Linear differential operators on contact manifolds

TL;DR

This paper extends the Heisenberg calculus to differential operators on contact manifolds, introducing a new invariant called the subsymbol and a refined filtration of the contact order.

Contribution

It introduces an intrinsic subsymbol invariant and a refined filtration for differential operators on contact manifolds, advancing the geometric analysis in this setting.

Findings

Defined an intrinsic subsymbol invariant.

Constructed a refined contact order filtration.

Linked second order operators to contact vector fields.

Abstract

We consider differential operators between sections of arbitrary powers of the determinant line bundle over a contact manifold. We extend the standard notions of the Heisenberg calculus: noncommutative symbolic calculus, the principal symbol, and the contact order to such differential operators. Our first main result is an intrinsically defined "subsymbol" of a differential operator, which is a differential invariant of degree one lower than that of the principal symbol. In particular, this subsymbol associates a contact vector field to an arbitrary second order linear differential operator. Our second main result is the construction of a filtration that strengthens the well-known contact order filtration of the Heisenberg calculus.