Null curve evolution in four-dimensional pseudo-Euclidean spaces

TL;DR

This paper introduces a Lie bracket structure on vector fields along null curves in 4D semi-Riemannian spaces, enabling the study of integrability and symmetries of evolution equations like the null localized induction equation.

Contribution

It defines a new Lie bracket framework for null curves in pseudo-Euclidean spaces and constructs a recursion operator for generating symmetries of related evolution equations.

Findings

Established a Lie algebra structure for vector fields along null curves.

Developed a recursion operator that produces infinite local symmetries.

Applied the framework to analyze integrability of the null localized induction equation.

Abstract

We define a Lie bracket on a certain set of local vector fields along a null curve in a 4-dimensional semi-Riemannian space form. This Lie bracket will be employed to study integrability properties of evolution equations for null curves in a pseudo-Euclidean space. In particular, a geometric recursion operator generating infinite many local symmetries for the null localized induction equation is provided.