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A singularity theorem for twistor spinors. (English) Zbl 1128.53026
This paper consists in a detailed study of spin structures on orbifolds. In particular, the authors prove that, given an oriented orbifold \(M\) and assuming that the set \(S\) of its singularities has codimension at least 4, then \(M\) is spin if and only if the manifold \(M\setminus S\) is spin. Furthermore, the spin structures on \(M\) are in \(1-1\) correspondence with the spin structures on \(M\setminus S\). Particular attention to compact spin orbifolds is given. In fact, the authors consider a compact Riemannian spin orbifold \(M\) admitting a non-trivial twistor spinor with zero at a point \(p_0\). Then, either this zero is unique and \(p_0\) is a singular point, or \(M\) is conformally equivalent to a round sphere. In addition, for any \(p\), the order \(\#\Gamma_p\) of the singularity group satisfies \(\#\Gamma_p\leq\#\Gamma_{p_0}\). The equality holds if and only if \(M\) is a quotient of a standard sphere. Several consequences of this theorem are then derived. Finally, the authors use the conformal compactification of a smooth manifold to produce explicit examples of orbifolds carrying twistor spinors.

MSC:
53C27 Spin and Spin\({}^c\) geometry
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
53A30 Conformal differential geometry (MSC2010)
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