The connection form arises when applying the exterior connection to a particular frame e. Upon applying the exterior connection to the eα, it is the unique k × k matrix (ωαβ) 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 = Σα eαξα. 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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