Editing G-structure on a manifold (section)

=== Principal bundles ===
Although the theory of [[principal bundle]]s plays an important role in the study of ''G''-structures, the two notions are different.  A ''G''-structure is a principal subbundle of the [[frame bundle#Tangent frame bundle|tangent frame bundle]], but the fact that the ''G''-structure bundle ''consists of tangent frames'' is regarded as part of the data.  For example, consider two Riemannian metrics on '''R'''<sup>''n''</sup>.  The associated O(''n'')-structures are isomorphic if and only if the metrics are isometric.  But, since '''R'''<sup>''n''</sup> is contractible, the underlying O(''n'')-bundles are always going to be isomorphic as principal bundles because the only bundles over contractible spaces are trivial bundles.

This fundamental difference between the two theories can be captured by giving an additional piece of data on the underlying ''G''-bundle of a ''G''-structure: the '''[[solder form]]'''.  The solder form is what ties the underlying principal bundle of the ''G''-structure to the local geometry of the manifold itself by specifying a canonical isomorphism of the tangent bundle of ''M'' to an [[associated bundle|associated vector bundle]].  Although the solder form is not a [[connection form]], it can sometimes be regarded as a precursor to one.

In detail, suppose that ''Q'' is the principal bundle of a ''G''-structure.  If ''Q'' is realized as a reduction of the frame bundle of ''M'', then the solder form is given by the [[pullback (differential geometry)|pullback]] of the [[frame bundle#Solder form|tautological form of the frame bundle]] along the inclusion.  Abstractly, if one regards ''Q'' as a principal bundle independently of its realization as a reduction of the frame bundle, then the solder form consists of a representation &rho; of ''G'' on '''R'''<sup>n</sup> and an isomorphism of bundles &theta; : ''TM'' &rarr; ''Q'' &times;<sub>&rho;</sub> '''R'''<sup>n</sup>.