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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 transport of structure ： ウィキペディア英語版 transport of structure In mathematics, transport of structure is the definition of a new structure on an object by reference to another object on which a similar structure already exists. Definitions by transport of structure are regarded as canonical. Since mathematical structures are often defined in reference to an underlying space, many examples of transport of structure involve spaces and mappings between them. For example, if ''V'' and ''W'' are vector spaces, and if $\backslash phi\; \backslash colon\; V\; \backslash to\; W$ is an isomorphism, and if $(\backslash cdot,\backslash cdot)$ is an inner product on $W$, then we can define an inner product $(\backslash cdot)$ on ''V'' by :$(v\_2)\; =\; (\backslash phi(v\_1),\; \backslash phi(v\_2))\backslash ;$. Although the equation makes sense even when $\backslash phi$ is not an isomorphism, it only defines an inner product on ''V'' when $\backslash phi$ is, since otherwise it will cause $()$ to be degenerate. The idea is that $\backslash phi$ allows us to consider ''V'' and ''W'' as "the same" vector space, and if we follow this analogy, we can transport an inner product from one to the other. A more involved example comes from differential topology, in which we have the notion of a smooth manifold. If ''M'' is such a manifold, and if ''X'' is any topological space which is homeomorphic to ''M'', we can consider ''X'' as a smooth manifold as well. That is, let $\backslash phi\; \backslash colon\; X\; \backslash to\; M$ be a homeomorphism; we must define coordinate charts on ''X'', which we will do by "pulling back" coordinate charts on ''M'' through $\backslash phi$. Recall that a coordinate chart on $M$ is an open set ''U'' together with an injective map :$c\; \backslash colon\; U\; \backslash to\; \backslash mathbb^n\backslash ;$ for some ''n''; to get such a chart on ''X'', we let :$U\text{'}\; =\; \backslash phi^(U)\backslash ;$ and $c\text{'}\; =\; c\; \backslash circ\; \backslash phi\backslash ;$. Furthermore, it is required that the charts cover ''M'', we must check that the transported charts cover ''X'', which follows immediately from the fact that $\backslash phi$ is a bijection. Finally, since ''M'' is a ''smooth'' manifold, we have that if ''U'' and ''V'', with their maps :$c\; \backslash colon\; U\; \backslash to\; \backslash mathbb^n\backslash ;$ and $d\; \backslash colon\; V\; \backslash to\; \backslash mathbb^n\backslash ;$, are two charts on ''M'', then the composition, the "transition map" :$d\; \backslash circ\; c^\; \backslash colon\; c(U\; \backslash cap\; V)\; \backslash to\; \backslash mathbb^n\backslash ;$ (a self-map of $\backslash mathbb^n$) is smooth. We must check this for our transported charts on ''X''. We have :$\backslash phi^(U)\; \backslash cap\; \backslash phi^(V)\; =\; \backslash phi^(U\; \backslash cap\; V)\backslash ;$, and therefore :$c\text{'}(U\text{'}\; \backslash cap\; V\text{'})\; =\; (c\; \backslash circ\; \backslash phi)(\backslash phi^(U\; \backslash cap\; V))\; =\; c(U\; \backslash cap\; V)\backslash ;$, and :$d\text{'}\; \backslash circ\; (c\text{'})^\; =\; (d\; \backslash circ\; \backslash phi)\; \backslash circ\; (c\; \backslash circ\; \backslash phi)^\; =\; d\; \backslash circ\; (\backslash phi\; \backslash circ\; \backslash phi^)\; \backslash circ\; c^\; =\; d\; \backslash circ\; c^\backslash ;$. Therefore the transition map for $U\text{'}$ and $V\text{'}$ is the same as that for ''U'' and ''V'', hence smooth. Therefore ''X'' is a smooth manifold via transport of structure. Although the second example involved considerably more checking, the principle was the same, and any experienced mathematician would have no difficulty performing the necessary verifications. Therefore when such an operation is indicated, it is invoked merely as "transport of structure" and the details left to the reader, if desired. The second example also illustrates why "transport of structure" is not always desirable. Namely, we can take ''M'' to be the plane, and we can take ''X'' to be an infinite one-sided cone. By "flattening" the cone we achieve a homeomorphism of ''X'' and ''M'', and therefore the structure of a smooth manifold on ''X'', but the cone is not "naturally" a smooth manifold. That is, we can consider ''X'' as a subspace of 3-space, in which context it is not smooth at the cone point. A more surprising example is that of exotic spheres, discovered by Milnor, which states that there are exactly 28 smooth manifolds which are homeomorphic (but by definition ''not'' diffeomorphic) to $S^7$, the 7-dimensional sphere in 8-space. Thus, transport of structure is most productive when there exists a canonical isomorphism between the two objects. 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「transport of structure」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.