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Trying to follow a calculation through a paper where, I think, something strange is happening with the indices in the product terms: Author defines the curvature tensor with covariant indices. Author places non-symmetric index last.

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.

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  • $\begingroup$ If you're familiar with connections on general bundles, we can write the connection as $\Gamma^\alpha{}_{\beta \mu}$, where $\mu$ is a 1-form index but $\alpha$ and $\beta$ are Lie algebra indices – in this case, $GL(n)$ Lie algebra indices. For a metric compatible connection we can reduce $GL(n)$ to $SO(1,n-1)$. The Lie algebra of rotation matrices consists of antisymmetric matrices, whilst the same is true of Lorentz transformations if we lower an index. $\endgroup$ – gj255 Feb 1 '17 at 12:37
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    $\begingroup$ Would Mathematics be a better home for this question? $\endgroup$ – Qmechanic Feb 1 '17 at 17:38
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    $\begingroup$ Just look at the formulas that you yourself have written down. In the formula for $\Gamma_{ijk}$ what is the relationship between $\Gamma_{ikj}$ and $\Gamma_{ijk}$? $\endgroup$ – Prahar Feb 1 '17 at 17:42
  • $\begingroup$ It's symmetric in the first two indices and the third is the odd one out. So $\Gamma_{ijk} = \Gamma_{jik}$, or $[ij,k]=[ji,k]$. $\endgroup$ – Bothorth Feb 1 '17 at 18:36
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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}

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