A Singularity Theorem for Twistor Spinors

Abstract

We study spin structures on orbifolds. In particular, we show that if the singular set has codimension greater than 2, an orbifold is spin if and only if its smooth part is. On compact orbifolds, we show that any non-trivial twistor spinor admits at most one zero which is singular unless the orbifold is conformally equivalent to a round sphere. We show the sharpness of our results through examples. The twistor operator (also called the Penrose operator) and the Dirac operator on a Riemannian spin manifold are obtained by composing the Levi-Civita covariant derivative with some natural linear maps. They are actually the two natural first order linear differential operators on spinors. The solutions of the corresponding P.D.E.'s (i.e., the kernels of these operators) are the twistor spinors and, respectively, the har- monic spinors, and they are both conformally covariant. Moreover, if we consider some appropriate weights, they appear to be conformally invariant objects (as sections of some weighted spinor bun- dle). However, it turns out that the norm of a twistor spinor defines a special metric in the class, for which the corresponding spinor is actually parallel, or, more generally, sum of two Killing spinors. This metric is then Einstein (of vanishing scalar curvature if and only if the corresponding spinor is parallel). Therefore, a dichotomy occurs: the study of the twistor spinors with- out zeros reduces to the study of parallel resp. Killing spinors, and the study of twistor spinors with zeros which seems to be a purely conformal problem. In both cases the corresponding manifolds and