How can we prove the covariant derivation of the Riemann-Christoffel tensor?

[Jinroh]

Given the definition of the covariant derivation of one order tensor :

$$\nabla _j v_k = \partial _j v_k - v_i \Gamma _{kj}^i$$

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


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

is :

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

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.

 

[Jinroh]

hum...

perhaps by moving this post in Special and General Relativity Room I will have an answer... ?

 

[sambo]

perhaps by moving this post in Special and General Relativity Room I will have an answer... ?

Maybe. I was looking at your problem last night, and I think I should be able to have something for you by this evening or tomorrow morning (party tonight, so no guarantees for today...)

However, while we're on the topic of soliciting responses, do you have any ideas about my question of a geometric interpretation of the Laplacian on a manifold? How's that for a shameless plug!