Editing Connection form (section)

===Compatible connections===
A connection is [[metric compatible|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''):
:<math>\Gamma(\gamma)_0^t e_\alpha(\gamma(0)) = \sum_\beta e_\beta(\gamma(t))g_\alpha^\beta(t) </math>
for some matrix ''g''<sub>α</sub><sup>β</sup> (which may also depend on ''t'').  Differentiation at ''t''=0 gives
:<math>\nabla_{\dot{\gamma}(0)} e_\alpha = \sum_\beta e_\beta \omega_\alpha^\beta(\dot{\gamma}(0))</math>
where the coefficients ω<sub>α</sub><sup>β</sup> are in the [[Lie algebra]] '''g''' of the Lie group ''G''.

With this observation, the connection form ω<sub>α</sub><sup>β</sup> defined by
:<math>D e_\alpha = \sum_\beta e_\beta\otimes \omega_\alpha^\beta(\mathbf e)</math>
is '''compatible with the structure''' if the matrix of one-forms ω<sub>α</sub><sup>β</sup>('''e''') takes its values in '''g'''.

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