Mediterranean Journal of Mathematics

, Volume 13, Issue 4, pp 2161–2169 | Cite as

Comparing Lie Derivatives on Real Hypersurfaces in Complex Projective Spaces

Article

Abstract

On a real hypersurface M in a complex projective space, we can consider the Levi-Civita connection and for any nonnull constant k the kth g-Tanaka–Webster connection. Therefore, we can also consider their associated Lie derivatives. We classify real hypersurfaces such that both the Lie derivatives associated with the Levi-Civita connection and the kth g-Tanaka–Webster connection either in the direction of the structure vector field ξ or in any direction of the maximal holomorphic distribution coincide when we apply them to the shape operator of M.

Keywords

k-th g-Tanaka–Webster connection Levi-Civita connection complex projective space real hypersurface Lie derivatives shape operator 

Mathematics Subject Classification

53C15 53B25 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Blair D.E.: Riemannian Geometry of Contact and Symplectic Manifolds. Progress in Mathematics, vol. 203. Birkhauser Boston Inc., Boston (2002)CrossRefGoogle Scholar
  2. 2.
    Cecil T.E., Ryan P.J.: Focal sets and real hypersurfaces in complex projective space. Trans. AMS 269, 481–499 (1982)MathSciNetMATHGoogle Scholar
  3. 3.
    Cho J.T.: CR-structures on real hypersurfaces of a complex space form. Publ. Math. Debr. 54, 473–487 (1999)MathSciNetMATHGoogle Scholar
  4. 4.
    Cho J.T.: Pseudo-Einstein CR-structures on real hypersurfaces in a complex space form. Hokkaido Math. J. 37, 1–17 (2008)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Kimura M.: Real hypersurfaces and complex submanifolds in complex projective space. Trans. AMS 296, 137–149 (1986)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Kimura M.: Sectional curvatures of holomorphic planes of a real hypersurface in \({P^n(\mathbb{C})}\). Math. Ann. 276, 487–497 (1987)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Lohnherr M., Reckziegel H.: On ruled real hypersurfaces in complex space forms. Geom. Dedicata 74, 267–286 (1999)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Maeda Y.: On real hypersurfaces of a complex projective space. J. Math. Soc. Jpn. 28, 529–540 (1976)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Okumura M.: On some real hypersurfaces of a complex projective space. Trans. AMS 212, 355–364 (1975)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Pérez, J.D.: Lie and generalized Tanaka–Webster derivatives on real hypersurfaces in complex projective space. Int. J. Math. 25, 1450115 (13 pages) (2014)Google Scholar
  11. 11.
    Pérez, J.D., Suh, Y.J.: Generalized Tanaka–Webster and covariant derivatives on a real hypersurface in a complex projective space. Monatshefte Math. (to appear). doi:10.1007/s0065-015-0777-9
  12. 12.
    Takagi R.: On homogeneous real hypersurfaces in a complex projective space. Osaka J. Math. 10, 495–506 (1973)MathSciNetMATHGoogle Scholar
  13. 13.
    Takagi R.: Real hypersurfaces in complex projective space with constant principal curvatures. J. Math. Soc. Jpn. 27, 43–53 (1975)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Takagi R.: Real hypersurfaces in complex projective space with constant principal curvatures II. J. Math. Soc. Jpn. 27, 507–516 (1975)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Tanaka N.: On non-degenerate real hypersurfaces, graded Lie algebras and Cartan connections. Jpn. J. Math. 2, 131–190 (1976)MathSciNetMATHGoogle Scholar
  16. 16.
    Tanno S.: Variational problems on contact Riemennian manifolds. Trans. AMS 314, 349–379 (1989)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Webster S.M.: Pseudohermitian structures on a real hypersurface. J. Differ. Geom. 13, 25–41 (1978)MathSciNetMATHGoogle Scholar

Copyright information

© Springer Basel 2015

Authors and Affiliations

  1. 1.Departamento de Geometria y TopologiaUniversidad de GranadaGranadaSpain

Personalised recommendations