Complex Equifocal Submanifolds and Infinite Dimensional Anti-Kaehlerian Isoparametric Submanifolds

Abstract

In a symmetric space of non-compact type, we have recently defined the notion of a complex equifocal submanifold. In this paper, we introduce the notion of an infinite dimensional anti-Kaehlerian isoparametric submanifold. We show that the investigation of complete real analytic complex equifocal submanifolds is reduced to that of infinite dimensional anti-Kaehlerian isoparametric submanifolds. Also, we show that an infinite dimensional anti-Kaehlerian isoparametric submanifold is multi-foliated by complex spheres (or complex affine subspaces) and that the main part of the focal set of the submanifold at each point consists of some complex hyperplanes in the normal space.