Editing Connection form (section)

===Change of frame===
Under a change of frame
:<math>e_\alpha' = \sum_\beta e_\beta g_\alpha^\beta</math>
where ''g'' is a ''G''-valued function defined on an open subset of ''M'', the connection form transforms via <!--Todo: incorporate index version above as well. -->
:<math>\omega_\alpha^\beta(\mathbf e\cdot g) = (g^{-1})_\gamma^\beta dg_\alpha^\gamma + (g^{-1})_\gamma^\beta \omega_\delta^\gamma(\mathbf e)g_\alpha^\delta.</math>
Or, using matrix products:
:<math>\omega({\mathbf e}\cdot g) = g^{-1}dg + g^{-1}\omega g.</math>
To interpret each of these terms, recall that ''g'' : ''M'' → ''G'' is a ''G''-valued (locally defined) function.  With this in mind,
:<math>\omega({\mathbf e}\cdot g) = g^*\omega_{\mathfrak g} + \text{Ad}_{g^{-1}}\omega(\mathbf e)</math>
where ω<sub>'''g'''</sub> is the [[Maurer-Cartan form]] for the group ''G'', here [[pullback (differential geometry)|pulled back]] to ''M'' along the function ''g'', and Ad is the [[adjoint representation]] of ''G'' on its Lie algebra.