On pinching problem of sectional curvature on minimal submanifolds in a symmetric space;
局部对称黎曼流形中极小子流形的截[面]曲率的pinching问题
Let Mmbe a compact submanifold with positive sectional curvature of a space form Nn(c).
设Mm是空间形式Nn(c)中具有正截[面]曲率的紧致子流形,证明了如果n-m≥2,Mm的平均曲率向量关于法联络平行且不为零,则在Mm中不存在稳定积分流,且Mm的同调群消没。
By using an inequality relation between a scalar curvature and the length of the second fundamental form,it is proved that sectional curvatures of a submanifold must be nonnegative (or positive).
利用数量曲率与第二基本形式长度之间的一个不等式关系,证明了其子流形的截[面]曲率一定非负(或者为正),并将此应用到紧致子流形上,得到一些结果。