Skip to main content
SpringerLink
Log in

Locally conformally flat Kähler and para-Kähler manifolds

  • Published:
Annals of Global Analysis and Geometry Aims and scope Submit manuscript

Abstract

We complete the classification of locally conformally flat Kähler and para-Kähler manifolds, describing all possible non-flat curvature models for Kähler and para-Kähler surfaces.

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. Afifi, Z.: Riemann extensions of affine connected spaces. Q. J. Math. Oxford Ser. (2) 5, 312–320 (1954)

    Article  MathSciNet  Google Scholar 

  2. Alekseevsky, D., Guilfoyle, B., Klingenberg, W.: On the geometry of spaces of oriented geodesics. Ann. Glob. Anal. Geom. 40, 389–409 (2011)

    Article  MathSciNet  Google Scholar 

  3. Anciaux, H.: Spaces of geodesics of pseudo-Riemannian space forms and normal congruences of hypersurfaces. Trans. Am. Math. Soc. 366, 2699–2718 (2014)

    Article  MathSciNet  Google Scholar 

  4. Anciaux, H., Romon, P.: A canonical structure on the tangent bundle of a pseudo- or para-Kähler manifold. Monatsh. Math. 174, 329–355 (2014)

    Article  MathSciNet  Google Scholar 

  5. Berger, M.: Quelques formules de variation pour une structure riemannienne. Ann. Sci. École Norm. Sup. 3(4), 285–294 (1970)

    Article  MathSciNet  Google Scholar 

  6. Boubel, Ch., Bérard-Bergery, L.: On pseudo-Riemannian manifolds whose Ricci tensor is parallel. Geom. Dedicata 86, 1–18 (2001)

    Article  MathSciNet  Google Scholar 

  7. Bryant, R.: Bochner–Kähler metrics. J. Am. Math. Soc. 14, 623–715 (2001)

    Article  Google Scholar 

  8. Calviño-Louzao, E., García-Río, E., Gilkey, P., Vázquez-Lorenzo, R.: The geometry of modified Riemannian extensions. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 465, 2023–2040 (2009)

    MathSciNet  MATH  Google Scholar 

  9. Calviño-Louzao, E., García-Río, E., Gilkey, P., Gutiérrez-Rodríguez, I., Vázquez-Lorenzo, R.: Affine surfaces which are Kähler, para-Kähler, or nilpotent Kähler. Results Math. 73, 24 (2018)

    Article  Google Scholar 

  10. D’Ascanio, D., Gilkey, P., Pisani, P.: Symmetric affine surfaces with torsion. J. Geom. Phys. 147, 103536 (2020)

    Article  MathSciNet  Google Scholar 

  11. Derdzinski, A.: Connections with skew-symmetric Ricci tensor on surfaces. Results Math. 52, 223–245 (2008)

    Article  MathSciNet  Google Scholar 

  12. García-Río, E., Vázquez-Lorenzo, R.: Four-dimensional Osserman symmetric spaces. Geom. Dedicata 88, 147–151 (2001)

    Article  MathSciNet  Google Scholar 

  13. Georgiou, N., Guilfoyle, B.: A new geometric structure on tangent bundles. arXiv:1806.05440v1 [math.DG]

  14. Guilfoyle, B., Klingenberg, W.: An indefinite Kähler metric on the space of oriented lines. J. Lond. Math. Soc. 2(75), 497–509 (2005)

    Article  Google Scholar 

  15. Narasimhan, R., Nievergelt, Y.: Complex analysis in one variable, 2nd edn. Birkhäuser Boston Inc, Boston (2001)

    Book  Google Scholar 

  16. Matsushita, Y.: Walker \(4\)-manifolds with proper almost complex structures. J. Geom. Phys. 55, 385–398 (2005)

    Article  MathSciNet  Google Scholar 

  17. Olszak, Z.: On conformally flat parakählerian manifolds. Math. Balkanica 5, 302–307 (1991)

    MathSciNet  MATH  Google Scholar 

  18. O’Neill, B.: Semi-Riemannian geometry: With applications to relativity. Pure Appl. Math. 103 (1983)

    Article  Google Scholar 

  19. Opozda, B.: Locally symmetric connections on surfaces. Results Math. 20, 725–743 (1991)

    Article  MathSciNet  Google Scholar 

  20. Patterson, E.M.: Symmetric Kähler spaces. J. Lond. Math. Soc. 30, 286–291 (1955)

    Article  Google Scholar 

  21. Patterson, E.M., Walker, A.G.: Riemann extensions. Q. J. Math. Oxford Ser. (2) 3, 19–28 (1952)

    Article  MathSciNet  Google Scholar 

  22. Tanno, S.: \(4\)-dimensional conformally flat Kähler manifolds. Tohoku Math. J. 24, 501–504 (1972)

    MathSciNet  MATH  Google Scholar 

  23. Walker, A.G.: Canonical form for a Riemannian space with a parallel field of null planes. Q. J. Math. Oxford Ser. 1, 69–79 (1950)

    Article  MathSciNet  Google Scholar 

  24. Yano, K., Ishihara, S.: Tangent and cotangent bundles. Marcel Dekker, New York (1973)

    MATH  Google Scholar 

Download references

Acknowledgements

It is a pleasure to acknowledge useful conversations on this subject with Professor P. Gilkey.

Author information

Authors and Affiliations

  1. Faculty of Mathematics, University of Santiago de Compostela, 15782, Santiago de Compostela, Spain

    M. Ferreiro-Subrido & E. García-Río

  2. Department of Mathematics, IES de Ribadeo Dionisio Gamallo, 27700, Ribadeo, Spain

    R. Vázquez-Lorenzo

Authors
  1. M. Ferreiro-Subrido
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. E. García-Río
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. R. Vázquez-Lorenzo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to E. García-Río.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Supported by projects PID2019-105138GB-C21(AEI/FEDER, Spain), and ED431C 2019/10, ED431F 2020/04 (Xunta de Galicia, Spain).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ferreiro-Subrido, M., García-Río, E. & Vázquez-Lorenzo, R. Locally conformally flat Kähler and para-Kähler manifolds. Ann Glob Anal Geom 59, 483–500 (2021). https://doi.org/10.1007/s10455-021-09755-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10455-021-09755-8

Keywords

Search

Navigation