A Characterization of Totally Umbilical Hypersurfaces of a Space Form by Geodesic Mapping

Abstract

The idea of considering the second fundamental form of a hypersurface as the first fundamental form of another hypersurface has found very useful applications in Riemannian and semi-Riemannian geometry especially when trying to characterize extrinsic hyperspheres and ovaloids. Recently T. Adachi and S. Maeda gave a characterization of totally umbilical hypersurfaces in a space form by circles. In our paper we give a characterization of totally umbilical hypersurfaces of a space form by means of geodesic mapping.