Hypersurfaces In A Real Space Form

In the present dissertation, we mainly study hypersurfaces in a real space form. We investigate three problems, and these problems are respectively discussed in chapter 2, chapter 3 and chapter 4. The main contents of this paper are as follows:In chapter 1, we first introduce the research backgrounds of Weingarten hypersurfaces and hypersurfaces in a real space form with a unit Killing vector field, then the related knowledge of these two types of hypersurfaces is given.In chapter 2, we resume the related basics of linear Weingarten hypersurfaces in the Euclidean space Rn+1. Then taking an operator introducted by Cheng-Yau in document as a tool and making use of the method in, we study a class of linear Weingarten hypersurfaces in Rn+1 and establish some rigid theorems for the totally umbilical hyper-surfaces.In chapter 3, we present some basic knowledge of compact hypersurfaces in the Eu-clidean space Rn+1 with a unit Killing vector field. Then we give an equivalent condition of the type of hypersurfaces which are isometric to the standard sphere Sn(c).In chapter 4, we study complete hypersurfaces in the hyperbolic space Hn+1(-1) with a unit Killing vector field. Under the assumption of the Killing vector field being the eigenvector of the shape operator A, we prove the hypersurfaces are totally umbilical or Riemannian product Sk(c1)×Hn-k(c2),1≤k≤n-1.