We know in general relativity (or rather in differential geometry) that you can have some locally flat coordinates (I think they are called Riemann normal coordinates) at a point $P$ in our manifold (space time). At this point $P$, the metric is Euclidean up to second order deviation, i.e. $$ g_{\tau \mu} \approx \eta_{\tau \mu} + B_{\tau \mu \ ,\lambda \sigma} \ x^\lambda x^\sigma + ... $$ where $B_{\tau \mu \ ,\lambda \sigma}$ are just the Taylor coefficient terms (second order in $g$).
Now I was led to believe that the Christoffel symbols should vanish at this point $P$ in locally flat coordinates, but under the definition of them, I get
\begin{split} \Gamma_{\rho \nu}^\lambda & \equiv \frac{1}{2} g^{\lambda \tau} (\partial_\rho g_{\nu \tau} + \partial_{\nu} g_{\rho \tau} - \partial_{\tau} g_{\rho \nu}) \\ & = \eta^{\lambda \tau}(B_{\tau \nu \ , \kappa \rho} + B_{\tau \rho \ , \kappa \nu} - B_{\rho \nu \ , \kappa \tau})\ x^\kappa + ... \end{split}
This is a non vanishing Christoffel symbol. If I am misunderstanding, then when exactly do the symbols vanish?
