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Affine Bundle
In mathematics, an affine bundle is a fiber bundle whose typical fiber, fibers, trivialization morphisms and transition functions are affine.. (page 60) Formal definition Let \overline\pi:\overline Y\to X be a vector bundle with a typical fiber a vector space \overline F. An affine bundle modelled on a vector bundle \overline\pi:\overline Y\to X is a fiber bundle \pi:Y\to X whose typical fiber F is an affine space modelled on \overline F so that the following conditions hold: (i) Every fiber Y_x of Y is an affine space modelled over the corresponding fibers \overline Y_x of a vector bundle \overline Y. (ii) There is an affine bundle atlas of Y\to X whose local trivializations morphisms and transition functions are affine isomorphisms. Dealing with affine bundles, one uses only affine bundle coordinates (x^\mu,y^i) possessing affine transition functions : y'^i= A^i_j(x^\nu)y^j + b^i(x^\nu). There are the bundle morphisms : Y\times_X\overline Y\longrightarrow Y,\qquad (y^i, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fiber Bundle
In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a product space B \times F is defined using a continuous surjective map, \pi : E \to B, that in small regions of E behaves just like a projection from corresponding regions of B \times F to B. The map \pi, called the projection or submersion of the bundle, is regarded as part of the structure of the bundle. The space E is known as the total space of the fiber bundle, B as the base space, and F the fiber. In the ''trivial'' case, E is just B \times F, and the map \pi is just the projection from the product space to the first factor. This is called a trivial bundle. Examples of non-trivial fiber bundles include the Möbius strip and Klein bottle, as well as nontrivial covering spaces. Fiber bundles, such as the tangent bundle of a mani ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reduction Of The Structure Group
In differential geometry, a ''G''-structure on an ''n''-manifold ''M'', for a given structure group ''G'', is a principal ''G''- subbundle of the tangent frame bundle F''M'' (or GL(''M'')) of ''M''. The notion of ''G''-structures includes various classical structures that can be defined on manifolds, which in some cases are tensor fields. For example, for the orthogonal group, an O(''n'')-structure defines a Riemannian metric, and for the special linear group an SL(''n'',R)-structure is the same as a volume form. For the trivial group, an -structure consists of an absolute parallelism of the manifold. Generalising this idea to arbitrary principal bundles on topological spaces, one can ask if a principal G-bundle over a group G "comes from" a subgroup H of G. This is called reduction of the structure group (to H). Several structures on manifolds, such as a complex structure, a symplectic structure, or a Kähler structure, are ''G''-structures with an additional integrability c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gennadi Sardanashvily
Gennadi Sardanashvily (russian: Генна́дий Алекса́ндрович Сарданашви́ли; March 13, 1950 – September 1, 2016) was a theoretical physicist, a principal research scientist of Moscow State University. Biography Gennadi Sardanashvily graduated from Moscow State University (MSU) in 1973, he was a Ph.D. student of the Department of Theoretical Physics ( MSU) in 1973–76, where he held a position in 1976. He attained his Ph.D. degree in physics and mathematics from MSU, in 1980, with Dmitri Ivanenko as his supervisor, and his D.Sc. degree in physics and mathematics from MSU, in 1998. Gennadi Sardanashvily was the founder and Managing Editor (2003 - 2013) of the International Journal of Geometric Methods in Modern Physics (IJGMMP). He was a member of Lepage Research Institute (Czech Republic). Research area Gennadi Sardanashvily research area is geometric method in classical and quantum mechanics and field theory, gravitation theory. H ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Affine Space
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] |
Vector Bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every point x of the space X we associate (or "attach") a vector space V(x) in such a way that these vector spaces fit together to form another space of the same kind as X (e.g. a topological space, manifold, or algebraic variety), which is then called a vector bundle over X. The simplest example is the case that the family of vector spaces is constant, i.e., there is a fixed vector space V such that V(x)=V for all x in X: in this case there is a copy of V for each x in X and these copies fit together to form the vector bundle X\times V over X. Such vector bundles are said to be ''trivial''. A more complicated (and prototypical) class of examples are the tangent bundles of smooth (or differentiable) manifolds: to every point of such a manifold w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fibered Manifold
In differential geometry, in the category of differentiable manifolds, a fibered manifold is a surjective submersion \pi : E \to B\, that is, a surjective differentiable mapping such that at each point y \in U the tangent mapping T_y \pi : T_ E \to T_B is surjective, or, equivalently, its rank equals \dim B. History In topology, the words fiber (Faser in German) and fiber space (gefaserter Raum) appeared for the first time in a paper by Herbert Seifert in 1932, but his definitions are limited to a very special case. The main difference from the present day conception of a fiber space, however, was that for Seifert what is now called the base space (topological space) of a fiber (topological) space E was not part of the structure, but derived from it as a quotient space of E. The first definition of fiber space is given by Hassler Whitney in 1935 under the name sphere space, but in 1940 Whitney changed the name to sphere bundle. The theory of fibered spaces, of which vecto ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fiber Bundle
In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a product space B \times F is defined using a continuous surjective map, \pi : E \to B, that in small regions of E behaves just like a projection from corresponding regions of B \times F to B. The map \pi, called the projection or submersion of the bundle, is regarded as part of the structure of the bundle. The space E is known as the total space of the fiber bundle, B as the base space, and F the fiber. In the ''trivial'' case, E is just B \times F, and the map \pi is just the projection from the product space to the first factor. This is called a trivial bundle. Examples of non-trivial fiber bundles include the Möbius strip and Klein bottle, as well as nontrivial covering spaces. Fiber bundles, such as the tangent bundle of a mani ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
General Linear Group
In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible, with identity matrix as the identity element of the group. The group is so named because the columns (and also the rows) of an invertible matrix are linearly independent, hence the vectors/points they define are in general linear position, and matrices in the general linear group take points in general linear position to points in general linear position. To be more precise, it is necessary to specify what kind of objects may appear in the entries of the matrix. For example, the general linear group over R (the set of real numbers) is the group of invertible matrices of real numbers, and is denoted by GL''n''(R) or . More generally, the general linear group of degree ''n'' over any ... [...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] |
Vector Bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every point x of the space X we associate (or "attach") a vector space V(x) in such a way that these vector spaces fit together to form another space of the same kind as X (e.g. a topological space, manifold, or algebraic variety), which is then called a vector bundle over X. The simplest example is the case that the family of vector spaces is constant, i.e., there is a fixed vector space V such that V(x)=V for all x in X: in this case there is a copy of V for each x in X and these copies fit together to form the vector bundle X\times V over X. Such vector bundles are said to be ''trivial''. A more complicated (and prototypical) class of examples are the tangent bundles of smooth (or differentiable) manifolds: to every point of such a manifold w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tangent Bundle
In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of manifold the tangent spaces and have no common vector. This is graphically illustrated in the accompanying picture for tangent bundle of circle , see tangent bundle#Examples, Examples section: all tangents to a circle lie in the plane of the circle. In order to make them disjoint it is necessary to align them in a plane perpendicular to the plane of the circle. of the tangent spaces of M . That is, : \begin TM &= \bigsqcup_ T_xM \\ &= \bigcup_ \left\ \times T_xM \\ &= \bigcup_ \left\ \\ &= \left\ \end where T_x M denotes the tangent space to M at the point x . So, an element of TM can be thought of as a ordered pair, pair (x,v), where x is a point in M and v is a tangent vector to M at x . There i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Section (fiber Bundle)
In the mathematical field of topology, a section (or cross section) of a fiber bundle E is a continuous right inverse of the projection function \pi. In other words, if E is a fiber bundle over a base space, B: : \pi \colon E \to B then a section of that fiber bundle is a continuous map, : \sigma \colon B \to E such that : \pi(\sigma(x)) = x for all x \in B . A section is an abstract characterization of what it means to be a graph. The graph of a function g\colon B \to Y can be identified with a function taking its values in the Cartesian product E = B \times Y , of B and Y : :\sigma\colon B\to E, \quad \sigma(x) = (x,g(x)) \in E. Let \pi\colon E \to B be the projection onto the first factor: \pi(x,y) = x . Then a graph is any function \sigma for which \pi(\sigma(x)) = x . The language of fibre bundles allows this notion of a section to be generalized to the case when E is not necessarily a Cartesian product. If \pi\colon E \to B is a fibre bundle, then a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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