Causal geometries and third-order ordinary differential equations

TL;DR

This paper explores the geometric structures associated with third-order ordinary differential equations, introducing causal geometries and linking invariants to conformal Lorentzian metrics and projective curvature.

Contribution

It defines causal geometries for third-order ODEs with non-zero Wuenschmann invariant and relates these to conformal structures and projective curvature, extending geometric understanding.

Findings

Causal geometries exist for third-order ODEs with non-zero Wuenschmann invariant.

Conditions are established for degenerate conformal Lorentzian metrics to correspond to third-order ODEs.

The Wuenschmann invariant relates to the projective curvature of the associated causal geometry.

Abstract

We discuss contact invariant structures on the space of solutions of a third-order ordinary differential equation. Associated to any third-order differential equation modulo contact transformations, Chern introduced a degenerate conformal Lorentzian metric on the space of 2-jets of functions of one variable. When the Wuenschmann invariant vanishes, the degenerate metric descends to a proper conformal Lorentzian metric on the space of solutions. In the general case, when the Wuenschmann invariant is not zero, we define the notion of a causal geometry, and show that the space of solutions supports one. The Wuenschmann invariant is then related to the projective curvature of the indicatrix curve cut out by the causal geometry in the projective tangent space. When the Wuenschmann vanishes, the causal structure is then precisely the sheaf of null geodesics of the Chern conformal structure. We then introduce a Lagrangian and associated Hamiltonian from which the degenerate conformal Lorentzian metric are constructed. Finally, necessary and sufficient conditions are given for a rank three degenerate conformal Lorentzian metric in four dimensions to correspond to a third-order differential equation.