Georg Friedrich Bernhard Riemann

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Georg Friedrich Bernhard Riemann

Birthplace: Jameln, Elbtalaue, Luechow-Dannenberg, Lower Saxony, Germany
Death: Died in Selasca, Verbania-Cusio-Ossola, Italy
Immediate Family:

Son of Friedrich Bernhard Riemann and Adelgunde Charlotte Amalie Riemann
Husband of Elise Riemann
Father of Ida Schilling
Brother of Marie Riemann; Ida Riemann; Clara Riemann; Wilhelm Riemann and Helene Riemann

Managed by: Martin Severin Eriksen
Last Updated:

About Georg Friedrich Bernhard Riemann

Dissertation: Grundlagen für eine allgemeine Theorie der Funktionen einer veränderlichen complexen Größe

Georg Friedrich Bernhard Riemann (September 17, 1826 – July 20, 1866) was an influential German mathematician who made lasting contributions to analysis and differential geometry, some of them enabling the later development of general relativity.

Riemann was born in Breselenz, a village near Dannenberg in the Kingdom of Hanover in what is the Federal Republic of Germany today. 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 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.


During 1840, Riemann went to Hanover to live with his grandmother and attend lyceum (middle school). After the death of his grandmother in 1842, he attended high school at the Johanneum Lüneburg. In high school, Riemann studied the Bible intensively, but he was often distracted by mathematics. To this end, he even tried to prove mathematically the correctness of the Book of Genesis.

His teachers were amazed by his adept ability to solve complicated mathematical operations, in which he often outstripped his instructor's knowledge.

In 1846, at the age of 19, he started studying philology and theology in order to become a priest and help with his family's finances.

During the spring of 1846, 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.

In 1847, Riemann moved to Berlin, where Jacobi, Dirichlet, Steiner, and Eisenstein were teaching. He stayed in Berlin for two years and returned to Göttingen in 1849.


Bernhard Riemann held his first lectures in 1854, which founded the field of Riemannian geometry and thereby set the stage for 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 Dirichlet's death, he was promoted to head the mathematics department at Göttingen. He was also the first to suggest using dimensions higher than merely three or four in order to describe physical reality[citation needed]—an idea that was ultimately vindicated with Einstein's contribution in the early 20th century. In 1862 he married Elise Koch and had a daughter.

Austro-Prussian War

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).

Meanwhile, in Göttingen his housekeeper tidied up some of the mess in his office, including much unpublished work. Riemann refused to publish incomplete work and some deep insights may have been lost forever.


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 manifold theory.

The theory of Riemann surfaces was elaborated by Felix Klein and 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 integral by means of Riemann sums, developed a theory of trigonometric series that are not Fourier series—a first step in generalized function theory—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 function and 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 principle from variational calculus to great effect; this was later seen to be a powerful heuristic rather than a rigorous method. Its justification took at least a generation. His work on monodromy and the hypergeometric 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 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 geometry of 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.

Higher dimensions

Riemann's idea was to introduce a collection of numbers at every point in space (i.e., a tensor) which 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 construction central to his geometry, known now as a Riemannian metric.

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, Bernhard (2004) (in English), Collected papers, Kendrick Press, Heber City, UT, MR2121437, ISBN 978-0-9740427-2-5

Bernhard Riemann, 1863

Born September 17, 1826

Breselenz, Kingdom of Hanover (modern-day Germany)

Died July 20, 1866 (aged 39)

Selasca, Kingdom of Italy

Residence Kingdom of Hannover

Nationality German

Fields Mathematics

Institutions Georg-August University of Göttingen

Alma mater Georg-August University of Göttingen

Berlin University

Doctoral advisor Carl Friedrich Gauss

Other academic advisors Ferdinand Eisenstein

Moritz Abraham Stern

Notable students Gustav Roch

Known for See list

Influences Johann Peter Gustav Lejeune Dirichlet

deutscher Mathematiker, der trotz seines kurzen Lebens auf vielen Gebieten der Analysis, Differentialgeometrie, mathematischen Physik und der analytischen Zahlentheorie bahnbrechend wirkte. Er gilt als einer der bedeutendsten Mathematiker.

Riemann wurde als Sohn eines lutherischen Pastors geboren und wuchs als eines von fünf Kindern unter beengten Verhältnissen auf. Seine Mutter, die Tochter des Hofrats Ebell in Hannover, war früh verstorben (1846) und sein Vater, Friedrich Bernhard Riemann, der aus Boizenburg stammte, an den Befreiungskriegen (Armee von Wallmoden) teilnahm und zuletzt in Quickborn Pastor war, starb 1855. Riemann hielt stets enge Verbindung zu seiner Familie. Er besuchte von 1840 bis 1842 das Gymnasium in Hannover, danach bis 1846 das Gymnasium Johanneum in Lüneburg, wobei er den katastrophalen Brand Hamburgs in der Ferne beobachten konnte. Schon früh fielen seine mathematischen Fähigkeiten auf. Ein Lehrer, der Rektor Schmalfuss, lieh ihm Legendres Zahlentheorie (Théorie des Nombres), ein schwieriges Werk von 859 Quartformat-Seiten, bekam sie aber schon eine Woche später zurück und fand, als er Riemann im Abitur über dieses Werk weit über das Übliche hinaus prüfte, dass Riemann sich dieses Buch vollständig zu eigen gemacht hatte.

Riemann sollte zunächst wie sein Vater Theologe werden und hatte dazu schon in Lüneburg neben Latein und Griechisch auch Hebräisch gelernt; dann aber wechselte er in Göttingen zur Mathematik. Von 1846 bis 1847 studierte er in Göttingen u. a. bei Moritz Stern, Johann Benedict Listing – einem Pionier der Topologie (1847 schrieb er ein Buch darüber) – und Carl Friedrich Gauß, der aber damals fast ausschließlich über Astronomie und nur noch selten über angewandte Themen wie seine Methode der kleinsten Quadrate las. 1847-1849 hörte Riemann in Berlin Vorlesungen von Peter Gustav Dirichlet über partielle Differentialgleichungen, bei Jacobi und Gotthold Eisenstein – mit dem er nähere Bekanntschaft schloss – über elliptische Funktionen, bei Steiner Geometrie. Nach Richard Dedekind beeindruckten ihn in dieser Zeit auch die Ereignisse der Revolution vom März 1848 – so hielt er als Teil des Studentenkorps einen Tag Wache vor dem königlichen Schloss. 1849 war er wieder in Göttingen und begann die Arbeit an seiner Dissertation zur Funktionentheorie, die er 1851 abschloss. Danach wurde er vorübergehend Assistent des Physikers Wilhelm Eduard Weber. 1854 habilitierte er.

Ab 1857 hatte er in Göttingen eine außerordentliche Professur. Im selben Jahr zogen seine zwei verbleibenden Schwestern zu ihm, für die er nach dem Tod seines Bruders trotz seines schmalen Gehalts sorgen musste – zur damaligen Zeit bestand das Gehalt eines Professors zum großen Teil aus Hörergeldern, und je anspruchsvoller die Vorlesung war, desto weniger Hörer stellten sich in aller Regel ein. Riemann erlitt aus Überarbeitung einen Zusammenbruch und begab sich zur Erholung nach Bad Harzburg zu Dedekind. 1858 besuchten ihn die italienischen Mathematiker Brioschi, Betti und Casorati in Göttingen, mit denen er sich anfreundete und denen er topologische Ideen vermittelte. Im selben Jahr besuchte er erneut Berlin und traf dort Kummer, Weierstraß und Kronecker. 1859 trat er die Nachfolge Dirichlets auf dem Lehrstuhl von Gauß in Göttingen an. 1860 reiste er nach Paris und traf Puiseux, Bertrand, Hermite, Briot und Bouquet. 1862 heiratete er Elise Koch, eine Freundin seiner Schwestern, mit der er eine Tochter, Ida, hatte, die 1863 in Pisa geboren wurde. Er hielt sich dann länger in Italien auf und traf seine italienischen Mathematikerfreunde wieder. Auf der Rückkehr von einer Italienreise 1862 verschlechterte sich sein Gesundheitszustand. Riemann litt an Tuberkulose. Auch längere Aufenthalte im milden Klima Italiens konnten die Krankheit nicht heilen. Auf der Flucht vor den 1866 in Göttingen aufeinander treffenden Heeren Hannovers und Preußens und auf neuerlicher Suche nach Erholung am Lago Maggiore, starb er im Alter von 39 Jahren auf seiner dritten Italienreise. Er wurde in Biganzolo begraben.

Der wissenschaftliche Nachlass von Riemann befindet sich in der Niedersächsischen Staats- und Universitätsbibliothek Göttingen. Ein Teil der privaten Briefe aus dem Besitz von Erich Bessel-Hagen kam an die Staatsbibliothek Berlin.

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Georg Friedrich Bernhard Riemann's Timeline

September 17, 1826
Elbtalaue, Luechow-Dannenberg, Lower Saxony, Germany
- 1851
Age 20
Göttingen, Lower Saxony, Germany
December 22, 1862
Age 36
Pisa, Pisa, Toscana, Italy
July 20, 1866
Age 39
Selasca, Verbania-Cusio-Ossola, Italy
July 1866
Age 39