follows. The difference of two adapted connections is a 1-form on M with values in the adjoint bundle AdQ. That is to say, the space AQ of adapted connections is an affine space for Ω1(AdQ). The torsion of an adapted connection defines a map to 2-forms with coefficients in TM. This map is linear; its linearization is called the algebraic torsion map. Given two adapted connections ∇ and ∇′, their torsion tensors T∇, T∇′ differ by τ(∇−∇′). Therefore, the image of T∇ in coker(τ) is independent from the choice of ∇. The image of T∇ in coker(τ) for any adapted connection ∇ is called the torsion of the G-structure. A G-structure is said to be torsion-free if its torsion vanishes. This happens precisely when Q admits a torsion-free adapted connection. Example: Torsion for almost complex structures An example of a G-structure is an almost complex structure, that is, a reduction of a structure group of an even-dimensional manifold to GL(n,C). Such a reduction is uniquely determined by a C∞-linear endomorphism J ∈ End(TM) such that J2 = −1. In this situation, the torsion can be computed explicitly as follows. An easy dimension count shows that , where Ω2,0(TM) is a space of forms B ∈ Ω2(TM) which satisfy Therefore, the torsion of an almost complex structure can be considered as an element in Ω2,0(TM). It is easy to check that the torsion of an almost complex structure is equal to its Nijenhuis tensor. Higher order G-structures Imposing integrability conditions on a particular G-structure (for instance, with the case of a symplectic form) can be dealt with via the process of prolongation. In such cases, the prolonged G-structure cannot be identified with a G-subbundle of the bundle of linear frames. In many cases, however, the prolongation is a principal bundle in its own right, and its structure group can be identified with a subgroup of a higher-order jet group. In which case, it is called a higher order G-structure [Kobayashi]. In general, Cartan's equivalence method applies to such cases. See also G2-structure Notes References Differential geometry Structures on manifolds