Isoparametric submanifolds in Hilbert spaces and holonomy maps

Naoyuki Koike

Published 2022-06-25Version 1

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, 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 with minimal regularizable fibres, where $\mathcal A_P^{H^s}$ is the Hilbert space of all $H^s$-connections of the bundle $P$ and $s$ is a positive number with $\displaystyle{s>\frac{1}{2}\,{\rm dim}\,B\,-\,1}$. From this fact, we can derive that each component of the inverse image of any equifocal submanifold in $G$ by the holonomy map ${\rm hol}_c:\mathcal A_P^{H^s}\to G$ is an isoparametric submanifold in the Hilbert space $\mathcal A_P^{H^s}$. As the result, we obtain a new systematic construction of isoparametric submanifolds in a Hilbert space.