Spinorial Representation of Submanifolds in a Product of Space Forms

Abstract

AbstractWe 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 $$\mathbb {S}^2\times \mathbb {R}$$ S 2 × R and we obtain new spinorial characterizations of immersions in $$\mathbb {S}^2\times \mathbb {R}^2$$ S 2 × R 2 and in $$\mathbb {H}^2\times \mathbb {R}.$$ H 2 × R . We then study the theory of $$H=1/2$$ H = 1 / 2 surfaces in $$\mathbb {H}^2\times \mathbb {R}$$ H 2 × R using this spinorial approach, obtain new proofs of some of its fundamental results and give a direct relation with the theory of $$H=1/2$$ H = 1 / 2 surfaces in $$\mathbb {R}^{1,2}$$ R 1 , 2 .