Riemann, Georg Friedrich Bernhard (1826–1866)

Georg Riemann showed that in spherical geometry (a form of non-Euclidean geometry), although a parallelogram may have two opposite angles of 90°, it does not necessarily follow that the other two angles are also 90°.

German mathematician whose system of non-Euclidean geometry, thought at the time to be a mere mathematical curiosity, was used by Albert Einstein to develop his general theory of relativity. Riemann made a breakthrough in conceptual understanding within several other areas of mathematics: the theory of functions, vector analysis, projective and differential geometry, and topology.

Riemann took into account the possible interaction between space and the bodies placed in it; until then, space had been treated as an entity in itself, and this new point of view was to become a central concept of 20th-century physics.

Riemann was born in Hannover and studied at Göttingen and Berlin. He spent his academic career at Göttingen, becoming professor in 1859.

Riemann's paper on the fundamental postulates of Euclidean geometry, written in the early 1850s but not published until 1867, was to open up the whole field of non-Euclidean geometry and become a classic in the history of mathematics.

He also published a paper on hypergeometric series, invented ‘spherical’ geometry as an extension of hyperbolic geometry, and in 1855–56 lectured on his theory of Abelian functions, one of his fundamental developments in mathematics.

The Riemann–Christoffel symbols (see also Elwin Bruno Christoffel) concern the theory of invariants.

He developed the Riemann surfaces to study complex function behaviour. These multiconnected, many-sheeted surfaces can be dissected by cross-cuts into a singly connected surface. By means of these surfaces he introduced topological considerations into the theory of functions of a complex variable, and into general analysis. He showed, for example, that all curves of the same class have the same Riemann surface.