Riemann,
Georg Friedrich Bernhard

Riemann, Georg Friedrich Bernhard
(1826-66), German mathematician, who developed a system of geometry that aided
the development of modern theoretical physics.

Riemann was born in Breselenz, and
educated at the universities of Göttingen and Berlin.
His doctoral thesis, “Foundations for a General Theory of Functions of a
Complex Variable,” submitted in 1851, was an outstanding contribution to
function theory. From 1857 until his death he was professor of mathematics at
the University of Göttingen. The significance of
Riemannian geometry lies in its use and extension of both Euclidean geometry
and the geometry of surfaces, leading to a number of generalized differential
geometries. Its most important effect was that it made
a geometrical application possible for some major abstractions of tensor
analysis, leading to the pattern and concepts for general relativity later used
by Albert Einstein in developing his theory of relativity. Riemannian geometry
is also necessary for treating electricity and magnetism in the framework of
general relativity.

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**Riemann,
(Georg Friedrich) Bernhard**

**b****. Sept. 17, 1826,
Breselenz, Hanover**

d. July 20, 1866, Selasca,
Italy

German
mathematician whose work widely influenced geometry and analysis. In addition,
his ideas concerning geometry of space had a profound effect on the development
of modern theoretical physics and provided the foundation for the concepts and
methods used later in relativity theory.

**Riemann** was the second of six children of a Lutheran pastor, who gave
him his first instruction. He obtained a good education with the encouragement
of a happy and devout family. At the local *Gymnasium *(high school), he
quickly progressed in mathematics beyond the guidance of his teachers,
mastering calculus and the *Théorie** des nombres *("Theory of Numbers") of Adrien-Marie Legendre. In 1846-51
he studied at the universities of Göttingen and
Berlin, where he was interested in problems concerning the theory of prime
numbers, elliptic functions, and geometry. Following studies in experimental
physics and *Naturphilosophie**, *which
sought to derive universal principles from all natural phenomena, he concluded
that mathematical theory could secure a connection between magnetism, light,
gravitation, and electricity, and he suggested field theories, in which the
space surrounding electrical charges may be mathematically described. Thus,
during his student days he began to develop original ideas that were to become
important to modern mathematical physics.

In 1851 he
obtained the doctorate at Göttingen with a
dissertation on the "Grundlagen für eine allgemeine
Theorie der Functionen einer veränderlichen complexen Grösse"
("Foundations for a General Theory of Functions of a Complex
Variable"). Function theory, which treats the relations between varying
complex numbers, is one of the major achievements of 19th-century mathematics. **Riemann**
based his treatment on geometrical ideas rather than algebraic calculation
alone. His work, which earned the rare praise of the renowned mathematician
Carl **Friedrich** Gauss, led to the idea of the **Riemann**
surface--a multilayered surface--on which a multivalued
function of a complex variable can be interpreted as a single-valued function.
This idea, in turn, contributed to methods in topology, which deals with
position and place instead of measure and quantity. His probationary essay (*Habilitationsschrift*) for admission to the faculty
in 1853 was "On the Representation of a Function by Means of a Trigonometrical Series."

While continuing
to develop unifying mathematical themes in the laws of physics, **Riemann**
also prepared in 1854 for his inaugural lecture at Göttingen,
required for admission to the faculty as a *Privatdozent**,
*an unpaid lecturer dependent entirely on student fees. He listed three
topics, from which Gauss, representing the faculty, chose "Über die Hypothesen, welche der Geometrie
zu Grunde liegen" ("On the Hypotheses That Form the
Foundations of Geometry"). Gauss himself had devoted long, profound
speculations to this difficult subject. In this lecture, one of the most
celebrated in the history of mathematics, **Riemann** developed a
comprehensive view of geometry. With a thorough understanding of the
limitations of ordinary, Euclidean geometry, which is based on the postulate of
parallels, he independently formulated a non-Euclidean
geometry. In so doing, he was apparently unaware that Nikolay
Lobachevsky and János Bolyai had already shown the possibility of devising a
consistent geometry without this postulate. **Riemann**'s non-Euclidean
geometry was an alternative to theirs and to that formulated by Gauss. He
postulated that, through a point outside a line, there are no parallels to that
line, a physical example of which can be seen in the fact that two ships on a
meridian must meet at a pole. He correctly perceived that his ideas would
benefit physics, as indeed they did when Einstein drew upon them to build his
model of space-time in relativity theory.

Beginning in
1855, **Riemann** received a small stipend that represented unusual academic
progress at the time and removed him from the ranks of the hardship cases. In
1857 he became professor extraordinarius (associate
professor) and in 1859 professor, succeeding the mathematician Peter Gustav Lejeune Dirichlet, who had
succeeded Gauss four years earlier. **Riemann** was beset by overwork,
deaths in his family, and his own faltering health. He continued, however, to
produce original papers, which, though few in number--some were published
posthumously--contained many rich ideas, such as his work on partial
differential equations. A measure of his influence is the extensive list of
methods, theorems, and concepts that bear his name: the **Riemann** approach
to function theory, the **Riemann**-Roch theorem
on algebraic functions, **Riemann** surfaces, the **Riemann** mapping
theorem, the **Riemann** integral, the **Riemann**-Lebesgue
lemma on trigonometrical integrals, the **Riemann**
method in the theory of trigonometrical series,
Riemannian geometry, **Riemann** curvature, **Riemann** matrices in the
theory of Abelian functions, the **Riemann** zeta
functions, the **Riemann** hypothesis, the **Riemann** method of solving
hyperbolic partial differential equations, and **Riemann**-Liouville integrals of fractional order. In 1859 he wrote
the paper "Über die Anzahle
der Primzahlen unter einer gegebenen
Grösse" ("On the Number of Primes in a
Given Magnitude"), in which he partially described the asymptotic
frequency of primes (positive integral numbers that have no other factors
except one and themselves, as 2, 3, 5, . . . ).

**Riemann**'s growing reputation finally earned him a permanent post in
1859 at Göttingen as the second successor to Gauss.
In 1862 he married Elise Koch, and, for a time, the conditions of his life
improved. Then he fell ill with pleurisy, which was complicated by
tuberculosis. His strength gradually ebbed, despite several visits to Italy for
recuperation. He died, in the Lutheran faith of his childhood, in 1866.