Hostname: page-component-5cf477f64f-mgq6s Total loading time: 0 Render date: 2025-04-04T11:35:19.831Z Has data issue: false hasContentIssue false
  • English
  • Français

REAL HYPERSURFACES WITH CYCLIC-PARALLEL STRUCTURE JACOBI OPERATORS IN A NONFLAT COMPLEX SPACE FORM

Part of: Local differential geometry Global differential geometry

Published online by Cambridge University Press:  21 October 2009

U-HANG KI  and
HIROYUKI KURIHARA
U-HANG KI
Affiliation:
Department of Mathematics, Kyungpook National University, Daegu 702-701, Korea (email: [email protected])
HIROYUKI KURIHARA*
Affiliation:
Department of Liberal Arts and Engineering Sciences, Hachinohe National College of Technology, Hachinohe, Aomori 039-1192, Japan (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

It is known that there are no real hypersurfaces with parallel structure Jacobi operators in a nonflat complex space form. In this paper, we classify real hypersurfaces in a nonflat complex space form whose structure Jacobi operator is cyclic-parallel.

Keywords

complex space formreal hypersurfacestructure Jacobi operatorcyclic-parallel

MSC classification

Secondary: 53B25: Local submanifolds 53C25: Special Riemannian manifolds (Einstein, Sasakian, etc.)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 81 , Issue 2 , April 2010 , pp. 260 - 273
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2009

References

[1]Berndt, J., ‘Real hypersurfaces with constant principal curvatures in complex hyperbolic spaces’, J. Reine Angew. Math. 395 (1989), 132–141.Google Scholar
[2]Berndt, J. and Tamaru, H., ‘Cohomogeneity one actions on noncompact symmetric spaces of rank one’, Trans. Amer. Math. Soc. 359 (2007), 3425–3438.CrossRefGoogle Scholar
[3]Cecil, T. E. and Ryan, P. J., ‘Focal sets and real hypersurfaces in complex projective space’, Trans. Amer. Math. Soc. 269 (1982), 481–499.Google Scholar
[4]Cho, J. T. and Ki, U.-H., ‘Real hypersurfaces in complex projective spaces in terms of Jacobi operators’, Acta Math. Hungar. 80 (1998), 155–167.CrossRefGoogle Scholar
[5]Ki, U.-H., Kurihara, H. and Takagi, R., ‘Jacobi operators along the structure flow on real hypersurfaces in a nonflat complex space form’, Tsukuba J. Math. 33 (2009), 39–56.Google Scholar
[6]Ki, U.-H. and Suh, Y. J., ‘On real hypersurfaces of a complex space form’, Math. J. Okayama Univ. 32 (1990), 207–221.Google Scholar
[7]Kimura, M., ‘Real hypersurfaces and complex submanifolds in complex projective space’, Trans. Amer. Math. Soc. 296 (1986), 137–149.Google Scholar
[8]Maeda, Y., ‘On real hypersurfaces in a complex projective space’, J. Math. Soc. Japan 28 (1976), 529–540.CrossRefGoogle Scholar
[9]Montiel, S. and Romero, A., ‘On some real hypersurfaces of a complex hyperbolic space’, Geom. Dedicata 20 (1986), 245–261.CrossRefGoogle Scholar
[10]Niebergall, R. and Ryan, P. J., ‘Real hypersurfaces in complex space forms’, in: Tight and Taut Submanifolds, (eds. Cecil, T. E. and Chern, S. S.) (Cambridge University Press, Cambridge, 1998), pp. 233–305.Google Scholar
[11]Okumura, M., ‘On some real hypersurfaces of a complex projective space’, Trans. Amer. Math. Soc. 212 (1975), 355–364.CrossRefGoogle Scholar
[12]Ortega, M., Pérez, J. D. and Santos, F. G., ‘Non-existence of real hypersurfaces with parallel structure Jacobi operator in nonflat complex space forms’, Rocky Mountain J. Math. 36 (2006), 1603–1613.CrossRefGoogle Scholar
[13]Pérez, J. D., Santos, F. G. and Suh, Y. J., ‘Real hypersurfaces in complex projective spaces whose structure Jacobi operator is D-parallel’, Bull. Belg. Math. Soc. Simon Stevin 13 (2006), 459–469.Google Scholar
[14]Takagi, R., ‘On homogeneous real hypersurfaces in a complex projective space’, Osaka J. Math. 19 (1973), 495–506.Google Scholar
[15]Takagi, R., ‘Real hypersurfaces in a complex projective space with constant principal curvatures’, J. Math. Soc. Japan 15 (1975), 43–53.Google Scholar
[16]Takagi, R., ‘Real hypersurfaces in a complex projective space with constant principal curvatures II’, J. Math. Soc. Japan 15 (1975), 507–516.Google Scholar