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In differential geometry, a ''G''-structure on an ''n''-manifold ''M'', for a given structure group〔Which is a Lie group mapping to the general linear group . This is often but not always a Lie subgroup; for instance, for a spin structure the map is a covering space onto its image.〕 ''G'', is a ''G''-subbundle of the tangent frame bundle F''M'' (or GL(''M'')) of ''M''. The notion of ''G''-structures includes many other structures on manifolds, some of them being defined by 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 -structure consists of an absolute parallelism of the manifold. Several structures on manifolds, such as a complex structure, a symplectic structure, or a Kähler structure, are ''G''-structures with an additional integrability condition. == Principal bundles and ''G''-structures == 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 R''n''. The associated O(''n'')-structures are isomorphic if and only if the metrics are isometric. But, since R''n'' is contractible, the underlying O(''n'')-bundles are always going to be isomorphic as principal bundles because the only bundles over contractible spaces 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「G-structure」の詳細全文を読む スポンサード リンク
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