Editing Connection form (section)

==Structure groups==
A more specific type of connection form can be constructed when the vector bundle ''E'' carries a [[associated bundle|structure group]].  This amounts to a preferred class of frames '''e''' on ''E'', which are related by a [[Lie group]] ''G''.  For example, in the presence of a [[metric (vector bundle)|metric]] in ''E'', one works with frames that form an [[orthonormal basis]] at each point.  The structure group is then the [[orthogonal group]], since this group preserves the orthonormality of frames.  Other examples include:
* The usual frames, considered in the preceding section, have structural group GL(''k'') where ''k'' is the fibre dimension of ''E''.
* The holomorphic tangent bundle of a [[complex manifold]] (or [[almost complex manifold]]).<ref name=Wells>Wells (1973).</ref>  Here the structure group is GL<sub>n</sub>('''C''') ⊂ GL<sub>2n</sub>('''R''').<ref>See for instance Kobayashi and Nomizu, Volume II.</ref>  In case a [[hermitian metric]] is given, then the structure group reduces to the [[unitary group]] acting on unitary frames.<ref name=Wells/>
* [[Spinor]]s on a manifold equipped with a [[spin structure]].  The frames are unitary with respect to an invariant inner product on the spin space, and the group reduces to the [[spin group]].
* Holomorphic tangent bundles on [[CR manifold]]s.<ref>See Chern and Moser.</ref>

In general, let ''E'' be a given vector bundle of fibre dimension ''k'' and ''G'' ⊂ GL(''k'') a given Lie subgroup of the general linear group of '''R'''<sup>k</sup>.  If (''e''<sub>α</sub>) is a local frame of ''E'', then a matrix-valued function (''g''<sub>i</sub><sup>j</sup>): ''M'' → ''G'' may act on the ''e''<sub>α</sub> to produce a new frame
:<math>e_\alpha' = \sum_\beta e_\beta g_\alpha^\beta.</math>
Two such frames are '''''G''-related'''.  Informally, the vector bundle ''E'' has the '''structure of a ''G''-bundle''' if a preferred class of frames is specified, all of which are locally ''G''-related to each other.  In formal terms, ''E'' is a [[fibre bundle]] with structure group ''G'' whose typical fibre is '''R'''<sup>k</sup> with the natural action of ''G'' as a subgroup of GL(''k'').

===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.

===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.