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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 $R=\text{d}\Gamma-[\Gamma\wedge\Gamma]$. 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 $$\partial_ag_{bc}=\Gamma_{bca}-\Gamma_{cba}.$$

Bonus question: How do you do index placement in stackexchange?

Ivan Burbano
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  • It was solved in https://physics.stackexchange.com/questions/699707/alternative-expression-for-riemann-curvature-tensor/699730#699730 – Ivan Burbano Mar 20 '22 at 05:59

1 Answers1

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The metric tensor $g_{fe}$ is used to lower the index $f$ and rename it to $e$: $$ g_{fe}\Gamma^f_{md}=\Gamma_{emd}\,. $$ and $g^{am}$ is used to raise $m$ and rename it to $a$. Try to apply this to the expression $\Gamma^f_{md}\Gamma^e_{cb}-\Gamma^f_{mc}\Gamma^e_{db}\,.$ It must also have some symmetry properties which should finally give the first expression.

Index placement I always to like this: ${R^a}_{bcd}$

Kurt G.
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  • Thank you very much for your answer. The problem I am having is precisely, how to turn an expression like $\Gamma_{eab}{\Gamma^{e}}{cd}$ into something like $\Gamma{aeb}{\Gamma^{e}}{ab}$. In doing so I obtain a derivative $\partial_b g{ae}{\Gamma^{e}}_{cd}$ that I don't know how to get rid of in the Riemann tensor. – Ivan Burbano Mar 19 '22 at 21:11
  • Thanks for the comments on index placement! I've used them to appropriately raise and lower all indices to make my question clearer! – Ivan Burbano Mar 19 '22 at 21:14
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    @IvanBurbano . Looks like you solved it. Note that a single Cristoffel symbol is not a tensor. That's why many authors (most?) don't use index placement for them. Only for real tensors like $R$. The GR book by Sean Carroll explains this well. – Kurt G. Mar 20 '22 at 10:03
  • The GR book by Sean Carroll explains this well. See also arXiv:gr-qc:9712019. – Kurt G. Mar 20 '22 at 10:10