Study transverse Dolbeault cohomology for almost complex structures.
problem Understanding cohomology of transverse structures on manifolds.
method Define transverse Dolbeault cohomology, extend transverse complex structure, introduce involutive limit distribution.
result Cohomology spaces of (p,0) for almost complex structures coincide with transverse Dolbeault cohomology.
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.
Smooth maps in o-minimal structures are mostly transverse.
problem Transversality of smooth definable maps in o-minimal structures.
method Definable smooth version of Thom transversality theorem, proving nowhere density of non-transverse maps, and a definable version of Trotman's theorem.
result Non-transverse maps are nowhere dense in the definable smooth topology.
New transverse links solve contact manifold problems.
problem Contact manifold structures.
method Transverse links in 3-sphere contact structures.
result All contact 3-manifolds are contact branched covers over a transverse link.
Paper develops theory of transverse generalized complex structures and proves a key lemma.
problem Proving equivalent conditions to the basic ddJ-lemma. method Describing transverse symplectic structure and relating the lemma to the Lefschetz map.
result Justified approach and proved equivalent conditions to the basic ddJ-lemma. 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.
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.
Study on stability of Sasaki structures under deformations.
problem Stability of Sasaki structures under transverse holomorphic deformations.
method Analysis of transverse Kähler holonomy groups and stability properties.
result Stability of ${\oldmathcal S}$ under certain conditions on Sasaki manifolds.
The transversal twistor space of a foliation F of an even codimension is the bundle ZF of the complex structures of the fibers of the transversal bundle of F. On ZF, there exists a foliation F' by covering spaces of the leaves of F, and any Bott connection of F produces an ordered pair (I,J) of transversal almost compl…
We classify positive transversal torus knots in tight contact structures up to transversal isotopy.
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.
Engel manifolds show transverse tori can be made to have various formal invariants.
problem Understanding transverse tori in Engel manifolds.
method Analogous to transverse knots, classify formal invariants and show their uniqueness.
result Engel manifolds can have infinitely many transverse isotopy classes of tori with specific invariants.
Study classifies twist knots with maximal self-linking number in S^3.
problem Classifying transverse-universal knots in S^3.
method Classification of transverse twist knots with maximal self-linking number.
result Obtained an infinite family of non-transverse-universal knots.
Classifies Legendrian and transverse torus knots in S3.
problem Classifying Legendrian and transverse torus knots in S3. method Complete coarse classification using Legendrian and transverse properties.
result Complete classification of torus knots in contact structures on S3. Study on transverse knots and their neighborhoods, proving unique standard neighborhoods and destabilization results.
problem Understanding the structure and properties of transverse knots and their neighborhoods.
method Proves unique standard neighborhoods and structure theorems for non-loose Legendrian knots through destabilization results.
result Finds a manifold with infinite tight contact structures, up to contactomorphism, without Giroux torsion.
We characterize the oriented Seifert-fibered three-manifolds which admit positive, transverse contact structures.
New theorem allows transverse links to be braided with rational book structure.
problem Transverse isotopy of links in contact 3-manifolds.
method Generalization of Pavelescu's argument for rational open book decomposition.
result Every transverse link can be isotoped to a braid with rational open book.
Study on Legendrian and transverse realizations of negative torus knots.
problem Classification of transverse and Legendrian realizations of negative torus knots.
method Analysis of contact structures, knot Floer homology, Legendrian surgeries.
result Classification of strongly non-loose transverse and Legendrian realizations.
Reeb flow made transverse to foliations without invariant measures.
problem Making Reeb flow transverse to foliations without invariant measures.
method Leafwise Brownian motion to construct transverse measures.
result Reeb flow has no contractible orbits when transverse to foliations without invariant measures.
Hodge numbers of Sasakian manifolds remain unchanged under deformations.
problem Invariance of Hodge numbers under deformations of Sasakian manifolds.
method Analysis of deformations of Sasakian structures and use of transversely elliptic operators.
result Hodge numbers are invariant under arbitrary deformations of the Sasakian structure.
Proves immediate transversality for conic singularities.
problem Transversality issues in Morse complexes with conic singularities.
method Proves immediate transversality for conic singularities in Morse complexes.
result Immediate transversality holds for conic singularities.
We consider manifolds equipped with a foliation F of codimension 4q, and an almost quaternionic structure Q on the transversal bundle of F. After discussing conditions of projectability and integrability of Q, we study the transversal twistor space ZF which, by definition, consists of the…
Study contact structures on lens spaces, classifying rational knots.
problem Classify rational knots in lens spaces.
method One-parametric convex surface theory to classify Legendrian and transverse rational unknots.
result Determine the contact mapping class group of lens spaces.
The study provides obstructions and examples for p-symplectic structures on complex manifolds.
problem Obstructing the existence of p-symplectic structures on compact complex manifolds. method Analyzing properties of (p,p)-transverse forms and 2p-forms. result Found obstructions and examples of p-symplectic structures on compact complex manifolds. The abstract discusses transforming Vaisman metrics into Kähler-Einstein structures.
problem Transforming Vaisman metrics into Kähler-Einstein structures.
method Using the transverse Kähler-Ricci flow on the canonical foliation of a closed Vaisman manifold.
result Direct proof of short time existence of transverse Kähler-Ricci flow on Vaisman manifolds.
This is a survey article on a known generalization of Dirac-type operators to transverse operators called basic Dirac operators on Riemannian foliations, which are smooth foliations that have a transverse geometric structure. Construction of these operators requires the additional structure of what is called a bundle-l…
We study Thom Transversality Theorem using a point of view, suggested by Gromov, which allows to avoid the use of Sard Theorem and gives finer informations on the structure of the set of non-transverse maps.
We show that a transverse link in a contact structure supported by an open book decomposition can be transversely braided. We also generalize Markov's theorem on when the closures of two braids represent (transversely) isotopic links.
A transverse knot exists in S^3 that covers all contact 3-manifolds.
problem Existence of a universal transverse knot covering all contact 3-manifolds.
method Contact covering via branched map φ: M → S^3.
result There exists a transverse knot K in S^3 such that every contact 3-manifold can be obtained as a contact covering branched along K.
We study transversality for Lipschitz-Fredholm maps in the context of bounded Fréchet manifolds. We show that the set of all Lipschitz-Fredholm maps of a fixed index between Fréchet spaces has the transverse stability property. We give a straightforward extension of the Smale transversality theorem by using the general…
Simple perturbation of Vafa-Witten equations leads to transversality.
problem Transversality of Vafa-Witten moduli space.
method Simple perturbation of Vafa-Witten equations, proving transversality for SU(2) or SO(3) structure groups. result For generic perturbation parameter, the full rank part of the moduli space satisfies transversality.
A transverse knot is a knot that is transverse to the planes of the standard contact structure on real 3-space. In this paper we prove the Markov Theorem for transverse braids, which states that two transverse closed braids that are isotopic as transverse knots are also isotopic as transverse braids. The methods of the…
Paper introduces entropy for Riemannian foliations and connects it to Ricci flow.
problem Entropy functional for Riemannian foliations and its relation to Ricci flow.
method Introduced entropy functional and related its gradient flow to transverse Ricci flow.
result Entropy functional is monotonic along transverse Ricci flow and related to it.
The paper confirms a specific type of Sasakian manifold's structure.
problem Characterizing Sasakian manifolds with nonnegative transverse bisectional curvature.
method Analyzing the Sasakian analogue of Yau's uniformization conjecture.
result 5-dimensional Sasakian manifolds with positive transverse bisectional curvature are CR-biholomorphic to the standard Heisenberg group.
Characterizes Hilbert schemes and their geometric properties.
problem Understanding transverse Hilbert schemes and their geometric properties.
method Characterization through bi-Poisson structures and hyperkähler geometry.
result Characterization of transverse Hilbert schemes and description of their hyperkähler geometry.
New theorems on Hodge numbers and Kähler structures derived from complex differential forms.
problem Deformation invariance and local stability of Hodge numbers and Kähler structures.
method Using the exponential operator and power series method, the approach focuses on d-closed extensions and foliated cases. result Local stabilities of transversely p-Kähler structures and new theorems on Hodge numbers. A knot type is exchange reducible if an arbitrary closed n-braid representative can be changed to a closed braid of minimum braid index by a finite sequence of braid isotopies, exchange moves and +/- destabilizations. In the manuscript [J Birman and NC Wrinkle, On transversally simple knots, preprint (1999)] a transver…
The paper proves transversality for special Lagrangian submanifolds in a 6D manifold.
problem Counting special Lagrangian submanifolds in higher dimensions.
method Proving transversality for the moduli space of perturbed special Lagrangian submanifolds using a Lagrange multipliers problem.
result The moduli space is generically a set of isolated points.
We study Riemannian foliations whose transverse Levi-Civita connection ∇ has special holonomy. In particular, we focus on the case where Hol(∇) is contained either in SU(n) or in Sp(n). We prove a Weitzenbock formula involving complex basic forms on Kähler foliations and we apply this formula for pointing…
Consider a transverse knot which is the binding of an open book for the ambient contact manifold. In this paper, we show that the transverse invariants defined by Lisca, Ozsvath, Stipsicz, and Szabo (LOSS) are nonvanishing for such transverse knots. This is true regardless of whether or not the ambient contact structur…
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.
We prove the existence of foliations transverse to pseudo-Anosov flows using veering triangulations.
problem Existence of foliations transverse to pseudo-Anosov flows in 3-manifolds.
method Combinatorial approach via veering triangulations and branched surfaces.
result Existence of transverse foliations for certain pseudo-Anosov flows.
The article details Fukaya's A∞-structure of Morse complexes.
problem Understanding the A∞-structure of Morse complexes.
method Detailed exposition of Fukaya's A∞-structure, emphasizing transversality arguments.
result The A∞-structure is homotopically independent of choices.
Final revision. To appear in the Journal of Differential Geometry. This paper studies knots that are transversal to the standard contact structure in R3, bringing techniques from topological knot theory to bear on their transversal classification. We say that a transversal knot type $\cTK$ is {\it transversally…
Let G be a simple Lie group of real rank one, and S the ideal boundary of the corresponding symmetric space of noncompact type (H^n_R, H^n_C, H^n_H or H^2_O). We show the finiteness of the possible values of the secondary characteristic classes of transversely homogeneous foliations on a fixed manifold whose transverse…
We study the transverse Poisson structure to adjoint orbits in a complex semi-simple Lie algebra. The problem is first reduced to the case of nilpotent orbits. We prove then that in suitably chosen quasi-homogeneous coordinates the quasi-degree of the transverse Poisson structure is -2. In the particular case of {\emph…
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.
Study the geometry of transversal lightlike submanifolds in Metallic Semi-Riemannian manifolds.
problem Exploring the geometry of submanifolds in Metallic Semi-Riemannian manifolds.
method Investigate the geometry of distributions and induced connections, providing necessary and sufficient conditions for the induced connection to be metric.
result Characterization of transversal lightlike submanifolds in Metallic semi-Riemannian manifolds.