A generalized Weierstrass representation of Lorentzian surfaces in $\mathbb{R}^{2,2}$ and applications

TL;DR

This paper introduces a generalized Weierstrass formula for Lorentzian surfaces in four-dimensional space using spinors, and explores their immersions in Anti-de Sitter space with new representation formulas.

Contribution

It provides a novel spinor-based representation for Lorentzian surfaces in ^{2,2} and their conformal immersion in Anti-de Sitter space, extending classical methods.

Findings

Derived a generalized Weierstrass formula for Lorentzian surfaces.

Presented a new spinor representation for immersions in Anti-de Sitter space.

Described conformal properties of flat Lorentzian surfaces in ^{2,2}.

Abstract

We give a generalized Weierstrass formula for a Lorentz surface conformally immersed in the four-dimensional space $\mathbb{R}^{2,2}$ using spinors and Lorentz numbers. We also study the immersions of a Lorentzian surface in {\bf the} Anti-de Sitter space (a pseudo-sphere in $\mathbb{R}^{2,2}$): we give a new spinor representation formula and deduce the conformal description of a flat Lorentzian surface in that space.