曲率tensorの計算やりなおし

$ {R_{ijk}}^m:=\frac{\partial{\Gamma_{ik}}^m}{\partial\bar e_j}-\frac{\partial{\Gamma_{ij}}^m}{\partial\bar e_k}+{\Gamma_{lj}}^m{\Gamma_{ik}}^l-{\Gamma_{lk}}^m{\Gamma_{ij}}^l

$ =\frac{\partial{\Gamma_{ik}}^m}{\partial\bar e_j}-\frac{\partial{\Gamma_{ij}}^m}{\partial\bar e_k}-{{\Gamma_j}^m}_l{\Gamma_{ik}}^l+{{\Gamma_k}^m}_l{\Gamma_{ij}}^l

$ =\frac{\partial\pmb\Gamma_{ik}^{\sf EE}}{\partial\bar e_j}\cdot\bar{\pmb e}_m+\pmb\Gamma_{ik}^{\sf EE}\cdot\pmb\Gamma_{jm}^{\sf E\bar E}-\frac{\partial\pmb\Gamma_{ij}^{\sf EE}}{\partial\bar e_k}\cdot\bar{\pmb e}_m-\pmb\Gamma_{ij}^{\sf EE}\cdot\pmb\Gamma_{km}^{\sf E\bar E}-\pmb\Gamma^{\sf E\bar E}_{jm}\cdot\pmb\Gamma^{\sf EE}_{ik}+\pmb\Gamma^{\sf E\bar E}_{km}\cdot\pmb\Gamma^{\sf EE}_{ij}

$ =0

こまったな。本当に0になってしまう