Spinorial representation of submanifolds in a product of space forms

TL;DR

This paper introduces a spinorial method to characterize submanifolds in products of constant curvature spaces, providing new proofs and characterizations, especially for surfaces with mean curvature 1/2 in hyperbolic products.

Contribution

It develops a novel spinorial characterization framework for immersions in product spaces of constant curvature, extending existing results and offering new insights into specific surface classes.

Findings

Spinorial characterization of immersions in product spaces.

New proofs for fundamental theorems in these geometries.

Characterizations of $H=1/2$ surfaces in $ ext{H}^2 imes ext{R}$.

Abstract

We present a method giving a spinorial characterization of an immersion in a product of spaces of constant curvature. As a first application we obtain a proof using spinors of the fundamental theorem of immersion theory in that 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 $\mathbb{R}^{1,2}.$