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Cartan Connection
In the mathematical field of differential geometry, a Cartan connection is a flexible generalization of the notion of an affine connection. It may also be regarded as a specialization of the general concept of a principal connection, in which the geometry of the principal bundle is tied to the geometry of the base manifold using a solder form. Cartan connections describe the geometry of manifolds modelled on homogeneous spaces. The theory of Cartan connections was developed by Élie Cartan, as part of (and a way of formulating) his method of moving frames (''repère mobile''). The main idea is to develop a suitable notion of the connection forms and curvature using moving frames adapted to the particular geometrical problem at hand. In relativity or Riemannian geometry, orthonormal frames are used to obtain a description of the Levi-Civita connection as a Cartan connection. For Lie groups, Maurer–Cartan frames are used to view the Maurer–Cartan form of the group as a Car ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra. The field has its origins in the study of spherical geometry as far back as antiquity. It also relates to astronomy, the geodesy of the Earth, and later the study of hyperbolic geometry by Lobachevsky. The simplest examples of smooth spaces are the plane and space curves and surfaces in the three-dimensional Euclidean space, and the study of these shapes formed the basis for development of modern differential geometry during the 18th and 19th centuries. Since the late 19th century, differential geometry has grown into a field concerned more generally with geometric structures on differentiable manifolds. A geometric structure is one which defines some notion of size, distance, shape, volume, or other rigidifying structu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemannian Geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a ''Riemannian metric'', i.e. with an inner product on the tangent space at each point that varies smoothly from point to point. This gives, in particular, local notions of angle, length of curves, surface 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 the differential geometry of surfaces in R3. Development of Riemannian geometry resulted in synthesis of diverse 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 dim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Affine Group
In mathematics, the affine group or general affine group of any affine space over a field is the group of all invertible affine transformations from the space into itself. It is a Lie group if is the real or complex field or quaternions. Relation to general linear group Construction from general linear group Concretely, given a vector space , it has an underlying affine space obtained by "forgetting" the origin, with acting by translations, and the affine group of can be described concretely as the semidirect product of by , the general linear group of : :\operatorname(V) = V \rtimes \operatorname(V) The action of on is the natural one (linear transformations are automorphisms), so this defines a semidirect product. In terms of matrices, one writes: :\operatorname(n,K) = K^n \rtimes \operatorname(n,K) where here the natural action of on is matrix multiplication of a vector. Stabilizer of a point Given the affine group of an affine space , the stabilizer of a point ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mean Curvature
In mathematics, the mean curvature H of a surface S is an ''extrinsic'' measure of curvature that comes from differential geometry and that locally describes the curvature of an embedded surface in some ambient space such as Euclidean space. The concept was used by Sophie Germain in her work on elasticity theory. Jean Baptiste Marie Meusnier used it in 1776, in his studies of minimal surfaces. It is important in the analysis of minimal surfaces, which have mean curvature zero, and in the analysis of physical interfaces between fluids (such as soap films) which, for example, have constant mean curvature in static flows, by the Young-Laplace equation. Definition Let p be a point on the surface S inside the three dimensional Euclidean space . Each plane through p containing the normal line to S cuts S in a (plane) curve. Fixing a choice of unit normal gives a signed curvature to that curve. As the plane is rotated by an angle \theta (always containing the normal line) that curvatur ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Erlangen Programme
In mathematics, the Erlangen program is a method of characterizing geometries based on group theory and projective geometry. It was published by Felix Klein in 1872 as ''Vergleichende Betrachtungen über neuere geometrische Forschungen.'' It is named after the University Erlangen-Nürnberg, where Klein worked. By 1872, non-Euclidean geometries had emerged, but without a way to determine their hierarchy and relationships. Klein's method was fundamentally innovative in three ways: :* Projective geometry was emphasized as the unifying frame for all other geometries considered by him. In particular, Euclidean geometry was more restrictive than affine geometry, which in turn is more restrictive than projective geometry. :* Klein proposed that group theory, a branch of mathematics that uses algebraic methods to abstract the idea of symmetry, was the most useful way of organizing geometrical knowledge; at the time it had already been introduced into the theory of equations in the form ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Felix Klein
Christian Felix Klein (; 25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and group theory. His 1872 Erlangen program, classifying geometries by their basic symmetry groups, was an influential synthesis of much of the mathematics of the time. Life Felix Klein was born on 25 April 1849 in Düsseldorf, to Prussian parents. His father, Caspar Klein (1809–1889), was a Prussian government official's secretary stationed in the Rhine Province. His mother was Sophie Elise Klein (1819–1890, née Kayser). He attended the Gymnasium in Düsseldorf, then studied mathematics and physics at the University of Bonn, 1865–1866, intending to become a physicist. At that time, Julius Plücker had Bonn's professorship of mathematics and experimental physics, but by the time Klein became his assistant, in 1866, Plücker's interest wa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Affine Subspace
In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related to parallelism and ratio of lengths for parallel line segments. In an affine space, there is no distinguished point that serves as an origin. Hence, no vector has a fixed origin and no vector can be uniquely associated to a point. In an affine space, there are instead '' displacement vectors'', also called ''translation'' vectors or simply ''translations'', between two points of the space. Thus it makes sense to subtract two points of the space, giving a translation vector, but it does not make sense to add two points of the space. Likewise, it makes sense to add a displacement vector to a point of an affine space, resulting in a new point translated from the starting point by that vector. Any vector space may be viewed as an affine spa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parallel Transport
In geometry, parallel transport (or parallel translation) is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection (a covariant derivative or connection (vector bundle), connection on the tangent bundle), then this connection allows one to transport vectors of the manifold along curves so that they stay ''parallel'' with respect to the connection. The parallel transport for a connection thus supplies a way of, in some sense, moving the local geometry of a manifold along a curve: that is, of ''connecting'' the geometries of nearby points. There may be many notions of parallel transport available, but a specification of one — one way of connecting up the geometries of points on a curve — is tantamount to providing a ''connection''. In fact, the usual notion of connection is the infinitesimal analog of parallel transport. Or, ''vice versa'', parallel transport is the local realization of a conne ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Klein Geometry
In mathematics, a Klein geometry is a type of geometry motivated by Felix Klein in his influential Erlangen program. More specifically, it is a homogeneous space ''X'' together with a transitive action on ''X'' by a Lie group ''G'', which acts as the symmetry group of the geometry. For background and motivation see the article on the Erlangen program. Formal definition A Klein geometry is a pair where ''G'' is a Lie group and ''H'' is a closed Lie subgroup of ''G'' such that the (left) coset space ''G''/''H'' is connected. The group ''G'' is called the principal group of the geometry and ''G''/''H'' is called the space of the geometry (or, by an abuse of terminology, simply the ''Klein geometry''). The space of a Klein geometry is a smooth manifold of dimension :dim ''X'' = dim ''G'' − dim ''H''. There is a natural smooth left action of ''G'' on ''X'' given by :g \cdot (aH) = (ga)H. Clearly, this action is transitive (take ), so that one may then regard ''X'' as a homogeneou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Einstein–Cartan Theory
In theoretical physics, the Einstein–Cartan theory, also known as the Einstein–Cartan–Sciama–Kibble theory, is a classical theory of gravitation similar to general relativity. The theory was first proposed by Élie Cartan in 1922. Einstein–Cartan theory is the simplest Poincaré gauge theory. Overview Einstein–Cartan theory differs from general relativity in two ways: (1) it is formulated within the framework of Riemann–Cartan geometry, which possesses a locally gauged Lorentz symmetry, while general relativity is formulated within the framework of Riemannian geometry, which does not; (2) an additional set of equations are posed that relate torsion to spin. This difference can be factored into by first reformulating general relativity onto a Riemann–Cartan geometry, replacing the Einstein–Hilbert action over Riemannian geometry by the Palatini action over Riemann–Cartan geometry; and second, removing the zero torsion constraint from the Palatini action, whic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cartan Formalism (physics)
The tetrad formalism is an approach to general relativity that generalizes the choice of basis for the tangent bundle from a coordinate basis to the less restrictive choice of a local basis, i.e. a locally defined set of four linearly independent vector fields called a ''tetrad'' or ''vierbein''. It is a special case of the more general idea of a ''vielbein formalism'', which is set in (pseudo-)Riemannian geometry. This article as currently written makes frequent mention of general relativity; however, almost everything it says is equally applicable to (pseudo-) Riemannian manifolds in general, and even to spin manifolds. Most statements hold simply by substituting arbitrary n for n=4. In German, "vier" translates to "four", and "viel" to "many". The general idea is to write the metric tensor as the product of two ''vielbeins'', one on the left, and one on the right. The effect of the vielbeins is to change the coordinate system used on the tangent manifold to one that is sim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conformal Connection
In conformal differential geometry, a conformal connection is a Cartan connection on an ''n''-dimensional manifold ''M'' arising as a deformation of the Klein geometry given by the celestial ''n''-sphere, viewed as the homogeneous space :O+(n+1,1)/''P'' where ''P'' is the stabilizer of a fixed null line through the origin in R''n''+2, in the orthochronous Lorentz group O+(n+1,1) in ''n''+2 dimensions. Normal Cartan connection Any manifold equipped with a conformal structure has a canonical conformal connection called the normal Cartan connection. Formal definition A conformal connection on an ''n''-manifold ''M'' is a Cartan geometry modelled on the conformal sphere, where the latter is viewed as a homogeneous space for O+(n+1,1). In other words it is an O+(n+1,1)-bundle equipped with * a O+(n+1,1)-connection (the Cartan connection) * a reduction of structure group to the stabilizer of a point in the conformal sphere (a null line in R''n''+1,1) such that the solder form In ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
