Some Problems On Hypersurfaces In The Euclidean Space Rⁿ⁺¹

Geometry investigates the size(intrinsic) and the camber(extrinsic) of geometric subjects(such as,convex sets,surfaces and submainfolds and etc.) in space, that is,investigates the volumes and curvatures of the subjects.The equalities or inequalities among volumes and curvatures also determine the properties of geometric elements.In differential geometric the study of the hypersurfaces(as special submanifolds) is very active in last century and fruitful research results have been obtained.The moving orthogonal frames method created by E.Cartan guides the study of general manifolds into a new era.It is an important tool for studying differential manifolds.The study on hpyersurfaces in the Euclidean space(?)ⁿ⁺¹ focuses on the r-th mean curvature H_r,r=1,…,n,the symmetric functions of the principal curvatures.We especially study the mean curvature H₁,the scalar curvature H₂, and the Gauss-Kronecker curvature H_n.When n=2,that is,in(?)³,they are the mean curvature H and the Gauss curvature K.For surface in(?)³,Gauss has found an amazing result:the Gauss Curvature is intrinsic.The well-known mathematician Liebmann proves that if the Gauss curvature of a surfaceΣin(?)³ is a constant then it must be a sphere.S.S.Chern gives a simplified proof of the Liebmann theorem by using the following lemmas.Lemma 3.2 Let M be a surface in(?)³.If every point x is umbilical,then either M is hyperplane or M is a hypersphere.Lemma 3.3 Let M be a compact surface embedded in(?)³,if the Gauss curvature K is constant,then K is greater than zero.These two lemmas has been extended to the case of higher dimensional hypersurfaces. Theorem 3.4 Let M be a complete connected hypersurface in the Euclidean space(?)ⁿ⁺¹.If every point x of M is umbilical,then M is either a hyperplane or a hypersphere.Theorem 3.5 Let M be an n-dimensional compact hypersurface embedded in the Euclidean space(?)ⁿ⁺¹,if the Gauss-Kronecker curvature H_n of M is constant, then H_n is greater than zero.In this paper,we follow S.S.Chern idea and give another Simplified proof of these two theorems by using the moving orthogonal frames method.