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Austere Matrices, Austere Submanifolds and Dupin Hypersurfaces

Jianquan Ge, Yi Zhou

Abstract

Motivated by Bryant's research on austere subspaces and Cartan's isoparametric hypersurfaces with 3 distinct principal curvatures, we construct three families of austere submanifolds with flat normal bundle in unit spheres. From these examples we find three irreducible proper Dupin hypersurfaces with 5 distinct principal curvatures of different multiplicities. Thus, we give a negative answer to an open question raised by Thorbergsson in 2000 which is instructive for the local classification of proper Dupin hypersurfaces. Moreover, as an application, we obtain an upper bound estimate for the dimension of austere subspaces.

Austere Matrices, Austere Submanifolds and Dupin Hypersurfaces

Abstract

Motivated by Bryant's research on austere subspaces and Cartan's isoparametric hypersurfaces with 3 distinct principal curvatures, we construct three families of austere submanifolds with flat normal bundle in unit spheres. From these examples we find three irreducible proper Dupin hypersurfaces with 5 distinct principal curvatures of different multiplicities. Thus, we give a negative answer to an open question raised by Thorbergsson in 2000 which is instructive for the local classification of proper Dupin hypersurfaces. Moreover, as an application, we obtain an upper bound estimate for the dimension of austere subspaces.
Paper Structure (10 sections, 19 theorems, 143 equations, 2 tables)

This paper contains 10 sections, 19 theorems, 143 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic properties of $\mathcal{B}_{n, \mathbb{F}}$, $n\geq 3$
  4. New examples of irreducible proper Dupin hypersyrfaces
  5. $\mathcal{B}_{4, \mathbb{F}}$ and $\mathcal{C}_{4, \mathbb{F}}$
  6. New examples of proper Dupin hypersurfaces
  7. Irreducibility of the proper Dupin hypersurface $\widetilde{\mathcal{B}}_{4, \mathbb{F}}$
  8. Weak irreducibility
  9. Compact austere submanifolds $\mathcal{C}_{4, \mathbb{F}}$ in $S(4, \mathbb{F})$
  10. Dimension upper bound of austere subspaces

Key Result

Proposition 3.1

$A\in\mathcal{B}_{n, \mathbb{F}}$ if and only if $\lambda_1, \cdots, \lambda_{p+1}$ are distinct.

Theorems & Definitions (39)

  • Proposition 3.1
  • proof
  • Example 3.2
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • Theorem 3.5
  • proof
  • Remark 3.6
  • Remark 3.7
  • ...and 29 more