Skip to main content
SpringerLink
Log in

Homogeneous hypersurfaces in complex hyperbolic spaces

  • Original Paper
  • Published:
Geometriae Dedicata Aims and scope Submit manuscript

Abstract

We study the geometry of homogeneous hypersurfaces and their focal sets in complex hyperbolic spaces. In particular, we provide a characterization of the focal set in terms of its second fundamental form and determine the principal curvatures of the homogeneous hypersurfaces together with their multiplicities.

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. Berndt J.: Real hypersurfaces with constant principal curvatures in complex hyperbolic space. J. Reine Angew. Math. 395, 132–141 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  2. Berndt J.: Homogeneous hypersurfaces in hyperbolic spaces. Math. Z. 229, 589–600 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  3. Berndt J., Brück M.: Cohomogeneity one actions on hyperbolic spaces. J. Reine Angew. Math. 541, 209–235 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  4. Berndt, J., Console, S., Olmos, C.: Submanifolds and Holonomy. Chapman & Hall/CRC Research Notes in Mathematics 434: Chapman & Hall/CRC, Boca Raton (2003)

  5. Berndt J., Díaz-Ramos J.C.: Real hypersurfaces with constant principal curvatures in complex hyperbolic spaces. J. Lond. Math. Soc. 74, 778–798 (2006)

    Article  MATH  Google Scholar 

  6. Berndt J., Díaz-Ramos J.C.: Real hypersurfaces with constant principal curvatures in the complex hyperbolic plane. Proc. Am. Math. Soc. 135, 3349–3357 (2007)

    Article  MATH  Google Scholar 

  7. Berndt J., Tamaru H.: Cohomogeneity one actions on noncompact symmetric spaces of rank one. Trans. Am. Math. Soc. 359, 3425–3438 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  8. Berndt, J., Tricerri, F., Vanhecke, L.: Generalized Heisenberg Groups and Damek–Ricci Harmonic Spaces. Lecture Notes in Mathematics 1598. Springer–Verlag, Berlin (1995)

  9. Cartan É.: Familles de surfaces isoparamétriques dans les espaces à courbure constante. Ann. Mat. Pura Appl., IV. Ser. 17, 177–191 (1938)

    Article  MathSciNet  Google Scholar 

  10. Cecil T.E., Ryan P.J.: Focal sets and real hypersurfaces in complex projective space. Trans. Am. Math. Soc. 269, 481–499 (1982)

    MathSciNet  MATH  Google Scholar 

  11. Hsiang W.-Y., Lawson H.B. Jr: Minimal submanifolds of low cohomogeneity. J. Differ. Geom. 5, 1–38 (1971)

    MathSciNet  MATH  Google Scholar 

  12. Lohnherr, M. (1998) On ruled real hypersurfaces of complex space forms. PhD thesis, University of Cologne

  13. Montiel S.: Real hypersurfaces of a complex hyperbolic space. J. Math. Soc. Jpn. 37, 515–535 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  14. Takagi R.: On homogeneous real hypersurfaces in a complex projective space. Osaka J. Math. 10, 495–506 (1973)

    MathSciNet  MATH  Google Scholar 

  15. Takagi R.: Real hypersurfaces in a complex projective space with constant principal curvatures. J. Math. Soc. Jpn. 27, 43–53 (1975)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University College Cork, Cork, Ireland

    Jürgen Berndt & José Carlos Díaz-Ramos

Authors
  1. Jürgen Berndt
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. José Carlos Díaz-Ramos
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to José Carlos Díaz-Ramos.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Berndt, J., Díaz-Ramos, J.C. Homogeneous hypersurfaces in complex hyperbolic spaces. Geom Dedicata 138, 129–150 (2009). https://doi.org/10.1007/s10711-008-9303-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10711-008-9303-8

Keywords

Mathematics Subject Classification (2000)

Search

Navigation