渐近平直时空
$\quad\\$
共形变换
$\quad\\$
共形变换$\quad$设流形$M$上有度规$g_{ab}$和$\widetilde{g}_{ab}$，若存在函数$\Omega\in C^\infty(M),\Omega>0$使得$\widetilde{g}_{ab}=\Omega^2 g_{ab}$，则称$\widetilde{g}_{ab}$共形于$g_{ab}$，$\Omega$称为此变换的共形因子
由于$\delta_c^{b}=\widetilde{g}_{ca}\widetilde{g}^{ab}=\Omega^2g_{ca}\widetilde{g}^{ab}=g_{ca}g^{ab}$，故$\widetilde{g}^{ab}=\Omega^{-2}g^{ab}$
$$\quad\\$$
符差张量$\quad$以$\nabla_a$和$\widetilde{\nabla}_a$代表分别与$g_{ab}$和$\widetilde{g}_{ab}$适配的导数算符，则有
$$0=\widetilde{\nabla}_a\widetilde{g}_{bc}=\nabla_a\widetilde{g}_{bc}-C^{d}_{ab}\widetilde{g}_{dc}-C^{d}_{ac}\widetilde{g}_{bd}\\\quad\\=\nabla_a\widetilde{g}_{bc}-C_{cab}-C_{bac}$$
轮换指标可得以下三个等式
$$\nabla_a\widetilde{g}_{bc}=C_{cab}+C_{bac}\\
\quad\\
\nabla_b\widetilde{g}_{ca}=C_{abc}+C_{cba}\\
\quad\\
\nabla_c\widetilde{g}_{ab}=C_{bca}+C_{acb}$$
由符差张量两个下指标的对称性可得$C_{cab}=\displaystyle\frac{1}{2}(\nabla_a\widetilde{g}_{bc}+\nabla_b\widetilde{g}_{ca}-\nabla_c\widetilde{g}_{ab})$，即$C^c_{ab}=\displaystyle\frac{1}{2}\widetilde{g}^{dc}(\nabla_a\widetilde{g}_{bd}+\nabla_b\widetilde{g}_{da}-\nabla_d\widetilde{g}_{ab})$，将$\widetilde{g}_{ab}$替换为$\Omega^2g_{ab}$，可得
$\nabla_a\widetilde{g}_{bc}=\nabla_a(\Omega^2g_{bc})=2\Omega g_{bc}\nabla_a\Omega$，代入符差张量表达式可得
$$C^c_{ab}=\Omega^{-1}g^{dc}(g_{bd}\nabla_a\Omega+g_{da}\nabla_b\Omega-g_{ab}\nabla_d\Omega)\\
\quad\\
=2\delta^c \sideset{_{(a}}{_{b)}}\nabla\ln{\Omega}-g_{ab}g^{cd}\nabla_d\ln{\Omega}$$
由共形因子的定义可知，共形变换不改变矢量场的类空、类时和类光性，但是却并不保证测地线不变
$$\quad\\$$
类光测地线共形不变$\quad$设$\gamma(\lambda)$在度规$g_{ab}$下为测地线，即其切矢$T_a=(\partial/\partial \lambda)^a$满足$T^a\nabla_aT^b=0$，则有
$$T^a\widetilde{\nabla}_aT^b=T^a(\nabla_aT^b+C^b_{ac}T^c)\\
\quad\\
=T^aT^c(2\delta^b \sideset{_{(a}}{_{c)}}\nabla\ln{\Omega}-g_{ac}g^{bd}\nabla_d\ln{\Omega})\\
\quad\\
=2T^bT^c\nabla_{c}\ln{\Omega}-(g_{ac}T^aT^c)g^{bd}\nabla_d\ln{\Omega}$$
若$\gamma(t)$为类光测地线，则有$g_{ac}T^aT^c=0$，此时
$$T^a\widetilde{\nabla}_aT^b=2T^bT^c\nabla_{c}\ln{\Omega}=\alpha T^b$$
$\gamma(t)$在度规$\widetilde{g}_{ab}$下成为非仿射参数化的测地线，故可选取参数$\widetilde{\lambda}$使得
$\widetilde{\gamma}(\widetilde{\lambda})[=\gamma(\lambda)]$成为仿射参数化的测地线，由$T^a=\widetilde{T}^a(\dd\widetilde{\lambda}/\dd\lambda)$有
$$\alpha T^b=T^a\widetilde{\nabla}_aT^b=\frac{\dd \widetilde{\lambda}}{\dd\lambda}\widetilde{T}^a\widetilde{\nabla}_a(\frac{\dd \widetilde{\lambda}}{\dd\lambda}\widetilde{T}^b)\\
\quad\\
=\left(\frac{\dd \widetilde{\lambda}}{\dd\lambda}\right)^2\widetilde{T}^a\widetilde{\nabla}_a\widetilde{T}^b+\widetilde{T}^b\frac{\dd \widetilde{\lambda}}{\dd\lambda}(\widetilde{T}^a\nabla_a)\left(\frac{\dd \widetilde{\lambda}}{\dd\lambda}\right)\\
\quad\\
=\left(\frac{\dd \widetilde{\lambda}}{\dd\lambda}\right)^2\widetilde{T}^a\widetilde{\nabla}_a\widetilde{T}^b+\widetilde{T}^b\frac{\dd \widetilde{\lambda}}{\dd\lambda}\frac{\dd}{\dd\widetilde{\lambda}}\frac{\dd \widetilde{\lambda}}{\dd\lambda}\\
\quad\\
=\left(\frac{\dd \widetilde{\lambda}}{\dd\lambda}\right)^2\widetilde{T}^a\widetilde{\nabla}_a\widetilde{T}^b+\frac{\dd ^2\widetilde{\lambda}}{\dd\lambda^2}\widetilde{T}^b$$
为使$\widetilde{T}^a\widetilde{\nabla}_a\widetilde{T}^b=0$，则需$\displaystyle\frac{\dd ^2\widetilde{\lambda}}{\dd\lambda^2}=\alpha\displaystyle\frac{\dd\widetilde{\lambda}}{\dd\lambda}$，$\displaystyle\dd\ln\left(\frac{\dd\widetilde{\lambda}}{\dd\lambda}\right)=\alpha\dd\lambda=\dd\,{\ln{\Omega^2}}$
即$$\frac{\dd\widetilde{\lambda}}{\dd\lambda}=c\Omega^2\qquad c\small为任意常数$$
类光测地线在这个意义下是共形不变的，但在度规$g_{ab}$下的一般测地线不具有形式$T^a\widetilde{\nabla}_aT^b=\alpha T^b$，因此在度规$\widetilde{g}_{ab}$下不是测地线
$$\quad\\$$
黎曼曲率张量$\quad$设$\widetilde{\nabla}_a$的曲率张量为$\widetilde{R}^{d}_{abc}$，$\nabla_a$的曲率张量为$R^{d}_{abc}$，以下讨论两者之间关系
$$\widetilde{\nabla}_a\widetilde{\nabla}_b\omega_c=\nabla_a\widetilde{\nabla}_b\omega_c-C^d_{ab}\widetilde{\nabla}_d\omega_c-C^d_{ac}\widetilde{\nabla}_b\omega_d\\
\quad\\=\nabla_a(\nabla_b\omega_c-C^d_{bc}\omega_d)-C^d_{ab}(\nabla_d\omega_c-C^e_{dc}\omega_e)-C^d_{ac}(\nabla_b\omega_d-C^e_{bd}\omega_e)$$
对指标$a,b$进行反对称化，则有
$$\widetilde{\nabla}\sideset{_{[a}}{_{b]}}{\widetilde{\nabla}}\omega_c-\nabla\sideset{_{[a}}{_{b]}}{\nabla}\omega_c\\
\quad\\
=-\omega_d\nabla\sideset{_{[a}}{_{b]c}^d}C-C^d_{c}\sideset{_{[b}}{_{a]}}\nabla\omega_d-C^d_{c}\sideset{_{[a}}{_{b]}}\nabla\omega_d+C^d_{c}\sideset{_{[a}}{_{b]d}^e}C\omega_e\\
\quad\\
=C^d_{c}\sideset{_{[a}}{_{b]d}^e}C\omega_e-\omega_d\nabla\sideset{_{[a}}{_{b]c}^d}C\\
\quad\\
=\frac{1}{2}[2\delta^d\sideset{_{(c}}{_{a)}}\nabla\ln{\Omega}-g_{ca}g^{df}\nabla_f\ln{\Omega}][2\delta^e\sideset{_{(b}}{_{d)}}\nabla\ln{\Omega}-g_{bd}g^{eg}\nabla_g\ln{\Omega}]\omega_e\\
-\frac{1}{2}\omega_d\nabla_a[2\delta^d\sideset{_{(b}}{_{c)}}\nabla\ln{\Omega}-g_{bc}g^{df}\nabla_f\ln{\Omega}]\\
-\frac{1}{2}[2\delta^d\sideset{_{(c}}{_{b)}}\nabla\ln{\Omega}-g_{cb}g^{df}\nabla_f\ln{\Omega}][2\delta^e\sideset{_{(a}}{_{d)}}\nabla\ln{\Omega}-g_{ad}g^{eg}\nabla_g\ln{\Omega}]\omega_e\\
+\frac{1}{2}\omega_d\nabla_b[2\delta^d\sideset{_{(a}}{_{c)}}\nabla\ln{\Omega}-g_{ac}g^{df}\nabla_f\ln{\Omega}]\\
\quad\\
=\omega_d\delta^d\sideset{_{[a}}{_{b]}}\nabla\nabla_c\ln{\Omega}-\omega_dg^{df}g_c\sideset{_{[a}}{_{b]}}\nabla\nabla_f\ln{\Omega}+(\nabla_{[a}\ln{\Omega})\delta^e_{b]}(\nabla_c\ln{\Omega})\omega_e\\
-(\nabla_{[a}\sideset{}{_{b]c}}{\ln{\Omega})g}g^{eg}(\nabla_g\ln{\Omega})\omega_e-g_{c}\sideset{_{[a\;}}{_{b]}^e}\delta g^{df}(\nabla_d\ln{\Omega})(\nabla_f\ln{\Omega})\omega_e\\
+(\nabla_{[a}\ln{\Omega})g_{b]c}g^{ef}(\nabla_f\ln{\Omega})\omega_e-(\nabla_{[a}\ln{\Omega})g_{b]c}g^{eg}(\nabla_g\ln{\Omega})\omega_e\\
\quad\\
=\omega_d\delta^d\sideset{_{[a}}{_{b]}}\nabla\nabla_c\ln{\Omega}-\omega_dg^{de}g_c\sideset{_{[a}}{_{b]}}\nabla\nabla_e\ln{\Omega}+(\nabla_{[a}\ln{\Omega})\delta^d_{b]}(\nabla_c\ln{\Omega})\omega_d\\
-(\nabla_{[a}\sideset{}{_{b]c}}{\ln{\Omega})g}g^{de}(\nabla_e\ln{\Omega})\omega_d-g_{c}\sideset{_{[a\;}}{_{b]}^d}\delta g^{ef}(\nabla_e\ln{\Omega})(\nabla_f\ln{\Omega})\omega_d$$
由于$\widetilde{\nabla}\sideset{_{[a}}{_{b]}}{\widetilde{\nabla}}\omega_c-\nabla\sideset{_{[a}}{_{b]}}{\nabla}\omega_c=\displaystyle\frac{1}{2}(\widetilde{R}^{d}_{abc}-R^{d}_{abc})\omega_d$，故有
$$\widetilde{R}^{d}_{abc}-R^{d}_{abc}=\\
\quad\\2\delta^d\sideset{_{[a}}{_{b]}}\nabla\nabla_c\ln{\Omega}-2g^{de}g_c\sideset{_{[a}}{_{b]}}\nabla\nabla_e\ln{\Omega}+2(\nabla_{[a}\ln{\Omega})\delta^d_{b]}\nabla_c\ln{\Omega}\\
-2(\nabla_{[a}\sideset{}{_{b]c}}{\ln{\Omega})g}g^{de}\nabla_e\ln{\Omega}-2g_{c}\sideset{_{[a\;}}{_{b]}^d}\delta g^{ef}(\nabla_e\ln{\Omega})\nabla_f\ln{\Omega}$$
$$\quad\\$$
里奇张量$\quad$对黎曼张量表达式中指标$b,d$缩并可得
$$\widetilde{R}_{ac}-R_{ac}\\
\quad\\
=[(1-n)\nabla_a\nabla_c\ln{\Omega}]-[g_{ca}g^{be}\nabla_b\nabla_e\ln{\Omega}-\nabla_a\nabla_c\ln{\Omega}]\\
+[(n-1)(\nabla_a\ln{\Omega})\nabla_c\ln{\Omega}]\\
+[g_{ac}g^{be}(\nabla_b\ln{\Omega})\nabla_e\ln{\Omega}-(\nabla_a\ln{\Omega})\nabla_c\ln{\Omega}]\\
-[(n-1)g_{ca}g^{ef}(\nabla_e\ln{\Omega})\nabla_f\ln{\Omega}]\\
\quad\\
=(2-n)\nabla_a\nabla_c\ln{\Omega}-g_{ac}g^{be}\nabla_b\nabla_e\ln{\Omega}\\
+(n-2)(\nabla_a\ln{\Omega})\nabla_c\ln{\Omega}+(2-n)g_{ac}g^{ef}(\nabla_e\ln{\Omega})\nabla_f\ln{\Omega}$$
$$\quad\\$$
标量曲率$\quad$设$\widetilde{\nabla}_a$的标量曲率为$\widetilde{R}\equiv \widetilde{g}^{ac}\widetilde{R}_{ac}$，$\nabla_a$的标量曲率为$R\equiv g^{ac}R_{ac}$，由里奇张量表达式对度规缩并可得
$$\widetilde{R}=\Omega^{-2}[R-2(n-1)g^{ac}\nabla_a\nabla_c\ln{\Omega}\\
-(n-1)(n-2)g^{ac}(\nabla_a\ln{\Omega})\nabla_c\ln{\Omega}]$$
$$\quad\\$$
外尔张量共形不变$\quad$设$\widetilde{\nabla}_a$的外尔张量为$\widetilde{C}^{d}_{abc}$，$\nabla_a$的外尔张量为$C^{d}_{abc}$，则有
$$\widetilde{C}^{d}_{abc}-C^{d}_{abc}\\
\quad\\
=\widetilde{g}^{de}\widetilde{C}_{abce}-g^{de}C_{abce}\\
\quad\\
=\widetilde{R}^{d}_{abc}-R^{d}_{abc}\\
+\frac{g^{de}}{n-2}[g_{ae}(\widetilde{R}_{cb}-R_{cb})+g_{bc}(\widetilde{R}_{ea}\\-R_{ea})-g_{ac}(\widetilde{R}_{eb}-R_{eb})-g_{be}(\widetilde{R}_{ca}-R_{ca})]\\
+\frac{g^{de}}{(n-1)(n-2)}(\Omega^2\widetilde{R}-R)(g_{ac}g_{eb}-g_{ae}g_{cb})\\
\quad\\
=2\delta^d\sideset{_{[a}}{_{b]}}\nabla\nabla_c\ln{\Omega}-2g^{de}g_c\sideset{_{[a}}{_{b]}}\nabla\nabla_e\ln{\Omega}+2(\nabla_{[a}\ln{\Omega})\delta^d_{b]}\nabla_c\ln{\Omega}\\
-2(\nabla_{[a}\sideset{}{_{b]c}}{\ln{\Omega})g}g^{de}\nabla_e\ln{\Omega}-2g_{c}\sideset{_{[a\;}}{_{b]}^d}\delta g^{ef}(\nabla_e\ln{\Omega})\nabla_f\ln{\Omega}\\
-2\delta^d\sideset{_{[a}}{_{b]}}\nabla\nabla_c\ln{\Omega}+\frac{2}{n-2}g_{c}\sideset{_{[a\;}}{_{b]}^d}\delta g^{ef}\nabla_e\nabla_f\ln{\Omega}\\
-2(\nabla_{[a}\ln{\Omega})\delta^d_{b]}\nabla_c\ln{\Omega}+2g_{c}\sideset{_{[a\;}}{_{b]}^d}\delta g^{ef}(\nabla_e\ln{\Omega})\nabla_f\ln{\Omega}\\+2g^{de}g_c\sideset{_{[a}}{_{b]}}\nabla\nabla_e\ln{\Omega}+\frac{2}{n-2}g_{c}\sideset{_{[a\;}}{_{b]}^d}\delta g^{ef}\nabla_e\nabla_f\ln{\Omega}\\
+2(\nabla_{[a}\sideset{}{_{b]c}}{\ln{\Omega})g}g^{de}\nabla_e\ln{\Omega}+2g_{c}\sideset{_{[a\;}}{_{b]}^d}\delta g^{ef}(\nabla_e\ln{\Omega})\nabla_f\ln{\Omega}\\
-\frac{4}{n-2}g_{c}\sideset{_{[a\;}}{_{b]}^d}\delta g^{ef}\nabla_e\nabla_f\ln{\Omega}-4g_{c}\sideset{_{[a\;}}{_{b]}^d}\delta g^{ef}(\nabla_e\ln{\Omega})\nabla_f\ln{\Omega}=0$$
外尔张量共形不变，故也称之为共形张量
$$\quad\\$$
共形平直$\quad$若度规$g_{ab}$共形于平直度规$\eta_{ab}$，则称$g_{ab}$共形平直，由于平直度规外尔张量为零，故共形平直的必要条件为外尔张量等于零
$$\quad\\$$
Gauss$\quad$2维广义黎曼空间局域共形平直
设$\alpha$为二维广义黎曼空间$(M,g_{ab})$上的谐和函数，$\nabla_a$为与度规$g_{ab}$适配的导数算符，$\varepsilon_{ab}$为与$g_{ab}$适配的体元，则有$\nabla^a\nabla_a\alpha=0$，令$\omega_a=\varepsilon_{ab}\nabla^b\alpha$，由于
$$\dd\boldsymbol{\omega}=2\nabla_{[c}(\varepsilon_{a]b}\nabla^b\alpha)\\
\quad\\=\varepsilon_{ab}\nabla_c\nabla^b\alpha-\varepsilon_{cb}\nabla_a\nabla^b\alpha\\
\quad\\
=\sqrt{|g|}[(e^1)_a(e^2)_b-(e^2)_a(e^1)_b]\nabla_\mu\nabla^\nu\alpha(e^\mu)_c(e_\nu)^b\\
-\sqrt{|g|}[(e^1)_c(e^2)_b-(e^2)_c(e^1)_b]\nabla_\sigma\nabla^\nu\alpha(e^\sigma)_a(e_\nu)^b\\
\quad\\
=\sqrt{|g|}[\nabla_\mu\nabla^{\nu=2}\alpha(e^\mu)_c(e^1)_a-\nabla_\mu\nabla^{\nu=1}\alpha(e^\mu)_c(e^2)_a]\\
-\sqrt{|g|}[\nabla_\sigma\nabla^{\nu=2}\alpha(e^\sigma)_a(e^1)_c-\nabla_\sigma\nabla^{\nu=1}\alpha(e^\sigma)_a(e^2)_c]\\
\quad\\
=\sqrt{|g|}[\nabla_{\mu=1}\nabla^{\nu=2}\alpha(e^1)_c(e^1)_a-\nabla_{\sigma=1}\nabla^{\nu=2}\alpha(e^1)_a(e^1)_c]\\
+\sqrt{|g|}[\nabla_{\sigma=2}\nabla^{\nu=1}\alpha(e^2)_a(e^2)_c-\nabla_{\mu=2}\nabla^{\nu=1}\alpha(e^2)_c(e^2)_a]\\
+\sqrt{|g|}[\nabla_{\mu=2}\nabla^{\nu=2}\alpha(e^2)_c(e^1)_a+\nabla_{\sigma=1}\nabla^{\nu=1}\alpha(e^1)_a(e^2)_c]\\
-\sqrt{|g|}[\nabla_{\mu=1}\nabla^{\nu=1}\alpha(e^1)_c(e^2)_a+\nabla_{\sigma=2}\nabla^{\nu=2}\alpha(e^2)_a(e^1)_c]=0$$
故$\omega_a$为闭，因而局域恰当，即存在局域定义的函数$\beta$使得
$\omega_a=(\dd\beta)_a=\nabla_a\beta$，又
$$\nabla^a\nabla_a\beta=\nabla^a\omega_a=\varepsilon_{ab}\nabla^a\nabla^b\alpha=\varepsilon_{[ab]}\nabla^{(a}\nabla^{b)}\alpha=0$$
即$\beta$也为谐和函数，由于$\nabla_a\alpha$与$\nabla_b\beta$为等$\alpha$线与等$\beta$线的法余矢，其内积
$$g^{ab}(\nabla_a\alpha)\nabla_b\beta\\
\quad\\
=g^{ab}(\nabla_a\alpha)\varepsilon_{bc}\nabla^c\alpha=\varepsilon_{bc}(\nabla^b\alpha)\nabla^c\alpha=\varepsilon_{[bc]}(\nabla^{(b}\alpha)\nabla^{c)}\alpha=0$$
可见等$\alpha$线与等$\beta$线处处正交，选取$\alpha,\beta$为局域坐标，则$g^{ab}$在此系分量
$$g^{12}=g^{21}=0\\
\quad\\
g^{22}=g^{ab}(\dd\beta)_a(\dd\beta)_b\\
\quad\\
=g^{ab}(\varepsilon_{ac}\nabla^c\alpha)(\varepsilon_{bd}\nabla^d\alpha)
=\varepsilon_{ac}\varepsilon^{ad}(\nabla^c\alpha)\nabla_d\alpha\\
\quad\\
=(-1)^s\delta_{c}^{d}(\nabla^c\alpha)\nabla_d\alpha=(-1)^sg^{cd}(\dd\alpha)_c(\dd\alpha)_d=(-1)^sg^{11}$$
令$\Omega^2(\alpha,\beta)=|g^{11}|$，则$g_{ab}$在${\alpha,\beta}$系中的线元为
$$\dd s^2=\pm\Omega^2[\dd\alpha^2+(-1)^S\dd\beta^2]$$
括号内为平直线元，因此$g_{ab}$至少在坐标域$\{\alpha,\beta\}$内共形平直
$$\quad$$
注$\quad$若$\alpha$为常数，则$\{\alpha,\beta\}$不能充当坐标系；若$g_{ab}$为洛伦兹度规且$\nabla^a\alpha$类光，则
$g^{ab}(\nabla_a\beta)\nabla_b\beta=-g^{ab}(\nabla_a\alpha)\nabla_b\alpha=0$，可知存在函数$\nabla^a\beta=\lambda\nabla^a\alpha$，此时
$\{\alpha,\beta\}$也不能充当坐标系.但可证明，$\forall p\in M$，存在$p$的邻域$U$，其上有函数$\alpha$，它在$U$上满足$g^{ab}\nabla_a\nabla_b=0，g^{ab}(\nabla_a\alpha)\nabla_b\alpha\neq 0$.以此$\alpha$为本证明的$\alpha$，则证明严密
$$\quad\\$$
Wyle-Schouten$\quad$
3维广义黎曼空间局域共形平直$\Leftrightarrow$Schouten张量为Codazzi张量
4维以上广义黎曼空间局域共形平直$\Leftrightarrow$Weyl张量为零