# 活动标架的结构方程

$\begin{cases} \rd \omega^i=-\omega^i_j\wedge\omega^j+\frac{1}{2}T_{kl}^i\omega^k\wedge\omega^l\\ \rd \omega^i_j=-\omega^i_k\wedge\omega^k_j+\Omega_j^i, \end{cases}$

$\Omega_j^i$称为曲率形式. 回忆, $(1,3)$型曲率张量定义为
$R(X,Y)Z=\nabla_X\nabla_YZ-\nabla_Y\nabla_XZ-\nabla_{[X,Y]}Z,$

$R(X,Y,Z,W)=g(R(X,Y)Z,W),$

\begin{align*}
\rd\omega^i(e_k,e_l)&=e_k\omega^i(e_l)-e_l\omega^i(e_k)-\omega^i([e_k,e_l])\\
&=\nabla_{e_k}(\omega^i(e_l))-\nabla_{e_l}(\omega^i(e_k))-\omega^i([e_k,e_l])\\
&=(\nabla_{e_k}\omega^i)(e_l)+\omega^i(\nabla_{e_k}e_l)-(\nabla_{e_l}\omega^i)(e_k)\\
&=-\omega^i_j(e_k)\omega^j(e_l)+\omega^i_j(e_l)\omega^j(e_k)+\omega^i(\nabla_{e_k}e_l)\\
&=-\omega^i_j\wedge\omega^j(e_k,e_l)+\omega^i(T(e_k,e_l))\\
&=\left(-\omega^i_j\wedge\omega^j+\frac{1}{2}T_{pq}^i\omega^p\wedge\omega^q\right)(e_k,e_l).
\end{align*}

\begin{align*}
\rd\omega^i_j(e_k,e_l)&=e_k(\omega^i_j(e_l))-e_l(\omega^i_j(e_k))-\omega^i_j([e_k,e_l])\\
&=\nabla_{e_k}(\omega^i(\nabla_{e_l}e_j))-\nabla_{e_l}(\omega^i(\nabla_{e_k}e_j))-\omega^i_j([e_k,e_l])\\
&=(\nabla_{e_k}\omega^i)(\nabla_{e_l}e_j)+\omega^i(\nabla_{e_k}(\nabla_{e_l}e_j))\\