A space in small domains of which the Euclidean geometry is approximately valid (up to infinitesimals of an order higher than the dimensions of the domains), though in the large such a space may be non-Euclidean. Such a space was named after B. Riemann, who in 1854 outlined the bases of the theory of such spaces (see Riemannian geometry). The simplest Riemannian spaces are Euclidean spaces and two other spaces of constant curvature closely related to it, in which the Lobachevskii geometry (also called hyperbolic geometry) and the Riemann geometry (also called elliptic geometry) hold, respectively.
A Riemannian space is also called a Riemannian manifold.
For references see Riemann tensor; Riemannian geometry.
Riemannian space. Material from the article "Riemannian space" in BSE-3 (originator), Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Riemannian_space&oldid=13360