Spinorially twisted Spin structures, III: CR structures

TL;DR

This paper introduces a spinorial framework for describing CR structures of any codimension on Riemannian manifolds, linking geometric properties to the existence of specific spinor fields and exploring their integrability conditions.

Contribution

It develops a novel spinorial characterization of CR structures of arbitrary codimension, including integrability conditions and special cases like pseudoconvex and Sasakian manifolds.

Findings

Characterization of CR structures via partially pure spinor fields

Spinorial conditions for integrability of CR structures

Description of hypersurfaces in Kaehler manifolds using spinors

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 Kaehler manifolds via partially pure Spin$^c$ spinors.