A completely integrable flow of star-shaped curves on the light cone in Lorentzian $\mathbb{R}^4$

TL;DR

This paper establishes a connection between differential invariants of light cone curves in Lorentzian space and conformal invariants, enabling the realization of coupled KdV system solutions as curve flows.

Contribution

It demonstrates the Poisson equivalence between invariants in Lorentzian light cone geometry and conformal invariants, and constructs solutions of coupled KdV equations via curve flows.

Findings

Poisson equivalence between Lorentzian and conformal invariants

Realization of coupled KdV solutions as curve flows

New geometric interpretation of integrable systems

Abstract

In this paper we prove that the space of differential invariants for curves with arc-length parameter in the light cone of Lorentzian $\mathbb{R}^4$, invariants under the centro-affine action of the Lorentzian group, is Poisson equivalent to the space of conformal differential invariants for curves in the M\"obius sphere. We use this relation to find realizations of solutions of a complexly coupled system of KdV equations as flows of curves in the cone.