Is it true that for any flat and torsion-free connection $\nabla : \mathfrak{X} (M) \times \mathfrak{X} (M) \rightarrow \mathfrak{X} (M) $ there exist a local systems of coordinates such that the christoffel symbols are 0?
Could you provide a reference where to find the result proved?
Thank you very much!
Michael Spivak, A Comprehensive Introduction to Differential Geometry, volume II, p. 383, theorem 19, proves that the vanishing of curvature tensor of a Riemannian manifold implies flatness. But in fact, if you look at the proof, he actually only uses the vanishing of curvature and torsion of the affine connection, and proves that there are coordinates in which parallel translation is just usual translation by a vector, i.e. the coordinate vector fields are parallel. It then follows that the Christoffel symbols are zero, as the operation $\nabla_X Y$ vanishes on those vector fields.
