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

TL;DR

This paper studies the module structure of second order linear differential operators on smooth manifolds, revealing equivalences across tensor-density degrees except at three critical values, and introduces a second order Lie derivative analogue.

Contribution

It characterizes the $Vect(M)$-module structures of second order differential operators and introduces a second order Lie derivative as an intertwining operator.

Findings

Module structures are equivalent for all tensor-density degrees except 0, 1/2, 1.

Identifies three critical values where structures differ.

Defines a second order analogue of the Lie derivative.

Abstract

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 tensor-densities except for three critical values: $\{0,{1\over 2},1\}$. Second order analogue of the Lie derivative appears as an intertwining operator between the spaces of second order differential operators on tensor-densities.