La géométrie métrique des variétés riemanniennes (variations sur la formule a 2 = b 2 + c 2 – 2 b c cos α). Berger, Marcel. Élie Cartan et les mathématiques. Une métrique semi-Riemannienne de l’indice 0 n’est qu’une métrique Rie- nentielle sur une variété Introduction à la Géométrie Riemannienne par l’étude des. qui avait organisé une conférence de géométrie sous-riemannienne `a . `a la dimension infinie le cadre de la géométrie sous-riemannienne.
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Any smooth manifold admits a Riemannian metricwhich often helps to solve problems of differential topology. What follows is an incomplete list of the most classical theorems in Riemannian geometry.
Riemannian geometry Bernhard Riemann. From Wikipedia, the free encyclopedia. Brans—Dicke theory Kaluza—Klein Quantum gravity. Riemannian geometry was first put forward in generality by Bernhard Riemann in the 19th century.
Variété pseudo-riemannienne — Wikipédia
Volume Cube cuboid Cylinder Pyramid Sphere. The formulations given are far from being very exact or the most general. Dislocations and Riemamnienne produce torsions and curvature. In other projects Wikimedia Commons.
Rie,annienne Plane Area Polygon. Background Introduction Mathematical formulation. Square Rectangle Rhombus Rhomboid. This gives, in particular, local notions of anglelength of curvessurface area and volume. Most of the results can be found in the classic monograph by Jeff Cheeger and D.
Riemannian geometry is the branch of differential geometry that studies Riemannian manifoldssmooth manifolds with a Riemannian metrici. There exists a close analogy of differential geometry with the mathematical structure of defects in regular crystals.
Other generalizations of Riemannian geometry include Finsler geometry. Background Principle of relativity Galilean relativity Galilean transformation Special relativity Doubly special relativity. Retrieved from ” https: Development of Riemannian geometry resulted in synthesis of riemanienne results concerning the geometry of surfaces and the behavior of geodesics on them, with techniques that can be applied to the study of differentiable manifolds of higher dimensions. From those, some other global quantities can be derived by integrating local contributions.
This page was last edited on 30 Decemberat In all of the following theorems we assume some local behavior of the space usually formulated using curvature assumption to derive some information about the global structure of the space, including either some information on the topological type of the manifold or on the behavior of points at “sufficiently large” distances.
Fundamental concepts Principle of relativity Theory of relativity Frame of reference Inertial frame of reference Rest frame Center-of-momentum frame Equivalence principle Mass—energy equivalence Special relativity Doubly special relativity de Sitter invariant special relativity World line Riemannian rieannienne. Principle of relativity Galilean relativity Galilean transformation Special relativity Doubly special relativity.
Light cone World line Minkowski diagram Biquaternions Minkowski space. Principle of relativity Theory of relativity Frame of reference Inertial frame of reference Rest frame Center-of-momentum frame Equivalence principle Mass—energy equivalence Special relativity Doubly special relativity de Sitter invariant special relativity World line Riemannian geometry.
Riemannian geometry – Wikipedia
Time dilation Mass—energy equivalence Length contraction Relativity of simultaneity Relativistic Doppler effect Thomas precession Ladder paradox Twin paradox. Elliptic geometry is also sometimes called “Riemannian geometry”. Introduction History Mathematical formulation Tests. Kaluza—Klein theory Quantum gravity Supergravity.
The choice is made depending on its importance and elegance of formulation. Riemannian geometry originated with the vision of Bernhard Riemann expressed in his inaugural lecture ” Ueber die Hypothesen, welche der Geometrie zu Grunde liegen ” “On the Hypotheses on which Geometry is Based”. It enabled the formulation of Einstein ‘s general riemanniennne of relativitymade profound impact on group theory and representation theoryas well as analysisand spurred the development of algebraic and differential topology.
It is a very broad and abstract generalization of the differential geometry of surfaces in R 3. Point Line segment ray Length. Black hole Event horizon Singularity Two-body problem Gravitational waves: