Cheng–Yau Operator and Gauss Map of Rotational Hypersurfaces in 4-Space

Abstract

We consider rotational hypersurface in the four-dimensional Euclidean space $$ {\mathbb {E}}^{4}$$ . We study the Gauss map $$\mathbf {G}$$ of rotational hypersurface in $${\mathbb {E}}^{4}$$ with respect to the so-called Cheng–Yau operator $$L_{1}$$ acting on the functions defined on the hypersurfaces. We obtain the classification theorem that the only rotational hypersurface with Gauss map $$\mathbf {G}$$ satisfying $$L_{1}\mathbf {G}=\mathbf {AG}$$ for some $$ 4\times 4$$ matrix $$\mathbf {A}$$ are the hyperplanes, right circular hypercones, circular hypercylinders, and hyperspheres.