Weyl structures for path geometries

Abstract

Path geometries provide a geometric encoding of systems of second order ODE, which serves as a model for the geometric theory of more general systems of ODE and for cone structures. They are an instance of the family of parabolic geometries, thus they are second order structures that are difficult to study using the usual tools of differential geometry. The general theory of parabolic geometries provides several efficient tools for the study of path geometries, but these use Cartan geometry methods and hence are not easily accessible. In this article, we build a bridge between these general methods and an elementary approach to path geometries. Motivated by the general theory of Weyl structures (but not using it), we first define a family of distinguished connections that is analogous to Webster-Tanaka connections in CR geometry. These are parametrized by (local) non-vanishing sections of a line bundle naturally associated to the geometry, and the dependence of this choice is described explicitly. We also discuss the Schouten tensor associated to such a choice and its dependence on the choice. We explain how these ingredients can be used to obtain an elementary approach to tractor calculus for path geometries and give examples of applications to the construction of invariant operators. A second major result that we prove is that in the case of path geometries, there is a smaller subclass of distinguished Weyl structures which does not seem to have an analog for any other type of parabolic geometries. This has interesting relations to the refinement of the de Rham complex induced by a path geometry via the machinery of BGG sequences. Again, all this is proved using elementary methods without reference to the general theory.