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Jacobi identity for covariant derivatives proof.

  1. Nov 24, 2013 #1
    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.
  2. jcsd
  3. Nov 24, 2013 #2
    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.
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