Skip to main content
SpringerLink
Log in

Generalized metrics and generalized twistor spaces

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

The twistor construction for Riemannian manifolds is extended to the case of manifolds endowed with generalized metrics (in the sense of generalized geometry à la Hitchin). The generalized twistor space associated to such a manifold is defined as the bundle of generalized complex structures on the tangent spaces of the manifold compatible with the given generalized metric. This space admits natural generalized almost complex structures whose integrability conditions are found in the paper. An interesting feature of the generalized twistor spaces discussed in it is the existence of intrinsic isomorphisms.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Atiyah, M.F., Hitchin, N.J., Singer, I.M.: Self-duality in four-dimensional Riemannian geometry. Proc. R. Soc. Lond, Ser. A 362, 425–461 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  2. Besse, A.: Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete 3.Folge, Band, vol. 10. Springer, Berlin, Heidelberg, New York (1987)

    Google Scholar 

  3. Bredthauer, A.: Generalized hyper-Kähler geometry and supersymmetry. Nuclear Phys. B 773, 172–183 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  4. Cavalcanti, G.: New aspects of the \(dd^c\)-lemma, Ph.D. thesis, New College, University of Oxford (2004). arXiv:math.DG/0501406

  5. Cavalcanti, G., Gualtieri, M.: Generalized complex structures on nilmanifolds. J. Symplectic Geom. 2, 393–410 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cavalcanti, G., Gualtieri, M.: Blow-up of generalized complex \(4\)-manifolds. J. Topol. 2, 840–864 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cavalcanti, G., Gualtieri, M.: Blowing up generalized Kähler 4-manifolds. Bull. Braz. Math. Soc. (N.S.) 42, 537–557 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. Courant, T.: Dirac manifolds. Trans. Am. Math. Soc. 319, 631–661 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  9. Davidov, J., Mushkarov, O.: Twistor spaces of generalized complex structures. J. Geom. Phys. 56, 1623–1636 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  10. Davidov, J., Mushkarov, O.: Twistorial construction of generalized Kähler manifolds. J. Geom. Phys. 57, 889–901 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  11. Deschamps, G.: Espace de twisteurs des structures complexes généralisées. Math. Z. 279, 703–721 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  12. Gromol, D., Klingenberg, W., Meyer, W.: Riemannsche Geometrie in Grossen, Lecture Notes in Mathematics, vol. 55. Springer, New York (1968)

  13. Eells, J., Salamon, S.: Twistorial constructions of harmonic maps of surfaces into four-manifolds. Ann. Scuola Norm. Sup. Pisa, ser. IV 12, 89–640 (1985)

    MathSciNet  MATH  Google Scholar 

  14. Glover, R., Sawon, J.: Generalized twistor spaces for hyperkähler manifolds. J. Lond. Math. Soc. 91, 321–324 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  15. Gualtieri, M.: Generalized complex geometry, Ph.D. thesis, St John’s College, University of Oxford (2003). arXiv:math.DG/0401221

  16. Gualtieri, M.: Generalized complex geometry. Ann. Math. 174, 75–123 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  17. Gualtieri, M.: Generalized Kähler geometry. Commun. Math. Phys. 331, 297–331 (2014)

    Article  MATH  Google Scholar 

  18. Hitchin, N.: Compact four-dimensional Einstein manifolds. J. Differ. Geom. 9, 435–441 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  19. Hitchin, N.: Generalized Calabi–Yau manifolds. Q. J. Math. 54, 281–308 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  20. Hitchin, N.: Instantons, Poisson structures and generalized Kähler geometry. Commun. Math. Phys. 265, 131–164 (2006)

    Article  MATH  Google Scholar 

  21. Hitchin, N.: Brackets, forms and invariant functionals. Asian J. Math. 10, 541–560 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  22. Hitchin, N.: Generalized geometry - an introduction, In: Cortes, V. (ed.) Handbook of pseudo-Riemannian geometry and suppersymmetry, IRMA Lectures in Mathematics and Theoretical Physics, vol. 16, pp. 185–208. European Mathematical Society, Zürich (2010)

  23. Pentilie, R.: Generalized quaternionic manifolds. Ann. Mat. Pura Appl. (4) 193, 633–641 (2014)

    Article  MathSciNet  Google Scholar 

  24. Penrose, R.: Twistor theory, its aims and achievements. In: Isham, C.J., Penrose, R., Sciama, D.W. (eds.) Quantum gravity, an Oxford Symposium, pp. 268–407. Clarendon Press, Oxford (1975)

    Google Scholar 

  25. Penrose, R., Ward, R.S.: Twistors for flat and curved space-time. In: Held, A. (ed.) General Relativity and Gravitation, vol. 2, pp. 283–328. Plenum Press, New York, London (1980)

    Google Scholar 

  26. Willmore, T.J.: Riemannian geometry. Clarendon Press, Oxford (1993)

    MATH  Google Scholar 

  27. Witt, F.: Special metric structures and closed forms, Ph.D. thesis, Oxvord University (2004). arXiv:math.DG/0502443

Download references

Acknowledgements

I would like to thank the referee whose remarks and comments helped to improve the final version of the paper.

Author information

Authors and Affiliations

  1. Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G.Bonchev St. Bl.8, 1113, Sofia, Bulgaria

    Johann Davidov

Authors
  1. Johann Davidov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Johann Davidov.

Additional information

The author is partially supported by the National Science Fund, Ministry of Education and Science of Bulgaria under contract DN 12/2.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Davidov, J. Generalized metrics and generalized twistor spaces. Math. Z. 291, 17–46 (2019). https://doi.org/10.1007/s00209-018-2071-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-018-2071-8

Keywords

Mathematics Subject Classification

Search

Navigation