A Pinching Theorem for Submanifolds of Locally Symmetric Space with Parallel Mean Curvature

设M为单位球面Sn+p(1)中的一个紧致子流形.∪M=∪x∈M∪Mx是M的单位切丛.陈卿引入函数f(x)=maxu,v∈∪Mx‖B(u,u)-B(v,v)‖2,其中B是M的第二基本形式.当M具有平行平均曲率向量时,陈卿通过研究函数f(x),得到一个Pinching定理.当考虑外围流形为局部对称空间时,我们应用Gauss方程,Ricci方程和外围空间的局部对称性质等方法得到:若f(x)满足一个Pinching条件,则M或是全脐的或是一个Veronese曲面.当p≥2时,所得的结果改进了陈卿研究的相应结果.Let Mn be a compact submanifold of unit sphere Sn+p(1)∪M=∪x∈M∪Mx is the unit tangent bundle on M.Chen Qing constructed a function f(x)=maxu,v∈∪Mx‖B(u,u)-B(v,v)‖2,where B is the second fundamental form of M.When M has parallel mean curvature vector,Chen Qing obtained a Pinching theorem through studying function f(x).When considering outer space which is locally symmetry,we apply Gauss equation,Ricci equation and the property of locally symmetry of the outer space,then we obtain the following theorem:if M satisfys a Pinching condition,M is a totally umbilical submanifold or a Veronese surface.When p≥2,What we obtain improves the corresponding theorem of Chen qin′s.福建省自然科学基金(Z0511001)资