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I'm trying to calculate the covariant derivative of a Riemann tensor, and I'm using the following way, but there is some problem in my calculations because my calculations do not match with the calculations mentioned in the research paper.

\begin{align} R_{ijkl} &= g_{im} R^{m}{}_{jkl}\tag1\\ \nabla^{a} R_{ijkl} &= g_{im} \nabla^{a} R^{m}{}_{jkl}\tag2\\ \nabla^{a} R_{ijkl} &= g_{im} g^{ab} \nabla_{b} R^{m}{}_{jkl}\tag3 \end{align} where $\nabla_{b} R^{m}_{jkl}$ I have defined as

\begin{equation} \nabla_{i} R^{j}_{abc} = R^{j}_{abc,i} + \Gamma^{j}_{ik} R^{k}_{abc} - \Gamma^{m}_{ia} R^{j}_{mbc} - \Gamma^{m}_{ib} R^{j}_{amc} - \Gamma^{m}_{ic} R^{j}_{abm} \end{equation}

Can anyone please help me in this regard? Where could be a problem? I'm sure that the definition of riemann= $R^{i}_{jkl}$ is the Riemann tensor.

Thank you.

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    $\begingroup$ Please look at this page mathematica.stackexchange.com/questions/289491/… $\endgroup$
    – chris
    Commented Apr 20 at 6:06
  • $\begingroup$ We need to see this in mathJax and also what that research paper claims. $\endgroup$
    – Kurt G.
    Commented Apr 24 at 4:24
  • $\begingroup$ I have pasted code from the Mathematica notebook, but it is not formatting well in the post. $\endgroup$
    – MMS
    Commented May 13 at 13:52
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    $\begingroup$ Are you using this: sites.math.washington.edu/~lee/Ricci $\endgroup$
    – Deane
    Commented May 14 at 17:43
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    $\begingroup$ Thanks @JackozeeHakkiuz, yes you are right. I was trying to calculate it for the Schwarzschild spacetime. I have obtained the required results! $\endgroup$
    – MMS
    Commented May 20 at 9:20

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