Geodesic foliations in Lorentz 3-manifolds

We study geodesic foliations on manifolds endowed with Lorentz metrics. The (local) theory works formally exactly as in the Riemannian case, if the induced metric on the leaves is non-degenerate. We consider here some local and global properties in the degenerate case. Received: October 24, 1994     

Spectral sequences and taut Riemannian foliations

Using a relation between the terms of the spectral sequence of a Riemannian foliation and its adiabatic limit, we obtain Bochner type techniques for this special setting and, as a consequence, in the special case of a Riemannian flow we obtain vanishing conditions for the top dimensional group of the basic cohomology \(H_{b}^{q}(\mathcal{F})\)-which is related to the property of being geodesible. We also extend a Weitzenböck type formula for the leafwise Laplacian and, for the particular class of compact foliations, we obtain a generalization of a result due to Ph. Tondeur, M. Min-Oo, and E. Ruh concerning the vanishing of the basic cohomology under the assumption that certain curvature operators are positive definite. In the final part we present an example.     

On the generic nonexistence of rational geodesic foliations in the torus, Mather sets and Gromov hyperbolic spaces

Given a rational homology classh in a two dimensional torusT ², we show that the set of Riemannian metrics inT ² with no geodesic foliations having rotation numberh isC ^k dense for everyk N. We also show that, generically in theC ² topology, there are no geodesic foliations with rational rotation number. We apply these results and Mather's theory to show the following: let (M, g) be a compact, differentiable Riemannian manifold with nonpositive curvature, if (M, g) satisfies the shadowing property, then (M, g) has no flat, totally geodesic, immersed tori. In particular,M has rank one and the Pesin set of the geodesic flow has positive Lebesgue measure. Moreover, if (M, g) is analytic, the universal covering ofM is a Gromov hyperbolic space.Partially supported by CNPq-GMD, FAPERJ, and the University of Freiburg.     

Broken hyperbolic structures and affine foliations on surfaces

In this paper, we introduce two new kinds of structures on a non-compact surface: broken hyperbolic structures and broken measured foliations. The space of broken hyperbolic structures contains the Teichmüller space of the surface as a subspace. The space of broken measured foliations is naturally identified with the space of affine foliations of the surface. We describe a topology on the union of the space of broken hyperbolic structures and of the space of broken measured foliations which generalizes Thurston's compactification of Teichmüller space.     

Hermitian metric rigidity on compact foliated manifolds

We prove a Hermitian metric rigidity theorem for leafwise symmetric Kaehler metrics on compact manifolds with smooth foliations. This provides applications to the study of the geometry of foliations as well as Kaehler manifolds that contain some symmetric geometry.     

Leaf closures of Riemannian foliations: A survey on topological and geometric aspects of Killing foliations

A smooth foliation is Riemannian when its leaves are locally equidistant. The closures of the leaves of a Riemannian foliation on a simply-connected manifold, or more generally of a Killing foliation, are described by flows of transverse Killing vector fields. This offers significant technical advantages in the study of this class of foliations, which nonetheless includes other important classes, such as those given by the orbits of isometric Lie group actions. Aiming at a broad audience, in this survey we introduce Killing foliations from the very basics, starting with a brief revision of the main objects appearing in this theory, such as pseudogroups, sheaves, holonomy and basic cohomology. We then review Molino’s structural theory for Riemannian foliations and present its transverse counterpart in the theory of complete pseudogroups of isometries, emphasizing the connections between these topics. We also survey some classical results and recent developments in the theory of Killing foliations. Finally, we review some topics in the theory of singular Riemannian foliations, including the recent proof of Molino’s conjecture, and discuss singular Killing foliations.     

Homotopy classification of contact foliations on open contact manifolds

We give a homotopy classification of foliations on open contact manifolds whose leaves are contact submanifolds of the ambient space. The result is an extension of Haefliger’s classification of foliations on open manifold in the contact setting. While proving the main theorem, we also prove a result on equidimensional isocontact immersions on open contact manifolds.     

The stretch of a foliation and geometric superrigidity

We consider compact smooth foliated manifolds with leaves isometrically covered by a fixed symmetric space of noncompact type. Such objects can be considered as compact models for the geometry of the symmetric space. Based on this we formulate and solve a geometric superrigidity problem for foliations that seeks the existence of suitable isometric totally geodesic immersions. To achieve this we consider the heat flow equation along the leaves of a foliation, a Bochner formula on foliations and a geometric invariant for foliations with leafwise Riemannian metrics called the stretch. We obtain as applications a metric rigidity theorem for foliations and a rigidity type result for Riemannian manifolds whose geometry is only partially symmetric.

     

Conformal geometry of foliations

In this article, we begin a systematic study of conformal properties of codimension-1 foliations. We first define and study local conformal invariants. A case of particular interest is that of harmonic foliations of the plane. Then we study existence of totally umbilical and “Dupin” foliations on compact 3-manifolds of constant curvature.      

Differential Algebra and Liouvillian first integrals of foliations

Intuitively, a complex Liouvillian function is one that is obtained from complex rational functions by a finite process of integrations, exponentiations and algebraic operations. In the framework of ordinary differential equations the study of equations admitting Liouvillian solutions is related to the study of ordinary differential equations that can be integrated by the use of elementary functions, that is, functions appearing in the Differential Calculus. A more precise and geometrical approach to this problem naturally leads us to consider the theory of foliations. This paper is devoted to the study of foliations that admit a Liouvillian first integral. We study holomorphic foliations (of dimension or codimension one) that admit a Liouvillian first integral. We extend results of Singer (1992) [20] related to Camacho and Scárdua (2001) [4], to foliations on compact manifolds, Stein manifolds, codimension-one projective foliations and germs of foliations as well.     

Foliations on ℂℙ2 of degree 2 with degenerate singularities

In this paper we construct non-singular, locally-closed, algebraic varieties which are sets of foliations on ??² of degree 2 with a certain degenerate singularity. We obtain the dimension and closure of these varieties. To do that we construct a stratification (based on GIT, see [7]) of the space of foliations with respect to the action by change of coordinates. We prove that the set of unstable foliations has two irreducible components. We have the following corollary: a foliation of degree 2 defined by a pencil of conics is unstable if and only if the pencil is unstable. Finallywe give another proof of the fact that there are only 4 foliations of degree 2 with a unique singular point (see [5]).     

The Weil–Petersson Kähler Form and Affine Foliations on Surfaces

The space of broken hyperbolic structures generalizes the usual Teichmüller space of a punctured surface, and the space of projectivized broken measured foliations – or, equivalently, the space of projectivized affine foliations of a punctured surface – likewise generalizes the space of projectivized measured foliations. Just as projectivized measured foliations provide Thurston's boundary for Teichmüller space, so too do projectivized broken measured foliations provide a boundary for the space of broken hyperbolic structures. In this paper, we naturally extend the Weil–Petersson Kähler two-form to a corresponding two-form on the space of broken hyperbolic structures as well as Thurston's symplectic form to a corresponding two-form on the space of broken measured foliations, and we show that the former limits in an appropriate sense to the latter. The proof in sketch follows earlier work of the authors for measured foliations and depends upon techniques from decorated Teichmüller theory, which is also applied here to a further study of broken hyperbolic structures.     

On existence of regular jacobi structures

We prove \(h\) -principle for locally conformal symplectic foliations and contact foliations on open manifolds. We then interpret the results in terms of regular Jacobi structures on manifolds.     

An invariant of nonpositively curved contact manifolds

We define an invariant of contact structures and foliations (on Riemannian manifolds of nonpositive sectional curvature) which is upper semi-continuous with respect to deformations and thus gives an obstruction to the topology of foliations which can be approximated by isotopies of a given contact structure.     

Higher dimensional tautological inequalities and applications

We study the degeneracy of holomorphic mappings tangent to holomorphic foliations on projective manifolds. Using Ahlfors currents in higher dimension, we obtain several strong degeneracy statements such as the proof of a generalized Green-Griffiths–Lang conjecture for threefolds with holomorphic foliations of codimension one.     

Tangential category of foliations

We present techniques to construct tangential homotopies of subsets of foliated manifolds and use these to obtain bounds and explicit computations for the tangential Lusternik-Schnirelmann category of foliations. For example, we show that this number is not greater than the dimension of the foliation, that it is an upper semi-continuous function on the space of p-dimensional foliations of a given manifold, and that it is equal to the dimension of the foliation for all codimension 1 foliations without holonomy on compact nilmanifolds.     

The automorphism groups of foliations with transverse linear connection

The category of foliations is considered. In this category morphisms are differentiable maps sending leaves of one foliation into leaves of the other foliation. We prove that the automorphism group of a foliation with transverse linear connection is an infinite-dimensional Lie group modeled on LF-spaces. This result extends the corresponding result of Macias-Virgós and Sanmartín Carbón for Riemannian foliations. In particular, our result is valid for Lorentzian and pseudo-Riemannian foliations.     

Properties of the basic cohomology of transversely Kähler foliations

In this paper we study the complex basic cohomology of transversely Hermitian foliations. We use the methods developed in [7] and prove that for transversely Kähler foliations the foliated version of the Frölicher spectral sequence collapses at the first level and that the minimal model for the complex basic cohomology is formal. To stress that these properties are particular to transversely Kähler foliations we construct examples of transversely Hermitian foliations for which these theorems do not hold.     

Singular Riemannian Foliations and Isoparametric Submanifolds

We will explain how polar actions on Riemannian manifolds lead in a natural way to a class of singular Riemannian foliations which turns out to generalize isoparametric foliations of Euclidean spaces and spheres. Both classical and recent results will be explained.     

The Noether–Fano inequalities for codimension one singular holomorphic foliations

The idea of the proof of the classical Noether–Fano inequalities can be adapted to the domain of codimension one singular holomorphic foliations of the projective space. We obtained criteria for proving that the degree of a foliation on the plane is minimal in the birational class of the foliation and for the non-existence of birational symmetries of generic foliations (except automorphisms). Moreover, we give several examples of birational symmetries of special foliations illustrating our results.