Spinorially Twisted Spin Structures, III: CR Structures

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.

Generalised Spin$^r$ Structures on Homogeneous Spaces

“…They call these structures spinorially twisted Spin structures. Their associated spinor bundles have been studied in depth in [10,15,14]. Albanese and Milivojević [2] call these structures generalised Spin k structures.…”