全纯映射

本书暂定概念：尚未进入正式概念库。下面是本书给出的定义原文。

定义 · 8.2 复流形上的微积分

函数（关于 $z^{\mu }$），则称 $f$ 为全纯映射(holomorphic map)。此定义与所选取的特殊坐标无关。事实上，设 $(U^{\prime },{\phi }^{\prime })$ 是另一个坐标卡，满足 $U\cap U^{\prime }\ne \varnothing$，并设 $z^{\prime \mu }=x^{\prime \lambda }+\mathrm{i}{y}^{\prime \lambda }$ 为其坐标。取一点 $p\in U\cap U^{\prime }$。若 $w^{\nu }=u^{\nu }+\mathrm{i}{v}^{\nu }$ 关于 $z$ 是全纯函数，则

$$\frac{\partial u^{\nu }}{\partial x^{\prime \lambda }}=\frac{\partial u^{\nu }}{\partial x^{\mu }}\frac{\partial x^{\mu }}{\partial x^{\prime \lambda }}+\frac{\partial u^{\nu }}{\partial y^{\mu }}\frac{\partial y^{\mu }}{\partial y^{\prime \lambda }}=\frac{\partial v^{\nu }}{\partial y^{\mu }}\frac{\partial y^{\mu }}{\partial y^{\prime \lambda }}+\frac{\partial v^{\nu }}{\partial x^{\mu }}\frac{\partial x^{\mu }}{\partial y^{\prime \lambda }}=\frac{\partial v^{\nu }}{\partial y^{\prime \lambda }}.$$

英文原文

function of $z^{\mu }$ , $f$ is called a holomorphic map. This definition is independent of the special coordinates chosen. In fact, let $(U^{\prime },{\phi }^{\prime })$ be another chart such that $U\cap U^{\prime }\ne \varnothing$ and $z^{\prime \mu }=x^{\prime \lambda }+\mathrm{i}{y}^{\prime \lambda }$ be the coordinates. Take a point $p\in U\cap U^{\prime }$ . If $w^{\nu }=u^{\nu }+\mathrm{i}{v}^{\nu }$ is a holomorphic function with respect to $z$ , then

$$\frac{\partial u^{\nu }}{\partial x^{\prime \lambda }}=\frac{\partial u^{\nu }}{\partial x^{\mu }}\frac{\partial x^{\mu }}{\partial x^{\prime \lambda }}+\frac{\partial u^{\nu }}{\partial y^{\mu }}\frac{\partial y^{\mu }}{\partial y^{\prime \lambda }}=\frac{\partial v^{\nu }}{\partial y^{\mu }}\frac{\partial y^{\mu }}{\partial y^{\prime \lambda }}+\frac{\partial v^{\nu }}{\partial x^{\mu }}\frac{\partial x^{\mu }}{\partial y^{\prime \lambda }}=\frac{\partial v^{\nu }}{\partial y^{\prime \lambda }}.$$

出处：8.2 复流形上的微积分 · node a-cb2858babb3919a189b66567 · 源节点 SHA-256 2bff81683b6fd46a…