Twistor and Killing spinors in Lorentzian geometry (Q2765842)

The paper under review is a nice survey about twistor and Killing spinors on Lorentzian manifolds. The main topics covered are the following. NEWLINENEWLINENEWLINEi) General facts: Integrability conditions for the twistor equation on a pseudo-Riemannian manifold, twistor spinors on conformally flat manifolds and some basic properties of the conformal vector field (``Dirac current'') associated to a twistor spinor on a Lorentzian manifold. NEWLINENEWLINENEWLINEii) Twistor spinors on 4-dimensional Lorentz manifolds: Classification results by W. Ehlers and J. Kundt (1962), C. Bohle (1998) and J. Lewandowski (1991). NEWLINENEWLINENEWLINEiii) Correspondence between Lorentzian twistor spinors with lightlike Dirac currents and Fefferman spaces: Fefferman spaces are certain Lorentzian circle bundles associated to a strictly pseudo-convex CR-manifold. They have been characterized [\textit{H. Baum}, Differ. Geom. Appl. 11, 69-96 (1999; Zbl 0930.53033)] in the spin case as Lorentzian spin manifolds admitting a nontrivial twistor spinor with lightlike Dirac current which satisfies certain algebraic and differential equations. NEWLINENEWLINENEWLINEiv) Correspondence between Lorentzian imaginary Killing spinors with timelike Dirac currents and Lorentzian Einstein-Sasaki structures. NEWLINENEWLINENEWLINEv) Lorentzian manifolds admitting parallel spinors: Wang's classification of Riemannian holonomy groups preserving a non-zero spinor has been extended to the case of simply connected irreducible pseudo-Riemannian manifolds [\textit{H. Baum} and \textit{I. Kath}, Ann. Global Anal. Geom. 17, 1-17 (1999; Zbl 0928.53027)]. From that result one obtains, as a corollary, that a parallel spinor on a simply connected irreducible Lorentzian spin manifold is necessarily zero. Nevertheless indecomposable (but not irreducible) Lorentzian spin manifolds admitting a nontrivial parallel spinor exist. Classifications are available in special dimensions (J. Figueroa-O'Farrill, R. Bryant) and for the case of symmetric spaces (H. Baum). NEWLINENEWLINENEWLINEvi) Lorentzian manifolds admitting real Killing spinors: Classification in terms of warped products over Riemannian manifolds admitting parallel spinors or Killing spinors.NEWLINENEWLINEFor the entire collection see [Zbl 0973.00041].