In this note, we discuss the interactions between differential topology and isoparametric foliations, surveying some recent progress and open problems.
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48 results for “foliated differential topology”
Geoffrey Martin's theorem proves normal forms for Lagrangian submanifolds in multisymplectic geometry.
problem Normal forms for Lagrangian submanifolds in multisymplectic geometry.
method Detailed, self-contained proof of normal form theorem, including necessary results in foliated differential topology.
result Geoffrey Martin's theorem provides a normal form for Lagrangian submanifolds in multisymplectic geometry.
Survey uses Milnor fibrations to classify first integrals of differential systems.
problem Classifying first integrals of differential systems using geometric-topological methods.
method Utilizing Milnor fibrations and connections with harmonic morphisms to provide topological and geometric descriptions.
result Geometric-topological classifications of first integrals for both isolated and non-isolated singularities.
Minimal hypersurfaces in spheres generated by isoparametric foliations are found.
problem Existence of minimal hypersurfaces in spheres generated by isoparametric foliations.
method Generalized rotational ansatz formed by the union of homothetic copies of isoparametric leaves, reducing the minimal surface equation to an ordinary differential equation.
result Closed embedded minimal hypersurfaces of topological type are found for any isoparametric hypersurface .
A foliation on a manifold M can be informally thought of as a partition of M into injectively immersed submanifolds, called leaves. In this thesis we study foliations whose leaves carry some specific geometric structures. The thesis consists of two parts. In the first part we classify foliations on open manifolds whose…
In this paper, we are concerned with interactions between isoparametric theory and differential topology. Two foliations are called equivalent if there exists a diffeomorphism between the foliated manifolds mapping leaves to leaves. Using differential topology, we obtain several results towards the classification probl…
New cohomological invariants defined for foliations.
problem Understanding cohomology of foliations under different metrics.
method Defining and analyzing basic and antibasic cohomologies, using Laplace operator.
result Antibasic cohomology groups are invariant under diffeomorphisms and metric changes.
The author studies regions foliated by 1D families of functions and their applications.
problem Understanding regions represented as foliated forms and natural smooth maps onto them.
method Investigates natural smooth maps respecting canonical projections and moment maps, focusing on foliated regions.
result Discusses the 1st derivative of functions and critical sets in foliated regions.
We construct infinite sequences of pseudo-Anosov homeomorphisms without fixed points and leaving invariant a sequence of orientable measured foliations on the same topological surface and the same stratum of the space of abelian differentials. The existence of such sequences show that all pseudo-Anosov homeomorphisms f…
This research classifies singular foliations and finds a universal deformation.
problem Classifying singular foliations on .
method Topological universal deformation through fixed invariants.
result Every equisingular deformation uniquely factors through the topological universal deformation.
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.
Constructing exponential families from statistical manifolds.
problem The central problem of constructing exponential families from statistical manifolds.
method Constructive approach proving every compact statistical manifold admits a foliation of Hessian manifolds.
result Compact orientable leaves are either finite quotients of flat torus or mapping torus with periodic monodromy.
We study the asymptotic geometry of Teichmueller geodesic rays. We show that when the transverse measures to the vertical foliations of the quadratic differentials determining two different rays are topologically equivalent, but are not absolutely continuous with respect to each other, then the rays diverge in Teichmue…
Proves a Thom isomorphism for foliated differential forms.
problem Transverse Lie algebra actions on foliated manifolds and vector bundles.
method Thom isomorphism theorem for differential forms in foliated settings.
result Established a new Thom isomorphism theorem.
We introduce a -valued invariant of a foliated manifold with a stable framing and with a partially flat vector bundle. This invariant can be expressed in terms of integration in differential -theory, or alternatively, in terms of -invariants of Dirac operators and local correction terms. In…
The paper studies Liouville structures for taut foliations and Anosov flows, proving their topological invariance.
problem Understanding the topological invariance of Liouville structures for taut foliations and Anosov flows.
method Combining smoothing schemes for topological conjugacies and a refinement of Vogel's uniqueness result.
result Liouville structures are topological invariants of taut foliations and orbit equivalent Anosov flows.
Kotschick and Morita recently discovered factorisations of characteristic classes of transversally symplectic foliations that yield new characteristic classes in foliated cohomology. We describe an alternative construction of such factorisations and construct examples of topologically trivial foliated vector bundles fo…
Study constructs transverse metrics using transformations commuting with elliptic operators.
problem Existence of transverse metrics in foliation theory.
method Applying the Average Method to construct a transverse metric.
result Pseudogroup of local transformations equicontinuous and quasi-analytic.
Study shows conditions for continuity of foliated homeomorphisms action on space of leaves.
problem Conditions for continuity of foliated homeomorphisms action on space of leaves.
method Investigated sufficient conditions for continuity of the homomorphism ψ: H(X, Δ) → H(Y) induced by the action of foliated homeomorphisms on the space of leaves.
result Similar results hold for a more general class of partitions of locally compact Hausdorff spaces.
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 is simply connected.
Defines Milnor number for foliations and shows its topological invariance.
problem Defining and proving invariance of Milnor number for non-isolated singularities of holomorphic foliations.
method Defining Milnor number as intersection number of sections; proving invariance via topological equivalences.
result Milnor number is invariant under topological equivalences.
Study topological properties of foliations induced by closed 1-forms on orbifolds.
problem Characterize the topology of foliation leaves induced by closed 1-forms on orbifolds.
method Establish criteria for the compactness of foliation leaves and extend a topological result to orbifolds.
result Criteria for the compactness and coexistence of foliation leaves are established.
The topological Molino's description of equicontinuous foliated spaces, studied by the first author and Moreira Galicia, gives conditions to reduce their study to the particular case where the holonomy pseudogroup can be represented by a pseudogroup on some local group generated by some of its local left translatio…
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…
Finite intersection numbers between horizontal foliations of quadratic differentials.
problem Intersection properties of horizontal foliations in quadratic differentials.
method Joint continuity of intersection number in -norm.
result Intersection number is finite and jointly continuous.
The current article studies certain problems related to complex cycles of holomorphic foliations with singularities in the complex plane. We focus on the case when polynomial differential one-form gives rise to a foliation by Riemann surfaces. In this setting, a complex cycle is defined as a nontrivial element of the f…
Study of foliations' geometric and topological structures.
problem Analyzing the geometric and topological properties of transversely affine foliations.
method Attach holonomy group and quotient stack, identify reparametrisations, classify them, and study the Kato-Nakayama space.
result Holonomy group controls the geometric part, while the Kato-Nakayama space captures the topological and dynamical aspects.
A meromorphic quadratic differential on a punctured Riemann surface induces horizontal and vertical measured foliations with pole-singularities. In a neighborhood of a pole such a foliation comprises foliated strips and half-planes, and its leaf-space determines a metric graph. We introduce the notion of an asymptotic …
This work deals with the topological classification of germs of singular foliations on . Working in a suitable class of foliations we fix the topological invariants given by the separatrix set, the Camacho-Sad indices and the projective holonomy representations and we compute the moduli space of topo…
It is known that all but finitely many leaves of a measured foliated 2-complex of thin type are quasi-isometric to an infinite tree with at most two topological ends. We show that if the foliation is cooriented, and the associated R-tree is self-similar, then a typical leaf has exactly one topological end. We also cons…
The paper extends Riemann-Hilbert correspondence to foliations.
problem Understanding representations of Lie algebroids and groupoids in foliated settings.
method Establishing an de Rham theorem and constructing an integration functor.
result An equivalence between -representations of -algebroids and -representations of Lie -groupoids for foliations.
The study characterizes infinite Riemann surfaces and their foliations using quadratic differentials.
problem Characterizing infinite Riemann surfaces and their foliations using quadratic differentials.
method Extending Hubbard-Masur theorem to infinite surfaces and analyzing Jenkins-Strebel differentials.
result Density of Jenkins-Strebel differentials and extension of Kerckhoff's formula for Teichmüller metric.
The paper studies Hodge structures on contact manifolds and their cohomology.
problem Analyzing Hodge structures transversal to Reeb foliations.
method Applying general results about Hodge structures to contact forms and Reeb vector fields.
result Differential complexes of basic forms are canonically isomorphic under certain conditions.
We consider hyperbolic and partially hyperbolic diffeomorphisms on compact manifolds. Associated with invariant foliation of these systems, we define some topological invariants and show certain relationships between these topological invariants and the geometric and Lyapunov growths of these foliations. As an applicat…
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.
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.
New findings on how foliations and Lie group actions interact.
problem Understanding interactions between foliations and Lie group actions.
method Functorial properties of holonomy groupoids and quotient manifolds.
result Holonomy groupoids of quotients by Lie group actions exhibit similar properties.
The present paper is a continuation of [13], [14] of the authors. Specifically, the paper considers the MD5-foliations associated to connected and simply connected MD5-groups such that their Lie algebras have 4-dimensional commutative derived ideal. In the paper, we give the topological classification of all considered…
The paper studies affine manifolds with linear foliations and their topological properties.
problem Characterizing the topology of affine manifolds with linear foliations.
method Analyzing the structure of affine manifolds and their foliations, proving topological properties.
result Compact affine manifolds with linear foliations are homeomorphic to the torus under certain conditions.
A meromorphic quadratic differential with poles of order two, on a compact Riemann surface, induces a measured foliation on the surface, with a spiralling structure at any pole that is determined by the complex residue of the differential at the pole. We introduce the space of such measured foliations, and prove that f…
We give a complete topological classification of germs of holomorphic foliations in the plane under rather generic conditions. The key point is the introduction of a new topological invariant called monodromy representation. This monodromy contains all the relevant dynamical information, in particular the projective ho…
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.
Foliations on modular curvesmath.NT
It is proved, that a foliation on a modular curve given by the vertical trajectories of holomorphic differential corresponding to the Hecke eigenform is either the Strebel foliation or the pseudo-Anosov foliation.
We describe the space of measured foliations induced on a compact Riemann surface by meromorphic quadratic differentials. We prove that any such foliation is realized by a unique such differential if we prescribe, in addition, the principal parts of at the poles. This generalizes a theorem of Hubbard and …
Minimal hyperbolic foliations on 3-manifolds have non-simply connected generic leaves.
problem Characterizing surfaces with minimal hyperbolic foliations.
method Analyzing codimension one foliations on closed 3-manifolds.
result Noncompact surfaces satisfying a specific condition are homeomorphic to the leaf of a minimal foliation with non-simply connected generic leaf.
We investigate compact Hausdorff foliations on compact Riemannian manifolds in the context of the Gromov-Hausdorff distance theory. We give some sufficient conditions for such foliations to be separated in the Gromov-Hausdorff topology.
Piecewise Euclidean structures (identified solid Euclidean polyhedra) on topological 3-dimensional manifolds and pseudo-manifolds are constructed so that they admit pseudo-foliations, a generalized type of foliation. The construction of non-manifold point neighborhoods is done to preserve as much of the geometric, and …
A k-submanifold L of an open n-manifold M is called weakly integrable (WI) [resp. strongly integrable (SI)] if there exists a submersion Φ:M\to R^{n-k} such that L\subset Φ^{-1}(0) [resp. L= Φ^{-1}(0)]. In this work we study the following problem, first stated in a particular case by Costa et al. (Invent. Math. 1988): …