Covariant derivatives in a rank 2 tensor

Question

I was trying to prove that for any second order tensor:

$$A^{\mu\nu}_{;\mu\nu}=A^{\mu\nu}_{;\nu\mu}$$

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?

Answer 1

Wikipedia gives the commutator of covariant derivatives acting on a $(2,0)$ tensor:

$$\tau^{ab}{}_{;cd}-\tau^{ab}{}_{;dc}=-R^a{}_{ecd}\tau^{eb} -R^b{}_{ecd}\tau^{ae}.$$

(Carroll has the generalization to an $(n,m)$ tensor in eq. 3.68.)

When one sets $cd$ to $ab$, the two terms on the right side of this cancel. No locally-flat coordinates are necessary when doing it this generally-covariant way.