Isoparametric submanifolds in Hilbert spaces and holonomy maps

TL;DR

This paper establishes that holonomy maps in certain principal bundles act as homothetic submersions with minimal fibers, leading to a new method for constructing isoparametric submanifolds in Hilbert spaces.

Contribution

It proves holonomy maps are homothetic submersions with minimal fibers and introduces a systematic way to construct isoparametric submanifolds in Hilbert spaces.

Findings

Holonomy map is a homothetic submersion with coefficient a.

Fibers of the holonomy map are minimal when s=0.

Inverse images of equifocal submanifolds are isoparametric in Hilbert space.

Abstract

Let $\pi:P\to B$ be a smooth $G$-bundle over a compact Riemannian manifold $B$ and $c$ a smooth loop in $B$ of constant seed $a(>0)$, where $G$ is compact semi-simple Lie group. In this paper, we prove that the holonomy map ${\rm hol}_c:\mathcal A_P^{H^s}\to G$ is a homothetic submersion of coefficient $a$, where $s$ is a non-negative integer, $\mathcal A_P^{H^s}$ is the Hilbert space of all $H^s$-connections of the bundle $P$. In particular, we prove that, if $s=0$, then ${\rm hol}_c$ has minimal regularizable fibres. From this fact, we can derive that each component of the inverse image of any equifocal submanifold in $G$ by the holonomy map ${\rm hol}_c:\mathcal A_P^{H^0}\to G$ is an isoparametric submanifold in $\mathcal A_P^{H^0}$. As the result, we obtain a new systematic construction of isoparametric submanifolds in a Hilbert space.