Jacobi identity for covariant derivatives proof.

[center o bass]

Suppose we have a torsion free connection. Does anyone here know of a slick way to prove that covariant derivatives satisfy the Jacobi identity? I.e. that

$$([\nabla_X,[\nabla_Y,\nabla_Z]] + [\nabla_Z,[\nabla_X,\nabla_Y]] +[\nabla_Y,[\nabla_Z,\nabla_X]])V = 0$$

without going into coordinate basis.

 

[center o bass]

Never mind. It's simply due to the properties of the commutator. The jacobi identity for lie brackets does not depend on on it being partial derivative operators; It can be any kind of operators.