# 克里斯托费尔符号

## 定义

${\displaystyle D_{l}g_{ik}={\frac {\partial g_{ik}}{\partial x^{l}}}-g_{mk}\Gamma _{il}^{m}-g_{im}\Gamma _{kl}^{m}=0}$

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

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

${\displaystyle [e_{k},e_{l}]={c_{kl}}^{m}e_{m}}$

## 和无指标符号的关系

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

${\displaystyle \left(\nabla _{X}Y\right)^{k}=X^{i}D_{i}Y^{k}=X^{i}\left({\frac {\partial Y^{k}}{\partial x^{i}}}+\Gamma _{im}^{k}Y^{m}\right)}$.

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

${\displaystyle \nabla _{X}Y-\nabla _{Y}X=[X,Y]}$

${\displaystyle \Gamma _{jk}^{i}=\Gamma _{kj}^{i}}$.

## 关系

${\displaystyle \Gamma _{ki}^{i}={\frac {1}{2}}g^{im}{\frac {\partial g_{im}}{\partial x_{k}}}={\frac {1}{2g}}{\frac {\partial g}{\partial x_{k}}}={\frac {\partial \ln {\sqrt {|g|}}}{\partial x_{k}}}}$

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

${\displaystyle D_{l}V^{m}={\frac {\partial V^{m}}{\partial x^{l}}}+\Gamma _{kl}^{m}V^{k}.}$

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

${\displaystyle D_{l}A^{ik}={\frac {\partial A^{ik}}{\partial x^{l}}}+\Gamma _{ml}^{i}A^{mk}+\Gamma _{ml}^{k}A^{im}}$.

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

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

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

## 黎曼曲率

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

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

${\displaystyle R_{iklm}+R_{imkl}+R_{ilmk}=0.}$
${\displaystyle D_{m}R_{ikl}^{n}+D_{l}R_{imk}^{n}+D_{k}R_{ilm}^{n}=0.}$

## Ricci曲率

Ricci张量由下式给出

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

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

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

${\displaystyle D_{l}R_{m}^{l}={\frac {1}{2}}{\frac {\partial R}{\partial x^{m}}}}$.

## 外尔张量

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

## 坐标变换

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

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