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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|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|Lobachevskii geometry]] (also called hyperbolic geometry) and the [[Riemann geometry|Riemann geometry]] (also called elliptic geometry) hold, respectively.



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A Riemannian space is also called a [[Riemannian manifold|Riemannian manifold]].

For references see [[Riemann tensor|Riemann tensor]]; [[Riemannian geometry|Riemannian geometry]].