# Kaluza-Klein Christoffel Symbols

You missed a term in expanding the upper-indexed metric. The full version is below: \begin{align} \tilde{\Gamma}^\lambda_{\mu\nu} & = \frac{1}{2} \tilde{g}^{\lambda X} \left(\partial_\mu \tilde{g}_{\nu X} + \partial_\nu \tilde{g}_{\mu X} - \partial_X \tilde{g}_{\mu\nu}\right) \\ & =\frac{1}{2} \tilde{g}^{\lambda\sigma} \left(\partial_\mu \tilde{g}_{\nu\sigma} + \partial_\nu \tilde{g}_{\mu\sigma} - \partial_\sigma \tilde{g}_{\mu\nu}\right) + \frac{1}{2} \tilde{g}^{\lambda5} \left(\partial_\mu \tilde{g}_{\nu5} + \partial_\nu \tilde{g}_{\nu5} - \partial_5 \tilde{g}_{\mu\nu}\right) \\ & = \frac{1}{2} g^{\lambda\sigma} \left(\partial_\mu \left(g_{\nu\sigma} + k A_\nu A_\sigma\right) + \partial_\nu \left(g_{\mu\sigma} + k A_\mu A_\sigma\right) - \partial_\sigma \left(g_{\mu\nu} + k A_\mu A_\nu\right)\right) \\ & \quad \qquad + \frac{1}{2} \left(-A^\lambda\right) \left(\partial_\mu \left(k A_\nu\right) + \partial_\nu \left(k A_\mu\right) - \partial_5 \left(g_{\mu\nu} + k A_\mu A_\nu\right)\right) \\ & = \frac{1}{2} g^{\lambda\sigma} \left(\partial_\mu g_{\nu\sigma} + \partial_\nu g_{\mu\sigma} - \partial_\sigma g_{\mu\nu}\right) + \frac{k}{2} g^{\lambda\sigma} \left(\partial_\mu \left(A_\nu A_\sigma\right) \partial_\nu \left(A_\mu A_\sigma\right) - \partial_\sigma \left(A_\mu A_\nu\right)\right) \\ & \quad \qquad - \frac{k}{2} A^\lambda \left(\partial_\mu A_\nu + \partial_\nu A_\mu\right) \\ & = \Gamma^\lambda_{\mu\nu} + \frac{k}{2} g^{\lambda\sigma} \left(A_\mu \left(\partial_\nu A_\sigma - \partial_\sigma A_\nu\right) + A_\nu \left(\partial_\mu A_\sigma - \partial_\sigma A_\mu\right) + A_\sigma \left(\partial_\mu A_\nu + \partial_\nu A_\mu\right)\right) \\ & \quad \qquad - \frac{k}{2} g^{\lambda\sigma} A_\sigma \left(\partial_\mu A_\nu + \partial_\nu A_\mu\right) \\ & = \Gamma^\lambda_{\mu\nu} + \frac{k}{2} \left(A_\mu F_{\nu\sigma} + A_\nu F_{\mu\sigma}\right). \end{align} This is because we have \begin{cases} \tilde{g}_{\mu\nu} = g_{\mu\nu} + k A_\mu A_\nu \\ \tilde{g}_{\mu5} = k A_\mu \\ \tilde{g}_{55} = k \end{cases} and also \begin{cases} \tilde{g}^{\mu\nu} = g^{\mu\nu} \\ \tilde{g}^{\mu5} = -A_\mu \\ \tilde{g}^{55} = \frac{1}{k} + A_\mu A^\mu. \end{cases}