The Homogeneous Slice Theorem for the Complete Complexification of a Proper Complex Equifocal Submanifold

Abstract

The notion of a complex equifocal submanifold in a Riemannian symmetric space of non-compact type has been recently introduced as a generalization of isoparametric hypersurfaces in the hyperbolic space. As its subclass, the notion of a proper complex equifocal submanifold has been introduced. Some results for a proper complex equifocal submanifold have been recently obtained by investigating the lift of its complete complexification to some path space. In this paper, we give a new construction of the complete complexification of a proper complex equifocal submanifold and, by using the construction, show that leaves of focal distributions of the complete complexification are the images by the normal exponential map of principal orbits of a certain kind of pseudo-orthogonal representation on the normal space of the corresponding focal submanifold.