Riemann was born on 17 September 1826 inBreselenz, a village nearDannenberg in theKingdom of Hanover. His father, Friedrich Bernhard Riemann, was a poorLutheran pastor in Breselenz who fought in theNapoleonic Wars. His mother, Charlotte Ebell, died in 1846. Riemann was the second of six children. Riemann exhibited exceptional mathematical talent, such as calculation abilities, from an early age but suffered from timidity and a fear of speaking in public, and had frail health.[7]
During 1840, Riemann went toHanover to live with his grandmother and attendlyceum (middle school years), because such a type of school was not accessible from his home village. After the death of his grandmother in 1842, he transferred to theJohanneum Lüneburg, a high school inLüneburg. There, Riemann studied theBible 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 studyingphilology andChristian theology in order to become a pastor and help with his family's finances.
Riemann held his first lectures in 1854, which founded the field ofRiemannian geometry and thereby set the stage forAlbert Einstein'sgeneral theory of relativity.[9] In 1857, there was an attempt to promote Riemann to extraordinary professor status at theUniversity 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 heldGauss'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 usingdimensions higher than merely three or four in order to describe physical reality.[10][9]
In 1862 he married Elise Koch; they had a daughter.
Riemann fled Göttingen when the armies ofHanover andPrussia clashed there in 1866.[11] He died oftuberculosis during his third journey to Italy in Selasca (now a hamlet ofVerbania onLake 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 theLord's Prayer with his wife and died before they finished saying the prayer.[12] 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.[11]
In 1853,Gauss asked Riemann, his student, to prepare aHabilitationsschrift on the foundations of geometry. Over many months, Riemann developed his theory ofhigher dimensions and delivered his lecture at Göttingen on 10 June 1854, entitledUeber die Hypothesen, welche der Geometrie zu Grunde liegen.[13][14][15] 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 isRiemannian geometry. Riemann found the correct way to extend inton dimensions thedifferential geometry of surfaces, which Gauss himself proved in histheorema egregium. The fundamental objects are called theRiemannian metric and theRiemann 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 thenon-Euclidean geometries.
The Riemann metric is a collection of numbers at every point in space (i.e., atensor) 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 amanifold, no matter how distorted it is.
In his dissertation, he established a geometric foundation forcomplex analysis throughRiemann surfaces, through which multi-valued functions like thelogarithm (with infinitely many sheets) or thesquare root (with two sheets) could becomeone-to-one functions. Complex functions areharmonic functions[citation needed] (that is, they satisfyLaplace's equation and thus theCauchy–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, where the surface has leaves coming together at branch points. For the Riemann surface has parameters (the "moduli").
His contributions to this area are numerous. The famousRiemann 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 or to the interior of the unit circle. The generalization of the theorem to Riemann surfaces is the famousuniformization theorem, which was proved in the 19th century byHenri Poincaré andFelix 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 theDirichlet 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 ofDavid Hilbert in the Calculus of Variations, the Dirichlet principle was finally established. Otherwise, Weierstrass was very impressed with Riemann, especially with his theory ofabelian functions. When Riemann's work appeared, Weierstrass withdrew his paper fromCrelle's Journal and did not publish it. They had a good understanding when Riemann visited him in Berlin in 1859. Weierstrass encouraged his studentHermann Amandus Schwarz to find alternatives to the Dirichlet principle in complex analysis, in which he was successful. An anecdote fromArnold Sommerfeld[16] 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 physicistHermann von Helmholtz assisted him in the work overnight and returned with the comment that it was "natural" and "very understandable".
Other highlights include his work on abelian functions andtheta 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 ofelliptic integrals. 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). ByFerdinand Georg Frobenius andSolomon Lefschetz the validity of this relation is equivalent with the embedding of (where is the lattice of the period matrix) in a projective space by means of theta functions. For certain values of, this is theJacobian variety of the Riemann surface, an example of an abelian manifold.
Many mathematicians such asAlfred Clebsch furthered Riemann's work on algebraic curves. These theories depended on the properties of a function defined on Riemann surfaces. For example, theRiemann–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 toDetlef Laugwitz,[17]automorphic functions 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 onminimal surfaces.
In his habilitation work onFourier series, where he followed the work of his teacher Dirichlet, he showed that Riemann-integrable functions are "representable" by Fourier series. Dirichlet had 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 theRiemann–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 forGeorg Cantor's work with Fourier series, which was the impetus forset theory.
He also worked withhypergeometric differential equations in 1857 using complex analytical methods and presented the solutions through the behaviour of closed paths about singularities (described by themonodromy matrix). The proof of the existence of such differential equations by previously known monodromy matrices is one of the Hilbert problems.
Riemann made some famous contributions to modernanalytic number theory. Ina single short paper, the only one he published on the subject of number theory, he investigated thezeta function that now bears his name, establishing its importance for understanding the distribution ofprime numbers. TheRiemann hypothesis 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 toLeonhard 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.
1868 –Über die Hypothesen, welche der Geometrie zugrunde liegen. Abh. Kgl. Ges. Wiss., Göttingen 1868. TranslationEMIS, pdfOn the hypotheses which lie at the foundation of geometry, translated byW.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 –Bernhard 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 editionsThe 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.
^Dudenredaktion; Kleiner, Stefan; Knöbl, Ralf (2015) [First published 1962].Das Aussprachewörterbuch [The Pronunciation Dictionary] (in German) (7th ed.). Berlin: Dudenverlag. pp. 229, 381, 398, 735.ISBN978-3-411-04067-4.
^Krech, Eva-Maria; Stock, Eberhard; Hirschfeld, Ursula; Anders, Lutz Christian (2009).Deutsches Aussprachewörterbuch [German Pronunciation Dictionary] (in German). Berlin: Walter de Gruyter. pp. 366, 520, 536, 875.ISBN978-3-11-018202-6.
^Mccleary, John.Geometry from a Differentiable Viewpoint. Cambridge University Press. p. 282.
^Watson, P. (2010). The German Genius: Europe's Third Renaissance, the Second Scientific Revolution and the Twentieth Century. United Kingdom: Simon & Schuster UK.
^Riemann, Bernhard; Jost, Jürgen (2016).On the Hypotheses Which Lie at the Bases of Geometry. Classic Texts in the Sciences (1st ed. 2016 ed.). Cham: Springer International Publishing : Imprint: Birkhäuser.ISBN978-3-319-26042-6.
