In an arbitrary Riemannian or Lorentzian manifold {M, g} the commutator of covariant derivatives acting on a smooth vector field v given by ;; ;; , is expressible in terms of the curvature tensor and the (undifferentiated) components of v . Derive this formula starting with the basic formula for covariant derivatives. Derive the corresponding result for a one form . Remark: For higher rank tensors you would get a curvature term of the corresponding type for each index.