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A characterization of the homogeneous minimal ruled real hypersurface in a complex hyperbolic space
Sadahiro MAEDA, Toshiaki ADACHI, and Young Ho KIM
Source: J. Math. Soc. Japan Volume 61, Number 1
(2009), 315-325.
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.
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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
Full-text: Open access
Links and Identifiers
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
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.
Mathematical Reviews (MathSciNet): MR1710088
Zentralblatt MATH: 0944.53029
Digital Object Identifier: doi:10.1007/s000130050402
T. Adachi and S. Maeda, Global behaviours of circles in a complex hyperbolic space, Tsukuba J. Math., 21 (1997), 29–42.
Mathematical Reviews (MathSciNet): MR1467219
Zentralblatt MATH: 0891.53036
J. Berndt, Real hypersurfaces with constant principal curvatures in complex hyperbolic space, J. Reine Angew. Math., 395 (1989), 132–141.
Mathematical Reviews (MathSciNet): MR983062
J. Berndt and H. Tamaru, Cohomogeneity one actions on noncompact symmetric spaces of rank one, Trans. Amer. Math. Soc., 359 (2007), 3425–3438.
Mathematical Reviews (MathSciNet): MR2299462
Zentralblatt MATH: 1117.53041
Digital Object Identifier: doi:10.1090/S0002-9947-07-04305-X
A. Comtet, On the Landau levels on the hyperbolic plane, Ann. Phys., 173 (1987), 185–209.
Mathematical Reviews (MathSciNet): MR870891
Zentralblatt MATH: 0635.58034
Digital Object Identifier: doi:10.1016/0003-4916(87)90098-4
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.
Mathematical Reviews (MathSciNet): MR1844424
Zentralblatt MATH: 1015.53039
Digital Object Identifier: doi:10.3836/tjm/1255958318
Project Euclid: euclid.tjm/1255958318
M. Kimura, Real hypersurfaces and complex submanifolds in complex projective space, Trans. Amer. Math. Soc., 296 (1986), 137–149.
Mathematical Reviews (MathSciNet): MR837803
Zentralblatt MATH: 0597.53021
Digital Object Identifier: doi:10.1090/S0002-9947-1986-0837803-2
JSTOR: links.jstor.org
M. Kimura, Sectional curvatures of holomorphic planes on a real hypersurface in ${\rm P}^n(C)$, Math. Ann., 276 (1987), 487–497.
Mathematical Reviews (MathSciNet): MR875342
Zentralblatt MATH: 0605.53023
Digital Object Identifier: doi:10.1007/BF01450843
M. Lohnherr and H. Reckziegel, On ruled real hypersurfaces in complex space forms, Geom. Dedicata, 79 (1999), 267–286.
Mathematical Reviews (MathSciNet): MR1669351
Zentralblatt MATH: 0932.53018
Digital Object Identifier: doi:10.1023/A:1005000122427
S. Maeda, Real hypersurfaces of complex projective spaces, Math. Ann., 263 (1983), 473–478.
Mathematical Reviews (MathSciNet): MR707242
Zentralblatt MATH: 0518.53025
Digital Object Identifier: doi:10.1007/BF01457054
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.
Mathematical Reviews (MathSciNet): MR2299725
Zentralblatt MATH: 1173.53320
Digital Object Identifier: doi:10.1007/s10711-006-9100-1
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.
Mathematical Reviews (MathSciNet): MR1466400
Digital Object Identifier: doi:10.1007/PL00004625
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.
Mathematical Reviews (MathSciNet): MR1486875
R. Takagi, On homogeneous real hypersurfaces in a complex projective space, Osaka J. Math., 10 (1973), 495–506.
Mathematical Reviews (MathSciNet): MR336660
Zentralblatt MATH: 0274.53062
Project Euclid: euclid.ojm/1200694557
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