Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 5 (2009), 079, 12 pages      math.DG/0406298      https://doi.org/10.3842/SIGMA.2009.079
Contribution to the Special Issue “Élie Cartan and Differential Geometry”

About Twistor Spinors with Zero in Lorentzian Geometry

Felipe Leitner
Universität Stuttgart, Institut für Geometrie und Topologie, Fachbereich Mathematik, Pfaffenwaldring 57, D-70550 Stuttgart, Germany

Received April 06, 2009, in final form July 10, 2009; Published online July 28, 2009

Abstract
We describe the local conformal geometry of a Lorentzian spin manifold (M,g) admitting a twistor spinor φ with zero. Moreover, we describe the shape of the zero set of φ. If φ has isolated zeros then the metric g is locally conformally equivalent to a static monopole. In the other case the zero set consists of null geodesic(s) and g is locally conformally equivalent to a Brinkmann metric. Our arguments utilise tractor calculus in an essential way. The Dirac current of φ, which is a conformal Killing vector field, plays an important role for our discussion as well.

Key words: Lorentzian spin geometry; conformal Killing spinors; tractors and twistors.

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