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arXiv · 1708.02773

Classification of isoparametric submanifolds admitting a reflective focal submanifold in symmetric spaces of non-compact type

Naoyuki Koike
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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 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.

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Naoyuki Koike. 2018-05-07. Classification of isoparametric submanifolds admitting a reflective focal submanifold in symmetric spaces of non-compact type. https://doi.org/10.18910/73746

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