里奇转动系数

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 · 7.8 非坐标基

设 $\nabla$ 是 $(M,g)$ 上的一个列维-奇维塔联络，它以度量相容性 ${\nabla }_{X}g=0$ 与挠率为零 ${\Gamma }_{\mu \nu }^{\lambda }-{\Gamma }_{\nu \mu }^{\lambda }=0$ 为特征。看看这两个条件在目前的处理方式中如何表述，是很有意思的。分量 ${\Gamma }_{\mu \nu }^{\lambda }$ 与 ${\Gamma }_{\beta \gamma }^{\alpha }$ 之间由(7.142)相联系。设 $(M,g)$ 为黎曼流形（若 $(M,g)$ 是洛伦兹的，则只需把下文所有的 ${\delta }_{\alpha \beta }$ 都换成 ${\eta }_{\alpha \beta }$）。若用 ${\delta }_{\alpha \delta }{\Gamma }_{\beta \gamma }^{\delta }$ 定义里奇转动系数(Ricci rotation coefficient) ${\Gamma }_{\alpha \beta \gamma }$，则度量相容性可表述为

$$\begin{array}{rl}{\Gamma }_{\alpha \beta \gamma }& ={\delta }_{\alpha \delta }{e}_{\lambda }^{\delta }{e}_{\beta }^{\mu }{\nabla }_{\mu }{e}_{\gamma }^{\lambda }=-{\delta }_{\alpha \delta }{e}_{\gamma }^{\lambda }{e}_{\beta }^{\mu }{\nabla }_{\mu }{e}_{\lambda }^{\delta }\\ & =-{\delta }_{\gamma \delta }{e}_{\lambda }^{\delta }{e}_{\beta }^{\mu }{\nabla }_{\mu }{e}_{\alpha }^{\lambda }=-{\Gamma }_{\gamma \beta \alpha }\end{array}\text{\hspace{1em}(7.154)}$$

英文原文

Let $\nabla$ be a Levi-Civita connection on $(M,g)$ , which is characterized by the metric compatibility ${\nabla }_{X}g=0$ , and the vanishing torsion ${\Gamma }_{\mu \nu }^{\lambda }-{\Gamma }_{\nu \mu }^{\lambda }=0$ . It is interesting to see how these conditions are expressed in the present approach. The components ${\Gamma }_{\mu \nu }^{\lambda }$ and ${\Gamma }_{\beta \gamma }^{\alpha }$ are related to each other by (7.142). Let $(M,g)$ be a Riemannian manifold (if $(M,g)$ is Lorentzian, we simply replace ${\delta }_{\alpha \beta }$ all below by ${\eta }_{\alpha \beta }$ ). If we define the Ricci rotation coefficient ${\Gamma }_{\alpha \beta \gamma }$ by ${\delta }_{\alpha \delta }{\Gamma }_{\beta \gamma }^{\delta }$ the metric compatibility is expressed as

$$\begin{array}{rl}{\Gamma }_{\alpha \beta \gamma }& ={\delta }_{\alpha \delta }{e}_{\lambda }^{\delta }{e}_{\beta }^{\mu }{\nabla }_{\mu }{e}_{\gamma }^{\lambda }=-{\delta }_{\alpha \delta }{e}_{\gamma }^{\lambda }{e}_{\beta }^{\mu }{\nabla }_{\mu }{e}_{\lambda }^{\delta }\\ & =-{\delta }_{\gamma \delta }{e}_{\lambda }^{\delta }{e}_{\beta }^{\mu }{\nabla }_{\mu }{e}_{\alpha }^{\lambda }=-{\Gamma }_{\gamma \beta \alpha }\end{array}\text{\hspace{1em}(7.154)}$$

出处：7.8 非坐标基 · node a-f5e163be1078d0de5a675ba0 · 源节点 SHA-256 311875b04045b756…