A matchbox manifold is a foliated space with totally disconnected transversals, and an equicontinuous matchbox manifold is the generalization of Riemannian foliations for smooth manifolds in this context. In this paper, we develop the Molino theory for all equicontinuous matchbox manifolds. Our work extends the Molino theory developed in the work of Alvarez Lopez and Moreira Galicia which required the hypothesis that the holonomy actions for these spaces satisfy the strong quasi-analyticity condition. The methods of this paper are based on the authors' previous works on the structure of weak solenoids, and provide many new properties of the Molino theory for the case of totally disconnected transversals, and examples to illustrate these properties. In particular, we show that the Molino space need not be uniquely well-defined, unless the global holonomy dynamical system is tame, a notion defined in this work. We show that examples in the literature for the theory of weak solenoids provide examples for which the strong quasi-analytic condition fails. Of particular interest is a new class of examples of equicontinuous minimal Cantor actions by finitely generated groups, whose construction relies on a result of Lubotzky. These examples have non-trivial Molino sequences, and other interesting properties.

Molino theory for matchbox manifolds

Remarks on the dynamics of the horocycle flow for homogeneous foliations by hyperbolic surfaces

The purpose of this paper is to show that any extension of a minimal Lie foliation on a compact manifold is a transversaly Riemannian g\h- foliation with trivial normal bundle. This result permits to classify the extensions of a minimal Lie foliation on a compact manifold from the Lie subgroups of its Lie group.

Extension d'un feuilletage de Lie minimal d'une vari\'{e}t\'{e} compacte

We show that the Hodge numbers of Sasakian manifolds are invariant under arbitrary deformations of the Sasakian structure. We also present an upper semi-continuity theorem for the dimensions of kernels of a smooth family of transversely elliptic operators on manifolds with homologically orientable transversely Riemannian foliations. We use this to prove that the $$\partial {\bar{\partial }}$$
 -lemma and being transversely Kahler are rigid properties under small deformations of the transversely holomorphic structure which preserve the foliation. We study an example which shows that this is not the case for arbitrary deformations of the transversely holomorphic foliation. Finally we point out an application of the upper-semi continuity theorem to K-contact manifolds.

/pdf/invariance-of-basic-hodge-numbers-under-deformations-of-2efxmwi8gy.pdf

Invariance of basic Hodge numbers under deformations of Sasakian manifolds

We show that the Hodge numbers of Sasakian manifolds are invariant under arbitrary deformations of the Sasakian structure. We also present an upper semi continuity Theorem for the dimensions of kernels of a smooth family of transversely elliptic operators on manifolds with transversely Riemannian foliations. We use this to prove that the $\partial\bar{\partial}$-lemma and being transversely K\"{a}hler are rigid properties under small deformations of the transversely holomorphic structure which preserve the foliation. Finally, we study an example which shows that this is not the case for arbitrary deformations of the transversely holomorphic foliation.

Sur l'existence des feuilletages de lie

On construit sur une variete compacte un feuilletage de Lie nilpotent et non abelien de codimension 4 qui n'admet pas de deformation resoluble non nilpotente

/pdf/sur-les-feuilletages-de-lie-nilpotents-4n8sn66vbe.pdf

Sur les Feuilletages de Lie Nilpotents

We introduce a notion of linear deformation of codimension one foliations into contact structures and describe some foliations which deform instantly into contact structures and some which do not Restricting ourselves to closed smooth manifolds, we obtain a neces- sary and su‰cient condition for a foliation defined by a closed nonsingular 1-form to be lin- early deformable into contact structures

/pdf/foliations-and-contact-structures-23iougimd4.pdf

Foliations and contact structures

We prove that a closed 3-dimensional manifold is a torus bundle over the circle if and only if it carries a closed nonsingular 1-form which is linearly deformable into contact forms

/pdf/fibrations-and-contact-structures-52z3h09v4f.pdf

Fibrations and contact structures

Nous donnons des exemples de feuilletages de Lie sur une variete compacte qui ne se deforment pas en des feuilletages de Lie a holonomie discrete.

https://afst.centre-mersenne.org/item/10.5802/afst.1119.pdf

Hamidou Dathe

Papers

Sur l'existence des feuilletages de lie

Sur les Feuilletages de Lie Nilpotents

Foliations and contact structures

Fibrations and contact structures

Exemples de feuilletages de Lie