totally umbilic hypersurfaces; circles; space forms
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.
 R. Miyaoka: The homogeneity of isoparametric hypersurfaces with six principal curvatures. Preprint Titech-Math. Vol. 80, 1998.
 S. T. Yau: Open problems in geometry. Part I
. Proc. Sympos. Pure Math. 54 (1993), 1–28. MR 1216573