Proving the symmetry of the Ricci tensor? Consider the Ricci tensor :
$R_{\mu\nu}=\partial_{\rho}\Gamma_{\nu\mu}^{\rho}
-\partial_{\nu}\Gamma_{\rho\mu}^{\rho}
+\Gamma_{\rho\lambda}^{\rho}\Gamma_{\nu\mu}^{\lambda}
-\Gamma_{\nu\lambda}^{\rho}\Gamma_{\rho\mu}^{\lambda}$
In the most general case, is this tensor symmetric ? If yes, how to prove it ?

This misunderstands the previous post by assuming different conventions about the ordering of indices on the curvature tensor. The previous post assumes the convention that 
$$2\nabla_{[j}\nabla_{k]}v^\ell = {R^\ell}_{ijk}v^i.$$
With this convention, the argument is correct. 
