Linear differential operators on contact manifolds

Charles H. Conley (1) , Valentin Ovsienko (2)

1 Department of Mathematics Univ. North Texas
2 ICJ - Institut Camille Jordan

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.

Mots clés: Contact geometry; differential operators; Heisenberg calculus; infinitesimal characters; differential invariants

Domaines: Physique mathématique [math-ph]; Géométrie différentielle [math.DG]; Théorie des représentations [math.RT]

Charles H. Conley, Valentin Ovsienko. Linear differential operators on contact manifolds. 2012. ⟨hal-00702324⟩