Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the Classification of Isoparametric Hypersurfaces with Constant Principal Curvatures in Compact 3-Manifolds

Minghao Li, Ling Yang

TL;DR

This work classifies transnormal systems on orientable compact 3-manifolds up to equivalence, with a focus on CPC transnormal systems whose foils have constant principal curvatures. It develops a covering-theoretic framework (essential covers and lifts) and uses mapping class group data to classify toric and Klein-bottled types, while detailing spherical and real-projective types through explicit geometric models such as lens spaces, twisted S^2-bundles, and RP^3. The main result is that, up to CPC equivalence, the ambient geometry is restricted to six Thurston geometries, and several equivalence classes contain no CPC representative, illustrating a sharp distinction between CPC transnormal systems and isoparametric foliations in dimension three. Moreover, the paper shows how lifting along universal covers can alter equivalence classes, underscoring subtle differences from cohomogeneity-one actions and highlighting the nuanced landscape of transnormal systems on 3-manifolds.

Abstract

Establishing detailed relationships between transnormal systems of different types and their behaviors under covering maps, this paper presents a classification of transnormal systems on compact 3-manifolds in the sense of equivalence. For CPC transnormal systems, we show that the ambient manifolds must be locally isometric to one of six standard geometries up to equivalence. We also find some equivalence classes containing no CPC transnormal system, highlighting a critical distinction between isoparametric foliations and CPC transnormal systems, which has not been previously addressed in the literature.

On the Classification of Isoparametric Hypersurfaces with Constant Principal Curvatures in Compact 3-Manifolds

TL;DR

This work classifies transnormal systems on orientable compact 3-manifolds up to equivalence, with a focus on CPC transnormal systems whose foils have constant principal curvatures. It develops a covering-theoretic framework (essential covers and lifts) and uses mapping class group data to classify toric and Klein-bottled types, while detailing spherical and real-projective types through explicit geometric models such as lens spaces, twisted S^2-bundles, and RP^3. The main result is that, up to CPC equivalence, the ambient geometry is restricted to six Thurston geometries, and several equivalence classes contain no CPC representative, illustrating a sharp distinction between CPC transnormal systems and isoparametric foliations in dimension three. Moreover, the paper shows how lifting along universal covers can alter equivalence classes, underscoring subtle differences from cohomogeneity-one actions and highlighting the nuanced landscape of transnormal systems on 3-manifolds.

Abstract

Establishing detailed relationships between transnormal systems of different types and their behaviors under covering maps, this paper presents a classification of transnormal systems on compact 3-manifolds in the sense of equivalence. For CPC transnormal systems, we show that the ambient manifolds must be locally isometric to one of six standard geometries up to equivalence. We also find some equivalence classes containing no CPC transnormal system, highlighting a critical distinction between isoparametric foliations and CPC transnormal systems, which has not been previously addressed in the literature.

Paper Structure

This paper contains 18 sections, 27 theorems, 66 equations, 1 figure, 6 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Covering and lifting on transnormal systems
  4. Basic definitions and properties
  5. Relationships among transnormal systems of various types
  6. Lifting of transnormal systems along covering maps
  7. Transnormal systems on three-dimensional manifolds
  8. Transnormal systems of toric type
  9. Transnormal systems of Klein-bottled type
  10. Transnormal systems of spherical type
  11. Case 1: $L_1$ and $L_2$ are both single points.
  12. Case 2: $L_1$ and $L_2$ are both circles, while $\mathcal{D}_1$ and $\mathcal{D}_2$ are both solid tori.
  13. Case 3: $L_1$ and $L_2$ are both circles, while $\mathcal{D}_1$ and $\mathcal{D}_2$ are both solid Klein bottles.
  14. Transnormal systems of real-projective type
  15. Case 1: $L$ is a single point.
  16. ...and 3 more sections

Key Result

Theorem 1.1

Suppose $\mathcal{F}$ is an embedded transnormal system of codimension one on a connected complete manifold $M$, then $M$ admits a transnormal function $f$ such that the transnormal system $\mathcal{F}_f$ induced by $f$ coincides with $\mathcal{F}$.

Figures (1)

  • Figure 1: $\mathcal{F}$ of toric type and its covering.

Theorems & Definitions (61)

  • Theorem 1.1: miyaoka2013transnormalli2024
  • Theorem 1.2: li2024
  • Definition 1.3
  • Definition 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Definition 2.1
  • Lemma 2.2: li2024
  • Lemma 2.3: li2024
  • Definition 3.1
  • ...and 51 more