Covariant Riemann tensor indices Trying to follow a calculation through a paper where, I think, something strange is happening with the indices in the product terms:


How does $\Gamma_{isl}\Gamma^s_{jk}$ in the second line become $-\Gamma_{jks}g^{st}\Gamma_{ilt}$ in the third line? This amounts to setting  $\Gamma_{isl} = -\Gamma_{ils}$, given the definition. But the first two indices are the symmetric ones and the author is swapping one with the last index (as in, $[is,l]=-[il,s]$). This shouldn't be anti-symmetric in general. 
 A: Got it. Had to look at the derivative terms. Use:
$$g_{ij,k}=\Gamma_{kij}+\Gamma_{kji}$$
and
\begin{align}
0&=(\delta_i^{\,j})_{,k}\\
&=(g_{is}g^{sj})_{,k}\\
&=g_{is,k}g^{sj}+g_{is}g^{sj}_{\,\,\,,k}\\
\therefore\,\,&g_{is,k}g^{sj}=-g_{is}g^{sj}_{\,\,\,,k}
\end{align}
Then:
\begin{align}
R_{ijkl}
&=g_{sl}(\partial_i\Gamma^{\,\,\,s}_{jk}-\partial_j\Gamma^{\,\,\,s}_{ik})+\Gamma_{isl}\Gamma^{\,\,\,s}_{jk}-\Gamma_{jsl}\Gamma^{\,\,\,s}_{ik}\\
&=g_{sl}[\partial_i(g^{st}\Gamma_{jkt})-\partial_j(g^{st}\Gamma_{ikt})]+\Gamma_{isl}\Gamma^{\,\,\,s}_{jk}-\Gamma_{jsl}\Gamma^{\,\,\,s}_{ik}\\
&=g_{sl}[(g^{st}_{\,\,\,,i}\Gamma_{jkt}+g^{st}\Gamma_{jkt,i})-(g^{st}_{\,\,\,,j}\Gamma_{ikt}+g^{st}\Gamma_{ikt,j})]+\Gamma_{isl}\Gamma^{\,\,\,s}_{jk}-\Gamma_{jsl}\Gamma^{\,\,\,s}_{ik}\\
&=[(-g_{sl,i}g^{st}\Gamma_{jkt}+\delta_l^{\,t}\Gamma_{jkt,i})-(-g_{sl,j}g^{st}\Gamma_{ikt}+\delta_l^{\,t}\Gamma_{ikt,j})]+g^{st}(\Gamma_{isl}\Gamma_{jkt}-\Gamma_{jsl}\Gamma_{ikt})\\
&=\Gamma_{jkl,i}-\Gamma_{ikl,j}+g^{st}(g_{sl,j}\Gamma_{ikt}-g_{sl,i}\Gamma_{jkt}+\Gamma_{isl}\Gamma_{jkt}-\Gamma_{jsl}\Gamma_{ikt})\\
&=\Gamma_{jkl,i}-\Gamma_{ikl,j}+g^{st}[\Gamma_{ikt}(g_{sl,j}-\Gamma_{jsl})-\Gamma_{jkt}(g_{sl,i}-\Gamma_{isl})]\\
&=\Gamma_{jkl,i}-\Gamma_{ikl,j}+g^{st}(\Gamma_{ikt}\Gamma_{jls}-\Gamma_{jkt}\Gamma_{ils})\\
&=\Gamma_{jkl,i}-\Gamma_{ikl,j}+g^{st}(\Gamma_{iks}\Gamma_{jlt}-\Gamma_{jks}\Gamma_{ilt})
\end{align}
