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Topic: Trivial bundle

	PlanetMath: fiber bundle
		Some examples of fiber bundles are vector bundles and covering spaces.
		Thus we have a category of fiber bundles over a fixed base with fixed structure group.
		This is version 7 of fiber bundle, born on 2002-10-31, modified 2003-06-24.
	planetmath.org /encyclopedia/FiberBundle.html (169 words)

	PlanetMath: locally trivial bundle
		Locally trivial bundles are useful because of their covering homotopy property and because each locally trivial bundle has an associated long exact sequence and Serre spectral sequence.
		Every fibre bundle is an example of a locally trivial bundle.
		This is version 7 of locally trivial bundle, born on 2002-12-11, modified 2006-02-15.
	planetmath.org /encyclopedia/LocallyTrivialBundle.html (112 words)

	Vector Bundle -- from Wolfram MathWorld (via CobWeb/3.1 planetlab2.cs.unc.edu) (Site not responding. Last check: 2007-11-06)
		, to be a vector bundle, all of the
		The simplest nontrivial vector bundle is a line bundle on the circle, and is analogous to the Möbius strip.
		The structure of the vector bundle, as in all bundles, is that it is locally
	mathworld.wolfram.com.cob-web.org:8888 /VectorBundle.html (325 words)

	PlanetMath: long exact sequence (locally trivial bundle) (via CobWeb/3.1 planetlab2.cs.unc.edu) (Site not responding. Last check: 2007-11-06)
		The covering homotopy property of a locally trivial bundle shows that this is well-defined.
		"long exact sequence (locally trivial bundle)" is owned by bwebste.
		This is version 3 of long exact sequence (locally trivial bundle), born on 2002-12-10, modified 2003-08-21.
	planetmath.org.cob-web.org:8888 /encyclopedia/LongExactSequenceLocallyTrivialBundle.html (121 words)

	What is a Vector Bundle?
		So, the tangent bundle is a vector bundle, as is the cotangent bundle (the cotangent space at a point is the dual space to the tangent space).
		A vector bundle is a family of vector spaces, one at each point of your base space, with one requirement: they must look locally like the product bundle, or trivial bundle UxV, where U is an open set in your base space and V is the model vector space for your vector bundle.
		thus in algebra the analog of a trivial bundle is a free module, and the analog of a vector bundle is a "locally 'free" module.
	www.physicsforums.com /showthread.php?t=29007 (942 words)

	Tangent bundle - Wikipedia, the free encyclopedia
		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 M whose transition functions are given by the Jacobian of the associated coordinate transformations.
		: that this tangent bundle is nontrivial is a consequence of the hairy ball theorem.
	en.wikipedia.org /wiki/Tangent_bundle (757 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		They prove >that the canonical bundle is not trivial >(Theorem 2.1).
		>I think by a trivial bundle they mean a >bundle where all the group transition >functions are the group identity element.
		They prove > that the canonical bundle is not trivial > (Theorem 2.1).
	www.math.niu.edu /~rusin/known-math/99/bundle (461 words)

	Springer Online Reference Works (Site not responding. Last check: 2007-11-06)
		The morphisms of an algebraic vector bundle are defined in the same manner as in topology.
		vector bundles on a compact complex algebraic surface was shown to be isomorphic to the moduli space of instanton unitary vector bundles on the corresponding
		An instanton vector bundle on a Riemannian manifold is a differential bundle together with a connection whose curvature form satisfies a certain system of non-linear differential equations.
	eom.springer.de /v/v096390.htm (1313 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		One example was the trivial bundle, where the lines are all bundled together in the simplest possible way.
		To understand what a connection is, we should think of a line bundle as a place for a sort of generalized function to take its values.
		The generalized function is called a cross-section of the bundle: the value at each point of the surface is not a number, but rather an element of the line associated to that point.
	www.math.temple.edu /~renault/candidates/porcupine2.html (531 words)

	Generalization and key examples (Site not responding. Last check: 2007-11-06)
		The most important example for us is the universal bundle over a Grassmannian, which parametrizes k-dimensional subspaces of an n-dimensional vector space.
		is the subbundle of the trivial bundle with the property that the fiber over each point in G(k,n) is the line in represented by that point.
		These are similar to vector bundles in many ways, but the fibers don't have to be vector spaces.
	www.math.uiuc.edu /~jacox/Ontalk/node5.html (197 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		Moreover E0 xB E is a trivial bundle since it admits the diagonal map as a section.
		Observe that the quo- tient of the bundle q1 by this action is canonically isomorphic to the bundle p : E !
		Comment One might bypass Theorem 3 by working work with a different auxiliary bundle, one whose fibres are finite non-Hausdorff spaces with one point representing each H-antipolar set in p-1(x), the closure of that point consisting of the points that represent the various subsets.
	hopf.math.purdue.edu /Feldman-Wilce/fibdegen.txt (5513 words)

	OSGi Alliance News: OSGi UIs and the Web
		Let us first layout the bundles I like to think in bundles because it is nicely concrete and down to earth.
		We then require a bundle to handle the grunts of the remote procedure calling and another bundle to prove that it works.
		Well, I guess it was trivial in the first place because the server only calculates the board and this is not rocket science.
	www.osgi.org /blog/2006/09/osgi-uis-and-web.html (1280 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		There is a more general theory for topological group extensions whihc are bundles I think due to at least two people independently G.
		However, > not every circle bundle admits a group structure, and the ones that > do will induce fibrations of classifying spaces > BS1 --> B\tilde G --> BG > classified by the transgression d_3 c_1(\tilde G) \in H3(BG;Z).
		> > P.P.S. An equivalent way to state this result is that the > obstruction to a circle bundle admitting the structure of a group > extension is the class > d_2 c_1(\tilde G) \in H2(BG; H1(G;Z)) > in the spectral sequence of the universal fibration G --> EG --> BG.
	www.lehigh.edu /~dmd1/js820.txt (562 words)

	Categorified Gauge Theory
		In particular, we define a "Lie 2-group" to be a category C where the set of objects and the set of morphisms are Lie groups, and source, target, identity and composition maps are homomorphisms of Lie groups.
		This turns out to be the same as a "Lie crossed module": a pair of Lie groups G and H with a homomorphism t: H -> G and an action of G on H satisfying the equations in the usual definition of crossed module.
		Just as a connection on a trivial G-bundle is the same as a Lie(G)-valued 1-form, a connection on a trivial C-2-bundle turns out to be a Lie(G)-valued 1-form together with a Lie(H)-valued 2-form.
	math.ucr.edu /home/baez/gauge (604 words)

	Tangent bundle on the sphere
		If TS² is trivial, then there exists at least one vector field without a zero.
		Thus, we would have a global basis for TS^2, which then is trivial.
		quick answer: if the tangent bundle on the sphere were trivial, then so would the cotangent bundle be trivial.
	www.physicsforums.com /showthread.php?t=106795 (1084 words)

	Fiber bundle - Wikipedia, the free encyclopedia (via CobWeb/3.1 planetlab2.cs.unc.edu) (Site not responding. Last check: 2007-11-06)
		B is called the base space of the bundle, E the total space, and F the fiber.
		The corresponding trivial bundle would be a torus, S
		Specifically, let G be a topological group which acts continuously on the fiber space F on the left.
	en.wikipedia.org.cob-web.org:8888 /wiki/Fiber_bundle (1512 words)

	[No title] (Site not responding. Last check: 2007-11-06)
		These are vector bundles whose fibres a* *re spaces of eigenvectors associated to linear operators which are parameterized by the base space.
		In this case the spectral line bundle exists and is tr* *ivial since there is a nonzero cross-section.
		CP 3 is a principal S1-fibre bundle and is an equivariant map from the free S3 action on S7 to the induced SO(3) action on CP 3.
	www.math.purdue.edu /research/atopology/Gottlieb/eigbndl.txt (5948 words)

	[No title]
		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.
		The vector bundle $\La^kT^M$ is in fact the value of a functor, which associates a bundle* over $M$ to each manifold $M$ and a vector bundle homomorphism over $f$ to each local diffeomorphism $f$ between manifolds of the same dimension.
		A systematic treatment of naturality in differential geometry requires to describe all natural bundles, and this is also one of the undertakings of this book.
	www.math.niu.edu /~rusin/known-math/01_incoming/GL_bundles (1439 words)

	Springer Online Reference Works (Site not responding. Last check: 2007-11-06)
		is associated with the tangent vector bundle to
		corresponding to the addition of a one-dimensional trivial bundle to a vector bundle.
		The term "classifying space" is not used solely in connection with fibre bundles.
	eom.springer.de /c/c022440.htm (536 words)

	Fiber bundle - Wikipedia, the free encyclopedia
		Every fiber bundle consists of a continuous surjective map
		Then E is a fiber bundle (of F) over B.
		A G-bundle is a fiber bundle with an equivalence class of G-atlases.
	en.wikipedia.org /wiki/Fiber_bundle (1502 words)

	Springer Online Reference Works (via CobWeb/3.1 planetlab2.cs.unc.edu) (Site not responding. Last check: 2007-11-06)
		is called a chart of the locally trivial bundle.
		forms the atlas of the locally trivial bundle.
		For example, a principal fibre bundle with a locally compact space and a Lie group
	eom.springer.de.cob-web.org:8888 /L/l060530.htm (90 words)

	Twisted N=4 SYM and Special Holonomy \| Musings
		is a rank-4 vector bundle associated to principal spin bundle of
		trivial, supersymmetry is generally broken (the above bundle has no nowhere-vanishing sections) unless
		, for which the bundle (1) has a trivial subbundle, and hence one or more unbroken supercharges, even on a general curved manifold
	golem.ph.utexas.edu /~distler/blog/archives/000690.html (650 words)

	Monodromies Of Algebraic Connections On The Trivial Bundle (ResearchIndex) (via CobWeb/3.1 planetlab2.cs.unc.edu) (Site not responding. Last check: 2007-11-06)
		In this note, we study monodromies of algebraic connections on the trivial vector bundle.
		We prove that on a smooth complex ane curve, any monodromy arises as the underlying local system of an algebraic connection on the trivial bundle.
		Dans cette note, nous etudions la monodromie de connexions algebriques sur le bre vectoriel trivial....
	citeseer.ist.psu.edu.cob-web.org:8888 /427218.html (268 words)

	Wang at AustMS 2006 \| The n-Category Café
		In the cases above a geometric cycle is (1) a vector bundle, (2) an equivariant vector bundle and (3) a representation of the group
		Unless I am missing something, this geometric cycle is usually called a module for a gerbe (which can be thought of as a twisted bundle, e.g.
		In fact, another way to realize a gerbe module as a special case of a more universal principle is to conceive it as a trivialization of a gerbe, but in an enlarged ambient category.
	golem.ph.utexas.edu /category/2006/10/wang_at_austms_2006.html (943 words)

	Citations: Classifying Sheaves in the Maruyama Boundary for Surfaces with Trivial Canonical Bundle University of ... (Site not responding. Last check: 2007-11-06)
		Maciocia, Classifying Sheaves in the Maruyama Boundary for Surfaces with Trivial Canonical Bundle University of Edinburgh Preprint (1992).
		The Determinant Line Bundle over Moduli Spaces of Instantons on..
		i) Let M be the moduli space of stable bundles of type (r; 0; Gammak) over a polarized torus (T; L) not necessarily principal) Then the ratio of the norms on PL and F PL over M k is locally constant.
	citeseer.ist.psu.edu.cob-web.org:8888 /context/771369/0 (203 words)

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