## Rocky Mountain Journal of Mathematics

### On characterizations of Hopf hypersurfaces in a nonflat complex space form with commuting operators

#### Abstract

Let $M$ be a real hypersurface in a complex space form $\mn$, $c \neq 0$. In this paper we prove that if $\rx\li = \li\rx$ holds on $M$, then $M$ is a Hopf hypersurface, where $\rx$ and $\li$ denote the structure Jacobi operator and the induced operator from the Lie derivative with respect to the structure vector field $\xi$, respectively. We characterize such Hopf hypersurfaces of $\mn$.

#### Article information

Source
Rocky Mountain J. Math., Volume 44, Number 6 (2014), 1923-1939.

Dates
First available in Project Euclid: 2 February 2015

https://projecteuclid.org/euclid.rmjm/1422885101

Digital Object Identifier
doi:10.1216/RMJ-2014-44-6-1923

Mathematical Reviews number (MathSciNet)
MR3310955

Zentralblatt MATH identifier
1315.53057

#### Citation

Kim, In-Bae; Lim, Dong Ho; Song, Hyunjung. On characterizations of Hopf hypersurfaces in a nonflat complex space form with commuting operators. Rocky Mountain J. Math. 44 (2014), no. 6, 1923--1939. doi:10.1216/RMJ-2014-44-6-1923. https://projecteuclid.org/euclid.rmjm/1422885101

#### References

• J. Berndt, Real hypersurfaces with constant principal curvatures in complex hyperbolic space, J. reine angew. Math. 395 (1989), 132–141.
• J.T. Cho and U-H. Ki, Real hypersurfaces of a complex projective space in terms of Jacobi operators, Acta Math. Hungar. 80 (1998), 155–167.
• U-H. Ki, I.-B. Kim and D.H. Lim, Characterizations of real hypersurfaces of type A in a complex space form, Bull. Korean Math. Soc. 47 (2010), 1–15.
• U-H. Ki and Y.J. Suh, On real hypersurfaces of a complex space form, J. Okayama Univ. 32 (1990), 207–221.
• S. Montiel and A. Romero, On some real hypersurfaces of a complex hyperbolic space, Geom. Ded. 20 (1986), 245–261.
• R. Niebergall and P.J. Ryan, Real hypersurfaces in complex space forms, in Tight and taut submanifolds, Cambridge University Press, 1998.
• M. Okumura, On some real hypersurfaces of a complex projective space, Trans. Amer. Math. Soc. 212 (1975), 355–364.
• 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.
• J.D. Pérez and F.G. Santos, Real hypersurfaces in complex projective space with recurrent structure Jacobi operator, Diff. Geom. Appl. 26 (2008), 218–223.
• R. Takagi, On homogeneous real hypersurfaces in a complex projective space, Osaka J. Math. 10 (1973), 495–506.