Spherical Theorems Of Hypersurface In Space Form

In classical differential geometry,the research on surfaces with constant Gauss curvature or constant average curvature in three-dimensional Euclidean space has al-ways attracted much attention.An important result is that if M is a complete immersed surface of R3 and the Gauss curvature of M is a non-zero constant,then M is a spherical surface.This is derived from Hilbert's theorem and Liebmann's theorem.Many math-ematicians have made various extensions to the spherical theorems of immersed hyper-surfaces in Euclidean space.The generalizations are mainly in two aspects.One is the generalization of curvature,that is to extend the average curvature to Gauss-Kronecker curvature.The second is the generalization of the space form,that is,the hypersurface of the Euclidean space is extended to the hypersurface of hyperbolic space,semi-open sphere and other space forms.This article is a summary article,which mainly summarizes the achievements and progress of the spherical theorem of hypersurfaces in space forms.The first chapter of this article is an introduction to some basic knowledge in differential manifolds and Riemannian geometry.The latter is divided into three parts to introduce sphericity the-orems.The second chapter describes the spherical theorems of Euclidean space.At the same time,in this part,we use Minkowski formula to give some proofs of the spherical theorems.The third chapter introduces the spherical theorems of hypersurfaces in hy-perbolic space.This part considers the spherical theorems under different relationships between Gauss-Kronecker curvatures.Using similar ideas,the spherical theorems of spherical space are introduced in Chapter 4.In this paper,the proofs of the sphericity theorems of Euclidean space in Chapter 3 and 4 give the proofs of the partial sphericity theorems of hyperbolic space and spherical space.This article introduces the spherical theorems of immersed hypersurfaces in dif-ferent space forms.Combining with the new results produced recently,I hope to give a more comprehensive summary.