April 2023 Isoparametric submanifolds in Hilbert spaces and holonomy maps
Naoyuki Koike
Author Affiliations +
Illinois J. Math. 67(1): 153-170 (April 2023). DOI: 10.1215/00192082-10450471


Let π:PB 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 holc:APHsG is a homothetic submersion of coefficient a, where s is a nonnegative integer and APHs is the Hilbert space of all Hs-connections of the bundle P. In particular, we prove that if s=0, then holc 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 holc:APH0G is an isoparametric submanifold in APH0. As the result, we obtain a new systematic construction of isoparametric submanifolds in a Hilbert space.


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Naoyuki Koike. "Isoparametric submanifolds in Hilbert spaces and holonomy maps." Illinois J. Math. 67 (1) 153 - 170, April 2023. https://doi.org/10.1215/00192082-10450471


Received: 5 August 2022; Revised: 6 January 2023; Published: April 2023
First available in Project Euclid: 23 February 2023

zbMATH: 07690189
MathSciNet: MR4570228
Digital Object Identifier: 10.1215/00192082-10450471

Primary: 53C40
Secondary: 53C05 , 53C29

Rights: Copyright © 2023 by the University of Illinois at Urbana–Champaign


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Vol.67 • No. 1 • April 2023
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