A Helicoidal Hypersurfaces Family in Five-Dimensional Euclidean Space

Abstract

A family of helicoidal hypersurfaces, denoted as $\mathfrak{x}(u,v,s,t)$, is introduced within the context of the five-dimensional Euclidean space $\mathbb{E}^{5}$. Matrices for the first and second fundamental forms, the Gauss map, and the shape operator matrix of $\mathfrak{x}$ 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 $\mathfrak{x}$. Several relationships between the mean and Gauss--Kronecker curvatures of $\mathfrak{x}$ are established. Additionally, the equation $\Delta \mathfrak{x}=\mathcal{A}\mathfrak{x}$ is demonstrated, where $\mathcal{A}$ is a $5\times 5$ matrix in $\mathbb{E}^{5}$.