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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 Harmonic morphism ： ウィキペディア英語版 Harmonic morphism In mathematics, a harmonic morphism is a (smooth) map $\backslash phi:(M^m,g)\backslash to\; (N^n,h)$ between Riemannian manifolds that pulls back real-valued harmonic functions on the codomain to harmonic functions on the domain. Harmonic morphisms form a special class of harmonic maps—i.e., those that are horizontally (weakly) conformal.〔(【引用サイトリンク】title=Harmonic Morphisms Between Riemannian Manifolds )〕 In local coordinates, $x$ on $M$ and $y$ on $N$, the harmonicity of $\backslash phi$ is expressed by the non-linear system :$\backslash tau(\backslash phi\; )=\backslash sum\_^m\; g^\backslash left(\backslash frac$ -\sum_^m\hat\Gamma^k_\frac +\sum_^n\Gamma^_\circ\phi \frac \frac \right)=0, where $\backslash phi^\backslash alpha=y\_\backslash alpha\backslash circ\backslash phi$ and $\backslash hat\backslash Gamma,\backslash Gamma$ are the Christoffel symbols on $M$ and $N$, respectively. The horizontal conformality is given by :$\backslash sum\_^mg^(x)\backslash frac(x)\backslash frac(x)=\backslash lambda^2(x)h^(\backslash phi(x)),$ where the conformal factor $\backslash lambda:M\backslash to\backslash mathbb\; R\_0^+$ is a continuous function called the dilation. Harmonic morphisms are therefore solutions to non-linear over-determined systems of partial differential equations, determined by the geometric data of the manifolds involved. For this reason, they are difficult to find and have no general existence theory, not even locally. == Complex Analysis == When the codomain of $\backslash phi:(M,g)\backslash to\; (N^2,h)$ is a surface, the system of partial differential equations that we are dealing with, is invariant under conformal changes of the metric $h$. This means that, at least for local studies, the codomain can be chosen to be the complex plane with its standard flat metric. In this situation a complex-valued function $\backslash phi=u+iv:(M,g)\backslash to\backslash mathbb\; C$ is a harmonic morphisms if and only if :$\backslash Delta\_M(\backslash phi)=\backslash Delta\_M\; (u)+i\backslash Delta\_M\; (v)=0$ and :$g(\backslash nabla\backslash phi,\backslash nabla\backslash phi\; )=\backslash |\backslash nabla\; u\backslash |^2-\backslash |\backslash nabla\; v\backslash |^2+2ig(\backslash nabla\; u,\backslash nabla\; v)=0.$ This means that we look for two real-valued harmonic functions $u,v:(M,g)\backslash to\backslash mathbb\; R$ with gradients $\backslash nabla\; u,\backslash nabla\; v$ that are orthogonal and of the same norm at each point. This shows that complex-valued harmonic morphisms $\backslash phi:(M,g)\backslash to\backslash mathbb\; C$ from Riemannian manifolds generalize holomorphic functions $f:(M,g,J)\backslash to\backslash mathbb\; C$ from Kähler manifolds and possess many of their highly interesting properties. The theory of harmonic morphisms can therefore be seen as a generalization of complex analysis.〔 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「Harmonic morphism」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.