The hypersurfaces in the Euclidean sphere with relative affine Gauss maps. (Chinese) Zbl 0567.53041

The notion of relative affine mappings was introduced by K. Yano and S. Ishihara [J. Differ. Geom. 10, 501-509 (1975; Zbl 0317.53044)]. The main purpose of this paper is to classify compact hypersurfaces of an m-sphere \(S^ m\subset E^{m+1}\) with relative affine Gauss map. They show that such a hypersurface is either a totally geodesic hypersurface or a generalized Clifford torus \[ M^ m_ k(\lambda)=S^ k(1/\sqrt{1+\lambda^ 2})\times S^{m-k}(\lambda /\sqrt{1+\lambda^ 2}),\quad \lambda >0,\quad 0\leq k\leq m \] .
Reviewer: B.-Y.Chen


53C40 Global submanifolds


Zbl 0317.53044