Spinorial Characterizations of Surfaces into 3-dimensional pseudo-Riemannian Space Forms

TL;DR

This paper provides a comprehensive spinorial framework for characterizing surfaces of various signatures immersed in 3D pseudo-Riemannian space forms, extending previous results and unifying different cases.

Contribution

It introduces a unified spinorial characterization for isometric immersions of surfaces with arbitrary signature into 3D pseudo-Riemannian space forms, generalizing prior work.

Findings

Unified spinorial characterization for Lorentzian and Riemannian surfaces.

Extension of previous Lorentzian results to other space forms.

Complete description for timelike and spacelike surface immersions.

Abstract

We give a spinorial characterization of isometrically immersed surfaces of arbitrary signature into 3-dimensional pseudo-Riemannian space forms. For Lorentzian surfaces, this generalizes a recent work of the first author in $\mathbb{R}^{2,1}$ to other Lorentzian space forms. We also characterize immersions of Riemannian surfaces in these spaces. From this we can deduce analogous results for timelike immersions of Lorentzian surfaces in space forms of corresponding signature, as well as for spacelike and timelike immersions of surfaces of signature (0,2), hence achieving a complete spinorial description for this class of pseudo-Riemannian immersions.