Spinorial Characterization of Surfaces into 3-dimensional homogeneous Manifolds

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