On Submanifold Of Local Symmetric Conformal Flat Space

In this thesis, we investigate the special submamfold of Riemannian manifold with parallel mean curvature metric.we investigate the submanifold of local symmetric conformal flat space with parallel unit mean curvature metric.And we obtain some new 6 -pinching theorem: Theorem 1: Let Nn+p be a local symmetric conformal flat n + p -dimension space with be a compact n -dimension submanifold with non-zero parallel unit mean curvature metric immersed in Nn+p ( p > 2 ) .IfHere, S is the square of the second fundamental form. Thus:(1) Mn is a hypersurface of geodesic n + 1 -dimension sumanifold of Nn+p .or(2 If n= p = 2,thus S = 4H2+2,and Mn is a surface of N4(1).Here, A(n,p) =max Theorem 2: Let Mn be a compact n -dimension submamfold with non-zero parallel mean curvature metric immersed in sphere Sn+p ( p > 2 ) ,ifhere,S is the square of the second fundamental form.Thus:(1) Mn is a hypersurface of geodesic n+1 -dimension sumanifold of Sn+p.or(2) if n= p = 2,thus S = 4H2+2Theorem 3: Let Nn+p be a local symmetric conformal flat n + p -dimension Riemannian manifold, Mn be a compact n-dimension submanifold with parallel unit mean curvature metric immersed in Nn+p ( p > 2 ) .Let Q be the infinimum of Ricci curva -ture at every point of Mn,if...