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 L-4. 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 L-4.