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Topic: Principal bundle

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	PlanetMath: principal bundle
		as a topological bundle is an equivariant trivialization.
		Cross-references: equivariant, local trivialization, isomorphism, section, locally trivial bundle, projection map, map, topological group, topological space
		This is version 5 of principal bundle, born on 2002-11-01, modified 2004-01-03.
	planetmath.org /encyclopedia/PrincipalBundle.html (72 words)

	Fiber bundle - Wikipedia, the free encyclopedia
		Fiber bundles generalize vector bundles, where the main example is the tangent bundle of a manifold.
		A sphere bundle is a fiber bundle whose fiber is an n-sphere.
		In the smooth category, a G-bundle is a smooth fiber bundle where G is a Lie group and the corresponding action on F is smooth and the transition functions are all smooth maps.
	en.wikipedia.org /wiki/Fiber_bundle (1166 words)

	Principal bundle - Wikipedia, the free encyclopedia
		In mathematics, a principal G-bundle is a special kind of fiber bundle for which the fibers are all G-torsors (also known as principal homogeneous spaces) for the action of a topological group G.
		Principal G-bundles are G-bundles in the sense that the group G also serves as the structure group of the bundle.
		Principal bundles provide a unifying framework for the theory of fiber bundles in the sense that all fiber bundles with structure group G determine a unique principal G-bundle from which the original bundle can be reconstructed.
	en.wikipedia.org /wiki/Principal_bundle (925 words)

	One Bag: Folding Clothes
		The easiest place to form the bundle is on a large flat surface, such as a bed.
		For sleeved garments, wrap one side of the garment around the bundle as far as it will go (part of the sleeve — which is wrapped straight across — will typically end up underneath the bundle); repeat with the other side.
		If you'd like an alternative description of the bundle wrapping technique, with different illustrations, consult Judith Gilford's book, which also recommends this method (though she calls it "bundle folding", which I find somewhat misleading, as the entire point of the technique is to eliminate folding).
	www.onebag.com /pack.html (1360 words)

	Fiber bundle
		In mathematics, in particular topology, a fiber bundle is a continuous surjective map, π from a topological space E to another topological space B, satisfying a further condition making it locally of a particularly simple form.
		For example in the case of a vector bundle, F is a vector space over the real numbers.
		An example of a principal bundle that occurs naturally in geometry is the bundle of all bases for the tangent space to a manifold, with G the general linear group ; restricting in Riemannian geometry to orthonormal bases, one would limit G to the orthogonal group.
	www.nebulasearch.com /encyclopedia/article/Fiber_bundle.html (573 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		By definition, a principal bundle with group S1 has fibre S1, structure group S1, with the group S1 acting on the fiber by left multiplication.
		In case you meant this, note that it is also true that if the base space is orientable, the total space of any principal S1 bundle will also be orientable, and here is the proof: at every point in the n-dimensional base space M, we have an n-form \omega that defines the orientation.
		It is straightforward to check that this is a fibre bundle with fibre S1 (and structure group Z/2, in case you care).
	www.lehigh.edu /~dmd1/ki97.txt (605 words)

	Fiber bundle (Site not responding. Last check: 2007-10-08)
		In mathematics, in particular topology, a fiber bundle is a continuous surjective map, π from a topological space E toanother topological space B, satisfying a further conditionmaking it locally of a particularly simple form.
		For example in the case of a vector bundle, F is a vectorspace over the real numbers.
		An example of a principal bundle that occurs naturally in geometry is thebundle of all bases for the tangent space to a manifold, with G the general lineargroup ; restricting in Riemannian geometry to orthonormalbases, one would limit G to the orthogonal group.
	www.therfcc.org /fiber-bundle-30091.html (465 words)

	Re: principal fibre bundle
		Just remember this: a principal bundle has a group as its fiber; a principle bundle has moral fiber.
		Actually, despite the above joke, you shouldn't think of the fiber of a principal G-bundle as being the group G; instead, you should think of it as a space that looks like G but has "forgotten its identity".
		Principal G-bundles for bigger groups G are harder to visualize, but mainly because bigger groups are harder to visualize.
	www.lns.cornell.edu /spr/2002-04/msg0041126.html (279 words)

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	www.asapportal.com (553 words)

	[No title]
		In this note we consider the sufficient condition that the homomorphism F of (1.1)is a monomorphism and several examples are given.
		Let E(B) be the group of based self homotopy equivalences of B, and we de- note by E# (B) the subgroup of E(B) consisting of classes that induce the ident* *ity automorphism of all homotopy groups.
		Proof.By the homotopy exact sequence of the given principal bundle, we have ssi(B) = ssi(Pk)(i n + 2), ssn+1(B) = ssn+1(P) ss0 (ss0 is a free subgroup o* *f ss) and ssi(B) = 0(i n).
	hopf.math.purdue.edu /Tsukiyama/forget8.txt (1236 words)

	Lagrangian Densities Invariant Under The Full . . . (ResearchIndex) (Site not responding. Last check: 2007-10-08)
		First order Lagrangian densities on the bundle of connections of a principal bundle invariant under the full Lie algebra of the infinitesimal automorphisms of the principal bundle are classified.
		Bundle of connections, infinitesimal automorphism of a principal bundle, invariant Lagrangian densities.
		M be the bundle of connections of P.
	citeseer.ist.psu.edu /perez93lagrangian.html (233 words)

	Symmetry of Einstein-Yang-Mills systems and dimensional reduction
		Although principal bundles proved to be an indispensable concept in discussing gauge fields and their interaction with matter, many people felt uneasy about an ad hoc introduction of such a very special geometrical structure.
		Keeping in mind the obvious shortcomings of our model it is nevertheless worthwhile to study it as a straightforward generalization of the principal bundle structure which proved to be already useful.
		In all these papers it was always assumed that E is a principal bundle, i.e.
	quantumfuture.net /quantum_future/einstein.htm (1330 words)

	NTU Info Centre: Classifying space (Site not responding. Last check: 2007-10-08)
		A more formal statement takes into account that G may be a topological group (not simply a discrete group), and that group actions of G are taken to be continuous; in the absence of continuous actions the classifying space concept can be dealt with, in homotopy terms, via the Eilenberg-MacLane space construction.
		In homotopy theory the definition of a topological space BG, the classifying space for principal G-bundles, is given, together with the space EG which is the total space of the universal bundle over BG.
		The abstract conditions being known for this (Brown's representability theorem) the result, as an existence theorem, is in affirmative and not too difficult.
	www.nowtryus.com /article:Classifying_space (750 words)

	Translation Map in Quantum Principal Bundles (ResearchIndex) (Site not responding. Last check: 2007-10-08)
		Abstract: The notion of a translation map in a quantum principal bundle is introduced.
		P, and that a quantum principal bundle is trivial if it admits a cross section which is an algebra map.
		The vertical automorphisms and gauge transformations of a quantum principal bundle are discussed.
	citeseer.ist.psu.edu /brzezinski94translation.html (311 words)

	Untitled Document (Site not responding. Last check: 2007-10-08)
		X with a structure group G and an associated Higgs bundle Z with a standard fiber G/H, the case of a matter bundle E whose standard fiber admits action only of an exact symmetry subgroup H of G is examined.
		In the presence of a fixed Higgs field h: X –>Z, matter fields are represented by sections of a matter bundle E_h associated with the corresponding reduced subbundle P_h of P.
		Z is the bundle associated with the principal H-bundle P
	webcenter.ru /~sardan/38.html (103 words)

	Table of Contents for Steenrod, N.: The Topology of Fibre Bundles. (PMS-14).
		Construction of a bundle from coordinate transformations 14
		The principal bundle and the principal map 35
		The Whitney characteristic classes of a sphere bundle 190
	pup.princeton.edu /TOCs/c1973.html (132 words)

	Quantum Principal Bundles
		The explanation is that in classical geometry homotopic bundles over the same space and with the same structure groups are always isomorphic.
		In particular, a given classical bundle could be homotopically equivalent to a truly quantum bundle (the base and the structure group remain the same-classical).
		Furthermore, there exist quantum bundles over a classical manifold M and with the classical structure group G that are not homotopically equivalent to any classical bundle.
	www.matem.unam.mx /~micho/qbun5.html (426 words)

	O'Reilly Media -- Bookstore: Programming .NET 3.5 Rough Cuts Version
		Bestselling author Jesse Liberty and industry expert Alex Horovitz uncover the common threads that unite the.NET 3.0 technologies, so you can benefit from the best practices and architectural patterns baked into the new Microsoft frameworks.
		The book offers a "Grand Tour" of.NET 3.0 that describes how the principal technologies can be used together, with Ajax, to build modern n-tier and service-oriented applications.
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	www.oreilly.com /catalog/059652756X (493 words)

	"Lie algebroid of a principal fibre bundle" (Site not responding. Last check: 2007-10-08)
		Three natural constructions of the Lie algebroid for a given principal fiber bundle are repeated from the paper [12].
		Two principal fibre bundles are locally isomorphic if and only if their Lie algebroids are isomorphic.
		The most if G is semisimple: two principal fibre bundles with semisimple structural Lie groups are locally isomorphic if and only if their associated Lie algebra bundles are isomorphic.
	im0.p.lodz.pl /~kubarski/forum/abs15.html (358 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		The idea of a fiber bundle over a manifold generalizes the concepts inherent in tangent vector fields to objects that can serve as models
		The principal bundle of frames serves as the natural framework for differential geometry (general relativity) and its
		principal bundle generalizations (with their natural bundle equivalences) definegauge theories, so central in contemporary physics.
	www.loyno.edu /~cbta/contents.html (1208 words)

	"Lie algebroid of a principal fibre bundle – three equivalent definitions" (Site not responding. Last check: 2007-10-08)
		Starting with some Atiyah's construction (1957) we define the so-called Lie functor for principal fibre bundles.
		This functor assigns some Lie algebroid (Pradines 1967) to each principal fibre bundle and plays an analogous role as the Lie functor for Lie groups.
		of a principal fibre bundle P(M,G) is constructed by using the Lie algebra X
	im0.p.lodz.pl /~kubarski/forum/abs12.html (150 words)

	resume.html (Site not responding. Last check: 2007-10-08)
		Connection of a vector bundle; local expression with respect to a trivialisation of the bundle.
		Definition of a covariant derivative on any associated bundle from a connection on the principal bundle.
		Spin 0 electromagnetism in a U(1)-principal bundle : gauge invariance of the square norm of $D\phi$.
	www.ulb.ac.be /sciences/ptm/pmif/Rencontres/resume.html (655 words)

	NTU Info Centre: Hopf bundle (Site not responding. Last check: 2007-10-08)
		In mathematics, the Hopf bundle (or Hopf fibration) is a particular fiber bundle with base space ''S''
		The Hopf bundle can actually be considered as a principal bundle when the fiber is identified with the circle group.
		As a consequence of Adams' theorem, these are the only fiber bundles with spheres as total space, base space, and fiber.
	www.nowtryus.com /article:Hopf_bundle (304 words)

	MPI MIS Leipzig - Preprint Nr. 22/2000
		Budzynski and Kondracki have introduced a notion of locally trivial quantum principal fibre bundle making use of an algebraic notion of covering, which allows a reconstruction of the bundle from local pieces.
		Following this approach, we construct covariant differential algebras and connections on locally trivial quantum principal fibre bundles by gluing together such locally given geometric objects.
		As an example, a U(1) quantum principal bundle over a glued quantum sphere as well as a connection in this bundle is constructed.
	www.mis.mpg.de /preprints/2000/prepr2200-abstr.html (136 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		Now a principal O(n)-bundle is > basically the same thing as an n-dimensional real vector bundle --- > there are obvious ways to go back and forth between these concepts.
		A > principal O(infinity)-bundle is thus very much like a real vector > bundle of arbitrary dimension, but where we don't care about adding > on arbitrarily many 1-dimensional trivial bundles.
		This has a > "canonical line bundle" over it, that is, a 1-dimensional real vector > bundle which to each point of RP^1 assigns the 1-dimensional subspace of > R^2 that is that point.
	www.math.niu.edu /~rusin/papers/known-math/98/bott_per (3343 words)

	Re: principal fibre bundle
		On Sun, 21 Apr 2002 04:17:06 +0000 (UTC), baez@galaxy.ucr.edu (John Baez) wrote: >So, for example, if you want to visualize a principal U(1) bundle >over the space M, you should just imagine a circle sitting over >each point of M in a smoothly varying way.
		This is the action of U(1) on the bundle.
		Moroever, this bundle is nontrivial - it has no global section: http://www.lns.cornell.edu/spr/2000-08/msg0027213.html http://www.mimuw.edu.pl/popularyzacja/delta/delta7/czesanie/czesanie.htm http://www.math.niu.edu/~rusin/known-math/95/hairy ark -- Arkadiusz Jadczyk http://www.cassiopaea.org/quantum_future/homepage.htm --
	www.lns.cornell.edu /spr/2002-04/msg0041167.html (292 words)

	[No title]
		What this means is that there's a principal G-bundle over BG called the "universal G-bundle", with the marvelous property that any principal G-bundle over any space X is a pullback of this one by some map f: X -> BG.
		In short: if we have a space X with a principal G-bundle P over it, and P is contractible, X must be BG.
		We get a principal U(1)-bundle like this: S^infinity -> CP^infinity Being a mathematical physicist, I'm using S^infinity here to stand for the unit sphere in some countable-dimensional Hilbert space, and the map sends each unit vector to the corresponding pure state, or unit vector mod phase.
	math.ucr.edu /home/baez/twf_ascii/week151 (2642 words)

	Fiber Bundle Encyclopedia Article, Definition, History, Biography (Site not responding. Last check: 2007-10-08)
		Looking For fiber bundle - Find fiber bundle and more at Lycos Search.
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		Look for fiber bundle - Find fiber bundle at one of the best sites the Internet has to offer!
	search.localcolorart.com /search/encyclopedia/Fiber_bundle (1311 words)

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