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In this paper we discuss the twistor equation in Lorentzian spin geometry. In particular, we explain the local conformal structure of Lorentzian manifolds, which admit twistor spinors inducing lightlike Dirac currents. Furthermore, we derive all local geometries with singularity free twistor spinors that occur up to dimension 7. (2000): 53C15, 53C50 Mathematics Subject Classification

We prove that there exist global solutions of the twistor equation on the Fefferman spaces of strictly pseudoconvex spin manifolds of arbitrary dimension and we study their properties.

In the present paper we study the geometry of doubly extended Lie groups with their natural biinvariant metric. We describe the curvature, the holonomy and the space of parallel spinors. This is completely done for all simply connected groups with biinvariant metric of Lorentzian signature (1, n − 1), of signature (2, n − 2) and of signature (p, q), where p + q ≤ 6. Furthermore, some special series with higher signature are discussed. * The second author was supported by the