The German mathematician Georg Friedrich Bernard Riemann (1826-1866) was one of the founders of algebraic geometry. His concept of geometric space cleared the way for the general theory of relativity.

# Georg Friedrich Bernard Riemann

On Sept. 17, 1826, Georg Riemann was born in Breselenz. Shortly afterward, the family moved to Quickborn, Holstein, where his father, a Lutheran minister, assumed the pastorate. Riemann senior quickly recognized his younger son's mathematical talent. When Georg was 10 years old, he was placed under a mathematics tutor who soon found himself outdistanced by his pupil.

Riemann had planned on a career in the Church in accordance with his father's wishes. In 1846 he entered the University of Göttingen as a student of theology and philology. But mathematics called, and he had probably already decided to change his mind, should his father consent. He may have strengthened his argument by a grand attempt to prove Genesis mathematically. The proof was hardly valid, but Riemann senior appreciated the effort and gave his blessing to the mathematical career. In 1847 Georg transferred to the University of Berlin, where such vigorous innovators as K. G. J. Jacobi, P. G. Lejeune-Dirichlet, J. Steiner, and F. G. M. Eisenstein had created a livelier atmosphere for learning. In 1849 Riemann returned to Göttingen to prepare for his doctoral examinations under Wilhelm Weber, the famous electrodynamicist.

## Riemann Surfaces

Riemann's doctoral dissertation was, in Karl Friedrich Gauss's words, the product of a "gloriously fertile originality." Its novel ideas were further developed in three papers published in 1857. Here is a crude explanation of the principal novelty:

A complex number may be represented by a point in a plane. A function (single-valued) of a complex variable is a rule which pairs each point in one plane with a *unique* point
in another plane. Imagine a fly wandering about the surface of a plate-glass window. As the fly moves from point to point, its shadow moves from point to point on the floor of the room. Each point which the fly occupies on the window determines a *unique* point that its shadow occupies on the floor.

Now suppose that the floor is a highly reflective surface. The incoming light strikes the floor and is reflected to the wall, and we see a second image of our wandering fly. Now each position of the fly on the window determines two shadows—one on the floor and one on the wall.

But that is not quite right. There are some positions in which the fly still casts only one shadow. These are the points which throw the shadow on the line of intersection between floor and wall. Let us call these points branch points.

Now suppose that we replace the plate-glass window with two parallel sheets of plate glass (like a double window for insulation against cold). We endow the sheets with the following magical properties: any object on the outside sheet will cast a shadow only on the floor, and any object on the inside sheet will cast a shadow only on the wall. Furthermore, we join the two sheets along the line of branch points, so that they now form a single surface. Our fly may crawl from one sheet to the other, but to each point that he occupies on the glass surface, there once again corresponds one unique location of his shadow.

This is what Riemann did for multiple-valued functions of a complex variable. His surfaces restore single-valuedness to functions and at the same time provide a method of representing these functions geometrically. Moreover, it turns out that the analytic properties of many functions are mirrored by the geometric (topological) properties of their associated Riemann surfaces.

## His Göttingen Lecture

After successfully defending his dissertation, Riemann applied for an opening at the Göttingen Observatory but did not get the job. He next set his sights on becoming a privatdozent (unpaid lecturer) at the university. There were two hurdles to surmount before he could obtain the lecture-ship: a probationary essay and a trial lecture before the assembled faculty. The former, a paper on trigonometric series, included the definition of the "Riemann integral" in almost the form that it appears in current textbooks. The essay was submitted in 1853.

For his trial lecture Riemann submitted three possible titles, fully expecting Gauss to abide by tradition and assign one of the first two. But the third topic was one with which Gauss himself had struggled for many years. He was curious to hear what Riemann had to say "On the Hypotheses Which Lie at the Foundations of Geometry."

The lecture that Riemann delivered to the Göttingen faculty on June 10, 1854, is one of the great masterpieces of mathematical creation and exposition. Riemann wove together and generalized three crucial discoveries of the 19th century: the extension of Euclidean geometry to n dimensions; the logical consistency of geometries that are not Euclidean; and the intrinsic geometry of a surface, in terms of its metric and curvature in the neighborhood of a point. In his synthesis Riemann demonstrated the existence of an infinite number of different geometries, each of which could be characterized by its peculiar differential form. Finally, he pointed out that the choice of a particular geometry to represent the structure of real physical space was a matter for physics, not mathematics.

The impact of the lecture was enormous but delayed. Riemann worked out some of the analytical machinery in a memoir of 1861 on the conduction of heat, but the lecture itself was not published until 1868. Twenty years later a respected historian noted simply that the paper "had excited much interest and discussion." By 1908 the same historian was calling it a "celebrated memoir" which had attracted "general attention to the subject of non-Euclidean geometry."

## "Riemann Hypothesis"

Riemann spent 3 years as a privatdozent. In 1857 he was appointed assistant professor, and 2 years later, when Dirichlet died, Riemann succeeded him in the chair of Gauss. After 1860 the honors, including international recognition, came thick and fast. He died on July 20, 1866, in Selasca, Italy.

Riemann's special genius was the penetrating vision that enabled him to see through a mass of obscuring detail and perceive the submerged foundations of a theory intuitively. This uncanny talent was most obvious in his geometric work, but the most remarkable instance occurs in analytic number theory. In an 1859 paper on prime numbers, Riemann proved several properties of what came to be called "Riemann's zeta function." Several other properties of the function he simply stated without proof. After his death a note was found, saying that he had deduced these properties "from the expression of it (the function) which, however, I did not succeed in simplifying enough to publish."

To this day no one has the slightest idea of what this "expression" might be. All but one of the properties have since been proved. The last one, now called the "Riemann hypothesis," still awaits its conqueror, despite the efforts of several generations of talented mathematicians.

## Further Reading on Georg Friedrich Bernard Riemann

The best biography of Riemann in English is in Eric T. Bell, *Men of Mathematics* (1937). He is discussed in the first volume of Ganesh Prasad, *Some Great Mathematicians of the Nineteenth Century: Their Lives and Their Works* (1933). The place of Riemannian geometry in relativity theory is discussed in Jagjit Singh, *Great Ideas of Modern Mathematics: Their Nature and Use* (1959). For a nontechnical introduction to non-Euclidean geometries see Richard Courant and Herbert Robbins, *What Is Mathematics?* (1941).

## Additional Biography Sources

*Riemann, topology, and physics,* Boston: Birkhauser, 1987.