Skip to main content
Log in

High-order Levi curvatures and classification results

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

Abstract

In this paper we write some differential formulas involving the high-order Levi curvatures of a real hypersurface in a complex space form. As application we get a classification result under suitable assumptions.

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

Access this article

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

Instant access to the full article PDF.

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)

    MATH  MathSciNet  Google Scholar 

  2. Bogges, A.: CR Manifolds and the Tangential Cauchy–Riemann Complex. Studies in Advanced Mathematics (1991)

  3. Borisenko, A.A.: On the global structure of Hopf hypersurfaces in a complex space form. Ill. J. Math. 45, 265–277 (2001)

    MATH  MathSciNet  Google Scholar 

  4. Dileo, G., Lotta, A.: Levi-parallel contact Riemannian manifolds. Math. Z. 274, 701–717 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  5. Klingenberg, W.: Real hypersurfaces in Kähler manifolds. Asian J. Math. 5, 1–17 (2001)

    MATH  MathSciNet  Google Scholar 

  6. Kimura, M.: Real hypersurfaces and complex submanifolds in complex projective space. Trans. Am. Math. Soc. 296, 137–149 (1986)

    Article  MATH  Google Scholar 

  7. Kon, M.: On a Hopf hypersurface of a complex space form. Differ. Geom. Appl. 28, 295–300 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  8. Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry. Wiley, London (1969)

    MATH  Google Scholar 

  9. Hounie, J., Lanconelli, E.: An Alexandrov type theorem for Reinhardt domains of \({\mathbb{C}}^2\). In: Recent progress on some problems in several complex variables and partial differential equations. Contemp. Math. 400, 129–146 (2006)

    Article  MathSciNet  Google Scholar 

  10. Lanconelli, E., Montanari, A.: Pseudoconvex fully nonlinear partial differential operators: strong comparison theorems. J. Differ. Equ. 202, 306–331 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  11. Maalaoui, A., Martino, V.: A symmetry result on submanifolds of space forms and applications. Mediterr. J. Math. 10, 507–517 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  12. Martino, V.: A symmetry result on Reinhardt domains. Differ. Integral Equ. 24, 495–504 (2011)

    MATH  MathSciNet  Google Scholar 

  13. Martino, V., Montanari, A.: On the characteristic direction of real hypersurfaces in \(\mathbb{C}^{N+1}\) and a symmetry result. Adv. Geom. 10, 371–377 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  14. Martino, V., Montanari, A.: Integral formulas for a class of curvature PDE’s and applications to isoperimetric inequalities and to symmetry problems. Forum Math. 22, 255–267 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  15. Martins, J.K.: Hopf hypersurfaces in space forms. Bull. Braz. Math. Soc. (N.S.) 35, 453–472 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  16. Miquel, V.: Compact Hopf hypersurfaces of constant mean curvature in complex space forms. Ann. Global Anal. Geom. 12, 211–218 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  17. Monti, R., Morbidelli, D.: Levi umbilical surfaces in complex space. J. Reine Angew. Math. 603, 113–131 (2007)

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  19. Montiel, S., Romero, A.: On some real hypersurfaces of a complex hyperbolic space. Geom. Dedicata 20, 245–261 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  20. Niebergall, R., Ryan, P.J.: (1997) Real hypersurfaces in complex space forms. In: Tight and Taut Submanifolds (Berkeley, CA, 1994), vol. 32, pp. 233–305. Mathematical Sciences Research Institute Publication, Cambridge University Press, Cambridge

  21. Okumura, M.: On some real hypersurfaces of a complex projective space. Trans. Am. Math. Soc. 212, 355–364 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  22. Ryan, P.J.: Hypersurfaces with parallel Ricci tensor. Osaka J. Math. 8, 251259 (1971)

    Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  25. Takagi, R.: Real hypersurfaces in a complex projective space with constant principal curvatures. II. J. Math. Soc. Japan 27, 507–516 (1975)

    Article  MATH  Google Scholar 

  26. Tralli, G.: Levi curvature with radial symmetry: a sphere theorem for bounded Reinhardt domains of \(\mathbb{C}^2\). Rend. Sem. Mat. Univ. Padova 124, 185–196 (2010)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Vittorio Martino.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Martino, V., Tralli, G. High-order Levi curvatures and classification results. Ann Glob Anal Geom 46, 351–359 (2014). https://doi.org/10.1007/s10455-014-9427-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10455-014-9427-z

Keywords

Mathematics Subject Classification (2010)

Navigation