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 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 of the 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 ρ of G on Rn and an isomorphism of bundles θ : TM → Q ×ρ Rn. Integrability conditions and flat G-structures Several structures on manifolds, such as a complex structure, a symplectic structure, or a Kähler structure, are G-structures (and thus can be obstructed), but need to satisfy an additional integrability condition. Without the corresponding integrability condition, the structure is instead called an "almost" structure, as in an almost complex structure, an almost symplectic structure, or an almost Kähler structure. Specifically, a symplectic manifold structure is a stronger concept than a G-structure for the symplectic group. A symplectic structure on a manifold is a 2-form ω on M that is non-degenerate (which is an -structure, or almost symplectic structure), together with the extra condition that dω = 0; this latter is called an integrability condition. Similarly, foliations correspond to G-structures coming from block matrices, together with integrability conditions so that the Frobenius theorem applies. 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. 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