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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Fiber bundle construction theorem ： ウィキペディア英語版 Fiber bundle construction theorem In mathematics, the fiber bundle construction theorem is a theorem which constructs a fiber bundle from a given base space, fiber and a suitable set of transition functions. The theorem also gives conditions under which two such bundles are isomorphic. The theorem is important in the associated bundle construction where one starts with a given bundle and surgically replaces the fiber with a new space while keeping all other data the same. ==Formal statement== Let ''X'' and ''F'' be topological spaces and let ''G'' be a topological group with a continuous left action on ''F''. Given an open cover of ''X'' and a set of continuous functions :$t\_\; :\; U\_i\; \backslash cap\; U\_j\; \backslash to\; G\backslash ,$ defined on each nonempty overlap, such that the ''cocycle condition'' :$t\_(x)\; =\; t\_(x)t\_(x)\; \backslash qquad\; \backslash forall\; x\; \backslash in\; U\_i\; \backslash cap\; U\_j\; \backslash cap\; U\_k$ holds, there exists a fiber bundle ''E'' → ''X'' with fiber ''F'' and structure group ''G'' that is trivializable over with transition functions ''t''''ij''. Let ''E''′ be another fiber bundle with the same base space, fiber, structure group, and trivializing neighborhoods, but transition functions ''t''′''ij''. If the action of ''G'' on ''F'' is faithful, then ''E''′ and ''E'' are isomorphic if and only if there exist functions :$t\_i\; :\; U\_i\; \backslash to\; G\backslash ,$ such that :$t\text{'}\_(x)\; =\; t\_i(x)^t\_(x)t\_j(x)\; \backslash qquad\; \backslash forall\; x\; \backslash in\; U\_i\; \backslash cap\; U\_j.$ Taking ''t''''i'' to be constant functions to the identity in ''G'', we see that two fiber bundles with the same base, fiber, structure group, trivializing neighborhoods, and transition functions are isomorphic. A similar theorem holds in the smooth category, where ''X'' and ''Y'' are smooth manifolds, ''G'' is a Lie group with a smooth left action on ''Y'' and the maps ''t''''ij'' are all smooth. 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「Fiber bundle construction theorem」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.