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Summary Article: Riemann, (Georg Friedrich) Bernhard (1826-1866)
From The Hutchinson Dictionary of Scientific Biography

Place: Germany

Subject: biography, maths and statistics

German mathematician whose work in geometry - both in combining the results of others and in his own crucial and innovative research - developed that branch of mathematics to a large degree. His concepts in non-Euclidean and topological space led to advances in complex algebraic function theory and in physics; later Albert Einstein was to make use of his work in his own theory of relativity. Riemann's study of analysis was also fundamental to further development. Despite the brevity of his life, he knew most of the other great mathematicians of his time, and was himself a famed and respected teacher.

Riemann was born on 17 September 1826 in Breselenz (in Hannover), one of the six children of a Lutheran pastor. From a very early age he showed considerable talent in mathematics. Nevertheless, when he leaft the Lyceum in Hanover in 1846 to enter Göttingen University, it was to study theology at the behest of his father. However, he soon obtained his father's permission to devote himself to mathematics and did so, being taught by the great Karl Gauss. Riemann moved to Berlin in 1847 and there came into contact with the equally prestigious teachers Lejeune Dirichlet and Karl Jacobi; he was particularly influenced by Dirichlet. Two years later he returned to Göttingen and submitted a thesis on complex function theory, for which he was awarded his doctorate in 1851.

During the next two years Riemann qualified as an unsalaried lecturer, preparing original work on Fourier series, and working as an assistant to the renowned physicist Wilhelm Weber. At Gauss's suggestion, Riemann also wrote a paper on the fundamental postulates of Euclidean geometry, a paper that was to open up the whole field of non-Euclidean geometry and become a classic in the history of mathematics. (Although Riemann first read this paper to the university on 10 June 1854, neither it nor the work on Fourier series was published until 1867 - a year after Riemann's death.) Riemann's first course of lectures concerned partial differential equations as applied to physics. The course was so admired by physicists that it was reprinted as long afterwards as in 1938 - 80 years later. He published a paper on hypergeometric series and in 1855-56 lectured on his (by now famous) theory of Abelian functions, one of his fundamental developments in mathematics, which he published in 1857. When Karl Gauss died in 1855, Dirichlet was appointed in his place, and Riemann was appointed assistant professor in 1857. On Dirichlet's untimely death in 1859 Riemann became professor. Three years later, Riemann married; but in the same year he fell seriously ill with tuberculosis and he spent much of the following four years trying to recover his health in Italy. But he died there, in Selasca, on 16 June 1866 at the age of 39.

Riemann's first work (his thesis, published when he was already 25) was a milestone in the theory of complex functions. Augustin Cauchy, who had struggled with general function theory for 35 years, had discovered most of the fundamental principles, and had made many daring advances -but some points of understanding were still lacking. Unlike Cauchy, Riemann based his theory on theoretical physics (potential theory) and geometry (conformal representation), and could thus develop the so-called ‘Riemann surfaces’ which were able to represent the branching behaviour of a complex algebraic function.

He developed these ideas further in a paper of 1857 that continued the exploration of Riemann surfaces as investigative tools to study complex function behaviour, these surfaces being given such properties that complex functions could map conformally onto them. In the theory of Abelian functions there is an integer p associated with the number of ‘double points’ - a double point may be represented on a graph as where a curve intersects itself. Riemann formed, by extension, a multiconnected many-sheeted surface that could be dissected by cross-cuts into a singly connected surface. By means of these surfaces he introduced topological considerations into the theory of functions of a complex variable, and into general analysis. He showed, for example, that all curves of the same class have the same Riemann surface. Extensions of this work become highly abstract - the genus p was in fact discovered by Niels Abel in a purely algebraic context; it is not simply the number of double points. Not until much more work had followed, by Henri Poincaré in particular, did Riemann's ideas in this field reach general understanding. However, his work considerably advanced the whole field of algebraic geometry.

Often his lectures and papers were highly philosophical, containing few formulae, but dealing in concepts. He took into account the possible interaction between space and the bodies placed in it; hitherto space had been treated as an entity in itself, and this new point of view - seized on by the theoretical physicist Hermann von Helmholtz among others - was to become a central concept of 20th-century physics.

Riemann's most profound paper on the foundation of geometry (presented in 1854), also had consequences for physics, for in it he developed the mathematical tools that later enabled Albert Einstein to develop his theory of relativity. The three creators of ‘hyperbolic geometry’ - Karl Gauss, Nikolai Lobachevsky, and János Bolyai (see Wolfgang and János Bolyai) - all died in the 1850s with their work unacknowledged, and it was Riemann's paper that initiated the revolution in geometry. In 1799 Gauss had claimed to have devised a geometry based on the rejection of Euclid's fifth postulate, which states that parallel lines meet at infinity. Another way of expressing this is to consider a parallelogram in which two opposite corners are right-angled - but in which the other two angles are less than 90°; this is ‘hyperbolic’ geometry.

One possible case for the two angles to be less than 90° would come about according to two-dimensional geometry on the surface of a sphere. Straight lines are in those circumstances sections of great circles, and a ‘rectangle’ drawn on the surface of the sphere may have angles of less than 90°. In this way, Riemann invented ‘spherical’ geometry, which had previously been overlooked (and which can be more disconcerting than hyperbolic geometry). In 1868 Eugenio Beltrami developed this, considering ‘pseudo-spheres’ - surfaces of constant negative curvature that can realize the conditions for a non-Euclidean geometry on them.

Many other ideas were contained in Riemann's paper. He took up the discussion - and returned again to it later in life - of the properties of topological variabilities (manifolds) with an arbitrary number n of dimensions, and presented a formula for its metric, the means of measuring length within it. He took an element of length ds along a curve, and defined it as:

ds2 = Σgij2 dxidx j

where i, j = 1, 2, ... n. The structure obtained by this rule is called a Riemann space. It is in fact exactly this sort of concept that Einstein used to deal with time as a ‘fourth dimension’, and to talk about the ‘curvature of space’. Euclidean geometry requires the ‘curvature’ to be zero. The definition of the curvature of a directed space curve at a point was introduced in this same paper, and implicitly introduces the concept of a tensor. From generalization of this work, and despite opposition at the time, n-dimensional geometries began to be used, especially to examine the properties of differential forms with more than three variables.

Riemann's career was short, and not prolific, but he had a profound and almost immediate effect on the development of mathematics. Everything he published was of the highest quality, and he made a breakthrough in conceptual understanding within several areas of mathematics: the theory of functions, vector analysis, projective and differential geometry, non-Euclidean geometry, and topology. 20th-century mathematics bears witness to the extreme fruitfulness of Riemann's ideas.

© RM, 2018. All rights reserved.

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