Wintgen ideal submanifolds: reduction theorems and a coarse classification
Wintgen ideal submanifolds: reduction theorems and a coarse classification
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Wintgen 理想子流形：约简定理和粗分类
DOI：
10.1007/s10455-017-9581-1
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发表时间：
2018
影响因子：
0.7
通讯作者：
Wang Changping
中科院分区：
文献类型：
--
作者：
Xie Zhenxiao;Li Tongzhu;Ma Xiang;Wang Changping
空间形式中的Wintgen理想子流形是那些在所谓的DDVV不等式中逐点相等的子流形，DDVV不等式将数量曲率、平均曲率和数量法曲率联系起来。作为共形不变对象，它们适合在莫比乌斯几何的框架下进行研究。本文继续我们以前的工作，在这个程序中，表明Wintgen理想子流形可以分为三类：可约的，不可约的最小的空间形式（莫比乌斯变换），和一般（不可约）的。可约的Wintgen理想子流形有一个特定的低维可积分布，这使我们能够得到最一般的约化定理，即它们是由低维空间形式中的极小Wintgen理想子流形生成的圆锥、圆柱或旋转曲面的莫比乌斯等价。
Wintgen ideal submanifolds in space forms are those ones attaining equality pointwise in the so-called DDVV inequality which relates the scalar curvature, the mean curvature and the scalar normal curvature. As conformal invariant objects, they are suitable to study in the framework of Möbius geometry. This paper continues our previous work in this program, showing that Wintgen ideal submanifolds can be divided into three classes: the reducible ones, the irreducible minimal ones in space forms (up to Möbius transformations), and the generic (irreducible) ones. The reducible Wintgen ideal submanifolds have a specific low-dimensional integrable distribution, which allows us to get the most general reduction theorem, saying that they are Möbius equivalent to cones, cylinders, or rotational surfaces generated by minimal Wintgen ideal submanifolds in lower-dimensional space forms.