RIEMANN, (GEORG FRIEDRICH) BERNHARD

born Sept. 17, 1826, Breselenz, Hanover died 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 Thorie des nombres (Theory of Numbers) of Adrien-Marie Legendre. In 184651 he studied at the universities of Gttingen 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 Gttingen with a dissertation on the Grundlagen fr eine allgemeine Theorie der Functionen einer vernderlichen complexen Grsse (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 surfacea multilayered surfaceon 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 Gttingen, 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 ber 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 Jnos 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 spacetime 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 numbersome were published posthumouslycontained 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 RiemannRoch theorem on algebraic functions, Riemann surfaces, the Riemann mapping theorem, the Riemann integral, the RiemannLebesgue 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 RiemannLiouville integrals of fractional order. In 1859 he wrote the paper ber die Anzahle der Primzahlen unter einer gegebenen Grsse (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 Gttingen 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.