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(adsbygoogle = window.adsbygoogle || []).push({});

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

without going into coordinate basis.

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# Jacobi identity for covariant derivatives proof.

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