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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 Musical isomorphism ： ウィキペディア英語版 Musical isomorphism In mathematics, the musical isomorphism (or canonical isomorphism) is an isomorphism between the tangent bundle and the cotangent bundle of a Riemannian manifold given by its metric. There are similar isomorphisms on symplectic manifolds. The term ''musical'' refers to the use of the symbols ♭ and ♯.〔http://mathoverflow.net/questions/69074/the-origin-of-the-musical-isomorphisms〕 It is also known as raising and lowering indices. ==Discussion== Let be a Riemannian manifold. Suppose is a local frame for the tangent bundle with dual coframe Then, locally, we may express the Riemannian metric (which is a -covariant tensor field which is symmetric and positive-definite) as (where we employ the Einstein summation convention). Given a vector field we define its flat by :$X^\backslash flat\; :=\; g\_\; X^i\; \backslash ,\; dx^j=X\_j\; \backslash ,\; dx^j.$ This is referred to as "lowering an index". Using the traditional diamond bracket notation for inner product defined by , we obtain the somewhat more transparent relation :$X^\backslash flat\; (Y)\; =\; \backslash langle\; X,\; Y\; \backslash rangle$ for all vectors and . Alternatively, given a covector field we define its sharp by :$\backslash omega^\backslash sharp\; :=g^\; \backslash omega\_i\; \backslash partial\_j\; =\; \backslash omega^j\; \backslash partial\_j$ where are the elements of the inverse matrix to . Taking the sharp of a covector field is referred to as "raising an index". In inner product notation, this reads :$\backslash left\; \backslash langle\; \backslash omega^\backslash sharp,\; Y\; \backslash right\; \backslash rangle\; =\; \backslash omega(Y),$ for an arbitrary covector and an arbitrary vector. Through this construction we have two inverse isomorphisms :$\backslash flat:TM\; \backslash to\; T^$ *M, \qquad \sharp:T^ *M \to TM. These are isomorphisms of vector bundles and hence we have, for each in , inverse vector space isomorphisms between and . The musical isomorphisms may also be extended to the bundles :$\backslash bigotimes\; ^k\; TM,\; \backslash qquad\; \backslash bigotimes\; ^k\; T^$ *M. It must be stated which index is to be raised or lowered. For instance, consider the tensor field . Raising the second index, we get the tensor field :$X^\backslash sharp\; =\; g^X\_\; \backslash ,\; dx^i\; \backslash otimes\; \backslash partial\; \_k.$ 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「Musical isomorphism」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.