# 克里斯托费尔符号

## 定义

$D_lg_{ik}=\frac{\partial g_{ik}}{\partial x^l} - g_{mk}\Gamma^m_{il} - g_{im}\Gamma^m_{kl}=0$

$\Gamma^i_{kl}=\frac{1}{2}g^{im} \left(\frac{\partial g_{mk}}{\partial x^l} + \frac{\partial g_{ml}}{\partial x^k} - \frac{\partial g_{kl}}{\partial x^m} \right)$

$\Gamma^i_{kl}=\frac{1}{2}g^{im} \left( \frac{\partial g_{mk}}{\partial x^l} + \frac{\partial g_{ml}}{\partial x^k} - \frac{\partial g_{kl}}{\partial x^m} + c_{mkl}+c_{mlk} - c_{klm} \right)$

$[e_k,e_l] = {c_{kl}}^m e_m$

## 和无指标符号的关系

XY向量场，其分量为$X^i$$Y^k$。则Y相对于X的共变导数的第k个分量为

$\left(\nabla_X Y\right)^k = X^i D_i Y^k = X^i \left(\frac{\partial Y^k}{\partial x^i} + \Gamma^k_{im} Y^m\right)$.

$\langle X,Y\rangle = g(X,Y) = X^i Y_i = g_{ik}X^i Y^k$.

$\nabla_X Y - \nabla_Y X = [X,Y]$

$\Gamma^i_{jk}=\Gamma^i_{kj}$.

## 关系

$\Gamma^i_{ki}=\frac{1}{2} g^{im}\frac{\partial g_{im}}{\partial x_k}=\frac{1}{2g} \frac{\partial g}{\partial x_k} = \frac{\partial \log \sqrt{|g|}}{\partial x_k}$

$g^{kl}\Gamma^i_{kl}=\frac{-1}{\sqrt{|g|}} \;\frac{\partial\sqrt{|g|}\,g^{ik}} {\partial x^k}.$

$D_l V^m = \frac{\partial V^m}{\partial x^l} + \Gamma^m_{kl} V^k.$

$D_m V^m = \frac{\partial V^m}{\partial x^m} + V^k \frac{\partial \log \sqrt{|g|}}{\partial x^k} = \frac{1}{\sqrt{|g|}} \frac{\partial (V^m\sqrt{|g|})}{\partial x^m}$.

$D_l A^{ik}=\frac{\partial A^{ik}}{\partial x^l} + \Gamma^i_{ml} A^{mk} + \Gamma^k_{ml} A^{im}$.

$D_k A^{ik}= \frac{1}{\sqrt{|g|}} \frac{\partial (A^{ik}\sqrt{|g|})}{\partial x^k}$.

$D^i\phi=g^{ik}\frac{\partial\phi}{\partial x^k}.$

$\Delta \phi=\frac{1}{\sqrt{|g|}} \frac{\partial}{\partial x^i}\left(g^{ik}\sqrt{|g|}\frac{\partial\phi}{\partial x^k}\right)$.

## 黎曼曲率

$R_{iklm}=\frac{1}{2}\left( \frac{\partial^2g_{im}}{\partial x^k \partial x^l} + \frac{\partial^2g_{kl}}{\partial x^i \partial x^m} - \frac{\partial^2g_{il}}{\partial x^k \partial x^m} - \frac{\partial^2g_{km}}{\partial x^i \partial x^l} \right) +g_{np} \left( \Gamma^n_{kl} \Gamma^p_{im} - \Gamma^n_{km} \Gamma^p_{il} \right)$.

$R_{iklm}=R_{lmik}$$R_{iklm}=-R_{kilm}=-R_{ikml}$.

$R_{iklm}+R_{imkl}+R_{ilmk}=0.$
$D_m R^n_{ikl} + D_l R^n_{imk} + D_k R^n_{ilm}=0.$

## Ricci曲率

Ricci张量由下式给出

$R_{ik}=\frac{\partial\Gamma^l_{ik}}{\partial x^l} - \frac{\partial\Gamma^l_{il}}{\partial x^k} + \Gamma^l_{ik} \Gamma^m_{lm} - \Gamma^m_{il}\Gamma^l_{km}.$

$R_{ik}=g^{lm}R_{limk}.$

$R=g^{ik}R_{ik}$.

$D_l R^l_m = \frac{1}{2} \frac{\partial R}{\partial x^m}$.

## 外尔张量

$C_{iklm}=R_{iklm} + \frac{1}{2}\left( - R_{il}g_{km} + R_{im}g_{kl} + R_{kl}g_{im} - R_{km}g_{il} \right) + \frac{1}{6} R \left( g_{il}g_{km} - g_{im}g_{kl} \right)$.

## 坐标变换

$\frac{\partial}{\partial y^i} = \frac{\partial x^k}{\partial y^i}\frac{\partial}{\partial x^k}$

$\overline{\Gamma^k_{ij}} = \frac{\partial x^p}{\partial y^i}\, \frac{\partial x^q}{\partial y^j}\, \Gamma^r_{pq}\, \frac{\partial y^k}{\partial x^r} + \frac{\partial y^k}{\partial x^m}\, \frac{\partial^2 x^m}{\partial y^i \partial y^j}$