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Wintgen ideal submanifolds with a low-dimensional integrable distribution |
Tongzhu LI1,*( ),Xiang MA2,Changping WANG3 |
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 |
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Abstract 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 M?bius geometry. We classify Wintgen ideal submanfiolds of dimension m≥3 and arbitrary codimension when a canonically defined 2-dimensional distribution ?2 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 ?2 generates a k-dimensional integrable distribution ?k<?Pub Caret?>and k<m, then similar reduction theorem holds true. This generalization when k = 3 has been proved in this paper.
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Keywords
Wintgen ideal submanifold
DDVV inequality
super-conformal surface
super-minimal surface
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Corresponding Author(s):
Tongzhu LI
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Issue Date: 30 December 2014
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