Hyperpolar homogeneous foliations on symmetric spaces of noncompact type - an overview

A foliation F on a Riemannian manifold M is homogeneous if its leaves coincide with the orbits of an isometric action on M. A foliation F is polar if it admits a section, that is, a connected closed totally geodesic submanifold of M which intersects each leaf of F, and intersects orthogonally at each point of intersection. A foliation F is hyperpolar if it admits a flat section. These notes are related to joint work with Jose Carlos Diaz-Ramos and Hiroshi Tamaru about hyperpolar homogeneous foliations on Riemannian symmetric spaces of noncompact type. Apart from the classification result which we proved in arXiv:0807.3517v2 [math.DG], we present here in more detail some relevant material about symmetric spaces of noncompact type, and discuss the classification in more detail for the special case M = SL_{r+1}(R)/SO_{r+1}.