Meaning of Riemannian geometry | Babel Free

Definitions

1. The branch of differential geometry that concerns Riemannian manifolds; an example of a geometry that involves Riemannian manifolds.
   uncountable, usually
2. Elliptical or spherical geometry.
   uncountable, usually

Equivalents

Examples

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

“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).”

“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.”

“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.”

“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.”

CEFR level