term reduction is misleading: it suggests that H is a subgroup of G, which is often the case, but need not be (for example for spin structures): it's properly called a lifting. More abstractly, "G-bundles over X" is a functor in G: Given a Lie group homomorphism H → G, one gets a map from H-bundles to G-bundles by inducing (as above). Reduction of the structure group of a G-bundle B is choosing an H-bundle whose image is B. The inducing map from H-bundles to G-bundles is in general neither onto nor one-to-one, so the structure group cannot always be reduced, and when it can, this reduction need not be unique. For example, not every manifold is orientable, and those that are orientable admit exactly two orientations. If H is a closed subgroup of G, then there is a natural one-to-one correspondence between reductions of a G-bundle B to H and global sections of the fiber bundle B/H obtained by quotienting B by the right action of H. Specifically, the fibration B → B/H is a principal H-bundle over B/H. If σ : X → B/H is a section, then the pullback bundle BH = σ−1B is a reduction of B. G-structures Every vector bundle of dimension has a canonical -bundle, the frame bundle. In particular, every smooth manifold has a canonical vector bundle, the tangent bundle. For a Lie group and a group homomorphism , a -structure is a reduction of the structure group of the frame bundle to . Examples The following examples are defined for real vector bundles, particularly the tangent bundle of a smooth manifold. Some -structures are defined terms of others: Given a Riemannian metric on an oriented manifold, a -structure for the 2-fold cover is a spin structure. (Note that the group homomorphism here is not an inclusion.) Principal bundles Although the theory of principal bundles plays an important role in the study of G-structures, the two notions are different. A G-structure is a principal subbundle of the tangent frame bundle, but the fact that the G-structure bundle consists of tangent frames is regarded as part of the data. For example, consider two Riemannian metrics on Rn. The associated O(n)-structures are isomorphic if and only if the metrics are isometric. But, since Rn is contractible, the underlying O(n)-bundles are always going to be isomorphic as principal bundles because the only bundles over contractible spaces