Given the definition of the covariant derivation of one order tensor :(adsbygoogle = window.adsbygoogle || []).push({});

[tex]\nabla _j v_k = \partial _j v_k - v_i \Gamma _{kj}^i[/tex]

How can we proove the covariant derivation of the Riemann-Christoffel tensor given by :

[tex]R_{i{\text{ }},{\text{ }}jk}^l = \partial _j \Gamma _{ik}^l - \partial _k \Gamma _{ij}^l + \Gamma _{ik}^r \Gamma _{jr}^l - \Gamma _{ij}^r \Gamma _{kr}^l[/tex]

is :

[tex]\nabla _t R_{i,rs}^l = \partial _{rt} \Gamma _{si}^l - \partial _{st} \Gamma _{ri}^l[/tex]

This could not be difficult but i'm afraid to make an error with the indices.

This proof will help me to define the Einstein tensor.

Thanks for your help.

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# Third Bianchi identity

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