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Topic: Vector bundle morphism

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	Vector bundle - Wikipedia, the free encyclopedia
		In mathematics, a vector bundle is a geometrical construct where to every point of a topological space (or manifold, or algebraic variety) we attach a vector space in a compatible way, so that all those vector spaces, "glued together", form another topological space (or manifold or variety).
		Vector bundles are special fiber bundles, loosely speaking those where the fibers are vector spaces.
		Smooth vector bundles are defined by requiring that E and X be smooth manifolds, π ;: E → X be a smooth map, and the local trivialization maps φ be diffeomorphisms.
	en.wikipedia.org /wiki/Vector_bundle (1009 words)

	PlanetMath: vector bundle
		A vector bundle is a fiber bundle having a vector space as a fiber and the general linear group of that vector space (or some subgroup) as structure group.
		In the algebraic category, that is, vector bundles over schemes, there is a very nice correspondence between vector bundles and locally free sheaves; when the dimension is one and the scheme is nice enough, there is a further correspondence with Cartier divisors.
		This is version 2 of vector bundle, born on 2002-11-01, modified 2004-02-16.
	planetmath.org /encyclopedia/Section3.html (624 words)

	Vector bundle - InformationBlast
		Vector bundles of rank 1 are called line bundles.
		The composition of two morphisms is again a morphism, and we obtain the category of vector bundles.
		Smooth vector bundles are defined by requiring that E and X be smooth manifolds, π ;: E → X be a smooth map, and the local trivialization maps φ be diffeomorphisms.
	www.informationblast.com /Vector_bundle.html (922 words)

	Vector bundle -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-14)
		The class of all vector bundles together with bundle morphisms forms a (A general concept that marks divisions or coordinations in a conceptual scheme) category.
		Two vector bundles on X, over the same field, have a Whitney sum, with fibre at any point the (A union of two disjoint sets in which every element is the sum of an element from each of the disjoint sets) direct sum of fibres.
		Vector bundles are special (A bundle of fibers (especially nerve fibers)) fiber bundles, loosely speaking those where the fibers are vector spaces.
	www.absoluteastronomy.com /encyclopedia/v/ve/vector_bundle.htm (1345 words)

	Vector bundle (Site not responding. Last check: 2007-10-14)
		In mathematics, a vector bundle is a geometrical constructwhere to every point of a topological space (or manifold, or algebraicvariety) we attach a vector space in a compatible way, so that allthose vectorspaces, "glued together", form another topological space (or manifold or variety).
		Vector bundles are special fiber bundles, loosely speaking those wherethe fibers are vector spaces.
		Smooth vector bundles are defined by requiring that E and X be smooth manifolds, π ;: E → X be a smooth map, and the local trivializationmaps φ be diffeomorphisms.
	www.therfcc.org /vector-bundle-206522.html (889 words)

	Pullback - Wikipedia, the free encyclopedia
		When the smooth map is a self-diffeomorphism, then the pullback, together with the pushforward, describe the transformation properties of the manifold under a change of coordinates.
		TM, are strict subspaces of the general tensor bundles*, closed under the exterior algebra.
		In traditional terms, the pullback describes the transformation properties of the covariant indices of a tensor; by contrast, the transformation of the contravariant indices is given by a pushforward.
	en.wikipedia.org /wiki/Pullback (914 words)

	Vector bundle (Site not responding. Last check: 2007-10-14)
		In mathematics a vector bundle is a geometrical construct where to point of a topological space (or manifold or algebraic variety) we attach a vector space in a compatible way so that those vector spaces "glued together" form another space (or manifold or variety).
		A typical is the tangent bundle of a differentiable manifold : to every point of the manifold attach the tangent space of the manifold that point.
		Smooth vector bundles are defined by requiring that E and X be smooth manifolds π : E → X be a smooth map and the trivialization maps φ be diffeomorphisms.
	www.freeglossary.com /Vector_bundle (1029 words)

	Vector bundle
		In mathematics, a vector bundle is a geometrical construct where to every point of a topological space (or manifold, or algebraic variety) we attach a vector space in a compatible way, so that all those vectorspaces, "glued together", form another topological space (or manifold or variety).
		Every vector bundle π : E → X is surjective, since vector spaces cannot be empty.
		Smooth vector bundles are defined by requiring that E and X be smooth manifolds, π : E → X be a smooth map, and the local trivialization maps φ be diffeomorphisms.
	www.sciencedaily.com /encyclopedia/vector_bundle (979 words)

	PlanetMath: vector bundle
		We talk about topological vector bundles (in the category of topological spaces), we talk about differentiable vector bundles, we talk about complex analytic (or holomorphic) vector bundles, and we talk about algebraic vector bundles.
		Sepcifically, if we want a topological vector bundle, we must supply a topological space for the base space, a topological space for the whole space, and the projection map must be continuous; this specifies a topology on each fiber.
		If we are in the category of schemes, each local trivialization must be an affine space over the affine ring of the neighborhood on the scheme, and the general linear group scheme must act on it through morphisms of schemes.
	www.planetmath.org /encyclopedia/VectorBundle.html (624 words)

	Dictionary of Meaning www.mauspfeil.net (Site not responding. Last check: 2007-10-14)
		A typical example is the tangent bundle of a manifold differentiable manifold: to every point of the manifold we attach the tangent space of the manifold at that point.
		'''Smooth vector bundles''' are defined by requiring that ''E'' and ''X'' be manifold smooth manifolds, π : ''E'' → ''X'' be a smooth map, and the local trivialization maps φ be diffeomorphisms.
		The definition should be of either smooth vector bundle or the general notion of vector bundle without restricting cases where the base space is a smooth manifold.
	www.mauspfeil.net /vector_bundle.html (1204 words)

	Re: exterior and geometric calculus
		Crudely speaking, a god-given vector >> bundle is a vector bundle that we can cook up on a >> manifold using no extra structure on our manifold.
		Jet bundles are god-given in the category-theoretic sense sketched above.
		There is the "nth jet" map from functions to n-jets, and the "Lie bracket" map from vector fields tensor vector fields to vector fields, and so on....
	www.lns.cornell.edu /spr/2001-12/msg0037548.html (585 words)

	[No title] (Site not responding. Last check: 2007-10-14)
		For example a non-holonomic space of G. Vr\u{a}nceanu \cite{Vr} is $(\theta,i)$, where the anchor $i:\theta \rightarrow \tau M$ is the inclusion morphism of a vector subbundle.
		Then the fibred manifold $R\xi =(RE,t,E)$ is a vector bundle and the AVB $(R\xi,\Delta)$ is called the \emph{R-anchored vector bundle}.
		The case of the transverse bundle of a foliation is considered as an example.
	im0.p.lodz.pl /konferencje/BC2000/abstracts/MarcelaPopescu.tex (386 words)

	PlanetMath: vector bundle
		The general linear group must also act continuously.
		Sections may be added and scaled by field elements by simply applying these operations to each fiber, so they form a vector space.
		; in fact, almost all the usual operations on vector spaces can be applied.
	www.planetmath.org /encyclopedia/TrivialVectorBundle.html (624 words)

	Abstract (Site not responding. Last check: 2007-10-14)
		of this morphism are compact and strongly positive.
		the first bundle is spanned by a positive vector (that is, a nonzero
		norms of the components (given by the splitting of the bundle) of
	www.math.umn.edu /~polacik/Publications/exposep-abst.htm (130 words)

	Pullback biography .ms (Site not responding. Last check: 2007-10-14)
		In mathematics, the pullback of smooth map f : M → N between differentiable manifolds is a smooth vector bundle morphism f* : TN → TM, for which the following digram commutes:
		The article on cotangent spaces has the precise definition.
		The pullback map gives rise to a contravariant functor from the category of smooth manifolds to the category of smooth vector bundles via the maps M ↦ TM and (f : M → N) ↦ (f : TN → TM).
	pullback.biography.ms (156 words)

	Vector bundle - InfoSearchPoint.com (Site not responding. Last check: 2007-10-14)
		A vector bundle is called tivial if there is a "global trivialization", i.e.
		F(U) always contains at least one element: the function s that maps every elememnt x of U to the zero element of the vector space π
		The collection of these vectorspace is a sheaf of vector spaces on X.
	www.infosearchpoint.com /display/Vector_bundle (914 words)

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