Conformal Killing spinors and the holonomy problem in Lorentzian geometry – a survey of new results.

*(English)*Zbl 1143.53046
Eastwood, Michael (ed.) et al., Symmetries and overdetermined systems of partial differential equations. Proceedings of the IMA summer program, Minneapolis, MN, USA, July 17–August 4, 2006. New York, NY: Springer (ISBN 978-0-387-73830-7/hbk). The IMA Volumes in Mathematics and its Applications 144, 251-264 (2008).

Let \((M,g)\) be a Lorentzian manifold, \(S\) the spinor bundle on \((M,g)\) (see [H. B. Lawson and M.-L. Michelsohn, Spin Geometry. Princeton Mathematical Series. 38. (Princeton), NJ: Princeton University Press (1989; Zbl 0688.57001)], for details on this notion) and \(P:\Gamma(S)\to \Gamma(\ker\mu)\) the twistor operator, where \(\mu:TM\otimes S\to S\) denotes the Clifford multiplication. Conformal Killing spinors \(\varphi\in \Gamma(S)\) are solutions of the conformally invariant twistor equation \(P\varphi=0\), which is equivalent to an overdetermined system of partial differential equations.

The paper is a survey of recent results about this kind of spinors in Lorentzian geometry. The author starts describing some basic properties of these spinors and states the local classification result of F. Leitner [The proceedings of the 24th winter school “Geometry and physics”, Srní, Czech Republic, January 17–24, 2004. Palermo: Circolo Matemático di Palermo. Supplemento ai Rendiconti del Circolo Matemático di Palermo. Serie II 75, 279–292 (2005; Zbl 1101.53040)] for Lorentzian manifolds with generic conformal Killing spinors: If \((M,g)\) is equipped with a generic conformal Killing spinor, then \((M,g)\) is locally conformal equivalent to one of five types of spaces, among them Lorentzian Einstein-Sasaki manifolds, Fefferman spaces and Brinkmann spaces with parallel spinors. The first two classes of these mentioned geometries are well known and well studied [see for example the author, Differ. Geom. Appl., 11, 69–96 (1999; Zbl 0930.53033); Proc. of the Workshop on Special Geometric Structures in String Theory, Bonn, Sept. (2001; Zbl 1046.53032)]; Ch. Bohle, [J. Geom. Phys. 45, No. 3–4, 285–308 (2003; Zbl 1027.53050)]; the author and F. Leitner, Math. Z. 247, 795–812 (2004; Zbl 1068.53031); S. Dragomir and G. Tomassini, Differential Geometry and Analysis on CR manifolds. Progress in Mathematics 246. Basel: Birkhäuser (2006; Zbl 1099.32008)].

These two geometries admit global solutions of the conformal Killing spinor equation. A Brinkmann space with parallel spinor has special holonomy, so in order to describe such a space one has to study holonomy groups for Lorentzian metrics. The author explains shortly how parallel spinors are related to holonomy groups of metrics and describes the possible holonomy groups of complete, simply-connected Riemannian and Lorentzian manifolds. In particular, the classification result obtained by Th. Leistner [Holonomy and parallel spinors in Lorentzian geometry. (Berlin): Logos-Verlag; Berlin: Humboldt-Univ., Mathematisch-Naturwissenschaftliche Fakultät II (Dissertation) (2003; Zbl 1088.53032)] is made obvious [see also A. S. Galaev, Int. J. Geom. Methods Mod. Phys. 3, No. 5-6, 1025–1045 (2006; Zbl 1112.53039), arXiv:math DG/0502575].

In the final section the author discusses the problem of finding global geometric models for given Lorentzian holonomy groups and describes her own results on the realization of Lorentzian holonomy groups by globally hyperbolic manifolds [see the author and O. Müller, Math. Z. 258, 185–211 (2008; Zbl 1139.53023)]. This includes a construction principle for globally hyperbolic Brinkmann spaces with parallel spinors and with complete Cauchy surfaces, inspired by a paper of Ch. Bär, P. Gauduchon and A. Moroianu [Math. Z. 249, 545–580 (2005; Zbl 1068.53030)].

For the entire collection see [Zbl 1126.35005].

The paper is a survey of recent results about this kind of spinors in Lorentzian geometry. The author starts describing some basic properties of these spinors and states the local classification result of F. Leitner [The proceedings of the 24th winter school “Geometry and physics”, Srní, Czech Republic, January 17–24, 2004. Palermo: Circolo Matemático di Palermo. Supplemento ai Rendiconti del Circolo Matemático di Palermo. Serie II 75, 279–292 (2005; Zbl 1101.53040)] for Lorentzian manifolds with generic conformal Killing spinors: If \((M,g)\) is equipped with a generic conformal Killing spinor, then \((M,g)\) is locally conformal equivalent to one of five types of spaces, among them Lorentzian Einstein-Sasaki manifolds, Fefferman spaces and Brinkmann spaces with parallel spinors. The first two classes of these mentioned geometries are well known and well studied [see for example the author, Differ. Geom. Appl., 11, 69–96 (1999; Zbl 0930.53033); Proc. of the Workshop on Special Geometric Structures in String Theory, Bonn, Sept. (2001; Zbl 1046.53032)]; Ch. Bohle, [J. Geom. Phys. 45, No. 3–4, 285–308 (2003; Zbl 1027.53050)]; the author and F. Leitner, Math. Z. 247, 795–812 (2004; Zbl 1068.53031); S. Dragomir and G. Tomassini, Differential Geometry and Analysis on CR manifolds. Progress in Mathematics 246. Basel: Birkhäuser (2006; Zbl 1099.32008)].

These two geometries admit global solutions of the conformal Killing spinor equation. A Brinkmann space with parallel spinor has special holonomy, so in order to describe such a space one has to study holonomy groups for Lorentzian metrics. The author explains shortly how parallel spinors are related to holonomy groups of metrics and describes the possible holonomy groups of complete, simply-connected Riemannian and Lorentzian manifolds. In particular, the classification result obtained by Th. Leistner [Holonomy and parallel spinors in Lorentzian geometry. (Berlin): Logos-Verlag; Berlin: Humboldt-Univ., Mathematisch-Naturwissenschaftliche Fakultät II (Dissertation) (2003; Zbl 1088.53032)] is made obvious [see also A. S. Galaev, Int. J. Geom. Methods Mod. Phys. 3, No. 5-6, 1025–1045 (2006; Zbl 1112.53039), arXiv:math DG/0502575].

In the final section the author discusses the problem of finding global geometric models for given Lorentzian holonomy groups and describes her own results on the realization of Lorentzian holonomy groups by globally hyperbolic manifolds [see the author and O. Müller, Math. Z. 258, 185–211 (2008; Zbl 1139.53023)]. This includes a construction principle for globally hyperbolic Brinkmann spaces with parallel spinors and with complete Cauchy surfaces, inspired by a paper of Ch. Bär, P. Gauduchon and A. Moroianu [Math. Z. 249, 545–580 (2005; Zbl 1068.53030)].

For the entire collection see [Zbl 1126.35005].

Reviewer: Mircea Craioveanu (Timişoara)

##### MSC:

53C29 | Issues of holonomy in differential geometry |

53C27 | Spin and Spin\({}^c\) geometry |

53C80 | Applications of global differential geometry to the sciences |

53C15 | General geometric structures on manifolds (almost complex, almost product structures, etc.) |