Skip to main content
SpringerLink
Log in
Book cover

Complex and Differential Geometry pp 205–233Cite as

Aspects of conformal holonomy

  • Conference paper
  • First Online:

  • 2155 Accesses

Part of the book series: Springer Proceedings in Mathematics ((PROM,volume 8))

Abstract

This is an expository article, which gives an overview about aspects of the theory of conformal holonomy. In particular, we announce a complete geometric description of compact Riemannian conformal manifolds with decomposable conformal holonomy representation. Furthermore, we discuss the relation to almost Einstein structures and generalised Fefferman constructions. Generically, the latter conformal geometries have irreducible conformal holonomy. Reduced conformal holonomy is related to the existence of solutions of certain overdetermined conformally covariant PDE systems. We explain this relation in a unified approach using BGG-sequences.

Mathematics Subject Classification (2010) Primary 53A30, 53C29. Secondary 32V05, 53C25.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. D. Alekseevskii. Groups of conformal transformations of Riemannian spaces. Mat. Sbornik 89 (131) 1972 (in Russian), English translation Math. USSR Sbornik 18 (1972), 285–301.

    Google Scholar 

  2. J. Alt. Fefferman constructions in conformal holonomy. Dissertation, Humbdolt University Berlin, 2008.

    Google Scholar 

  3. S. Armstrong. Definite signature conformal holonomy: a complete classification. J. Geom. Phys. 57 (2007), no. 10, 2024–2048.

    Article  MATH  MathSciNet  Google Scholar 

  4. S. Armstrong. Free 3-distributions: holonomy, Fefferman constructions and dual distributions. arXiv:0708.3027 (2008).

    Google Scholar 

  5. S. Armstrong, F. Leitner. Decomposable conformal holonomy in Riemannian signature. Preprint 2010.

    Google Scholar 

  6. T.N. Bailey, M. Eastwood, A.R. Gover. Thomas‘s structure bundle for conformal, projective and related structures. Rocky Mountain J. Math. 24 (1994), no. 4, 1191–1217.

    Article  MATH  MathSciNet  Google Scholar 

  7. H. Baum. Lorentzian twistor spinors and CR-geometry. J. Diff. Geom. and its Appl. 11 (1999), no. 1, p. 69–96.

    Article  MATH  MathSciNet  Google Scholar 

  8. H. Baum, Th. Friedrich, R. Grunewald, I. Kath. Twistor and Killing spinors on Riemannian manifolds, Teubner-Text Nr. 124, Teubner-Verlag Stuttgart-Leipzig, 1991.

    Google Scholar 

  9. H. Baum, A. Juhl. Conformal Differential Geometry. Q-curvatrue and conformal holonomy. Oberwolfach Seminars, Vol. 40, Birkhäuser-Verlag, 2010.

    Google Scholar 

  10. O. Biquard. Metriques d‘Einstein asymptotiquement symetriques. Asterisque No. 265 (2000).

    Google Scholar 

  11. T. Branson. Q-curvature and spectral invariants. Rend. Circ. Mat. Palermo (2) Suppl. No. 75 (2005), 11–55.

    MathSciNet  Google Scholar 

  12. R. Bryant. Conformal geometry and 3-plane fields on 6-manifolds. in Developments of Cartan Geometry and Related Mathematical Problems, RIMS Symposium Proceedings, vol. 1502 (July, 2006), pp. 1–15, Kyoto University

    Google Scholar 

  13. A. Čap. Two constructions with parabolic geometries. Rend. Circ. Mat. Palermo (2) Suppl. No. 79 (2006), 11–37.

    Google Scholar 

  14. A. Čap, A.R. Gover. CR-tractors and the Fefferman space. Indiana Univ. Math. J. 57 (2008), no. 5, 2519–2570.

    Article  MATH  MathSciNet  Google Scholar 

  15. A. Čap, K. Sagerschnig. On Nurowski‘s Conformal Structure Associated to a Generic Rank Two Distribution in Dimension Five. J. Geom. Phys. 59 (2009) 901–912.

    Article  MathSciNet  Google Scholar 

  16. A. Čap, J. Slovák. Parabolic Geometries I: Background and General Theory. Mathematical Surveys and Monographs, 154. American Mathematical Society, Providence, RI, 2009.

    Google Scholar 

  17. A. Čap, J. Slovák. V. Souček. Bernstein-Gelfand-Gelfand sequences. Ann. of Math. (2) 154 (2001), no. 1, pp. 97–113.

    Article  MATH  MathSciNet  Google Scholar 

  18. E. Cartan. Les systemes de Pfaff a cinq variables et les equations aux derivees partielles du second ordre. Ann. Ec. Normale 27 (1910), 109–192.

    MathSciNet  Google Scholar 

  19. B. Doubrov, J. Slovak. Inclusions of parabolic geometries on a manifold. Pure and Applied Mathematics Quaterly 6, 3 (2010), Special Issue: In honor of Joseph J. Kohn, Part 1 of 2, 755–780.

    Google Scholar 

  20. M. Eastwood. Higher symmetries of the Laplacian. Ann. of Math. (2) 161 (2005), no. 3, 1645–1665.

    Article  MATH  MathSciNet  Google Scholar 

  21. C. Fefferman. Monge-Ampère equations, the Bergman kernel and geometry of pseudoconvex domains, Ann. Math. 103 (1976), p. 395–416.

    Article  MathSciNet  Google Scholar 

  22. C. Fefferman, and C.R. Graham. Conformal invariants in: The mathematical heritage of Élie Cartan (Lyon, 1984). Astérisque 1985, Numero Hors Serie, 95–116.

    Google Scholar 

  23. C. Fefferman, and C.R. Graham. The ambient metric. e-preprint: arXiv:0710.0919 (2007).

    Google Scholar 

  24. A.R. Gover. Almost Einstein and Poincaré-Einstein manifolds in Riemannian signature. J. Geom. Phys. 60 (2010), p. 182–204.

    Article  MATH  MathSciNet  Google Scholar 

  25. A.R. Gover, F. Leitner. A class of compact Poincare-Einstein manifolds: properties and construction. to appear in Commun. Contemp. Math.; e-preprint: arXiv:0808.2097 (2008).

    Google Scholar 

  26. C.R. Graham. On Sparling‘s characterisation of Fefferman metrics. Amer. J. Math. 109, pp. 853–874,1987.

    Article  MATH  MathSciNet  Google Scholar 

  27. C.R. Graham, K. Hirachi. The ambient obstruction tensor and Q-curvature. AdS/CFT correspondence: Einstein metrics and their conformal boundaries, 59–71, IRMA Lect. Math. Theor. Phys., 8, Eur. Math. Soc., Zürich, 2005.

    Google Scholar 

  28. C.R. Graham, R. Jenne, L.J. Mason, G.A. Sparling, Conƒormally invariant powers oƒ the Lapiacian. I. Existence, J. London Math. Soc. (2) 46 (1992), 557–565.

    Article  MATH  MathSciNet  Google Scholar 

  29. M. Hammerl. Natural prolongations oƒ BGG-operators. Dissertation, Universität Wien (2009).

    Google Scholar 

  30. M. Hammerl, K. Sagerschnig. Conƒormal structures associated to generic rank 2 distributions on 5-maniƒolds—characterization and Killing-field decomposition. SIGMA Symmetry Integrability Geom. Methods Appl. 5 (2009), Paper 081, 29 pp.

    MathSciNet  Google Scholar 

  31. O. Hijazi, S. Raulot. Branson‘s Q-curvature in Riemannian and spin geometry. SIGMA Symmetry Integrability Geom. Methods Appl. 3 (2007), Paper 119, 14 pp.

    MathSciNet  Google Scholar 

  32. S. Kobayashi, K. Nomizu. Foundations oƒ diƒƒerential geometry I & II, John Wiley & Sons, New York, 1963/69.

    Google Scholar 

  33. R.S. Kulkarni. Conƒormal structures and Möbius structures. Conformal geometry (Bonn, 1985/1986), 1–39, Aspects Math., E12, Vieweg, Braunschweig, 1988.

    Google Scholar 

  34. J. M. Lee. The Feƒferman metric and pseudo-Hermitian invariants. Trans. Amer. Math. Soc. 296 (1986), no. 1, 411–429.

    MATH  MathSciNet  Google Scholar 

  35. J.M. Lee, T.H. Parker. The Yamabe problem. Bull. Amer. Math. Soc. (N.S.) 17 (1987), no. 1, 37–91.

    Article  MATH  MathSciNet  Google Scholar 

  36. F. Leitner. Normal conƒormai Killing ƒorms. e-print: arXiv:math.DG/0406316 (2004).

    Google Scholar 

  37. F. Leitner. Conƒormal Killing ƒorms with normalisation condition. Rend. Circ. Mat. Palermo (2) Suppl. No. 75 (2005), 279–292.

    MathSciNet  Google Scholar 

  38. F. Leitner. About complex structures in conƒormal tractor calculus. e-print: arXiv:math.DG/0510637, (2005).

    Google Scholar 

  39. F. Leitner. On transversaiiy symmetric pseudo-Einstein and Feƒferman-Einstein spaces. Math. Z. 256 (2007), no. 2, 443–459.

    Article  MATH  MathSciNet  Google Scholar 

  40. F. Leitner. Applications oƒ Cartan and Tractor Calculus to Conƒormal and CR-Geometry. Habilitationsschrift, http://elib.uni-stuttgart.de/opus/volltexte/2009/ 3922, University of Stuttgart (2007).

  41. F. Leitner. A remark on unitary conƒormal holonomy. IMA Volumes in Mathematics and its Applications: Symmetries and Overdetermined Systems oƒ Partial Di//ërentiai Equations, Editors: Michael Eastwood and Willard Miller, Jr., Springer New York, Volume 144 (2007), p. 445–461.

    Google Scholar 

  42. F. Leitner. Conƒormal holonomy oƒ bi-invariant metrics. Conform. Geom. Dyn. 12 (2008), 18–31.

    Article  MATH  MathSciNet  Google Scholar 

  43. F. Leitner. The collapsing sphere product oƒ Poincaré-Einstein spaces. J. Geom. Phys. 60 no. 10, 1558–1575.

    Article  MATH  MathSciNet  Google Scholar 

  44. F. Leitner. Multiple almost Einstein structures with intersecting scale singularities. Preprint (2008).

    Google Scholar 

  45. J. Lewandowski. Twistor equation in a curved spacetime. Class. Quant. Grav., 8, pp. 11–17, 1991.

    Article  MathSciNet  Google Scholar 

  46. J. Maldacena. The large N limit oƒ superconƒormal field theories and supergravity. Adv. Theor. Math. Phys. 2 (1998), 231–252.

    MATH  MathSciNet  Google Scholar 

  47. P. Nurowski. Differential equations and conƒormal structures. J. Geom. Phys. 55 (2005), no. 1, 19–49.

    Article  MATH  MathSciNet  Google Scholar 

  48. M. Obata. Conƒormal transƒormations oƒ Riemannian maniƒolds. J. Diff. Geom. 4 (1970) 311–333.

    MATH  MathSciNet  Google Scholar 

  49. R. Penrose, W. Rindler. Spinors and Space-time II. Cambr. Univ. Press, 1986.

    Google Scholar 

  50. G. deRham. Sur la reductibilite d‘un espace de Riemann. Comment. Math. Helv. 26, (1952). 328–344.

    Article  MathSciNet  Google Scholar 

  51. U. Semmelmann. Conformal Killing forms on Riemannian manifolds. Habilitations-schrift, LMU München, 2001.

    Google Scholar 

  52. G.A.J. Sparling. Twistor theory and the characterisation of Fefferman‘s conformal structures. Preprint Univ. Pittsburg, 1985.

    Google Scholar 

  53. T.Y. Thomas. On conformal geometry. Proc. Natl. Acad. Sci. USA 12, 352–359 (1926).

    Article  MATH  Google Scholar 

  54. E. Witten. Anti de Sitter space and holography. Adv. Theor. Math. Phys. 2 (1998), no. 2, 253–291.

    MATH  MathSciNet  Google Scholar 

  55. H. Wu. On the deRham decomposition theorem. Ill. J. Math 8 (1964), 291–311.

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

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

    Felipe Leitner

Authors
  1. Felipe Leitner
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Felipe Leitner .

Editor information

Editors and Affiliations

  1. , Institut für Algebraische Geometrie, Leibniz Universität Hannover, Welfengarten 1, Hannover, 30167, Germany

    Wolfgang Ebeling

  2. FB Mathematik, Inst. Mathematik, Universität Hannover, Welfengarten 1, Hannover, 30167, Germany

    Klaus Hulek

  3. , Institut für Differentialgeometrie, Leibniz Universität Hannover, Welfengarten 1, Hannover, 30167, Germany

    Knut Smoczyk

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Leitner, F. (2011). Aspects of conformal holonomy. In: Ebeling, W., Hulek, K., Smoczyk, K. (eds) Complex and Differential Geometry. Springer Proceedings in Mathematics, vol 8. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-20300-8_13

Download citation

Publish with us

Policies and ethics

Search

Navigation