Two-dimensional Riemannian and Lorentzian geometries from second order ODEs

TL;DR

This paper provides an alternative geometric derivation of two-dimensional Riemannian and Lorentzian metrics from second order ODEs, linking them to Hamilton-Jacobi equations and Cartan connections, with parallels to General Relativity.

Contribution

It introduces a new geometric approach to derive 2D metrics from second order ODEs, connecting torsion conditions and Cartan connections, expanding on recent results.

Findings

Derived 2D metrics from second order ODEs using geometric methods

Linked torsion tensor conditions to Null Surface Formulation of GR

Established connection between ODEs, Cartan connections, and 2D geometries

Abstract

In this note we give an alternative geometrical derivation of the results recently presented by Garcia-Godinez, Newman and Silva-Ortigoza in [1] on the class of all two-dimensional riemannian and lorentzian metrics from 2nd order ODEs which are in duality with the two dimensional Hamilton-Jacobi equation. We show that, as it happens in the Null Surface Formulation of General Relativity, the Wuschmann-like condition can be obtained as a requirement of a vanishing torsion tensor. Furthermore, from these second order ODEs we obtain the associated Cartan connections.