Wintgen ideal submanifolds: reduction theorems and a coarse classification
type: Item
instance of: scholarly article
title: Wintgen ideal submanifolds: reduction theorems and a coarse classification
zbMATH Open document ID: 1397.53076
author: Zhenxiao Xie
author: Tongzhu Li
author: Xiang Ma
author: Changping Wang
published in: Annals of Global Analysis and Geometry
publication date: 2018-05-16T00:00:00Z
review text: Wintgen ideal submanifolds in space forms are those ones attaining equality pointwise in the so-called DDVV inequality \(s\leq c+|H|^2 -s_N\), see [Zbl 1054.53075; Zbl 0515.53044; Zbl 0421.53003], which relates the scalar curvature, the mean curvature and the scalar normal curvature. Those submanifolds attaining the equality at every point are called \textit{Wintgen ideal submanifolds}. As conformal invariant objects, they are suitable to study in the framework of Möbius geometry. This paper continues authors's previous work [Zbl 1335.53015] 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 the authors 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.
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 53A30
Mathematics Subject Classification ID: 53A55
zbMATH DE Number: 6869153
zbMATH Keywords: Wintgen ideal submanifolds
zbMATH Keywords: DDVV inequality
zbMATH Keywords: Möbius geometry
zbMATH Keywords: conformal Gauss map
zbMATH Keywords: minimal submanifolds
reviewed by: Vladimir Yu. Rovenskij
MaRDI profile type: Publication
full work available at URL: s10455-017-9581-1
OpenAlex ID: W2769105614
cites work: Conformal geometry of surfaces in \(S^4\) and quaternions
cites work: Classification of Wintgen ideal surfaces in Euclidean 4-space with equal Gauss and normal curvatures
cites work: On the DDVV conjecture and the comass in calibrated geometry. I
cites work: A class of austere submanifolds
cites work: All superconformal surfaces in \({\mathbb R^4}\) in terms of minimal surfaces
cites work: Submanifolds of codimension two attaining equality in an extrinsic inequality
cites work: Q4814455
cites work: Q5446805
cites work: On the completeness of nullity foliations
cites work: A proof of the DDVV conjecture and its equality case
cites work: Normal curvature of surfaces in space forms
cites work: Wintgen ideal submanifolds: new examples, frame sequence and Möbius homogeneous classification
cites work: Wintgen ideal submanifolds of codimension two, complex curves, and Möbius geometry
cites work: Deformation of hypersurfaces preserving the Möbius metric and a reduction theorem
cites work: Wintgen ideal submanifolds with a low-dimensional integrable distribution
cites work: Möbius isotropic submanifolds in \(\mathbb{S}^n\)
cites work: Normal scalar curvature conjecture and its applications
cites work: The Möbius Geometry of Wintgen Ideal Submanifolds
cites work: Isometric immersions into spaces of constant curvature
cites work: Q3057322
cites work: Ideal Lagrangian submanifolds
cites work: Moebius geometry of submanifolds in \(\mathbb{S}^n\)
cites work: Q3854271
cites work: Wintgen ideal submanifolds with vanishing Möbius form
cites work: Möbius geometry of three-dimensional Wintgen ideal submanifolds in \(\mathbb S^5\)
DOI: 10.1007/S10455-017-9581-1
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description: scientific article; zbMATH DE number 6869153