Prolongation on Contact Manifolds

TL;DR

This paper introduces a notion of finite-type for linear partial differential operators on contact manifolds, linking their solutions to parallel sections of a constructed partial connection, thus revealing finite-dimensional solution spaces.

Contribution

It defines finite-type operators on contact manifolds, constructs associated partial connections, and relates solution spaces to the holonomy of these connections, advancing geometric analysis.

Findings

Finite-dimensional solution spaces for finite-type operators.

Explicit construction of partial connections linked to differential operators.

Representation-theoretic determination of vector bundle structures in parabolic contact geometries.

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, and for any such operator we construct a partial connection on a (finite rank) vector bundle with the property that sections in the null space of the operator correspond bijectively, and via an explicit map, with sections parallel for the partial connection. It follows that the solution space is finite dimensional and bounded by the corank of the holonomy algebra of the partial connection. For finite-type operators on a parabolic contact geometry, the structure of the vector bundle, in particular its rank, can be easily determined by representation theory.