Spherical hypersurfaces with 2-type Gauss map
In the theory of submanifolds of finite Chen-type, the study of submanifolds with finite type Gauss map was initiated by \textit{B.-Y. Chen} and \textit{P. Piccinni} [Bull. Aust. Math. Soc. 35, 161-186 (1987; Zbl 0672.53044)], where they classified submanifolds of Euclidean space \(E^m\) with 1-type Gauss map. In the paper under review, the authors present theorems on spherical hypersurfaces with 2-type Gauss map. Among other things, they prove the following interesting Theorem. If a compact spherical hypersurface has a 2-type Gauss map, then it is non-Einsteinian and has nonconstant mean curvature and nonconstant length of the second fundamental form. The results obtained in this paper give support for a conjecture of the first author which claims that the only spherical hypersurfaces with finite type Gauss map are the hypersurfaces with 1-type Gauss map. For submanifolds of finite Chen-type and submanifolds with finite type Gauss map, a useful reference is the survey paper of \textit{B.-Y. Chen} [Soochow J. Math. 22, 117-337 (1996; Zbl 0867.53001)].
scientific article; zbMATH DE number 1127646