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The Equivalence of the Existences of Transnormal and Isoparametric Functions on Compact Manifolds

Minghao Li, Ling Yang

TL;DR

The paper proves that on connected complete Riemannian manifolds, the existence of a transnormal function is equivalent to the manifold supporting an embedded transnormal system of codimension 1, and that on compact manifolds these transnormal cases coincide with isoparametric ones under suitable metrics. It classifies the global structure of transnormal systems via seven types distinguished by foils and the normal sphere bundle, showing that manifolds are, up to diffeomorphism, normal disk bundle unions or vector bundles, and that one can construct metrics and functions realizing these foliations as transnormal or isoparametric. The work provides two constructive directions: from LDDBDs or vector bundles to transnormal/isoparametric functions, and from transnormal systems to corresponding transnormal functions, with additional results guaranteeing isoparametricity under compactness and, in rank-one cases, harmonic leaves. These findings unify the topological constraints of transnormal and isoparametric functions and extend prior results by incorporating broader geometric decompositions and metric adjustments. In particular, the results generalize known theorems and clarify when a given transnormal function can or cannot be promoted to an isoparametric one via metric changes.

Abstract

Through exploring the embedded transnormal systems of codimension 1, we show the existence of a transnormal function on a connected complete Riemannian manifold requires the underlying manifold to have a vector bundle structure or a linear double disk bundle decomposition. Conversely, any smooth manifold with either of these structures can be endowed with a Riemannian metric so that it admits a transnormal function, which, under suitable compactness conditions, can become isoparametric. As a corollary, for compact manifolds, the existences of transnormal and isoparametric functions impose the same topological constraints.

The Equivalence of the Existences of Transnormal and Isoparametric Functions on Compact Manifolds

TL;DR

The paper proves that on connected complete Riemannian manifolds, the existence of a transnormal function is equivalent to the manifold supporting an embedded transnormal system of codimension 1, and that on compact manifolds these transnormal cases coincide with isoparametric ones under suitable metrics. It classifies the global structure of transnormal systems via seven types distinguished by foils and the normal sphere bundle, showing that manifolds are, up to diffeomorphism, normal disk bundle unions or vector bundles, and that one can construct metrics and functions realizing these foliations as transnormal or isoparametric. The work provides two constructive directions: from LDDBDs or vector bundles to transnormal/isoparametric functions, and from transnormal systems to corresponding transnormal functions, with additional results guaranteeing isoparametricity under compactness and, in rank-one cases, harmonic leaves. These findings unify the topological constraints of transnormal and isoparametric functions and extend prior results by incorporating broader geometric decompositions and metric adjustments. In particular, the results generalize known theorems and clarify when a given transnormal function can or cannot be promoted to an isoparametric one via metric changes.

Abstract

Through exploring the embedded transnormal systems of codimension 1, we show the existence of a transnormal function on a connected complete Riemannian manifold requires the underlying manifold to have a vector bundle structure or a linear double disk bundle decomposition. Conversely, any smooth manifold with either of these structures can be endowed with a Riemannian metric so that it admits a transnormal function, which, under suitable compactness conditions, can become isoparametric. As a corollary, for compact manifolds, the existences of transnormal and isoparametric functions impose the same topological constraints.

Paper Structure

This paper contains 3 sections, 10 theorems, 58 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Global structure of transnormal systems
  3. The construction of transnormal and isoparametric functions

Key Result

Theorem 1

Let $\mathcal{F}$ be an embedded transnormal system of codimension 1 on a complete connected Riemannian manifold $M$, $N_{\text{SR}}$ and $N_{\text{S}}$ be the numbers of SR-foils and S-foils in $\mathcal{F}$, respectively, $N_\text{C}:=N_{\text{SR}}+N_{\text{S}}$ be the number of foils whose normal be the diameter of $\mathcal{F}$. Then $N_C$ is at most two. More precisely:

Figures (1)

  • Figure 1: Illustrations of the seven types.

Theorems & Definitions (19)

  • Definition 1.1
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Corollary 4
  • Corollary 5
  • Lemma 2.1: bolton1973transnormal
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • ...and 9 more