Prolongation on contact manifolds

Abstract

On contact manifolds we describe a notion of (contact) finite type for linear partial differential operators satisfying a natural condition on their leading terms. A large class of linear differential operators are of finite type in this sense but are not well understood by currently available techniques. We resolve this in the following sense. For any such D we construct a partial connection ∇H on a (finiterank) vector bundle with the property that sections in the null space of D correspond bijectively, and via an explicit map, with sections parallel for the partial connection. It follows that the solution space of D is finite dimensional and bounded by the corank of the holonomy algebra of ∇H. The treatment is via a uniformprocedure, even though in most cases no normal Cartan connection is available.