Marcel Nicolau

On the differentiability of first integrals of two dimensional flows

By using techniques of differential geometry we answer the following open problem proposed by Chavarriga, Giacomini, Gine, and Llibre in 1999. For a given two dimensional flow, what is the maximal order of differentiability of a first integral on a canonical region in function of the order of differentiability of the flow? Moreover, we prove that for every planar polynomial differential system there exist finitely many invariant curves and singular points γ i , i = 1, 2, , l, such that R 2 \ (∪ l i=1 γ i ) has finitely many connected open components, and that on each of these connected sets the system has an analytic first integral. For a homogeneous polynomial differential system in R 3 , there exist finitely many invariant straight lines and invariant conical surfaces such that their complement in R 3 is the union of finitely many open connected components, and that on each of these connected open components the system has an analytic first integral.

Déformations des feuilletages transversalement holomorphes à type différentiable fixé

Let F be a transversely holomorphic foliation on a compact manifold. We show the existence of a versal space for those deformations of F which keep fixed its differentiable type if F is Hermitian or if F has complex codimension one and admits a transverse projectable connection. We also prove the existence of a versal space of deformations for the complex structures on a Lie group invariant by a cocompact subgroup.

On deformations of transversely homogeneous foliations

Abstract A transversely homogeneous foliation is a foliation whose transverse model is a homogeneous space G/H. In this paper we consider the class of transversely homogeneous foliations F on a manifold M which can be defined by a family of 1-forms on M fulfilling the Maurer–Cartan equation of the Lie group G. This class includes as particular cases Lie foliations and certain homogeneous spaces foliated by points. We develop, for the foliations belonging to this class, a deformation theory for which both the foliation F and the model homogeneous space G/H are allowed to change. As the main result we show that, under some cohomological assumptions, there exist a versal space of deformations of finite dimension for the foliations of the class and when the manifold M is compact. Some concrete examples are discussed.

Foliations : dynamics, geometry and topology

Fundamentals of Foliation Theory.- Foliation Dynamics.- Deformation of Locally Free Actions and Leafwise Cohomology.- Transversal Dirac Operators on Distributions, Foliations, and G-Manifolds.