Space of Second-Order Linear Differential Operators as a Module over the Lie Algebra of Vector Fields

Abstract

The space of linear differential operators on a smooth manifoldMhas 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 tensor densities except for three critical values; {0,12,1}. A second-order analogue of the Lie derivative appears as an intertwining operator between the spaces of second-order differential operators on tensor densities.