 Riemannian geometry

Geom.1. Also called elliptic geometry. the branch of nonEuclidean geometry that replaces the parallel postulate of Euclidean geometry with the postulate that in a plane every pair of distinct lines intersects. Cf. hyperbolic geometry.2. the differential geometry of a metric space that generalizes a Euclidean space.[191520]
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also called elliptic geometryone of the nonEuclidean geometries (Euclidean geometry) that completely rejects the validity of Euclid's fifth postulate and modifies his second postulate. Simply stated, Euclid's fifth postulate is: through a point not on a given line there is only one line parallel to the given line. In Riemannian geometry, there are no lines parallel to the given line. Euclid's second postulate is: a straight line of finite length can be extended continuously without bounds. In Riemannian geometry, a straight line of finite length can be extended continuously without bounds, but all straight lines are of the same length. The tenets of Riemannian geometry, however, admit the other three Euclidean postulates (compare hyperbolic geometry).Although some of the theorems of Riemannian geometry are identical to those of Euclidean, most differ. In Euclidean geometry, for example, two parallel lines are taken to be everywhere equidistant. In elliptic geometry, parallel lines do not exist. In Euclidean, the sum of the angles in a triangle is two right angles; in elliptic, the sum is greater than two right angles. In Euclidean, polygons of differing areas can be similar; in elliptic, similar polygons of differing areas do not exist.The first published works on nonEuclidean geometries appeared about 1830. Such publications were unknown to the German mathematician Bernhard Riemann (Riemann, Bernhard) who, in 1866, extended the concepts from two to three or more dimensions. Another German mathematician, Felix Klein (Klein, Felix), later discriminated between elliptical space (polar) and doubleelliptical space (antipodal).* * *
Universalium. 2010.
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Riemannian geometry — [rē män′ē ən] n. [after G. F. B. Riemann (1826 66), Ger mathematician] a form of non Euclidean geometry in which there are no parallel lines, since its figures can be conceived as constructed on a curved surface where all straight lines intersect … English World dictionary
Riemannian geometry — Elliptic geometry is also sometimes called Riemannian geometry. Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric , i.e. with an inner product on the tangent… … Wikipedia
Riemannian geometry — noun the branch of differential geometry that studies Riemannian manifolds … Wiktionary
Riemannian geometry — noun (mathematics) a non Euclidean geometry that regards space as like a sphere and a line as like a great circle Bernhard Riemann pioneered elliptic geometry • Syn: ↑elliptic geometry • Topics: ↑mathematics, ↑math, ↑maths … Useful english dictionary
Riemannian geometry — Riemann′ian geom′etry n. math. the branch of non Euclidean geometry that replaces the parallel postulate of Euclidean geometry with the postulate that in a plane every pair of distinct lines intersects • Etymology: 1915–20; after G.F.B. Riemann … From formal English to slang
Riemannian geometry — noun Etymology: G. F. B. Riemann Date: 1896 a non Euclidean geometry in which straight lines are geodesics and in which the parallel postulate is replaced by the postulate that every pair of straight lines intersects … New Collegiate Dictionary
Fundamental theorem of Riemannian geometry — In Riemannian geometry, the fundamental theorem of Riemannian geometry states that on any Riemannian manifold (or pseudo Riemannian manifold) there is a unique torsion free metric connection, called the Levi Civita connection of the given metric … Wikipedia
Gauss's lemma (Riemannian geometry) — In Riemannian geometry, Gauss s lemma asserts that any sufficiently small sphere centered at a point in a Riemannian manifold is perpendicular to every geodesic through the point. More formally, let M be a Riemannian manifold, equipped with its… … Wikipedia
List of formulas in Riemannian geometry — This is a list of formulas encountered in Riemannian geometry.Christoffel symbols, covariant derivativeIn a smooth coordinate chart, the Christoffel symbols are given by::Gamma {ij}^m=frac12 g^{km} left( frac{partial}{partial x^i} g {kj}… … Wikipedia
Isometry (Riemannian geometry) — In the study of Riemannian geometry in mathematics, a local isometry from one (pseudo )Riemannian manifold to another is a map which pulls back the metric tensor on the second manifold to the metric tensor on the first. When such a map is also a… … Wikipedia