Regular cross sections of Borel flows

Any free Borel flow is shown to admit a cross section with only two possible distances between adjacent points. Non smooth flows are proved to be Lebesgue orbit equivalent if and only if they admit the same number of invariant ergodic probability measures.Comment: Minor improvements in expositio

Singular riemannian foliations with sections, transnormal maps and basic forms

A singular riemannian foliation F on a complete riemannian manifold M is said to admit sections if each regular point of M is contained in a complete totally geodesic immersed submanifold (a section) that meets every leaf of F orthogonally and whose dimension is the codimension of the regular leaves of F. We prove that the algebra of basic forms of M relative to F is isomorphic to the algebra of those differential forms on a section that are invariant under the generalized Weyl pseudogroup of this section. This extends a result of Michor for polar actions. It follows from this result that the algebra of basic function is finitely generated if the sections are compact. We also prove that the leaves of F coincide with the level sets of a transnormal map (generalization of isoparametric map) if M is simply connected, the sections are flat and the leaves of F are compact. This result extends previous results due to Carter and West, Terng, and Heintze, Liu and Olmos.Comment: Preprint IME-USP; The final publication is available at springerlink.com http://www.springerlink.com/content/q48682633730t831