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Isoparametric foliations and bounded geometry

Manuel Krannich, Alexander Lytchak, Marco Radeschi

TL;DR

The paper addresses finiteness of isoparametric foliations on closed manifolds under bounded geometry and finite fundamental group, excluding dimension $n=5$ for diffeomorphism types. It recasts isoparametric foliations as manifold submetries to intervals (equivalently as double disc bundle decompositions) and proves a global finiteness theorem via compactness arguments and convergence of both metrics and gluing maps. Key contributions include establishing the finite foliated-diffeomorphism types under the stated bounds, and exhibiting infinite families of inequivalent foliations when the geometry-bounded hypotheses are dropped. The findings illuminate how curvature, diameter, volume, and fundamental group interact with foliation structure, and provide a framework potentially extendable to related transverse geometric decompositions. The work also highlights sharpness through explicit counterexamples and connects to broader questions about finite-type foliations on manifolds.

Abstract

We prove that there are only finitely many isoparametrically foliated closed connected Riemannian manifolds with bounded geometry, fixed dimension $n\neq5$, and finite fundamental group, up to foliated diffeomorphism. In addition, we construct various infinite families of isoparametric foliations that are mutually not foliated diffeomorphic, for instance on a fixed sphere.

Isoparametric foliations and bounded geometry

TL;DR

The paper addresses finiteness of isoparametric foliations on closed manifolds under bounded geometry and finite fundamental group, excluding dimension for diffeomorphism types. It recasts isoparametric foliations as manifold submetries to intervals (equivalently as double disc bundle decompositions) and proves a global finiteness theorem via compactness arguments and convergence of both metrics and gluing maps. Key contributions include establishing the finite foliated-diffeomorphism types under the stated bounds, and exhibiting infinite families of inequivalent foliations when the geometry-bounded hypotheses are dropped. The findings illuminate how curvature, diameter, volume, and fundamental group interact with foliation structure, and provide a framework potentially extendable to related transverse geometric decompositions. The work also highlights sharpness through explicit counterexamples and connects to broader questions about finite-type foliations on manifolds.

Abstract

We prove that there are only finitely many isoparametrically foliated closed connected Riemannian manifolds with bounded geometry, fixed dimension , and finite fundamental group, up to foliated diffeomorphism. In addition, we construct various infinite families of isoparametric foliations that are mutually not foliated diffeomorphic, for instance on a fixed sphere.

Paper Structure

This paper contains 9 sections, 18 theorems, 11 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Transnormal functions, manifold submetries, and double disc bundles
  4. Connectedness of fibers and existence of singular fibers
  5. Foliated diffeomorphism, concordance, and isotopy
  6. Proof of the main result
  7. Proof of \ref{['T:true-main']}
  8. Examples of infinite families of foliations
  9. Topological background

Key Result

Theorem 1.1

For a fixed dimension $n\neq 5$ and constants $\kappa, \nu, r>0$, the class of isoparametrically foliated closed connected Riemannian manifolds $(M, g, \mathcal{F})$ with contains finitely many foliated diffeomorphism types. In dimension $n=5$, it contains only finitely many foliated homeomorphism types.

Theorems & Definitions (42)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Remark 1.7
  • Conjecture 1.8
  • Remark 1.9
  • Lemma 2.1
  • ...and 32 more