On The Problems Of Submanifolds In A Locally Symmetric Conformally Flat And Mobius Submanifolds In Unit Sphere

The article studies the Pinching problem and classification of Euclidean spaceand Mobius submanifolds by studying the theory and method. It consists of four parts.The first part briefly introduces the research background of Riemann manifold andMobius submanifolds.The second part makes a conclusion of compact on submanifolds in a locallysymmetric and conformally flat riemannian Maniflod with parallel mean curvaturevector in tight submanifolds problem,which leads to the pinching theoremThe third part studies the Pinching problem of Mobius submanifold in Sn+1andthe invariant structure equation,which comes to two pinching theorem ofPara-Blaschke.The fourth part is about the para-Blaschke isoparametric hypersurfaces, whichgives a completely classification of them by the feature of para-blaschke.