The Pinching Problem Of Submanifolds In Locally Symmetric Space

The theory of submanifolds is a developed subject of differential geometry.If we give some restrician to the intrinsic quantities of submanifolds, such as second fundamental form, scalar curvature,Ricci curvature or sectional curvature,then we can get some new property of the submanifolds.The procedure is called pinching problem of submanifolds. In 1968,J.simons got the integral formulaof the minimal submanifolds of unit sphere S^n+p(1).After that time,many geometrician had got lots of results on the pinching problem of submanifolds.We study a pinching theorem for submanifolds of locally symmetric space in this paper.This paper has three chapters.In the first chapter, we give a brief introduction of the property of submanifoldsin locally symmetric space,which prepares for the proof of the following main results.In the second chapter, we study a pinching theorem for submanifolds of locally symmetric space with parallel mean curvature .Let M be a compact submanifold of locally symmetric space N^n+p,we apply Gauss equation,Ricci equation and the property of locally symmetry of the outer space ,through studying f(x) = (?) .then we get a pinching theorem.When p≥2,what we obtain improves the corresponding theorem of article[1].In the third chapter, we study complete minimal submanifolds of locallysymmetric space, and we obtain a pinching theorem about the Ricci curvature of the minimal submanifolds, which generalizes the result of Norio Ejiri's from sphere space to locally symmetric space.