A splitting theorem for equifocal submanifolds in simply connected compact symmetric spaces

Ewert, Heiko

doi:10.1090/S0002-9939-98-04328-7

A splitting theorem for equifocal submanifolds in simply connected compact symmetric spaces
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Proc. Amer. Math. Soc. 126 (1998), 2443-2452 Request permission

Abstract:

A submanifold in a symmetric space is called equifocal if it has a globally flat abelian normal bundle and its focal data is invariant under normal parallel transportation. This is a generalization of the notion of isoparametric submanifolds in Euclidean spaces. To each equifocal submanifold, we can associate a Coxeter group, which is determined by the focal data at one point. In this paper we prove that an equifocal submanifold in a simply connected compact symmetric space is a non-trivial product of two such submanifolds if and only if its associated Coxeter group is decomposable. As a consequence, we get a similar splitting result for hyperpolar group actions on compact symmetric spaces. These results are an application of a splitting theorem for isoparametric submanifolds in Hilbert spaces by Heintze and Liu.

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