Spinorial Representation of Submanifolds in a Product of Space Forms

Abstract

We present a method giving a spinorial characterization of an immersion into a product of spaces of constant curvature. As a first application we obtain a proof using spinors of the fundamental theorem of immersion theory for such target spaces. We also study special cases: we recover previously known results concerning immersions in mathbb {S}^2times mathbb {R} and we obtain new spinorial characterizations of immersions in mathbb {S}^2times mathbb {R}^2 and in mathbb {H}^2times mathbb {R}. We then study the theory of H=1/2 surfaces in mathbb {H}^2times mathbb {R} using this spinorial approach, obtain new proofs of some of its fundamental results and give a direct relation with the theory of H=1/2 surfaces in mathbb {R}^{1,2}.