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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 Integration along fibers ： ウィキペディア英語版 Integration along fibers In differential geometry, the integration along fibers of a ''k''-form yields a $(k-m)$-form where ''m'' is the dimension of the fiber, via "integration". More precisely, let $\backslash pi:\; E\; \backslash to\; B$ be a fiber bundle over a manifold with compact oriented fibers. If $\backslash alpha$ is a ''k''-form on ''E'', then let: : $(\backslash pi\_$ * \alpha)_b(w_1, \dots, w_) = \int_ \beta where $\backslash beta$ is the induced top-form on the fiber $\backslash pi^(b)$; i.e., an $m$-form given by :$\backslash beta(v\_1,\; \backslash dots,\; v\_m)\; =\; \backslash alpha(\backslash widetilde,\; \backslash dots,\; \backslash widetilde\backslash text\; w\_i.$ (To see $b\; \backslash mapsto\; (\backslash pi\_$ * \alpha)_b is smooth, work it out in coordinates; cf. an example below.) $\backslash pi\_$ * is then a linear map $\backslash Omega^k(E)\; \backslash to\; \backslash Omega^(B)$, which is in fact surjective. By Stokes' formula, if the fibers have no boundaries, the map descends to de Rham cohomology: :$\backslash pi\_$ *: \operatorname^k(E) \to \operatorname^(B). This is also called the fiber integration. Now, suppose $\backslash pi$ is a sphere bundle; i.e., the typical fiber is a sphere. Then there is an exact sequence $0\; \backslash to\; K\; \backslash to\; \backslash Omega^$ *(E) \overset\to \Omega^ *(B) \to 0, ''K'' the kernel, which leads to a long exact sequence, using $\backslash operatorname^k(B)\; \backslash simeq\; \backslash operatorname^(K)$: :$\backslash dots\; \backslash rightarrow\; \backslash operatorname^k(B)\; \backslash overset\backslash to\; \backslash operatorname^(B)\; \backslash overset\; \backslash rightarrow\; \backslash operatorname^(E)\; \backslash overset\; \backslash rightarrow\; \backslash operatorname^(B)\; \backslash rightarrow\; \backslash dots$, called the Gysin sequence. == Example == Let $\backslash pi:\; M\; \backslash times\; (1)\; \backslash to\; M$ be an obvious projection. For simplicity, assume $M\; =\; \backslash mathbb^n$ with coordinates $x\_j$ and consider a ''k''-form: :$\backslash alpha\; =\; f\; \backslash ,\; dx\_\; \backslash wedge\; \backslash dots\; \backslash wedge\; dx\_\; +\; g\; \backslash ,\; dt\; \backslash wedge\; dx\_\; \backslash wedge\; \backslash dots\; \backslash wedge\; dx\_\; \backslash wedge\; \backslash dots\; \backslash wedge\; dx\_\; \backslash wedge\; \backslash dots\; \backslash wedge\; dx\_.$ From this the next formula follows easily: if $\backslash alpha$ is any ''k''-form on $M\; \backslash times\; I,$ :$\backslash pi\_$ *(d \alpha) = \alpha_1 - \alpha_0 - d \pi_ *(\alpha) where $\backslash alpha\_i$ is the restriction of $\backslash alpha$ to $M\; \backslash times\; \backslash$. This formula is a special case of Stokes' formula. As an application of this, let $f:\; M\; \backslash times\; (1)\; \backslash to\; N$ be a smooth map (thought of as a homotopy). Then the composition $h\; =\; \backslash pi\_$ * \circ f^ * is a homotopy operator: :$d\; \backslash circ\; h\; +\; h\; \backslash circ\; d\; =\; f\_1^$ * - f_0^ *: \Omega^k(N) \to \Omega^k(M), which implies $f\_1,\; f\_0$ induces the same map on cohomology. For example, let ''U'' be an open ball with center at the origin and let $f\_t:\; U\; \backslash to\; U,\; x\; \backslash mapsto\; tx$. Then $\backslash operatorname^k(U)\; =\; \backslash operatorname^k(pt)$, the fact known as the Poincaré lemma. 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.