New examples of rigid Lie foliations with dense leaves found.
problem Infinitesimal rigidity of Lie foliations with dense leaves.
method Construction of specific Lie foliations.
result First examples of infinitesimally rigid Riemannian foliations with dense leaves.
Lie foliations with symmetric leaves are smoothly conjugate to homogeneous ones.
problem Characterizing Lie foliations with symmetric leaves.
method Analyzing the rigidity of Lie foliations with locally symmetric leaves.
result Lie foliations with symmetric leaves are smoothly conjugate to homogeneous ones.
Study rigid Lie affine foliations on compact manifolds.
problem Cohomological criterion for rigidity of Lie foliations.
method Detailed study of cohomology groups, Morse-Novikov cohomology.
result Many examples of rigid Lie affine foliations on compact manifolds.
In this paper we try to generalize the Haefliger theorem on completly solvable Lie foliations. We prove that: every completely solvable Lie foliation on a compact manifold is the inverse image of a homogenus foliation. Every manifold in this paper is compact and our Lie group G is connexe and simply connexe.
In this work, we study Lie groupoids equipped with multiplicative foliations and the corresponding infinitesimal data. We determine the infinitesimal counterpart of a multiplicative foliation in terms of its core and sides together with a partial connection satisfying special properties, giving rise to the concept of I…
Establishes necessary and sufficient conditions for smooth triviality of Lie subalgebras and Lie ideals, and proves Moser's trick for foliations.
problem Smooth triviality of Lie subalgebras and Lie ideals
method Establishing necessary and sufficient conditions and proving Moser's trick for foliations
result Direct proof of Moser's trick for foliations
Study conformal foliations on Lie groups, finding new families and harmonic morphisms.
problem Classifying conformal foliations on Lie groups with minimal leaves.
method Analyzing left-invariant foliations generated by specific subgroups.
result New multi-dimensional families of Lie groups with conformal foliations.
The study classifies nilpotent Lie foliations with cohomological obstructions.
problem Understanding rigidity of nilpotent Lie foliations under solvable deformations.
method Development of a cohomological framework and algebraic criterion for rigidity.
result Established a necessary and sufficient algebraic criterion for rigidity in generalized Heisenberg groups.
New families of Lie groups with special foliations discovered.
problem Characterizing left-invariant foliations on semi-Riemannian Lie groups.
method Classifying foliations generated by specific subgroups.
result Constructing new families of Lie groups with conformal minimal foliations.
The paper extends Riemann-Hilbert correspondence to foliations.
problem Understanding representations of Lie algebroids and groupoids in foliated settings.
method Establishing an A∞ de Rham theorem and constructing an integration functor. result An equivalence between ∞-representations of L∞-algebroids and ∞-representations of Lie ∞-groupoids for foliations. This paper extends foliation concepts to singular foliations using Lie ∞-algebroids.
problem Cohomological obstruction to volume forms in singular foliations.
method Replacing singular foliations with universal Lie ∞-algebroids to define modular class. result Geometric meaning of modular class as an obstruction to universal Lie ∞-algebroids. Study on Lie groups' conformal foliations and harmonic morphisms.
problem Characterizing minimal conformal foliations on Lie groups.
method Analyzing left-invariant metrics and foliations on Lie groups.
result Minimal conformal foliations on Lie groups are fibres of harmonic morphisms.
Study derived Lie ∞-groupoids and algebroids in higher differential geometry.
problem Addressing problems in higher differential geometry using derived Lie ∞-groupoids and algebroids.
method Construct CFO structures, study L∞-algebroids, homotopical algebras, and homotopy-coherent representations.
result Construct Atiyah classes for L∞-algebroids pairs and study singular foliations and their holonomies.
New Lie groupoid and algebroid constructed for octonionic Hopf foliation.
problem No known Lie group action generates the singular octonionic Hopf foliation.
method Constructs a G2-equivariant Lie groupoid and Lie algebroid.
result Minimal Lie algebroid and groupoid generate the singular octonionic Hopf foliation.
Study the pullbacks and blowups of Lie algebroids and related structures.
problem Understanding the relationship between Lie algebroids, singular foliations, and Dirac structures under maps.
method Examine pullbacks and blowups of Lie algebroids and related structures under maps with constant rank or transversality assumptions.
result Establish the relation between the blowup of a Lie algebroid and its singular foliation.
If A is a Lie algebroid over a foliated manifold (M,F), a foliation of A is a Lie subalgebroid B with anchor image TF and such that A/B is locally equivalent with Lie algebroids over the slice manifolds of F. We give several examples and, for foliated Lie algebroids, we discu…
Paper studies symmetries in singular foliations using Lie ∞-morphisms.
problem Understanding symmetries in singular foliations.
method Analyzes Lie ∞-morphisms induced by Lie algebra actions on singular foliations. result Deduces geometrical consequences, including examples of non-extendable symmetries.
The paper classifies foliations formed by generic coadjoint orbits of specific Lie groups.
problem Classifying foliations formed by generic coadjoint orbits of certain Lie groups.
method Analyzing Lie groups with specific dimensions and nilradicals, proving foliations in the coadjoint representation.
result The family of generic coadjoint orbits forms a measurable foliation in the Lie groups considered.
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.
In this paper we study the Lie groupoids which appear in foliation theory. A foliation groupoid is a Lie groupoid which integrates a foliation, or, equivalently, whose anchor map is injective. The first theorem shows that, for a Lie groupoid G, the following are equivalent: - G is a foliation groupoid, - G has discrete…
The abstract describes a foliation of orbits for a specific class of Lie groups.
problem Classifying foliations formed by generic coadjoint orbits of Lie groups.
method Geometric description and topological classification of foliations.
result The family of generic coadjoint orbits forms a measurable foliation.
We associate a Lie ∞-algebroid to every resolution of a singular foliation, where we consider a singular foliation as a locally generated O-submodule of vector fields on the underlying manifold closed under Lie bracket. Here O can be the ring of smooth, holomorphic, or real analytic funct…
A singular (or Hermann) foliation on a smooth manifold M can be seen as a subsheaf of the sheaf X of vector fields on M. We show that if this singular foliation admits a resolution (in the sense of sheaves) consisting of sections of a graded vector bundle of finite type, then one can lift the Lie brack…
The paper studies Morse theory for Lie algebra actions on Riemannian foliations.
problem Morse theory for Lie algebra actions on Riemannian foliations.
method Equivariant Morse-Bott theory on leaf space.
result Established foliated versions of Morse-Bott lemma and handle presentation theorem.
We combine classic stability results for foliations with recent results on deformations of Lie groupoids and Lie algebroids to provide a cohomological characterization for rigidity of compact foliations on compact manifolds.
We describe a local model for any Singular Riemannian Foliation in a neighbourhood of a closed saturated submanifold of a regular stratum. Moreover we construct a Lie groupoid which controls the transverse geometry of the linear approximation of the Singular Riemannian Foliation around these submanifolds. We also discu…
This research extends Lie algebra actions to singular foliations.
problem Understanding symmetries in singular foliations without additional assumptions.
method Equivalence of categories between Lie-Rinehart algebras and Lie ∞-algebroids. result Universal Lie ∞-algebroids for singular foliations. Constructs a Lie groupoid integrating singular foliations.
problem Integrating singular foliations into higher Lie groupoids.
method Recursive use of bi-submersions and geometric resolutions.
result Finite-dimensional Lie groupoid integrating singular foliations.
Minimal conformal foliations on Lie groups are shown to be fibres of harmonic morphisms.
problem Characterizing minimal conformal foliations on Lie groups.
method Analyzing left-invariant semi-Riemannian metrics and harmonic morphisms.
result Minimal conformal foliations of codimension two are fibres of complex-valued harmonic morphisms.
The study classifies natural almost Hermitian structures on specific Lie groups.
problem Classifying natural almost Hermitian structures on conformally foliated Lie groups.
method Examining 4-dimensional Riemannian Lie groups with a 2-dimensional conformal foliation and minimal leaves, constructing new examples of multi-dimensional structures.
result Constructing several new multi-dimensional examples of almost Kähler, integrable, and Kähler structures.
In this paper we study (smooth and holomorphic) foliations which are invariant under transverse actions of Lie groups.
Computes L∞-algebroid for linear foliations on vector spaces.
problem Invariants of singular foliations on vector spaces induced by Lie subalgebras.
method Explicitly constructs projective resolutions and computes L∞-algebroid structure. result Provides invariants and constant-rank replacements of singular foliations.
Motivated by questions of deformations/moduli in foliation theory, we investigate the structure of some groups of diffeomorphisms preserving a foliation. We give an example of a C∞ foliation whose diffeomorphism group is not a Lie group in any reasonable sense. On the positive side, we prove that the automorphi…
We study Lie foliations on compact manifolds, in case the Lie group is compact. Our main results improve Tischler classical result on the existence of fibration and, as an application, we study the case the manifold has an amenable fundamental group.
A pre-Lie algebroid is an anchored bundle provided with an almost Lie bracket such that the anchor is compatible with the Lie bracket of vector fields. We firstly show how most geometrical structures intensively studied in the framework of Lie algebroid can easily be extended in the pre-Lie algebroid context. The princ…
Foliate systems are those which preserve some (possibly singular) foliation of phase space, such as systems with integrals, systems with continuous symmetries, and skew product systems. We study numerical integrators which also preserve the foliation. The case in which the foliation is given by the orbits of an action …
This thesis treats two main topics: calibrated symplectic foliations, and local Lie groupoids. Calibrated symplectic foliations are one possible generalization of taut foliations of 3-manifolds to higher dimensions. Their study has been popular in recent years, and we collect several interesting results. We then show h…
Godbillon-Vey classes for Lie subalgebroids
problem Extending Godbillon-Vey class to singular foliations
method Defining Godbillon-Vey class for Lie subalgebroids
result Presenting several examples
Study finds conditions for finite-dimensional Lie group structure in basic automorphism groups of certain Cartan foliations.
problem Characterizing the basic automorphism groups of Cartan foliations.
method Analyzes sufficient conditions and estimates dimensions for basic automorphism groups of Cartan foliations covered by fibrations.
result Identifies sufficient conditions for the existence of a finite-dimensional Lie group structure in basic automorphism groups.
Every singular foliation has an associated topological groupoid, called holonomy groupoid (see arXiv:math/0612370). In this note we exhibit some functorial properties of this assignment: if a foliated manifold (M,FM) is the quotient of a foliated manifold (P,FP) along a surjective submersion w…
Study of Hermitian metrics on Lie algebroids over complex spaces.
problem Exploring Hermitian metrics on Lie algebroids over analytic spaces.
method Introduced Hermitian metrics on holomorphic Lie algebroids and examined associated characteristics foliations.
result Extended equivariant de Rham cohomology to Hermitian Lie algebroids.
Regularisation method studies Lie algebroids via foliated structures.
problem Geometric structures on Lie algebroids.
method Regularisation procedure applied to various Lie algebroids.
result Proof of Weinstein conjecture for specific Lie algebroids.
Extends Cheeger's method to Lie groupoid actions on manifolds.
problem Smooth Lie group actions on manifolds with singularities.
method Extension of Cheeger's deformation techniques to Lie groupoid actions.
result Explicit sectional curvature description of the deformation.
Study Lie foliation of Walker manifolds in pseudo-Riemannian geometry.
problem Characterize Lie foliations of Walker manifolds.
method Analyzes Walker manifolds with null parallel distributions to integrate to Lie foliations and studies their properties.
result Walker manifolds with null parallel distributions always integrate to Lie foliations, and the transverse holonomy coincides with the image of the holonomy morphism.
New proof of Lie algebroid action equivalence and integrability.
problem Equivalence between Lie algebroid action and integrability.
method Double Lie groupoids and multiplicative foliations.
result Simple characterization of double Lie groupoids inducing Lie groupoid structures.
The paper classifies natural almost Hermitian structures on Lie groups with minimal conformal leaves.
problem Classifying natural almost Hermitian structures on Lie groups with minimal conformal leaves.
method Analyzing Lie groups with a 2-dimensional conformal foliation and classifying structures based on Lie algebra properties.
result 16 multi-dimensional almost Kähler families, 18 integrable families, and 11 Kähler families were constructed.
Extends foliation results to singular cases.
problem Understanding foliations near singular leaves.
method Proves semi-local Levi-Malcev theorem for holonomy Lie algebroid.
result Formal semi-local triviality for all 2-connected and a wide class of 1-connected leaves.
Paper introduces Lie algebroid index theory and a generalized Riemann-Roch theorem.
problem Generalizing Riemann-Roch theorem for manifolds with regular foliations.
method Developed Lie algebroid index theory and applied it to obtain a generalized Riemann-Roch theorem.
result Obtained a generalized Riemann-Roch theorem for manifolds with regular foliations.