  Simply connected 3-dimensional homogeneous manifolds $E(\kappa, \tau)$, with
4-dimensional isometry group, have a canonical Spin$^c$ structure carrying
parallel or Killing spinors. The restriction to any hypersurface of these
parallel or Killing spinors allows to characterize isometric immersions of
surfaces into $E(\kappa, \tau)$. As application, we get an elementary proof of
a Lawson type correspondence for constant mean curvature surfaces in $E(\kappa,
\tau)$. Real hypersurfaces of the complex projective space and the complex
hyperbolic space are also characterized via Spin$^c$ spinors.
