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 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. Connections on G-structures Let Q be a G-structure on M. A principal connection on the principal bundle Q induces a connection on any associated vector bundle: in particular on the tangent bundle. A linear connection ∇ on TM arising in this way is said to be compatible with Q. Connections compatible with Q are also called adapted connections. Concretely speaking, adapted connections can be understood in terms of a moving frame. Suppose that Vi is a basis of local sections of TM (i.e., a frame on M) which defines a section of Q. Any connection ∇ determines a system of basis-dependent 1-forms ω via ∇X Vi = ωij(X)Vj where, as a matrix of 1-forms, ω ∈ Ω1(M)⊗gl(n). An adapted connection is one for which ω takes its values in the Lie algebra g of G. Torsion of a G-structure Associated to any G-structure is a notion of torsion, related to the torsion of a connection. Note that a given G-structure may admit many different compatible connections which in turn can have different torsions, but in spite of this it is possible to give an independent notion of torsion of the G-structure as