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Concept
G-structure on a manifold
Résumé
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.
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).
In the following, let be a topological space, topological groups and a group homomorphism .
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.
Given a map , where is the classifying space for -bundles, a reduction of the structure group is a map and a homotopy .
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.
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