On the Addedum to the Chapter VI, vol. II, the first proposition states that to connections have the same geodesics if and only if their difference tensor is antisymmetric.
When proving that if the geodesics are the same the tensor is antisymmetric, he uses the following equality $$ \nabla_{X_p}X = 0 $$
where the vector field $X$ is arbitrary. I don't understand why this shoud hold, since at a first moment this is only true when $X = \dot\gamma$ for a geodesic $\gamma$ (which is a geodesic for both connections).
It seems to me this is the same as stating that, at any point, every field has an integral curve that is a geodesic for a short interval around that point, which doesn't feel to me like it should be true.