Cheng–yau operator and gauss map of rotational hypersurfaces in 4-space

Abstract

We consider rotational hypersurface in the four-dimensional Euclidean space E4. We study the Gauss map G of rotational hypersurface in E4 with respect to the so-called Cheng–Yau operator L1 acting on the functions defined on the hypersurfaces. We obtain the classification theorem that the only rotational hypersurface with Gauss map G satisfying L1G = AG for some 4 × 4 matrix A are the hyperplanes, right circular hypercones, circular hypercylinders, and hyperspheres.