Geodesic
A characterization of a certain real hypersurface of type $({\rm A}_2)$ in a complex projective space
Czechoslovak Mathematical Journal, Tome 67 (2017) no. 1, pp. 271-278
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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 $c$ $(\ne 0)$ 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.
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 $c$ $(\ne 0)$ 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.
DOI : 10.21136/CMJ.2017.0546-15
Classification : 53B25, 53C40
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 
@article{10_21136_CMJ_2017_0546_15,
     author = {Kim, Byung Hak and Kim, In-Bae and Maeda, Sadahiro},
     title = {A characterization of a certain real hypersurface of type $({\rm A}_2)$ in a complex projective space},
     journal = {Czechoslovak Mathematical Journal},
     pages = {271--278},
     year = {2017},
     volume = {67},
     number = {1},
     doi = {10.21136/CMJ.2017.0546-15},
     mrnumber = {3633011},
     zbl = {06738517},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2017.0546-15/}
}
TY  - JOUR
AU  - Kim, Byung Hak
AU  - Kim, In-Bae
AU  - Maeda, Sadahiro
TI  - A characterization of a certain real hypersurface of type $({\rm A}_2)$ in a complex projective space
JO  - Czechoslovak Mathematical Journal
PY  - 2017
SP  - 271
EP  - 278
VL  - 67
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2017.0546-15/
DO  - 10.21136/CMJ.2017.0546-15
LA  - en
ID  - 10_21136_CMJ_2017_0546_15
ER  - 
%0 Journal Article
%A Kim, Byung Hak
%A Kim, In-Bae
%A Maeda, Sadahiro
%T A characterization of a certain real hypersurface of type $({\rm A}_2)$ in a complex projective space
%J Czechoslovak Mathematical Journal
%D 2017
%P 271-278
%V 67
%N 1
%U http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2017.0546-15/
%R 10.21136/CMJ.2017.0546-15
%G en
%F 10_21136_CMJ_2017_0546_15
Kim, Byung Hak; Kim, In-Bae; Maeda, Sadahiro. A characterization of a certain real hypersurface of type $({\rm A}_2)$ in a complex projective space. Czechoslovak Mathematical Journal, Tome 67 (2017) no. 1, pp. 271-278. doi: 10.21136/CMJ.2017.0546-15

[1] Adachi, T., Maeda, S., Udagawa, S.: Circles in a complex projective space. Osaka J. Math. 32 (1995), 709-719. | MR | JFM

[2] Adachi, T., Maeda, S.: Global behaviours of circles in a complex hyperbolic space. Tsukuba J. Math. 21 (1997), 29-42. | DOI | MR | JFM

[3] Adachi, T.: Geodesics on real hypersurfaces of type $( A_2)$ in a complex space form. Monatsh. Math. 153 (2008), 283-293. | DOI | MR | JFM

[4] Berndt, J., Tamaru, H.: Cohomogeneity one actions on noncompact symmetric spaces of rank one. Trans. Am. Math. Soc. 359 (2007), 3425-3438. | DOI | MR | JFM

[5] Ki, U.-H., Kim, I.-B., Lim, D. H.: Characterizations of real hypersurfaces of type A in a complex space form. Bull. Korean Math. Soc. 47 (2010), 1-15. | DOI | MR | JFM

[6] Maeda, Y.: On real hypersurfaces of a complex projective space. J. Math. Soc. Japan 28 (1976), 529-540. | DOI | MR | JFM

[7] Maeda, S., Adachi, T.: Characterizations of hypersurfaces of type $ A_2$ in a complex projective space. Bull. Aust. Math. Soc. 77 (2008), 1-8. | DOI | MR | JFM

[8] Maeda, S., Adachi, T., Kim, Y. H.: A characterization of the homogeneous minimal ruled real hypersurface in a complex hyperbolic space. J. Math. Soc. Japan 61 (2009), 315-325. | DOI | MR | JFM

[9] Niebergall, R., Ryan, P. J.: 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 | JFM

[10] Takagi, R.: On homogeneous real hypersurfaces in a complex projective space. Osaka J. Math. 10 (1973), 495-506. | MR | JFM

Cité par Sources :