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 noneuclidean 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.