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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 Bismut connection ： ウィキペディア英語版 Bismut connection In mathematics, the Bismut connection $\backslash nabla$ is the unique connection on a complex Hermitian manifold that satisfies the following conditions, # It preserves the metric $\backslash nabla\; g\; =0$ # It preserves the complex structure $\backslash nabla\; J=0$ # The torsion $T(X,Y)$ contracted with the metric, i.e. $T(X,Y,Z)=g(T(X,Y),Z)$, is totally skew-symmetric. Bismut has used this connection when proving a local index formula for the Dolbeault operator on non-Kähler manifolds. Bismut connection has applications in type II and heterotic string theory. The explicit construction goes as follows. Let $\backslash langle-,-\backslash rangle$ denote the pairng of two vectors using the metric that is Hermitian w.r.t the complex structure, i.e. $\backslash langle\; X,JY\backslash rangle=-\backslash langle\; JX,Y\backslash rangle$. Further let $\backslash nabla$ be the Levi-Civita connection. Define first a tensor $T$ such that $T(Z,X,Y)=-\backslash frac12\backslash langle\; Z,(\backslash nabla\_J)Y\backslash rangle$. It is easy to see that this tensor is anti-symmetric in the first and last entry, i.e. the new connection $\backslash nabla+T$ still preserves the metric. In concrete terms, the new connection is given by $\backslash Gamma^\_-\backslash frac12\; J^\_\backslash nabla\_J^\_$ with $\backslash Gamma^\_$ being the Levi-Civita connection. It is also easy to see that the new connection preserves the complex structure. However, the tensor $T$ is not yet totall anti-symmetric, in fact the anti-symmetrization will lead to the Nijenhuis tensor. Denote the anti-symmetrization as $T(Z,X,Y)+\backslash textrmX,Y,Z=T(Z,X,Y)+S(Z,X,Y)$, with $S$ given explicitly as :$S(Z,X,Y)=-\backslash frac12\backslash langle\; X,J(\backslash nabla\_J)Z\backslash rangle-\backslash frac12\backslash langle\; Y,J(\backslash nabla\_J)X\backslash rangle.$ We show that $S$ still preserves the complex structure (that it preserves the metric is easy to see), i.e. $S(Z,X,JY)=-S(JZ,X,Y)$. :$\backslash begin$ S(Z,X,JY)+S(JZ,X,Y)&=-\frac12\langle JX, \big(-(\nabla_J)Z-(J\nabla_ZJ)Y+(J\nabla_YJ)Z+(\nabla_J)Y\big)\rangle\\ &=-\frac12\langle JX, Re\big((1-iJ)()\big)\rangle.\end So if $J$ is integrable, then above term vanishes, and the connection :$\backslash Gamma^\_+T^\_+S^\_.$ gives the Bismut connection. 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「Bismut connection」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.