arXiv subjects

Gael Meigniez

Publications and source records attributed to Gael Meigniez.

8 recordsLinked to original sources

arXiv2025

Rigidity of Lie foliations with locally symmetric leaves

We prove that if the leaves of a minimal Lie foliation are locally isometric to a symmetric space of non-compact type without a Poincare disk factor, then the foliation is smoothly conjugate to a homogeneous Lie foliation up to finite covering. This result generalizes and strengthens Zimmer's theorem, which characterizes minimal Lie foliations with leaves isometric to a symmetric space of non-compact type without real rank one factors as pullbacks of homogeneous foliations. As applications, we extend Zimmer's arithmeticity theorem for holonomy groups and establish a rigidity theorem for Riemannian foliations with locally symmetric leaves.

math.DG↗

arXiv2021

Matching Cells

A (complete) matching of the cells of a triangulated manifold can be thought as a combinatorial or discrete version of a nonsingular vector field. We give several methods for constructing such matchings.

math.GT↗

arXiv2021

Conformal Symplectic structures, Foliations and Contact Structures

This paper presents two existence h-principles, the first for conformal symplectic structures on closed manifolds, and the second for leafwise conformal symplectic structures on foliated manifolds with non empty boundary. The latter h-principle allows to linearly deform certain codimension-$1$ foliations to contact structures. These results are essentially applications of the Borman-Eliashberg-Murphy h-principle for overtwisted contact structures and of the Eliashberg-Murphy symplectization of cobordisms, together with tools pertaining to foliated Morse theory, which are elaborated here.

math.SG↗

arXiv2021

On the Haefliger-Thurston conjecture

The classifying space for the framed Haefliger structures of codimension $q$ and class $C^r$ is $(2q-1)$-connected, for $1\le r\le\infty$. The corollaries deal with the existence of foliations, with the homology and the perfectness of the diffeomorphism groups, with the existence of foliated products, and of foliated bundles.

math.GT↗

arXiv2021

A compactly generated pseudogroup which is not realizable

We exhibit a pseudogroup of smooth local transformations of the real line which is compactly generated, but not realizable as the holonomy pseudogroup of a foliation of codimension 1 on a compact manifold. The proof relies on a description of all foliations with the same dynamic as the Reeb component.

math.GT↗

arXiv2020

Quasi-Complementary Foliations and the Mather-Thurston Theorem

We establish a form of the h-principle for the existence of foliations quasi-complementary to a given one; the same methods also provide a proof of the classical Mather-Thurston theorem.

math.GT↗

arXiv2016

Haefliger structures and symplectic/contact structures

For some geometries including symplectic and contact structures on an n-dimensional manifold, we introduce a two-step approach to Gromov's h-principle. From formal geometric data, the first step builds a transversely geometric Haefliger structure of codimension n. This step works on all manifolds, even closed. The second step, which works only on open manifolds and for all geometries, regularizes the intermediate Haefliger structure and produces a genuine geometric structure. Both steps admit relative parametric versions. The proofs borrow ideas from W. Thurston, like jiggling and inflation. Actually, we are using a more primitive jiggling due to R. Thom.

math.GT↗

arXiv2012

Regularization and minimization of Haefliger structures of codimension one

We prove the existence of a minimal (all leaves dense) foliation of codimension one, on every closed manifold of dimension at least 4 whose Euler characteristic is null, in every homotopy class of hyperplanes distributions, in every homotopy class of Haefliger structures, in every differentiability class, under the obvious embedding assumption. The proof uses only elementary means, and reproves Thurston's existence theorem in all dimensions. A parametric version is also established.

math.GT↗