Geometriae Dedicata

, Volume 119, Issue 1, pp 219–234

Proofs of Conjectures about Singular Riemannian Foliations

Authors

    • Instituto de Matemática e EstatísticaUniversidade de São Paulo (USP)
Original Article

DOI: 10.1007/s10711-006-9073-0

Cite this article as:
Alexandrino, M.M. Geom Dedicata (2006) 119: 219. doi:10.1007/s10711-006-9073-0

Abstract

A singular foliation on a complete Riemannian manifold M is said to be Riemannian if each geodesic that is perpendicular at one point to a leaf remains perpendicular to every leaf it meets. We prove that if the distribution of normal spaces to the regular leaves is integrable, then each leaf of this normal distribution can be extended to be a complete immersed totally geodesic submanifold (called section), which meets every leaf orthogonally. In addition the set of regular points is open and dense in each section. This result generalizes a result of Boualem and solves a problem inspired by a remark of Palais and Terng and a work of Szenthe about polar actions. We also study the singular holonomy of a singular Riemannian foliation with sections (s.r.f.s. for short) and in particular the tranverse orbit of the closure of each leaf. Furthermore we prove that the closures of the leaves of a s.r.f.s on M form a partition of M which is a singular Riemannian foliation. This result proves partially a conjecture of Molino.

Keywords

Singular Riemannian foliationsPseudogroupsEquifocal submanifoldsPolar actionsIsoparametric submanifolds

Mathematics Subject Classifications (2000)

Primary 53C12Secondary 57R30

Copyright information

© Springer Science+Business Media B.V. 2006