Extrinsic homogeneity of curvature-adapted isoparametric submanifolds in symmetric spaces of non-compact type

Naoyuki Koike

Published 2017-08-09Version 1

In this paper, we study a full irreducible complete curvature-adapted isoparametric submanifold of codimension greater than one in a symmetric space of non-compact type. First we prove that, if the submanifold is of real analytic and satisfies a certain condition for its focal structure, then it is extrinsically homogeneous. Its proof is performed by applying the extrinsic homogeneity theorem for a certain kind of infinite dimensional anti-Kaehler isoparametric submanifold to the lift of the complexification of the original submanifold to a path space through a submersion. Furthermore, we prove that the submanifold is a principal orbit of a Hermann type action on the symmetric space by using the extrinsic homogeneity theorem. Secondly, we prove that, if the submanifold is of codimension greater than two and satisfies a certain stronger condition for its focal structure, then it is a principal orbit of the isotropy action of the ambient symmetric space, where we need not to assume the real analyticity of the submanifold. Its proof is performed by constructing the topological Tits building associated to the submanifold and using it.