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There is the usual expression for the Riemann tensor

$$R_{abcd}=\partial_c\Gamma_{adb}-\partial_d\Gamma_{acb}+\Gamma_{ace}{\Gamma^e}_{db}-\Gamma_{ade}{\Gamma^e}_{cb}.$$ However, in the last page of https://www.mathi.uni-heidelberg.de/~walcher/teaching/wise1516/geo_phys/SigmaAndLGModels.pdf, another expression is used: $$R_{abcd}=\partial_c\Gamma_{adb}-\partial_d\Gamma_{acb}+\Gamma_{ead}{\Gamma^e}_{cb}-\Gamma_{eac}{\Gamma^e}_{db}.$$

How does one obtain the first expression from the second? I've never seen the second expression before. The first one is obvious from the expression $R=\text{d}\Gamma+\frac{1}{2}[\Gamma\wedge\Gamma]$. However, the second one is less intuitive. In orthogonal coordinates it would be easy to obtain since $$\Gamma_{abc}=-\Gamma_{bac}$$

However, is there a way to see this two expression are equivalent without using orthogonality? I think it will have to do with the metricity condition $$\partial_ag_{bc}=\Gamma_{bca}-\Gamma_{cba}$$

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    $\begingroup$ I think the second equation is correct and the first is incorrect. The first equation with raised $a$ is correct, but lowering it doesn't commute with the derivative. $\endgroup$
    – benrg
    Mar 20, 2022 at 5:03
  • $\begingroup$ Yes! You are 100% correct! Thank you so much! I'll add the computation as an answer to the computation just in case anyone else is interested in the future. But if you'd like to repost the comment as an answer, I'll gladly accept it $\endgroup$ Mar 20, 2022 at 5:47

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Following @benrg comment, I made the mistake of passing the metric past a partial derivative. In case anyone is interested, the computation is $$R_{abcd}=g_{ae}(\partial_c{\Gamma^e}_{db}-\partial_d{\Gamma^e}_{cb}+{\Gamma^e}_{cf}{\Gamma^f}_{db}-{\Gamma^e}_{df}{\Gamma^f}_{cb})=\partial_c{\Gamma}_{adb}-\partial_d{\Gamma}_{acb}+({\Gamma}_{acf}-\partial_cg_{ae}){\Gamma^f}_{db}-({\Gamma}_{adf}-\partial_dg_{af}){\Gamma^f}_{cb}=\partial_c{\Gamma}_{adb}-\partial_d{\Gamma}_{acb}-{\Gamma}_{fca}{\Gamma^f}_{db}+{\Gamma}_{fda}{\Gamma^f}_{cb},$$ where in the second step we integrated by parts the derivative terms (and did some index renaming), while on the third we used the torsion-free condition mentioned in the question.

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