This means that locally, each covering map is ' isomorphic ' to a projection in the sense that there is a homeomorphism from the pre-image of an evenly covered neighbourhood U, to U X F, where F is the fiber, satisfying the local trivialization condition.

An open set U in X admits a local trivialization if and only if there exists a local section on U. Given a local trivialization one can define an associated local section by

Given an equivariant local trivialization (

The surjectivity of the differential is a necessary condition for the existence of a local trivialization.

The frame bundle F ( E ) can be given a natural topology and bundle structure determined by that of E. Let ( U < sub > i </ sub >, φ < sub > i </ sub >) be a local trivialization of E. Then for each x ∈ U < sub > i </ sub > one has a linear isomorphism φ < sub > i, x </ sub >: E < sub > x </ sub > → R < sup > k </ sup >.

A local section of a fiber bundle is a continuous map s: U → E where U is an open set in B and π ( s ( x )) = x for all x in U. If ( U, φ ) is a local trivialization of E, where φ is a homeomorphism from π < sup >− 1 </ sup >( U ) to U × F ( where F is the fiber ), then local sections always exist over U in bijective correspondence with continuous maps from U to F. The ( local ) sections form a sheaf over B called the sheaf of sections of E.

There is a local trivialization of the bundle E consisting of an open cover U < sub > i </ sub > of X, and a collection of fibre maps

Now let F ′ be a specified topological space, equipped with a continuous left action of G. Then the bundle associated to E with fibre F ′ is a bundle E ′ with a local trivialization subordinate to the cover U < sub > i </ sub > whose transition functions are given by

where the G-valued functions g < sub > ij </ sub >( u ) are the same as those obtained from the local trivialization of the original bundle E.

( Using another local trivialization, and passing to a common refinement if necessary, the g < sub > ij </ sub > transform via the same coboundary.

Hence the ostensible fibre bundle T < sup > k </ sup > M admits a local trivialization in each coordinate neighborhood.

If ( U, φ ) is a local trivialization of E then ( f < sup >− 1 </ sup > U, ψ ) is a local trivialization of f < sup >*</ sup > E where

Since going into ecumenical communion with their respective Anglican body, bishops in the ELCA or the ELCIC not only approve the " rostering " of all ordained pastors, diaconal ministers, and associates in ministry, but they serve as the principal celebrant of all pastoral ordination and installation ceremonies, diaconal consecration ceremonies, as well as serving as the " chief pastor " of the local synod, upholding the teachings of Martin Luther as well as the documentations of the Ninety-Five Theses and the Augsburg Confession.

In 1831, a legislative assembly was established by local consent at a meeting of principal inhabitants held at Pedro St. James Castle on December 5 of that year.

The principal function of the county was to impose royal control in the areas of taxation, security and the administration of justice at the local level.

The Beneventan rite has not survived in its complete form, although most of the principal feasts and several feasts of local significance are extant.

Leto was identified from the fourth century onwards with the principal local mother goddess of Anatolian Lycia, as the region became Hellenized.

Little says that the principal objective in the local wars, at least among the Mende, was plunder, not the acquisition of territory.

Sherwood Forest is home to the famous Major Oak, which, according to local folklore, was Robin Hood's principal hideout.

Slightly over half the members of each national park authority are appointees from the principal local authorities covered by the park ; the remainder are appointed by the Secretary of State for Environment, Food and Rural Affairs, some to represent local parish councils, others selected to represent the " national interest ".

Telo Martius became one of the two principal Roman dye manufacturing centres, producing the purple colour used in imperial robes, made from the local sea snail called murex, and from the acorns of the oak trees.

After his election as princeps ( prince, principal leader ) of the Asturians by the local magnates in the Visigothic manner, Pelagius made his capital at Cangas de Onís.

Over a two-week shooting schedule, Buchanan staged principal photography for Mars Needs Women in his hometown, Dallas, with the Southland Life Insurance Building visible as the Martians drive among the humans ; other prominent local landmarks including the Southern Methodist University were also featured.

The boroughs are the principal local authorities in London and are responsible for running most local services in their areas, such as schools, social services, waste collection and roads.

was formed in 1925 and won its first seat in 1966. by 2012 had 1 of 4 Welsh seats in the European Parliament, 3 of 40 Welsh seats in the Parliament of the United Kingdom, 11 of 60 seats in the National Assembly for Wales, and 206 of 1, 264 principal local authority councillors.

The principal sources of income for the local economy are gold mining and fisheries.

The principal local authority in this area is Cornwall Council, but until March 2009 the parish was in the area of North Cornwall District Council.

This has in turn been a factor in the ongoing factionalism within the two principal parties at national and local levels.

The principal hypothesis is that they dispersed and were absorbed by either the local Croatans on Hatteras Island, or another Algonquian people ; it has yet to be established if they did assimilate with one or other of the native populations.

The current principal is Professor Gerry McCormac and the current chairman of court is a local businessman Alan Simpson.

The network's principal directors and other officers are appointed by the State, and so are the top officials at local conventional television stations in mainland China ; nearly all of them are restricted to broadcasting within their own province or municipality.

His temples held the image of a phallus ; in Lavinium, this was the principal focus for his month-long festival, when according to St. Augustine, the " dishonourable member " was placed " on a little trolley " and taken in procession around the local crossroad shrines, then to the local forum for its crowning by an honourable matron.

In the 14th – 17th centuries, the region was dominated by the Abkhaz, Ubykh and Adyghe tribes, the current location of the city of Sochi known as Ubykhia was part of historical Circassia, and was controlled by the native people of the local mountaineer clans of the north-west Caucasus, nominally under the sovereignty of the Ottoman Empire, which was their principal trading partner in the Muslim world.

In some counties, the sheriff's principal deputy is the warden of the county jail or other local correctional institution.

For a de Broglie – Bohm theory on curved space with spin, the spin space becomes a vector bundle over configuration space and the potential in Schrödinger's equation becomes a local self-adjoint operator acting on that space.

Geodesic flow is a local R-action on tangent bundle TM of a manifold M defined in the following way

It serves as a transfer station for passengers on local trains on the Amsterdam-Amersfoort and Schiphol Airport-Almere lines, which bundle up just before Weesp and diverge right after.

Proving that this form is, indeed, symplectic can be done by noting that being symplectic is a local property: since the cotangent bundle is locally trivial, this definition need only be checked on.

A bundle map from the base space itself ( with the identity mapping as projection ) to E is called a section of E. Fiber bundles can be generalized in a number of ways, the most common of which is requiring that the transition between the local trivial patches should lie in a certain topological group, known as the structure group, acting on the fiber F.

A fiber bundle consists of the data ( E, B, π, F ), where E, B, and F are topological spaces and π: E → B is a continuous surjection satisfying a local triviality condition outlined below.

If we want to ignore the physics inside the conductor and only describe the physics in the outside region, it becomes natural to mathematically describe the quantum electron by a section in a complex line bundle with an " external " connection rather than an external EM field ( by incorporating local gauge transformations we have already acknowledged that quantum mechanics defines the notion of a ( locally ) flat wavefunction ( zero momentum density ) but not that of unit wavefunction ).

Included with iDisk in the MobileMe bundle was Backup, a software utility that allowed users to back up local files to an iDisk.

: Choosing local coordinates x < sup > a </ sup >, and writing :, where the are local basis vectors for the tangent bundle, we have locally

This is not obvious from the right-hand side of Kosmann's local expression, as the right-hand side seems to depend on the metric through the spin connection ( covariant derivative ), the dualisation of vector fields ( lowering of the indices ) and the Clifford multiplication on the spinor bundle.

The local version of the cross section theorem then states that the equivariant local trivializations of a principal bundle are in one-to-one correspondence with local sections.

Fibrations do not necessarily have the local Cartesian product structure that defines the more restricted fiber bundle case, but something weaker that still allows " sideways " movement from fiber to fiber.

* Fiber bundles have local trivializations such Cartesian product structures exist locally on B, and this is usually enough to show that a fiber bundle is a fibration.

More precisely, if there are local trivializations over a " numerable open cover " of B, the bundle is a fibration.

The symbol of a differential operator of order n on a smooth manifold X is defined in much the same way using local coordinate charts, and is a function on the cotangent bundle of X, homogeneous of degree n on each cotangent space.

( In general, differential operators transform in a rather complicated way under coordinate transforms ( see jet bundle ); however, the highest order terms transform like tensors so we get well defined homogeneous functions on the cotangent spaces that are independent of the choice of local charts.

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 (