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January 2020 Classification of isoparametric submanifolds admitting a reflective focal submanifold in symmetric spaces of non-compact type
Naoyuki Koike
Osaka J. Math. 57(1): 207-246 (January 2020).

Abstract

In this paper, we assume that all isoparametric submanifolds have flat section. The main purpose of this paper is to prove that, if a full irreducible complete isoparametric submanifold of codimension greater than one in a symmetric space of non-compact type admits a reflective focal submanifold and if it is of real analytic, then it is a principal orbit of a Hermann type action on the symmetric space. A hyperpolar action on a symmetric space of non-compact type admits a reflective singular orbit if and only if it is a Hermann type action. Hence is not extra the assumption that the isoparametric submanifold admits a reflective focal submanifold. Also, we prove that, if a full irreducible complete isoparametric submanifold of codimension greater than one in a symmetric space of non-compact type satisfies some additional conditions, then it is a principal orbit of the isotropy action of the symmetric space, where we need not impose that the submanifold is of real analytic. We use the building theory in the proof.

Citation

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Naoyuki Koike. "Classification of isoparametric submanifolds admitting a reflective focal submanifold in symmetric spaces of non-compact type." Osaka J. Math. 57 (1) 207 - 246, January 2020.

Information

Published: January 2020
First available in Project Euclid: 15 January 2020

zbMATH: 07196624
MathSciNet: MR4052637

Subjects:
Primary: 53C40
Secondary: 53C35

Rights: Copyright © 2020 Osaka University and Osaka City University, Departments of Mathematics

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Vol.57 • No. 1 • January 2020
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