Editing Connection form (section)

==Principal bundles==
The connection form, as introduced thus far, depends on a particular choice of frame.  In the first definition, the frame is just a local basis of sections.  To each frame, a connection form is given with a transformation law for passing from one frame to another.  In the second definition, the frames themselves carry some additional structure provided by a Lie group, and changes of frame are constrained to those that take their values in it.  The language of principal bundles, pioneered by [[Charles Ehresmann]] in the 1940s, provides a manner of organizing these many connection forms and the transformation laws connecting them into a single intrinsic form with a single rule for transformation.  The disadvantage to this approach is that the forms are no longer defined on the manifold itself, but rather on a larger principal bundle.

===The principal connection for a connection form===
Suppose that ''E'' → ''M'' is a vector bundle with structure group ''G''.  Let {''U''} be an open cover of ''M'', along with ''G''-frames on each ''U'', denoted by '''e'''<sub>U</sub>.  These are related on the intersections of overlapping open sets by
:<math>{\mathbf e}_V={\mathbf e}_U\cdot h_{UV}</math>
for some ''G''-valued function ''h''<sub>UV</sub> defined on ''U'' ∩ ''V''.

Let F<sub>G</sub>''E'' be the set of all ''G''-frames taken over each point of ''M''.  This is a principal ''G''-bundle over ''M''.  In detail, using the fact that the ''G''-frames are all ''G''-related, F<sub>G</sub>''E'' can be realized in terms of gluing data among the sets of the open cover:
:<math>F_GE = \left.\coprod_U U\times G\right/\sim</math>
where the [[equivalence relation]] <math>\sim</math> is defined by
:<math>((x,g_U)\in U\times G) \sim ((x,g_V) \in V\times G) \iff {\mathbf e}_V={\mathbf e}_U\cdot h_{UV} \text{ and } g_U = h_{UV}^{-1}(x) g_V. </math>

On F<sub>G</sub>''E'', define a [[connection (principal bundle)|principal ''G''-connection]] as follows, by specifying a '''g'''-valued one-form on each product ''U'' × ''G'', which respects the equivalence relation on the overlap regions.  First let
:<math>\pi_1:U\times G \to U,\quad \pi_2 : U\times G \to G</math>
be the projection maps.  Now, for a point (''x'',''g'') ∈ ''U'' × ''G'', set
:<math>\omega_{(x,g)} = Ad_{g^{-1}}\pi_1^*\omega(\mathbf e_U)+\pi_2^*\omega_{\mathbf g}.</math>
The 1-form ω constructed in this way respects the transitions between overlapping sets, and therefore descends to give a globally defined 1-form on the principal bundle F<sub>G</sub>''E''.  It can be shown that ω is a principal connection in the sense that it reproduces the generators of the right ''G'' action on F<sub>G</sub>''E'', and equivariantly intertwines the right action on T(F<sub>G</sub>''E'') with the adjoint representation of ''G''.

===Connection forms associated to a principal connection===
Conversely, a principal ''G''-connection ω in a principal ''G''-bundle ''P''→''M'' gives rise to a collection of connection forms on ''M''.  Suppose that '''e''' : ''M'' → ''P'' is a local section of ''P''.  Then the pullback of ω along '''e''' defines a '''g'''-valued one-form on ''M'':
:<math>\omega({\mathbf e}) = {\mathbf e}^*\omega.</math>
Changing frames by a ''G''-valued function ''g'', one sees that ω('''e''') transforms in the required manner by using the Leibniz rule, and the adjunction:
:<math>\langle X, ({\mathbf e}\cdot g)^*\omega\rangle = \langle [d(\mathbf e\cdot g)](X), \omega\rangle</math>
where ''X'' is a vector on ''M'', and ''d'' denotes the [[pushforward (differential)|pushforward]].