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

SIGMA 5 (2009), 079, 12 pages      math.DG/0406298
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

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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  1. Bailey T.N., Eastwood M.G., Gover A.R., Thomas's structure bundle for conformal, projective and related structures, Rocky Mountain J. Math. 24 (1994), 1191-1217.
  2. Baum H., Spin-Strukturen und Dirac-Operatoren über pseudoriemannschen Mannigfaltigkeiten, Teubner-Texte zur Mathematik, Vol. 41, Teubner-Verlag, Leipzig, 1981.
  3. Baum H., Twistor and Killing spinors in Lorentzian geometry, in Global Analysis and Harmonic Analysis (Marseille-Luminy, 1999), Sémin. Congr., Vol. 4, Soc. Math. France, Paris, 2000, 35-52.
  4. Baum H., Conformal Killing spinors and the holonomy problem in Lorentzian geometry - a survey of new results, in Symmetries and Overdetermined Systems of Partial Differential Equations, IMA Vol. Math. Appl., Vol. 144, Springer, New York, 2008, 251-264.
  5. Baum H., Friedrich Th., Grunewald R., Kath I., Twistor and Killing spinors on Riemannian manifolds, Teubner-Texte zur Mathematik, Vol. 124, Teubner-Verlag, Stuttgart - Leipzig, 1991.
  6. Baum H., Leitner F., The twistor equation in Lorentzian spin geometry, Math. Z. 247 (2004), 795-812, math.DG/0305063.
  7. Cap A., Slovák J., Weyl structures for parabolic geometries, Math. Scand. 93 (2003), 53-90, math.DG/0001166.
  8. Cap A., Slovák J., Soucek V., Invariant operators on manifolds with almost Hermitian symmetric structures. I. Invariant differentiation, Acta Math. Univ. Comenian. (N.S.) 66 (1997), 33-69, dg-ga/9503003.
  9. Cap A., Slovák J., Soucek V., Bernstein-Gelfand-Gelfand sequences, Ann. of Math. (2) 154 (2001), 97-113, math.DG/0001164.
  10. D'Ambra G., Gromov M., Lectures on transformation groups: geomerty and dynamics, in Surveys in Differential Geometry (Cambridge, MA, 1990), Lehigh Univ., Bethlehem, PA, 1991, 19-111.
  11. Frances Ch., Causal conformal vector fields, and singularities of twistor spinors, Ann. Global Anal. Geom. 32 (2007), 277-295.
  12. Friedrich H., Twistor connection and normal conformal Cartan connection, General Relativity and Gravitation 8 (1977), 303-312.
  13. Kobayashi S., Transformation groups in differential geometry, Springer-Verlag, New York - Heidelberg, 1972.
  14. Lawson H.B., Michelsohn M.-L., Spin geometry, Princeton Mathematical Series, Vol. 38, Princeton University Press, Princeton, NJ, 1989.
  15. Leitner F., Zeros of conformal vector fields and twistor spinors in Lorentzian geometry, SFB288-Preprint No. 439, Berlin, 1999.
  16. Leitner F., The twisor equation in Lorentzian geometry, Dissertation HU Berlin, 2001, available at\&id=10473.
  17. Leitner F., Normal conformal Killing forms, math.DG/0406316.
  18. Leitner F., Conformal Killing forms with normalisation condition, Rend. Circ. Mat. Palermo (2) Suppl. (2005), no. 75, 279-292.
  19. Leitner F., Twistor spinors with zero on Lorentzian 5-space, Comm. Math. Phys. 275 (2007), 587-605, math.DG/0602622.
  20. Leitner F., Applications of Cartan and tractor calculus to conformal and CR-geometry, Habilitationsschrift Uni Stuttgart, 2007, available at
  21. Lewandowski J., Twistor equation in a curved spacetime, Classical Quantum Gravity 8 (1991), L11-L17.
  22. Lichnerowicz A., Killing spinors, twistor spinors and Hijazi inequality, J. Geom. Phys. 5 (1988), 1-18.
  23. Penrose R., MacCallum M.A.H., Twistor theory: an approach to the quantisation of fields and space-time, Phys. Rep. C 6 (1973), 241-315.
  24. Penrose R., Rindler W., Spinors and space-time, Vol. 2, Spinor and twistor methods in space-time geometry, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 1986.
  25. Thomas T., The differential invariants of generalized spaces, Cambridge University Press, Cambridge, 1934.

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