正规坐标系

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 · 7.4 列维-奇维塔联络

其中 $\{e_{\mu }\}$ 是 $p$ 处的坐标基。从 $p$ 出发的任一条测地线都由 $X\in T_{p}M$ 给定。取 $p$ 附近的一点 $q$。连接 $p$ 与 $q$ 的测地线有很多条，但存在唯一一条测地线 $c_q$ 满足 $c_q(1)=q$。设 $X_q\in T_{p}M$ 为该测地线在 $p$ 处的切向量。只要 $q$ 离 $p$ 不太远，$q$ 就唯一地确定 $X_q=X_q^{\mu }{e}_{\mu }\in T_{p}M$，于是 $\phi :q\to X_q^{\mu }$ 便构成 $p$ 邻域内一个良好的坐标系。这个坐标系称为以 $p$ 为基点、以 $\{e_{\mu }\}$ 为基的正规坐标系(normal coordinate system)。显然有 $\phi (p)=0$。我们由 $\mathrm{EXP}:X_q\mapsto q$ 定义映射 $\mathrm{EXP}:T_{p}M\to M$。按定义有

$$\phi \left(\mathrm{EXP}{X}_q^{\mu }{e}_{\mu }\right)=X_q^{\mu }.\text{\hspace{1em}(7.72)}$$

英文原文

where $\{e_{\mu }\}$ is the coordinate basis at $p$ . Any geodesic emanating from $p$ is specified by giving $X\in T_{p}M$ . Take a point $q$ near $p$ . There are many geodesics which connect $p$ and $q$ . However, there exists a unique geodesic $c_q$ such that $c_q(1)=q$ . Let $X_q\in T_{p}M$ be the tangent vector of this geodesic at $p$ . As long as $q$ is not far from $p$ , $q$ uniquely specifies $X_q=X_q^{\mu }{e}_{\mu }\in T_{p}M$ and $\phi :q\to X_q^{\mu }$ serves as a good coordinate system in the neighbourhood of $p$ . This coordinate system is called the normal coordinate system based on $p$ with basis $\{e_{\mu }\}$ . Obviously $\phi (p)=0$ . We define a map $\mathrm{EXP}:T_{p}M\to M$ by $\mathrm{EXP}:X_q\mapsto q$ . By definition, we have

$$\phi \left(\mathrm{EXP}{X}_q^{\mu }{e}_{\mu }\right)=X_q^{\mu }.\text{\hspace{1em}(7.72)}$$

出处：7.4 列维-奇维塔联络 · node a-63dc7c3c70a9ccd2b6c8e888 · 源节点 SHA-256 b85924804c53a118…