Spin structures on real projective quadrics

Abstract

Projective quadrics are known to be conformal compactifications of Euclidean spaces. In particular, the (projective) real quadric Q p,q = (S p × S q )/ Z 2 is associated, in this manner, with the flat space R p+q endowed with a metric tensor of signature ( p, q). For p and q positive, the quadric Q p,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.