Riemannian geometry

Article By:

Chern, S. S. Formerly, Mathematical Sciences Research Institute, University of California, Berkeley, California.

Last reviewed:August 2020

DOI:https://doi.org/10.1036/1097-8542.590000

The geometry of Riemannian manifolds. Riemannian geometry was initiated by B. Riemann in 1854, following the pioneering work of C. F. Gauss on surface theory in 1827. Riemann introduced a coordinate space in which the infinitesimal distance between two neighboring points is specified by a quadratic differential form, given below. Such a space is a natural generalization of Euclidean geometry and Gauss's geometry of surfaces in three-dimensional euclidean space, as well as the non-Euclidean geometries: hyperbolic geometry (previously discovered by J. Bolyai and N. I. Lobachevsky) and elliptic geometry. A Riemannian manifold is a topological space that further generalizes this notion. Riemannian geometry derives great importance from its application in the general theory of relativity, where the universe is considered to be a Riemannian manifold. See also: Differential geometry; Euclidean geometry; Non-Euclidean geometry; Relativity

The content above is only an excerpt.