Differential Geometry of the Family of Helical Hypersurfaces with a Light-like Axis in Minkowski Spacetime L4

Abstract

We investigate the class of helical hypersurfaces parametrized by x=x(u,v,w), characterized by a light-like axis in Minkowski spacetime L4. We determine the matrices that represent the fundamental forms, Gauss map, and shape operator of x. Furthermore, employing the Cayley–Hamilton theorem, we compute the curvatures associated with x. We explore the conditions under which the curvatures of x possess the property of being umbilical. Moreover, we provide the Laplace–Beltrami operator for the family of helical hypersurfaces with a light-like axis in L4.