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4-Dimensional Isoparametric Hypersurfaces of Index 2 in the Pseudo-Riemannian Space Forms

Yuta Sasahara

Abstract

We study isoparametric hypersurfaces, whose principal curvatures are all constant, in the pseudo-Riemannian space forms. In this paper, we investigate three topics.Firstly, according to Petrov's classification theorem, we give a classification of hypersurfaces of index 2 with respect to a pair of a shape operator and a metric. Therefore, we can define types of isoparametric hypersurfaces of index 2 concerning the classification. Secondly, we give several examples of certain types. Thirdly, we show that there exist no isoparametric hypersurfaces of index 2 whose shape operators have complex principal curvatures in certain cases.

4-Dimensional Isoparametric Hypersurfaces of Index 2 in the Pseudo-Riemannian Space Forms

Abstract

We study isoparametric hypersurfaces, whose principal curvatures are all constant, in the pseudo-Riemannian space forms. In this paper, we investigate three topics.Firstly, according to Petrov's classification theorem, we give a classification of hypersurfaces of index 2 with respect to a pair of a shape operator and a metric. Therefore, we can define types of isoparametric hypersurfaces of index 2 concerning the classification. Secondly, we give several examples of certain types. Thirdly, we show that there exist no isoparametric hypersurfaces of index 2 whose shape operators have complex principal curvatures in certain cases.
Paper Structure (15 sections, 31 theorems, 69 equations, 3 tables)

This paper contains 15 sections, 31 theorems, 69 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. A Shape Operator
  4. Petrov's Classification Theorem
  5. A Classification of Isoparametric Hypersurfaces of Index 1
  6. A Classification of Isoparametric Hypersurfaces of Index 2
  7. 4-Dimensional Isoparametric Hypersurfaces of Index 2 in the Pseudo-Riemannian Space Forms
  8. Hahn's Examples
  9. Other Examples
  10. Non-existence
  11. Type I
  12. Type II
  13. Type III
  14. Type IV
  15. Non-existence of Index 1

Key Result

Proposition 3.1

There exist the family $\{p_{i, k, u}\}$ of vectors of $V$ and the family $\{q_{j, l, v}\}$ of vectors of $V^{\mathbb{C}}$ such that is an ordered basis of $V^{\mathbb{C}}$ and then the matrix representation of $A^{\mathbb{C}}$ with respect to $\mathcal{B}$ is where $s_{i}$ is a positive integer for all $i\in \{1,\ldots, a\}$, and where the sequence of positive integers $\{m_{i, k}\}_{k=1}^{s_{i

Theorems & Definitions (62)

  • Proposition 3.1: The Jordan Normal Form Theorem
  • Theorem 3.2: Petrov's Classification Theorem
  • proof : (Outline of the Proof of Theorem \ref{['theo:Petrov']})
  • Lemma 3.3
  • proof
  • Example 3.4
  • Proposition 3.5
  • proof
  • Example 3.6
  • Example 4.1: MR753432
  • ...and 52 more