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Spin structures on real projective quadrics

par Cahen, Michel ;Gutt, Simone ;Trautman, Andrzej
Référence Journal of geometry and physics, 10, 2, page (127-154)
Publication Publié, 1993-03

Article révisé par les pairs

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Résumé :

Projective quadrics are known to be conformal compactifications of Euclidean spaces. In particular, the (projective) real quadric Qp,q = (Sp × Sq)/Z2 is associated, in this manner, with the flat space Rp+q endowed with a metric tensor of signature (p, q). For p and q positive, the quadric Qp,q is orientable iff p + q is even. The quadric has two natural metrics, invariant with respect to the action of O(p + 1) × O(q + 1): a proper Riemannian one and a pseudo-Riemannian metric of signature (p,q). This paper contains an explicit description of spin structures on real, even-dimensional quadrics for both metrics, whenever these structures exist. In particular, it is shown that, for p and q even positive, the proper (pseudo-Riemannian) metric gives rise to two inequivalent spin structures iff p + q ≡ 2 (mod 4) (p + q ≡ 0 (mod 4)). If p and q are odd and > 1, then there is no spin structure for their metric whenever p + q ≡ 0 (mod 4); otherwise, there are two spin structures for each of the metrics. There always exist spin structures on real quadrics with a Lorentzian metric, i.e., when p and q are odd and p or q = 1. © 1993.