Abstract
Let 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 is a homothetic submersion of coefficient a, where s is a nonnegative integer and is the Hilbert space of all -connections of the bundle P. In particular, we prove that if , then 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 is an isoparametric submanifold in . As the result, we obtain a new systematic construction of isoparametric submanifolds in a Hilbert space.
Citation
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
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