Christoffel 定義為 Γijk=12(∂∂xigjk+∂∂xjgki−∂∂xkgij)\Gamma_{ijk} = \frac{1}{2}( \frac{\partial}{\partial x^i}g_{jk} + \frac{\partial}{\partial x^j}g_{ki} - \frac{\partial}{\partial x^k}g_{ij} )Γijk=21(∂xi∂gjk+∂xj∂gki−∂xk∂gij) Proof. [local-0] (<=) ggg 是常數表示微分為 000，因此 Christoffel Γijk=0\Gamma_{ijk} = 0Γijk=0 (=>) 因為 Γijk+Γjki=12(∂∂xigjk+∂∂xjgki−∂∂xkgij)+12(∂∂xjgki+∂∂xkgij−∂∂xigjk)=∂∂xjgki\begin{aligned} \Gamma_{ijk} + \Gamma_{jki} & = \frac{1}{2}( \frac{\partial}{\partial x^i}g_{jk} + \frac{\partial}{\partial x^j}g_{ki} - \frac{\partial}{\partial x^k}g_{ij} ) + \frac{1}{2}( \frac{\partial}{\partial x^j}g_{ki} + \frac{\partial}{\partial x^k}g_{ij} - \frac{\partial}{\partial x^i}g_{jk} ) \\ & = \frac{\partial}{\partial x^j}g_{ki} \end{aligned}Γijk+Γjki=21(∂xi∂gjk+∂xj∂gki−∂xk∂gij)+21(∂xj∂gki+∂xk∂gij−∂xi∂gjk)=∂xj∂gki 對兩邊取積分可知 gki=Cg_{ki} = Cgki=C
(<=) ggg 是常數表示微分為 000，因此 Christoffel Γijk=0\Gamma_{ijk} = 0Γijk=0 (=>) 因為 Γijk+Γjki=12(∂∂xigjk+∂∂xjgki−∂∂xkgij)+12(∂∂xjgki+∂∂xkgij−∂∂xigjk)=∂∂xjgki\begin{aligned} \Gamma_{ijk} + \Gamma_{jki} & = \frac{1}{2}( \frac{\partial}{\partial x^i}g_{jk} + \frac{\partial}{\partial x^j}g_{ki} - \frac{\partial}{\partial x^k}g_{ij} ) + \frac{1}{2}( \frac{\partial}{\partial x^j}g_{ki} + \frac{\partial}{\partial x^k}g_{ij} - \frac{\partial}{\partial x^i}g_{jk} ) \\ & = \frac{\partial}{\partial x^j}g_{ki} \end{aligned}Γijk+Γjki=21(∂xi∂gjk+∂xj∂gki−∂xk∂gij)+21(∂xj∂gki+∂xk∂gij−∂xi∂gjk)=∂xj∂gki 對兩邊取積分可知 gki=Cg_{ki} = Cgki=C