Proofs of Conjectures about Singular Riemannian Foliations

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.

On Closed Geodesics in the Leaf Space of Singular Riemannian Foliations

“…DEFINITION 2.9. Two W -loops are equivalent if there exists a subdivision common to the loops represented by (c i , w i ) and (c i ,w i ) and elements g i ∈ W defined in a neighbourhood of the path c i such that (2) of Definition 2.9. In order to understand this claim, consider W , the group generated by the reflection in the line {x = 0} in ‫ޒ‬ 2 .…”

“…Typical examples of (singular) Riemannian foliations are the partition by orbits of an isometric action, by leaf closures of a Riemannian foliation (see [2] and [18]), examples constructed by suspension of homomorphisms (see [1] and [2]) and examples constructed by changes of metric and surgery (see [3]). …”

Geometric resolution of singular Riemannian foliations

“…Horizontal sections. We refer to [Bou95], [Ale04], [Ale06] for more on polar singular Riemannian foliations. Let F be a singular Riemannian foliation on a Riemannian manifold M. A global (local) horizontal section through x is a smooth immersed submanifold x ∈ N ⊂ M that intersects all leaves of F (all leaves in a neighborhood of x), such that all intersections are orthogonal.…”

“…Recall, that a singular Riemannian foliation F is locally polar if and only if the restriction of F to the regular part M 0 has integrable horizontal distribution ([Ale06]). Moreover, a locally polar singular Riemannian foliation on a complete Riemannian manifold is polar.…”

“…For polar singular Riemannian foliations we refer to Subsection 2.4 (cf. [Ter85], [Bou95], [Ale04], [Ale06]) and for singular Riemannian foliations without horizontal conjugate points we refer to [BS58], [LT07b] [LT07a].…”