# Georg Friedrich Bernhard Riemann

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## Riemann, Georg Friedrich Bernhard

Born Sept 17, 1826, in Breselenz, Lower Saxony; died July 20, 1866, in Selasca, near Intra, Italy. German mathematician.

Riemann entered the University of Göttingen in 1846. There he attended the lectures of K. Gauss, many of whose ideas he later developed. Between 1847 and 1849 he attended the lectures of K. Jacobi on mechanics and of P. Dirichlet on number theory at the University of Berlin. In 1849 he returned to Göttingen, where he became friends with a colleague of Gauss, the physicist W. Weber, who aroused in him a deep interest in the problems of mathematical natural science. In 1851, Riemann defended his doctoral dissertation, “Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grosse” (Foundations for a General Theory of Functions of a Complex Variable). He became a privatdocent in 1854 and a professor in 1857 at the University of Göttingen. His lectures were the basis for a number of textbooks—on mathematical physics, on the theory of gravitation, electricity, and magnetism, and on the theory of elliptic functions—that were published after his death by his students. Riemann died of tuberculosis.

Riemann’s works had a considerable influence on the development of mathematics in the second half of the 19th century and in the 20th century. In his doctoral dissertation he initiated the geometric orientation of the theory of analytic functions. He introduced what are known as Riemann surfaces, which are important in the study of multiple-valued functions. He elaborated the theory of conformai mappings and set forth, in connection with this, the fundamental ideas of topology. His other achievements included the investigation of the conditions for the existence of analytic functions within domains of various types (the Dirichlet principle). The methods worked out by Riemann found extensive application in his subsequent work in such areas as the theory of algebraic functions and integrals, the analytic theory of differential equations (in particular, equations that define hypergeometric functions), and analytic number theory. In analytic number theory, for example, he pointed out the relation between the distribution of prime numbers and the properties of the zeta function—in particular, the distribution of the function’s zeros in the complex domain; the Riemann hypothesis about the zeros of the zeta function is still unproved.

In a number of works Riemann investigated the representation of functions by trigonometric series. In connection with this, he defined the necessary and sufficient conditions for integrability in the sense of Riemann (*see*INTEGRAL). This achievement was of importance for set theory and the theory of functions of a real variable. Riemann also proposed methods of integrating partial differential equations—for example, by making use of Riemann invariants and the Riemann function.

Very well known is Riemann’s lecture of 1854, “Über die Hypothesen, welche der Geometrie zu Grunde liegen” (On the Hypotheses That Form the Foundations of Geometry), which was published in 1867. In this lecture he presented a general idea of mathematical space—he used the term “manifold”—including function and topological spaces. Riemann here considered geometry in the broad sense as the study of continuous *n*-dimensional manifolds, that is, sets of any like objects. Generalizing Gauss’ results on the intrinsic geometry of a surface, he put forth the general concept of linear element, that is, the differential of the distance between the points of a manifold; he thereby defined what came to be known as Finsler spaces. Riemann considered in particular detail Riemannian spaces, which generalize the spaces of Euclidean, Lobachevskian, and Riemannian geometry (*see*NON-EUCLIDEAN GEOMETRY) and are characterized by a special type of linear element; he developed the notion of the curvature of such spaces. In discussing the application of his ideas to physical space, Riemann raised the question of the “ground of its metric relations,” as if anticipating what was done in the general theory of relativity.

The ideas and methods proposed by Riemann opened up new paths in the development of mathematics and found application in mechanics and physics.

### WORKS

*Gesammelte mathematische Werke und wissenschaftlicher Nachlass*, 2nd ed. New York, 1953.

*In Russian translation:*

*Sochineniia.*Moscow-Leningrad, 1948.

### REFERENCE

Klein, F.*Lektsii o razvitii matematiki v XIX stoletii*, part 1. Moscow-Leningrad, 1937. (Translated from German.)