In differential geometry, a G-structure on an n-manifold M, for a given structure group G, is a principal G-subbundle of the tangent frame bundle FM (or GL(M)) of M. The notion of G-structures includes various classical structures that can be defined on manifolds, which in some cases are tensor fields. For example, for the orthogonal group, an O(n)-structure defines a Riemannian metric, and for the special linear group an SL(n,R)-structure is the same as a volume form. For the trivial group, an {e}-structure consists of an absolute parallelism of the manifold. Generalising this idea to arbitrary principal bundles on topological spaces, one can ask if a principal -bundle over a group "comes from" a subgroup of . This is called reduction of the structure group (to ). Several structures on manifolds, such as a complex structure, a symplectic structure, or a Kähler structure, are G-structures with an additional integrability condition. Reduction of the structure group One can ask if a principal -bundle over a group "comes from" a subgroup of . This is called reduction of the structure group (to ), and makes sense for any map , which need not be an inclusion map (despite the terminology). Definition In the following, let be a topological space, topological groups and a group homomorphism . In terms of concrete bundles Given a principal -bundle over , a reduction of the structure group (from to ) is a -bundle and an isomorphism of the associated bundle to the original bundle. In terms of classifying spaces Given a map , where is the classifying space for -bundles, a reduction of the structure group is a map and a homotopy . Properties and examples Reductions of the structure group do not always exist. If they exist, they are usually not essentially unique, since the isomorphism is an important part of the data. As a concrete example, every even-dimensional real vector space is isomorphic to the underlying real space of a complex vector space: it admits a linear complex structure. A real vector bundle admits an almost complex structure if and only if it is isomorphic to the underlying real bundle of a complex vector bundle. This is then a reduction along the inclusion GL(n,C) → GL(2n,R) In terms of transition maps, a G-bundle can be reduced if and only if the transition maps can be taken to have values in H. Note that the