MathSciDoc: An Archive for Mathematician ∫

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 .

No keywords uploaded!