Hi. I'm trying to understand a derivation of the Bianchi idenity which starts from the torsion tensor in a torsion free space;(adsbygoogle = window.adsbygoogle || []).push({});

$$ 0 = T(X,Y) = \nabla_X Y - \nabla_Y X - [X,Y]$$

according to the author, covariant differentiation of this identity with respect to a vector Z yields

$$$ 0 = \nabla_Z \{\nabla_X Y - \nabla_Y X - [X,Y]\} = \nabla_Z\nabla_X Y - \nabla_Z \nabla_Y X - \{ \nabla_{[X,Y]}Z + [Z,[X,Y]] \}$$.

The two first terms are obvious, but how does he arrive at the third term?

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# Covariant derivative of a commutator (deriving Bianchi identity)

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