Hopf Real Hypersurfaces in the Indefinite Complex Projective Space

  • Makoto Kimura
  • Miguel OrtegaEmail author


We wish to attack the problems that Anciaux and Panagiotidou posed in (Differ Geom Appl 42:1–14,, 2015), for non-degenerate real hypersurfaces in indefinite complex projective space. We will slightly change these authors’ point of view, obtaining cleaner equations for the almost-contact metric structure. To make the theory meaningful, we construct new families of non-degenerate Hopf real hypersurfaces whose shape operator is diagonalisable, and one Hopf example with degenerate metric and non-diagonalisable shape operator. Next, we obtain a rigidity result. We classify those real hypersurfaces which are \(\eta \)-umbilical. As a consequence, we characterize some of our new examples as those whose Reeb vector field \(\xi \) is Killing.


Real hypersurface indefinite complex projective space Hopf real hypersurface 

Mathematics Subject Classification

Primary 53B25 53C50 Secondary 53C42 53B30 



  1. 1.
    Anciaux, H., Panagiotidou, K.: Hopf hypersurfaces in pseudo-Riemannian complex and para-complex space forms. Differ. Geom. Appl. 42, 1–14 (2015). MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Barros, M., Romero, A.: Indefinite Kähler Manifolds. Math. Ann. 261, 55–62 (1982)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bejancu, A., Duggal, K.L.: Real hypersurfaces of indefinite Kaehler manifolds. Int. J. Math. Math. Sci. 16(3), 545–556 (1993)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Berndt, J.: Real hypersurfaces with constant principal curvatures in complex hyperbolic space. J. Reine Angew. Math. 395, 132–141 (1989)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Cecil, T.E., Ryan, P.J.: Geometry of Hypersurfaces. Springer Monographs in Mathematics. Springer, New York (2015). CrossRefGoogle Scholar
  6. 6.
    Jin, D.H.: Lightlike real hypersurfaces with totally umbilical screen distributions. Commun. Korean Math. Soc. 25(3), 443–450 (2010)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Kimura, M.: Real hypersurfaces and complex submanifolds in complex projective space. Trans. Am. Math. Soc. 296(1), 137–149 (1986)MathSciNetCrossRefGoogle Scholar
  8. 8.
    O’Neill, Semi-Riemannian Geometry: With Applications to Relativity, Pure and Applied Mathematics, vol. 103. Academic Press, New York (1983)Google Scholar
  9. 9.
    Montiel, S.: Real hypersurfaces of a complex hyperbolic space. J. Math. Soc. Jpn. 37(3), 515–535 (1985)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Takagi, R.: On homogeneous real hypersurfaces in a complex projective space. Osaka J. Math. 10, 495–506 (1973)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Takagi, R.: Real hypersurfaces in a complex projective space with constant principal curvatures. J. Math. Soc. Jpn. 27, 43–53 (1975)MathSciNetCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Department of Mathematics Faculty of ScienceIbaraki UniversityMitoJapan
  2. 2.Departamento de Geometría y Topología Facultad de Ciencias Instituto de Matemáticas IEMathUGRUniversidad de GranadaGranadaSpain

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