Riemannian geometry

English Etymology From en Riemannian + en geometry, after German mathematician Bernhard Riemann. Noun - + en mathematics geometry The branch of differential geometry that concerns Riemannian manifolds; an example of a geometry that involves Riemannian manifolds. en 2001 Miklós Farkas Dynamical Models in Biology 171 https://books.google.com.au/books?id=I-sUVMhhQCAC&pg=PA171&dq=%22Riemannian+geometry%22&hl=en&sa=X&ved=0ahUKEwjG2cP7wPzMAhXG6KYKHe-gAK44ZBDoAQj6AjBO#v=onepage&q=%22Riemannian%20geometry%22&f=false Riemannian geometry was introduced by Riemann in 1854 as the n-dimensional generalization of the theory of curved surfaces of the 3D Euclidean space. en 2005 John F. Hawley; Katherine A. Holcomb Foundations of Modern Cosmology 2nd https://books.google.com.au/books?id=s5MUDAAAQBAJ&pg=PA235&dq=%22Riemannian+geometries%22&hl=en&sa=X&ved=0ahUKEwi5xP26xfzMAhVEl5QKHRDvD3oQ6AEISzAJ#v=onepage&q=%22Riemannian%20geometries%22&f=false 235 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). en 2010 Saul Stahl Geometry from Euclid to Knots 22 https://books.google.com.au/books?id=jLk7lu3bA1wC&pg=PA22&dq=%22Riemannian+geometry%22&hl=en&sa=X&ved=0ahUKEwjG2cP7wPzMAhXG6KYKHe-gAK44ZBDoAQivAjA_#v=onepage&q=%22Riemannian%20geometry%22&f=false 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. en 2010 Ilka Agricola Chapter 9: Non-integrable geometries, torsion, and holonomy Vicente Cortés Handbook of Pseudo-Riemannian Geometry and Supersymmetry https://books.google.com.au/books?id=AL56H5pF3akC&pg=PA278&dq=%22Riemannian+geometries%22&hl=en&sa=X&ved=0ahUKEwi5xP26xfzMAhVEl5QKHRDvD3oQ6AEIMDAE#v=onepage&q=%22Riemannian%20geometries%22&f=false 278 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. en 2013 Andrew McInerney First Steps in Differential Geometry: Riemannian, Contact, Symplectic 195 https://books.google.com.au/books?id=nNK4BAAAQBAJ&pg=PA194&dq=%22Riemannian+geometry%22&hl=en&sa=X&ved=0ahUKEwiOwY2GtvzMAhVF4aYKHbMZCDAQ6AEIsQMwRw#v=onepage&q=%22Riemannian%20geometry%22&f=false 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. The geometry of elliptical and spherical shapes. en elliptic geometry spherical geometry Translations branch of differential geometry Chinese: Mandarin: cmn 黎曼幾何 límàn jǐhé German: de Riemannsche Geometrie f Japanese: ja リーマン幾何学 Polish: pl geometria riemannowska f en eponyms en Non-Euclidean geometry