Isoparametric submanifolds in a Riemannian Hilbert manifold

Abstract

In this paper, we introduce the notion of a regularizable submanifold in a Riemannian Hilbert manifold. This submanifold is defined as a curvature-invariant submanifold such that its shape operators and its normal Jacobi operators are regularizable, where ``the operators are regularizable'' means that the operators are compact and that their regularized traces and the usual traces of their squares exist. Furthermore, we introduce the notion of an isoparametric submanifold in a Riemannian Hilbert manifold. This submanifold is defined as a regularizable submanifold with flat section and trivial normal holonomy group satisfying the constancy of the regularized mean curvatures in the radial direction of the parallel submanifolds. For a curvature-adapted regularizable submanifold $M$ with trivial normal holonomy group in a locally symmetric Riemannian Hilbert manifold, we prove that if, for any parallel normal vecrtor field $\widetilde{\xi}$ of $M$, the shape operaors $A_{\widetilde{\xi}_x}$ and the normal Jacobi operator $\widetilde R(\widetilde{\xi}_x)$ are independent of the base point $x(\in M)$ (up to orthogonal equivalent), then it is isoparametric under some additional conditions. Also, we define the notion of an equifocal submanifold in a Riemannian Hilbert manifold. We prove that the principal orbits of a certain kind of Hilbert Lie group action on the Riemannian Hilbert manifold $\mathcal A_P^{H^s}$ consisting of all $H^s$-connections of a $G$-bundle $P$ over a compact Riemannian manifold $B$ are equifocal, where $G$ is a semi-simple Lie group and $s>\frac{1}{2}\,{\rm dim}\,B-1$.