Editing G-structure on a manifold (section)

== Isomorphism of ''G''-structures ==
The set of [[diffeomorphism]]s of ''M'' that preserve a  ''G''-structure is called the ''[[automorphism group]]'' of that structure. For an O(''n'')-structure they are the group of [[isometry|isometries]] of the Riemannian metric and for an SL(''n'','''R''')-structure volume preserving maps.

Let ''P'' be a ''G''-structure on a manifold ''M'', and ''Q'' a ''G''-structure on a manifold ''N''.  Then an '''isomorphism''' of the ''G''-structures is a diffeomorphism ''f'' : ''M'' &rarr; ''N'' such that the [[pushforward (differential)|pushforward]] of linear frames ''f''<sub>*</sub> : ''FM'' &rarr; ''FN'' restricts to give a mapping of ''P'' into ''Q''.  (Note that it is sufficient that ''Q'' be contained within the image of ''f''<sub>*</sub>.)  The ''G''-structures ''P'' and ''Q'' are '''locally isomorphic''' if ''M'' admits a covering by open sets ''U'' and a family of diffeomorphisms ''f''<sub>U</sub> : ''U'' &rarr; ''f''(''U'') &sub; ''N'' such that ''f''<sub>U</sub> induces an isomorphism of ''P''|<sub>U</sub> &rarr; ''Q''|<sub>''f''(''U'')</sub>.

An '''automorphism''' of a ''G''-structure is an isomorphism of a ''G''-structure ''P'' with itself.  Automorphisms arise frequently<ref>{{harvnb|Kobayashi|1972}}</ref> in the study of [[transformation group]]s of geometric structures, since many of the important geometric structures on a manifold can be realized as ''G''-structures.

A wide class of [[Cartan's equivalence method|equivalence problems]] can be formulated in the language of ''G''-structures.  For example, a pair of Riemannian manifolds are (locally) equivalent if and only if their bundles of [[orthonormal frame]]s are (locally) isomorphic ''G''-structures.  In this view, the general procedure for solving an equivalence problem is to construct a system of invariants for the ''G''-structure which are then sufficient to determine whether a pair of ''G''-structures are locally isomorphic or not.