# Institute of Mathematics

## 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

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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: bhkim@khu.ac.kr; In-Bae Kim, Department of Mathematics, Hankuk University of Foreign Studies, 107 Imun-ro, Imun-dong, Dongdaemun-gu, Seoul 130-791, Korea, e-mail: ibkim@hufs.ac.kr; Sadahiro Maeda, Department of Mathematics, Saga University, 1 Honjo-machi, Saga 840-8502, Japan, e-mail: smaeda@ms.saga-u.ac.jp

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