Bernhard Riemann

Georg Friedrich Bernhard Riemann was a German mathematician whose work reshaped analysis, number theory, and differential geometry. In real analysis, he is famous for giving the first rigorous definition of the integral—now called the Riemann integral—and for his studies of Fourier series. In complex analysis, he introduced Riemann surfaces, which allowed a natural geometric approach to the subject. His 1859 paper on the prime-counting function, which first stated the Riemann hypothesis, is a cornerstone of analytic number theory. His pioneering work in differential geometry provided the mathematical foundation for general relativity. Many regard him as one of the greatest mathematicians ever.

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 had fought in the Napoleonic Wars. His mother, Charlotte Ebell, died in 1846. He was the second of six children. From an early age, he showed exceptional mathematical talent, including strong calculation skills, but he was timid, afraid of public speaking, and had frail health.

In 1840, Riemann moved to Hanover to live with his grandmother and attend middle school, since no such school existed near his home. After his grandmother died in 1842, he transferred to the Johanneum Lüneburg, a high school in Lüneburg. There, he studied the Bible intensively but was often distracted by mathematics. His teachers were amazed by his ability to perform complex mathematical operations, sometimes surpassing their own knowledge. In 1846, at age 19, he began studying philology and Christian theology to become a pastor and help his family financially. That spring, his father saved enough money to send him to the University of Göttingen, where Riemann planned to study theology. Once there, he began studying mathematics under Carl Friedrich Gauss, specifically Gauss’s lectures on the method of least squares. Gauss advised Riemann to abandon theology for mathematics. With his father’s approval, Riemann transferred to the University of Berlin in 1847, where he studied under Carl Gustav Jacob Jacobi, Peter Gustav Lejeune Dirichlet, Jakob Steiner, and Gotthold Eisenstein. He stayed in Berlin for two years and returned to Göttingen in 1849.

Riemann gave his first lectures in 1854, which founded Riemannian geometry and later set the stage for Einstein’s general relativity. In 1857, an attempt to promote him to extraordinary professor at Göttingen failed, but he finally received a regular salary. In 1859, after Dirichlet’s death, Riemann took over Gauss’s chair and became head of the mathematics department at Göttingen. He was also the first to suggest using dimensions beyond three or four to describe physical reality.

In 1862, he married Elise Koch; they had one daughter. When the armies of Hanover and Prussia clashed in Göttingen in 1866, Riemann fled. He died of tuberculosis during his third trip to Italy, in Selasca (now part of Verbania on Lake Maggiore), and was buried in the cemetery in Biganzolo, Verbania. Riemann was a devout Christian, the son of a Protestant minister, and saw his mathematical work as another way to serve God. He held closely to his Christian faith throughout his life, considering it most important. As he died, he was reciting the Lord’s Prayer with his wife and passed away before they finished. Meanwhile, back in Göttingen, his housekeeper discarded some of his office papers, including much unpublished work. Riemann refused to publish incomplete results, so some deep insights may have been lost.

Riemann’s published works opened research areas that combined analysis with geometry, later becoming major parts of Riemannian geometry, algebraic geometry, and complex manifold theory. The theory of Riemann surfaces was further developed by Felix Klein and especially Adolf Hurwitz. This area of mathematics is foundational to topology and continues to find new applications in mathematical physics. In 1853, Gauss asked Riemann to prepare a Habilitationsschrift on the foundations of geometry. Over many months, Riemann developed his theory of higher dimensions and delivered his lecture at Göttingen on 10 June 1854, titled *Ueber die Hypothesen, welche der Geometrie zu Grunde liegen*. It was not published until twelve years later, in 1868, by Dedekind, two years after Riemann’s death. Its early reception was slow, but it is now recognized as one of the most important works in geometry. The subject it founded is Riemannian geometry. Riemann found the correct way to extend the differential geometry of surfaces—which Gauss had proved in his theorema egregium—into n dimensions. The fundamental objects are the Riemannian metric and the Riemann curvature tensor. For two-dimensional surfaces, curvature at each point reduces to a single number (scalar), with surfaces of constant positive or negative curvature serving as models of non-Euclidean geometries. The Riemann metric is a collection of numbers.