Journal of Geometry

, Volume 103, Issue 2, pp 247–261 | Cite as

Curvatures properties of Lie hypersurfaces in the complex hyperbolic space

  • Tatsuyoshi Hamada
  • Yuji Hoshikawa
  • Hiroshi TamaruEmail author


A Lie hypersurface in the complex hyperbolic space is a homogeneous real hypersurface without focal submanifolds. The set of all Lie hypersurfaces in the complex hyperbolic space is bijective to a closed interval, which gives a deformation of homogeneous hypersurfaces from the ruled minimal one to the horosphere. In this paper, we study intrinsic geometry of Lie hypersurfaces, such as Ricci curvatures, scalar curvatures, and sectional curvatures.

Mathematics Subject Classification (2010)

53C40 53C30 53C35 22E25 


Real hypersurfaces complex hyperbolic spaces solvable Lie groups Ricci curvatures scalar curvatures sectional curvatures 


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  1. 1.
    Berndt J.: Homogeneous hypersurfaces in hyperbolic spaces. Math. Z. 229, 589–600 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Berndt J., Díaz-Ramos J.C.: Homogeneous hypersurfaces in complex hyperbolic spaces. Geom. Dedicata 138, 129–150 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Berndt J., Tamaru H.: Homogeneous codimension one foliations on noncompact symmetric spaces. J. Differ. Geom. 63, 1–40 (2003)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Berndt J., Tamaru H.: Cohomogeneity one actions on noncompact symmetric spaces of rank one. Trans. Am. Math. Soc. 359, 3425–3438 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Berndt, J., Tricerri, F., Vanhecke, L.: Generalized Heisenberg groups and Damek-Ricci harmonic spaces. In: Lecture Notes in Mathematics, vol. 1598. Springer, Berlin (1995)Google Scholar
  6. 6.
    Besse, A.L.: Einstein manifolds. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 10. Springer, Berlin (1987)Google Scholar
  7. 7.
    Hamada T.: Real hypersurfaces of complex space forms in terms of Ricci *-tensor. Tokyo J. Math. 25, 473–483 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Helgason, S.: Differential geometry, Lie groups, and symmetric spaces. In: Graduate Studies in Mathematics, vol. 34. American Mathematical Society, Providence (2001)Google Scholar
  9. 9.
    Kobayashi, S., Nomizu, K.: Foundations of differential geometry, vol. II. Reprint of the 1969 original. Wiley Classics Library, A Wiley-Interscience Publication. Wiley, New York (1996)Google Scholar
  10. 10.
    Lauret J.: Degenerations of Lie algebras and geometry of Lie groups. Differ. Geom. Appl. 18, 177–194 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Lohnherr M., Reckziegel H.: On ruled real hypersurfaces in complex space forms. Geom. Dedicata 74, 267–286 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Milnor J.: Curvatures of left invariant metrics on Lie groups. Adv. Math. 21, 293–329 (1976)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Niebergall, R., Ryan, P.J.: Real hypersurfaces in complex space forms. In: Cecil, T.E., Chern S.-S. (eds.) Tight and Taut Submanifolds. Math. Sci. Res. Inst. Publ., vol. 32, pp. 233–305. Cambridge University Press, Cambridge (1997)Google Scholar
  14. 14.
    Pyateskii-Shapiro, I.I.: Automorphic functions and the geometry of classical domains. Translated from the Russian: Mathematics and its Applications, vol. 8, Gordon and Breach Science Publishers, New York (1969)Google Scholar
  15. 15.
    Ryan, P.: Intrinsic properties of real hypersurfaces in complex space forms. In: Chen, W.H., et al. (eds.) Geometry and Topology of Submanifolds, X, pp. 266–273. World Scientific, River Edge (2000)Google Scholar
  16. 16.
    Tamaru H.: Parabolic subgroups of semisimple Lie groups and Einstein solvmanifolds. Math. Ann. 351, 51–66 (2011)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Basel 2012

Authors and Affiliations

  • Tatsuyoshi Hamada
    • 1
    • 2
  • Yuji Hoshikawa
    • 3
    • 4
  • Hiroshi Tamaru
    • 3
    Email author
  1. 1.Department of Applied MathematicsFukuoka UniversityFukuokaJapan
  2. 2.JST, CRESTChiyoda-kuJapan
  3. 3.Department of MathematicsHiroshima UniversityHigashi-HiroshimaJapan
  4. 4.Takamatsu-Kita Junior High SchoolTakamatsuJapan

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