A helicoidal hypersurfaces family in five-dimensional euclidean space

Abstract

A family of helicoidal hypersurfaces, denoted as r(u, v, s, t), is introduced within the context of the five-dimensional Euclidean space E-5. Matrices for the first and second fundamental forms, the Gauss map, and the shape operator matrix of r are derived. Furthermore, by employing the Cayley-Hamilton theorem to define the curvatures of these hypersurfaces, the curvatures are computed specifically for the helicoidal hypersurfaces family r. Several relationships between the mean and Gauss-Kronecker curvatures of r are established. Additionally, the equation triangle r = Ar is demonstrated, where A is a 5 x 5 matrix in E-5.