  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}.
