Connection Form - Structure Groups - Compatible Connections

Compatible Connections

A connection is compatible with the structure of a G-bundle on E provided that the associated parallel transport maps always send one G-frame to another. Formally, along a curve γ, the following must hold locally (that is, for sufficiently small values of t):

for some matrix g_αβ (which may also depend on t). Differentiation at t=0 gives

where the coefficients ω_αβ are in the Lie algebra g of the Lie group G.

With this observation, the connection form ω_αβ defined by

is compatible with the structure if the matrix of one-forms ω_αβ(e) takes its values in g.

The curvature form of a compatible connection is, moreover, a g-valued two-form.