An Inequality Of Submanifolds With Parallel Mean Curvature And The Pinching Problem

In this paper,we mainly study the Pinching problem of compact submanifolds with parallel mean curvature vector in Riemannian manifolds, we get a series of new results as follows:Theorem I. Let Mⁿ be a compact oriented submanifold in a manifold with constant curvature c, let h be the parallel mean curvature vector and K(x) be the function assigns to each point of Mⁿ the infinimum of the sectional curvatures of Mⁿ at that point.Then (?) a∈[-1, 1],we havewhereθ_{p, h} is denned in introduction equality(1.3).Theorem II. Let Mⁿ be a compact oriented hypersurface in a manifold N_cⁿ⁺¹ with constant curvature c,if Mⁿ has nonnegative sectional curvature,then Mⁿ is totally umbilical or the sectional curvature K = 0.Theorem III. Let Mⁿ be a pseudo-umbilical submanifold in a manifold N_c^n+p with constant curvature c,(i).If the sectional curvature of Mⁿ is everywhere not less than , then Mⁿ is totally umbilical or the sectional curvature .(ii).Suppose , then |Φ|² = 0,and Mⁿ is totally umbilical orTheorem IV. Let Mⁿ be a compact oriented submanifold in a unit sphere S^n+p with parallel mean curvature vector, suppose , ,then Mⁿ is a standard sphere Sⁿ(r), ; or the equality holds. in the latter case,we give some incomplete classifications of Mⁿ.