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Some Remarks on Isoparametric Functions in Closed 4-Manifolds

Minghao Li

TL;DR

The paper investigates transnormal and isoparametric functions on closed Riemannian 4-manifolds, linking foliation geometry to global topology via double disk bundle decompositions. It develops a foundational algebraic-Topological framework to express $\pi_1(M)$ in terms of base data and gluing maps, then performs a detailed case analysis by the dimensions of the base manifolds to deduce strong curvature and fundamental-group constraints. The main contributions show that closed 4-manifolds admitting such foliations cannot carry negatively curved metrics and, in various configurations, must have constrained fundamental groups or be structured as sphere bundles or mapping tori with restricted monodromy. These results illuminate how isoparametric foliations impose nontrivial global geometric restrictions and extend the understanding of the interplay between foliation theory and global Riemannian geometry, while noting that the corresponding conclusions fail in dimension three due to distinct topological phenomena.

Abstract

We study transnormal and isoparametric functions on closed Riemannian 4-manifolds and establish fundamental restrictions on their topology and geometry. In particular, we show that such manifolds cannot be endowed with negatively curved metrics, contrasting with known results in the compact simply connected case. Moreover, in certain cases, we provide a description of their fundamental groups. These findings contribute to a better understanding of the global structure of isoparametric foliations.

Some Remarks on Isoparametric Functions in Closed 4-Manifolds

TL;DR

The paper investigates transnormal and isoparametric functions on closed Riemannian 4-manifolds, linking foliation geometry to global topology via double disk bundle decompositions. It develops a foundational algebraic-Topological framework to express in terms of base data and gluing maps, then performs a detailed case analysis by the dimensions of the base manifolds to deduce strong curvature and fundamental-group constraints. The main contributions show that closed 4-manifolds admitting such foliations cannot carry negatively curved metrics and, in various configurations, must have constrained fundamental groups or be structured as sphere bundles or mapping tori with restricted monodromy. These results illuminate how isoparametric foliations impose nontrivial global geometric restrictions and extend the understanding of the interplay between foliation theory and global Riemannian geometry, while noting that the corresponding conclusions fail in dimension three due to distinct topological phenomena.

Abstract

We study transnormal and isoparametric functions on closed Riemannian 4-manifolds and establish fundamental restrictions on their topology and geometry. In particular, we show that such manifolds cannot be endowed with negatively curved metrics, contrasting with known results in the compact simply connected case. Moreover, in certain cases, we provide a description of their fundamental groups. These findings contribute to a better understanding of the global structure of isoparametric foliations.
Paper Structure (12 sections, 10 theorems, 13 equations)

This paper contains 12 sections, 10 theorems, 13 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Homotopy groups.
  4. Double disk bundle decomposition
  5. Proofs of the Main Theorems
  6. Auxiliary results on the fundamental group
  7. Discussion of double disk bundle decompositions
  8. Case 1: $d_{\pm}\le 2$.
  9. Case 2: $d_{+}\le 2$ and $d_{-}=3$.
  10. Case 3: $d_{+}=d_{-}=3$, and both disk bundles are trivial.
  11. Case 4: $d_{+}=d_{-}=3$, and both disk bundles are twisted.
  12. Failure of Theorem \ref{['thm: exclu of hyperbolic']} in Dimension Three

Key Result

Theorem A

For each closed smooth manifold $M$, the following statements are equivalent:

Theorems & Definitions (14)

  • Theorem A: li2025equiv
  • Theorem B: ge2015differentiable
  • Theorem C
  • Theorem D
  • Definition 2.1
  • Lemma 2.2: Seifert-van Kampen Theorem
  • Lemma 2.3: do1992-riemannian
  • Lemma 2.4: Preissmann's theorem
  • Definition 2.5
  • Lemma 2.6
  • ...and 4 more