ABSTRACT
In this chapter we discuss polar actions on Riemannian symmetric spaces of noncompact type. Compared with the compact case, the situation is much more involved. For example, on any Riemannian symmetric space of noncompact type and rank greater than one there exist polar actions which are not hyperpolar, which is not true in the compact case. As in the compact case we will distinguish between rank one and higher rank. The theory for polar actions on Riemannian symmetric spaces of noncompact type has been developed to a large extent by the Berndt and Tamaru in a series of papers. The theoretical background for it includes the structure theory of parabolic subalgebras of semisimple Lie algebras and horospherical decompositions of Riemannian symmetric spaces of noncompact type. We will explain all this in the first two sections of this chapter.
