Hypersphere satisfying ∆^j x=Ax in 4-space

In this thesis, the sphere surface with radius r, which provides the property ∆^I x=Ax, A∈Mat(3,3), in 3-dimensional Euclidean space will be examined. The fundamental form I of the sphere and the Gauss map will be obtained. The sphere will find the Laplace-Beltrami operator, which depends on the fundamental form I. The relations of the Laplace-Beltrami operator, Gauss map, and mean curvature of the sphere surface will be given. Moreover, the Laplace-Beltrami operator will be calculated for the fundamental form II of the sphere. The operations for the sphere surface, the proof techniques and formulas in 3-dimensional differential geometry will be carried to 4-dimensional and the calculations will be done on the hypersphere. In 4-dimensional Euclidean space, the hypersphere of radius r, which satisfies the property ∆^I x=Ax, A∈Mat(4,4), will be studied. The fundamental form I and Gauss map of the hypersphere will be obtained. The Laplace-Beltrami operator of the hypersphere depending on the fundamental form I will be found. In addition, the relations of the Laplace-Beltrami operator, Gauss map, and mean curvature of the hypersphere will be investigated. Finally, the Laplace-Beltrami operator will be obtained for the fundamental form II of the hypersphere.