On The Characterizations Of Hyper Spheres And Infinitesimal Isometry Of Surfaces

This dissertation is consist of two parts.In the first part,we discussed the characterizations of Hypersphere in Euclidean space,summarized some theorems about the characterizations of Hypersphere,we obtained some equivalent theorems on the characterizations of Hypersphere in E³, E⁴, E⁵, E⁶ and given simple proof, generalized several theorems.Theorem A:Let M(?)Eⁿ⁺¹ be a compact hypersurface with an orthogonal system of lines of curvature.Suppose:â‘ k_i> 0, i = 1,2, ,n;â‘¡the boundary ?M of M consists of umbilical points;â‘¢H 1 = C1( H = C);Then M is a part of a hypersphere.Theorem B:Let M(?)Eⁿ⁺¹ be a compact hypersurface with an orthogonal system of lines of curvature.Suppose:â‘ k_i> 0, i = 1,2, ,n;â‘¡H 2 = C2( Ri jij= C′) ;Then M is a part of a hypersphere.Theorem C:Let M(?)Eⁿ⁺¹ be a compact hypersurface with an orthogonal system of lines of curvature.Suppose:â‘ k_i> 0, i = 1,2, ,n;â‘¡the boundary ?M of M consists of umbilical points;â‘¢there is,on M ,f i:M→R being positive functions such that the matrix (b k_l )( n-1 )×( n-1 ) is positively semi-definite for pairwise different i , j₁ , j₂ , , jn-1 , , ;bk_l = f jk₊ f jl(?)f ik_≠l2( );bk_k= f i + fjkThen M is a part of a hypersphere.In the second part, the conception of Infinitesimal Isometry of Surfaces in Eⁿis introduced, we generalized some results about Infinitesimal Isometry in E³, E⁴,E5 by theory of partial differential equation (solution of partial differential equations) and obtained a theorem about Infinitesimal Isometry of Surfaces in Eⁿ.Theorem D:Let M: D→En, D(?)R2 a bounded domain,be a surface. n: M→N(M) is a section, formΙΙm ( nm, ?) is positively definite for m∈M, vector n m and mean curvature vector is not orthogonal at m .Let V is Infinitesimal Isometry of M , V m belong to the vector space spanned by Tm (M) and n m for m∈M.If m∈?M, V m orthogonal to Tm (M).then V =0 on M .