Singular Riemannian foliations on simply connected spaces

A singular foliation on a complete Riemannian manifold is said to be Riemannian if each geodesic that is perpendicular at one point to a leaf remains perpendicular to every leaf it meets. The singular foliation is said to admit sections if each regular point is contained in a totally geodesic complete immersed submanifold that meets every leaf orthogonally and whose dimension is the codimension of the regular leaves. A typical example of such a singular foliation is the partition by orbits of a polar action, e.g. the orbits of the adjoint action of a compact Lie group on itself.We prove that a singular Riemannian foliation with compact leaves that admits sections on a simply connected space has no exceptional leaves, i.e., each regular leaf has trivial normal holonomy. We also prove that there exists a convex fundamental domain in each section of the foliation and in particular that the space of leaves is a convex Coxeter orbifold.     

Equifocality of a singular Riemannian foliation

A singular foliation on a complete Riemannian manifold 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.

     

Progress in the theory of singular Riemannian foliations

A singular foliation is called a singular Riemannian foliation (SRF) if every geodesic that is perpendicular to one leaf is perpendicular to every leaf it meets. A typical example is the partition of a complete Riemannian manifold into orbits of an isometric action.In this survey, we provide an introduction to the theory of SRFs, leading from the foundations to recent developments. Sketches of proofs are included and useful techniques are emphasized. We study the local structure of SRFs in general and under curvature conditions in particular. We also review the solution of the Palais–Terng problem on integrability of the horizontal distribution. Important special classes of SRFs, like polar and variationally complete foliations and their relations, are treated. A characterization of SRFs whose leaf space is an orbifold is given. Moreover, desingularizations of SRFs are studied and applications, e.g., to Molino?s conjecture, are presented.     

Singular riemannian foliations with sections,transnormal maps and basic forms

A singular riemannian foliation 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 orthogonally and whose dimension is the codimension of the regular leaves of . We prove that the algebra of basic forms of M relative to 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 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 are compact. This result extends previous results due to Carter and West, Terng, and Heintze, Liu and Olmos. Marcos M. Alexandrino and Claudio Gorodski have been partially supported by FAPESP and CNPq.     

Integrable Riemannian Submersion with Singularities

A map of a Riemannian manifold into an euclidian space is said to be transnormal if its restrictions to neighbourhoods of regular level sets are integrable Riemannian submersions. Analytic transnormal maps can be used to describe isoparametric submanifolds in spaces of constant curvature and equifocal submanifolds with flat sections in simply connected symmetric spaces. These submanifolds are also regular leaves of singular Riemannian foliations with sections. We prove that regular level sets of an analytic transnormal map on a real analytic complete Riemannian manifold are equifocal submanifolds and leaves of a singular Riemannian foliation with sections.     

Polar foliations and isoparametric maps

A singular Riemannian foliation F{\mathcal{F}} on a complete Riemannian manifold M is called a polar foliation if, for each regular point p, there is an immersed submanifold Σ, called section, that passes through p and that meets all the leaves and always perpendicularly. A typical example of a polar foliation is the partition of M into the orbits of a polar action, i.e., an isometric action with sections. In this article we prove that the leaves of F{\mathcal{F}} coincide with the level sets of a smooth map H: M → Σ, if M is simply connected. In particular, the orbits of a polar action on a simply connected space are level sets of an isoparametric map. This result extends previous results due to the author and Gorodski, Heintze, Liu and Olmos, Carter and West, and Terng.     

Desingularization of singular Riemannian foliation

Let F{\mathcal{F}} be a singular Riemannian foliation on a compact Riemannian manifold M. By successive blow-ups along the strata of F{\mathcal{F}} we construct a regular Riemannian foliation [^(F)]{\hat{\mathcal{F}}} on a compact Riemannian manifold [^(M)]{\hat{M}} and a desingularization map [^(r)]:[^(M)]? M{\hat{\rho}:\hat{M}\rightarrow M} that projects leaves of [^(F)]{\hat{\mathcal{F}}} into leaves of F{\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 F{\mathcal{F}} are compact, then, for each small ${\epsilon >0 }${\epsilon >0 }, we can find [^(M)]{\hat{M}} and [^(F)]{\hat{\mathcal{F}}} so that the desingularization map induces an e{\epsilon}-isometry between M/F{M/\mathcal{F}} and [^(M)]/[^(F)]{\hat{M}/\hat{\mathcal{F}}}. This implies in particular that the space of leaves M/F{M/\mathcal{F}} is a Gromov-Hausdorff limit of a sequence of Riemannian orbifolds {([^(M)]_n/[^(F)]_n)}{\{(\hat{M}_{n}/\hat{\mathcal{F}}_{n})\}}.     

Minimal sets of Cartan foliations

A foliation that admits a Cartan geometry as its transversal structure is called a Cartan foliation. We prove that on a manifold M with a complete Cartan foliation ?, there exists one more foliation (M, \(\mathcal{O}\)), which is generally singular and is called an aureole foliation; moreover, the foliations ? and \(\mathcal{O}\) have common minimal sets. By using an aureole foliation, we prove that for complete Cartan foliations of the type ?/? with a compactly embedded Lie subalgebra ? in ?, the closure of each leaf forms a minimal set such that the restriction of the foliation onto this set is a transversally locally homogeneous Riemannian foliation. We describe the structure of complete transversally similar foliations (M, ?). We prove that for such foliations, there exists a unique minimal set ?, and ? is contained in the closure of any leaf. If the foliation (M, ?) is proper, then ? is a unique closed leaf of this foliation.     

Attractors and an analog of the Lichnérowicz conjecture for conformal foliations

We prove that each codimension q ≥ 3 conformal foliation (M,F) either is Riemannian or has a minimal set that is an attractor. If (M,F) is a proper conformal foliation that is not Riemannian then there exists a closed leaf that is an attractor. We do not assume that M is compact. Moreover, if M is compact then a non-Riemannian conformal foliation (M,F) is a (Conf(S ^q ), S ^q )-foliation with a finite family of attractors, and each leaf of this foliation belongs to the basin of at least one attractor.     

The singularities of the wave trace of the basic Laplacian of a Riemannian foliation

We apply techniques of microlocal analysis to the study of the transverse geometry of Riemannian foliations in order to analyze spectral invariants of the basic Laplacian acting on functions on a Riemannian foliation with a bundle-like metric. In particular, we consider the trace of the basic wave operator when the mean curvature form is basic. We extend the concept of basic functions to distributions and demonstrate the existence of the basic wave kernel. The singularities of the trace of this basic wave kernel occur at the lengths of certain geodesic arcs which are orthogonal to the closures of the leaves of the foliation. In cases when the foliation has regular closure, a complete representation of the trace of the basic wave kernel can be computed for t≠0. Otherwise, a partial trace formula over a certain set of lengths of well-behaved geodesic arcs is obtained.     

Foliations with leaves of nonpositive curvature

We answer a question of Gromov ([G2]) in the codimension 1 case: ifF is a codimension 1 foliation of a compact manifoldM with leaves of negative curvature, thenπ ₁(M) has exponential growth. We also prove a result analogous to Zimmer’s ([Z2]): ifF is a codimension 1 foliation on a compact manifold with leaves of nonpositive curvature, and ifπ ₁(M) has subexponential growth, then almost every leaf is flat. We give a foliated version of the Hopf theorem on surfaces without conjugate points. Partially supported by NSF Grant #DMS 9403870.     

The wave trace of the basic Laplacian of a Riemannian foliation near a non-zero period

Given a compact boundaryless Riemannian manifold that admits a Riemannian foliation, recall that the space of leaf closures is a singular stratified space. Associated to this space is an operator called the basic Laplacian defined on the space of smooth functions that are constant on the leaves (and, hence, the closures of the leaves of the foliation). The corresponding basic spectrum is, under certain assumptions, an infinite subset of the spectrum of the ordinary laplacian. If the metric is bundle-like with respect to the foliation, the trace of the basic wave operator can be analyzed, and invariants of the basic spectrum can be computed. These invariants include the lengths of certain geodesic arcs which are orthogonal to the leaf closures, and from them, basic wave trace asymptotic expansions are derived. Using the connection between Riemannian foliations and manifolds being acted upon by a compact Lie group of isometries, $G$ , the wave trace for the $G$ -invariant spectrum of a $G$ -manifold is sketched out as a related result.     

Indice local pour des ensembles symétriques par arcs

Let M be a complete Riemannian manifold with sectional curvature and dimension . Given a unit vector and a point we prove the existence of a complete geodesic through x whose tangent vector never comes close to v. As a consequence we show the existence of a bounded geodesic through every point in a complete negatively pinched manifold with finite volume and dimension . Received April 13, 1998; in final form July 23, 1999 / Published online October 11, 2000     

Harmonic forms and near-minimal singular foliations

For a closed 1-form with Morse singularities, Calabi discovered a simple global criterion for the existence of a Riemannian metric in which is harmonic. For a codimension 1 foliation , Sullivan gave a condition for the existence of a Riemannian metric in which all the leaves of are minimal hypersurfaces. The conditions of Calabi and Sullivan are strikingly similar. If a closed form has no singularities, then both criteria are satisfied and, for an appropriate choice of metric, is harmonic and the associated foliation is comprised of minimal leaves. However, when has singularities, the foliation is not necessarily minimal.? We show that the Calabi condition enables one to find a metric in which is harmonic and the leaves of the foliation are minimal outside a neighborhood U of the -singular set. In fact, we prove the best possible result of this type: we construct families of metrics in which, as U shrinks to the singular set, the taut geometry of the foliation outside U remains stable. Furthermore, all compact leaves missing U are volume minimizing cycles in their homology classes. Their volumes are controlled explicitly. Received: January 24, 2000     

On mean curvature flow of singular Riemannian foliations: Noncompact cases

In this paper we investigate the mean curvature flow (MCF) of a regular leaf of a closed generalized isoparametric foliation as initial datum, generalizing previous results of Radeschi and the first author. We show that, under bounded curvature conditions, any finite time singularity is a singular leaf, and the singularity is of type I. The new techniques also allow us to discuss the existence of basins of attraction, how cylinder structures can affect convergence of basic MCF of immersed submanifolds and assure convergence of MCF of non-closed leaves of generalized isoparametric foliation on compact manifold.     

Pseudo-Riemannian geodesic foliations by circles

We investigate under which assumptions an orientable pseudo-Riemannian geodesic foliations by circles is generated by an S ¹-action. We construct examples showing that, contrary to the Riemannian case, it is not always true. However, we prove that such an action always exists when the foliation does not contain lightlike leaves, i.e. a pseudo-Riemannian Wadsley’s Theorem. As an application, we show that every Lorentzian surface all of whose spacelike/timelike geodesics are closed, is finitely covered by ${S^1 \times \mathbb{R}}$ . It follows that every Lorentzian surface contains a nonclosed geodesic.     

Basic forms for transversely integrable singular Riemannian foliations

Basic forms for a transversely integrable singular Riemannian foliation with compact leaves are in one-to-one correspondence with ``Weyl"-invariant differential forms on a generalized section of the foliation.

     

Minimal singular Riemannian foliations

We prove that a singular foliation on a compact manifold admitting an adapted Riemannian metric for which all leaves are minimal must be regular. To cite this article: V. Miquel, R.A. Wolak, C. R. Acad. Sci. Paris, Ser. I 342 (2006).