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In differential geometry, a G-structure on an n-manifold M, for a given structure group G, is a 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.

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  • G-structure on a manifold
  • G-结构
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  • 在微分几何中,对一个给定的结构群 G,n 维流形 M 上一个 G-结构是 M 的切标架丛 FM(或 GL(M))的一个 G-。 G-结构的概念包括了许多流形上其它结构,其中一些是用张量场定义的。例如,对正交群,一个 O(n)-结构定义了一个黎曼度量;而对特殊线性群,一个 SL(n,R)-结构就是一个体积形式;对平凡群,一个 {e}-结构由流形的一个绝对平行化组成。 一些流形上的结构,比如複结构,,或 凯勒结构,都是 G-结构带上附加的。 物理学中的术语是规范群。
  • In differential geometry, a G-structure on an n-manifold M, for a given structure group G, is a 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.
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  • In differential geometry, a G-structure on an n-manifold M, for a given structure group G, is a 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.
  • 在微分几何中,对一个给定的结构群 G,n 维流形 M 上一个 G-结构是 M 的切标架丛 FM(或 GL(M))的一个 G-。 G-结构的概念包括了许多流形上其它结构,其中一些是用张量场定义的。例如,对正交群,一个 O(n)-结构定义了一个黎曼度量;而对特殊线性群,一个 SL(n,R)-结构就是一个体积形式;对平凡群,一个 {e}-结构由流形的一个绝对平行化组成。 一些流形上的结构,比如複结构,,或 凯勒结构,都是 G-结构带上附加的。 物理学中的术语是规范群。
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