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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 Weitzenböck identity ： ウィキペディア英語版 Weitzenböck identity In mathematics, in particular in differential geometry, mathematical physics, and representation theory a Weitzenböck identity, named after Roland Weitzenböck, expresses a relationship between two second-order elliptic operators on a manifold with the same leading symbol. (The origins of this terminology seem doubtful, however, as there does not seem to be any evidence that such identities ever appeared in Weitzenböck's work.) Usually Weitzenböck formulae are implemented for ''G''-invariant self-adjoint operators between vector bundles associated to some principal ''G''-bundle, although the precise conditions under which such a formula exists are difficult to formulate. Instead of attempting to be completely general, then, this article presents three examples of Weitzenböck identities: from Riemannian geometry, spin geometry, and complex analysis. ==Riemannian geometry== In Riemannian geometry there are two notions of the Laplacian on differential forms over an oriented compact Riemannian manifold ''M''. The first definition uses the divergence operator δ defined as the formal adjoint of the de Rham operator ''d'': ::$\backslash int\_M\; \backslash langle\; \backslash alpha,\backslash delta\backslash beta\backslash rangle\; :=\; \backslash int\_M\backslash langle\; d\backslash alpha,\backslash beta\backslash rangle$ where α is any ''p''-form and β is any (''p'' + 1)-form, and $\backslash langle\; -,-\backslash rangle$ is the metric induced on the bundle of (''p'' + 1)-forms. The usual form Laplacian is then given by ::$\backslash Delta\; =\; d\backslash delta\; +\backslash delta\; d.$ On the other hand, the Levi-Civita connection supplies a differential operator ::$\backslash nabla:\backslash Omega^pM\backslash rightarrow\; T^$ *M\otimes\Omega^pM where Ωp''M'' is the bundle of ''p''-forms and ''T'' *M is the cotangent bundle of ''M''. The Bochner Laplacian is given by ::$\backslash Delta\text{'}=\backslash nabla^$ *\nabla where $\backslash nabla^$ * is the adjoint of $\backslash nabla$. The Weitzenböck formula then asserts that ::$\backslash Delta\text{'}\; -\; \backslash Delta\; =\; A$ where ''A'' is a linear operator of order zero involving only the curvature. The precise form of ''A'' is given, up to an overall sign depending on curvature conventions, by ::$A=\backslash frac\backslash langle\; R(\backslash theta,\backslash theta)\backslash \#,\backslash \#\backslash rangle\; +\; \backslash operatorname(\backslash theta,\backslash \#)\; \backslash ,$ where : *''R'' is the Riemann curvature tensor, : * Ric is the Ricci tensor, : * $\backslash theta:T^$ *M\otimes\Omega^pM\rightarrow\Omega^M is the map that takes the wedge product of a 1-form and p-form and gives a (p+1)-form, : * $\backslash \#:\backslash Omega^M\backslash rightarrow\; T^$ *M\otimes\Omega^pM is the universal derivation inverse to θ on 1-forms. 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「Weitzenböck identity」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.