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 su...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.展开更多
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）.
文摘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.
