Spinorial characterizations of surfaces into three-dimensional homogeneous manifolds

We give spinorial characterizations of isometrically immersed surfaces into three-dimensional homogeneous manifolds with four-dimensional isometry group in terms of the existence of a particular spinor field. This generalizes works by Friedrich for R3R3 and Morel for S3S3 and H3H3. The main argument is the interpretation of the energy–momentum tensor of such a spinor field as the second fundamental form up to a tensor depending on the structure of the ambient space.