggest using dimensions higher than merely three or four in order to describe physical reality. In 1862 he married Elise Koch and they had a daughter Ida Schilling who was born on 22 December 1862.


Protestant family and death in Italy

Riemann fled Göttingen when the armies of Hanover and Prussia clashed there in 1866. He died of tuberculosis during his third journey to Italy in Selasca (now a hamlet of Verbania on Lake Maggiore) where he was buried in the cemetery in Biganzolo (Verbania).
Riemann was a dedicated Christian, the son of a Protestant minister, and saw his life as a mathematician as another way to serve God. During his life, he held closely to his Christian faith and considered it to be the most important aspect of his life. At the time of his death, he was reciting the Lord's Prayer with his wife and died before they finished saying the prayer. Meanwhile, in Göttingen his housekeeper discarded some of the papers in his office, including much unpublished work. Riemann refused to publish incomplete work, and some deep insights may have been lost forever. Riemann's tombstone in
Biganzolo Verbania (, , ) is the most populous ''comune'' (municipality) and the capital city of the province of Verbano-Cusio-Ossola in the Piedmont region of northwest Italy. It is situated on the shore of Lake Maggiore, about north-west of Milan and ab ...
(Italy) refers to :


Riemannian geometry

Riemann's published works opened up research areas combining analysis with geometry. These would subsequently become major parts of the theories of Riemannian geometry,
algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
, and
complex manifold In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic. The term complex manifold is variously used to mean a com ...
theory. The theory of Riemann surfaces was elaborated by Felix Klein and particularly
Adolf Hurwitz Adolf Hurwitz (; 26 March 1859 – 18 November 1919) was a German mathematician who worked on algebra, analysis, geometry and number theory. Early life He was born in Hildesheim, then part of the Kingdom of Hanover, to a Jewish family and died ...
. This area of mathematics is part of the foundation of topology and is still being applied in novel ways to mathematical physics. In 1853, Gauss asked Riemann, his student, to prepare a '' Habilitationsschrift'' on the foundations of geometry. Over many months, Riemann developed his theory of
higher dimensions In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordinat ...
and delivered his lecture at Göttingen in 1854 entitled ''Ueber die Hypothesen, welche der Geometrie zu Grunde liegen''. It was not published until twelve years later in 1868 by Dedekind, two years after his death. Its early reception appears to have been slow, but it is now recognized as one of the most important works in geometry. The subject founded by this work is Riemannian geometry. Riemann found the correct way to extend into ''n'' dimensions the
differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
of surfaces, which Gauss himself proved in his '' theorema egregium''. The fundamental objects are called the Riemannian metric and the Riemann curvature tensor. For the surface (two-dimensional) case, the curvature at each point can be reduced to a number (scalar), with the surfaces of constant positive or negative curvature being models of the non-Euclidean geometries. The Riemann metric is a collection of numbers at every point in space (i.e., a tensor) which allows measurements of speed in any trajectory, whose integral gives the distance between the trajectory's endpoints. For example, Riemann found that in four spatial dimensions, one needs ten numbers at each point to describe distances and curvatures on a
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
, no matter how distorted it is.


Complex analysis

In his dissertation, he established a geometric foundation for
complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
through Riemann surfaces, through which multi-valued functions like the logarithm (with infinitely many sheets) or the square root (with two sheets) could become one-to-one functions. Complex functions are harmonic functions (that is, they satisfy
Laplace's equation In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties. This is often written as \nabla^2\! f = 0 or \Delta f = 0, where \Delta = \nab ...
and thus the Cauchy–Riemann equations) on these surfaces and are described by the location of their singularities and the topology of the surfaces. The topological "genus" of the Riemann surfaces is given by g=w/2-n+1, where the surface has n leaves coming together at w branch points. For g>1 the Riemann surface has (3g-3) parameters (the " moduli"). His contributions to this area are numerous. The famous Riemann mapping theorem says that a simply connected domain in the complex plane is "biholomorphically equivalent" (i.e. there is a bijection between them that is holomorphic with a holomorphic inverse) to either \mathbb or to the interior of the unit circle. The generalization of the theorem to Riemann surfaces is the famous uniformization theorem, which was proved in the 19th century by
Henri Poincaré Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "The ...
and Felix Klein. Here, too, rigorous proofs were first given after the development of richer mathematical tools (in this case, topology). For the proof of the existence of functions on Riemann surfaces he used a minimality condition, which he called the Dirichlet principle. Karl Weierstrass found a gap in the proof: Riemann had not noticed that his working assumption (that the minimum existed) might not work; the function space might not be complete, and therefore the existence of a minimum was not guaranteed. Through the work of
David Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many a ...
in the Calculus of Variations, the Dirichlet principle was finally established. Otherwise, Weierstrass was very impressed with Riemann, especially with his theory of
abelian function In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular func ...
s. When Riemann's work appeared, Weierstrass withdrew his paper from ''
Crelle's Journal ''Crelle's Journal'', or just ''Crelle'', is the common name for a mathematics journal, the ''Journal für die reine und angewandte Mathematik'' (in English: ''Journal for Pure and Applied Mathematics''). History The journal was founded by Augus ...
'' and did not publish it. They had a good understanding when Riemann visited him in Berlin in 1859. Weierstrass encouraged his student
Hermann Amandus Schwarz Karl Hermann Amandus Schwarz (; 25 January 1843 – 30 November 1921) was a German mathematician, known for his work in complex analysis. Life Schwarz was born in Hermsdorf, Silesia (now Jerzmanowa, Poland). In 1868 he married Marie Kumme ...
to find alternatives to the Dirichlet principle in complex analysis, in which he was successful. An anecdote from
Arnold Sommerfeld Arnold Johannes Wilhelm Sommerfeld, (; 5 December 1868 – 26 April 1951) was a German theoretical physicist who pioneered developments in atomic and quantum physics, and also educated and mentored many students for the new era of theoretica ...
shows the difficulties which contemporary mathematicians had with Riemann's new ideas. In 1870, Weierstrass had taken Riemann's dissertation with him on a holiday to Rigi and complained that it was hard to understand. The physicist Hermann von Helmholtz assisted him in the work over night and returned with the comment that it was "natural" and "very understandable". Other highlights include his work on abelian functions and theta functions on Riemann surfaces. Riemann had been in a competition with Weierstrass since 1857 to solve the Jacobian inverse problems for abelian integrals, a generalization of
elliptic integrals In integral calculus, an elliptic integral is one of a number of related functions defined as the value of certain integrals, which were first studied by Giulio Fagnano and Leonhard Euler (). Their name originates from their originally arising in ...
. Riemann used theta functions in several variables and reduced the problem to the determination of the zeros of these theta functions. Riemann also investigated period matrices and characterized them through the "Riemannian period relations" (symmetric, real part negative). By
Ferdinand Georg Frobenius Ferdinand Georg Frobenius (26 October 1849 – 3 August 1917) was a German mathematician, best known for his contributions to the theory of elliptic functions, differential equations, number theory, and to group theory. He is known for the famous ...
and Solomon Lefschetz the validity of this relation is equivalent with the embedding of \mathbb^n/\Omega (where \Omega is the lattice of the period matrix) in a projective space by means of theta functions. For certain values of n, this is the Jacobian variety of the Riemann surface, an example of an abelian manifold. Many mathematicians such as
Alfred Clebsch Rudolf Friedrich Alfred Clebsch (19 January 1833 – 7 November 1872) was a German mathematician who made important contributions to algebraic geometry and invariant theory. He attended the University of Königsberg and was habilitated at Berlin. ...
furthered Riemann's work on algebraic curves. These theories depended on the properties of a function defined on Riemann surfaces. For example, the Riemann–Roch theorem (Roch was a student of Riemann) says something about the number of linearly independent differentials (with known conditions on the zeros and poles) of a Riemann surface. According to
Detlef Laugwitz Detlef Laugwitz (1932–2000) was a German people, German mathematician and historian, who worked in differential geometry, history of mathematics, functional analysis, and non-standard analysis. Biography He was born on 11 May 1932 in Breslau, W ...
,
Detlef Laugwitz Detlef Laugwitz (1932–2000) was a German people, German mathematician and historian, who worked in differential geometry, history of mathematics, functional analysis, and non-standard analysis. Biography He was born on 11 May 1932 in Breslau, W ...
: ''Bernhard Riemann 1826–1866''. Birkhäuser, Basel 1996,
automorphic function In mathematics, an automorphic function is a function on a space that is invariant under the action of some group, in other words a function on the quotient space. Often the space is a complex manifold and the group is a discrete group. Factor ...
s appeared for the first time in an essay about the Laplace equation on electrically charged cylinders. Riemann however used such functions for conformal maps (such as mapping topological triangles to the circle) in his 1859 lecture on hypergeometric functions or in his treatise on minimal surfaces.


Real analysis

In the field of real analysis, he discovered the Riemann integral in his habilitation. Among other things, he showed that every piecewise continuous function is integrable. Similarly, the Stieltjes integral goes back to the Göttinger mathematician, and so they are named together the Riemann–Stieltjes integral. In his habilitation work on
Fourier series A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or ''p ...
, where he followed the work of his teacher Dirichlet, he showed that Riemann-integrable functions are "representable" by Fourier series. Dirichlet has shown this for continuous, piecewise-differentiable functions (thus with countably many non-differentiable points). Riemann gave an example of a Fourier series representing a continuous, almost nowhere-differentiable function, a case not covered by Dirichlet. He also proved the Riemann–Lebesgue lemma: if a function is representable by a Fourier series, then the Fourier coefficients go to zero for large ''n''. Riemann's essay was also the starting point for Georg Cantor's work with Fourier series, which was the impetus for set theory. He also worked with hypergeometric differential equations in 1857 using complex analytical methods and presented the solutions through the behavior of closed paths about singularities (described by the monodromy matrix). The proof of the existence of such differential equations by previously known monodromy matrices is one of the Hilbert problems.


Number theory

Riemann made some famous contributions to modern
analytic number theory In mathematics, analytic number theory is a branch of number theory that uses methods from mathematical analysis to solve problems about the integers. It is often said to have begun with Peter Gustav Lejeune Dirichlet's 1837 introduction of Diric ...
. In a single short paper, the only one he published on the subject of number theory, he investigated the
zeta function In mathematics, a zeta function is (usually) a function analogous to the original example, the Riemann zeta function : \zeta(s) = \sum_^\infty \frac 1 . Zeta functions include: * Airy zeta function, related to the zeros of the Airy function * A ...
that now bears his name, establishing its importance for understanding the distribution of prime numbers. The
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 ...
was one of a series of conjectures he made about the function's properties. In Riemann's work, there are many more interesting developments. He proved the functional equation for the zeta function (already known to Leonhard Euler), behind which a theta function lies. Through the summation of this approximation function over the non-trivial zeros on the line with real portion 1/2, he gave an exact, "explicit formula" for \pi(x). Riemann knew of Pafnuty Chebyshev's work on the
Prime Number Theorem In mathematics, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by precisely quantifying ...
. He had visited Dirichlet in 1852.


Writings

Riemann's works include: * 1851 – '' Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse'', Inauguraldissertation, Göttingen, 1851. * 1857 – '' Theorie der Abelschen Functionen'', Journal fur die reine und angewandte Mathematik, Bd. 54. S. 101–155. * 1859 – ''Über die Anzahl der Primzahlen unter einer gegebenen Größe'', in: ''Monatsberichte der Preußischen Akademie der Wissenschaften.'' Berlin, November 1859, S. 671ff. With Riemann's conjecture. '' Über die Anzahl der Primzahlen unter einer gegebenen Grösse.'' (Wikisource)
Facsimile of the manuscript
with Clay Mathematics. * 1867 – '' Über die Darstellbarkeit einer Function durch eine trigonometrische Reihe'', Aus dem dreizehnten Bande der Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen. * 1868
''Über die Hypothesen, welche der Geometrie zugrunde liegen''.
Abh. Kgl. Ges. Wiss., Göttingen 1868. Translatio
EMIS, pdf
'On the hypotheses which lie at the foundation of geometry'', translated by W.K.Clifford, Nature 8 1873 183 – reprinted in Clifford's Collected Mathematical Papers, London 1882 (MacMillan); New York 1968 (Chelsea) http://www.emis.de/classics/Riemann/. Also in Ewald, William B., ed., 1996 “From Kant to Hilbert: A Source Book in the Foundations of Mathematics”, 2 vols. Oxford Uni. Press: 652–61. * 1876 – ''Berhard Riemann´s Gesammelte Mathematische Werke und wissenschaftlicher Nachlass. herausgegeben von Heinrich Weber unter Mitwirkung von Richard Dedekind'', Leipzig, B. G. Teubner 1876, 2. Auflage 1892, Nachdruck bei Dover 1953 (with contributions by Max Noether and Wilhelm Wirtinger, Teubner 1902). Later editions ''The collected works of Bernhard Riemann: the complete German texts. Eds. Heinrich Weber; Richard Dedekind; M Noether; Wilhelm Wirtinger; Hans Lewy. Mineola, New York: Dover Publications, Inc., 1953, 1981, 2017 * 1876 – ''Schwere, Elektrizität und Magnetismus'', Hannover: Karl Hattendorff. * 1882 – ''Vorlesungen über Partielle Differentialgleichungen'' 3. Auflage. Braunschweig 1882. * 1901 – ''Die partiellen Differential-Gleichungen der mathematischen Physik nach Riemann's Vorlesungen''. PDF on Wikimedia Commons. On archive.org: * 2004 –


See also

*
List of things named after Bernhard Riemann The German mathematician Bernhard Riemann (1826–1866) is the eponym of many things. "Riemann" (by field) *Riemann bilinear relations * Riemann conditions *Riemann form * Riemann function *Riemann–Hurwitz formula * Riemann matrix * Riemann op ...
* Non-Euclidean geometry * On the Number of Primes Less Than a Given Magnitude, Riemann's 1859 paper introducing the complex zeta function


References


Further reading

* . * . *


External links

*
The Mathematical Papers of Georg Friedrich Bernhard Riemann

Riemann's publications at emis.de
*

* ttps://web.archive.org/web/20160318034045/http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Geom/ Bernhard Riemann's inaugural lecture*
Richard Dedekind (1892), Transcripted by D. R. Wilkins, Riemanns biography.
{{DEFAULTSORT:Riemann, Georg Friedrich Bernhard 1826 births 1866 deaths 19th-century deaths from tuberculosis 19th-century German mathematicians Differential geometers Foreign Members of the Royal Society German Lutherans Tuberculosis deaths in Italy People from the Kingdom of Hanover University of Göttingen alumni University of Göttingen faculty Infectious disease deaths in Piedmont