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

  • SADAHIRO MAEDA (a1), TOSHIAKI ADACHI (a2) and YOUNG HO KIM (a3)

Abstract

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.

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References

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1.Adachi, T., Geodesics on real hypersurfaces of type A2 in a complex space form, Mon. Math. 153 (2008), 283293.
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), 1117.
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), 303310.
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), 373408.
5.Berndt, J., Real hypersurfaces with constant principal curvatures in complex hyperbolic space, J. Reine Angew. Math. 395 (1989), 132141.
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), 133152.
7.Ferus, D. and Schirrmacher, S., Submanifolds in Euclidean space with simple geodesics, Math. Ann. 260 (1982), 5762.
8.Kimura, M., Real hypersurfaces and complex submanifolds in complex projective space, Trans. Amer. Math. Soc. 296 (1986), 137149.
9.Maeda, Y., On real hypersurfaces of a complex projective space, J. Math. Soc. Japan 28 (1976), 529540.
10.Maeda, S., Real hypersurfaces of complex projective spaces, Math. Ann. 263 (1983), 473478.
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).
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), 537542.
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), 7180.
14.Montiel, S., Real hypersurfaces of a complex hyperbolic space, J. Math. Soc. Japan 37 (1985), 515535.
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), 233305.
16.Sakamoto, K., Planar geodesic immersions, Tôhoku Math. J. 29 (1977), 2556.
17.Takagi, R., On homogeneous real hypersurfaces in a complex projective space, Osaka J. Math. 10 (1973), 495506.
18.Takagi, R., Real hypersurfaces in a complex projective space with constant principal curvatures, J. Math. Soc. Japan 27 (1975), 4353.

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