Space of second order linear differential operators as a module over the Lie algebra of vector fields

The space of linear differential operators on a smooth manifold M has a natural one-parameter family of Diff(M) (and Vect(M))-module structures, defined by their action on the space of tensor densities. It is shown that, in the case of second order differential operators, the Vect(M)-module structures are equivalent for any degree of tensordensities except for three critical values: {0, 1 2 , 1}. Second order analogue of the Lie derivative appears as an intertwining operator between the spaces of second order differential operators on tensor-densities.