|
Adjoint Bundle
In mathematics, an adjoint bundle is a vector bundle naturally associated to any principal bundle. The fibers of the adjoint bundle carry a Lie algebra structure making the adjoint bundle into a (nonassociative) algebra bundle. Adjoint bundles have important applications in the theory of connections as well as in gauge theory. Formal definition Let ''G'' be a Lie group with Lie algebra \mathfrak g, and let ''P'' be a principal ''G''-bundle over a smooth manifold ''M''. Let :\mathrm: G\to\mathrm(\mathfrak g)\sub\mathrm(\mathfrak g) be the (left) adjoint representation of ''G''. The adjoint bundle of ''P'' is the associated bundle :\mathrm P = P\times_\mathfrak g The adjoint bundle is also commonly denoted by \mathfrak g_P. Explicitly, elements of the adjoint bundle are equivalence classes of pairs 'p'', ''X''for ''p'' ∈ ''P'' and ''X'' ∈ \mathfrak g such that : \cdot g,X= ,\mathrm_(X)/math> for all ''g'' ∈ ''G''. Since the structure group of the adjoint bundle consi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector-valued Differential Form
In mathematics, a vector-valued differential form on a manifold ''M'' is a differential form on ''M'' with values in a vector space ''V''. More generally, it is a differential form with values in some vector bundle ''E'' over ''M''. Ordinary differential forms can be viewed as R-valued differential forms. An important case of vector-valued differential forms are Lie algebra-valued forms. (A connection form is an example of such a form.) Definition Let ''M'' be a smooth manifold and ''E'' → ''M'' be a smooth vector bundle over ''M''. We denote the space of smooth sections of a bundle ''E'' by Γ(''E''). An ''E''-valued differential form of degree ''p'' is a smooth section of the tensor product bundle of ''E'' with Λ''p''(''T'' ∗''M''), the ''p''-th exterior power of the cotangent bundle of ''M''. The space of such forms is denoted by :\Omega^p(M,E) = \Gamma(E\otimes\Lambda^pT^*M). Because Γ is a strong monoidal functor, this can also be interpreted as :\Gamma(E\otimes\Lambd ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wiley Interscience
John Wiley & Sons, Inc., commonly known as Wiley (), is an American multinational publishing company founded in 1807 that focuses on academic publishing and instructional materials. The company produces books, journals, and encyclopedias, in print and electronically, as well as online products and services, training materials, and educational materials for undergraduate, graduate, and continuing education students. History The company was established in 1807 when Charles Wiley opened a print shop in Manhattan. The company was the publisher of 19th century American literary figures like James Fenimore Cooper, Washington Irving, Herman Melville, and Edgar Allan Poe, as well as of legal, religious, and other non-fiction titles. The firm took its current name in 1865. Wiley later shifted its focus to scientific, technical, and engineering subject areas, abandoning its literary interests. Wiley's son John (born in Flatbush, New York, October 4, 1808; died in East Orange, New Jers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Foundations Of Differential Geometry
''Foundations of Differential Geometry'' is an influential 2-volume mathematics book on differential geometry written by Shoshichi Kobayashi and Katsumi Nomizu. The first volume was published in 1963 and the second in 1969, by Interscience Publishers. Both were published again in 1996 as Wiley Classics Library. The first volume considers manifolds, fiber bundles, tensor analysis, connections in bundles, and the role of Lie groups. It also covers holonomy, the de Rham decomposition theorem and the Hopf–Rinow theorem. According to the review of James Eells, it has a "fine expositional style" and consists of a "special blend of algebraic, analytic, and geometric concepts". Eells says it is "essentially a textbook (even though there are no exercises)". An advanced text, it has a "pace geared to a neterm graduate course". The second volume considers submanifolds of Riemannian manifolds, the Gauss map, and the second fundamental form. It continues with geodesics on Riemannian man ... [...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] |
Frame Bundle
In mathematics, a frame bundle is a principal fiber bundle F(''E'') associated to any vector bundle ''E''. The fiber of F(''E'') over a point ''x'' is the set of all ordered bases, or ''frames'', for ''E''''x''. The general linear group acts naturally on F(''E'') via a change of basis, giving the frame bundle the structure of a principal GL(''k'', R)-bundle (where ''k'' is the rank of ''E''). The frame bundle of a smooth manifold is the one associated to its tangent bundle. For this reason it is sometimes called the tangent frame bundle. Definition and construction Let ''E'' → ''X'' be a real vector bundle of rank ''k'' over a topological space ''X''. A frame at a point ''x'' ∈ ''X'' is an ordered basis for the vector space ''E''''x''. Equivalently, a frame can be viewed as a linear isomorphism :p : \mathbf^k \to E_x. The set of all frames at ''x'', denoted ''F''''x'', has a natural right action by the general linear group GL(''k'', R) of invertible ''k'' × ''k'' matrices: a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conjugation (group Theory)
In mathematics, especially group theory, two elements a and b of a group are conjugate if there is an element g in the group such that b = gag^. This is an equivalence relation whose equivalence classes are called conjugacy classes. In other words, each conjugacy class is closed under b = gag^. for all elements g in the group. Members of the same conjugacy class cannot be distinguished by using only the group structure, and therefore share many properties. The study of conjugacy classes of non-abelian groups is fundamental for the study of their structure. For an abelian group, each conjugacy class is a set containing one element (singleton set). Functions that are constant for members of the same conjugacy class are called class functions. Definition Let G be a group. Two elements a, b \in G are conjugate if there exists an element g \in G such that gag^ = b, in which case b is called of a and a is called a conjugate of b. In the case of the general linear group \operat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gauge Transformation
In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie groups). The term ''gauge'' refers to any specific mathematical formalism to regulate redundant degrees of freedom in the Lagrangian of a physical system. The transformations between possible gauges, called ''gauge transformations'', form a Lie group—referred to as the ''symmetry group'' or the ''gauge group'' of the theory. Associated with any Lie group is the Lie algebra of group generators. For each group generator there necessarily arises a corresponding field (usually a vector field) called the ''gauge field''. Gauge fields are included in the Lagrangian to ensure its invariance under the local group transformations (called ''gauge invariance''). When such a theory is quantized, the quanta of the gauge fields are called ''gauge bosons' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Connection (principal Bundle)
In mathematics, and especially differential geometry and gauge theory, a connection is a device that defines a notion of parallel transport on the bundle; that is, a way to "connect" or identify fibers over nearby points. A principal ''G''-connection on a principal G-bundle ''P'' over a smooth manifold ''M'' is a particular type of connection which is compatible with the action of the group ''G''. A principal connection can be viewed as a special case of the notion of an Ehresmann connection, and is sometimes called a principal Ehresmann connection. It gives rise to (Ehresmann) connections on any fiber bundle associated to ''P'' via the associated bundle construction. In particular, on any associated vector bundle the principal connection induces a covariant derivative, an operator that can differentiate sections of that bundle along tangent directions in the base manifold. Principal connections generalize to arbitrary principal bundles the concept of a linear connection on the f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Curvature Form
In differential geometry, the curvature form describes curvature of a connection on a principal bundle. The Riemann curvature tensor in Riemannian geometry can be considered as a special case. Definition Let ''G'' be a Lie group with Lie algebra \mathfrak g, and ''P'' → ''B'' be a principal ''G''-bundle. Let ω be an Ehresmann connection on ''P'' (which is a \mathfrak g-valued one-form on ''P''). Then the curvature form is the \mathfrak g-valued 2-form on ''P'' defined by :\Omega=d\omega + omega \wedge \omega= D \omega. (In another convention, 1/2 does not appear.) Here d stands for exterior derivative, cdot \wedge \cdot/math> is defined in the article "Lie algebra-valued form" and ''D'' denotes the exterior covariant derivative. In other terms, :\,\Omega(X, Y)= d\omega(X,Y) + omega(X),\omega(Y)/math> where ''X'', ''Y'' are tangent vectors to ''P''. There is also another expression for Ω: if ''X'', ''Y'' are horizontal vector fields on ''P'', thenProof: \sigma\Omega(X ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Algebra-valued Form
In differential geometry, a Lie-algebra-valued form is a vector-valued differential form, differential form with values in a Lie algebra. Such forms have important applications in the theory of connection (principal bundle), connections on a principal bundle as well as in the theory of Cartan connections. Formal definition A Lie-algebra-valued differential k-form on a manifold, M, is a smooth section (fiber bundle), section of the fibre bundle, bundle (\mathfrak \times M) \otimes \wedge^k T^*M, where \mathfrak is a Lie algebra, T^*M is the cotangent bundle of M and \wedge^k denotes the k^ Exterior algebra, exterior power. Wedge product Since every Lie algebra has a bilinear Lie_algebra#Definition_of_a_Lie_algebra, Lie bracket operation, the wedge product of two Lie-algebra-valued forms can be composed with the bracket operation to obtain another Lie-algebra-valued form. For a \mathfrak-valued p-form \omega and a \mathfrak-valued q-form \eta, their wedge product [\omega\wedge\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tensorial Form
In mathematics, a vector-valued differential form on a manifold ''M'' is a differential form on ''M'' with values in a vector space ''V''. More generally, it is a differential form with values in some vector bundle ''E'' over ''M''. Ordinary differential forms can be viewed as R-valued differential forms. An important case of vector-valued differential forms are Lie algebra-valued forms. (A connection form is an example of such a form.) Definition Let ''M'' be a smooth manifold and ''E'' → ''M'' be a smooth vector bundle over ''M''. We denote the space of smooth sections of a bundle ''E'' by Γ(''E''). An ''E''-valued differential form of degree ''p'' is a smooth section of the tensor product bundle of ''E'' with Λ''p''(''T'' ∗''M''), the ''p''-th exterior power of the cotangent bundle of ''M''. The space of such forms is denoted by :\Omega^p(M,E) = \Gamma(E\otimes\Lambda^pT^*M). Because Γ is a strong monoidal functor, this can also be interpreted as :\Gamma(E\otimes\Lambda ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
.svg "Click for more on -> General Linear Group")
