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We prove that a transversely product component of the singular set of a holomorphic foliation on ℙn is necessarily a Kupka component. [ABSTRACT FROM AUTHOR]

The purpose of this Note is to establish a geometric realization of the cohomological isomorphism in the case of a transversely oriented Killing foliation on a compact smooth manifold through equivariant basic Chern characters. [ABSTRACT FROM AUTHOR]

We provide a significant extension of the Hyperboloidal Foliation Method introduced by the authors in 2014 in order to establish global existence results for systems of quasilinear wave equations posed on a curved space, when wave equations and Klein–Gordon equations are coupled. This method is based on a ( 3 + 1 ) foliation (of the interior of a future light cone in Minkowski spacetime) by spacelike and asymptotically hyperboloidal hypersurfaces. In the new formulation of the method, we succeed to cover wave-Klein–Gordon systems containing “strong interaction” terms at the level of the metric, and then generalize our method in order to establish a new existence theory for the Einstein equations of general relativity. Following pioneering work by Lindblad and Rodnianski on the Einstein equations in wave coordinates, we establish the nonlinear stability of Minkowski spacetime for self-gravitating massive scalar fields. [ABSTRACT FROM AUTHOR]

Given a foliation on a manifold with suitable curvature form, the Euler class of its tangent bundle is explicitly computed whenever it admits an umbilic leaf. If the leaf is compact, then topological obstructions arise by considering foliated manifolds with certain trivial cohomology group. The results fully generalize to distributions tangent to at least one compact umbilic submanifold. [ABSTRACT FROM AUTHOR]

Abstract: Iakovos Androulidakis and Georges Skandalis have defined a holonomy groupoid for any singular foliation. This groupoid, whose topology is usually quite bad, is the starting point for the study of longitudinal pseudodifferential calculus on such foliation and its associated index theory. These studies can be highly simplified under the assumption of the holonomy groupoid being longitudinally smooth. In this note, we rephrase the period bounding lemma that asserts that a vector field on a compact manifold admits a strictly positive lower bound for its periodic orbits in order to prove that the holonomy groupoid is always longitudinally smooth. [Copyright &y& Elsevier]

Abstract: We prove in this Note that there is, for some foliated bundles, a bijective correspondence between Garnettʼs harmonic measures and measures on the fiber that are stationary for some probability measure on the holonomy group. As a consequence, we show the uniqueness of the harmonic measure in the case of some foliations transverse to projective fiber bundles. [Copyright &y& Elsevier]

Abstract: We realize the Reeb foliation of as a family of Legendrian submanifolds of the unit . Moreover, we construct a deformation of the standard contact in , via a family of contact submanifolds, into this realization. [Copyright &y& Elsevier]

Abstract: We prove that the basic signature of a homologically orientable Riemannian foliation with even codimension, is a foliated homotopy invariant. [Copyright &y& Elsevier]

Abstract: In this Note we prove the equivalence between the Riemannian foliation and each of the following conditions: 1) the lifted foliation on the bundle of r-transverse jets is Riemannian for ; 2) the foliation on the slashed is Riemannian and vertically exact for ; 3) there exists a positively admissible transverse Lagrangian on , the r-transverse slashed jet bundle of a foliated bundle , for . [Copyright &y& Elsevier]

Abstract: We prove that Lehmannʼs foliation cannot be deformed into a non-nilpotent one. [Copyright &y& Elsevier]

Abstract: On the slit tangent bundle of Finsler manifolds, we introduce a class of metrics and study the relation between Levi-Civita connection, Vaisman connection, vertical foliation, and Reinhart spaces. We show that the Levi-Civita and the Vaisman connections induce the same connections in the structural bundle if and only if the base manifold is Landsbergian. Moreover every foliated Reinhart manifold reduces to a Riemannian manifold. [Copyright &y& Elsevier]

Abstract: For any Lie groupoid with a twisting, we define an analytic index morphism using the Connes tangent groupoid. This morphism agrees with the one of the Lie groupoid when the twisting is trivial. We discuss a longitudinal index theorem, geometric cycles, push-forward maps and Baum–Connes assembly maps for foliations with a twisting on the space of leaves. [ABSTRACT FROM AUTHOR]

Abstract: Let be a codimension one foliation of an aspherical manifold M. Assume that has no vanishing cycles. If there is an aspherical dense leaf of , then each leaf of is aspherical. If is minimal and the universal covering of a leaf of is not k-connected, then the universal coverings of no leaves are k-connected. To cite this article: T. Yokoyama, C. R. Acad. Sci. Paris, Ser. I 347 (2009). [Copyright &y& Elsevier]

Abstract: In this Note one proves a kind of Arnold–Liouville theorem for a 2-isotropic foliation, notion introduced in a previous Note (Turiel, 2008) for any closed r-form, and one studies some properties of the singular integer nets associated to it. To cite this article: F.-J. Turiel, C. R. Acad. Sci. Paris, Ser. I 347 (2009). [Copyright &y& Elsevier]

Abstract: We present two results on the classification of exceptional CDQL webs on and on 2-dimensional complex tori. These are the exceptional k-webs formed by k global foliations, of which are linear. To cite this article: J.V. Pereira, L. Pirio, C. R. Acad. Sci. Paris, Ser. I 346 (2008). [Copyright &y& Elsevier]

Abstract: This Note is partly an announcement. We describe the geodesic foliation of the translation surface associated to a triangular billiard in terms of real and holomorphic foliations defined by quadratic homogeneous vector fields on . Our technique allows, in particular, to state the existence of real foliations by curves on arising from homogeneous quadratic vector fields and having a dense set of periodic orbits. To cite this article: F. Valdez, C. R. Acad. Sci. Paris, Ser. I 346 (2008). [Copyright &y& Elsevier]

Abstract: In this Note we introduce, for a closed form of any degree, the notion of 2-isotropic foliation, which plays the same role as that of the Lagrangian foliation in the symplectic case. After constructing an affine structure on its leaves we describe the tubular neighborhood of a ‘Lagrangian’ transverse section. To cite this article: F.-J. Turiel, C. R. Acad. Sci. Paris, Ser. I 346 (2008). [Copyright &y& Elsevier]

Abstract: From the fact that a foliation by curves of degree greater than one, with isolated singularities, in the complex projective plane is uniquely determined by its subscheme of singular points (the singular subscheme of the foliation), we pose the problem of existence of proper closed subschemes Z of the singular subscheme which still determine the foliation in a unique way. We prove the existence of such subschemes Z for foliations with reduced singular subscheme. Unlike the degree degZ of such subschemes is not sharp for the posed problem, we show that it is so in the sense that Z defines the so-called polar net of the foliation. To cite this article: A. Campillo, J. Olivares, C. R. Acad. Sci. Paris, Ser. I 344 (2007). [Copyright &y& Elsevier]

Abstract: In this Note we consider examples of partially hyperbolic actions: restrictions of Weyl chamber flows on (). We show that generic restrictions of rank at least two are locally rigid. Our approach combines the geometry of the invariant foliations for the action and the algebraic properties of the group . The method is applicable to restrictions of Weyl chamber flows on other homogeneous spaces. To cite this article: D. Damjanović, A. Katok, C. R. Acad. Sci. Paris, Ser. I 344 (2007). [Copyright &y& Elsevier]

Abstract: We prove a result of Riemann–Hilbert kind for germs of foliation in . This result has the following consequence: the cobordism class of a second kind foliation in is onto the equisingularity class of its separatrix. To cite this article: Y. Genzmer, C. R. Acad. Sci. Paris, Ser. I 344 (2007). [Copyright &y& Elsevier]

Abstract: The equisingularity type of a generic polar of a curve is not determined by the type of C. We expand the category of curves by associating to C the corresponding Hamiltonian foliation. In this way, we work in the ‘less rigid’ frame of the space of foliations having C as invariant curve. We characterize ‘kind types of equisingularity’ for C for which we completely determine the generic polar equisingularity type. The Hamiltonian case, corresponding to the classical theory of polar curves, can be special and sometimes does not provide the generic type. To cite this article: N. Corral, C. R. Acad. Sci. Paris, Ser. I 344 (2007). [Copyright &y& Elsevier]

Abstract: We study Levi-flat foliations which complex part is an holomorphic foliation with an isolated singularity at the origin of or . In dimension 2, we show that, if the complex part is non-degenerate or non-degenerate after one blow up, then the Levi-flat foliation and its complex part are both described by a first integral which is submersive on . In dimension 3, we show that a linear vector field is never the complex part of a codimension 3 Levi-flat foliation. To cite this article: L. Landuré, C. R. Acad. Sci. Paris, Ser. I 343 (2006). [Copyright &y& Elsevier]

Abstract: A definition of reducibility for algebraic codimension two foliations given by closed 2-forms is proposed. Reducible foliations are characterised on theirs Galois groupoids. We apply this to the foliation given by the first Painlevé equation. Its Galois groupoid is computed and this proves its irreducibility. To cite this article: G. Casale, C. R. Acad. Sci. Paris, Ser. I 343 (2006). [Copyright &y& Elsevier]

Abstract: Associated to a smooth foliation are defined the basic and the foliated cohomologies. These cohomologies are related to the de Rham cohomology by the de Rham spectral sequence of , , constructed by filtering the de Rham complex of the manifold. For a Riemannian foliation on a compact manifold the second term of this spectral sequence, , is finite dimensional and a topological invariant. In this Note we prove these two results, fitness dimension and topological invariance, for the cohomology of singular Riemannian foliations. The proof uses the previous theorems for the regular case and the structure of singular Riemannian foliations described by P. Molino. For the basic cohomology these results have been proved by R. Wolak. To cite this article: X.M. Masa, A. Rodríguez-Fernández, C. R. Acad. Sci. Paris, Ser. I 342 (2006). [Copyright &y& Elsevier]

Abstract: We prove the arithmeticity of totally geodesic Riemannian foliations, with a dense leaf, on complete finite volume Riemannian manifolds when the leaves are isometrically covered by an irreducible symmetric space of noncompact type and rank at least 2. To cite this article: R. Quiroga-Barranco, C. R. Acad. Sci. Paris, Ser. I 342 (2006). [Copyright &y& Elsevier]

Abstract: 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). [Copyright &y& Elsevier]

Abstract: Let be a G-Lie foliation on a compact manifold M. If is not unimodular then either M or the closures of the leaves fiber over . To cite this article: E. Macias-Virgós, P. Martín-Méndez, C. R. Acad. Sci. Paris, Ser. I 340 (2005). [Copyright &y& Elsevier]

We construct secondary classes for super-foliations of codimension and . We indicate how to generalize this construction for any regular super-foliations on super-manifolds. We interpret the secondary classes as classes of foliated flat connections. To cite this article: C. Laurent-Gengoux, C. R. Acad. Sci. Paris, Ser. I 339 (2004). [Copyright &y& Elsevier]

The notion of 2-calibrated structure, generalizing contact structures, smooth taut foliations, etc., is defined. Approximately holomorphic geometry as introduced by S. Donaldson for symplectic manifolds is extended to 2-calibrated manifolds. An estimated transversality result that enables to study the geometry of such manifolds is presented. To cite this article: A. Ibort, D. Marti´nez Torres, C. R. Acad. Sci. Paris, Ser. I 338 (2004). [Copyright &y& Elsevier]

We prove that there is no Levi-flat immersion of class C1 of a Riemann surface foliation of class C1 of a 3-dimensional compact manifold in the complex projective plane, if the foliation carries a harmonic current which is absolutely continuous with respect to Lebesgue measure, with a density bounded from above and below. This comes as a corollary of a rigidity result for Levi-flat immersions of class C1 of Riemann surface foliations having this regularity into complex surfaces of non negative Ricci curvature. To cite this article: B. Deroin, C. R. Acad. Sci. Paris, Ser. I 337 (2003). [Copyright &y& Elsevier]

We describe the classification of germs of holomorphic saddle-node vector fields Z at (0,0)∈C2, up to analytical change of local coordinates. The classification of saddle-node foliations FZ, obtained by Martinet and Ramis, led to functional moduli. We reduce the conjugacy equation between two vector fields inducing the same foliation to a homological equation. We derive then a complete set of additional functional invariants for vector fields. To cite this article: L. Teyssier, C. R. Acad. Sci. Paris, Ser. I 336 (2003). [Copyright &y& Elsevier]

Riemann–Poisson manifolds were introduced by the author in C. R. Acad. Sci. Paris, Ser. I 333 (2001) 763–768, and studied in detail in preprint math.DG/0206102. Ka¨hler–Riemann foliations form an interesting subset of the Riemannian foliations with remarkable properties (see Ann. Global Anal. Geom. 21 (2002) 377–399). In this Note we will show that to give a regular Riemann–Poisson structure on a manifold P is equivalent to give a Ka¨hler–Riemann foliation on P such that the leafwise symplectic form is invariant with respect to all local foliation-preserving perpendicular vector fields. Finally, we give some examples of such manifolds. To cite this article: M. Boucetta, C. R. Acad. Sci. Paris, Ser. I 336 (2003). [Copyright &y& Elsevier]

We announce the following result: every maximal globally hyperbolic 3-dimensional spacetime with compact Cauchy surface, and with nonpositive constant curvature admits a unique time function whose fibers are constant mean curvature surfaces. We discuss the extension of this result in higher dimensions. To cite this article: T. Barbot et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003). [Copyright &y& Elsevier]

Abstract: Let be a transversely orientable codimension one minimal foliation without vanishing cycles of a manifold M and . We show that if the fundamental group of each leaf of has polynomial growth of degree k, then the foliation is without holonomy. [Copyright &y& Elsevier]

Abstract: We define the class of weakly regular spacetimes with -symmetry, and investigate their global geometrical structure. We formulate the initial value problem for the Einstein vacuum equations with weak regularity, and establish the existence of a global foliation by the level sets of the area R of the orbits of symmetry, so that each leaf can be regarded as an initial hypersurface. Except for the flat Kasner spacetimes which are known explicitly, R takes all positive values. Our weak regularity assumptions only require that the gradient of R is continuous while the metric coefficients belong to the Sobolev space (or have even less regularity). [ABSTRACT FROM AUTHOR]