Compact Spacelike Submanifolds In A De Sitter Space

Research on geometry of spacelike submanifolds in pseudo-Riemannian manifold has been always a focus to lots of mathematicians and physicians since the research contents are umbilically linked with theoretical physics,Riemannian geometry,com-plex geometry and so forth and have considerable practical significance.In this paper,we apply the self-adjiont second order differential operator ? introduced by S.Y.Cheng and S.T.Yau[15]to the research on compact spacelike submanifolds in a de Sitter s-pace,and give a corresponding integral equality.Assuming that the normal bundles of these submanifolds are flat,by some lemmas,we get an integral inequality and then get some rigidity results.The paper is divided into the following parts.In chapter 1,we will roughly introduce the background,significance,development of the research on submanifolds in de Sitter spaces at home and abroad,main contents in this paper as well.In chapter 2,we will present some fundamental conceptions concerned,recall Cheng-Yau's self-adjiont operator and also give several lemmas which will be used afterwards.In chapter 3,we will give some fundamental formulas about submanifolds in de Sitter space,then combine these formulas with chapter 2 to get an integral inequality.In chapter 4,in light of the integral inequality obtained in chapter 3 and relevant Lemmas,we will give main conclusions and their simple proofs.In chapter 5,allowing for the entire paper,we will put forward several relevant questions for further possible research.