Connection (vector Bundle) - Local Expression

Local Expression

Let E → M be a vector bundle of rank k, and let U be an open subset of M over which E is trivial. Given a local smooth frame (e₁, …,e_k) of E over U, any section σ of E can be written as (Einstein notation assumed). A connection on E restricted to U then takes the form

where

Here ωα_β defines a k × k matrix of one-forms on U. In fact, given any such matrix the above expression defines a connection on E restricted to U. This is because ωα_β determines a one-form ω with values in End(E) and this expression defines ∇ to be the connection d+ω, where d is the trivial connection on E over U defined by differentiating the components of a section using the local frame. In this context ω is sometimes called the connection form of ∇ with respect to the local frame.

If U is a coordinate neighborhood with coordinates (xi) then we can write

Note the mixture of coordinate and fiber indices in this expression. The coefficient functions ω_iα_β are tensorial in the index i (they define a one-form) but not in the indices α and β. The transformation law for the fiber indices is more complicated. Let (f₁, …,f_k) be another smooth local frame over U and let the change of coordinate matrix be denoted t (i.e. f_α = e_βtβ_α). The connection matrix with respect to frame (f_α) is then given by the matrix expression

Here dt is the matrix of one-forms obtained by taking the exterior derivative of the components of t.

The covariant derivative in the local coordinates and with respect to the local frame field (e_α) is given by the expression