On Compact Submanifolds In A Locally Symmetric And Conformally Flat Riemannian Manifold

In this paper,we study the problem on the reduction of codimensions for compact submanifolds in locally symmetric and conformally flat Riemannian manifolds with parallel unit mean curvature vector field. Some pinching constants for the reduction of codimension are obtained.The article has five sections. The first and the second sections may be viewed as preliminary sections. In the section three we study the submanifolds in locally symmetric and conformally flat Riemannian manifolds of which the sectionalcurvature is satisfied 1/2<δ≤K≤1 ,and improve the work of Cai yan did in [3]. Our theorem states that:Theorem 1. Let N^n+p be a n+p (p≥2) dimensional locally symmetric and conformally flat Riemannian manifold, of which the sectional curvature satisfy 1/2<δ≤K≤1. Let Mⁿ be a compact submanifold in N^n+p with parallel unit mean cuavture vector. If the length of the second fundamental form of Mⁿ satisfied:where M = max{(?),1+(1/2)sgn(p-2)}.Then either(1) Mⁿ is a hypersurface of Nⁿ⁺¹,which is a totally geodesic submanifoldin N^n+p or(2) n=p=2, M² is a Clifford minimal surface in constant curved manifold N⁴ ,In the forth section, we study the submanifolds in locally sysmmetric and conformally flat manifold which has bounded Ricci curvature, and improve Zhang's theorems in [2]. We get:Theorem 2. Let N^n+p be a n+p (p≥2) dimensional locally symmetric and conformally flat manifold which has bounded Ricci curvature. Let Mⁿ be a compact submanifold in N^n+p with parallel unit mean cuavture vector. If the length of the second fundamental form of M" satisfied:S<(n/M(n+p-2))t_Twhere M = max{(?),1+1/2sgn(p-2)} , t_T=3t_c -T_c(K/n+p-1), T_c and t_c arerespectively the up and bound of the Ricci curvature of N^n+p, K is the scalorcurvature of N^n+p.Then, either(1)Mⁿ is a hypersurface of Nⁿ⁺¹ ,which is a totally geodesic submanifoldin N^n+p or(2) n=p=2, M² is a Clifford minimal surface in constant curved manifold N⁴ ,In the section 5,we give an example to show that the pinching constant in theorem 2 is the best when n≥8.