Isoparametric submanifolds in Hilbert spaces and holonomy maps

Abstract

Let $\mathrm{\pi }:P\to B$ 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 ${\mathrm{hol}}_c:{\mathcal{A}}_P^{H^s}\to G$ is a homothetic submersion of coefficient a, where s is a nonnegative integer and ${\mathcal{A}}_P^{H^s}$ is the Hilbert space of all $H^s$-connections of the bundle P. In particular, we prove that if $s=0$, then ${\mathrm{hol}}_c$ has minimal regularizable fibers. From this fact, we can derive that each component of the inverse image of any equifocal submanifold in G by the holonomy map ${\mathrm{hol}}_c:{\mathcal{A}}_P^{H^0}\to G$ 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.