is incorrect because for fixed c and d, ( ∇ ∇ c ∇ ∇ d − − ∇ ∇ d ∇ ∇ c ) is, in general, a first order differential operator rather than a zeroth order operator which defines a tensor coefficient. Substituting a general tetrad basis in the abstract formula we find the proper definition of the curvature in abstract index notation, however: