共形群

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 · 7.6 等距变换与共形变换

$M$ 上全体共形变换构成一个群，称为共形群(conformal group)，记作 $\mathrm{Conf}(M)$。下面定义两个向量 $X=X^{\mu }{\partial }_{\mu }$、$Y=Y^{\mu }{\partial }_{\mu }\in T_{p}M$ 之间的夹角 $\theta$：

$$\cos \theta =\frac{g_p(X,Y)}{\sqrt{g_p(X,X)g_p(Y,Y)}}=\frac{g_{\mu \nu }{X}^{\mu }{Y}^{\nu }}{\sqrt{g_{\zeta \eta }{X}^{\zeta }{X}^{\eta }{g}_{\kappa \lambda }{Y}^{\kappa }{Y}^{\lambda }}}.\text{\hspace{1em}(7.98)}$$

英文原文

The set of conformal transformations on $M$ is a group, the conformal group denoted by $\mathrm{Conf}(M)$ . Let us define the angle $\theta$ between two vectors $X=X^{\mu }{\partial }_{\mu }$ , $Y=Y^{\mu }{\partial }_{\mu }\in T_{p}M$ by

$$\cos \theta =\frac{g_p(X,Y)}{\sqrt{g_p(X,X)g_p(Y,Y)}}=\frac{g_{\mu \nu }{X}^{\mu }{Y}^{\nu }}{\sqrt{g_{\zeta \eta }{X}^{\zeta }{X}^{\eta }{g}_{\kappa \lambda }{Y}^{\kappa }{Y}^{\lambda }}}.\text{\hspace{1em}(7.98)}$$

出处：7.6 等距变换与共形变换 · node a-ac8d6adb3a87d1460093053a · 源节点 SHA-256 42d0021bb6ac6655…