Spinor representation of Lorentzian surfaces in R^{2,2}

TL;DR

This paper establishes a spinor-based representation for Lorentzian surfaces in R^{2,2}, linking isometric immersions to solutions of a Dirac equation and deriving explicit formulas for these surfaces.

Contribution

It introduces a novel spinor representation formula for Lorentzian surfaces in R^{2,2} and applies it to classify flat surfaces with flat normal bundle and Gauss map.

Findings

Explicit spinor representation formula for Lorentzian surfaces in R^{2,2}

Classification of flat Lorentzian surfaces with flat normal bundle

Dependence of surfaces on functions of one variable or holomorphic functions

Abstract

We prove that an isometric immersion of a simply connected Lorentzian surface in $\mathbb{R}^{2,2}$ is equivalent to a normalised spinor field solution of a Dirac equation on the surface. Using the quaternions and the Lorentz numbers, we also obtain an explicit representation formula of the immersion in terms of the spinor field. We then apply the representation formula in $\mathbb{R}^{2,2}$ to give a new spinor representation formula for Lorentzian surfaces in 3-dimensional Minkowski space. Finally, we apply the representation formula to the local description of the flat Lorentzian surfaces with flat normal bundle and regular Gauss map in $\mathbb{R}^{2,2},$ and show that these surfaces locally depend on four real functions of one real variable, or on one holomorphic function together with two real functions of one real variable, depending on the sign of a natural invariant.