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 be an orthogonal
subset of a set of simple roots associated with the symmetric
space M. Then determines a horospherical decomposition
M = Fs
×ErankM−||×N, where Fs
is the Riemannian product
of || 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 Fs
, a foliation
by parallel affine subspaces on the Euclidean space ErankM−||,
and the horocycle subgroup N of the parabolic subgroup of the
isometry group of M determined by .