On Submanifolds With Parallel Mean Curvature Vector In Locally Symmetric Spaces

Let Nn+p be a locally symmetric and complete Riemannian manifold, whose sectional curvature KN satisties 1/2 < δ ≤KN ≤1.Let Mn be a compact submanifold with parallel mean curvature vector in Nn+p. In this paper, we give a pinching theorem for the square of the length of the second fundamental form of Mn, which generalizes the results obtained by W.D.Song [1]and C.Z.OuYang[7] respectively.Theorem 1 Let Nn+p(n > 2,p > 2) be an (n + p) -dimensional locally symmetric and complete Riemannian manifold , whose sectional curvature KN satisfies 1/2 < δ ≤ KN ≤ 1 .Suppose that Mn is a compact submanifold with parallel mean curvature vector h in Nn+p ,and the mean curvature H = |h| of Mnsatisfies|H| ≥ n2(p- 1),S is the square of the length of the second fundamental form of Mn.If |φ|2 = S - nH2 satisfiesthen MN is a totally umbilical submanifold in Sn+p(1), where Theorem 2 Let Mn(n > 2) be a compact submanifold with parallel mean curvature vector h in a unit sphere Sn+p(1)(p > 2). The mean curvature H = |h| of Mnsatisfies |H|≥ n2(p - 1), 5 is the square of the length of the second fundamental form of Mn.If |φ|2 = S - nH2satisfiesthenMn is a totally umbilical submanifold in Sn+p(1).