Some Pinching Problems About Sub Manifolds In Some Riemannian Manifolds

The paper describes some pinching problems about four species sub manifolds in Rieman manifold.1 Many questions of compact minimal sub manifolds in Riemanian manifold with quasi-constant curvature are studied. When it is total geodesic riemannia sub manifold. Four integrating adequate conditions are given.2 Le Npn+p be a locally symmetric complete and connected pseudo-Riemanian manifold whose sectional curvature kN satieties c1 kN c2 let M"be a maximal space like sub manifold in Npn+p, An.estimate is given for the square of the length of the second fundamental form of Mn. It is total geodesic sub manifold .3 The pinching problems are discussed on the compact space-like sub manifold with unit parallel mean curvature vector in de sitter space a sufficient condition for it to be total geodesic sub manifold.4 A compact sub manifold with positive curvature in the constant curved manifold s weaken the condition s of the theorems and obtained better conclusion for it to be a totally umbilical sub manifold.