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Topic: Local trivialization

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	Fiber bundle - Wikipedia, the free encyclopedia
		In mathematics, in particular in topology, a fiber bundle (or fibre bundle) is a space which locally looks like a product of two spaces but may possess a different global structure.
		A local section of a fiber bundle is a continuous map f : U → E where U is an open set in B and π(f(x))=x for all x in U.
		Fiber bundles often come with a group of symmetries which describe the matching conditions between overlapping local trivialization charts.
	en.wikipedia.org /wiki/Fiber_bundles (1279 words)

	[No title] (Site not responding. Last check: 2007-10-30)
		There is some local basis over a neighborhood small enough to be within a local trivialization of \begin_inset Formula \(E \) \end_inset, and that local trivialization produces a basis at each point.
		Perhaps local triviality is a necessary additional condition, but it may well be that the free action does imply a smooth quotient.
		Local triviality is clear from the structure of the tangent bundle.
	www.lehigh.edu /dlj0/Desktop/dlj0/courses/424sp97-6.lyx (4067 words)

	PlanetMath: principal bundle
		-bundle which is topologically trivial is also isomorphic to
		as a topological bundle is an equivariant trivialization.
		Cross-references: equivariant, local trivialization, isomorphic, isomorphism, section, locally trivial bundle, projection map, map, topological group, topological space
	planetmath.org /encyclopedia/PrincipalBundle.html (73 words)

	Orthodoxy & the Environment - Holiness of Place: Images from Ireland
		The parish sits on the edge of the Bog of Allen, and for centuries the local people have dug turf from the bog for their winter heating.
		This wood from the bog is an icon for the local place, for the holy that is in it.
		It is local, it is important to the physical world and the culture of a large area.
	www.orthodoxresearchinstitute.org /articles/misc/flanagan_ireland.htm (1144 words)

	More on Geodesic Equation
		In metric geometry, a geodesic is a curve which is everywhere locally a distance minimizer.
		The shortest path from point A to point B on a sphere is given by the shorter piece of the great circle passing through A and B. Note that if A and B are antipodal points (like the North pole and the South pole), then there are many shortest paths between them.
		The local existence and uniqueness theorem for geodesics states that geodesics exist, and are unique; this is a variant of the Frobenius theorem.
	www.artilifes.com /geodesic-equation.htm (1291 words)

	mat531 weeks123
		Analogously, the {U_a} in general are called "trivializing neighborhoods" and H_a is called a "local trivialization."
		The tangent bundle pi : TM --> M of an n-dimensional smooth manifold with atlas {U_a} is clearly an n-dimensional coordinate vector bundle.
		Two coordinate vector bundles over the same base are equivalent if the two sets of local trivializations are compatible; an equivalence class is a vector bundle.
	www.math.sunysb.edu /~tony/archive/top2/weeks123.html (1042 words)

	Vector bundles (Site not responding. Last check: 2007-10-30)
		For every there exists a neighborhood U of p and a homeomorphism such that (h preserves fibers.) Furthermore must be a k-vector space isomorphism for all p.
		The pair (U,h) is called a local trivialization.
		A bundle of the form is called a trivial bundle.
	www.math.uiuc.edu /~jacox/Ontalk/node2.html (106 words)

	TGD abstracts
		The general topological structure of the configuration space is described and the difficulties associated with the geometrization attempt relying on the local coset space structure are described.
		The U(1) algebra is localized with respect to light cone boundary and the function basis are naturally grouped into unitary representations of Lorentz group but possess no four-momentum.
		This local choice has interpretation as a fixing of the plane of non-physical polarizations and rise to degeneracy which is a good candidate for the ground state degeneracy caused by the vacuum extremals.
	www.physics.helsinki.fi /~matpitka/tabstracts.html (14412 words)

	[No title] (Site not responding. Last check: 2007-10-30)
		The minima will be in the class of Cartesian currents, but those will be in the same homology class as the collection of smooth graphs over which the volume is minimized.
		The examples in this section are all for trivial bundles B=M\times F, so there are obvious global minimizers, simply the constant graphs M\times \{y\}, y\in F. The real homology classes of graphs all have integral periods, and must project to the volume element [M]\in H_{n}(M,\R).
		It should go through nicely, though, to show local uniqueness in a coordinatiable nbhd where the vertical range is restricted.
	www.lehigh.edu /dlj0/yesterday/Desktop/dlj0/yesterday/courses/js-part-reg.txt (5093 words)

	Frobenius Algebras and 2D QFT \| The String Coffee Table
		In trying to understand the deeper ‘meaning’ of it all I played around with the idea that this prescription is really, to some extent at least, the ‘dual’ incarnation of the application of a certain 2-functor to the worldsheet.
		I claim that any local trivialization of a 2-functor on some sort of 2-category of surface elements gives rise to a dual triangulation of the surface whose edges are labeled by (possibly a generalization of) a Frobenius algebra object and whose vertices are labeled by (possibly a generalization of) product and coproduct operations.
		There is more data in a locally trivialized 2-functor, and it seems to correctly reproduce the main structure of bulk field insertions as appearing above.
	golem.ph.utexas.edu /string/archives/000699.html (1123 words)

	Not Even Wrong » Blog Archive » P. University Press
		This 6-manifold N is in turn an associated SO(4)-bundle over M with fiber S^2, trivially reducible to an SO(3)-bundle since the generators of the SU(2) with opposite “chirality” to I (let’s call it SU(2)_L) all act trivially on the fiber.
		On a trivial gauge bundle, E is the semidirect product of the vector algebra and the gauge algebra over M. On a non-trivial gauge bundle, infinitesimal gauge transformations still form a “vertical” ideal of E, but the “horizontal” vector algebra does not form a subalgebra.
		He also doesn’t particularly emphasize that the fixed auxiliary connexion is only a device for identifying the “horizontal” portion of an algebra element in a particular local trivialization, and that the actual graded Lie algebra of derivations on \Lambda is independent of this connexion.
	www.math.columbia.edu /~woit/wordpress/?p=438 (9752 words)

	RADIO FREE EUROPE/ RADIO LIBERTY
		Depicting Europe as a whole as a wartime arena of assassinations shrinks the historical responsibility for the terrible occurrence whose significance must be faced by all European nations, Romania included" (Mediafax, 30 July 2003).
		The significance of the Romanian president's comparative trivialization did not escape the attention of local observers.
		Cornea quite rightly insisted also on the fact that the reference to Iliescu's father having suffered like a Jew during the war, and having died as a result shortly thereafter, was another instance of "trivialization" by comparison.
	www.rferl.org /reports/eepreport/2003/11/24-261103.asp (1995 words)

	Background independence \| The String Coffee Table
		To find a local, non-gauge operator which moves the lightcone I have to resort to handwaving, but I think that all we need to do is to give the scalar field a mass.
		Assume we have picked a zero section, and a local trivialization the worry is that you have now obscured yourself from the global meaning of say the connection, along with all the nonuniqueness that can entail.
		Also, perturbative string theory is well-known to be background-indepent in the second sense (at least under infinitesimal shifts of classical solutions) and is trivially background indepent (of the metric) in the first sense (since the metric tensor is not fixed, otherwise there would be just a single classical solution for it, which there is not).
	golem.ph.utexas.edu /string/archives/000621.html (10342 words)

	Discrete Reduction
		The flow arising from a Discrete Lagrange-d'Alembert variational principle is shown to preserve a discrete analogue of a modified symplectic form and momentum map.
		This approach is equivalent, in the sense of the reconstructed flow, to the discrete Hamel equations in a local trivialization.
		In addition, we illustrate the technique by applying the construction procedure to the double spherical pendulum.
	www.cds.caltech.edu /~mleok/integrators.htm (182 words)

	Topics: T (Site not responding. Last check: 2007-10-30)
		is flat, so there are local degrees of freedom only if one admits torsion.
		Amenable topological group: G is amenable if there is a positive G-invariant functional A (a "measure") on the Banach space of bounded Borel measurable functions on G s.t.
		For example, SO(3) is not amenable wrt the discrete topology, but it is as a compact Lie group; SO(n,1) is not amenable; Local gauge groups also have topologies different from their natural ones wrt which they are amenable.
	www.phy.olemiss.edu /~luca/Topics/t.html (1678 words)

	Homepage of Matti Pitkänen
		Conservation laws suggests that in the case of non-vacuum extremals the dynamics of the local automorphism associated with the hyper-octonionic spinor field is dictated by field equations of some kind.
		Conservation laws suggests that in the case of non-vacuum extremals the dynamics of the the local G_2 automorphism is dictated by field equations of some kind.
		The assignment is not unique but determined only up to an element of the local octonionic automorphism group G_2 acting in HO and fixing the local choices of the preferred imaginary unit of the hyper-octonionic tangent plane.
	www.physics.helsinki.fi /~matpitka/newtgd.html (20486 words)

	Salon.com News \| In San Francisco: Theatrical Bush hate
		As always, the exact size of the protest was a subject of dispute.
		In one of those absurd estimates that give credence to left-wing paranoia theories that local authorities are taking their marching orders from Attorney General Ashcroft, police estimated the size of the march at 55,000.
		I'm no professional crowd estimator, but I've been to plenty of games at Candlestick Park (football capacity: 60,000) and the idea that a sellout crowd there would completely fill four-lane Market Street all the way from the Ferry Building, on the Bay, to the Civic Center, about two miles away, is patently ridiculous.
	archive.salon.com /news/feature/2003/01/20/san_francisco/index_np.html (518 words)

	4.1 The geometry of spacelike 2-surfaces
		is defined only on the local trivialization domains of
		Note, however, that in general a GHP spin frame can be specified only locally, but not globally on the whole
		The only closed orientable 2-surface with globally trivial tangent bundle is the torus.
	www.univie.ac.at /EMIS/journals/LRG/Articles/lrr-2004-4/articlesu6.html (1260 words)

	Topics in Quantum Dynamics
		Physicists are being taught in the course of their education rather early that every attempt at a sharp localization of a relativistic particle results in creation and annihilation processes.
		There is a distinguished family of local trivializations, a family parameterized by
		One can demand even more - it is reasonable to demand a special kind of stability: that it should be always possible to extend the system and its evolution in a trivial way, by adding extra degrees of freedom that do not couple to our system.
	www.quantumfuture.net /quantum_future/papers/9406204/topics.htm (5212 words)

	[No title]
		All we need now is for some grad student to come along and play around with specific examples and see what interesting things can be said about them.
		There is a limit to how interesting they can be, because they're utterly trivial except in 2d spacetime.
		But anyway, to answer your question: there are non-TQFTs like 2d Yang-Mills theory which can be described via spin foam models and can be completely understood.
	www.math.niu.edu /~rusin/known-math/00_incoming/foam (2744 words)

	University of Iowa Press - Browse (Site not responding. Last check: 2007-10-30)
		It complicates our ideas about 'regionalism,' links nineteenth- and twentieth-century writers, places issues of race and ethnicity at the center, makes us rethink the academy's usual (and limiting) preoccupation with fiction as the only important genre, and effectively deconstructs the scholarly trivialization of local color writing as a 'minor' American tradition.
		It looks back but also at the present, asking us how our thinking about the 'local' shapes and affects major issues today, such as the environment, race relations, gender dynamics, and attitudes toward sexuality.”—Elizabeth Ammons, author of Conflicting Stories: American Women Writers at the Turn of the Twentieth Century
		These lively essays reveal the generational continuum of women's regional literature, which has always offered a voice to women and their concerns.
	www.uiowa.edu /uiowapress/innbrebou.htm (456 words)

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