Hyperpolar Homogeneous Foliations on Symmetric Spaces of Noncompact Type

Abstract. A foliation F on a Riemannian manifold M is hyperpolar if it admits a flat section, that is, a connected closed flat submanifold of M that intersects each leaf of F orthogonally. In this article we classify the hyperpolar homogeneous foliations on every Riemannian symmetric space M of noncompact type. These foliations are constructed as follows. Let Φ be an orthogonal subset of a set of simple roots associated with the symmetric space M . Then Φ determines a horospherical decomposition M = F s Φ × E × NΦ, where F s Φ is the Riemannian product of |Φ| symmetric spaces of rank one. Every hyperpolar homogeneous foliation on M is isometrically congruent to the product of the following objects: a particular homogeneous codimension one foliation on each symmetric space of rank one in F s Φ , a foliation by parallel affine subspaces on the Euclidean space E, and the horocycle subgroup NΦ of the parabolic subgroup of the isometry group of M determined by Φ.