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


SIGMA 5 (2009), 081, 29 pages      arXiv:0908.0483      https://doi.org/10.3842/SIGMA.2009.081
Contribution to the Special Issue “Élie Cartan and Differential Geometry”

Conformal Structures Associated to Generic Rank 2 Distributions on 5-Manifolds – Characterization and Killing-Field Decomposition

Matthias Hammerl and Katja Sagerschnig
Faculty of Mathematics, University of Vienna, Nordbergstrasse 15, 1090 Vienna, Austria

Received April 09, 2009, in final form July 28, 2009; Published online August 04, 2009; Misprints in Theorem B are corrected November 09, 2009

Abstract
Given a maximally non-integrable 2-distribution D on a 5-manifold M, it was discovered by P. Nurowski that one can naturally associate a conformal structure [g]D of signature (2,3) on M. We show that those conformal structures [g]D which come about by this construction are characterized by the existence of a normal conformal Killing 2-form which is locally decomposable and satisfies a genericity condition. We further show that every conformal Killing field of [g]D can be decomposed into a symmetry of D and an almost Einstein scale of [g]D.

Key words: generic distributions; conformal geometry; tractor calculus; Fefferman construction; conformal Killing fields; almost Einstein scales.

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References

  1. Alt J., Fefferman constructions in conformal holonomy, Thesis, Humboldt-Universität zu Berlin, 2008.
  2. Armstrong S., Definite signature conformal holonomy: a complete classification, J. Geom. Phys. 57 (2007), 2024-2048, math.DG/0503388.
  3. Bryant R.L., Metrics with exceptional holonomy, Ann. of Math. (2) 126 (1987), 525-576.
  4. Burns D. Jr., Diederich K., Shnider S., Distinguished curves in pseudoconvex boundaries, Duke Math. J. 44 (1977), 407-431.
  5. Calderbank D.M.J., Diemer T., Differential invariants and curved Bernstein-Gelfand-Gelfand sequences, J. Reine Angew. Math. 537 (2001), 67-103, math.DG/0001158.
  6. Cap A., Parabolic geometries, CR-tractors, and the Fefferman construction, Differential Geom. Appl. 17 (2002), 123-138, 2002.
  7. Cap A., Correspondence spaces and twistor spaces for parabolic geometries, J. Reine Angew. Math. 582 (2005), 143-172, math.DG/0102097.
  8. Cap A., Two constructions with parabolic geometries, Rend. Circ. Mat. Palermo (2) Suppl. (2006), no. 79, 11-37, math.DG/0504389.
  9. Cap A., Infinitesimal automorphisms and deformations of parabolic geometries, J. Eur. Math. Soc. (JEMS) 10 (2008), 415-437, math.DG/0508535.
  10. Cap A., Gover A.R., Tractor bundles for irreducible parabolic geometries, in Global Analysis and Harmonic Analysis (Marseille-Luminy, 1999), Sémin. Congr., Vol. 4, Soc. Math. France, Paris, 2000, 129-154.
  11. Cap A., Gover A.R., Tractor calculi for parabolic geometries, Trans. Amer. Math. Soc. 354 (2002), 1511-1548.
  12. Cap A., Gover A.R., A holonomy characterisation of Fefferman spaces, ESI Preprint 1875, 2006, math.DG/0611939.
  13. Cap A., Gover A.R., CR-tractors and the Fefferman space, Indiana Univ. Math. J. 57 (2008), 2519-2570, math.DG/0611938.
  14. Cap A., Sagerschnig K., On Nurowski's conformal structure associated to a generic rank 2 distributions in dimension 5, J. Geom. Phys., to appear, arXiv:0710.2208.
  15. Cap A., Slovák J., Parabolic geometries I: Background and general theory, Vol. 1, Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2009.
  16. Cap A., Slovák J., Soucek V., Bernstein-Gelfand-Gelfand sequences, Ann. of Math. (2) 154 (2001), 97-113, math.DG/0001164.
  17. Cap A., Zádník V., On the geometry of chains, J. Differential Geom. 82 (2009), 1-33, math.DG/0504469.
  18. Cartan É., Les espaces á connexion conforme, Ann. Soc. Pol. Math. (1923), no. 2, 172-202, 1923.
  19. Cartan É., Les systèmes de Pfaff, à cinq variables et les équations aux dérivées partielles du second ordre, Ann. Sci. École Norm. Sup. (3) 27 (1910), 109-192.
  20. Doubrov B., Slovák J., Inclusions between parabolic geometries, arXiv:0807.3360.
  21. Eastwood M.G., Michor P.W., Some remarks on the Plücker relations, In The Proceedings of the 19th Winter School "Geometry and Physics" (Srní, 1999), Rend. Circ. Mat. Palermo (2) Suppl. (2000), no. 63, 85-88, math.AG/9905090.
  22. Fefferman C.L., Monge-Ampère equations, the Bergman kernel, and geometry of pseudoconvex domains, Ann. of Math. (2) 103 (1976), 395-416.
  23. Gover A.R., Laplacian operators and Q-curvature on conformally Einstein manifolds, Math. Ann. 336 (2006), 311-334, math.DG/0506037.
  24. Gover A.R., Almost Einstein and Poincaré-Einstein manifolds in Riemannian signature, arXiv:0803.3510.
  25. Gover A.R., Silhan J., The conformal Killing equation on forms - prolongations and applications, Differential Geom. Appl. 26 (2008), 244-266, math.DG/0601751.
  26. Graham C.R., On Sparling's characterization of Fefferman metrics, Amer. J. Math. 109 (1987), 853-874.
  27. Hammerl M., Natural prolongations of BGG-operators, Thesis, University of Vienna, submitted.
  28. Hammerl M., Homogeneous Cartan geometries, Arch. Math. (Brno) 43 (2007), suppl., 431-442, math.DG/0703627.
  29. Hammerl M., Invariant prolongation of BGG-operators in conformal geometry, Arch. Math. (Brno) 44 (2008), 367-384, arXiv:0811.4122.
  30. Hitchin N., Stable forms and special metrics, in Global Differential Geometry: the Mathematical Legacy of Alfred Gray (Bilbao, 2000), Contemp. Math., Vol. 288, Amer. Math. Soc., Providence, RI, 2001, 70-89, math.DG/0107101.
  31. Kostant B., Lie algebra cohomology and the generalized Borel-Weil theorem, Ann. of Math. (2) 74 (1961), 329-387.
  32. Leistner T., Nurowski P., Conformal structures with G2(2)-ambient metrics, arXiv:0904.0186.
  33. Leitner F., Conformal Killing forms with normalisation condition, Rend. Circ. Mat. Palermo (2) Suppl. (2005), no. 75, 279-292.
  34. Leitner F., A remark on conformal su(p,q)-holonomy, math.DG/0604393.
  35. Leitner F., A remark on unitary conformal holonomy, in Symmetries and Overdetermined Systems of Partial Differential Equations, IMA Vol. Math. Appl., Vol. 144, Springer, New York, 2008, 445-460.
  36. Nurowski P., Differential equations and conformal structures, J. Geom. Phys. 55 (2005), 19-49, math.DG/0406400.
  37. Nurowski P., Conformal structures with explicit ambient metrics and conformal G2 holonomy, in Symmetries and Overdetermined Systems of Partial Differential Equations, IMA Vol. Math. Appl., Vol. 144, Springer, New York, 2008, 515-526, math.DG/0701891.
  38. Sagerschnig K., Split octonions and generic rank two distributions in dimension five, Arch. Math. (Brno) 42 (2006), suppl., 329-339.
  39. Sagerschnig K., Weyl structures for generic rank two distributions in dimension five, Thesis, University of Vienna, 2008.
  40. Silhan J., Cohomologies of real Lie algebras, available at http://bart.math.muni.cz/~silhan/lie/lac/formR.php.
  41. Silhan J., A real analog of Kostant's version of the Bott-Borel-Weil theorem, J. Lie Theory 14 (2004), 481-499.

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