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The Möbius Geometry of Wintgen Ideal Submanifolds

  • Xiang Ma
  • Zhenxiao Xie
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 106)

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. They are Möbius invariant objects. The mean curvature sphere defines a conformal Gauss map into a Grassmann manifold. We show that any Wintgen ideal submanifold of dimension greater than or equal to 3 has a Riemannian submersion structure over a Riemann surface with the fibers being round spheres. Then the conformal Gauss map is shown to be a super-conformal and harmonic map from the underlying Riemann surface. Some of our previous results are surveyed in the final part.

Keywords

Riemann Surface Fundamental Form Curvature Vector Round Sphere Curvature Sphere 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Japan 2014

Authors and Affiliations

  1. 1.School of Mathematical SciencesPeking UniversityBeijingPeople’s Republic of China
  2. 2.Department of MathematicsChina University of Mining and Technology (Beijing)BeijingPeople’s Republic of China

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