Bernhard Riemann

Bernhard Riemann was born in 1826 in Breselenz near Hannover, the son of a Lutheran pastor. He entered the University of Göttingen in 1846 to study theology at his father's wish, but soon transferred to mathematics under Carl Friedrich Gauss; after a stay at Berlin under Jacobi and Dirichlet he returned to Göttingen and completed his doctorate in 1851 under Gauss's supervision on the foundations of complex-function theory. His habilitation followed in 1854 with the lecture *Über die Hypothesen welche der Geometrie zu Grunde liegen* — the talk that generalised geometry to n-dimensional spaces of intrinsic curvature and prepared, without anyone realising at the time, the mathematical machinery of general relativity.

The 1854 lecture defined intrinsic curvature without any reference to embedding in a higher-dimensional ambient space — geometry from the inside, the way an ant on a surface might infer its own curvature from triangle-angle deficits without ever leaving the surface. Riemann introduced what would become the metric tensor (the rule for measuring infinitesimal distances ds² = g_{μν} dx^μ dx^ν) and what would become the Riemann curvature tensor (the obstruction to vectors returning to themselves under parallel transport around a closed loop). The lecture was hand-written, delivered on 10 June 1854 to an audience of three — Gauss and two assessors. Gauss reportedly walked home that afternoon "in unusual silence and emotion." The text was published only posthumously in 1868, and the geometric apparatus it founded became canonically known as Riemannian geometry.

Riemann's other achievements would have made an ordinary mathematician immortal. The 1859 paper *Über die Anzahl der Primzahlen unter einer gegebenen Grösse* introduced what is now called the Riemann zeta function and stated the Riemann hypothesis — the central open problem of analytic number theory. He founded the rigorous theory of complex manifolds (Riemann surfaces) in his 1857 work on Abelian functions; the Riemann integral remains the standard rigorous definition of integration in real analysis. He died of tuberculosis on 20 July 1866 at age 39 in Selasca on Lake Maggiore, Italy. His work sat largely unused by physicists for half a century, until Marcel Grossmann in 1912 introduced Einstein to the Riemann-Christoffel-Levi-Civita machinery — the toolkit that, three years later, Einstein used to write the field equations of general relativity.