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Classification of isoparametric submanifolds admitting a reflective focal submanifold in symmetric spaces of non-compact type. (English) Zbl 1439.53057
Summary: 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 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 to impose that the submanifold is of real analytic. We use the building theory in the proof.

MSC:
53C40 Global submanifolds
53C35 Differential geometry of symmetric spaces
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Full Text: Euclid
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