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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Riemannian geometry is the branch of differential geometry that studies Riemannian manifoldssmooth manifolds with a Riemannian metrici.
This gives, in particular, local notions of anglelength of curvessurface area and volume. From those, some other global quantities can be derived by integrating local contributions. 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 is a very broad and abstract generalization of riemannienbe differential geometry of surfaces in R 3. Development of Riemannian geometry resulted in synthesis of diverse results concerning the geometry of geoketrie and the behavior of geodesics on them, with techniques that can be applied to the study of differentiable manifolds of higher dimensions.
It enabled the ridmannienne of Einstein ‘s general theory of relativitymade profound impact on group theory and representation theoryas well as analysisand spurred the development of algebraic and differential topology. Riemannian geometry was first put forward in generality by Bernhard Riemann in the 19th century. It deals with a broad range of geometries whose metric properties vary from point to point, including the standard types of Non-Euclidean geometry.
Any smooth manifold admits a Riemannian metricwhich often helps to solve problems of differential topology. It also serves as an entry level for the more complicated structure of pseudo-Riemannian manifoldswhich in four dimensions are the main objects of the theory of general relativity.
Variété pseudo-riemannienne — Wikipédia
Other generalizations of Riemannian geometry include Finsler geometry. There exists a close analogy of differential geometry with the mathematical structure of defects in regular crystals. Dislocations and Disclinations produce torsions and curvature. What follows is an incomplete list of the most classical theorems in Riemannian geometry.
The choice is made depending on its riejannienne and elegance of formulation. Most of the results can be found in the classic monograph by Jeff Cheeger and D.
The formulations given are far from being very exact or the most general. This list is oriented to those who already know the basic definitions and want to know what these definitions are about. 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” riemannnienne.
From Wikipedia, the free encyclopedia.
Gelmetrie geometry is also sometimes called “Riemannian geometry”. Projecting a sphere to a plane. Point Line segment ray Length. Volume Cube cuboid Cylinder Pyramid Sphere.
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.
Principle of relativity Galilean relativity Galilean transformation Special relativity Doubly special relativity. Time dilation Mass—energy equivalence Length contraction Relativity of simultaneity Relativistic Doppler effect Thomas precession Ladder paradox Twin paradox.
Light cone World line Minkowski diagram Biquaternions Minkowski space. Equivalence principle Riemannian geometry Penrose diagram Geodesics Mach’s principle. Black hole Event horizon Singularity Two-body problem Gravitational waves: Brans—Dicke theory Kaluza—Klein Quantum gravity.
Retrieved from ” https: Riemannian geometry Bernhard Riemann. Views Read Edit View history. In other projects Wikimedia Commons.
Two-dimensional Plane Area Polygon. Altitude Hypotenuse Pythagorean theorem. Square Rectangle Rhombus Rhomboid. Introduction History Mathematical formulation Tests.
Géométrie riemannienne en dimension 4 : Séminaire Arthur Besse /79 in SearchWorks catalog
Fundamental concepts Principle of relativity Theory of relativity Frame of rimeannienne 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.
Phenomena Gravitoelectromagnetism Kepler problem Gravity Gravitational field Gravity well Gravitational lensing Gravitational waves Gravitational riemanienne Redshift Blueshift Time dilation Gravitational time dilation Shapiro time delay Gravitational potential Gravitational compression Gravitational collapse Frame-dragging Geodetic effect Gravitational singularity Event horizon Naked singularity Black hole White hole.
Kaluza—Klein theory Quantum gravity Supergravity. Background Principle of relativity Galilean relativity Galilean transformation Special relativity Doubly special relativity. Background Introduction Mathematical formulation.