Annals of Global Analysis and Geometry

, Volume 32, Issue 3, pp 209–223

Singular riemannian foliations with sections, transnormal maps and basic forms

Original Paper
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Abstract

A singular riemannian foliation \(\mathcal{F}\) 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 Σ that meets every leaf of \(\mathcal{F}\) orthogonally and whose dimension is the codimension of the regular leaves of \(\mathcal{F}\). We prove that the algebra of basic forms of M relative to \(\mathcal{F}\) is isomorphic to the algebra of those differential forms on Σ that are invariant under the generalized Weyl pseudogroup of Σ. 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 \(\mathcal{F}\) 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 \(\mathcal{F}\) are compact. This result extends previous results due to Carter and West, Terng, and Heintze, Liu and Olmos.

Keywords

Singular riemannian foliations Basic forms Basic functions Pseudogroups Equifocal submanifolds Polar actions Isoparametric submanifolds 

Mathematics Subject Classifications (2000)

Primary 53C12 Secondary 57R30 

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Copyright information

© Springer Science + Business Media B.V. 2006

Authors and Affiliations

  1. 1.Instituto de Matemática e EstatísticaUniversidade de São Paulo (USP)São PauloBrazil

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