where [ e a , e b ] = f a b c e c . Note that the expression ( ∇ ∇ c ∇ ∇ d − − ∇ ∇ d ∇ ∇ c − − f c d e ∇ ∇ e ) is indeed a zeroth order operator, hence (the (c d)-component of) a tensor. Since it agrees with the coordinate expression for the curvature when specialised to a coordinate tetrad it is clear, even without using the abstract definition of the curvature, that it defines the same tensor as the coordinate basis expression.