Spinorial Characterization of Surfaces into 3-dimensional homogeneous Manifolds

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 ${\mathbb{R}}^3$ and B. Morel for ${\text{Ss}}^3$ and ${\text{HH}}^3$. 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