Abstract
We develop a spinorial description of CR structures of arbitrary codimension. More precisely, we characterize almost CR structures of arbitrary codimension on (Riemannian) manifolds by the existence of a Spin\(^{c, r}\) structure carrying a partially pure spinor field. We study various integrability conditions of the almost CR structure in our spinorial setup, including the classical integrability of a CR structure as well as those implied by Killing-type conditions on the partially pure spinor field. In the codimension one case, we develop a spinorial description of strictly pseudoconvex CR manifolds, metric contact manifolds, and Sasakian manifolds. Finally, we study hypersurfaces of Kähler manifolds via partially pure Spin\(^c\) spinors.
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Acknowledgements
The authors are grateful to Oussama Hijazi for his encouragement and valuable comments. The authors thank Helga Baum and the Institute of Mathematics of the University of Humboldt-Berlin for their hospitality and support. The first author would also like to thank the hospitality and support of the International Centre for Theoretical Physics and the Institut des Hautes Études Scientifiques. The second author gratefully acknowledges the support and hospitality of the Centro de Investigación en Matemáticas A.C. (CIMAT). Rafael Herrera was partially supported by grants of CONACyT, LAISLA (CONACyT-CNRS), and the IMU Berlin Einstein Foundation Program. Iván Téllez was supported by a CONACyT scholarship.
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Herrera, R., Nakad, R. & Téllez, I. Spinorially Twisted Spin Structures, III: CR Structures. J Geom Anal 28, 3223–3277 (2018). https://doi.org/10.1007/s12220-017-9958-1
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DOI: https://doi.org/10.1007/s12220-017-9958-1