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Harmonic morphism : ウィキペディア英語版
Harmonic morphism
In mathematics, a harmonic morphism is a (smooth) map \phi:(M^m,g)\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 \phi is expressed by the non-linear system
:\tau(\phi )=\sum_^m g^\left(\frac
-\sum_^m\hat\Gamma^k_\frac
+\sum_^n\Gamma^_\circ\phi
\frac
\frac \right)=0,
where \phi^\alpha=y_\alpha\circ\phi and \hat\Gamma,\Gamma are the Christoffel symbols on M and N, respectively. The horizontal conformality is given by
:\sum_^mg^(x)\frac(x)\frac(x)=\lambda^2(x)h^(\phi(x)),
where the conformal factor \lambda:M\to\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 \phi:(M,g)\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 \phi=u+iv:(M,g)\to\mathbb C is a harmonic morphisms if and only if
:\Delta_M(\phi)=\Delta_M (u)+i\Delta_M (v)=0
and
:g(\nabla\phi,\nabla\phi )=\|\nabla u\|^2-\|\nabla v\|^2+2ig(\nabla u,\nabla v)=0.
This means that we look for two real-valued harmonic functions u,v:(M,g)\to\mathbb R with gradients \nabla u,\nabla v that are orthogonal and of the same norm at each point. This shows that complex-valued harmonic morphisms \phi:(M,g)\to\mathbb C from Riemannian manifolds generalize holomorphic functions f:(M,g,J)\to\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)
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