The **connection form** arises when applying the exterior connection to a particular frame **e**. Upon applying the exterior connection to the *e*_{α}, it is the unique *k* × *k* matrix (ω_{α}β) of one-forms on *M* such that

In terms of the connection form, the exterior connection of any section of *E* can now be expressed, for suppose that ξ = Σ_{α} e_{α}ξα. Then

Taking components on both sides,

where it is understood that *d* and ω refer to the exterior derivative and a matrix of 1-forms, respectively, acting on the components of ξ. Conversely, a matrix of 1-forms ω is *a priori* sufficient to completely determine the connection locally on the open set over which the basis of sections **e** is defined.

Read more about Connection Form: Structure Groups, Principal Bundles

### Other articles related to "connection, connection form, form, forms, connection forms":

... Then an Ehresmann

**connection**H on E is said to be a principal (Ehresmann)

**connection**if it is invariant with respect to the G action on E in the sense that for any e∈E and g∈G ... The

**connection form**v of the Ehresmann

**connection**may then be viewed as a 1-

**form**ω on E with values in g defined by ω(X)=ι(v(X)) ... Thus reinterpreted, the

**connection form**ω satisfies the following two properties It transforms equivariantly under the G action for all h∈G, where Rh* is the pullback under the ...

... The approach of Cartan and Weyl, using

**connection**1-

**forms**on the frame bundle of M, gives a third way to understand the Riemannian

**connection**... subspace is defined by a differential 1-

**form**on the orthonormal frame bundle, the

**connection form**... This enabled the curvature properties of the surface to be encoded in differential

**forms**on the frame bundle and formulas involving their exterior derivatives ...

**Connection Form**s Associated To A Principal Connection

... Conversely, a principal G-

**connection**ω in a principal G-bundle P→M gives rise to a collection of

**connection forms**on M ... Then the pullback of ω along e defines a g-valued one-

**form**on M Changing frames by a G-valued function g, one sees that ω(e) transforms in the required manner by using the Leibniz rule, and the ...

### Famous quotes containing the words form and/or connection:

“A novel which survives, which withstands and outlives time, does do something more than merely survive. It does not stand still. It accumulates round itself the understanding of all these persons who bring to it something of their own. It acquires associations, it becomes a *form* of experience in itself, so that two people who meet can often make friends, find an approach to each other, because of this one great common experience they have had ...”

—Elizabeth Bowen (1899–1973)

“Children of the same family, the same blood, with the same first associations and habits, have some means of enjoyment in their power, which no subsequent connections can supply; and it must be by a long and unnatural estrangement, by a divorce which no subsequent *connection* can justify, if such precious remains of the earliest attachments are ever entirely outlived.”

—Jane Austen (1775–1817)