- Bernhard Riemann
name =Bernhard Riemann
caption =Bernhard Riemann, 1863
September 17, 1826
death_date =death date and age|1866|7|20|1826|9|17
Georg-August University of Göttingen
Georg-August University of Göttingen Berlin University
Carl Friedrich Gauss
Ferdinand Eisenstein Moritz Abraham Stern
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
influences = nowrap|
Johann Peter Gustav Lejeune Dirichlet
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 and differential geometry, some of them paving the way for the later development of general relativity.
Riemann was born in
Breselenz, a village near Dannenbergin the Kingdom of Hanover in what is today Germany. His father, Friedrich Bernhard Riemann, was a poor Lutheran pastor in Breselenz who fought in the Napoleonic 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.
In high school, Riemann studied the
Bibleintensively, but his mind often drifted back to mathematics. To this end, he even tried to prove mathematically the correctness of the Book 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 to Hanoverto live with his grandmother and attend lyceum(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 studying philologyand theologyin 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 renowned University of Göttingen, where he first met Carl Friedrich Gauss, and attended his lectures on the method of least squares.
Bernhard Riemann held his first lectures in 1854, which not only founded the field of
Riemannian geometrybut set the stage for Einstein's general 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 Dirichlet's death, he was promoted to head the mathematics department at Göttingen. He was also the first to propose the theory of higher dimensionsFact|date=February 2007, which greatly simplified the laws of physics. In 1862 he married Elise Koch and had a daughter.He died of tuberculosison his third journey to Italyin Selasca (now a hamlet of Ghiffaon Lake Maggiore).
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, and complex manifoldtheory. The theory of Riemann surfaces was elaborated by Felix Kleinand particularly Adolf Hurwitz. This area of mathematics is part of the foundation of topology, and is still being applied in novel ways to mathematical physics.
Riemann made major contributions to
real analysis. He defined the Riemann integralby means of Riemann sums, developed a theory of trigonometric seriesthat are not Fourier series—a first step in generalized functiontheory—and studied the Riemann-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 the Riemann zeta functionand established its importance for understanding the distribution of prime numbers. He made a series of conjectures about properties of the zeta function, one of which is the well-known Riemann hypothesis.
He applied the
Dirichlet principlefrom variational calculusto great effect; this was later seen to be a powerful heuristicrather than a rigorous method. Its justification took at least a generation. His work on monodromyand the hypergeometric functionin 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 of higher 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 the differential geometryof surfaces, which Gauss himself proved in his " theorema egregium". The fundamental object is called the Riemann 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 known non-Euclidean geometries.
Riemann's idea was to introduce a collection of numbers at every point in
spacethat 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 a manifold, no matter how distorted it is. This is the famous metric 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.
Riemann zeta function
Riemann mapping theorem
Riemann-von Mangoldt formula
Riemann theta function
Riemann-Siegel theta function
Riemann's differential equation
*Riemannian metric tensor
Riemann curvature tensor
Riemann series theorem
*Riemann's 1859 paper introducing the complex zeta function
The Music of the Primes"
John Derbyshire, "" (John Henry Press, 2003) ISBN 0-309-08549-7
* [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/
* [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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