A global invariant for path structures and second order differential equations

TL;DR

This paper introduces a global invariant for path structures derived from Cartan connections, with explicit formulas for second order differential equations on a torus, advancing geometric analysis of differential equations.

Contribution

It defines a new global invariant for path structures using Cartan connections and provides explicit computations for second order differential equations on a torus.

Findings

Derived a formula for the global invariant on T^2

Connected the invariant to reductions of path structures

Applied the invariant to second order differential equations

Abstract

We study a global invariant for path structures. The invariant is obtained as a secondary invariant from a Cartan connection on a canonical bundle associated to a path structure. It is computed in examples which are defined in terms of reductions of the path structure. In particular we give a formula for this global invariant for second order differential equations defined on a torus $T^2$.