Associativity of covariant derivatives

Question

I'm having trouble proving that covariant differentiation is an associative operation. Essentially I'll have to show

$$\nabla_\mu( \nabla_\nu \nabla_\sigma) = (\nabla_\mu\nabla_\nu) \nabla_\sigma. $$

But is it enough to show that both LHS and RHS yield the same result when acted up on a scalar or a contravariant vector?. Will this hold for any general tensor? Is there any other method to show this ?

Answer 1

If you see the covariant derivation as a bundle-map between the relevant vector bundles of tensors, associativity is automatically guaranteed by the usual associativity of composition of functions.

Answer 2

Frankly it boils down to function/operator composition, which is associative. Take a general tensor $T^\alpha_\beta$. $$(\nabla_\nu \nabla_\sigma)T^\alpha_\beta = \nabla_\nu (\nabla_\sigma T^\alpha_\beta )$$ so $$\nabla_\mu( \nabla_\nu \nabla_\sigma)T^\alpha_\beta = \nabla_\mu(\nabla_\nu (\nabla_\sigma T^\alpha_\beta ))$$ Similarly for the right hand side

$$ (\nabla_\mu\nabla_\nu) (\nabla_\sigma T^\alpha_\beta)= \nabla_\mu(\nabla_\nu (\nabla_\sigma T^\alpha_\beta ))$$ So both sides of your equation are the same, when expanded.