Study of affine and projective structures on foliated complex manifolds.
problem Formalizing and analyzing affine and projective structures on foliations.
method Formalizing concepts, providing local normal forms, proving index formulae, classifying structures.
result Compact algebraic manifolds of even dimension do not admit foliated projective structures.
No projective structure found on foliations of elliptic curves.
problem Existence of transversely projective structures on foliations.
method Analyzing foliations on the product of two elliptic curves.
result Generic nonsingular turbulent foliations do not have transversely projective structures.
The study examines conditions for Poisson structures on orientable manifolds with specific foliations.
problem Conditions for the existence of Poisson structures on orientable manifolds with given foliations.
method Analyzes necessary and sufficient conditions for the existence of Poisson structures with a specified characteristic foliation.
result Describes obstructions and conditions for unimodular and transversally constant Poisson structures.
Paper proves h-principles for symplectic structures and foliations.
problem Existence of conformal symplectic structures and foliations.
method Application of h-principles and foliated Morse theory.
result Linear deformation of foliations to contact structures.
The study of tautness in foliations across different smoothness classes.
problem Understanding tautness in various smoothness classes of foliations.
method Examining tautness in C0, C1,0, and C∞,0 foliations, constructing examples. result Smoothly taut C∞,0 foliations can be approximated by various contact structures. Study on foliation automorphisms, finding non-Lie groups and ILH Lie groups.
problem Understanding the structure of diffeomorphism groups of foliations.
method Investigation of diffeomorphism groups of foliations, proving properties of automorphism groups.
result Found examples of foliations with non-Lie automorphism groups and proved properties of ILH Lie groups for certain foliations.
According to a theorem of Eliashberg and Thurston a C2-foliation on a closed 3-manifold can be C0-approximated by contact structures unless all leaves of the foliation are spheres. Examples on the 3-torus show that every neighbourhood of a foliation can contain non-diffeomorphic contact structures. In this paper …
Develops deformation theory for symplectic foliations using L∞-algebras.
problem Deformation of symplectic foliations.
method Uses L∞-algebras to control deformation problems. result Establishes a correspondence between small deformations and Maurer-Cartan elements of L∞-algebra. Study simplifies classification of foliations with specific geometric structures.
problem Classifying foliations with transverse similarity structures.
method Conceptual approach and proof of important results.
result Holonomy classification of Weyl structures on compact manifolds.
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.
Survey on holomorphic structures on complex manifolds.
problem Holomorphic foliations with transverse holomorphic Cartan geometries.
method Analyzes G-structures and Cartan geometries on compact complex manifolds.
result Foliated holomorphic Cartan geometries on compact complex manifolds.
The paper shows how to approximate foliations by contact structures on 3-manifolds.
problem Approximating foliations by contact structures on 3-manifolds.
method Showing that any co-orientable foliation can be C0-approximated by contact structures. result Existence of taut C0-foliations implies universally tight contact structures. The study examines attractors in Cartan foliations and their properties.
problem Existence of attractors in Cartan foliations.
method Reduction of the problem to the action of Lie groups and holonomy groups.
result Conditions for the existence of attractors in reductive Cartan foliations.
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.
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…
3D foliations can be approximated by contact structures.
problem Approximating taut, transversely oriented foliations in 3-manifolds.
method Showing C0 approximations of taut foliations by contact structures. result Taut foliations not product foliations can be approximated by contact structures.
The paper introduces new metric structures on g-foliations and uses a flow to deform them.
problem Developing new flexible metric structures on g-foliations. method Introducing new metric structures and using the partial Ricci flow to deform them.
result Deformation retraction of new structures with positive partial Ricci curvature onto classical structures.
Study how pairs of 1D foliations can be deformed into contact structures.
problem Understanding deformations of pairs of 1D foliations.
method Linear deformations of pairs of codimension one foliations into contact pairs.
result Main result provides applications and insights into foliation deformations.
We extend the Eliashberg-Thurston theorem on approximations of taut oriented C2-foliations of 3-manifolds by both positive and negative contact structures to a large class of taut oriented C1,0-foliations, where by C1,0 foliation, we mean a foliation with continuous tangent plane field. These C1,0-fol…
The paper studies a new structure on contact manifolds and its foliation properties.
problem Investigating a new structure on contact manifolds.
method Examining weak nearly Sasakian structures and their foliations.
result The weak nearly Sasakian structure foliates into two types of totally geodesic foliations.
Study the structure of Kähler foliations with negative Ricci curvature.
problem Characterize the structure of Kähler foliations with negative Ricci curvature.
method Prove a de Rham type theorem decomposition on the leaf space.
result Characterize each factor in the decomposition of the leaf space.
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.
We show the equivalence of several notions in the theory of taut foliations and the theory of tight contact structures. We prove equivalence, in certain cases, of existence of tight contact structures and taut foliations.
Proof of contact structure from taut foliation for certain knots.
problem Contact structures from taut foliations for specific knots.
method Alternative proof using pseudo-Anosov monodromy and fractional Dehn twist coefficient.
result Fibered knots with certain properties support contact structures that are taut foliation perturbations.
The study characterizes pseudo-Riemannian foliations and their graphs.
problem Characterizing pseudo-Riemannian foliations and their graphs.
method Analyzing geodesics and metrics on foliation graphs.
result A unique pseudo-Riemannian metric exists on foliation graphs.
The study explores conformal symplectic foliations on closed manifolds, proving their existence in dimensions 5 and above.
problem Existence of conformal symplectic foliations on closed manifolds.
method Analysis of symplectic and conformal symplectic codimension-one foliations on closed manifolds of dimension at least 5.
result Construction of conformal symplectic foliations on closed, simply-connected, almost contact manifolds in dimension 5.
The study introduces new foliations and structures on complex manifolds.
problem Understanding transverse Kähler structures on complex manifolds.
method Introducing holomorphic foliations and developing differential graded and bigraded algebras.
result Obtains quasi-isomorphic complexes to de Rham and Dolbeault complexes, similar to compact Kähler manifolds.
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.
New surgery method for foliated spheres preserves key numbers.
problem Creating groups from foliated sphere concordance classes.
method Define surgery procedure, find obstruction, use surgery to preserve characteristic numbers.
result Groups structure on concordance classes of foliated spheres.
We prove h-principle for locally conformal symplectic foliations and contact foliations on open manifolds. We interpret the result on h principle of contact foliations in terms of the regular Jacobi structures.
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.
Study Godbillon-Vey class for regular Jacobi foliations.
problem Characterizing foliations in Jacobi manifolds.
method Explicitly defined and computed Godbillon-Vey class for regular foliations.
result Expressed Godbillon-Vey class in terms of Jacobi structures.
Study on Riemannian foliations, decomposing cohomology based on J structure.
problem Decomposition of basic cohomology for specific Riemannian foliations.
method Analysis of codimension four taut Riemannian foliations using J structure.
result Obtained estimates for the dimension of basic J-anti-invariant subgroup.
Research proves the connectedness of classifying spaces for Haefliger structures.
problem Existence and properties of foliations and diffeomorphism groups.
method Analyzes the connectedness of classifying spaces for Haefliger structures.
result Proves (2q−1)-connectedness for framed Haefliger structures. New foliations constructed from contact pairs, revealing flexible taut foliations.
problem Characterizing taut foliations in 3D.
method Construction of codimension-one foliations from pairs of contact structures in 3D.
result Foliations constructed are taut, providing new insights into the L-space conjecture.
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 …
The aim of this paper is to extend the notion of commutativity of vector fields to the category of singular foliations, using Nambu structures, i.e. integrable multi-vector fields. We will classify the relationship between singular foliations and Nambu structures, and show some basic results about commuting Nambu struc…
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.
Study on symplectic Dirac operators on foliations, estimating eigenvalues.
problem Estimating eigenvalues of transversely symplectic Dirac operators.
method Analysis of transversely symplectic structures and use of Weitzenbock formula.
result Estimation of lower bounds for eigenvalues of transversely symplectic Dirac operators.
Study jets of flat partial connections in foliations.
problem Characterize and understand flat partial connections in foliations.
method Define and apply jets to flat partial connections in smooth foliations and locally free sheaves, focusing on codimension one and arbitrary codimension foliations.
result Define and apply jets to characterize transversely affine and projective structures in foliations.
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.
A regular Poisson manifold can be described as a foliated space carrying a tangentially symplectic form. Examples of foliations are produced here that are not induced by any Poisson structure although all the basic obstructions vanish.
The space of broken hyperbolic structures generalizes the Teichmüller space of a punctured surface, and the space of projectivized broken measured foliations (equivalently, the space of projectivized affine foliations) generalizes the space of projectivized measured foliations. Just as projectivized measured foliations…
Constructing a conformal product structure on S3 using the Reeb foliation.
problem First example of conformal product structure on compact simply connected manifold.
method Reeb foliation
result Conformal product structure on S3 Proves geometric invariance of signature and cohomology for Riemannian foliations.
problem Defining and proving invariance of geometric invariants for Riemannian foliations.
method Analyzes basic signature and Lichnerowicz cohomology under homotopy equivalence.
result Foliated homotopy invariance of basic signature and cohomology.
Introduces zebra structures on surfaces for directional foliation.
problem Finding canonical representatives of homotopy classes on surfaces.
method Introduces zebra structures and uses triangulations with edge connections to prove existence.
result Canonical representatives exist for homotopy classes on surfaces with specific triangulations.
Study foliations in PSL(4,R)-Teichmüller theory, proving two invariant foliations.
problem Structure of domains of discontinuity in PSL(4,R)-Hitchin representations.
method Detailed study and proof of foliations by projective line segments and convex domains.
result Foliated component has exactly two group-invariant foliations.
Let M be a fibered 3-manifold with multiple boundary components. We show that the fiber structure of M transforms to closely related transversely oriented taut foliations realizing all rational multislopes in some open neighborhood of the multislope of the fiber. Each such foliation extends to a taut foliation in t…