Czechoslovak Mathematical Journal, Vol. 67, No. 1, pp. 271-278, 2017

A characterization of a certain real hypersurface of type $({\rm A}_2)$ in a complex projective space

Byung Hak Kim, In-Bae Kim, Sadahiro Maeda

Received October 12, 2015.  First published February 24, 2017.

Abstract:  In the class of real hypersurfaces $M^{2n-1}$ isometrically immersed into a nonflat complex space form $\widetilde{M}_n(c)$ of constant holomorphic sectional curvature $c$ $(\ne0)$ which is either a complex projective space $\mathbb{C}P^n(c)$ or a complex hyperbolic space $\mathbb{C}H^n(c)$ according as $c > 0$ or $c < 0$, there are two typical examples. One is the class of all real hypersurfaces of type (A) and the other is the class of all ruled real hypersurfaces. Note that the former example are Hopf manifolds and the latter are non-Hopf manifolds. In this paper, inspired by a simple characterization of all ruled real hypersurfaces in $\widetilde{M}_n(c)$, we consider a certain real hypersurface of type $({\rm A}_2)$ in $\mathbb{C}P^n(c)$ and give a geometric characterization of this Hopf manifold.
Keywords:  ruled real hypersurface; nonflat complex space form; real hypersurfaces of type $({\rm A}_2)$ in a complex projective space; geodesics; structure torsion; Hopf manifold
Classification MSC:  53B25, 53C40
DOI:  10.21136/CMJ.2017.0546-15

[1] T. Adachi, S. Maeda, S. Udagawa: Circles in a complex projective space. Osaka J. Math. 32 (1995), 709-719. MR 1367901 | Zbl 0857.53034
[2] T. Adachi, S. Maeda: Global behaviours of circles in a complex hyperbolic space. Tsukuba J. Math. 21 (1997), 29-42. MR 1467219 | Zbl 0891.53036
[3] T. Adachi: Geodesics on real hypersurfaces of type $( A_2)$ in a complex space form. Monatsh. Math. 153 (2008), 283-293. DOI 10.1007/s00605-008-0521-9 | MR 2394551 | Zbl 1151.53049
[4] J. Berndt, H. Tamaru: Cohomogeneity one actions on noncompact symmetric spaces of rank one. Trans. Am. Math. Soc. 359 (2007), 3425-3438. DOI 10.1090/S0002-9947-07-04305-X | MR 2299462 | Zbl 1117.53041
[5] U.-H. Ki, I.-B. Kim, D. H. Lim: Characterizations of real hypersurfaces of type A in a complex space form. Bull. Korean Math. Soc. 47 (2010), 1-15. DOI 10.4134/BKMS.2010.47.1.001 | MR 2604227 | Zbl 1191.53039
[6] Y. Maeda: On real hypersurfaces of a complex projective space. J. Math. Soc. Japan 28 (1976), 529-540. DOI 10.2969/jmsj/02830529 | MR 0407772 | Zbl 0324.53039
[7] S. Maeda, T. Adachi: Characterizations of hypersurfaces of type $ A_2$ in a complex projective space. Bull. Aust. Math. Soc. 77 (2008), 1-8. DOI 10.1017/S0004972708000014 | MR 2411863 | Zbl 1137.53311
[8] S. Maeda, T. Adachi, Y. H. Kim: A characterization of the homogeneous minimal ruled real hypersurface in a complex hyperbolic space. J. Math. Soc. Japan 61 (2009), 315-325. DOI 10.2969/jmsj/06110315 | MR 2272881 | Zbl 1159.53012
[9] R. Niebergall, P. J. Ryan: Real hypersurfaces in complex space forms. Tight and Taut Submanifolds T. E. Cecil et al. Math. Sci. Res. Inst. Publ. 32, Cambridge Univ. Press, Cambridge (1998), 233-305. MR 1486875 | Zbl 0904.53005
[10] R. Takagi: On homogeneous real hypersurfaces in a complex projective space. Osaka J. Math. 10 (1973), 495-506. MR 0336660 | Zbl 0274.53062

Affiliations:   Byung Hak Kim, Department of Applied Mathematics and Institute of National Sciences, Kyung Hee University, 26 Kyungheedae-ro, Hoegi-dong, Dongdaemun-gu, Yongin 446-701, Korea, e-mail:; In-Bae Kim, Department of Mathematics, Hankuk University of Foreign Studies, 107 Imun-ro, Imun-dong, Dongdaemun-gu, Seoul 130-791, Korea, e-mail:; Sadahiro Maeda, Department of Mathematics, Saga University, 1 Honjo-machi, Saga 840-8502, Japan, e-mail:

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