Hypersurfaces of Spinc Manifolds and Lawson Type Correspondence

Abstract

Simply connected three-dimensional homogeneous manifolds $${\mathbb{E}(\kappa, \tau)}$$ , with four-dimensional isometry group, have a canonical Spinc 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 $${\mathbb{E}(\kappa, \tau)}$$ . As application, we get an elementary proof of a Lawson type correspondence for constant mean curvature surfaces in $${\mathbb{E}(\kappa, \tau)}$$ . Real hypersurfaces of the complex projective space and the complex hyperbolic space are also characterized via Spinc spinors.