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I'm currently working through

Any help would be much appreciated.

*chapter 7*on*Riemannian geometry*in Nakahara's book*"Geometry, topology & physics"*and I'm having a bit of trouble reproducing his calculation for the metric compatibility in a non-coordinate basis, using the*Ricci rotation coefficients*##\Gamma_{\alpha\beta\gamma}\equiv\delta_{\alpha\delta}\Gamma^{\delta}_{\;\beta\gamma}## that he defines in*section 7.8.4 ("Levi-Civita connection in a non-coordinate basis"*). Here's his calculation: $$\Gamma_{\alpha\beta\gamma}=\delta_{\alpha\delta}e^{\delta}_{\;\lambda}e_{\beta}^{\;\mu}\nabla_{\mu}e_{\gamma}^{\;\lambda}\\ \qquad\quad=-\delta_{\alpha\delta}e_{\gamma}^{\;\lambda}e_{\beta}^{\;\mu}\nabla_{\mu}e^{\delta}_{\;\lambda}\\ \qquad\quad =-\delta_{\gamma\delta}e^{\delta}_{\;\lambda}e_{\beta}^{\;\mu}\nabla_{\mu}e_{\alpha}^{\;\lambda}\\ =-\Gamma_{\gamma\beta\alpha}\quad$$ where ##\Gamma^{\delta}_{\;\beta\gamma}=e^{\delta}_{\;\lambda}e_{\beta}^{\;\mu}\nabla_{\mu}e_{\gamma}^{\;\lambda}## has been used. He states that this is found using that ##\nabla_{\mu}g=0##, but I can't seem to reproduce the result. I assume that between lines 1 and 2 he simply uses that $$e_{\gamma}^{\;\lambda}e_{\beta}^{\;\mu}\nabla_{\mu}e^{\delta}_{\;\lambda}=-\delta_{\gamma\delta}e^{\delta}_{\;\lambda}e_{\beta}^{\;\mu}\nabla_{\mu}e_{\alpha}^{\;\lambda}+\delta_{\gamma\delta}e_{\beta}^{\;\mu}\nabla_{\mu}\left(e_{\alpha}^{\;\lambda}e^{\delta}_{\;\lambda}\right)=-\delta_{\gamma\delta}e^{\delta}_{\;\lambda}e_{\beta}^{\;\mu}\nabla_{\mu}e_{\alpha}^{\;\lambda}$$ since ##e_{\alpha}^{\;\lambda}e^{\delta}_{\;\lambda}=\delta_{\alpha\delta}=diag\lbrace 1,1,1,1\rbrace## and so ##\nabla_{\mu}\left(e_{\alpha}^{\;\lambda}e^{\delta}_{\;\lambda}\right)=0##. I don't see how he gets from the second to the third line using ##\nabla_{\mu}g=0\Rightarrow \partial_{\mu}g_{\nu\lambda}-g_{\kappa\lambda}\Gamma^{\kappa}_{\;\mu\nu}-g_{\nu\kappa}\Gamma^{\kappa}_{\;\mu\lambda}=0## though, as when I naively use this result I end up with additional terms and no overall minus sign.Any help would be much appreciated.

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