Contact Path Geometries

TL;DR

This paper studies contact path geometries on contact manifolds, characterizes their invariants, and constructs a Cartan connection for a special subclass with vanishing contact torsion, revealing links to quaternionic contact structures.

Contribution

It introduces a normalized Cartan connection for contact path geometries with vanishing contact torsion and explores their geometric properties and relations to quaternionic structures.

Findings

Contact torsion invariants classify contact path geometries.

A Cartan connection is constructed for geometries with vanishing contact torsion.

Vanishing secondary contact torsion implies a split quaternionic contact structure.

Abstract

Contact path geometries are curved geometric structures on a contact manifold comprising smooth families of paths modeled on the family of all isotropic lines in the projectivization of a symplectic vector space. Locally such a structure is equivalent to the graphs in the space of independent and depedent variables of the family of solutions of a system of an odd number of second order ODE's subject to a single maximally non-integrable constraint. A subclass of contact path geometries is distinguished by the vanishing of an invariant contact torsion. For this subclass the equivalence problem is solved by constructing a normalized Cartan connection using the methods of Tanaka-Morimoto-\v{C}ap-Schichl. The geometric meaning of the contact torsion is described. If a secondary contact torsion vanishes then the locally defined space of contact paths admits a split quaternionic contact structure (analogous to the quaternionic contact structures studied by O. Biquard).