Proofs of Conjectures about singular riemannian foliations

Details Proofs of Conjectures about singular riemannian foliations Proofs of Conjectures about singular riemannian foliations

2005-09-18

arxiv.org/abs/math/0509405v1

Geom. Dedicata 119(2006) No1, 219-234

Mathematics

Differential Geometry

Latex2e; The final publication is available at springerlink.com http://www.springerlink.com/content/q48682633730t831/

Scientific paper

10.1007/s10711-006-9073-0

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.

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