Geometric Characterizations of Euclidean Spheres Using the Tangential Component of the Position Vector

Abstract

In this study, we explore the geometric properties of spheres on a compact hypersurface in Euclidean space Rn+1\mathbb{R}^{n+1} using the support function θ and the tangential component ψT\psi_T of the position vector field ψ\psi. The first characterization extends existing results by removing constraints on the tangential component ψT\psi_T, and applies alternative proof techniques to obtain new insights. In the second characterization, we focus on the specific role of the support function θ in defining these geometric properties, providing a more comprehensive understanding of the structure of spheres within the hypersurface. This work contributes to the broader field of differential geometry by offering novel approaches to the study of hypersurfaces and their associated curvature characteristics.