Hypersphere and the Third Laplace-Beltrami Operator

Abstract

In this work, we examine the differential geometric objects of the hypersphere 𝐡� in four dimensional Euclidean geometry 𝔼�4. Giving some notions of four dimension, we consider the 𝑖�th curvature formulas of the hypersurfaces of 𝔼�4. In addition, we reveal the hypersphere satisfying ∆𝐈�𝐈�𝐈�𝐡� = 𝒜�𝐡� for some 4×4 matrix 𝒜�.