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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.

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You are perfectly right. In a curved Riemannian manifold, a theorem says that "it is always possible to make a coordinate transformation such that in the neighbourhood of some specified point P the line element takes the Euclidean form." Such coordinates are then in particular characterized by the fact that partial derivatives of the metric components relative to these new coordinates vanish at point P. But since Christoffel symbols can be expressed using precisely these metric derivatives, the previous condition on partial derivatives is equivalent to the vanishing of Christoffel symbols at P. The corresponding coordinates are indeed generally referred as "geodesic coordinates" about P.

A precise description of the above is given in "Hobson, Efstathiou, Lasenby, General relativity", §2.11 and 3.11.

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Here are some useful formulas for geodesic coordinates on Riemann manifold. I have not seen them explicitly said any introductory book, although they are frequently used in calculations.

They show how the Riemann tensor influences the the metric and Christoffel symbols in the neighbourhood of the the point at which the Christoffel symbols are 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
$$ {\omega^a}_{b\mu}(x)=- \frac 12 {R^a}_{b\mu\tau}(0)x^\tau+O(|x|^2). $$

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