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Article

Adachi, Toshiaki ; Maeda, Sadahiro
Characterization of totally umbilic hypersurfaces in a space form by circles. (English). Czechoslovak Mathematical Journal, vol. 55 (2005), issue 1, pp. 203-207
MSC: 53B25, 53C40 | MR 2121667 | Zbl 1081.53020
Keywords:
totally umbilic hypersurfaces; circles; space forms
Summary:
In this paper we characterize totally umbilic hypersurfaces in a space form by a property of the extrinsic shape of circles on hypersurfaces. This characterization corresponds to characterizations of isoparametric hypersurfaces in a space form by properties of the extrinsic shape of geodesics due to Kimura-Maeda.
References:
[1] M. Kimura and S. Maeda: Geometric meaning of isoparametric hypersurfaces in a real space form. Canadian Math. Bull. 43 (2000), 74–78. DOI 10.4153/CMB-2000-011-3 | MR 1749951
[2] R. Miyaoka: The homogeneity of isoparametric hypersurfaces with six principal curvatures. Preprint Titech-Math. Vol.  80, 1998.
[3] S. T. Yau: Open problems in geometry. Part  I. Proc. Sympos. Pure Math. 54 (1993), 1–28. MR 1216573
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