# Covariant Derivative Commutation

Hello,

Can anyone tell me the general formula for commuting covariant derivatives, I mean, given a (r,s)-tensor field what is the formula to commute covariant derivatives?

I found a formula here page 25, Eq.6.18 but it doesn't seem right, since for a vector field one would write:

$(\nabla_{\alpha} \nabla_{\beta}- \nabla_{\beta}\nabla_{\alpha})U^{\gamma}=R^{\gamma}\hspace{.5 mm}_{\delta \alpha \beta}U^{\delta}$

And according to the formula in the link it would be, for a vector field

$(\nabla_{\alpha} \nabla_{\beta}- \nabla_{\beta}\nabla_{\alpha})U^{\gamma}=-R^{\gamma}\hspace{.5 mm}_{\delta \alpha \beta}U^{\delta}$

Thank you

Matterwave