Bernhard Riemann

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Lore & Background

Riemann was born on 17 September 1826 in Breselenz, a village near Dannenberg in the Kingdom of Hanover. His father, Friedrich Bernhard Riemann, was a poor Lutheran pastor who fought in the Napoleonic Wars. His mother, Charlotte Ebell, died in 1846. Riemann was the second of six children. He exhibited exceptional mathematical talent from an early age but suffered from timidity and a fear of speaking in public, and had frail health. In 1846, at age 19, he started studying philology and Christian theology at the University of Göttingen, but Carl Friedrich Gauss recommended he give up theological work and enter the mathematical field. Riemann transferred to the University of Berlin in 1847, where Carl Gustav Jacob Jacobi, Peter Gustav Lejeune Dirichlet, Jakob Steiner, and Gotthold Eisenstein were teaching. He returned to Göttingen in 1849. Riemann held his first lectures in 1854, which founded the field of Riemannian geometry. In 1859, following Dirichlet's death, he was promoted to head the mathematics department at the University of Göttingen. He married Elise Koch in 1862; they had a daughter. Riemann fled Göttingen when the armies of Hanover and Prussia clashed there in 1866. He died of tuberculosis during his third journey to Italy in Selasca (now a hamlet of Verbania on Lake Maggiore), where he was buried in the cemetery in Biganzolo. At the time of his death, he was reciting the Lord's Prayer with his wife and died before they finished saying the prayer. Meanwhile, in Göttingen his housekeeper discarded some of the papers in his office, including much unpublished work. Riemann refused to publish incomplete work, and some deep insights may have been lost.

Reader's Guide

Riemann's significance lies in his transformative contributions across multiple mathematical fields. In real analysis, he provided the first rigorous formulation of the integral, now called the Riemann integral, and advanced the study of Fourier series. In complex analysis, he introduced Riemann surfaces, through which multi-valued functions like the logarithm or the square root could become one-to-one functions. His 1859 paper on the prime-counting function, containing the original statement of the Riemann hypothesis, is regarded as a foundational paper of analytic number theory. In differential geometry, his 1854 lecture 'Ueber die Hypothesen, welche der Geometrie zu Grunde liegen' founded Riemannian geometry, extending Gauss's theorema egregium to n dimensions and defining the Riemannian metric and Riemann curvature tensor. Riemann found that in four spatial dimensions, one needs ten numbers at each point to describe distances and curvatures on a manifold. He was also the first to suggest using dimensions higher than merely three or four in order to describe physical reality. His work on abelian functions and theta functions on Riemann surfaces included a competition with Weierstrass to solve the Jacobian inverse problems for abelian integrals. Despite his short life and frail health, his ideas opened new research areas combining analysis with geometry, which became major parts of Riemannian geometry, algebraic geometry, and complex manifold theory.

Did You Know?

• Riemann initially studied philology and Christian theology at the University of Göttingen to become a pastor, before Carl Friedrich Gauss recommended he switch to mathematics.
• Riemann's 1854 lecture founding Riemannian geometry was not published until twelve years later in 1868 by Dedekind, two years after his death.
• At the time of his death, Riemann was reciting the Lord's Prayer with his wife and died before they finished saying the prayer.
• Riemann's housekeeper discarded some of the papers in his office after his death, including much unpublished work, because Riemann refused to publish incomplete work.
• Riemann was the first to suggest using dimensions higher than merely three or four in order to describe physical reality.