Wintgen ideal submanifolds with a low-dimensional integrable distribution  

作　　者：Tongzhu LI Xiang MA Changping WANG 

机构地区：[1]Department of Mathematics, Beijing Institute of Technology, Beijing 100081, China [2]LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, China, [3]College of Mathematics and Computer Science, Fujian Normal University, Fuzhou 350108, China

出　　处：《Frontiers of Mathematics in China》2015年第1期111-136,共26页中国高等学校学术文摘·数学（英文）

基　　金：The authors thank Dr. Zhenxiao Xie for helpful discussion which deepen the understanding of the main results, and they are grateful to the referees for their critical viewpoints and suggestions, which improve the exposition and correct many errors. This work was supported by the National Natural Science Foundation of China （Grant No. 11171004）; Xiang Ma was partially supported by the National Natural Science Foundation of China （Grant No. 10901006）; and Changping Wang was partially supported by the National Natural Science Foundation of China （Grant No. 11331002）.

摘　　要：Submanifolds in space forms satisfy the well-known DDVV inequality. A submanifold attaining equality in this inequality pointwise is called a Wintgen ideal submanifold. As conformal invariant objects, Wintgen ideal submanifolds are investigated in this paper using the framework of MSbius geometry. We classify Wintgen ideal submanfiolds of dimension rn ≥ 3 and arbitrary codimension when a canonically defined 2-dimensional distribution D2 is integrable. Such examples come from cones, cylinders, or rotational submanifolds over super-minimal surfaces in spheres, Euclidean spaces, or hyperbolic spaces, respectively. We conjecture that if D2 generates a k-dimensional integrable distribution Dk and k 〈 m, then similar reduction theorem holds true. This generalization when k = 3 has been proved in this paper.Submanifolds in space forms satisfy the well-known DDVV inequality. A submanifold attaining equality in this inequality pointwise is called a Wintgen ideal submanifold. As conformal invariant objects, Wintgen ideal submanifolds are investigated in this paper using the framework of MSbius geometry. We classify Wintgen ideal submanfiolds of dimension rn ≥ 3 and arbitrary codimension when a canonically defined 2-dimensional distribution D2 is integrable. Such examples come from cones, cylinders, or rotational submanifolds over super-minimal surfaces in spheres, Euclidean spaces, or hyperbolic spaces, respectively. We conjecture that if D2 generates a k-dimensional integrable distribution Dk and k 〈 m, then similar reduction theorem holds true. This generalization when k = 3 has been proved in this paper.

关 键 词：Wintgen ideal submanifold DDVV inequality super-conformalsurface super-minimal surface 

分 类 号：O186.12[理学—数学] O153.3[理学—基础数学]