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I have been doing a few proofs and stumbled across this little problem.

Trying to show the symmetry of the Ricci tensor by using the Riemann tensor definition

##R^m_{\ ikp} = \partial_k \Gamma^m_{\ ip} - \partial_p \Gamma^m_{\ ki} + \Gamma^a_{\ ip} \Gamma^m_{\ ak} - \Gamma^a_{\ ik} \Gamma^m_{\ ap}##

Now set m = k

##R^m_{\ \,imp} = R_{ip} = \partial_m \Gamma^m_{\ ip} - \partial_p \Gamma^m_{\ mi} + \Gamma^a_{\ ip} \Gamma^m_{\ am} - \Gamma^a_{\ im} \Gamma^m_{\ ap}##

Checking every term for symmetry (i <-> p)

1. symmetric because ##\Gamma^{m}_{\ ip} = \Gamma^{m}_{\ pi}##

2. see below

3. symmetric because ##\Gamma^{m}_{\ ip} = \Gamma^{m}_{\ pi}##

4. symmetric because ##\Gamma^a_{\ im} \Gamma^m_{\ ap} = \Gamma^m_{\ pa} \Gamma^a_{\ mi}## (interchanging a & m, and just rotate) and ##\Gamma^{a}_{\ im} = \Gamma^{a}_{\ mi}##

Now for the 2nd term

##\partial_p \Gamma^m_{\ mi} == \partial_i \Gamma^m_{\ mp}##

This is a contracted Christoffel symbol...

using ##\Gamma^k_{ij} = \frac{1}{2} g^{kl} \left(\partial_i g_{jl} + \partial_j g_{il} - \partial_l g_{ij}\right)##

leads to

##\Gamma^{m}_{\ mi} = \frac{1}{2} g^{ml} \frac{\partial g_{ml}}{\partial {x^i}} ##

Now the derivative

##\partial_p \Gamma^m_{\ mi} = \frac{1}{2} \left( \frac{\partial g^{ml}}{\partial x^p} \frac{\partial g_{ml}}{\partial {x^i}} + g^{ml} \frac{\partial g_{ml}}{\partial {x^i} \partial {x^p}}\right)##

The 2nd term in this expression is symmetric (i <-> p), because order of partial differentiation doesn't matter.

For the first term I am not sure though.

Is ##\frac{\partial g^{ml}}{\partial x^p} \frac{\partial g_{ml}}{\partial {x^i}} == \frac{\partial g^{ml}}{\partial x^i} \frac{\partial g_{ml}}{\partial {x^p}}##?

The inverse is something totally different, but you also contract it, this is the point where I am slightly confused.

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I know there are other ways to "derive" the symmetry of the Ricci tensor, but I wanted to try this one :)

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Thank you in advance!

It's probably so obvious that I don't see it

Hope you guys had a nice Christmas ;)

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# Symmetry of Ricci Tensor

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