Isomorphism of G-structures
The set of diffeomorphisms 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 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 → N such that the pushforward of linear frames f* : FM → FN restricts to give a mapping of P into Q. (Note that it is sufficient that Q be contained within the image of f*.) The G-structures P and Q are locally isomorphic if M admits a covering by open sets U and a family of diffeomorphisms fU : U → f(U) ⊂ N such that fU induces an isomorphism of P|U → Q|f(U).
An automorphism of a G-structure is an isomorphism of a G-structure P with itself. Automorphisms arise frequently in the study of transformation groups of geometric structures, since many of the important geometric structures on a manifold can be realized as G-structures.
A flat G-structure is a G-structure P having a global section (V1,...,Vn) consisting of commuting vector fields. A G-structure is integrable (or locally flat) if it is locally isomorphic to a flat G-structure.
A wide class of 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 frames 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.
Read more about this topic: G-structure