# Christoffel symbol

In flat space, it is possible to find a coordinate system in which the Christoffel symbols are equal to zero at every point on the flat manifold. However, I was wondering if it is possible to find a coordinate system on a curved Riemannian manifold in which a specific point on that manifold has its Christoffel symbols equal to zero. I was thinking that maybe if the coordinate system is chosen such that the coordinate lines going through that aforementioned point are geodesics, then maybe the Christoffel symbol at that point can be equal to zero.

In any number of dimensions, we can find coordinates $$x^\mu$$ in which $$g_{\mu\nu}(x)= \delta_{\mu\nu}- \frac 13 R_{\mu\sigma \nu\tau}(0) x^\sigma x^\tau + O(|x|^3),$$ $${\Gamma^{\lambda}}_{\mu\nu}(x)= -\frac 13 (R_{\lambda\nu\mu\tau}(0)+R_{\lambda\mu\nu\tau}(0))x^\tau+ O(|x|^2).$$
Similarly we we can construct a local vielbein frame $${\bf e}_a$$ and co-frame $${\bf e}^{*a}$$ in which we have $$e_a^\mu(x)= \delta_{a \mu}+ \frac 16 R_{a \sigma \mu\tau}(0) x^\sigma x^\tau +O(x^2)$$ $$e^{*a}_\mu(x)= \delta_{a \mu}- \frac 16 R_{a \sigma \mu\tau}(0) x^\sigma x^\tau +O(x^2).$$ The spin connection associated to this frame is
$${\omega^a}_{b\mu}(x)=- \frac 12 {R^a}_{b\mu\tau}(0)x^\tau+O(|x|^2).$$