- Bernhard Riemann
Infobox Scientist

name =Bernhard Riemann

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image_width =225px

caption =Bernhard Riemann, 1863

birth_date =September 17 ,1826

birth_place =Breselenz ,Germany

death_date =death date and age|1866|7|20|1826|9|17

death_place =Selasca,Italy

residence =

citizenship =

nationality =

ethnicity =

field =Mathematician

work_institutions =Georg-August University of Göttingen

alma_mater =Georg-August University of Göttingen Berlin University

doctoral_advisor =Carl Friedrich Gauss

academic_advisors =Ferdinand Eisenstein Moritz Abraham Stern

doctoral_students =

notable_students =Gustav Roch

known_for =Riemann hypothesis Riemann integral Riemann sphere Riemann differential equation Riemann mapping theorem Riemann curvature tensor Riemann sum Riemann-Stieltjes integral Cauchy-Riemann equations Riemann-Hurwitz formula Riemann-Lebesgue lemma Riemann-von Mangoldt formula Riemann problem Riemann series theorem Hirzebruch-Riemann-Roch theorem Riemann Xi function

author_abbrev_bot =

author_abbrev_zoo =

influences = nowrap|Johann Peter Gustav Lejeune Dirichlet

influenced =

awards =

religion =Lutheran

footnotes =**Georg Friedrich Bernhard Riemann**(pronounced "REE mahn" or in IPA2|'ri:man;September 17 ,1826 –July 20 ,1866 ) was a German mathematician who made important contributions to analysis anddifferential geometry , some of them paving the way for the later development ofgeneral relativity .**Biography****Early life**Riemann was born in

Breselenz , a village nearDannenberg in the Kingdom of Hanover in what is todayGermany . His father, Friedrich Bernhard Riemann, was a poor Lutheran pastor in Breselenz who fought in theNapoleonic Wars . His mother died before her children were grown. Riemann was the second of six children, shy, and suffered from numerous nervous breakdowns. Riemann exhibited exceptional mathematical skills, such as fantastic calculation abilities, from an early age, but suffered from timidity and a fear of speaking in public.**Middle life**In high school, Riemann studied the

Bible intensively, but his mind often drifted back to mathematics. To this end, he even tried to prove mathematically the correctness of theBook of Genesis . His teachers were amazed by his genius and his ability to solve extremely complicated mathematical operations. He often outstripped his instructor's knowledge. In 1840, Riemann went toHanover to live with his grandmother and attendlyceum (middle school). After the death of his grandmother in 1842, he attended high school at the [*http://de.wikipedia.org/wiki/Johanneum_L%C3%BCneburg Johanneum Lüneburg*] . In 1846, at the age of 19, he started studyingphilology andtheology in order to become a priest and help with his family's finances.In 1847, his father (Friedrich Riemann), after gathering enough money to send Riemann to university, allowed him to stop studying theology and start studying

mathematics . He was sent to the renownedUniversity of Göttingen , where he first metCarl Friedrich Gauss , and attended his lectures on themethod of least squares .In 1847, Riemann moved to

Berlin , where Jacobi, Dirichlet, and Steiner were teaching. He stayed in Berlin for two years and returned to Göttingen in 1849.**Later life**Bernhard Riemann held his first lectures in 1854, which not only founded the field of

Riemannian geometry but set the stage for Einstein'sgeneral relativity . 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 Dirichlet's death, he was promoted to head the mathematics department at Göttingen. He was also the first to propose the theory ofhigher dimensions Fact|date=February 2007, which greatly simplified the laws of physics. In 1862 he married Elise Koch and had a daughter.He died oftuberculosis on his third journey toItaly in Selasca (now a hamlet ofGhiffa onLake Maggiore ).**Influence**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 , andcomplex manifold theory. The theory ofRiemann surface s was elaborated byFelix Klein and particularlyAdolf Hurwitz . This area of mathematics is part of the foundation oftopology , and is still being applied in novel ways tomathematical physics .Riemann made major contributions to

real analysis . He defined theRiemann integral by means ofRiemann sum s, developed a theory oftrigonometric series that are notFourier series —a first step ingeneralized function theory—and studied theRiemann-Liouville differintegral .He made some famous contributions to modern

analytic number theory . In a single short paper (the only one he published on the subject of number theory), he introduced theRiemann zeta function and established its importance for understanding the distribution ofprime numbers . He made a series of conjectures about properties of the zeta function, one of which is the well-knownRiemann hypothesis .He applied the

Dirichlet principle fromvariational calculus to great effect; this was later seen to be a powerfulheuristic rather than a rigorous method. Its justification took at least a generation. His work onmonodromy and thehypergeometric function in the complex domain made a great impression, and established a basic way of working with functions by "consideration only of their singularities".**Euclidean geometry versus Riemannian geometry**In 1853, Gauss asked his student Riemann to prepare a "

Habilitationsschrift " on the foundations of geometry. Over many months, Riemann developed his theory ofhigher dimensions . When he finally delivered his lecture at Göttingen in 1854, the mathematical public received it with enthusiasm, and it is one of the most important works in geometry. It was titled "Über die Hypothesen welche der Geometrie zu Grunde liegen" (loosely: "On the foundations of geometry"; more precisely, "On the hypotheses which underlie geometry"), and was published in 1868.The subject founded by this work is

Riemannian geometry . Riemann found the correct way to extend into "n" dimensions thedifferential geometry of surfaces, which Gauss himself proved in his "theorema egregium ". The fundamental object is called theRiemann curvature tensor . For the surface case, this can be reduced to a number (scalar), positive, negative or zero; the non-zero and constant cases being models of the knownnon-Euclidean geometries .**Higher dimensions**Riemann's idea was to introduce a collection of numbers at every point in

space that would describe how much it was bent or curved. Riemann found that in four spatial dimensions, one needs a collection of ten numbers at each point to describe the properties of amanifold , no matter how distorted it is. This is the famousmetric tensor .**Writings in English***1868.“On the hypotheses which lie at the foundation of geometry” 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.

**ee also**

*Riemann hypothesis

*Riemann zeta function

*Riemann integral

*Riemann sum

*Riemann lemma

*Riemannian manifold

*Riemann mapping theorem

*Riemann-Hilbert problem

*Riemann-Hurwitz formula

*Riemann-von Mangoldt formula

*Riemann surface

*Riemann-Roch theorem

*Riemann theta function

*Riemann-Siegel theta function

*Riemann's differential equation

*Riemann matrix

*Riemann sphere

*Riemannian metric tensor

*Riemann curvature tensor

*Cauchy-Riemann equations

*Hirzebruch-Riemann-Roch theorem

*Riemann-Lebesgue lemma

*Riemann-Stieltjes integral

*Riemann-Liouville differintegral

*Riemann series theorem

*Riemann's 1859 paper introducing the complex zeta function

* "Prime Obsession "

*"The Music of the Primes "**Bibliography***

John Derbyshire , "" (John Henry Press, 2003) ISBN 0-309-08549-7**External links***MathGenealogy|id=18232

* [*http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Papers.html The Mathematical Papers of Georg Friedrich Bernhard Riemann*]

* All publications of Riemann can be found at: http://www.emis.de/classics/Riemann/

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* [*http://www.fh-lueneburg.de/u1/gym03/englpage/chronik/riemann/riemann.htm Bernhard Riemann - one of the most important mathematicians*]

* [*http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Geom/WKCGeom.html Bernhard Riemann's inaugural lecture*]

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