Singular Riemannian foliations and $\mathcal{I}$-Poisson manifolds

A singular foliation on a complete riemannian manifold is said to be riemannian if every geodesic that is perpendicular at one point to a leaf remains perpendicular to every leaf it meets. In this paper, we study singular riemannian foliations with sections. A section is a totally geodesic complete immersed submanifold that meets each leaf orthogonally and whose dimension is the codimension of the regular leaves. We prove here that the restriction of the foliation to a slice of a leaf is diffeomorphic to an isoparametric foliation on an open set of an euclidean space. This result provides local information about the singular foliation and in particular about the singular stratification of the foliation. It allows us to describe the plaques of the foliation as level sets of a transnormal map (a generalization of an isoparametric map). We also prove that the regular leaves of a singular riemannian foliation with sections are locally equifocal. We use this property to define a singular holonomy. Then we establish some results about this singular holonomy and illustrate them with a couple of examples.

Singular Riemannian foliations on simply connected spaces

A singular foliation $\mathcal {F}$ gives a partition of a manifold $M$ into leaves whose dimension may vary. Associated to a singular foliation are two complexes, that of the diffeological differential forms on the leaf space $M / \mathcal {F}$ and that of the basic differential forms on $M$. We prove the pullback by the quotient map provides an isomorphism of these complexes in the following cases: when $\mathcal {F}$ is a regular foliation, when points in the leaves of the same dimension assemble into an embedded (more generally, diffeological) submanifold of $M$, and, as a special case of the latter, when $\mathcal {F}$ is induced by a linearizable Lie groupoid or is a singular Riemannian foliation.

Progress in the theory of singular Riemannian foliations

A singular foliation on a complete Riemannian manifold M M is said to be Riemannian if each geodesic that is perpendicular to a leaf at one point remains perpendicular to every leaf it meets. We prove that the regular leaves are equifocal, i.e., the end point map of a normal foliated vector field has constant rank. This implies that we can reconstruct the singular foliation by taking all parallel submanifolds of a regular leaf with trivial holonomy. In addition, the end point map of a normal foliated vector field on a leaf with trivial holonomy is a covering map. These results generalize previous results of the authors on singular Riemannian foliations with sections.

Using variational methods together with symmetries given by singular Riemannian foliations with positive dimensional leaves, we prove the existence of an infinite number of sign-changing solutions to Yamabe type problems, which are constant along the leaves of the foliation, and one positive solution of minimal energy among any other solution with these symmetries. In particular, we find sign-changing solutions to the Yamabe problem on the round sphere with new qualitative behavior when compared to previous results, that is, these solutions are constant along the leaves of a singular Riemannian foliation which is not induced neither by a group action nor by an isoparametric function. To prove the existence of these solutions, we prove a Sobolev embedding theorem for general singular Riemannian foliations, and a Principle of Symmetric Criticality for the associated energy functional to a Yamabe type problem.

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.

In this paper, we first prove that any closed simply connected 4-manifold that admits a decomposition into two disk bundles of rank greater than 1 is diffeomorphic to one of the standard elliptic 4-manifolds: $$\mathbb {S}^4$$ , $$\mathbb {CP}^2$$ , $$\mathbb {S}^2\times \mathbb {S}^2$$ , or $$\mathbb {CP}^2$$ # $$\pm \mathbb {CP}^2$$ . As an application we prove that any closed simply connected 4-manifold admitting a nontrivial singular Riemannian foliation is diffeomorphic to a connected sum of copies of standard $$\mathbb {S}^4$$ , $$\pm \mathbb {CP}^2$$ and $$\mathbb {S}^2\times \mathbb {S}^2$$ . A classification of singular Riemannian foliations of codimension 1 on all closed simply connected 4-manifolds is obtained as a byproduct. In particular, there are exactly 3 non-homogeneous singular Riemannian foliations of codimension 1, complementing the list of cohomogeneity one 4-manifolds.

In this paper we prove the conjecture of Molino that for every singular Riemannian foliation $(M,{\mathcal{F}})$, the partition $\overline{{\mathcal{F}}}$ given by the closures of the leaves of ${\mathcal{F}}$ is again a singular Riemannian foliation.

In this paper we introduce the concept of singular Finsler foliation, which generalizes the concepts of Finsler actions, Finsler submersions and (regular) Finsler foliations. We show that if $\mathcal{F}$ is a singular Finsler foliation on a Randers manifold $(M,Z)$ with Zermelo data $(\mathtt{h},W),$ then $\mathcal{F}$ is a singular Riemannian foliation on the Riemannian manifold $(M,\mathtt{h} )$. As a direct consequence we infer that the regular leaves are equifocal submanifolds (a generalization of isoparametric submanifolds) when the wind $W$ is an infinitesimal homothety of $\mathtt{h}$ (e.,g when $W$ is killing vector field or $M$ has constant Finsler curvature). We also present a slice theorem that relates local singular Finsler foliations on Finsler manifolds with singular Finsler foliations on Minkowski spaces.

Let $${\mathcal{F}}$$ be a singular Riemannian foliation on a compact Riemannian manifold M. By successive blow-ups along the strata of $${\mathcal{F}}$$ we construct a regular Riemannian foliation $${\hat{\mathcal{F}}}$$ on a compact Riemannian manifold $${\hat{M}}$$ and a desingularization map $${\hat{\rho}:\hat{M}\rightarrow M}$$ that projects leaves of $${\hat{\mathcal{F}}}$$ into leaves of $${\mathcal{F}}$$ . This result generalizes a previous result due to Molino for the particular case of a singular Riemannian foliation whose leaves were the closure of leaves of a regular Riemannian foliation. We also prove that, if the leaves of $${\mathcal{F}}$$ are compact, then, for each small $${\epsilon >0 }$$ , we can find $${\hat{M}}$$ and $${\hat{\mathcal{F}}}$$ so that the desingularization map induces an $${\epsilon}$$ -isometry between $${M/\mathcal{F}}$$ and $${\hat{M}/\hat{\mathcal{F}}}$$ . This implies in particular that the space of leaves $${M/\mathcal{F}}$$ is a Gromov-Hausdorff limit of a sequence of Riemannian orbifolds $${\{(\hat{M}_{n}/\hat{\mathcal{F}}_{n})\}}$$ .

In the present work we study $A$-foliations, i.e. singular Riemannian foliations with regular leaf aspherical. The main result is that, for a simply-connected closed $(n+2)$-manifold $M$, an $A$-foliation with regular leaves of codimension $2$ in $M$ is homogeneous. In other words it is given by a smooth effective action of the torus $\mathbb{T }^n$ on $M$ by isometries. We will give some conditions to compare two simply-connected, closed manifolds with $A$-foliations, up to foliated homeomorphism, via their leaf spaces.

We give a necessary and sufficient condition for a submanifold with parallel focal structure to give rise to a global foliation of the ambient space by parallel and focal manifolds. We show that this is a singular Riemannian foliation with complete orthogonal transversals. For this object we construct an action on the transversals that generalizes the Weyl group action for polar actions.

A singular riemannian foliation \(\mathcal{F}\) on a complete riemannian manifold M is said to admit sections if each regular point of M is contained in a complete totally geodesic immersed submanifold Σ that meets every leaf of \(\mathcal{F}\) orthogonally and whose dimension is the codimension of the regular leaves of \(\mathcal{F}\). We prove that the algebra of basic forms of M relative to \(\mathcal{F}\) is isomorphic to the algebra of those differential forms on Σ that are invariant under the generalized Weyl pseudogroup of Σ. This extends a result of Michor for polar actions. It follows from this result that the algebra of basic function is finitely generated if the sections are compact. We also prove that the leaves of \(\mathcal{F}\) coincide with the level sets of a transnormal map (generalization of isoparametric map) if M is simply connected, the sections are flat and the leaves of \(\mathcal{F}\) are compact. This result extends previous results due to Carter and West, Terng, and Heintze, Liu and Olmos.

We describe a local model for any singular Riemannian foliation in a neighborhood of a closed saturated submanifold of a singular stratum. Moreover, we construct a Lie groupoid that controls the transverse geometry of the linear approximation of the singular Riemannian foliation around these submanifolds. We also discuss the closure of this Lie groupoid and its Lie algebroid.