Hermann type actions on a pseudo-Riemannian symmetric space

Abstract

In this paper, we first investigate the shape operators of certain kind of orbits of the isotropy action of a semi-simple pseudo-Riemannian symmetric space. The investigation is performed by investigating the complexified action. Next, by using the fact obtained by the investigation, we show that certain kind of principal orbits of a Hermann type action on a semi-simple pseudo-Riemannian symmetric space are curvature-adapted proper complex equifocal submanifolds and that their shape operators are semisimple. It follows from this fact that the principal orbits are isoparametric submanifolds with flat section. Also, we derive an interesting structure of a semi-simple pesudo-Riemannian symmetric space (in particular, the complexification of a Riemannian symmetric space) from two special Hermann type actions on the space.