Proofs of Conjectures about singular riemannian foliations
arXiv:math/0509405 · doi:10.1007/s10711-006-9073-0
Abstract
We prove that if the normal distribution of a singular riemannian foliation 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 transverse orbit of the closure of each leaf. Furthermore we prove that the closure 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.
Latex2e; The final publication is available at springerlink.com http://www.springerlink.com/content/q48682633730t831/
References in corpus (2)
Cited by in corpus (15)
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- Progress in the Theory of Singular Riemannian Foliations
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- Low dimensional Singular Riemannian Foliations in spheres
- On smoothness of isometries between orbit spaces
- On polar foliations and fundamental group
- Polar foliations and isoparametric maps
- Curvature explosion in quotients and applications
- On closed geodesics in the leaf spaces of singular Riemannian foliations
- Smoothness of isometric flows on orbit spaces and applications to the theory of foliations
- Introduction to Lie groups, isometric and adjoint actions and some generalizations
- Geometric resolution of singular Riemannian foliations
- Equifocality of a singular riemannian foliation
- Taut Submanifolds and Foliations