A Few Comments on Conformal Spin Structures and Conformal U(1)-Spin Structures on a Pseudo-Riemannian 2r-Dimensional Manifold V

Abstract

This self-contained paper wants to precise the study of a real conformal spin structure in a strict sense over a pseudo-Riemannian or Riemannian 2r-dimensional manifold V already made in previous publications (Angles in Studia Scientiarum Mathematicarum Hungarica 23:115–139, 1988; Angles in Progress in Mathematical Physics, vol 50. Birkhauser, Boston, 2008). We give a fundamental diagram (A) concerning U(1)-spin geometry, a notion which has been initiated in Atiyah et al. (Topology 3(suppl 1):3–38 (Pergamon Press), 1964), in a special case. The obstruction class for the existence of a conformal spin structure in a strict sense over V is studied. Necessary and sufficient conditions for the existence of such a structure are recalled, using groups called conformal spinoriality groups in a strict sense. The notion of a conformal U(1)-spin structure over a pseudo-Riemannian or Riemannian 2r-dimensional manifold V is defined and studied. Two fundamental diagrams (B) and (C), relative to the conformal U(1)-spin geometry are given. We study the obstruction class for the existence of a conformal U(1)-spin structure over V. New fiber bundles are defined.