Covariant Derivative Commutation

PLuz

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:

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

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


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

Thank you

Matterwave

I see only a difference in minus signs between the two expressions which can be accounted for in the convention used to define the Riemann tensor. I believe this formula is correct (at least for a coordinate basis, I cannot be sure if there are more terms for a non-coordinate basis).

dextercioby

For non-coordinate (aka anholonomic) basis, compute the Riemann tensor (it doesn't matter which sign convention you use) with the connection coefficients as provided in MTW page 210, formula 8.24b.