Hostname: page-component-76fb5796d-wq484 Total loading time: 0 Render date: 2024-04-26T23:45:46.144Z Has data issue: false hasContentIssue false
  • English
  • Français

CHARACTERISATIONS OF GEODESIC HYPERSPHERES IN A NON-FLAT COMPLEX SPACE FORM

Published online by Cambridge University Press:  02 August 2012

SADAHIRO MAEDA ,
TOSHIAKI ADACHI  and
YOUNG HO KIM
SADAHIRO MAEDA
Affiliation:
Department of Mathematics, Saga University, Saga 840-8502, Japan e-mail: smaeda@ms.saga-u.ac.jp
TOSHIAKI ADACHI
Affiliation:
Department of Mathematics, Nagoya Institute of Technology, Gokiso, Nagoya 466-8555, Japan e-mail: adachi@nitech.ac.jp
YOUNG HO KIM
Affiliation:
Department of Mathematics, Teachers College, Kyungpook National University, Taegu 702-701, Korea e-mail: yhkim@knu.ac.kr
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.

Totally η-umbilic real hypersurfaces are the simplest examples of real hypersurfaces in a non-flat complex space form. Geodesic hyperspheres in this ambient space are typical examples of such real hypersurfaces. We characterise every geodesic hypersphere by observing the extrinsic shapes of its geodesics and using the derivative of its contact form.

Keywords

Primary 53B25Secondary 53C40
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 55 , Issue 1 , January 2013 , pp. 217 - 227
Copyright
Copyright © Glasgow Mathematical Journal Trust 2012

References

REFERENCES

1.Adachi, T., Geodesics on real hypersurfaces of type A2 in a complex space form, Mon. Math. 153 (2008), 283–293.Google Scholar
2.Adachi, T. and Maeda, S., A congruence theorem of geodesics on some naturally reductive Riemannian homogeneous manifolds, C. R. Math. Rep. Acad. Sci. Canada 26 (2004), 11–17.Google Scholar
3.Adachi, T., Maeda, S. and Kimura, M., A characterization of all homogeneous real hypersurfaces in a complex projective space by observing the extrinsic shape of geodesics, Archiv der Math. 73 (1999), 303–310.Google Scholar
4.Adachi, T., Maeda, S. and Yamagishi, M., Length spectrum of geodesic spheres in a non-flat complex space form, J. Math. Soc. Japan 54 (2002), 373–408.CrossRefGoogle Scholar
5.Berndt, J., Real hypersurfaces with constant principal curvatures in complex hyperbolic space, J. Reine Angew. Math. 395 (1989), 132–141.Google Scholar
6.Chen, B. Y. and Maeda, S., Hopf hypersurfaces with constant principal curvatures in complex projective or complex hyperbolic spaces, Tokyo J. Math. 24 (2001), 133–152.Google Scholar
7.Ferus, D. and Schirrmacher, S., Submanifolds in Euclidean space with simple geodesics, Math. Ann. 260 (1982), 57–62.Google Scholar
8.Kimura, M., Real hypersurfaces and complex submanifolds in complex projective space, Trans. Amer. Math. Soc. 296 (1986), 137–149.CrossRefGoogle Scholar
9.Maeda, Y., On real hypersurfaces of a complex projective space, J. Math. Soc. Japan 28 (1976), 529–540.CrossRefGoogle Scholar
10.Maeda, S., Real hypersurfaces of complex projective spaces, Math. Ann. 263 (1983), 473–478.Google Scholar
11.Maeda, S. and Kimura, M., Sectional curvatures of some homogeneous real hypersurfaces in a complex projective space, Complex Analysis and Mathematical Physics, in Proceedings of the 8th international workshop on complex structures and vector fields, (World Scientific, Bulgaria, 2007).Google Scholar
12.Maeda, S. and Ogiue, K., Characterizations of geodesic hyperspheres in a complex projective space by observing the extrinsic shape of geodesics, Math. Z. 225 (1997), 537–542.Google Scholar
13.Maeda, S. and Okumura, K., Three real hypersurfaces some of whose geodesics are mapped to circles with the same curvature in a nonflat complex space form, Geom. Dedicata 156 (2012), 71–80.Google Scholar
14.Montiel, S., Real hypersurfaces of a complex hyperbolic space, J. Math. Soc. Japan 37 (1985), 515–535.Google Scholar
15.Niebergall, R. and Ryan, P. J., Real hypersurfaces in complex space forms, in Tight and taut submanifolds (Cecil, T. E. and Chern, S. S., Editors) (Cambridge University Press, New York, 1997), 233–305.Google Scholar
16.Sakamoto, K., Planar geodesic immersions, Tôhoku Math. J. 29 (1977), 25–56.Google Scholar
17.Takagi, R., On homogeneous real hypersurfaces in a complex projective space, Osaka J. Math. 10 (1973), 495–506.Google Scholar
18.Takagi, R., Real hypersurfaces in a complex projective space with constant principal curvatures, J. Math. Soc. Japan 27 (1975), 43–53.Google Scholar