Classifies neighborhoods around specific leaf structures.
problem Classifying singular foliations with given leaf and transverse singular foliation.
method Analyzes the structure of singular foliations and their leaves.
result Developed a method to classify neighborhoods around specific leaf structures.
Survey and extend work on singular foliations in diffeology.
problem Understanding singular foliations and their properties in diffeological settings.
method Survey Stefan and Sussmann's work, introduce transverse equivalence, and define basic cohomology.
result Transverse equivalence of singular foliations preserves leaf spaces diffeologically but not conversely.
The paper resolves singular foliations through a series of blowups.
problem Singular foliations that cannot be resolved directly.
method Applying Nash modifications to the universal Lie ∞-algebroid of a singular foliation.
result Any singular foliation becomes a Debord foliation after one blowup.
Singular Riemannian Foliations are particular types of foliations on Riemannian manifolds, in which leaves locally stay at a constant distance from each other. Singular Riemannian Foliations in round spheres play a special role, since they provide "infinitesimal information" about general Singular Riemannian Foliations…
Develops Chern-Weil theory for singular foliations.
problem Chern-Weil theory for Haefliger-singular foliations.
method Constructs explicit forms representing characteristic classes in de Rham cohomology.
result Theory applies to general smooth Haefliger structures up to homotopy.
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. Survey on the topology of singular foliations in complex 2-space.
problem Understanding the topology of singular foliations in complex 2-space.
method Overview and survey of existing research.
result Overview of current knowledge on foliation singularities.
Desingularizes singular foliations with a locally compact groupoid.
problem Handling singularities in foliations.
method Blow-up construction of smooth manifolds and groupoids.
result Locally compact locally Hausdorff groupoid desingularizes singular foliations.
In this paper we study singular riemannian foliations that have sections,i.e., totally geodesic complete immersed submanifolds that meet each leaf orthogonally and whose dimensions are the codimensions of the regular leaves. We prove here that the restriction of the foliation to a slice of a leaf is diffeomorphic to an…
A singular foliation in the sense of Androulidakis and Skandalis is an involutive and locally finitely generated module of compactly supported vector fields on a manifold. An automorphism of a singular foliation is a diffeomorphism that preserves the module. In this note, we give an alternative proof of the (surprising…
Introduces a new equivalence for singular foliations and their groupoids.
problem Preserving transverse geometry in singular foliations.
method Introduces a new equivalence relation for singular foliations and connects it to holonomy groupoids.
result Establishes a connection between singular foliations and their associated holonomy groupoids.
The paper solves a 5-manifold foliation problem using a Sasaki-Ricci flow.
problem Solving foliation singularities on Sasakian 5-manifolds.
method Applying the Sasaki-Ricci flow to resolve cyclic quotient foliation singularities.
result Proves a Sasaki analogue of the analytic minimal model program.
In this paper we introduce the concept of singular Finsler foliation, which generalizes the concepts of Finsler actions, Finsler submersions and (regular) Finsler foliations. We show that if F is a singular Finsler foliation on a Randers manifold (M,Z) with Zermelo data (h,W), then $\mathcal{F}…
Investigates singular Finsler foliations on (α,β)-spaces and their relation to Riemannian foliations.
problem Understanding conditions for singular Finsler foliations to be singular Riemannian foliations.
method Analyzes (α,β)-spaces and verifies conditions for SFFs to be SRFs, extending Molino's conjecture. result Equifocality of regular leaves for SFFs under certain conditions.
Study uses blow-up method to analyze foliations in Riemannian geometry.
problem Analyzing dynamics and structure of singular Riemannian foliations.
method Blow-up desingularization method applied to Molino sheaf and leaf closures.
result Strong constraint on foliation leaves for non-vanishing Euler characteristic.
Study on cohomology of singular foliations with localization results.
problem Understanding cohomology of singular Riemannian foliations.
method Introduced equivariant basic cohomology and proved its properties.
result Equivariant basic cohomology localizes to closed leaves.
Positive curvature forces foliation leaf spaces to have boundaries.
problem Understanding boundaries in foliated leaf spaces with positive curvature.
method Analyzing singular Riemannian foliations with positive sectional curvature.
result Polar foliations of positively curved manifolds have leaf spaces with nonempty boundaries.
Classifies foliations with a hypersurface as the singular leaf and open leaves.
problem Classifying foliations with a hypersurface as the singular leaf and open leaves.
method Analyzes the transverse order k foliations, showing that a loop in the singular leaf induces a well-defined holonomy transformation.
result A complete classification of these foliations and concrete descriptions of their associated groupoids and algebras.
Lecture notes on singular foliations, smooth and holomorphic.
problem Understanding singular foliations in geometry.
method Review of foundations, recent tools from non-commutative geometry, and homotopic notions.
result Introduction of various homotopic notions and open questions.
Linearizability of singular foliations is preserved under a specific equivalence relation.
problem Preserving properties of singular foliations under equivalence relations.
method Characterization of tubular neighborhood embeddings using Euler-like vector fields.
result Linearizability along a leaf is a Morita invariant.
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.
In order to understand the linearization problem around a leaf of a singular foliation, we extend the familiar holonomy map from the case of regular foliations to the case of singular foliations. To this aim we introduce the notion of holonomy transformation. Unlike the regular case, holonomy transformations can not be…
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.
In this paper, we use the methods of subriemannian geometry to study the dual foliation of the singular Riemannian foliation induced by isometric Lie group actions on a complete Riemannian manifold M. We show that under some conditions, the dual foliation has only one leaf.
We determine the structure of the fundamental group of the regular leaves of a closed singular Riemannian foliation on a compact, simply connected Riemannian manifold. We also study closed singular Riemannian foliations whose leaves are homeomorphic to aspherical or to Bieberbach manifolds. These foliations, which we c…
In this paper we prove the conjecture of Molino that for every singular Riemannian foliation (M,F), the partition Fˉ given by the closures of the leaves of F is again a singular Riemannian foliation.
Characterizes elliptic operators on singular foliations.
problem Understanding elliptic operators on singular foliations.
method Using Nash algebroids and symplectic leaves of dual Lie algebroids.
result Characterization of longitudinally elliptic differential operators.
A singular foliation on a complete riemannian manifold M is said to be riemannian if each geodesic that is perpendicular at one point to a leaf remains perpendicular to every leaf it meets. We prove that the regular leaves are equifocal, i.e., the end point map of a normal foliated vector field has constant rank. This …
Smooth algebra analysis for one-dimensional singular foliations.
problem Analyzing smooth algebras of one-dimensional singular foliations.
method Analyzing natural ideals and using Dixmier-Malliavin theorem.
result Smooth algebras of one-dimensional singular foliations are pairwise nonisomorphic.
Explains complex analytic invariants of vector fields and foliations.
problem Integrating theories of singular varieties and foliations.
method Expository discussion of invariants.
result Introduces connections between complex analytic singular varieties and foliations.
The paper proves isomorphisms between two complexes related to singular foliations.
problem Understanding the isomorphisms between two complexes associated with singular foliations.
method Analyzing the quotient map and proving isomorphisms in specific cases.
result Isomorphisms between the complexes of differential forms on the leaf space and basic differential forms on the manifold.
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.
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.
The singular set of a foliation is always connected under certain conditions.
problem Understanding the connectedness of singular sets in foliations.
method Analyzing the normal sheaf and dimension properties of the singular set.
result The union of irreducible components of dimension k−1 in the singular set is connected. In this paper we survey on some recent results on Riemannian orbifolds and singular Riemannian foliations and combine them to conclude the existence of closed geodesics in the leaf space of some classes of singular Riemannian foliations (s.r.f.), namely s.r.f. that admit sections or have no horizontal conjugate points.…
Study of singular foliations of b^k-type and their geometric properties.
problem Classify and understand singular foliations of b^k-type.
method Introduce and study singular foliations of b^k-type, classify them, and relate them to geometric structures.
result Obstructed by a characteristic class when extending k-th order foliations to (k+1)-th order foliations.
Study on the topology of leaves in singular Riemannian foliations.
problem Characterizing the topology of leaves in singular Riemannian foliations.
method Analyzing the fundamental groups and using nilpotent spaces.
result Leaves of singular Riemannian foliations are finitely covered by nilpotent spaces when M is simply connected. Simplified proof of foliation closure theorem for linear foliations.
problem Proving the closure of linear foliations on Riemannian manifolds.
method Direct geometric approach, focusing on projectable foliations and compatible connections.
result Smoothness of the closure of linear foliations directly proven.
Defines and calculates foliation homology from flows.
problem Homology of foliations defined by flows.
method Definition and calculation of foliation homology.
result Homology naturally associated with Seifert fibration.
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.
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…
New solutions found for Yamabe problem on spheres with foliations.
problem Yamabe problem on spheres with singular Riemannian foliations.
method Variational methods, symmetries from foliations, Sobolev embedding theorem, Principle of Symmetric Criticality.
result Existence of sign-changing and positive solutions with specific symmetries.
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.
This research classifies singular foliations and finds a universal deformation.
problem Classifying singular foliations on (C2,0). method Topological universal deformation through fixed invariants.
result Every equisingular deformation uniquely factors through the topological universal deformation.
We prove that singular Riemannian foliations in Euclidean spheres can be defined by polynomial equations.
Classifies singular foliations of a specific type and studies their extensions.
problem Classifying singular foliations of a particular type and understanding their extensions.
method Introduces and classifies singular foliations of bk+1-type, proving they are encoded by k-th order foliations. result Singular foliations of bk+1-type are encoded by k-th order foliations, and these groupoids fiber over certain character stacks. The volume of a k-dimensional foliation F in a Riemannian manifold Mn is defined as the mass of image of the Gauss map, which is a map from M to the Grassmann bundle of k-planes in the tangent bundle. Generalizing a construction by Gluck and Ziller, "singular" foliations by 3-spheres are constructed on…
We study codimension one smooth foliations with Morse type singularities on closed ma-nifolds. We obtain a description of the manifold in case the number of centers in greater then the number of saddles. This result relies on and extends previous results of Reeb (for foliations having only centers) and Ells-Kuiper (for…