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Oriented Vector Bundle
In mathematics, an orientation of a real vector bundle is a generalization of an orientation of a vector space; thus, given a real vector bundle π: ''E'' →''B'', an orientation of ''E'' means: for each fiber ''E''''x'', there is an orientation of the vector space ''E''''x'' and one demands that each trivialization map (which is a bundle map) :\phi_U : \pi^(U) \to U \times \mathbf^n is fiberwise orientation-preserving, where R''n'' is given the standard orientation. In more concise terms, this says that the structure group of the frame bundle of ''E'', which is the real general linear group ''GL''n(R), can be reduced to the subgroup consisting of those with positive determinant. If ''E'' is a real vector bundle of rank ''n'', then a choice of metric on ''E'' amounts to a reduction of the structure group to the orthogonal group ''O''(''n''). In that situation, an orientation of ''E'' amounts to a reduction from ''O''(''n'') to the special orthogonal group ''SO''(''n''). A vecto ... [...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] |
Unit Sphere Bundle
In Riemannian geometry, the unit tangent bundle of a Riemannian manifold (''M'', ''g''), denoted by T1''M'', UT(''M'') or simply UT''M'', is the unit sphere bundle for the tangent bundle T(''M''). It is a fiber bundle over ''M'' whose fiber at each point is the unit sphere in the tangent bundle: :\mathrm (M) := \coprod_ \left\, where T''x''(''M'') denotes the tangent space to ''M'' at ''x''. Thus, elements of UT(''M'') are pairs (''x'', ''v''), where ''x'' is some point of the manifold and ''v'' is some tangent direction (of unit length) to the manifold at ''x''. The unit tangent bundle is equipped with a natural projection :\pi : \mathrm (M) \to M, :\pi : (x, v) \mapsto x, which takes each point of the bundle to its base point. The fiber ''π''−1(''x'') over each point ''x'' ∈ ''M'' is an (''n''−1)-sphere S''n''−1, where ''n'' is the dimension of ''M''. The unit tangent bundle is therefore a sphere bundle over ''M'' with fiber S''n''−1. The definit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Algebra
Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices. Linear algebra is central to almost all areas of mathematics. For instance, linear algebra is fundamental in modern presentations of geometry, including for defining basic objects such as lines, planes and rotations. Also, functional analysis, a branch of mathematical analysis, may be viewed as the application of linear algebra to spaces of functions. Linear algebra is also used in most sciences and fields of engineering, because it allows modeling many natural phenomena, and computing efficiently with such models. For nonlinear systems, which cannot be modeled with linear algebra, it is often used for dealing with first-order approximations, using the fact that the differential of a multivariate function at a point is the linear ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orientation Sheaf
In the mathematical field of algebraic topology, the orientation sheaf on a manifold ''X'' of dimension ''n'' is a locally constant sheaf ''o''''X'' on ''X'' such that the stalk of ''o''''X'' at a point ''x'' is :o_ = \operatorname_n(X, X - \) (in the integer coefficients or some other coefficients). Let \Omega^k_M be the sheaf of differential ''k''-forms on a manifold ''M''. If ''n'' is the dimension of ''M'', then the sheaf :\mathcal_M = \Omega^n_M \otimes \mathcal_M is called the sheaf of (smooth) densities on ''M''. The point of this is that, while one can integrate a differential form only if the manifold is oriented, one can always integrate a density, regardless of orientation or orientability; there is the integration map: :\textstyle \int_M: \Gamma_c(M, \mathcal_M) \to \mathbb. If ''M'' is oriented; i.e., the orientation sheaf of the tangent bundle of ''M'' is literally trivial, then the above reduces to the usual integration of a differential form. See also * Orientati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orientation Bundle
In the mathematical field of algebraic topology, the orientation sheaf on a manifold ''X'' of dimension ''n'' is a locally constant sheaf ''o''''X'' on ''X'' such that the stalk of ''o''''X'' at a point ''x'' is :o_ = \operatorname_n(X, X - \) (in the integer coefficients or some other coefficients). Let \Omega^k_M be the sheaf of differential ''k''-forms on a manifold ''M''. If ''n'' is the dimension of ''M'', then the sheaf :\mathcal_M = \Omega^n_M \otimes \mathcal_M is called the sheaf of (smooth) densities on ''M''. The point of this is that, while one can integrate a differential form only if the manifold is oriented, one can always integrate a density, regardless of orientation or orientability; there is the integration map: :\textstyle \int_M: \Gamma_c(M, \mathcal_M) \to \mathbb. If ''M'' is oriented; i.e., the orientation sheaf of the tangent bundle of ''M'' is literally trivial, then the above reduces to the usual integration of a differential form. See also *Orientatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integration Along The Fiber
In differential geometry, the integration along fibers of a ''k''-form yields a (k-m)-form where ''m'' is the dimension of the fiber, via "integration". It is also called the fiber integration. Definition Let \pi: E \to B be a fiber bundle over a manifold with compact oriented fibers. If \alpha is a ''k''-form on ''E'', then for tangent vectors ''w''''i'''s at ''b'', let : (\pi_* \alpha)_b(w_1, \dots, w_) = \int_ \beta where \beta is the induced top-form on the fiber \pi^(b); i.e., an m-form given by: with \widetilde lifts of w_i to E, :\beta(v_1, \dots, v_m) = \alpha(v_1, \dots, v_m, \widetilde, \dots, \widetilde). (To see b \mapsto (\pi_* \alpha)_b is smooth, work it out in coordinates; cf. an example below.) Then \pi_* is a linear map \Omega^k(E) \to \Omega^(B). By Stokes' formula, if the fibers have no boundaries(i.e. ,\int0), the map descends to de Rham cohomology: :\pi_*: \operatorname^k(E; \mathbb) \to \operatorname^(B; \mathbb). This is also called the fiber i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reduced Cohomology
In mathematics, reduced homology is a minor modification made to homology theory in algebraic topology, motivated by the intuition that all of the homology groups of a single point should be equal to zero. This modification allows more concise statements to be made (as in Alexander duality) and eliminates many exceptional cases (as in the homology groups of spheres). If ''P'' is a single-point space, then with the usual definitions the integral homology group :''H''0(''P'') is isomorphic to \mathbb (an infinite cyclic group), while for ''i'' ≥ 1 we have :''H''''i''(''P'') = . More generally if ''X'' is a simplicial complex or finite CW complex, then the group ''H''0(''X'') is the free abelian group with the connected components of ''X'' as generators. The reduced homology should replace this group, of rank ''r'' say, by one of rank ''r'' − 1. Otherwise the homology groups should remain unchanged. An ''ad hoc'' way to do this is to think of a 0-th homology class not a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thom Isomorphism
In mathematics, the Thom space, Thom complex, or Pontryagin–Thom construction (named after René Thom and Lev Pontryagin) of algebraic topology and differential topology is a topological space associated to a vector bundle, over any paracompact space. Construction of the Thom space One way to construct this space is as follows. Let :p: E \to B be a rank ''n'' real vector bundle over the paracompact space In mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite. These spaces were introduced by . Every compact space is paracompact. Every paracompact Hausdorff space is normal ... ''B''. Then for each point ''b'' in ''B'', the Fiber (mathematics)#Fiber in naive set theory, fiber E_b is an n-dimensional real vector space. Choose an orthogonal structure on E, a smoothly varying inner product on the fibers; we can do this using partitions of unity. Let D(E) be the unit ball bundle with respect to our orth ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thom Space
In mathematics, the Thom space, Thom complex, or Pontryagin–Thom construction (named after René Thom and Lev Pontryagin) of algebraic topology and differential topology is a topological space associated to a vector bundle, over any paracompact space. Construction of the Thom space One way to construct this space is as follows. Let :p: E \to B be a rank ''n'' real vector bundle over the paracompact space ''B''. Then for each point ''b'' in ''B'', the fiber E_b is an n-dimensional real vector space. Choose an orthogonal structure on E, a smoothly varying inner product on the fibers; we can do this using partitions of unity. Let D(E) be the unit ball bundle with respect to our orthogonal structure, and let S(E) be the unit sphere bundle, then the Thom space T(E) is the quotient T(E) := D(E)/S(E) of topological spaces. T(E) is a pointed space with the image of S(E) in the quotient as basepoint. If ''B'' is compact, then T(E) is the one-point compactification of ''E''. For example ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grassmannian
In mathematics, the Grassmannian is a space that parameterizes all -Dimension, dimensional linear subspaces of the -dimensional vector space . For example, the Grassmannian is the space of lines through the origin in , so it is the same as the projective space of one dimension lower than . When is a real or complex vector space, Grassmannians are compact space, compact smooth manifolds. In general they have the structure of a smooth algebraic variety, of dimension k(n-k). The earliest work on a non-trivial Grassmannian is due to Julius Plücker, who studied the set of projective lines in projective 3-space, equivalent to and parameterized them by what are now called Plücker coordinates. Hermann Grassmann later introduced the concept in general. Notations for the Grassmannian vary between authors; notations include , , , or to denote the Grassmannian of -dimensional subspaces of an -dimensional vector space . Motivation By giving a collection of subspaces of some vecto ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Determinant Bundle
In mathematics, a line bundle expresses the concept of a line that varies from point to point of a space. For example, a curve in the plane having a tangent line at each point determines a varying line: the ''tangent bundle'' is a way of organising these. More formally, in algebraic topology and differential topology, a line bundle is defined as a ''vector bundle'' of rank 1. Line bundles are specified by choosing a one-dimensional vector space for each point of the space in a continuous manner. In topological applications, this vector space is usually real or complex. The two cases display fundamentally different behavior because of the different topological properties of real and complex vector spaces: If the origin is removed from the real line, then the result is the set of 1×1 invertible real matrices, which is homotopy-equivalent to a discrete two-point space by contracting the positive and negative reals each to a point; whereas removing the origin from the complex plan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orientation Of A Vector Space
The orientation of a real vector space or simply orientation of a vector space is the arbitrary choice of which ordered bases are "positively" oriented and which are "negatively" oriented. In the three-dimensional Euclidean space, right-handed bases are typically declared to be positively oriented, but the choice is arbitrary, as they may also be assigned a negative orientation. A vector space with an orientation selected is called an oriented vector space, while one not having an orientation selected, is called . In mathematics, orientability is a broader notion that, in two dimensions, allows one to say when a cycle goes around clockwise or counterclockwise, and in three dimensions when a figure is left-handed or right-handed. In linear algebra over the real numbers, the notion of orientation makes sense in arbitrary finite dimension, and is a kind of asymmetry that makes a reflection impossible to replicate by means of a simple displacement. Thus, in three dimensions, it is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
