A characterization of a certain real hypersurface of type ( A 2 ) in a complex projective space

Byung Hak Kim; In-Bae Kim; Sadahiro Maeda

Czechoslovak Mathematical Journal (2017)

  • Volume: 67, Issue: 1, page 271-278
  • ISSN: 0011-4642

Abstract

top
In the class of real hypersurfaces M 2 n - 1 isometrically immersed into a nonflat complex space form M ˜ n ( c ) of constant holomorphic sectional curvature c ( ≠ 0 ) which is either a complex projective space ℂ P n ( c ) or a complex hyperbolic space 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 M ˜ n ( c ) , we consider a certain real hypersurface of type ( A 2 ) in ℂ P n ( c ) and give a geometric characterization of this Hopf manifold.

How to cite

top

Kim, Byung Hak, Kim, In-Bae, and Maeda, Sadahiro. "A characterization of a certain real hypersurface of type $({\rm A}_2)$ in a complex projective space." Czechoslovak Mathematical Journal 67.1 (2017): 271-278. <http://eudml.org/doc/287912>.

@article{Kim2017,
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$$(\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.},
author = {Kim, Byung Hak, Kim, In-Bae, Maeda, Sadahiro},
journal = {Czechoslovak Mathematical Journal},
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},
language = {eng},
number = {1},
pages = {271-278},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {A characterization of a certain real hypersurface of type $(\{\rm A\}_2)$ in a complex projective space},
url = {http://eudml.org/doc/287912},
volume = {67},
year = {2017},
}

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
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 67
IS - 1
SP - 271
EP - 278
AB - 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.
LA - eng
KW - ruled real hypersurface; nonflat complex space form; real hypersurfaces of type $({\rm A}_2)$ in a complex projective space; geodesics; structure torsion; Hopf manifold
UR - http://eudml.org/doc/287912
ER -

References

top
  1. Adachi, T., Maeda, S., Udagawa, S., Circles in a complex projective space, Osaka J. Math. 32 (1995), 709-719. (1995) Zbl0857.53034MR1367901
  2. Adachi, T., Maeda, S., 10.21099/tkbjm/1496163159, Tsukuba J. Math. 21 (1997), 29-42. (1997) Zbl0891.53036MR1467219DOI10.21099/tkbjm/1496163159
  3. Adachi, T., 10.1007/s00605-008-0521-9, Monatsh. Math. 153 (2008), 283-293. (2008) Zbl1151.53049MR2394551DOI10.1007/s00605-008-0521-9
  4. Berndt, J., Tamaru, H., 10.1090/S0002-9947-07-04305-X, Trans. Am. Math. Soc. 359 (2007), 3425-3438. (2007) Zbl1117.53041MR2299462DOI10.1090/S0002-9947-07-04305-X
  5. Ki, U.-H., Kim, I.-B., Lim, D. H., 10.4134/BKMS.2010.47.1.001, Bull. Korean Math. Soc. 47 (2010), 1-15. (2010) Zbl1191.53039MR2604227DOI10.4134/BKMS.2010.47.1.001
  6. Maeda, Y., 10.2969/jmsj/02830529, J. Math. Soc. Japan 28 (1976), 529-540. (1976) Zbl0324.53039MR0407772DOI10.2969/jmsj/02830529
  7. Maeda, S., Adachi, T., 10.1017/S0004972708000014, Bull. Aust. Math. Soc. 77 (2008), 1-8. (2008) Zbl1137.53311MR2411863DOI10.1017/S0004972708000014
  8. Maeda, S., Adachi, T., Kim, Y. H., 10.2969/jmsj/06110315, J. Math. Soc. Japan 61 (2009), 315-325. (2009) Zbl1159.53012MR2272881DOI10.2969/jmsj/06110315
  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. (1998) Zbl0904.53005MR1486875
  10. Takagi, R., On homogeneous real hypersurfaces in a complex projective space, Osaka J. Math. 10 (1973), 495-506. (1973) Zbl0274.53062MR0336660

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.