Advertisement

Czechoslovak Mathematical Journal

, Volume 60, Issue 4, pp 1025–1036 | Cite as

Real hypersurfaces in a complex projective space with pseudo-\( \mathbb{D} \)-parallel structure Jacobi operator

  • Hyunjin LeeEmail author
  • Juan de Dios Pérez
  • Young Jin Suh
Article

Abstract

We introduce the new notion of pseudo-\( \mathbb{D} \)-parallel real hypersurfaces in a complex projective space as real hypersurfaces satisfying a condition about the covariant derivative of the structure Jacobi operator in any direction of the maximal holomorphic distribution. This condition generalizes parallelness of the structure Jacobi operator. We classify this type of real hypersurfaces.

Keywords

real hypersurface structure Jacobi operator 

MSC 2010

53C15 53B25 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    D. E. Blair: Riemannian geometry of contact and symplectic manifolds. Progress in Mathematics, Vol 203, Birkhäuser Boston Inc. Boston, Ma (2002).Google Scholar
  2. [2]
    J. T. Cho and U-H. Ki: Jacobi operators on real hypersurfaces of a complex projective space. Tsukuba J. Math. 22 (1998), 145–156.zbMATHMathSciNetGoogle Scholar
  3. [3]
    J. T. Cho and U-H. Ki: Real hypersurfaces of a complex projective space in terms of the Jacobi operators. Acta Math. Hungar. 80 (1998), 155–167.zbMATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    U-H. Ki, H. J. Kim and A. A. Lee: The Jacobi operator of real hypersurfaces in a complex space form. Commun. Korean Math. Soc. 13 (1998), 545–560.zbMATHMathSciNetGoogle Scholar
  5. [5]
    M. Kimura: Sectional curvatures of holomorphic planes on a real hypersurface in Pn(ℂ). Math. Ann. 276 (1987), 487–497.zbMATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    M. Lohnherr and H. Reckziegel: On ruled real hypersurfaces in complex space forms. Geom. Dedicata 74 (1999), 267–286.zbMATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    R. Niebergall and P. J. Ryan: Real hypersurfaces in complex space forms, in Tight and Taut Submanifolds. MSRI Publications 32 (1997), 233–305.MathSciNetGoogle Scholar
  8. [8]
    M. Okumura: On some real hypersurfaces of a complex projective space. Trans. A.M.S. 212 (1975), 355–364.zbMATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    M. Ortega, J.D. Pérez and F.G. Santos: Non-existence of real hypersurfaces with parallel structure Jacobi operator in nonflat complex space forms. Rocky Mountain J. Math. 36 (2006), 1603–1613.zbMATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    J.D. Pérez, F.G. Santos and Y. J. Suh: Real hypersurfaces in complex projective space whose structure Jacobi operator is D-parallel. Bull. Belg. Math. Soc. Simon Stevin 13 (2006), 459–469.zbMATHMathSciNetGoogle Scholar
  11. [11]
    R. Takagi: On homogeneous real hypersurfaces in a complex projective space. Osaka J. Math. 10 (1973), 495–506.zbMATHMathSciNetGoogle Scholar
  12. [12]
    R. Takagi: Real hypersurfaces in complex projective space with constant principal curvatures. J. Math. Soc. Japan 27 (1975), 43–53.zbMATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    R. Takagi: Real hypersurfaces in complex projective space with constant principal curvatures II. J. Math. Soc. Japan 27 (1975), 507–516.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Institute of Mathematics of the Academy of Sciences of the Czech Republic, Praha, Czech Republic 2010

Authors and Affiliations

  • Hyunjin Lee
    • 1
    • 2
    Email author
  • Juan de Dios Pérez
    • 3
    • 4
  • Young Jin Suh
    • 1
    • 5
  1. 1.TaeguKorea
  2. 2.School of Electrical Engineering and Computer ScienceKyungpook National UniversityTaeguRepublic of Korea
  3. 3.GranadaSpain
  4. 4.Departamento de Geometria y TopologiaUniversidad de GranadaGranadaSpain
  5. 5.Department of MathematicsKyungpook National UniversityTaeguRepublic of Korea

Personalised recommendations