Riemannian geometry

plural Riemannian geometries

1. uncountableusuallygeometryThe branch of differential geometry that concerns Riemannian manifolds; an example of a geometry that involves Riemannian manifolds.

   Riemannian geometry was introduced by Riemann in 1854 as the n-dimensional generalization of the theory of curved surfaces of the 3D Euclidean space.

   2001, Miklós Farkas, Dynamical Models in Biology, page 171

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   Such geometries are called Riemannian geometries; they are characterized by invariant distances (for example, the space-time interval) that depend at most on the squares of the coordinate distances (∆x or ∆t).

   2005, John F. Hawley, Katherine A. Holcomb, Foundations of Modern Cosmology, 2nd edition, page 235

   Riemannian geometry has found many applications in science, the most spectacular of these being the theory of relativity.[…]Every Riemannian geometry has geodesics, which are defined as the shortest curves joining two points.

   2010, Saul Stahl, Geometry from Euclid to Knots, page 22

   At the beginning of the seventies, A. Gray generalized the classical holonomy concept by introducing a classification principle for non-integrable special Riemannian geometries [and] discovered in this context nearly Kähler manifolds in dimension six and nearly parallel G₂-manifolds in dimension seven.

   2010, Ilka Agricola, “Chapter 9: Non-integrable geometries, torsion, and holonomy”, in Vicente Cortés, editor, Handbook of Pseudo-Riemannian Geometry and Supersymmetry, page 278

   The concepts of Riemannian geometry are familiar: length, angle, distance, and curvature, among others. Historically tied to the origins of differential geometry, and with such familiar concepts, Riemannian geometry is often presented in textbooks as being synonymous with differential geometry itself, instead of as one differential-geometric structure among many.

   2013, Andrew McInerney, First Steps in Differential Geometry: Riemannian, Contact, Symplectic, page 195

2. uncountableusuallyThe geometry of elliptical and spherical shapes.

   Synonyms: elliptic geometry, spherical geometry