Hypersurfaces of revolution family supplying $\mathrm{\Delta }\mathfrak{r}=\mathcal{A}\mathfrak{r}$ in pseudo-Euclidean space ${\mathbb{E}}_3^7$

In this study, we introduce a family of hypersurfaces of revolution characterized by six parameters in the seven-dimensional pseudo-Euclidean space ${\mathbb{E}}_3^7$. These hypersurfaces exhibit intriguing geometric properties, and our aim is to analyze them in detail. To begin, we calculate the matrices corresponding to the fundamental form, Gauss map, and shape operator associated with this hypersurface family. These matrices provide essential information about the local geometry of the hypersurfaces, including their curvatures and tangent spaces. Using the Cayley-Hamilton theorem, we employ matrix algebra techniques to determine the curvatures of the hypersurfaces. This theorem allows us to express the characteristic polynomial of a matrix in terms of the matrix itself, enabling us to compute the curvatures effectively. In addition, we establish equations that describe the interrelation between the mean curvature and the Gauss-Kronecker curvature of the hypersurface family. These equations provide insights into the geometric behavior of the surfaces and offer a deeper understanding of their intrinsic properties. Furthermore, we investigate the relationship between the Laplace-Beltrami operator, a differential operator that characterizes the geometry of the hypersurfaces, and a specific $7\times 7$ matrix denoted as $\mathcal{A}$. By studying this relation, we gain further insights into the geometric structure and differential properties of the hypersurface family. Overall, our study contributes to the understanding of hypersurfaces of revolution in ${\mathbb{E}}_3^7$, offering mathematical insights and establishing connections between various geometric quantities and operators associated with this family.