Christoffel symbol:
\begin{align}
\Gamma_{\lambda \, {\color{red}\mu} {\color{red}\nu}} &= \frac{1}{2} (g_{\lambda {\color{red}\mu},{\color{red}\nu}} + g_{\lambda {\color{red}\nu},{\color{red}\mu}} - \partial_{\lambda} g_{{\color{red}\mu \nu}}) \\
\Gamma^{\rho}_{{\color{red}\mu} {\color{red} \nu}} &= \frac{1}{2} g^{\rho \lambda} (g_{\lambda {\color{red}\mu},{\color{red}\nu}} + g_{\lambda {\color{red}\nu},{\color{red}\mu}} - \partial_{\lambda} g_{{\color{red}\mu \nu}}) = g^{\rho \lambda} \Gamma_{\lambda \, {\color{red}\mu } {\color{red}\nu}} .