非齐次坐标

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 · 8.1 复流形

：若存在复数 $a\ne 0$ 使得 $w=az$，则称 $z\sim w$。于是 ${\mathbb{CP}}^n\equiv ({\mathbb{C}}^{n+1}-\{0\})/\sim$。这 $(n+1)$ 个数 $z^0,z^1,\dots ,z^n$ 称为齐次坐标(homogeneous coordinate)，记作 $[z^0,z^1,\dots ,z^n]$，其中 $(z^0,\dots ,z^n)$ 与 $(\lambda z^0,\dots ,\lambda z^n)(\lambda \ne 0)$ 视为等同。一个坐标卡 $U_{\mu }$ 是 ${\mathbb{C}}^{n+1}-\{0\}$ 的子集，满足 $z^{\mu }\ne 0$。在坐标卡 $U_{\mu }$ 中，非齐次坐标(inhomogeneous coordinate)定义为 ${\xi }_{(\mu )}^{\nu }=z^{\nu }/z^{\mu }$（$\nu \ne \mu$）。在 $U_{\mu }\cap U_{\nu }\ne \varnothing$ 中，坐标变换 ${\psi }_{\mu \nu }:{\mathbb{C}}^n\to {\mathbb{C}}^n$ 为

$${\xi }_{(\nu )}^{\lambda }\mapsto {\xi }_{(\mu )}^{\lambda }=\frac{z^{\nu }}{z\mu }{\xi }_{(\nu )}^{\lambda }.\text{\hspace{1em}(8.6)}$$

英文原文

by $z\sim w$ if there exists a complex number $a\ne 0$ such that $w=az$ . Then ${\mathbb{CP}}^n\equiv ({\mathbb{C}}^{n+1}-\{0\})/\sim$ . The $(n+1)$ numbers $z^0,z^1,\dots ,z^n$ are called the homogeneous coordinates, which is denoted by $[z^0,z^1,\dots ,z^n]$ where $(z^0,\dots ,z^n)$ is identified with $(\lambda z^0,\dots ,\lambda z^n)(\lambda \ne 0)$ . A chart $U_{\mu }$ is a subset of ${\mathbb{C}}^{n+1}-\{0\}$ such that $z^{\mu }\ne 0$ . In a chart $U_{\mu }$ , the inhomogeneous coordinates are defined by ${\xi }_{(\mu )}^{\nu }=z^{\nu }/z^{\mu }$ ( $\nu \ne \mu$ ). In $U_{\mu }\cap U_{\nu }\ne \varnothing$ , the coordinate transformation ${\psi }_{\mu \nu }:{\mathbb{C}}^n\to {\mathbb{C}}^n$ is

$${\xi }_{(\nu )}^{\lambda }\mapsto {\xi }_{(\mu )}^{\lambda }=\frac{z^{\nu }}{z\mu }{\xi }_{(\nu )}^{\lambda }.\text{\hspace{1em}(8.6)}$$

出处：8.1 复流形 · node a-dd1a54778d165ad36fa3e5f6 · 源节点 SHA-256 b3bdd0ce9a2b73b1…