Right Conoids Demonstrating a Time-like Axis within Minkowski Four-Dimensional Space

Abstract

In the four-dimensional Minkowski space, hypersurfaces classified as right conoids with a time-like axis are introduced and studied. The computation of matrices associated with the fundamental form, the Gauss map, and the shape operator specific to these hypersurfaces is included in our analysis. The intrinsic curvatures of these hypersurfaces are determined to provide a deeper understanding of their geometric properties. Additionally, the conditions required for these hypersurfaces to be minimal are established, and detailed calculations of the Laplace–Beltrami operator are performed. Illustrative examples are provided to enhance our comprehension of these concepts. Finally, the umbilical condition is examined to determine when these hypersurfaces become umbilic, and also the Willmore functional is explored.