Construction of a New Hypersurface Family Using the Spherical Product in Minkowski Geometry

The spherical product of two curves, composed of a total of n components, gives rise to spherical product surfaces in Euclidean space ${\mathbb{E}}^n$, frequently resulting in surfaces of revolution, including superquadrics, which often exhibit inherent symmetry. When $(n-1)$-planar curves are considered, this construction enables the generation of hypersurfaces in n-dimensional spaces. Building upon this geometric framework, we conduct the first-ever investigation of spherical product hypersurfaces in the context of Minkowski geometry. We define these hypersurfaces in four-dimensional Minkowski space ${\mathbb{E}}_1^4$ and derive explicit expressions for their Gaussian and mean curvatures. We also determine the conditions under which such hypersurfaces are flat or minimal. Furthermore, we reinterpret certain hyperquadrics as specific instances of spherical product hypersurfaces in ${\mathbb{E}}_1^4$, supported by visual illustrations. Finally, we extend the construction to arbitrary-dimensional Minkowski spaces, providing a unified formulation for spherical product hypersurfaces across higher-dimensional Lorentzian geometries.