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

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.