On the spinorial representation of spacelike surfaces into 4-dimensional Minkowski space

TL;DR

This paper establishes a spinorial framework for representing spacelike surfaces in four-dimensional Minkowski space, linking isometric immersions to solutions of a Dirac equation and providing new proofs for classifications of flat surfaces.

Contribution

It introduces a spinor-based representation of spacelike surfaces in Minkowski space and applies it to classify flat surfaces with specific geometric properties.

Findings

Equivalent spinor field characterization of isometric immersions

Representation of surfaces via solutions to a Dirac equation

Spinorial proofs of classification results for flat surfaces

Abstract

We prove that an isometric immersion of a simply connected Riemannian surface M in four-dimensional Minkowski space, with given normal bundle E and given mean curvature vector H \in \Gamma(E), is equivalent to a normalized spinor field \varphi \in \Gamma(\Sigma E \otimes \Sigma M) solution of a Dirac equation D\varphi=H\cdot\varphi on the surface. Using the immersion of the Minkowski space into the complex quaternions, we also obtain a representation of the immersion in terms of the spinor field. We then use these results to describe the flat spacelike surfaces with flat normal bundle and regular Gauss map in four-dimensional Minkowski space, and also the flat surfaces in three-dimensional hyperbolic space, giving spinorial proofs of results by J.A. Galvez, A. Martinez and F. Milan.