Twistor and Killing Spinors in Lorentzian Geometry
نویسندگان
چکیده
— This paper is a survey of recent results concerning twistor and Killing spinors on Lorentzian manifolds based on lectures given at CIRM, Luminy, in June 1999, and at ESI, Wien, in October 1999. After some basic facts about twistor spinors we explain a relation between Lorentzian twistor spinors with lightlike Dirac current and the Fefferman spaces of strictly pseudoconvex spin manifolds which appear in CR-geometry. Secondly, we discuss the relation between twistor spinors with timelike Dirac current and Lorentzian Einstein Sasaki structures. Then, we indicate the local structure of all Lorentzian manifolds carrying real Killing spinors. In particular, we show a global Splitting Theorem for complete Lorentzian manifolds in the presence of Killing spinors. Finally, we review some facts about parallel spinors in Lorentzian geometry. Résumé (Twisteurs et spineurs de Killing en géométrie lorentzienne). — Le présent papier est un article de synthèse basé sur les exposés donnés au CIRM, Luminy, en juin 1999, et à l’ESI, Vienne, en octobre 1999, concernant des nouveaux résultats sur les spineurs twisteurs et les spineurs de Killing lorentziens. Après quelques préliminaires sur les spineurs twisteurs, on met en évidence des relations entre les spineurs twisteurs lorentziens admettant un courant de Dirac isotrope et les espaces de Fefferman des variétés spinorielles strictement pseudoconvexes qui apparaissent dans la géométrie CR. De plus, on décrit la relation entre les spineurs twisteurs admettant un courant de Dirac de type temps et les structures de Sasaki-Einstein lorentziennes. On indique aussi la structure locale des variétés lorentziennes admettant des spineurs de Killing réels. En particulier, on obtient un théorème de < splitting > global pour les variétés lorentziennes complètes qui admettent des spineurs de Killing. Enfin, on fait le point sur la théorie des spineurs parallèles en géométrie lorentzienne. 2000 Mathematics Subject Classification. — 58G30, 53C50, 53A50.
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