Editing Connection form (section)

===Connection forms===
The '''connection form''' arises when applying the exterior connection to a particular frame '''e'''.  Upon applying the exterior connection to the ''e''<sub>''α''</sub>, it is the unique ''k'' × ''k'' matrix (''ω''<sub>''α''</sub><sup>''β''</sup>) of [[one-form]]s on ''M'' such that
:<math>D e_\alpha = \sum_{\beta=1}^k e_\beta\otimes\omega^\beta_\alpha.</math>
In terms of the connection form, the exterior connection of any section of ''E'' can now be expressed. For example, suppose that ''ξ'' = Σ<sub>''α''</sub> ''e''<sub>''α''</sub>''ξ''<sup>''α''</sup>.  Then
:<math>D\xi = \sum_{\alpha=1}^k D(e_\alpha\xi^\alpha(\mathbf e)) = \sum_{\alpha=1}^k e_\alpha\otimes d\xi^\alpha(\mathbf e) + \sum_{\alpha=1}^k\sum_{\beta=1}^k e_\beta\otimes\omega^\beta_\alpha \xi^\alpha(\mathbf e).</math>

Taking components on both sides,
:<math>D\xi(\mathbf e) = d\xi(\mathbf e)+\omega \xi(\mathbf e) = (d+\omega)\xi(\mathbf e)</math>
where it is understood that ''d'' and ω refer to the component-wise derivative with respect to the frame '''e''', 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.

====Change of frame====
In order to extend ''ω'' to a suitable global object, it is necessary to examine how it behaves when a different choice of basic sections of ''E'' is chosen.  Write ''ω''<sub>''α''</sub><sup>''β''</sup> = ''ω''<sub>''α''</sub><sup>''β''</sup>('''e''') to indicate the dependence on the choice of '''e'''.

Suppose that '''e'''{{prime}} is a different choice of local basis.  Then there is an invertible ''k'' × ''k'' matrix of functions ''g'' such that
:<math>{\mathbf e}' = {\mathbf e}\, g,\quad \text{i.e., }\,e'_\alpha = \sum_\beta e_\beta g^\beta_\alpha.</math>
Applying the exterior connection to both sides gives the transformation law for ''ω'':
:<math>\omega(\mathbf e\, g) = g^{-1}dg+g^{-1}\omega(\mathbf e)g.</math>
Note in particular that ''ω'' fails to transform in a [[tensor]]ial manner, since the rule for passing from one frame to another involves the derivatives of the transition matrix ''g''.

====Global connection forms====
If {''U''<sub>''p''</sub>} is an open covering of ''M'', and each ''U''<sub>''p''</sub> is equipped with a trivialization '''e'''<sub>''p''</sub> of ''E'', then it is possible to define a global connection form in terms of the patching data between the local connection forms on the overlap regions.  In detail, a '''connection form''' on ''M'' is a system of matrices ''ω''('''e'''<sub>''p''</sub>) of 1-forms defined on each ''U''<sub>''p''</sub> that satisfy the following compatibility condition
:<math>\omega(\mathbf e_q) = (\mathbf e_p^{-1}\mathbf e_q)^{-1}d(\mathbf e_p^{-1}\mathbf e_q)+(\mathbf e_p^{-1}\mathbf e_q)^{-1}\omega(\mathbf e_p)(\mathbf e_p^{-1}\mathbf e_q).</math>
This ''compatibility condition'' ensures in particular that the exterior connection of a section of ''E'', when regarded abstractly as a section of ''E'' ⊗ Ω<sup>1</sup>''M'', does not depend on the choice of basis section used to define the connection.