Twisted Hypersurfaces in Euclidean 5-Space

Abstract

The twisted hypersurfaces x with the (0,0,0,0,1) rotating axis in five-dimensional Euclidean space E5 is considered. The fundamental forms, the Gauss map, and the shape operator of x are calculated. In E5, describing the curvatures by using the Cayley–Hamilton theorem, the curvatures of hypersurfaces x are obtained. The solutions of differential equations of the curvatures of the hypersurfaces are open problems. The umbilically and minimality conditions to the curvatures of x are determined. Additionally, the Laplace–Beltrami operator relation of x is given.