In mathematics , an algebrabundle is a fiber bundle whose fiber s are algebra over a field algebra s and local trivialization s respect the algebra structure. It follows that the transition function s are algebra isomorphism s. Since algebras are also vector space s, every algebrabundle is a vector bundle . Examples include the tensor bundle , exterior bundle , and symmetric bundle associated to a given vector bundle , as well as the Clifford bundle associated to any Riemannian vector bundle. See also Lie algebrabundle References 1. W. Greub, S. Halperin and R. Vanstone, Connections, Curvature and Cohomology, Vo. 2, Academic Press, New Yark, 1973 2. C. Chidambara and B.S. Kiranagi, On Cohomology of Associative algebra bundles, J. Ramanujan Math. Soc., Vol. 9 1 , 1994. pp. 1 12 3. B.S. Kiranagi and R. Rajendra, Revisiting Hochschild Cohomology for Algebra Bundles, Journal of Algebra and Its Applications Vol. 7, No. 6 2008 685 715. DEFAULTSORT AlgebraBundle Category Vector bundles geometry stub ... more details
In Mathematics , a weak Lie algebrabundle math xi xi, p, X, theta , math is a vector bundle math xi , math over a base space X together with a morphism math theta xi otimes xi rightarrow xi math which induces a Lie algebra structure on each fibre math xi x , math . A Lie algebrabundle math xi xi, p, X , math is a vector bundle in which each fibre is a Lie algebra and for every x in X , there is an open set math U math containing x , a Lie algebra L and a homeomorphism math phi U times L to p 1 U , math such that math phi x x times L rightarrow p 1 x , math is a Lie algebra isomorphism. Any Lie algebrabundle is a weak Lie algebrabundle but the converse need not be true in general. As an example of a weak Lie algebrabundle that is not a strong Lie algebrabundle, consider the total space math mathfrak so 3 times mathbb R math over the real line math mathbb R math . Let .,. denote the Lie bracket of So 3 Lie algebra math mathfrak so 3 math and deform it by the real parameter as math X,Y x x cdot X,Y math for math X,Y in mathfrak so 3 math and math x in mathbb R math . Lie s third theorem states that every bundle of Lie algebras can locally be integrated to a bundle of Lie groups. However globally the total space might fail to be Hausdorff space Hausdorff . ref A. Weinstein, A.C. ..., A decomposition theorem of Lie algebra Bundles, Communications in Algebra 18 6 , 1990, 1869 1877 . B.S.Kiranagi, G.Prema and C.Chidambara, Rigidity theorem for Lie algebra Bundles, Communications in Algebra 20 6 , 1992, pp. 1549 1556. See also Algebrabundle Adjoint bundle Category Differential topology ..., 1966, pp.133 151 B.S.Kiranagi, Lie Algebra bundles, Bull. Sci. Math., 2 sup e sup serie, 102 1978 , 57 62. B.S.Kiranagi, Semi simple Lie algebra bundles, Bull. Math de la Sci. Math de la R.S.de Roumaine .... Austral. Math Soc., 28 1983 , 401 409. B.S.Kiranagi and G.Prema, Cohomology of Lie algebra bundles ..., Lie algebra bundles defined by Jordan algebra bundles, Bull. Math. Soc.Sci.Math.Rep.Soc. Roum., Noun ... more details
Wiktionary bundleBundle or Bundling may refer to In marketing Product bundling , a marketing strategy that involves offering several products for sale as one combined product Bundling fundraising , when donations from many individuals are collected by one person and presented to the recipient Bundling political economy , a similar concept to product bundling that occurs in electoral republics In mathematics Bundle mathematics , a generalization of a fiber bundle dropping the condition of a local product structure Fiber bundle , a topology space that looks locally like a product space seealso Category Fiber bundles In medicine Bundle of His , a collection of heart muscle cells specialized for electrical conduction Bundle of Kent , an extra conduction pathway between the atria and ventricles in the heart Other uses Bundle adjustment , a photogrammetry computer vision technique Bundle conductor power engineering Bundle NEXTSTEP , a type of directory in NEXTSTEP and Mac OS X Bundle of rights property law Bundle theory philosophy Bundled payment , a method for reimbursing health care providers Bundles album Bundles album , a 1975 album by Soft Machine, including a song of the same title Bundling packaging , the process of using straps to bundle up items Bundling tradition , the traditional practice of wrapping one person in a bed accompanied by his her courter Optical fiber bundle , a cable consisting of a collection of fiber optics Bundle Brent Eileen Bundle Brent , an Agatha Christie character bundled headphones Need explanation disambig Category Dutch loanwords de Bundle eo Vikipedio Projekto matematiko Paka o fr Bundle ko ... more details
The term algebra is defined below after first defining a ring . ring In mathematics , a ring is an associative ring with a map A A which is an antiautomorphism , and an Semigroup with involution involution . More precisely, is required to satisfy the following properties math x y x y math math x y y ... over any ring. algebra A algebra A is a ring that is an associative algebra over another ring R , with the agreeing ... x mu y x lambda y mu lambda x mu y math A homomorphism math f colon A to B math is algebra homomorphism ... on the complex numbers. A operation on a algebra is an operation on an algebra over a ring that behaves ... familiar example of a algebra is the field of complex numbers C where is just complex conjugation . More generally, the conjugation involution in any Cayley Dickson algebra such as the complex numbers ... example is the Matrix ring matrix algebra of n × n matrix mathematics matrices over C with given ... space is also a star algebra. In Hecke algebra , an involution is important to the Kazhdan ... ring of an elliptic curve becomes a algebra over the integers, where the involution is given by taking ... see Milne s lecture notes on abelian varieties . Hopf algebra Examples Involutive Hopf algebras are important ... familiar example being The group Hopf algebra a group ring , with involution given by math g mapsto ... elements form a Jordan algebra The skew Hermitian elements form a Lie algebra If 2 is invertible ... symmetrizing and anti symmetrizing , so the algebra decomposes as a direct sum of symmetric and anti symmetric Hermitian and skew Hermitian elements. This decomposition is as a vector space, not as an algebra, because the idempotents are operators, not elements of the algebra. Skew structures ... algebra characteristic is 2, in which case it s identical to the original , as math ... B algebra C algebra von Neumann algebra Baer ring operator algebra This article is no longer a stub, but there is more to be said about algebras which are not B or C algebras. DEFAULTSORT Algebra Category ... more details
A algebra or, more explicitly, a closed algebra is the name occasionally used in physics ref John A. Holbrook, David W. Kribs, and Raymond Laflamme. Noiseless Subsystems and the Structure of the Commutant in Quantum Error Correction. Quantum Information Processing . Volume 2, Number 5, p. 381&ndash 419. Oct 2003. ref for a finite dimensional C algebra . The dagger, , is used in the name because physicists typically use that symbol to denote a hermitian adjoint and are often not worried about the subtleties associated with an infinite number of dimensions. Mathematicians usually use the asterisk, , to denote the hermitian adjoint. algebra feature prominently in quantum mechanics , and especially quantum information science . References references math stub Category C algebras ... more details
about the branch of mathematics pp move indef sprotect small yes Algebra is the branch of mathematics ... , topology , combinatorics , and number theory , algebra is one of the main branches of pure mathematics . Elementary algebra is often part of the curriculum in secondary education and introduces ... be done for a variety of reasons, including equation solving . Algebra is much broader than elementary algebra and studies what happens when different rules of operations are used and when operations ... algebra . History Main History of algebra See also Timeline of algebra File Image Al Kit b al mu ta ar ... Greece Greeks created a geometric algebra where terms were represented by sides of geometric ... Khwarizmi s Algebra made use of lettered diagrams but all coefficients in the equations used in the Algebra ... , sometimes called the father of algebra , was an Alexandria n Greek mathematics Greek mathematician ... 1 4460 2221 8 ref While the word algebra comes from the Arabic language al jabr , wikt ... later wrote The Compendious Book on Calculation by Completion and Balancing , which established algebra ... Al Khwarizmi The Beginnings of Algebra author Roshdi Rashed publisher Saqi Books date November 2009 isbn 0 86356 430 5 ref The roots of algebra can be traced to the ancient Babylonian mathematics Babylonians ... ref http library.thinkquest.org 25672 diiophan.htm Diophantus, Father of Algebra ref as well ... level. ref http www.algebra.com algebra about history History of Algebra ref For example, the first ... been known as the father of algebra but in more recent times there is much debate over whether al Khwarizmi ... point to the fact that the algebra found in Al Jabr is slightly more elementary than the algebra found .... ref supported by geometric proofs, while treating algebra as an independent discipline in its own right. ref Gandz and Saloman 1936 , The sources of al Khwarizmi s algebra , Osiris i, p. 263 277 In a sense, Khwarizmi is more entitled to be called the father of algebra than Diophantus because Khwarizmi ... more details
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 an algebrabundle . Adjoint bundles has important applications in the theory of connection mathematics connections as well as in gauge theory . Formal definition Let G be a Lie group with Lie algebra math mathfrak g math , and let P be a principal bundle principal G bundle over a smooth manifold M . Let math mathrm Ad G to mathrm Aut mathfrak g sub mathrm GL mathfrak g math be the adjoint representation of G . The adjoint bundle of P is the associated bundle math mathrm Ad P P times mathrm Ad mathfrak g math The adjoint bundle is also commonly denoted by math mathfrak g P math . Explicitly, elements of the adjoint bundle are equivalence class es of pairs p , x for p P and x math mathfrak g math such that math p cdot g,x p, mathrm Ad g 1 x math for all g G . Since the structure group of the adjoint bundle consists of Lie algebra automorphism s, the fibers naturally carry a Lie algebra structure making the adjoint bundle into a bundle of Lie algebras over M . Properties Vector valued differential form Differential forms on M with values in Ad sub P sub are in one to one corresponding with tensorial form horizontal, G equivariant Lie algebra valued form s on P . A prime example is the curvature form curvature of any connection principal bundle connection on P which may be regarded as a 2 form on M with values in Ad sub P sub . The space of sections of the adjoint bundle is naturally an infinite dimensional Lie algebra. It may be regarded as the Lie algebra of the infinite dimensional Lie group of gauge transformation s of P which can be thought of as sections of the bundle P × sub sub G where is the action of G on itself by conjugation group theory conjugation . Category Vector bundles Category Lie algebras geometry stub unref date December 2007 zh ... more details
In mathematics , a Clifford bundle is an algebrabundle whose fibers have the structure of a Clifford algebra and whose local trivialization s respect the algebra structure. There is a natural Clifford bundle associated to any pseudo Riemannian manifold pseudo Riemannian manifold M which is called the Clifford bundle of M . General construction Let V be a real number real or complex number complex vector space together with a symmetric bilinear form , . The Clifford algebra C V is a natural unital algebra unital associative algebra associative algebra over a field algebra generated by V subject ... bundle of M math C ell T M cong Lambda T M . math This is an isomorphism of vector bundles not algebra ... convention choice of sign in the definition of a Clifford algebra. In general, one can take v sup ... C V as a quotient of the tensor algebra of V by the ideal ring theory ideal generated by the above ... vector bundle . Let E be a smooth vector bundle over a smooth manifold M , and let g be a smooth symmetric bilinear form on E . The Clifford bundle of E is the fiber bundle whose fibers are the Clifford ... structure topology of C E is determined by that of E via an associated bundle construction. One ... that is, when E , g is a Riemannian or pseudo Riemannian vector bundle. For concreteness, suppose that E , g is a Riemannian vector bundle. The Clifford bundle of E can be constructed as follows. Let C sub n sub R be the Clifford algebra generated by R sup n sup with the Euclidean metric ... n sup . The Clifford bundle of E is then given by math C ell E F E times rho C ell n mathbb R math where F E is the orthonormal frame bundle of E . It is clear from this construction that the structure ... E is a bundle of superalgebra Z sub 2 sub graded algebras over M . The Clifford bundle C E can ... bundle E is orientability orientable then one can reduce the structure group of C E from O n to SO n in the natural manner. Clifford bundle of a Riemannian manifold If M is a Riemannian manifold ... more details
In differential geometry , a field of mathematics , a normal bundle is a particular kind of vector bundle , complementary to the tangent bundle , and coming from an embedding or immersion mathematics immersion ... p math . Just as the total space of the tangent bundle to a manifold is constructed from all tangent space s to the manifold, the total space of the normal bundle math mathrm N S math to math S math is defined as math mathrm N S coprod p in S mathrm N p S math . The conormal bundle is defined as the dual bundle to the normal bundle. It can be realised naturally as a sub bundle of the cotangent bundle ... math for instance an embedding , one can define a normal bundle of N in M , by at each point of N , taking the quotient space linear algebra quotient space of the tangent space on M by the tangent ... math V to V W math . Thus the normal bundle is in general a quotient of the tangent bundle of the ambient space restricted to the subspace. Formally, the normal bundle to N in M is a quotient bundle of the tangent bundle on M one has the short exact sequence of vector bundles on N math 0 ... of the tangent bundle on M to N properly, the pullback math i TM math of the tangent bundle on M to a vector bundle on N via the map math i math . Stable normal bundle abstraction Abstract manifolds have a canonical tangent bundle, but do not have a normal bundle only an embedding or immersion of a manifold in another yields a normal bundle. However, since every compact manifold can be embedded in math mathbf R N math , by the Whitney embedding theorem , every manifold admits a normal bundle, given ... , and hence induce the same normal bundle. The resulting class of normal bundles it is a class of bundles and not a specific bundle because N could vary is called the stable normal bundle . Dual to tangent bundle The normal bundle is dual to the tangent bundle in the sense of K theory by the above ... mathbf R N math , the tangent bundle of the ambient space is trivial since math mathbf R N math ... more details
, automatically passing to a quotient group simply loses information. Therefore a Spin bundle always gives rise to an associated bundle with fibers math mathbb R n math , since Spin acts on math mathbb ... question on the transition data, in passing to a Spin bundle. The obstruction to the lifting is known to be the second Stiefel Whitney class . See also DEFAULTSORT Spinor Bundle Category Vector ... more details
In mathematics , a frame bundle is a principal fiber bundle F E associated to any vector bundle E . The fiber ... x sub . The general linear group acts naturally on F E via a change of basis , giving the frame bundle the structure of a principal GL sub k sub 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 ... action transitive This follows from the standard linear algebra result that there is a unique invertible ... . The space F sub x sub is said to be a GL sub k sub R torsor . The frame bundle of E , denoted by F ... . The frame bundle F E can be given a natural topology and bundle structure determined by that of E ... maps sup &minus 1 sup U sub i sub F E . With all of the above data the frame bundle F E becomes a principal fiber bundle over X with structure group GL sub k sub R and local trivializations U sub ... . The above all works in the smooth category as well if E is a smooth vector bundle over a smooth manifold M then the frame bundle of E can be given the structure of a smooth principal bundle over M . Associated vector bundles A vector bundle E and its frame bundle F E are associated bundle s. Each one determines the other. The frame bundle F E can be constructed from E as above, or more abstractly using the fiber bundle construction theorem . With the latter method, F E is the fiber bundle with same ... bundle math mathrm F E times rho V , math associated to F E which is given by product F E × ... classes by p , v . The vector bundle E is naturally isomorphic to the bundle F E × sub ... E sub x sub is a frame at x . One can easily check that this map is well defined . Any vector bundle associated to E can be given by the above construction. For example, the dual bundle of E is given .... Tensor bundle s of E can be constructed in a similar manner. Tangent frame bundle The tangent ... more details
Image Tangent bundle.svg right thumb Informally, the tangent bundle of a manifold in this case a circle ... overlapping manner bottom . ref group note name disjoint In mathematics , the tangent bundle of a differentiable ... for tangent bundle of circle S sup 1 sup , see Tangent bundle Examples Examples section all tangents ... maps each tangent space T sub x sub M to the single point x . The tangent bundle to a manifold is the prototypical example of a vector bundle a fiber bundle whose fibers are vector space s . A Section fiber bundle section of TM is a vector field on M , and the dual bundle to TM is the cotangent bundle , which is the disjoint union of the cotangent space s of M . By definition, a manifold M is Parallelizable manifold parallelizable if and only if the tangent bundle is trivial bundle trivial . By definition, a manifold M is Framed mathematics framed if and only if the tangent bundle TM is stably trivial, meaning that for some trivial bundle E the Vector bundle Whitney sum nowrap 1 TM &oplus ... bundle is to provide a domain and range for the derivative of a smooth function. Namely, if math ... The tangent bundle comes equipped with a natural topology not the disjoint union topology and differential ... bundle is said to be trivial . Trivial tangent bundles usually occur for manifolds equipped ... . The tangent bundle of the unit circle is trivial because it is a Lie group under multiplication and its ... bundles are Lie groups manifolds which have a trivial tangent bundle are called parallelizable . Just ... smooth maps between open subsets of R sup 2 n sup . The tangent bundle is an example of a more general construction called a vector bundle which is itself a specific kind of fiber bundle . Explicitly, the tangent bundle to an n dimensional manifold M may be defined as a rank n vector bundle over .... Examples The simplest example is that of R sup n sup . In this case the tangent bundle is trivial. Another simple example is the unit circle , S sup 1 sup see picture above . The tangent bundle of the circle ... more details
Unreferenced date August 2008 In mathematics , the tensor bundle of a manifold is the direct sum of vector bundles direct sum of all tensor products of the tangent bundle and the cotangent bundle of that manifold. To do calculus on the tensor bundle a connection mathematics connection is needed. Category Vector bundles geometry stub ru ... more details
Unreferenced date December 2009 In mathematics , a surface bundle is a fiber bundlebundle in which the fiber is a surface . When the base space is a circle the total space is 3 manifold three dimensional and is often called a surface bundle over the circle . See also Mapping torus DEFAULTSORT Surface Bundle Category Geometric topology es Surface Bundle ... more details
In mathematics , the exterior bundle of a manifold M is the subbundle of the tensor bundle consisting of all antisymmetric covariant tensors. It has special significance, because one can define a Connection mathematics connection independent derivative on it, namely the exterior derivative . Section fiber bundle Section s of the exterior bundle are differential forms on M . DEFAULTSORT Exterior Bundle Category Vector bundles Category Differential forms geometry stub ... more details
In mathematics , the canonical bundle of a non singular algebraic variety math V math of dimension math n math is the line bundle Using , to force PNG rendering, else formula won t show up used again below math , Omega n omega math which is the n sup th sup exterior power of the cotangent bundle on V . Over the complex number s, it is the determinant bundle of holomorphic n forms on V . This is the Duality mathematics dualising object for Serre duality on V . It may equally well be considered as an invertible sheaf . The canonical class is the divisor class of a Cartier divisor K on V giving rise to the canonical bundle &mdash it is an equivalence class for linear equivalence on V , and any divisor in it may be called a canonical divisor . An anticanonical divisor is any divisor &minus K with K canonical. The anticanonical bundle is the corresponding inverse bundle sup &minus 1 sup . The adjunction formula main Adjunction formula algebraic geometry Suppose that X is a smooth variety and that D is a smooth divisor on X . The adjunction formula relates the canonical bundles of X and D ... The best studied case is that of curves. Here, the canonical bundle is the same as the holomorphic cotangent bundle . A global section of the canonical bundle is therefore the same as an everywhere ... curve, and K sub C sub is the trivial bundle. The global sections of the trivial bundle form ... case If C has genus two or more, then the canonical class is big line bundle big , so the image ... of commutative algebra . The field started with Max Noether s theorem the dimension of the space of quadrics ... language the canonical bundle is normally generated the symmetric power s of the space of sections of the canonical bundle map onto the sections of its tensor powers. ref springer title Noether ... line bundle , then the canonical ring is the homogeneous coordinate ring of the image of the canonical ... bundle Differential form Notes Reflist Category Vector bundles Category Algebraic varieties ko ... more details
In mathematics, the inverse bundle of a fibre bundle is its inverse with respect to the Whitney sum operation. Let math E rightarrow M math be a fibre bundle . A bundle math E rightarrow M math is called the em inverse bundle em of math E math if their Whitney sum is a trivial bundle, namely if math E oplus E cong M times mathbb R n. , math Any vector bundle over a compact space compact Hausdorff space Hausdorff base has an inverse bundle. References Citation last Hatcher first Allen author link Allen Hatcher title http www.math.cornell.edu hatcher VBKT VBpage.html Vector Bundles & K Theory edition 2.0 year 2003 Category Differential topology Category Algebraic topology Category Vector bundles ... more details
Unreferenced date November 2006 In mathematics , the dual bundle of a vector bundle E X is a vector bundle E X whose fibers are the dual space s to the fibers of E . The dual bundle can be constructed using the associated bundle construction by taking the dual representation of the structure group . Specifically, given a local trivialization of E with Topology transition functions t sub ij sub , a local trivialization of E is given by the same open cover of X with transition functions t sub ij sub t sub ij sub sup T sup sup &minus 1 sup the inverse matrix inverse of the transpose . The dual bundle E is then constructed using the fiber bundle construction theorem . For example, the dual to the tangent bundle of a differentiable manifold is the cotangent bundle . If the base space X is paracompact and Hausdorff space Hausdorff then a finite rank vector bundle E and its dual E are isomorphic as vector bundles. However, just as for vector space s, there is no natural isomorphism canonical choice of isomorphism unless E is equipped with an inner product . DEFAULTSORT Dual Bundle Category Vector bundles ... more details
In mathematics , tautological bundle is a term for a particularly natural vector bundle occurring over a Grassmannian , and more specially over projective space . Canonical bundle as a name dropped out of favour, on the grounds that canonical is heavily overloaded as it is, in mathematical terminology, and worse confusion with the canonical class in algebraic geometry could scarcely be avoided. Grassmannians by definition are the parameter spaces for linear subspace s, of a given dimension, in a given vector space W . If G is a Grassmannian, and V sub g sub is the subspace of W corresponding to g in G , this is already almost the data required for a vector bundle namely a vector space for each point g , varying continuously. All that can stop the definition of the tautological bundle from this indication, is the pedantic difficulty that the V sub g sub are going to intersect. Fixing this up is a routine application of the disjoint union device, so that the bundle projection is from a Fiber bundle total space made up of identical copies of the V sub g sub , that now do not intersect. With this, we have the bundle. The projective space case is included see tautological line bundle . By convention and use P V may usefully carry the tautological bundle in the dual space sense. That is, with V sup sup the dual space, points of P V carry the vector subspaces of V sup sup that are their kernels ... line bundle is one tautological bundle, and the other, just described, is of rank n . Properties ... line bundle is a generator. In the case of projective space, where the tautological bundle is a line bundle equivalence class , the associated invertible sheaf of sections is math mathcal O 1 math , the tensor inverse of the hyperplane bundle or Serre twist sheaf math mathcal O 1 math in other words the hyperplane bundle is the generator of the Picard group having positive degree and the tautological bundle is the generator of negative degree. See also Hopf bundle References Citation last1 ... more details
The vertical bundle of a smooth fiber bundle is the subbundle of the tangent bundle that consists of all vectors which are tangent to the fibers. More precisely, if &pi E &rarr M is a smooth fiber bundle over a smooth manifold M and e &isin E with &pi e x &isin M , then the vertical space V sub e sub E at e is the tangent space T sub e sub E sub x sub to the fiber E sub x sub containing e . That is, V sub e sub E T sub e sub E sub &pi e sub . The vertical space is therefore a subspace of T sub e sub E , and the union of the vertical spaces is a subbundle V E of T E this is the vertical bundle of E . The vertical bundle is the kernel mathematics kernel of the pushforward differential differential d &pi T E &rarr &pi sup 1 sup T M where &pi sup 1 sup T M is the pullback bundle symbolically, V sub e sub E ker d&pi sub e sub . Since d&pi sub e sub is surjective at each point e , it yields a canonical identification of the quotient bundle T E V E with the pullback &pi sup 1 sup T M . An Ehresmann connection on E is a choice of a complementary subbundle to V E in T E , called the horizontal bundle of the connection. Example A simple example of a smooth fiber bundle is a Cartesian product of two manifold s. Consider the bundle B sub 1 sub M × N , pr sub 1 sub with bundle projection pr sub 1 sub M × N &rarr M x , y &rarr x . The vertical bundle is then V B sub 1 sub M × T N , which is a subbundle of T M × N . If we take the other projection pr sub 2 sub M × N &rarr N x , y &rarr y to define the fiber bundle B sub 2 sub M × N , pr sub 2 sub then the vertical bundle will be V B sub 2 sub T M × N . In both cases, the product structure gives a natural choice of horizontal bundle, and hence an Ehresmann connection the horizontal bundle of B sub 1 sub is the vertical bundle of B sub 2 sub and vice versa. References cite book author Kobayashi, Shoshichi and Nomizu, Katsumi title Foundations of Differential Geometry, Vol. 1 publisher Wiley Interscience ... more details
cleanup date June 2009 In algebraic geometry , a conic bundle is an algebraic variety that appears as a solution of a Cartesian equation of the form math X 2 aXY b Y 2 P T . , math Theoretically, it can be considered as a Severi Brauer surface , or more precisely as a Ch telet surface . This can be a double covering of a ruled surface . Through an isomorphism, it can be associated with a symbol math a, P math in the second Galois cohomology of the field math k math . In fact, it is a surface with a well understood divisor group and simplest cases share with Del Pezzo surface s the property of being a rational surface . But many problems of contemporary mathematics remain open, notably for those examples which are not rational the question of unirationality . A naive point of view To write correctly a conic bundle, one must first reduce the quadratic form of the left hand side. Thus, after a harmless change, it has a simple expression like math X 2 aY 2 P T . , math In a second step, it should be placed in a projective space in order to complete the surface at infinity . To do this, we write the equation in homogeneous coordinates and expresses the first visible part of the fiber math X 2 aY 2 P T Z 2. , math That is not enough to complete the fiber as non singular clean and smooth , and then glue it to infinity by a change of classical maps Seen from infinity, i.e. through the change math T mapsto T frac 1 T math , the same fiber excepted the fibers math T 0 math and math T 0 math , written as the set of solutions math X 2 aY 2 P T Z 2 math where math P T math appears naturally as the reciprocal polynomial of math P math . Details are below about the map change math x y z ... P T T 2m P frac 1 T math , and the conic bundle F sub a , P sub as follows Definition math F a,P ... bundle over P sub 1, k sub . See also Algebraic surface Intersection number algebraic geometry ... 2 cite book author David Eisenbud year 1999 title Commutative Algebra with a View Toward Algebraic ... more details
for an entire tribe . A tribal medicine bundle is usually much larger and contains special .... See also Sacred bundle References http www.saskschools.ca gregory firstnations beliefs.html Plains ... Bundle By Dorthea Calverley Category American Indian relics ... more details
In mathematics , in the field of differential topology , given &pi E &rarr M , a smooth fiber bundle over a smooth manifold M , then the vertical bundle V E of E is the subbundle of the tangent bundle T E consisting of the vectors which are tangent to the fibers of E over M . A horizontal bundle is then a particular choice of a subbundle of T E which is complementary to V E , in other words provides a complementary subspace in each fiber. In full generality, the horizontal bundle concept is one way to formulate the notion of an Ehresmann connection on a fiber bundle . However, the concept is usually applied in more specific contexts. More precisely, if e &isin E with &pi e x &isin M , then the vertical space V sub e sub E at e is the tangent space T sub e sub E sub x sub to the fiber E sub x sub through e . A horizontal bundle then determines an horizontal space H sub e sub E such that T sub e sub E is the direct sum of vector spaces direct sum of V sub e sub E and H sub e sub E . If E is a principal bundle principal G bundle then the horizontal bundle is usually required to be G invariant see Connection principal bundle for further details. In particular, this is the case when E is the frame bundle , i.e., the set of all ordered basis frame s for the tangent spaces of the manifold, and G GL sub n sub . References cite book author Kobayashi, Shoshichi and Nomizu, Katsumi title Foundations of Differential Geometry, Vol. 1 publisher Wiley Interscience year 1996 New edition isbn 0471157333 Category Differential geometry Category Differential topology Category Connection mathematics geometry stub zh ... more details
In mathematics , an orientation mathematics oriented circle bundle is an oriented fiber bundle where the fiber is the circle math scriptstyle mathbf S 1 math , or, more precisely, a principal bundle principal U 1 bundle . It is homotopically equivalent to a complex line bundle . In physics , circle bundles are the natural geometric setting for electromagnetism . A circle bundle is a special case of a fiber bundle Sphere bundles sphere bundle . As 3 manifolds In this 3 manifold sub branch of low dimensional topology , a way of getting circle bundles is the classical construction of Seifert fiber space s, but G. Peter Scott consider a generalization of them as a spaces foliated by circles over 2 dimensional orbifolds. Relationship to electrodynamics The Maxwell equation s correspond to an electromagnetic field represented by a 2 form F , with math scriptstyle pi F math being cohomologous to zero. In particular, there always exists a 1 form A such that math scriptstyle pi F dA. math Given a circle bundle P over M and its projection math pi P to M math one has the homomorphism math scriptstyle pi H 2 M, mathbb Z to H 2 P, mathbb Z math where math scriptstyle pi math is the pullback . Each homomorphism corresponds to a Dirac monopole the integer cohomology group s correspond to the quantization of the electric charge . Examples The Hopf fibration s are examples of non trivial circle bundles. Classification The isomorphism class es of circle bundles over a manifold M are in one to one correspondence with the elements of the second integral cohomology group math scriptstyle H 2 M, mathbb Z math of M . This isomorphism is realized by the Euler class . Equivalently, the isomorphism ... are, by the associated bundle construction, equivalent to smooth complex line bundle s because the transition ... bundle or real two plane bundle is the same as the first Chern class of the line bundle. See also Wang sequence . References MathWorld title Circle Bundle urlname CircleBundle Citation last Chern ... more details
Unreferenced date December 2009 In mathematics , a bundle map or bundle morphism is a morphism in the category mathematics category of fiber bundle s. There are two distinct, but closely related, notions of bundle map, depending on whether the fiber bundles in question have a common fiber bundle base .... Bundle maps over a common base Let sub E sub E M and sub F sub F M be fiber bundles over a space M . Then a bundle map from E to F over M is a continuous map E F such that math pi F .... Then a continuous map E F is called a bundle map from E to F if there is a continuous map ... of fibers of E since sub E sub is surjective, f is uniquely determined by . For a given f , such a bundle map is said to be a bundle map covering f . Relation between the two notions It follows immediately from the definitions that a bundle map over M in the first sense is the same thing as a bundle map covering the identity map of M . Conversely, general bundle maps can be reduced to bundle maps over a fixed base space using the notion of a pullback bundle . If sub F sub F N is a fiber bundle over N and f M N is a continuous map, then the pullback of F by f is a fiber bundle f sup sup F over M whose fiber over x is given by f sup sup F sub x sub . F sub f x sub . It then follows that a bundle map from E to F covering f is the same thing as a bundle map from E to f sup sup F over M . Variants and generalizations There are two kinds of variation of the general notion of a bundle ..., to the notion of a smooth bundle map between smooth fiber bundles over a smooth manifold . Second, one can consider fiber bundles with extra structure in their fibers, and restrict attention to bundle maps which preserve this structure. This leads, for example, to the notion of a vector bundle homomorphism between vector bundle s, in which the fibers are vector spaces, and a bundle map is required to be a linear map on each fiber. In this case, such a bundle map covering f may also be viewed ... more details
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