It's a standard fact of GR that at a given point in space-time, we can construct a coordinate system such that the metric tensor takes the form of Minkowski spacetime and its first derivatives vanish. Equivalently, we can make the Christoffel symbols vanish at point. Moreover, the fact that, in general, there's no coordinate system that lets us do this simultaneously for the entire manifold is the essence of curvature. But, is it possible to construct a coordinate system such that the metric is Minkowskian along an entire geodesic? Since geodesics are the generalization in GR of inertial motion, it seems intuitively that it should be possible to do so. According to the equivalence principle, an arbitrarily small 'laboratory' falling along a geodesic should be unable to determine if its in an inertial (in the SR sense) frame in flat spacetime or falling along a geodesic in curved spacetime. It seems to me that finding a coordinate system in which the Christoffel symbols along the geodesic vanish would be mathematical realization of this.