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Local Wintgen ideal submanifolds

Marcos Dajczer, Theodoros Vlachos

TL;DR

This work addresses the local parametric classification of Wintgen ideal submanifolds, i.e., submanifolds attaining equality in the DDVV inequality pointwise, in space forms. It leverages the pointwise structure of the second fundamental form to separate nongeneric and generic cases, providing complete parametrizations for the nongeneric cases and proving that generic submanifolds locally factor as a composition $f=\Psi\circ j$, with $\Psi$ a high-multiplicity hypersurface obtained via conformal Gauss parametrization and $j$ a minimal immersion with relative nullity $\nu=n-2$. The results unify and extend known parametrizations (including elliptic, Dupin, and minimal submanifolds) and show how the composition framework reduces the local classification to tractable two-step data, applicable across conformal space forms. Overall, the paper delivers a concrete, geometry-driven local classification that connects intrinsic curvature, mean curvature, and normal bundle data through Dupin principal normals and polar/conformal constructions, enabling explicit local models and constructions of Wintgen ideal submanifolds.

Abstract

This paper is dedicated to the local parametric classification of Wintgen ideal submanifolds in space forms. These submanifolds are characterized by the pointwise attainment of equality in the DDVV inequality, which relates the scalar curvature, the length of the mean curvature vector field and the normal curvature tensor.

Local Wintgen ideal submanifolds

TL;DR

This work addresses the local parametric classification of Wintgen ideal submanifolds, i.e., submanifolds attaining equality in the DDVV inequality pointwise, in space forms. It leverages the pointwise structure of the second fundamental form to separate nongeneric and generic cases, providing complete parametrizations for the nongeneric cases and proving that generic submanifolds locally factor as a composition , with a high-multiplicity hypersurface obtained via conformal Gauss parametrization and a minimal immersion with relative nullity . The results unify and extend known parametrizations (including elliptic, Dupin, and minimal submanifolds) and show how the composition framework reduces the local classification to tractable two-step data, applicable across conformal space forms. Overall, the paper delivers a concrete, geometry-driven local classification that connects intrinsic curvature, mean curvature, and normal bundle data through Dupin principal normals and polar/conformal constructions, enabling explicit local models and constructions of Wintgen ideal submanifolds.

Abstract

This paper is dedicated to the local parametric classification of Wintgen ideal submanifolds in space forms. These submanifolds are characterized by the pointwise attainment of equality in the DDVV inequality, which relates the scalar curvature, the length of the mean curvature vector field and the normal curvature tensor.
Paper Structure (6 sections, 7 theorems, 77 equations)

This paper contains 6 sections, 7 theorems, 77 equations.

Table of Contents

  1. The pointwise structure
  2. The nongeneric cases
  3. The three cases
  4. The elliptic submanifolds
  5. The case of minimal submanifolds
  6. The generic case

Key Result

Proposition 1

An isometric immersion $f\colon M^n\to\mathbb{Q}^{n+m}_c,n\geq 3$ and $m\geq 2$, is a Wintgen ideal submanifold if and only if at any point of $M^n$ there is an orthonormal tangent base $\{e_i\}_{1\leq i\leq n}$ and an orthonormal normal base $\{\eta_a\}_{1\leq a\leq m}$ such that the shape operator

Theorems & Definitions (9)

  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Lemma 4
  • Lemma 5
  • Remark 6
  • Lemma 7
  • Theorem 8
  • Remark 9