On the geometry of conformal geodesics equations

TL;DR

This paper investigates the relationship between third-order ODE systems and conformal geodesics, providing explicit formulas and a functorial construction linking conformal and Cartan geometries.

Contribution

It introduces a functor from conformal geometries to Cartan geometries associated with third-order ODEs and derives explicit formulas for conformal geodesic equations.

Findings

Constructed a functor linking conformal and Cartan geometries.

Derived explicit formulas for conformal geodesic equations.

Established conditions under which third-order ODEs describe conformal geodesics.

Abstract

We answer to the question whether a system of the 3rd order ODEs describes geodesics of a conformal structure. We construct a functor from a category of conformal geometries to a category of Cartan geometries associated to the 3rd order ODEs systems. Explicit formulas which define the family of all equations on conformal geodesics are given in the last section of the article.