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Generally the Christoffel symbol of the first kind is defined as

$$\Gamma_{\lambda\mu\nu}=\frac12\,(\partial_\nu g_{\lambda\mu}+\partial_\mu g_{\lambda\nu}-\partial_\lambda g_{\mu\nu}) \tag{1}$$

and the Christoffel symbol of the second kind is defined as

$$\Gamma^\rho{}_{\mu\nu}=g^{\rho\lambda}\Gamma_{\lambda\mu\nu}=\frac12\,g^{\rho\lambda}\,(\partial_\nu g_{\lambda\mu}+\partial_\mu g_{\lambda\nu}-\partial_\lambda g_{\mu\nu}) \tag{2}.$$

Is the following definition of $\Gamma_{\mu\nu}{}^\rho$ correct or should it be something else?

\begin{align} \Gamma_{\mu\nu}{}^\rho=g^{\rho\lambda}\Gamma_{\mu\nu\lambda}&=\frac12\,g^{\rho\lambda}\,(\partial_\lambda g_{\mu\nu}+\partial_\nu g_{\mu\lambda}-\partial_\mu g_{\nu\lambda}) \tag{3} \\ &=g^{\rho\lambda}\,\partial_\nu g_{\lambda\mu}-\Gamma^\rho{}_{\mu\nu}. \end{align}

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    $\begingroup$ Be aware that different authors use different ordering conventions for the indices of the Christoffel symbols. $\endgroup$
    – Qmechanic
    Apr 20, 2023 at 7:36
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    $\begingroup$ No, you have to be rather careful about which convention a book or a paper is using. Whether the index that is raised is in the leftmost slot or the rightmost slot is merely convention, definition of the Christoffel symbol. We never do the weird thing of lowering that index and raising another one. It would not be physically sensible that way. The only sensible transformation of those symbols is to lower the only raised index, so that we can study the symmetry properties. Other than that, we tend to leave the raised and lowered indices as they normally are. $\endgroup$ Apr 20, 2023 at 7:39
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    $\begingroup$ Note that the Christoffel Symbols are not a tensor $\endgroup$
    – Wihtedeka
    Apr 20, 2023 at 7:58
  • $\begingroup$ Thanks everyone! I understand it now. $\endgroup$
    – vyali
    Apr 24, 2023 at 15:37

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