Deriving Christoffel symbols from the metric covariant derivative

[elenuccia]

How can I derive the Christoffel symbol from the vanishing of the covariant derivative of the metric tensor? can somebody write the calculation, I read that I have to do some permutation and resumming but I don't get the result! Thank you!

 

[Fredrik]

Did you obtain a result like

$$g_{\rho\sigma,\mu}=\Gamma^\lambda_{\mu\rho}g_{\lambda\sigma}+\Gamma^\lambda_{\mu\sigma}g_{\rho\lambda}$$​

already? In that case, consider the quantity

$$g_{\mu\sigma,\rho}+g_{\mu\rho,\sigma}-g_{\rho\sigma,\mu}$$​

and I think you'll be able to figure out the rest. Don't forget that a Levi-Civita connection is torsion free. This implies that the Christoffel symbol is symmetric in the lower indices.