Isoparametric Submanifolds of Hyperbolic Spaces
- BINGLE WU
In a symmetric space of non-compact type, the notion of a complex equifocal submanifold (with flat section) is defined. It is conjectured that this notion coincides with the notion of an isoparametric submanifold with flat section. As a subclass of a complex equifocal submanifold, the notion of a proper complex equifocal submanifold is defined. In this paper, we show that all irreducible proper complex equifocal submanifolds of codimension greater than one in a symmetric space of non-compact type are extrinsically homogeneous and hence they occur as principal orbits of complex hyperpolar actions on the symmetric space. The proof is performed by showing the homogeneity of the lifted submanifold of the complexification of the original submanifold to an infinite dimensional anti-Kaehlerian space.