Introduces singular Finsler foliations and their properties.
problem Generalizing Finsler actions, submersions, and foliations.
method Definition and analysis of singular Finsler foliations on Randers manifolds.
result Singular Finsler foliations on Randers manifolds are singular Riemannian foliations on the Riemannian manifold induced by the metric.
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.
The paper proves properties of Finsler submanifolds and analytic maps.
problem Analyzing properties of Finsler submanifolds and their analytic maps.
method Proving properties of regular fibers of analytic maps and Finsler submersions.
result Regular fibers of an analytic map are equifocal under certain conditions.
Using the definition of entropy of a family of increasing distances on a compact metric set given in [10] we introduce a notion of Finsler entropy for smooth distributions and Stefan-Sussmann foliations. This concept generalizes most of classical topological entropy on a compact Riemannian manifold : the entropy of a f…
The study defines Finsler metrics on special surfaces and constructs geodesic currents.
problem Defining and studying Finsler metrics on specific geometric structures.
method Defined compatible Finsler distances, studied geodesics, and constructed Liouville currents.
result Constructs a Liouville current for each metric, encoding curve lengths.
We extend the notion of the geometric entropy of foliation to foliated manifolds equipped with leafwise Finsler structure. We study the relation between the geometric entropy and the topological entropy of the holonomy pseudogroup. The case of foliated manifold with leafwise Randers structure. In this case the estimate…
On the slit tangent manifold TM0 of a Finsler space (M,F) there are given some natural foliations as vertical foliation and some other fundamental foliations produced by the vertical and horizontal Liouville vector fields, see [A. Bejancu, H. R. Farran, Finsler Geometry and Natural Foliations on the Tangent Bundle…
On the slit tangent manifold of a Finsler manifold M are given the vertical and the Liouville foliations. In this paper we define some new types of vertical forms with respect to the Liouville foliation on TM^0. We define a cohomology group of TM^0 using these new forms. We prove a de Rham type theorem.
New method uses broken scattering to uniquely identify Finsler manifolds.
problem Identifying Finsler manifolds from scattering data.
method Uses broken scattering relation to compare geodesics.
result Two reversible Finsler manifolds with the same broken scattering relation are isometric.
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…
Here, a Finsler manifold (M, F) is considered with corresponding curvature tensor, regarded as 2-forms on the bundle of non-zero tangent vectors. Certain subspaces of the tangent spaces of M determined by the curvature are introduced and called k-nullity foliations of the curvature operator. It is shown that if the dim…
Study describes periodic controls in step 2 sub-Finsler problems on Carnot groups.
problem Optimal control problems on step 2 Carnot groups with convex control sets.
method Describes Casimirs and symplectic foliations; shows extremal controls are periodic.
result Extremal controls are either constant or periodic.
The nullity distributions of the two curvature tensors \, $\overast{R}$ and $\overast{P}$ of the Chern connection of a Finsler manifold are investigated. The completeness of the nullity foliation associated with the nullity distribution NR∗ is proved. Two counterexamples are given: the first shows that $\N_{R…
The paper explores traveling along broken geodesics in Finsler submersions.
problem Analyzing the attainable sets of analytic vector fields in Finsler submersions.
method Investigates the dual leaves and attainable sets of horizontal broken geodesics.
result Proves that in compact Finsler manifolds with positive flag curvature, the attainable sets coincide with orbits.
The abstract develops weighted Ricci curvature in Lorentz-Finsler geometry and extends singularity theorems.
problem Extending singularity theorems in weighted Lorentz-Finsler geometry.
method Generalizing Jacobi, Riccati, and Raychaudhuri equations; applying generalized Bishop inequality.
result Weighted Lorentz-Finsler singularity theorems extended.
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.
Study of perimeter measures in Heisenberg group with sub-Finsler metric.
problem Isoperimetric problem in sub-Finsler Heisenberg group.
method Reduction of Minkowski content to Lebesgue surface area, study of Finsler normed planes, use of CC-geodesics.
result Evidence supports Pansu's conjecture in sub-Finsler case, but with lower isoperimetric ratio.
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.
Paper examines conditions for singular square metrics to have constant curvature.
problem Conditions for constant curvature in singular square metrics.
method Analyzes Finsler metrics, introduces singular square metrics, provides necessary and sufficient conditions.
result Necessary and sufficient conditions for constant Ricci or flag curvature in singular square metrics.
Alternative proof of automorphism property of singular foliations.
problem Understanding the automorphisms of singular foliations.
method Alternative proof using exponential of elements in singular foliations.
result Time-one flow of elements in singular foliations is an automorphism.
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.
We study singularities of geodesics flows in two-dimensional generalized Finsler spaces (pseudo-Finsler spaces). Geodesics are defined as extremals of a certain auxiliary functional whose non-isotropic extremals coincide with extremals of the action functional. This allows to consider isotropic lines as (unparametrized…
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. We study extrinsic geometry of a codimension-one foliation F of a closed Finsler space (M,F), in particular, of a Randers space (M,α+β). Using a unit vector field ν orthogonal (in the Finsler sense) to the leaves of F we define a new Riemannian metric g on M, which for Randers case depends n…
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…
Local models and Lie groupoids for Riemannian foliations.
problem Understanding the geometry of Riemannian foliations locally.
method Construction of Lie groupoids and algebroids controlling foliation transverse geometry.
result Local models and Lie groupoids provide control over foliation geometry.
New equivalence for singular foliations preserves transverse geometry.
problem Transverse geometry of singular foliations.
method Introducing a new notion of equivalence for singular foliations and showing compatibility with holonomy groupoids.
result Unified invariants and equivalence of singular foliations.
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.
3D foliation study finds Poisson structure with specific singularities.
problem Characterizing Poisson structures on Bott-Morse foliations in 3D.
method Analyzes Bott-Morse foliations, computes Poisson bivectors and symplectic forms.
result Linear, singular Poisson structure of rank 2 with Bott-Morse singularities found.
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.
In this paper, we identify the Bott connection on the natural foliation of the projective sphere bundle of a Finsler manifold to the Chern connection of this manifold. As a consequence, the symmetrization of the Bott connection turns out to be the Cartan connection of the Finsler manifold. Following Liu-Zhang \cite{Liu…
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.
Formula for foliations' singularities in complex projective spaces.
problem Counting singularities of foliations on complex projective spaces.
method Global residue formula for logarithmic indices of foliations with isolated singularities.
result Formula for the number of singularities in the complement of the invariant divisor on complex projective spaces.
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.
New method for integrating singular foliations via paths.
problem Characterize holonomy and fundamental groupoids of singular foliations.
method Quotient of an infinite dimensional space of paths, extending classical construction for regular foliations.
result Characterization of holonomy and fundamental groupoids of singular foliations.
Study on MCF of foliations, focusing on singular cases.
problem Investigate mean curvature flow of foliations with singularities.
method Generalizes previous results on MCF of foliations, focusing on singular cases.
result Finite time singularities are singular leaves of type I under bounded curvature conditions.
Homotopy theory applied to singular foliations leads to new results.
problem Existence and uniqueness of universal L∞-algebroids for singular foliations. method Applied homotopy theory to left semi-model categories and L∞-algebroids. result Recovery of results similar to Laurent-Gengoux and al. about universal L∞-algebroids. 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.
Local study of foliation deformation cohomology.
problem Understanding deformations of singular foliations.
method Introducing and studying local deformation cohomology.
result Local deformation cohomology for singular foliations and related structures.
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…