2005/08/18 by Daniel J. F. Fox, Fox, Daniel J. F.
Mathematics · #32L25 #53A55 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Holomorphic and Operator Theory #math.DG #msc:32L25 #msc:53A55
paper · pdf · doi:10.48550/arxiv.math/0508343
36 pages
arxiv created 2005/08/18 · openalex publication_date 2005/08/18 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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-Čap-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).