Journal of the Mathematical Society of Japan

A characterization of the homogeneous minimal ruled real hypersurface in a complex hyperbolic space

Sadahiro MAEDA, Toshiaki ADACHI, and Young Ho KIM

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Abstract

It is well-known that there exist no homogeneous ruled real hypersurfaces in a complex projective space. On the contrary there exists the unique homogeneous ruled real hypersurface in a complex hyperbolic space. Moreover, it is minimal. We characterize geometrically this minimal homogeneous real hypersurface by properties of extrinsic shapes of some curves.

Article information

Source
J. Math. Soc. Japan Volume 61, Number 1 (2009), 315-325.

Dates
First available in Project Euclid: 9 February 2009

Permanent link to this document
http://projecteuclid.org/euclid.jmsj/1234189038

Digital Object Identifier
doi:10.2969/jmsj/06110315

Mathematical Reviews number (MathSciNet)
MR2272881

Zentralblatt MATH identifier
1159.53012

Subjects
Primary: 53B25: Local submanifolds [See also 53C40]
Secondary: 53C40: Global submanifolds [See also 53B25]

Keywords
complex hyperbolic spaces real hypersurfaces totally geodesic complex hypersurfaces homogeneous ruled real hypersurfaces geodesics horocycle-circles integral curves of the chracteristic vector field real hyperbolic planes

Citation

MAEDA, Sadahiro; ADACHI, Toshiaki; KIM, Young Ho. A characterization of the homogeneous minimal ruled real hypersurface in a complex hyperbolic space. J. Math. Soc. Japan 61 (2009), no. 1, 315--325. doi:10.2969/jmsj/06110315. http://projecteuclid.org/euclid.jmsj/1234189038.


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References

  • T. Adachi, M. Kimura and S. Maeda, A characterization of all homogeneous real hypersurfaces in a complex projective space by observing the extrinsic shape of geodesics, Arch. Math. (Basel), 73 (1999), 303–310.
  • T. Adachi and S. Maeda, Global behaviours of circles in a complex hyperbolic space, Tsukuba J. Math., 21 (1997), 29–42.
  • J. Berndt, Real hypersurfaces with constant principal curvatures in complex hyperbolic space, J. Reine Angew. Math., 395 (1989), 132–141.
  • J. Berndt and H. Tamaru, Cohomogeneity one actions on noncompact symmetric spaces of rank one, Trans. Amer. Math. Soc., 359 (2007), 3425–3438.
  • A. Comtet, On the Landau levels on the hyperbolic plane, Ann. Phys., 173 (1987), 185–209.
  • B. Y. Chen and S. Maeda, Hopf hypersurfaces with constant principal curvatures in complex projective or complex hyperbolic spaces, Tokyo J. Math., 24 (2001), 133–152.
  • M. Kimura, Real hypersurfaces and complex submanifolds in complex projective space, Trans. Amer. Math. Soc., 296 (1986), 137–149.
  • M. Kimura, Sectional curvatures of holomorphic planes on a real hypersurface in ${\rm P}^n(C)$, Math. Ann., 276 (1987), 487–497.
  • M. Lohnherr and H. Reckziegel, On ruled real hypersurfaces in complex space forms, Geom. Dedicata, 79 (1999), 267–286.
  • S. Maeda, Real hypersurfaces of complex projective spaces, Math. Ann., 263 (1983), 473–478.
  • S. Maeda and T. Adachi, Integral curves of characteristic vector fields of real hypersurfaces in nonflat complex space forms, Geom. Dedicata, 123 (2006), 65–72.
  • S. Maeda and K. Ogiue, Characterizations of geodesic hyperspheres in a complex projective space by observing the extrinsic shape of geodesics, Math. Z., 225 (1997), 537–542.
  • R. Niebergall and P. J. Ryan, Real hypersurfaces in complex space forms, Tight and Taut Submanifolds, (eds. T. E. Cecil and S. S. Chern), Cambridge University Press, 1998, pp.,233–305.
  • R. Takagi, On homogeneous real hypersurfaces in a complex projective space, Osaka J. Math., 10 (1973), 495–506.