Spinorial Characterization of Surfaces into 3-dimensional homogeneous Manifolds

TL;DR

This paper provides a spinorial method to characterize surfaces immersed in 3D homogeneous manifolds with 4D isometry groups, extending previous results for Euclidean and constant curvature spaces.

Contribution

It introduces a generalized Killing spinor framework to describe isometric immersions into these manifolds, broadening the understanding of surface geometry via spinor fields.

Findings

Characterization of surfaces using generalized Killing spinors

Extension of Friedrich's and Morel's results to new manifolds

Interpretation of energy-momentum tensor as second fundamental form

Abstract

We give a spinorial characterization of isometrically immersed surfaces into 3-dimensional homogeneous manifolds with 4-dimensional isometry group in terms of the existence of a particular spinor, called generalized Killing spinor. This generalizes results by T. Friedrich for $\R^3$ and B. Morel for $\Ss^3$ and $\HH^3$. The main argument is the interpretation of the energy-momentum tensor of a genralized Killing spinor as the second fondamental form up to a tensor depending on the structure of the ambient space