Singular riemannian foliations with sections, transnormal maps and basic forms

Abstract

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.

Preprint IME-USP; The final publication is available at springerlink.com http://www.springerlink.com/content/q48682633730t831/

References in corpus (2)

• Isoparametric submanifolds and a Chevalley-type restriction theorem
• Proofs of Conjectures about singular riemannian foliations

Cited by in corpus (8)

• Progress in the Theory of Singular Riemannian Foliations
• On smoothness of isometries between orbit spaces
• Polar foliations and isoparametric maps
• Introduction to Lie groups, isometric and adjoint actions and some generalizations
• A-Foliations of codimension two on compact simply-connected manifolds
• Equifocal submanifolds with non-flat section and topological Tits buildings
• On equifocal submanifolds with non-flat section in symmetric spaces of rank two
• A splitting theorem for equifocal submanifolds with non-flat section