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

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