Spherical Submanifold In Codimension Compression Theorem

In 1968, J. Simons studied the n-dimensional compact minimal submanifold M" in an (n+p)-dimensional unit sphere Sn+p, and proved the famous Simons pinching theorem. In 1971, Chern-do Carmo-Kobayashi researched the geometric structures of the n-dimensional compact minimal submanifold M" in a unit sphere Sn+p under the pinching condition, here S is the squared norm of the second fundamental form. H. B.Lawson also studied the case of p=1 independently. In 1991, Y. B. Shen and A. M. Li successfully improved the constant of the pinching problem tomax . As a generalization of the pinching problem for minimalsubmanifolds, M. Okumura and S. T. Yau ever researched a Simons' type pinching problem for submanifold with parallel mean curvature in a sphere, and obtained partial results. In 1993, H. W. Xu proved the pinching theorem for submanifolds with parallel mean curvature in a sphere.In the present paper, we mainly study the codimension reduction problems for submanifolds with parallel unit mean curvature vector in a sphere. We prove the codimension reduction theorems for compact submanifolds with parallel unit mean curvature vector in a sphere under pointwise and global pinching conditions respectively. We obtain thefollowingTheorem 3.1. Let Mn be an w-dimensional compact submanifold with parallel unit mean curvature vector in an (w+p)-dimensional unit sphere Sn+p (p>1). Let 5 and H be the squared norm of the second fundamental form and mean curvature of M respectively. If S2,p>1). Let S be the squared norm of the second fundamental form of M, and H0 the maximum of the mean curvature of M If H0), where B(n, p, H0) is apositive explicit constant depending only on n, p and H0 , then exists a totally geodesic submanifold Sn+1, such that Mn lies inSn+1.+...