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.
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Alexandrino, M.M. Proofs of Conjectures about Singular Riemannian Foliations. Geom Dedicata 119, 219–234 (2006). https://doi.org/10.1007/s10711-006-9073-0
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DOI: https://doi.org/10.1007/s10711-006-9073-0
Keywords
- Singular Riemannian foliations
- Pseudogroups
- Equifocal submanifolds
- Polar actions
- Isoparametric submanifolds