# derivation of the Christoffel symbol

by elenuccia
Tags: christoffel, derivation, symbol
 P: 1 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!
 Emeritus Sci Advisor PF Gold P: 8,864 Did you obtain a result like $$g_{\rho\sigma,\mu}=\Gamma^\lambda_{\mu\rho}g_{\lambda\sigma}+\Gamma^\la mbda_{\mu\sigma}g_{\rho\lambda}$$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.

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