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Isoparametric submanifolds in a Riemannian Hilbert manifold

Naoyuki Koike
Nov 2023
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. Furthemore, we introduce the notions of an isoparametric submanifold and weakly isoparametric submanifold in a Riemannian Hilbert manifold. These submanifolds are defined as a regularizable submanifold with flat section and trivial normal holonomy group satisfying certain conditions. 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 $\widetilde{\xi}$, 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 prove that the principal orbits of 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 weakly isoparametric.
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