Isoparametric foliations and bounded geometry

TL;DR

This paper proves finiteness results for isoparametric foliations on certain Riemannian manifolds with bounded geometry and constructs infinite families of non-diffeomorphic such foliations on spheres.

Contribution

It establishes finiteness of isoparametric foliations under specific geometric and topological constraints and provides constructions of infinite non-diffeomorphic examples.

Findings

Finitely many isoparametric foliations on certain manifolds with bounded geometry.

Construction of infinite non-diffeomorphic isoparametric foliations on spheres.

Results hold for manifolds with fixed dimension (excluding 5) and finite fundamental group.

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.