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

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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 (Anglès in Studia Scientiarum Mathematicarum Hungarica 23:115–139, 1988; Anglès in Progress in Mathematical Physics, vol 50. Birkhäuser, 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.

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  1. Center CAIROS, Institut de Mathématiques de Toulouse, CNRS UMR 5219, Université Paul Sabatier, 118 Route de Narbonne, 31062, Toulouse Cedex 9, France

    Pierre Anglès

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  1. Pierre Anglès
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Correspondence to Pierre Anglès.

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This paper is dedicated to the memory of my friends Pertti Lounesto, Jaime Keller and Artibano Micali

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Anglès, P. A Few Comments on Conformal Spin Structures and Conformal U(1)-Spin Structures on a Pseudo-Riemannian 2r-Dimensional Manifold V . Adv. Appl. Clifford Algebras 27, 165–183 (2017). https://doi.org/10.1007/s00006-016-0648-z

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