Geodesic
Characterization of totally umbilic hypersurfaces in a space form by circles
Czechoslovak Mathematical Journal, Tome 55 (2005) no. 1, pp. 203-207.

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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.
Classification : 53B25, 53C40
Keywords: totally umbilic hypersurfaces; circles; space forms 
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Adachi, Toshiaki; Maeda, Sadahiro. Characterization of totally umbilic hypersurfaces in a space form by circles. Czechoslovak Mathematical Journal, Tome 55 (2005) no. 1, pp. 203-207. http://geodesic.mathdoc.fr/item/CMJ_2005__55_1_a14/