Connection Form

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_αβ) 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_αξα. 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.