I was trying to prove that for any second order tensor:
considering the torsion free property and locally flat coordinates. Considering the point where all the Christoffel symbols vanish and applying the covariant derivatives one at a time we see that all the terms with Christoffel symbols vanish and the only term left is the one that only involves the partial derivatives and we know that partial derivatives commute. But I thought about the Riemann tensor definition and using that approach it would imply that the two derivatives of the Christoffel symbols cancel each other out. What am I missing?