Annals of Global Analysis and Geometry

, Volume 53, Issue 3, pp 377–403 | Cite as

Wintgen ideal submanifolds: reduction theorems and a coarse classification

  • Zhenxiao Xie
  • Tongzhu Li
  • Xiang Ma
  • Changping Wang
Article
  • 49 Downloads

Abstract

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.

Keywords

Wintgen ideal submanifolds DDVV inequality Möbius geometry Conformal Gauss map Minimal submanifolds 

Mathematics Subject Classification

53A30 53A55 53C42 

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Copyright information

© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  1. 1.Department of MathematicsChina University of Mining and Technology (Beijing)BeijingChina
  2. 2.Department of MathematicsBeijing Institute of TechnologyBeijingChina
  3. 3.LMAM, School of Mathematical SciencesPeking UniversityBeijingChina
  4. 4.School of Mathematics and Computer Science and FJKLMAAFujian Normal UniversityFuzhouChina

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