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 \(\mathbb {S}^2\times \mathbb {R}\) and we obtain new spinorial characterizations of immersions in \(\mathbb {S}^2\times \mathbb {R}^2\) and in \(\mathbb {H}^2\times \mathbb {R}.\) We then study the theory of \(H=1/2\) surfaces in \(\mathbb {H}^2\times \mathbb {R}\) 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 \(\mathbb {R}^{1,2}\).