ビアンキ恒等式を、ひたすら計算で示す。

この記事について

この記事は、人に見せるというよりも自分の計算を整理するために書いたようなものなので、人に読んでもらうということを全く考えていません。本を読んでてビアンキ恒等式が証明なしに出てきて、それを自分で何とか示したので、記録として残しておきたくなったのです。メモ程度のものだと思ってください。それでも読みたいという方がいらっしゃったら、ぜひ読んであげてください。

前提

数式にはアインシュタインの規約を用いる.
　$\delta$：クロネッカーのデルタ, $g$：計量, $\mathrm{\Gamma }$：接続, $R$：曲率, $\mathrm{\nabla }$：共変微分
$$\begin{gathered} {\mathrm{\Gamma }}_{\nu \lambda }^{\mu }={\mathrm{\Gamma }}_{\lambda \nu }^{\mu } \\ {\mathrm{\nabla }}_{\lambda }{g}_{\mu \nu }=0 \\ {\mathrm{\Gamma }}_{\nu \lambda }^{\mu }=\frac{1}{2}{g}^{\mu \kappa }({\partial }_{\lambda }{g}_{\kappa \nu }+{\partial }_{\nu }{g}_{\kappa \lambda }-{\partial }_{\kappa }{g}_{\lambda \nu }) \\ {R}_{\mu \nu \lambda \kappa }=\frac{1}{2}({\partial }_{\nu }{\partial }_{\lambda }{g}_{\mu \kappa }+{\partial }_{\mu }{\partial }_{\kappa }{g}_{\nu \lambda }-{\partial }_{\mu }{\partial }_{\lambda }{g}_{\nu \kappa }-{\partial }_{\nu }{\partial }_{\kappa }{g}_{\mu \lambda })+g_{\eta \tau }({\mathrm{\Gamma }}_{\mu \kappa }^{\eta }{\mathrm{\Gamma }}_{\nu \lambda }^{\tau }-{\mathrm{\Gamma }}_{\mu \lambda }^{\eta }{\mathrm{\Gamma }}_{\nu \kappa }^{\tau }) \\ {R}_{\mu \nu \lambda \kappa }=R_{\lambda \kappa \mu \nu } \\ {R}_{\mu \nu \lambda \kappa }=-R_{\nu \mu \lambda \kappa } \\ {R}_{\mu \nu \lambda \kappa }=-R_{\mu \nu \kappa \lambda } \end{gathered}$$

目標

$${\mathrm{\nabla }}_{\lambda }{R}_{\mu \nu \kappa \eta }+{\mathrm{\nabla }}_{\eta }{R}_{\mu \nu \lambda \kappa }+{\mathrm{\nabla }}_{\kappa }{R}_{\mu \nu \eta \lambda }=0$$
を示します！！！！！

ひたすら計算して証明

$${\mathrm{\nabla }}_{\lambda }{R}_{\mu \nu \kappa \eta }+{\mathrm{\nabla }}_{\eta }{R}_{\mu \nu \lambda \kappa }+{\mathrm{\nabla }}_{\kappa }{R}_{\mu \nu \eta \lambda }=B$$
とおく. $B=0$を示す.

$$B={\mathrm{\nabla }}_{\lambda }{R}_{\mu \nu \kappa \eta }+{\mathrm{\nabla }}_{\eta }{R}_{\mu \nu \lambda \kappa }+{\mathrm{\nabla }}_{\kappa }{R}_{\mu \nu \eta \lambda }$$
$$\begin{gathered} ={\partial }_{\lambda }{R}_{\mu \nu \kappa \eta }-{\mathrm{\Gamma }}_{\mu \lambda }^{\tau }{R}_{\tau \nu \kappa \eta }-{\mathrm{\Gamma }}_{\nu \lambda }^{\tau }{R}_{\mu \tau \kappa \eta }-{\mathrm{\Gamma }}_{\kappa \lambda }^{\tau }{R}_{\mu \nu \tau \eta }-{\mathrm{\Gamma }}_{\eta \lambda }^{\tau }{R}_{\mu \nu \kappa \tau } \\ +{\partial }_{\eta }{R}_{\mu \nu \lambda \kappa }-{\mathrm{\Gamma }}_{\mu \eta }^{\tau }{R}_{\tau \nu \lambda \kappa }-{\mathrm{\Gamma }}_{\nu \eta }^{\tau }{R}_{\mu \tau \lambda \kappa }-{\mathrm{\Gamma }}_{\lambda \eta }^{\tau }{R}_{\mu \nu \tau \kappa }-{\mathrm{\Gamma }}_{\kappa \eta }^{\tau }{R}_{\mu \nu \lambda \tau } \\ +{\partial }_{\kappa }{R}_{\mu \nu \eta \lambda }-{\mathrm{\Gamma }}_{\mu \kappa }^{\tau }{R}_{\tau \nu \eta \lambda }-{\mathrm{\Gamma }}_{\nu \kappa }^{\tau }{R}_{\mu \tau \eta \lambda }-{\mathrm{\Gamma }}_{\eta \kappa }^{\tau }{R}_{\mu \nu \tau \lambda }-{\mathrm{\Gamma }}_{\lambda \kappa }^{\tau }{R}_{\mu \nu \eta \tau } \end{gathered}$$

$$\begin{gathered} ={\partial }_{\lambda }{R}_{\mu \nu \kappa \eta }-{\mathrm{\Gamma }}_{\mu \lambda }^{\tau }{R}_{\tau \nu \kappa \eta }-{\mathrm{\Gamma }}_{\nu \lambda }^{\tau }{R}_{\mu \tau \kappa \eta } \\ +{\partial }_{\eta }{R}_{\mu \nu \lambda \kappa }-{\mathrm{\Gamma }}_{\mu \eta }^{\tau }{R}_{\tau \nu \lambda \kappa }-{\mathrm{\Gamma }}_{\nu \eta }^{\tau }{R}_{\mu \tau \lambda \kappa } \\ +{\partial }_{\kappa }{R}_{\mu \nu \eta \lambda }-{\mathrm{\Gamma }}_{\mu \kappa }^{\tau }{R}_{\tau \nu \eta \lambda }-{\mathrm{\Gamma }}_{\nu \kappa }^{\tau }{R}_{\mu \tau \eta \lambda } \\ -{\mathrm{\Gamma }}_{\kappa \lambda }^{\tau }(R_{\mu \nu \tau \eta }+R_{\mu \nu \eta \tau })-{\mathrm{\Gamma }}_{\lambda \eta }^{\tau }(R_{\mu \nu \tau \kappa }+R_{\mu \nu \kappa \tau })-{\mathrm{\Gamma }}_{\eta \kappa }^{\tau }(R_{\mu \nu \tau \lambda }+R_{\mu \nu \lambda \tau }) \end{gathered}$$

$$\begin{gathered} ={\partial }_{\lambda }{R}_{\mu \nu \kappa \eta }-{\mathrm{\Gamma }}_{\mu \lambda }^{\tau }{R}_{\tau \nu \kappa \eta }-{\mathrm{\Gamma }}_{\nu \lambda }^{\tau }{R}_{\mu \tau \kappa \eta } \\ +{\partial }_{\eta }{R}_{\mu \nu \lambda \kappa }-{\mathrm{\Gamma }}_{\mu \eta }^{\tau }{R}_{\tau \nu \lambda \kappa }-{\mathrm{\Gamma }}_{\nu \eta }^{\tau }{R}_{\mu \tau \lambda \kappa } \\ +{\partial }_{\kappa }{R}_{\mu \nu \eta \lambda }-{\mathrm{\Gamma }}_{\mu \kappa }^{\tau }{R}_{\tau \nu \eta \lambda }-{\mathrm{\Gamma }}_{\nu \kappa }^{\tau }{R}_{\mu \tau \eta \lambda } \end{gathered}$$

次に,
$${\partial }_{\lambda }{R}_{\mu \nu \kappa \eta }$$

$$={\partial }_{\lambda }(\frac{1}{2}({\partial }_{\nu }{\partial }_{\kappa }{g}_{\mu \eta }+{\partial }_{\mu }{\partial }_{\eta }{g}_{\nu \kappa }-{\partial }_{\mu }{\partial }_{\kappa }{g}_{\nu \eta }-{\partial }_{\nu }{\partial }_{\eta }{g}_{\mu \kappa })+g_{\tau \theta }({\mathrm{\Gamma }}_{\mu \eta }^{\tau }{\mathrm{\Gamma }}_{\nu \kappa }^{\theta }-{\mathrm{\Gamma }}_{\mu \kappa }^{\tau }{\mathrm{\Gamma }}_{\nu \eta }^{\theta }))$$

$$\begin{gathered} =\frac{1}{2}({\partial }_{\lambda }{\partial }_{\nu }{\partial }_{\kappa }{g}_{\mu \eta }+{\partial }_{\lambda }{\partial }_{\mu }{\partial }_{\eta }{g}_{\nu \kappa }-{\partial }_{\lambda }{\partial }_{\mu }{\partial }_{\kappa }{g}_{\nu \eta }-{\partial }_{\lambda }{\partial }_{\nu }{\partial }_{\eta }{g}_{\mu \kappa }) \\ +({\partial }_{\lambda }{g}_{\tau \theta })({\mathrm{\Gamma }}_{\mu \eta }^{\tau }{\mathrm{\Gamma }}_{\nu \kappa }^{\theta }-{\mathrm{\Gamma }}_{\mu \kappa }^{\tau }{\mathrm{\Gamma }}_{\nu \eta }^{\theta })+g_{\tau \theta }{\partial }_{\lambda }({\mathrm{\Gamma }}_{\mu \eta }^{\tau }{\mathrm{\Gamma }}_{\nu \kappa }^{\theta }-{\mathrm{\Gamma }}_{\mu \kappa }^{\tau }{\mathrm{\Gamma }}_{\nu \eta }^{\theta }) \end{gathered}$$
を用いると,
$${\partial }_{\lambda }{R}_{\mu \nu \kappa \eta }+{\partial }_{\eta }{R}_{\mu \nu \lambda \kappa }+{\partial }_{\kappa }{R}_{\mu \nu \eta \lambda }$$

$$\begin{gathered} =\frac{1}{2}({\partial }_{\lambda }{\partial }_{\nu }{\partial }_{\kappa }{g}_{\mu \eta }+{\partial }_{\lambda }{\partial }_{\mu }{\partial }_{\eta }{g}_{\nu \kappa }-{\partial }_{\lambda }{\partial }_{\mu }{\partial }_{\kappa }{g}_{\nu \eta }-{\partial }_{\lambda }{\partial }_{\nu }{\partial }_{\eta }{g}_{\mu \kappa }) \\ +\frac{1}{2}({\partial }_{\eta }{\partial }_{\nu }{\partial }_{\lambda }{g}_{\mu \kappa }+{\partial }_{\eta }{\partial }_{\mu }{\partial }_{\kappa }{g}_{\nu \lambda }-{\partial }_{\eta }{\partial }_{\mu }{\partial }_{\lambda }{g}_{\nu \kappa }-{\partial }_{\eta }{\partial }_{\nu }{\partial }_{\kappa }{g}_{\mu \lambda }) \\ +\frac{1}{2}({\partial }_{\kappa }{\partial }_{\nu }{\partial }_{\eta }{g}_{\mu \lambda }+{\partial }_{\kappa }{\partial }_{\mu }{\partial }_{\lambda }{g}_{\nu \eta }-{\partial }_{\kappa }{\partial }_{\mu }{\partial }_{\eta }{g}_{\nu \lambda }-{\partial }_{\kappa }{\partial }_{\nu }{\partial }_{\lambda }{g}_{\mu \eta }) \\ +({\partial }_{\lambda }{g}_{\tau \theta })({\mathrm{\Gamma }}_{\mu \eta }^{\tau }{\mathrm{\Gamma }}_{\nu \kappa }^{\theta }-{\mathrm{\Gamma }}_{\mu \kappa }^{\tau }{\mathrm{\Gamma }}_{\nu \eta }^{\theta })+g_{\tau \theta }{\partial }_{\lambda }({\mathrm{\Gamma }}_{\mu \eta }^{\tau }{\mathrm{\Gamma }}_{\nu \kappa }^{\theta }-{\mathrm{\Gamma }}_{\mu \kappa }^{\tau }{\mathrm{\Gamma }}_{\nu \eta }^{\theta }) \\ +({\partial }_{\eta }{g}_{\tau \theta })({\mathrm{\Gamma }}_{\mu \kappa }^{\tau }{\mathrm{\Gamma }}_{\nu \lambda }^{\theta }-{\mathrm{\Gamma }}_{\mu \lambda }^{\tau }{\mathrm{\Gamma }}_{\nu \kappa }^{\theta })+g_{\tau \theta }{\partial }_{\eta }({\mathrm{\Gamma }}_{\mu \kappa }^{\tau }{\mathrm{\Gamma }}_{\nu \lambda }^{\theta }-{\mathrm{\Gamma }}_{\mu \lambda }^{\tau }{\mathrm{\Gamma }}_{\nu \kappa }^{\theta }) \\ +({\partial }_{\kappa }{g}_{\tau \theta })({\mathrm{\Gamma }}_{\mu \lambda }^{\tau }{\mathrm{\Gamma }}_{\nu \eta }^{\theta }-{\mathrm{\Gamma }}_{\mu \eta }^{\tau }{\mathrm{\Gamma }}_{\nu \lambda }^{\theta })+g_{\tau \theta }{\partial }_{\kappa }({\mathrm{\Gamma }}_{\mu \lambda }^{\tau }{\mathrm{\Gamma }}_{\nu \eta }^{\theta }-{\mathrm{\Gamma }}_{\mu \eta }^{\tau }{\mathrm{\Gamma }}_{\nu \lambda }^{\theta }) \end{gathered}$$

$$\begin{gathered} =({\partial }_{\lambda }{g}_{\tau \theta })({\mathrm{\Gamma }}_{\mu \eta }^{\tau }{\mathrm{\Gamma }}_{\nu \kappa }^{\theta }-{\mathrm{\Gamma }}_{\mu \kappa }^{\tau }{\mathrm{\Gamma }}_{\nu \eta }^{\theta })+g_{\tau \theta }{\partial }_{\lambda }({\mathrm{\Gamma }}_{\mu \eta }^{\tau }{\mathrm{\Gamma }}_{\nu \kappa }^{\theta }-{\mathrm{\Gamma }}_{\mu \kappa }^{\tau }{\mathrm{\Gamma }}_{\nu \eta }^{\theta }) \\ +({\partial }_{\eta }{g}_{\tau \theta })({\mathrm{\Gamma }}_{\mu \kappa }^{\tau }{\mathrm{\Gamma }}_{\nu \lambda }^{\theta }-{\mathrm{\Gamma }}_{\mu \lambda }^{\tau }{\mathrm{\Gamma }}_{\nu \kappa }^{\theta })+g_{\tau \theta }{\partial }_{\eta }({\mathrm{\Gamma }}_{\mu \kappa }^{\tau }{\mathrm{\Gamma }}_{\nu \lambda }^{\theta }-{\mathrm{\Gamma }}_{\mu \lambda }^{\tau }{\mathrm{\Gamma }}_{\nu \kappa }^{\theta }) \\ +({\partial }_{\kappa }{g}_{\tau \theta })({\mathrm{\Gamma }}_{\mu \lambda }^{\tau }{\mathrm{\Gamma }}_{\nu \eta }^{\theta }-{\mathrm{\Gamma }}_{\mu \eta }^{\tau }{\mathrm{\Gamma }}_{\nu \lambda }^{\theta })+g_{\tau \theta }{\partial }_{\kappa }({\mathrm{\Gamma }}_{\mu \lambda }^{\tau }{\mathrm{\Gamma }}_{\nu \eta }^{\theta }-{\mathrm{\Gamma }}_{\mu \eta }^{\tau }{\mathrm{\Gamma }}_{\nu \lambda }^{\theta }) \end{gathered}$$
よって,
$$\begin{gathered} B \\ =({\partial }_{\lambda }{g}_{\tau \theta })({\mathrm{\Gamma }}_{\mu \eta }^{\tau }{\mathrm{\Gamma }}_{\nu \kappa }^{\theta }-{\mathrm{\Gamma }}_{\mu \kappa }^{\tau }{\mathrm{\Gamma }}_{\nu \eta }^{\theta })+g_{\tau \theta }{\partial }_{\lambda }({\mathrm{\Gamma }}_{\mu \eta }^{\tau }{\mathrm{\Gamma }}_{\nu \kappa }^{\theta }-{\mathrm{\Gamma }}_{\mu \kappa }^{\tau }{\mathrm{\Gamma }}_{\nu \eta }^{\theta }) \\ +({\partial }_{\eta }{g}_{\tau \theta })({\mathrm{\Gamma }}_{\mu \kappa }^{\tau }{\mathrm{\Gamma }}_{\nu \lambda }^{\theta }-{\mathrm{\Gamma }}_{\mu \lambda }^{\tau }{\mathrm{\Gamma }}_{\nu \kappa }^{\theta })+g_{\tau \theta }{\partial }_{\eta }({\mathrm{\Gamma }}_{\mu \kappa }^{\tau }{\mathrm{\Gamma }}_{\nu \lambda }^{\theta }-{\mathrm{\Gamma }}_{\mu \lambda }^{\tau }{\mathrm{\Gamma }}_{\nu \kappa }^{\theta }) \\ +({\partial }_{\kappa }{g}_{\tau \theta })({\mathrm{\Gamma }}_{\mu \lambda }^{\tau }{\mathrm{\Gamma }}_{\nu \eta }^{\theta }-{\mathrm{\Gamma }}_{\mu \eta }^{\tau }{\mathrm{\Gamma }}_{\nu \lambda }^{\theta })+g_{\tau \theta }{\partial }_{\kappa }({\mathrm{\Gamma }}_{\mu \lambda }^{\tau }{\mathrm{\Gamma }}_{\nu \eta }^{\theta }-{\mathrm{\Gamma }}_{\mu \eta }^{\tau }{\mathrm{\Gamma }}_{\nu \lambda }^{\theta }) \\ -({\mathrm{\Gamma }}_{\mu \lambda }^{\tau }{R}_{\tau \nu \kappa \eta }+{\mathrm{\Gamma }}_{\nu \lambda }^{\tau }{R}_{\mu \tau \kappa \eta }+{\mathrm{\Gamma }}_{\mu \eta }^{\tau }{R}_{\tau \nu \lambda \kappa }+{\mathrm{\Gamma }}_{\nu \eta }^{\tau }{R}_{\mu \tau \lambda \kappa }+{\mathrm{\Gamma }}_{\mu \kappa }^{\tau }{R}_{\tau \nu \eta \lambda }+{\mathrm{\Gamma }}_{\nu \kappa }^{\tau }{R}_{\mu \tau \eta \lambda }) \end{gathered}$$
次に,
$$g_{\tau \theta }{\partial }_{\lambda }({\mathrm{\Gamma }}_{\mu \eta }^{\tau }{\mathrm{\Gamma }}_{\nu \kappa }^{\theta }-{\mathrm{\Gamma }}_{\mu \kappa }^{\tau }{\mathrm{\Gamma }}_{\nu \eta }^{\theta })+g_{\tau \theta }{\partial }_{\eta }({\mathrm{\Gamma }}_{\mu \kappa }^{\tau }{\mathrm{\Gamma }}_{\nu \lambda }^{\theta }-{\mathrm{\Gamma }}_{\mu \lambda }^{\tau }{\mathrm{\Gamma }}_{\nu \kappa }^{\theta })+g_{\tau \theta }{\partial }_{\kappa }({\mathrm{\Gamma }}_{\mu \lambda }^{\tau }{\mathrm{\Gamma }}_{\nu \eta }^{\theta }-{\mathrm{\Gamma }}_{\mu \eta }^{\tau }{\mathrm{\Gamma }}_{\nu \lambda }^{\theta })$$
$$\begin{gathered} =g_{\tau \theta }({\partial }_{\lambda }{\mathrm{\Gamma }}_{\mu \eta }^{\tau }){\mathrm{\Gamma }}_{\nu \kappa }^{\theta }+g_{\tau \theta }{\mathrm{\Gamma }}_{\mu \eta }^{\tau }({\partial }_{\lambda }{\mathrm{\Gamma }}_{\nu \kappa }^{\theta })-g_{\tau \theta }({\partial }_{\lambda }{\mathrm{\Gamma }}_{\mu \kappa }^{\tau }){\mathrm{\Gamma }}_{\nu \eta }^{\theta }-g_{\tau \theta }{\mathrm{\Gamma }}_{\mu \kappa }^{\tau }({\partial }_{\lambda }{\mathrm{\Gamma }}_{\nu \eta }^{\theta }) \\ +g_{\tau \theta }({\partial }_{\eta }{\mathrm{\Gamma }}_{\mu \kappa }^{\tau }){\mathrm{\Gamma }}_{\nu \lambda }^{\theta }+g_{\tau \theta }{\mathrm{\Gamma }}_{\mu \kappa }^{\tau }({\partial }_{\eta }{\mathrm{\Gamma }}_{\nu \lambda }^{\theta })-g_{\tau \theta }({\partial }_{\eta }{\mathrm{\Gamma }}_{\mu \lambda }^{\tau }){\mathrm{\Gamma }}_{\nu \kappa }^{\theta }-g_{\tau \theta }{\mathrm{\Gamma }}_{\mu \lambda }^{\tau }({\partial }_{\eta }{\mathrm{\Gamma }}_{\nu \kappa }^{\theta }) \\ +g_{\tau \theta }({\partial }_{\kappa }{\mathrm{\Gamma }}_{\mu \lambda }^{\tau }){\mathrm{\Gamma }}_{\nu \eta }^{\theta }+g_{\tau \theta }{\mathrm{\Gamma }}_{\mu \lambda }^{\tau }({\partial }_{\kappa }{\mathrm{\Gamma }}_{\nu \eta }^{\theta })-g_{\tau \theta }({\partial }_{\kappa }{\mathrm{\Gamma }}_{\mu \eta }^{\tau }){\mathrm{\Gamma }}_{\nu \lambda }^{\theta }-g_{\tau \theta }{\mathrm{\Gamma }}_{\mu \eta }^{\tau }({\partial }_{\kappa }{\mathrm{\Gamma }}_{\nu \lambda }^{\theta }) \end{gathered}$$
$$\begin{gathered} =g_{\tau \theta }{\mathrm{\Gamma }}_{\nu \kappa }^{\theta }({\partial }_{\lambda }{\mathrm{\Gamma }}_{\mu \eta }^{\tau }-{\partial }_{\eta }{\mathrm{\Gamma }}_{\mu \lambda }^{\tau })+g_{\tau \theta }{\mathrm{\Gamma }}_{\nu \lambda }^{\theta }({\partial }_{\eta }{\mathrm{\Gamma }}_{\mu \kappa }^{\tau }-{\partial }_{\kappa }{\mathrm{\Gamma }}_{\mu \eta }^{\tau })+g_{\tau \theta }{\mathrm{\Gamma }}_{\nu \eta }^{\theta }({\partial }_{\kappa }{\mathrm{\Gamma }}_{\mu \lambda }^{\tau }-{\partial }_{\lambda }{\mathrm{\Gamma }}_{\mu \kappa }^{\tau }) \\ +g_{\tau \theta }{\mathrm{\Gamma }}_{\mu \eta }^{\tau }({\partial }_{\lambda }{\mathrm{\Gamma }}_{\nu \kappa }^{\theta }-{\partial }_{\kappa }{\mathrm{\Gamma }}_{\nu \lambda }^{\theta })+g_{\tau \theta }{\mathrm{\Gamma }}_{\mu \kappa }^{\tau }({\partial }_{\eta }{\mathrm{\Gamma }}_{\nu \lambda }^{\theta }-{\partial }_{\lambda }{\mathrm{\Gamma }}_{\nu \eta }^{\theta })+g_{\tau \theta }{\mathrm{\Gamma }}_{\mu \lambda }^{\tau }({\partial }_{\kappa }{\mathrm{\Gamma }}_{\nu \eta }^{\theta }-{\partial }_{\eta }{\mathrm{\Gamma }}_{\nu \kappa }^{\theta }) \end{gathered}$$
となる. さらに,
$$g_{\tau \theta }{\mathrm{\Gamma }}_{\nu \kappa }^{\theta }({\partial }_{\lambda }{\mathrm{\Gamma }}_{\mu \eta }^{\tau }-{\partial }_{\eta }{\mathrm{\Gamma }}_{\mu \lambda }^{\tau })$$
$$=g_{\tau \theta }{\mathrm{\Gamma }}_{\nu \kappa }^{\theta }({\partial }_{\lambda }(g^{\tau \varphi }{g}_{\rho \varphi }{\mathrm{\Gamma }}_{\mu \eta }^{\rho })-{\partial }_{\eta }(g^{\tau \varphi }{g}_{\rho \varphi }{\mathrm{\Gamma }}_{\mu \lambda }^{\rho }))$$
$$=g_{\tau \theta }{\mathrm{\Gamma }}_{\nu \kappa }^{\theta }(({\partial }_{\lambda }{g}^{\tau \varphi })g_{\rho \varphi }{\mathrm{\Gamma }}_{\mu \eta }^{\rho }+g^{\tau \varphi }{\partial }_{\lambda }(g_{\rho \varphi }{\mathrm{\Gamma }}_{\mu \eta }^{\rho })-({\partial }_{\eta }{g}^{\tau \varphi })g_{\rho \varphi }{\mathrm{\Gamma }}_{\mu \lambda }^{\rho }-g^{\tau \varphi }{\partial }_{\eta }(g_{\rho \varphi }{\mathrm{\Gamma }}_{\mu \lambda }^{\rho }))$$
$$\begin{gathered} =g_{\tau \theta }{g}_{\rho \varphi }{\mathrm{\Gamma }}_{\nu \kappa }^{\theta }(({\partial }_{\lambda }{g}^{\tau \varphi }){\mathrm{\Gamma }}_{\mu \eta }^{\rho }-({\partial }_{\eta }{g}^{\tau \varphi }){\mathrm{\Gamma }}_{\mu \lambda }^{\rho }) \\ +{\mathrm{\Gamma }}_{\nu \kappa }^{\theta }{\delta }_{\theta }^{\varphi }({\partial }_{\lambda }(\frac{1}{2}({\partial }_{\eta }{g}_{\varphi \mu }+{\partial }_{\mu }{g}_{\varphi \eta }-{\partial }_{\varphi }{g}_{\eta \mu }))-{\partial }_{\eta }(\frac{1}{2}({\partial }_{\lambda }{g}_{\varphi \mu }+{\partial }_{\mu }{g}_{\varphi \lambda }-{\partial }_{\varphi }{g}_{\lambda \mu }))) \end{gathered}$$
$$\begin{gathered} =g_{\tau \theta }{g}_{\rho \varphi }{\mathrm{\Gamma }}_{\nu \kappa }^{\theta }(({\partial }_{\lambda }{g}^{\tau \varphi }){\mathrm{\Gamma }}_{\mu \eta }^{\rho }-({\partial }_{\eta }{g}^{\tau \varphi }){\mathrm{\Gamma }}_{\mu \lambda }^{\rho }) \\ +\frac{1}{2}{\mathrm{\Gamma }}_{\nu \kappa }^{\theta }({\partial }_{\lambda }{\partial }_{\mu }{g}_{\theta \eta }+{\partial }_{\eta }{\partial }_{\theta }{g}_{\lambda \mu }-{\partial }_{\lambda }{\partial }_{\theta }{g}_{\eta \mu }-{\partial }_{\eta }{\partial }_{\mu }{g}_{\theta \lambda }) \end{gathered}$$
$$\begin{gathered} =g_{\tau \theta }{g}_{\rho \varphi }{\mathrm{\Gamma }}_{\nu \kappa }^{\theta }(({\partial }_{\lambda }{g}^{\tau \varphi }){\mathrm{\Gamma }}_{\mu \eta }^{\rho }-({\partial }_{\eta }{g}^{\tau \varphi }){\mathrm{\Gamma }}_{\mu \lambda }^{\rho }) \\ +{\mathrm{\Gamma }}_{\nu \kappa }^{\theta }(R_{\mu \theta \eta \lambda }-g_{\tau \rho }({\mathrm{\Gamma }}_{\theta \eta }^{\tau }{\mathrm{\Gamma }}_{\mu \lambda }^{\rho }-{\mathrm{\Gamma }}_{\mu \eta }^{\tau }{\mathrm{\Gamma }}_{\theta \lambda }^{\rho })) \end{gathered}$$
$$\begin{gathered} =g_{\tau \theta }{g}_{\rho \varphi }{\mathrm{\Gamma }}_{\nu \kappa }^{\theta }(({\partial }_{\lambda }{g}^{\tau \varphi }){\mathrm{\Gamma }}_{\mu \eta }^{\rho }-({\partial }_{\eta }{g}^{\tau \varphi }){\mathrm{\Gamma }}_{\mu \lambda }^{\rho }) \\ +{\mathrm{\Gamma }}_{\nu \kappa }^{\tau }{R}_{\mu \tau \eta \lambda }-g_{\tau \rho }{\mathrm{\Gamma }}_{\nu \kappa }^{\theta }({\mathrm{\Gamma }}_{\theta \eta }^{\tau }{\mathrm{\Gamma }}_{\mu \lambda }^{\rho }-{\mathrm{\Gamma }}_{\mu \eta }^{\tau }{\mathrm{\Gamma }}_{\theta \lambda }^{\rho }) \end{gathered}$$
以上より,
$$B$$
$$\begin{gathered} =({\partial }_{\lambda }{g}_{\tau \theta })({\mathrm{\Gamma }}_{\mu \eta }^{\tau }{\mathrm{\Gamma }}_{\nu \kappa }^{\theta }-{\mathrm{\Gamma }}_{\mu \kappa }^{\tau }{\mathrm{\Gamma }}_{\nu \eta }^{\theta })+({\partial }_{\eta }{g}_{\tau \theta })({\mathrm{\Gamma }}_{\mu \kappa }^{\tau }{\mathrm{\Gamma }}_{\nu \lambda }^{\theta }-{\mathrm{\Gamma }}_{\mu \lambda }^{\tau }{\mathrm{\Gamma }}_{\nu \kappa }^{\theta })+({\partial }_{\kappa }{g}_{\tau \theta })({\mathrm{\Gamma }}_{\mu \lambda }^{\tau }{\mathrm{\Gamma }}_{\nu \eta }^{\theta }-{\mathrm{\Gamma }}_{\mu \eta }^{\tau }{\mathrm{\Gamma }}_{\nu \lambda }^{\theta }) \\ +g_{\tau \theta }{\mathrm{\Gamma }}_{\nu \kappa }^{\theta }({\partial }_{\lambda }{\mathrm{\Gamma }}_{\mu \eta }^{\tau }-{\partial }_{\eta }{\mathrm{\Gamma }}_{\mu \lambda }^{\tau })+g_{\tau \theta }{\mathrm{\Gamma }}_{\nu \lambda }^{\theta }({\partial }_{\eta }{\mathrm{\Gamma }}_{\mu \kappa }^{\tau }-{\partial }_{\kappa }{\mathrm{\Gamma }}_{\mu \eta }^{\tau })+g_{\tau \theta }{\mathrm{\Gamma }}_{\nu \eta }^{\theta }({\partial }_{\kappa }{\mathrm{\Gamma }}_{\mu \lambda }^{\tau }-{\partial }_{\lambda }{\mathrm{\Gamma }}_{\mu \kappa }^{\tau }) \\ +g_{\tau \theta }{\mathrm{\Gamma }}_{\mu \eta }^{\tau }({\partial }_{\lambda }{\mathrm{\Gamma }}_{\nu \kappa }^{\theta }-{\partial }_{\kappa }{\mathrm{\Gamma }}_{\nu \lambda }^{\theta })+g_{\tau \theta }{\mathrm{\Gamma }}_{\mu \kappa }^{\tau }({\partial }_{\eta }{\mathrm{\Gamma }}_{\nu \lambda }^{\theta }-{\partial }_{\lambda }{\mathrm{\Gamma }}_{\nu \eta }^{\theta })+g_{\tau \theta }{\mathrm{\Gamma }}_{\mu \lambda }^{\tau }({\partial }_{\kappa }{\mathrm{\Gamma }}_{\nu \eta }^{\theta }-{\partial }_{\eta }{\mathrm{\Gamma }}_{\nu \kappa }^{\theta }) \\ -({\mathrm{\Gamma }}_{\mu \lambda }^{\tau }{R}_{\tau \nu \kappa \eta }+{\mathrm{\Gamma }}_{\nu \lambda }^{\tau }{R}_{\mu \tau \kappa \eta }+{\mathrm{\Gamma }}_{\mu \eta }^{\tau }{R}_{\tau \nu \lambda \kappa }+{\mathrm{\Gamma }}_{\nu \eta }^{\tau }{R}_{\mu \tau \lambda \kappa }+{\mathrm{\Gamma }}_{\mu \kappa }^{\tau }{R}_{\tau \nu \eta \lambda }+{\mathrm{\Gamma }}_{\nu \kappa }^{\tau }{R}_{\mu \tau \eta \lambda }) \end{gathered}$$
$$\begin{gathered} =({\partial }_{\lambda }{g}_{\tau \theta })({\mathrm{\Gamma }}_{\mu \eta }^{\tau }{\mathrm{\Gamma }}_{\nu \kappa }^{\theta }-{\mathrm{\Gamma }}_{\mu \kappa }^{\tau }{\mathrm{\Gamma }}_{\nu \eta }^{\theta })+({\partial }_{\eta }{g}_{\tau \theta })({\mathrm{\Gamma }}_{\mu \kappa }^{\tau }{\mathrm{\Gamma }}_{\nu \lambda }^{\theta }-{\mathrm{\Gamma }}_{\mu \lambda }^{\tau }{\mathrm{\Gamma }}_{\nu \kappa }^{\theta })+({\partial }_{\kappa }{g}_{\tau \theta })({\mathrm{\Gamma }}_{\mu \lambda }^{\tau }{\mathrm{\Gamma }}_{\nu \eta }^{\theta }-{\mathrm{\Gamma }}_{\mu \eta }^{\tau }{\mathrm{\Gamma }}_{\nu \lambda }^{\theta }) \\ +g_{\tau \theta }{\mathrm{\Gamma }}_{\nu \kappa }^{\theta }({\partial }_{\lambda }{\mathrm{\Gamma }}_{\mu \eta }^{\tau }-{\partial }_{\eta }{\mathrm{\Gamma }}_{\mu \lambda }^{\tau })-{\mathrm{\Gamma }}_{\nu \kappa }^{\tau }{R}_{\mu \tau \eta \lambda } \\ +g_{\tau \theta }{\mathrm{\Gamma }}_{\nu \lambda }^{\theta }({\partial }_{\eta }{\mathrm{\Gamma }}_{\mu \kappa }^{\tau }-{\partial }_{\kappa }{\mathrm{\Gamma }}_{\mu \eta }^{\tau })-{\mathrm{\Gamma }}_{\nu \lambda }^{\tau }{R}_{\mu \tau \kappa \eta } \\ +g_{\tau \theta }{\mathrm{\Gamma }}_{\nu \eta }^{\theta }({\partial }_{\kappa }{\mathrm{\Gamma }}_{\mu \lambda }^{\tau }-{\partial }_{\lambda }{\mathrm{\Gamma }}_{\mu \kappa }^{\tau })-{\mathrm{\Gamma }}_{\nu \eta }^{\tau }{R}_{\mu \tau \lambda \kappa } \\ +g_{\tau \theta }{\mathrm{\Gamma }}_{\mu \eta }^{\tau }({\partial }_{\lambda }{\mathrm{\Gamma }}_{\nu \kappa }^{\theta }-{\partial }_{\kappa }{\mathrm{\Gamma }}_{\nu \lambda }^{\theta })-{\mathrm{\Gamma }}_{\mu \eta }^{\tau }{R}_{\tau \nu \lambda \kappa } \\ +g_{\tau \theta }{\mathrm{\Gamma }}_{\mu \kappa }^{\tau }({\partial }_{\eta }{\mathrm{\Gamma }}_{\nu \lambda }^{\theta }-{\partial }_{\lambda }{\mathrm{\Gamma }}_{\nu \eta }^{\theta })-{\mathrm{\Gamma }}_{\mu \kappa }^{\tau }{R}_{\tau \nu \eta \lambda } \\ +g_{\tau \theta }{\mathrm{\Gamma }}_{\mu \lambda }^{\tau }({\partial }_{\kappa }{\mathrm{\Gamma }}_{\nu \eta }^{\theta }-{\partial }_{\eta }{\mathrm{\Gamma }}_{\nu \kappa }^{\theta })-{\mathrm{\Gamma }}_{\mu \lambda }^{\tau }{R}_{\tau \nu \kappa \eta } \end{gathered}$$
$$\begin{gathered} =({\partial }_{\lambda }{g}_{\tau \theta })({\mathrm{\Gamma }}_{\mu \eta }^{\tau }{\mathrm{\Gamma }}_{\nu \kappa }^{\theta }-{\mathrm{\Gamma }}_{\mu \kappa }^{\tau }{\mathrm{\Gamma }}_{\nu \eta }^{\theta })+({\partial }_{\eta }{g}_{\tau \theta })({\mathrm{\Gamma }}_{\mu \kappa }^{\tau }{\mathrm{\Gamma }}_{\nu \lambda }^{\theta }-{\mathrm{\Gamma }}_{\mu \lambda }^{\tau }{\mathrm{\Gamma }}_{\nu \kappa }^{\theta })+({\partial }_{\kappa }{g}_{\tau \theta })({\mathrm{\Gamma }}_{\mu \lambda }^{\tau }{\mathrm{\Gamma }}_{\nu \eta }^{\theta }-{\mathrm{\Gamma }}_{\mu \eta }^{\tau }{\mathrm{\Gamma }}_{\nu \lambda }^{\theta }) \\ +g_{\tau \theta }{g}_{\rho \varphi }{\mathrm{\Gamma }}_{\nu \kappa }^{\theta }(({\partial }_{\lambda }{g}^{\tau \varphi }){\mathrm{\Gamma }}_{\mu \eta }^{\rho }-({\partial }_{\eta }{g}^{\tau \varphi }){\mathrm{\Gamma }}_{\mu \lambda }^{\rho })-g_{\tau \rho }{\mathrm{\Gamma }}_{\nu \kappa }^{\theta }({\mathrm{\Gamma }}_{\theta \eta }^{\tau }{\mathrm{\Gamma }}_{\mu \lambda }^{\rho }-{\mathrm{\Gamma }}_{\mu \eta }^{\tau }{\mathrm{\Gamma }}_{\theta \lambda }^{\rho }) \\ +g_{\tau \theta }{g}_{\rho \varphi }{\mathrm{\Gamma }}_{\nu \lambda }^{\theta }(({\partial }_{\eta }{g}^{\tau \varphi }){\mathrm{\Gamma }}_{\mu \kappa }^{\rho }-({\partial }_{\kappa }{g}^{\tau \varphi }){\mathrm{\Gamma }}_{\mu \eta }^{\rho })-g_{\tau \rho }{\mathrm{\Gamma }}_{\nu \lambda }^{\theta }({\mathrm{\Gamma }}_{\theta \kappa }^{\tau }{\mathrm{\Gamma }}_{\mu \eta }^{\rho }-{\mathrm{\Gamma }}_{\mu \kappa }^{\tau }{\mathrm{\Gamma }}_{\theta \eta }^{\rho }) \\ +g_{\tau \theta }{g}_{\rho \varphi }{\mathrm{\Gamma }}_{\nu \eta }^{\theta }(({\partial }_{\kappa }{g}^{\tau \varphi }){\mathrm{\Gamma }}_{\mu \lambda }^{\rho }-({\partial }_{\lambda }{g}^{\tau \varphi }){\mathrm{\Gamma }}_{\mu \kappa }^{\rho })-g_{\tau \rho }{\mathrm{\Gamma }}_{\nu \eta }^{\theta }({\mathrm{\Gamma }}_{\theta \lambda }^{\tau }{\mathrm{\Gamma }}_{\mu \kappa }^{\rho }-{\mathrm{\Gamma }}_{\mu \lambda }^{\tau }{\mathrm{\Gamma }}_{\theta \kappa }^{\rho }) \\ +g_{\tau \theta }{g}_{\rho \varphi }{\mathrm{\Gamma }}_{\mu \eta }^{\theta }(({\partial }_{\lambda }{g}^{\tau \varphi }){\mathrm{\Gamma }}_{\nu \kappa }^{\rho }-({\partial }_{\kappa }{g}^{\tau \varphi }){\mathrm{\Gamma }}_{\nu \lambda }^{\rho })-g_{\tau \rho }{\mathrm{\Gamma }}_{\mu \eta }^{\theta }({\mathrm{\Gamma }}_{\theta \kappa }^{\tau }{\mathrm{\Gamma }}_{\nu \lambda }^{\rho }-{\mathrm{\Gamma }}_{\nu \kappa }^{\tau }{\mathrm{\Gamma }}_{\theta \lambda }^{\rho }) \\ +g_{\tau \theta }{g}_{\rho \varphi }{\mathrm{\Gamma }}_{\mu \kappa }^{\theta }(({\partial }_{\eta }{g}^{\tau \varphi }){\mathrm{\Gamma }}_{\nu \lambda }^{\rho }-({\partial }_{\lambda }{g}^{\tau \varphi }){\mathrm{\Gamma }}_{\nu \eta }^{\rho })-g_{\tau \rho }{\mathrm{\Gamma }}_{\mu \kappa }^{\theta }({\mathrm{\Gamma }}_{\theta \lambda }^{\tau }{\mathrm{\Gamma }}_{\nu \eta }^{\rho }-{\mathrm{\Gamma }}_{\nu \lambda }^{\tau }{\mathrm{\Gamma }}_{\theta \eta }^{\rho }) \\ +g_{\tau \theta }{g}_{\rho \varphi }{\mathrm{\Gamma }}_{\mu \lambda }^{\theta }(({\partial }_{\kappa }{g}^{\tau \varphi }){\mathrm{\Gamma }}_{\nu \eta }^{\rho }-({\partial }_{\eta }{g}^{\tau \varphi }){\mathrm{\Gamma }}_{\nu \kappa }^{\rho })-g_{\tau \rho }{\mathrm{\Gamma }}_{\mu \lambda }^{\theta }({\mathrm{\Gamma }}_{\theta \eta }^{\tau }{\mathrm{\Gamma }}_{\nu \kappa }^{\rho }-{\mathrm{\Gamma }}_{\nu \eta }^{\tau }{\mathrm{\Gamma }}_{\theta \kappa }^{\rho }) \end{gathered}$$
次に,
$$g_{\tau \rho }{\mathrm{\Gamma }}_{\nu \kappa }^{\theta }{\mathrm{\Gamma }}_{\theta \eta }^{\tau }{\mathrm{\Gamma }}_{\mu \lambda }^{\rho }+g_{\tau \rho }{\mathrm{\Gamma }}_{\mu \lambda }^{\theta }{\mathrm{\Gamma }}_{\theta \eta }^{\tau }{\mathrm{\Gamma }}_{\nu \kappa }^{\rho }$$
$$={\mathrm{\Gamma }}_{\nu \kappa }^{\theta }{\mathrm{\Gamma }}_{\mu \lambda }^{\rho }(g_{\tau \rho }{\mathrm{\Gamma }}_{\theta \eta }^{\tau }+g_{\tau \theta }{\mathrm{\Gamma }}_{\rho \eta }^{\tau })$$
$$={\mathrm{\Gamma }}_{\nu \kappa }^{\theta }{\mathrm{\Gamma }}_{\mu \lambda }^{\rho }({\partial }_{\eta }{g}_{\theta \rho }-{\mathrm{\nabla }}_{\eta }{g}_{\theta \rho })$$
$$={\mathrm{\Gamma }}_{\nu \kappa }^{\theta }{\mathrm{\Gamma }}_{\mu \lambda }^{\rho }{\partial }_{\eta }{g}_{\theta \rho }$$
を用いると,
$$\begin{gathered} -g_{\tau \rho }{\mathrm{\Gamma }}_{\nu \kappa }^{\theta }({\mathrm{\Gamma }}_{\theta \eta }^{\tau }{\mathrm{\Gamma }}_{\mu \lambda }^{\rho }-{\mathrm{\Gamma }}_{\mu \eta }^{\tau }{\mathrm{\Gamma }}_{\theta \lambda }^{\rho })-g_{\tau \rho }{\mathrm{\Gamma }}_{\nu \lambda }^{\theta }({\mathrm{\Gamma }}_{\theta \kappa }^{\tau }{\mathrm{\Gamma }}_{\mu \eta }^{\rho }-{\mathrm{\Gamma }}_{\mu \kappa }^{\tau }{\mathrm{\Gamma }}_{\theta \eta }^{\rho })-g_{\tau \rho }{\mathrm{\Gamma }}_{\nu \eta }^{\theta }({\mathrm{\Gamma }}_{\theta \lambda }^{\tau }{\mathrm{\Gamma }}_{\mu \kappa }^{\rho }-{\mathrm{\Gamma }}_{\mu \lambda }^{\tau }{\mathrm{\Gamma }}_{\theta \kappa }^{\rho }) \\ -g_{\tau \rho }{\mathrm{\Gamma }}_{\mu \eta }^{\theta }({\mathrm{\Gamma }}_{\theta \kappa }^{\tau }{\mathrm{\Gamma }}_{\nu \lambda }^{\rho }-{\mathrm{\Gamma }}_{\nu \kappa }^{\tau }{\mathrm{\Gamma }}_{\theta \lambda }^{\rho })-g_{\tau \rho }{\mathrm{\Gamma }}_{\mu \kappa }^{\theta }({\mathrm{\Gamma }}_{\theta \lambda }^{\tau }{\mathrm{\Gamma }}_{\nu \eta }^{\rho }-{\mathrm{\Gamma }}_{\nu \lambda }^{\tau }{\mathrm{\Gamma }}_{\theta \eta }^{\rho })-g_{\tau \rho }{\mathrm{\Gamma }}_{\mu \lambda }^{\theta }({\mathrm{\Gamma }}_{\theta \eta }^{\tau }{\mathrm{\Gamma }}_{\nu \kappa }^{\rho }-{\mathrm{\Gamma }}_{\nu \eta }^{\tau }{\mathrm{\Gamma }}_{\theta \kappa }^{\rho }) \end{gathered}$$
$$\begin{gathered} =-(g_{\tau \rho }{\mathrm{\Gamma }}_{\nu \kappa }^{\theta }{\mathrm{\Gamma }}_{\theta \eta }^{\tau }{\mathrm{\Gamma }}_{\mu \lambda }^{\rho }+g_{\tau \rho }{\mathrm{\Gamma }}_{\mu \lambda }^{\theta }{\mathrm{\Gamma }}_{\theta \eta }^{\tau }{\mathrm{\Gamma }}_{\nu \kappa }^{\rho })-(g_{\tau \rho }{\mathrm{\Gamma }}_{\nu \lambda }^{\theta }{\mathrm{\Gamma }}_{\theta \kappa }^{\tau }{\mathrm{\Gamma }}_{\mu \eta }^{\rho }+g_{\tau \rho }{\mathrm{\Gamma }}_{\mu \eta }^{\theta }{\mathrm{\Gamma }}_{\theta \kappa }^{\tau }{\mathrm{\Gamma }}_{\nu \lambda }^{\rho })-(g_{\tau \rho }{\mathrm{\Gamma }}_{\nu \eta }^{\theta }{\mathrm{\Gamma }}_{\theta \lambda }^{\tau }{\mathrm{\Gamma }}_{\mu \kappa }^{\rho }+g_{\tau \rho }{\mathrm{\Gamma }}_{\mu \kappa }^{\theta }{\mathrm{\Gamma }}_{\theta \lambda }^{\tau }{\mathrm{\Gamma }}_{\nu \eta }^{\rho }) \\ +(g_{\tau \rho }{\mathrm{\Gamma }}_{\nu \kappa }^{\theta }{\mathrm{\Gamma }}_{\mu \eta }^{\tau }{\mathrm{\Gamma }}_{\theta \lambda }^{\rho }+g_{\tau \rho }{\mathrm{\Gamma }}_{\mu \eta }^{\theta }{\mathrm{\Gamma }}_{\nu \kappa }^{\tau }{\mathrm{\Gamma }}_{\theta \lambda }^{\rho })+(g_{\tau \rho }{\mathrm{\Gamma }}_{\nu \lambda }^{\theta }{\mathrm{\Gamma }}_{\mu \kappa }^{\tau }{\mathrm{\Gamma }}_{\theta \eta }^{\rho }+g_{\tau \rho }{\mathrm{\Gamma }}_{\mu \kappa }^{\theta }{\mathrm{\Gamma }}_{\nu \lambda }^{\tau }{\mathrm{\Gamma }}_{\theta \eta }^{\rho })+(g_{\tau \rho }{\mathrm{\Gamma }}_{\nu \eta }^{\theta }{\mathrm{\Gamma }}_{\mu \lambda }^{\tau }{\mathrm{\Gamma }}_{\theta \kappa }^{\rho }+g_{\tau \rho }{\mathrm{\Gamma }}_{\mu \lambda }^{\theta }{\mathrm{\Gamma }}_{\nu \eta }^{\tau }{\mathrm{\Gamma }}_{\theta \kappa }^{\rho }) \end{gathered}$$
$$=-{\mathrm{\Gamma }}_{\nu \kappa }^{\theta }{\mathrm{\Gamma }}_{\mu \lambda }^{\rho }{\partial }_{\eta }{g}_{\theta \rho }-{\mathrm{\Gamma }}_{\nu \lambda }^{\theta }{\mathrm{\Gamma }}_{\mu \eta }^{\rho }{\partial }_{\kappa }{g}_{\theta \rho }-{\mathrm{\Gamma }}_{\nu \eta }^{\theta }{\mathrm{\Gamma }}_{\mu \kappa }^{\rho }{\partial }_{\lambda }{g}_{\theta \rho }+{\mathrm{\Gamma }}_{\nu \kappa }^{\theta }{\mathrm{\Gamma }}_{\mu \eta }^{\rho }{\partial }_{\lambda }{g}_{\theta \rho }+{\mathrm{\Gamma }}_{\nu \lambda }^{\theta }{\mathrm{\Gamma }}_{\mu \kappa }^{\rho }{\partial }_{\eta }{g}_{\theta \rho }+{\mathrm{\Gamma }}_{\nu \eta }^{\theta }{\mathrm{\Gamma }}_{\mu \lambda }^{\rho }{\partial }_{\kappa }{g}_{\theta \rho }$$
$$=({\partial }_{\lambda }{g}_{\tau \theta })({\mathrm{\Gamma }}_{\mu \eta }^{\tau }{\mathrm{\Gamma }}_{\nu \kappa }^{\theta }-{\mathrm{\Gamma }}_{\mu \kappa }^{\tau }{\mathrm{\Gamma }}_{\nu \eta }^{\theta })+({\partial }_{\eta }{g}_{\tau \theta })({\mathrm{\Gamma }}_{\mu \kappa }^{\tau }{\mathrm{\Gamma }}_{\nu \lambda }^{\theta }-{\mathrm{\Gamma }}_{\mu \lambda }^{\tau }{\mathrm{\Gamma }}_{\nu \kappa }^{\theta })+({\partial }_{\kappa }{g}_{\tau \theta })({\mathrm{\Gamma }}_{\mu \lambda }^{\tau }{\mathrm{\Gamma }}_{\nu \eta }^{\theta }-{\mathrm{\Gamma }}_{\mu \eta }^{\tau }{\mathrm{\Gamma }}_{\nu \lambda }^{\theta })$$
したがって,
$$B$$
$$\begin{gathered} =2({\partial }_{\lambda }{g}_{\tau \theta })({\mathrm{\Gamma }}_{\mu \eta }^{\tau }{\mathrm{\Gamma }}_{\nu \kappa }^{\theta }-{\mathrm{\Gamma }}_{\mu \kappa }^{\tau }{\mathrm{\Gamma }}_{\nu \eta }^{\theta })+2({\partial }_{\eta }{g}_{\tau \theta })({\mathrm{\Gamma }}_{\mu \kappa }^{\tau }{\mathrm{\Gamma }}_{\nu \lambda }^{\theta }-{\mathrm{\Gamma }}_{\mu \lambda }^{\tau }{\mathrm{\Gamma }}_{\nu \kappa }^{\theta })+2({\partial }_{\kappa }{g}_{\tau \theta })({\mathrm{\Gamma }}_{\mu \lambda }^{\tau }{\mathrm{\Gamma }}_{\nu \eta }^{\theta }-{\mathrm{\Gamma }}_{\mu \eta }^{\tau }{\mathrm{\Gamma }}_{\nu \lambda }^{\theta }) \\ +g_{\tau \theta }{g}_{\rho \varphi }{\mathrm{\Gamma }}_{\nu \kappa }^{\theta }(({\partial }_{\lambda }{g}^{\tau \varphi }){\mathrm{\Gamma }}_{\mu \eta }^{\rho }-({\partial }_{\eta }{g}^{\tau \varphi }){\mathrm{\Gamma }}_{\mu \lambda }^{\rho }) \\ +g_{\tau \theta }{g}_{\rho \varphi }{\mathrm{\Gamma }}_{\nu \lambda }^{\theta }(({\partial }_{\eta }{g}^{\tau \varphi }){\mathrm{\Gamma }}_{\mu \kappa }^{\rho }-({\partial }_{\kappa }{g}^{\tau \varphi }){\mathrm{\Gamma }}_{\mu \eta }^{\rho }) \\ +g_{\tau \theta }{g}_{\rho \varphi }{\mathrm{\Gamma }}_{\nu \eta }^{\theta }(({\partial }_{\kappa }{g}^{\tau \varphi }){\mathrm{\Gamma }}_{\mu \lambda }^{\rho }-({\partial }_{\lambda }{g}^{\tau \varphi }){\mathrm{\Gamma }}_{\mu \kappa }^{\rho }) \\ +g_{\tau \theta }{g}_{\rho \varphi }{\mathrm{\Gamma }}_{\mu \eta }^{\theta }(({\partial }_{\lambda }{g}^{\tau \varphi }){\mathrm{\Gamma }}_{\nu \kappa }^{\rho }-({\partial }_{\kappa }{g}^{\tau \varphi }){\mathrm{\Gamma }}_{\nu \lambda }^{\rho }) \\ +g_{\tau \theta }{g}_{\rho \varphi }{\mathrm{\Gamma }}_{\mu \kappa }^{\theta }(({\partial }_{\eta }{g}^{\tau \varphi }){\mathrm{\Gamma }}_{\nu \lambda }^{\rho }-({\partial }_{\lambda }{g}^{\tau \varphi }){\mathrm{\Gamma }}_{\nu \eta }^{\rho }) \\ +g_{\tau \theta }{g}_{\rho \varphi }{\mathrm{\Gamma }}_{\mu \lambda }^{\theta }(({\partial }_{\kappa }{g}^{\tau \varphi }){\mathrm{\Gamma }}_{\nu \eta }^{\rho }-({\partial }_{\eta }{g}^{\tau \varphi }){\mathrm{\Gamma }}_{\nu \kappa }^{\rho }) \end{gathered}$$
となる. 次に,
$$\begin{gathered} g_{\tau \theta }{g}_{\rho \varphi }{\mathrm{\Gamma }}_{\mu \eta }^{\theta }(({\partial }_{\lambda }{g}^{\tau \varphi }){\mathrm{\Gamma }}_{\nu \kappa }^{\rho }-({\partial }_{\kappa }{g}^{\tau \varphi }){\mathrm{\Gamma }}_{\nu \lambda }^{\rho }) \\ +g_{\tau \theta }{g}_{\rho \varphi }{\mathrm{\Gamma }}_{\mu \kappa }^{\theta }(({\partial }_{\eta }{g}^{\tau \varphi }){\mathrm{\Gamma }}_{\nu \lambda }^{\rho }-({\partial }_{\lambda }{g}^{\tau \varphi }){\mathrm{\Gamma }}_{\nu \eta }^{\rho }) \\ +g_{\tau \theta }{g}_{\rho \varphi }{\mathrm{\Gamma }}_{\mu \lambda }^{\theta }(({\partial }_{\kappa }{g}^{\tau \varphi }){\mathrm{\Gamma }}_{\nu \eta }^{\rho }-({\partial }_{\eta }{g}^{\tau \varphi }){\mathrm{\Gamma }}_{\nu \kappa }^{\rho }) \end{gathered}$$
$$\begin{gathered} =g_{\tau \theta }{g}_{\rho \varphi }{\mathrm{\Gamma }}_{\mu \eta }^{\theta }({\partial }_{\lambda }{g}^{\tau \varphi }){\mathrm{\Gamma }}_{\nu \kappa }^{\rho }-g_{\tau \theta }{g}_{\rho \varphi }{\mathrm{\Gamma }}_{\mu \eta }^{\theta }({\partial }_{\kappa }{g}^{\tau \varphi }){\mathrm{\Gamma }}_{\nu \lambda }^{\rho } \\ +g_{\tau \theta }{g}_{\rho \varphi }{\mathrm{\Gamma }}_{\mu \kappa }^{\theta }({\partial }_{\eta }{g}^{\tau \varphi }){\mathrm{\Gamma }}_{\nu \lambda }^{\rho }-g_{\tau \theta }{g}_{\rho \varphi }{\mathrm{\Gamma }}_{\mu \kappa }^{\theta }({\partial }_{\lambda }{g}^{\tau \varphi }){\mathrm{\Gamma }}_{\nu \eta }^{\rho } \\ +g_{\tau \theta }{g}_{\rho \varphi }{\mathrm{\Gamma }}_{\mu \lambda }^{\theta }({\partial }_{\kappa }{g}^{\tau \varphi }){\mathrm{\Gamma }}_{\nu \eta }^{\rho }-g_{\tau \theta }{g}_{\rho \varphi }{\mathrm{\Gamma }}_{\mu \lambda }^{\theta }({\partial }_{\eta }{g}^{\tau \varphi }){\mathrm{\Gamma }}_{\nu \kappa }^{\rho } \end{gathered}$$
$$\begin{gathered} =g_{\tau \theta }{g}_{\rho \varphi }{\mathrm{\Gamma }}_{\nu \kappa }^{\rho }(({\partial }_{\lambda }{g}^{\tau \varphi }){\mathrm{\Gamma }}_{\mu \eta }^{\theta }-({\partial }_{\eta }{g}^{\tau \varphi }){\mathrm{\Gamma }}_{\mu \lambda }^{\theta }) \\ +g_{\tau \theta }{g}_{\rho \varphi }{\mathrm{\Gamma }}_{\nu \lambda }^{\rho }(({\partial }_{\eta }{g}^{\tau \varphi }){\mathrm{\Gamma }}_{\mu \kappa }^{\theta }-({\partial }_{\kappa }{g}^{\tau \varphi }){\mathrm{\Gamma }}_{\mu \eta }^{\theta }) \\ +g_{\tau \theta }{g}_{\rho \varphi }{\mathrm{\Gamma }}_{\nu \eta }^{\rho }(({\partial }_{\kappa }{g}^{\tau \varphi }){\mathrm{\Gamma }}_{\mu \lambda }^{\theta }-({\partial }_{\lambda }{g}^{\tau \varphi }){\mathrm{\Gamma }}_{\mu \kappa }^{\theta }) \end{gathered}$$
より,
$$B$$
$$\begin{gathered} =2({\partial }_{\lambda }{g}_{\tau \theta })({\mathrm{\Gamma }}_{\mu \eta }^{\tau }{\mathrm{\Gamma }}_{\nu \kappa }^{\theta }-{\mathrm{\Gamma }}_{\mu \kappa }^{\tau }{\mathrm{\Gamma }}_{\nu \eta }^{\theta })+2({\partial }_{\eta }{g}_{\tau \theta })({\mathrm{\Gamma }}_{\mu \kappa }^{\tau }{\mathrm{\Gamma }}_{\nu \lambda }^{\theta }-{\mathrm{\Gamma }}_{\mu \lambda }^{\tau }{\mathrm{\Gamma }}_{\nu \kappa }^{\theta })+2({\partial }_{\kappa }{g}_{\tau \theta })({\mathrm{\Gamma }}_{\mu \lambda }^{\tau }{\mathrm{\Gamma }}_{\nu \eta }^{\theta }-{\mathrm{\Gamma }}_{\mu \eta }^{\tau }{\mathrm{\Gamma }}_{\nu \lambda }^{\theta }) \\ +2g_{\tau \theta }{g}_{\rho \varphi }{\mathrm{\Gamma }}_{\nu \kappa }^{\theta }(({\partial }_{\lambda }{g}^{\tau \varphi }){\mathrm{\Gamma }}_{\mu \eta }^{\rho }-({\partial }_{\eta }{g}^{\tau \varphi }){\mathrm{\Gamma }}_{\mu \lambda }^{\rho }) \\ +2g_{\tau \theta }{g}_{\rho \varphi }{\mathrm{\Gamma }}_{\nu \lambda }^{\theta }(({\partial }_{\eta }{g}^{\tau \varphi }){\mathrm{\Gamma }}_{\mu \kappa }^{\rho }-({\partial }_{\kappa }{g}^{\tau \varphi }){\mathrm{\Gamma }}_{\mu \eta }^{\rho }) \\ +2g_{\tau \theta }{g}_{\rho \varphi }{\mathrm{\Gamma }}_{\nu \eta }^{\theta }(({\partial }_{\kappa }{g}^{\tau \varphi }){\mathrm{\Gamma }}_{\mu \lambda }^{\rho }-({\partial }_{\lambda }{g}^{\tau \varphi }){\mathrm{\Gamma }}_{\mu \kappa }^{\rho }) \end{gathered}$$
最後に,
$$({\partial }_{\lambda }{g}_{\tau \theta }){\mathrm{\Gamma }}_{\mu \eta }^{\tau }{\mathrm{\Gamma }}_{\nu \kappa }^{\theta }+g_{\tau \theta }{g}_{\rho \varphi }{\mathrm{\Gamma }}_{\nu \kappa }^{\theta }({\partial }_{\lambda }{g}^{\tau \varphi }){\mathrm{\Gamma }}_{\mu \eta }^{\rho }$$
$$={\mathrm{\Gamma }}_{\mu \eta }^{\tau }{\mathrm{\Gamma }}_{\nu \kappa }^{\theta }({\partial }_{\lambda }{g}_{\tau \theta }+g_{\rho \theta }{g}_{\tau \varphi }({\partial }_{\lambda }{g}^{\rho \varphi }))$$
$$={\mathrm{\Gamma }}_{\mu \eta }^{\tau }{\mathrm{\Gamma }}_{\nu \kappa }^{\theta }(g_{\rho \theta }{g}^{\rho \varphi }({\partial }_{\lambda }{g}_{\tau \varphi })+g_{\rho \theta }{g}_{\tau \varphi }({\partial }_{\lambda }{g}^{\rho \varphi }))$$
$$={\mathrm{\Gamma }}_{\mu \eta }^{\tau }{\mathrm{\Gamma }}_{\nu \kappa }^{\theta }{g}_{\rho \theta }({\partial }_{\lambda }{\delta }_{\tau }^{\rho })$$
$$=0$$
が成り立つので,
$$B$$
$$\begin{gathered} =2(({\partial }_{\lambda }{g}_{\tau \theta }){\mathrm{\Gamma }}_{\mu \eta }^{\tau }{\mathrm{\Gamma }}_{\nu \kappa }^{\theta }+g_{\tau \theta }{g}_{\rho \varphi }{\mathrm{\Gamma }}_{\nu \kappa }^{\theta }({\partial }_{\lambda }{g}^{\tau \varphi }){\mathrm{\Gamma }}_{\mu \eta }^{\rho }) \\ +2(({\partial }_{\eta }{g}_{\tau \theta }){\mathrm{\Gamma }}_{\mu \kappa }^{\tau }{\mathrm{\Gamma }}_{\nu \lambda }^{\theta }+g_{\tau \theta }{g}_{\rho \varphi }{\mathrm{\Gamma }}_{\nu \lambda }^{\theta }({\partial }_{\eta }{g}^{\tau \varphi }){\mathrm{\Gamma }}_{\mu \kappa }^{\rho }) \\ +2(({\partial }_{\kappa }{g}_{\tau \theta }){\mathrm{\Gamma }}_{\mu \lambda }^{\tau }{\mathrm{\Gamma }}_{\nu \eta }^{\theta }+g_{\tau \theta }{g}_{\rho \varphi }{\mathrm{\Gamma }}_{\nu \eta }^{\theta }({\partial }_{\kappa }{g}^{\tau \varphi }){\mathrm{\Gamma }}_{\mu \lambda }^{\rho }) \\ -2(({\partial }_{\lambda }{g}_{\tau \theta }){\mathrm{\Gamma }}_{\mu \kappa }^{\tau }{\mathrm{\Gamma }}_{\nu \eta }^{\theta }+g_{\tau \theta }{g}_{\rho \varphi }{\mathrm{\Gamma }}_{\nu \eta }^{\theta }({\partial }_{\lambda }{g}^{\tau \varphi }){\mathrm{\Gamma }}_{\mu \kappa }^{\rho }) \\ -2(({\partial }_{\eta }{g}_{\tau \theta }){\mathrm{\Gamma }}_{\mu \lambda }^{\tau }{\mathrm{\Gamma }}_{\nu \kappa }^{\theta }+g_{\tau \theta }{g}_{\rho \varphi }{\mathrm{\Gamma }}_{\nu \kappa }^{\theta }({\partial }_{\eta }{g}^{\tau \varphi }){\mathrm{\Gamma }}_{\mu \lambda }^{\rho }) \\ -2(({\partial }_{\kappa }{g}_{\tau \theta }){\mathrm{\Gamma }}_{\mu \eta }^{\tau }{\mathrm{\Gamma }}_{\nu \lambda }^{\theta }+g_{\tau \theta }{g}_{\rho \varphi }{\mathrm{\Gamma }}_{\nu \lambda }^{\theta }({\partial }_{\kappa }{g}^{\tau \varphi }){\mathrm{\Gamma }}_{\mu \eta }^{\rho }) \end{gathered}$$
$$=0$$

以上より, $B=0$が示された！やったあ！

終わりに

もしこの記事をここまで読んでくださった方がいらっしゃったら、本当にありがとうございました！こんな記事にもなってない記事投稿して申し訳ございませんでした(土下座)！！！