On Submanifolds With Parallel Mean Curvature Vector

By means of careful estimates in the Simons-type Formula for the submanifold with parallel mean curvature vector in a conformally flat Riemannian manifold, we get the following pinching resultsTheoreml Let Mn be a compact submanifold with non-zero parallel mean curvature vector in a conformally flat Riemannian manifold. Denote the infimum and the supremum of the Ricci curvature of Nn+p at a point x by a(x) and b(x) repec-tively.Denote by K, S, H the scalar curvature of Nn+p,the length square of the second fundamental form of Mn and the mean curvature of Mn respectively.LetWhen p > 2, n > 2, we have(1) If 2 < n ≤ 7 and A0 + (5 - nH2)[nC0 -3/2S] ≥ 0, then a(x) = b(x) = const,Nn+p is a constant curvature manifold with curvature a/n+p-2, and Mn is its totallvumbilical submanifold.(2) If n ≥ 8 and A0 + (S - nH2)[nC0- S] ≥ 0, then a(x) = b(x) =const, Nn+p is a constant curvature manifold with curvaturea/n+p-2,and Mn is itstotally umbilical submanifold, or Mn lies in Mn lies in an (n+1)-dimensional totallygeodesic submanifold in Nn+p, S is constant and satisfy: Theorem2 Under the same hypothesis as in Theorem l,if p=2 and A0 + (S- 0, then a(x) = b(x) = const, Nn+2 is a constant curvaturemanifold with curvature a/n, and Mn is its totally umbilical submanifold, or Mn lies inMn lies in an (n+1)-dimensional totally geodesic submanifold in Nn+2, S is constantand satisfy: A0+ .If Nn+p is a unit sphere Sn+p(1),then A0 = 0, C0 = 1.Thus we have from Theorml and Theorm2Corollaryl Let Mn be a compact submanifold with a non-zero parallel mean curvature vector in a unit sphere Sn+p,n > 2,p > 2(1) If S ≤2/3n (2 < n≤ 7) or S < 2n - 1 (n ≥ 8),then thenS = nH2 is constant,and Mn is a small sphere Sn(r),where r =(2) If S = 2n - 1 (n ≥ 8), then M is either a small sphere Sn(r0),or a ring surfaceCorollary 2. Let Mn be a compact submanifold with a non-zero parallel mean curvature vector in a unit sphere Sn+2(1),n > 2(1) If S < 2 n -1,then 5 = nH2 is constant, and Mn is a small sphere Sn(r),where r =n/n+S;(2) If S =2n-1 (n ≥ 8), then Mn is either a small sphere Sn(r0),or a ringsurface S1(r)× Sn-1(1- r2), where r02 =Remark Theoreml and Theorem2 correct some mistakes in papers[10,11]. Corol-lary1 and Corrollary2 improve the results in papers [1,3,4,5].