The paper examines a modified Godbillon-Vey class for Reeb foliations and finds it non-trivial for some foliations.
problem Characterizing foliations using the modified Godbillon-Vey class.
method Defined and analyzed the modified Godbillon-Vey class for Reeb foliations.
result The modified Godbillon-Vey class can distinguish non-diffeomorphic foliations and is non-trivial for some foliations.
New problems defined for maps onto graphs related to Reeb spaces.
problem Constructing explicit functions that yield specific graphs as Reeb spaces.
method Defining new classes of maps onto graphs and considering problems for these classes.
result New insights and challenges in constructing functions for specific Reeb spaces.
The study shows infinitely many Reeb orbits on star-shaped hypersurfaces with growth rate like prime numbers.
problem Growth rate of Reeb orbits on star-shaped hypersurfaces.
method Analyzing fiberwise star-shaped hypersurfaces in cotangent bundles with topological conditions.
result The number of Reeb orbits with period at most T grows at least like T/log(T).
I describe a general scheme which associates conjugacy classes of tori in the contactomorphism group to transverse almost complex structures on a compact contact manifold. Moreover, to tori of Reeb type whose Lie algebra contains a Reeb vector field one can associate a Sasaki cone. Thus, for contact structures of K-con…
Study of special subgroups of automorphism groups of Kronrod-Reeb graphs for Morse functions on 2-torus.
problem Characterizing subgroups of automorphism groups of Kronrod-Reeb graphs.
method Analysis of diffeomorphisms preserving Morse functions on 2-torus.
result Full description of special classes of automorphism groups.
New contact structures detected by contact homology.
problem Detecting pseudo-Anosov flows in contact structures.
method Introducing pseudo-Anosov contact structures and using contact homology.
result Contact homology detects pseudo-Anosov flows and contact structures properties.
Unimodularity criteria for Poisson structures on foliated manifolds are established.
problem Understanding unimodularity in Poisson structures on foliated manifolds.
method Explicit formula for bigraded decomposition of modular vector fields and analysis of the Reeb class.
result Unimodularity of transverse Poisson structure is a necessary condition for semilocal unimodularity.
We give a dynamical characterisation of odd-dimensional balls within the class of all contact manifolds whose boundary is a standard even-dimensional sphere. The characterisation is in terms of the non-existence of short periodic Reeb orbits.
New method realizes planar graphs as Reeb graphs of algebraic functions.
problem Realizing planar graphs as Reeb graphs of algebraic functions.
method Generic embedding and elementary procedures.
result Generically embedded planar graphs are homeomorphic to Reeb graphs of algebraic functions.
The study finds a special type of smooth function on connected sums of manifolds.
problem Finding smooth functions that are Morse on preimages of non-extrema values.
method Investigates internally Morse (I-Morse) and neat with respect to Reeb graph (N-Reeb) functions.
result Constructs an IN-Morse-Reeb function on a connected sum of given manifolds.
No complete Einstein hypersurfaces found in a specific type of Sasakian manifold.
problem Existence of Einstein hypersurfaces in Sasakian manifolds.
method Non-existence proof for complete, Einstein hypersurfaces tangent to the Reeb vector field.
result No complete Einstein hypersurfaces found in the specified Sasakian manifold.
New proof for surface groups using Reeb graphs and Morse functions.
problem Classifying epimorphisms onto free groups for surface groups.
method Constructing a correspondence between epimorphisms and systems of hypersurfaces, using Morse functions and Reeb graphs.
result Any epimorphism onto a free group for a surface can be represented by a Reeb epimorphism.
The paper uses symplectic homology to study 3D Besse manifolds with vanishing first Chern class.
problem Identifying 3D Besse manifolds with vanishing first Chern class.
method Computing the first Chern class, analyzing periodic Reeb orbits, and using symplectic homology.
result Classifies 3D Besse manifolds with vanishing first Chern class.
The Weinstein conjecture is extended to a new class of manifolds.
problem Existence of Reeb 2-curves in locally conformally symplectic manifolds.
method Extended Gromov-Witten theory and elliptic curve counts.
result Partial verification of the conjecture in higher dimensions.
Deformations of the Reeb flow of a Sasakian manifold as transversely Kähler flows may not admit compatible Sasakian metrics anymore. We show that the triviality of the (0,2)-component of the basic Euler class characterizes the existence of compatible Sasakian metrics for given small deformations of the Reeb flow as tra…
The study finds at least two closed orbits for Reeb flows on certain contact manifolds.
problem Existence of at least two closed orbits for Reeb flows on contact manifolds.
method Analysis of equivariant symplectic homology and contact finite quotients.
result Existence of at least two geometrically distinct closed orbits under specified conditions.
The study finds multiple closed Reeb orbits on specific contact forms.
problem Finding geometrically distinct closed Reeb orbits on prequantization bundles.
method Analyzes contact forms on prequantization bundles with specific index requirements.
result Establishes multiplicity results for closed Reeb orbits under certain conditions.
New method constructs smooth functions for given Reeb graphs.
problem Construct smooth functions with specific Reeb graphs.
method Explicit construction on 3D closed manifold, new ideas for surfaces.
result Smooth function on 3D manifold induces Reeb graph isomorphic to given graph.
Sharp systolic inequality for invariant tight contact forms on S1-bundles over S2.
problem Finding the shortest period of closed Reeb orbits on invariant tight contact forms.
method Proving a sharp systolic inequality based on the Euler class of the bundle.
result A behavior of the systolic inequality depends on the Euler class of the bundle.
Study of fold maps and Reeb spaces via surgery operations.
problem Constructing explicit fold maps for complex manifolds.
method Surgery operations on fundamental fold maps to study Reeb spaces.
result Detailed study of cohomology rings of Reeb spaces for specific cases.
The paper extends previous work on Reeb graphs of smooth functions on 3D manifolds to non-orientable cases.
problem Extending the understanding of Reeb graphs to non-orientable 3D manifolds.
method Constructing smooth functions on non-orientable 3D manifolds whose Reeb graphs match prescribed graphs.
result Explicit construction of smooth functions on non-orientable 3D manifolds with prescribed Reeb graphs.
New techniques reveal tight contact manifolds with vanishing contact homology.
problem Understanding closed tight contact manifolds with vanishing contact homology.
method Developed algebraic tools and techniques to study holomorphic curves in surgery cobordisms.
result First known examples of closed tight contact manifolds with vanishing contact homology.
We show that, for any regular Poisson manifold, there is an injective natural linear map from the first leafwise cohomology space into the first Poisson cohomology space which maps the Reeb class of the symplectic foliation to the modular class of the Poisson manifold. The Riemannian interpretation of those classes wil…
The paper studies branched surfaces and their properties.
problem Global topologies of branched surfaces and their maps.
method Explicit construction of maps and study of topological properties.
result New insights into the geography of branched surfaces and their maps.
Optimal Reeb graphs identified for polygon decomposition.
problem Investigating the topological structure of planar polygon decomposition.
method Using oriented Reeb graphs with a marked vertex for height functions.
result Described all possible optimal Reeb graphs for specific polygon configurations.
The paper characterizes contact 3-manifolds with closed Reeb orbits.
problem Characterizing contact 3-manifolds with closed Reeb orbits.
method Analyzing the action spectrum and minimal periods of Reeb orbits.
result A contact form with an action spectrum of rank 1 is uniquely determined by the minimal periods of its closed Reeb orbits.
Study on real hypersurfaces in Kaehler manifolds with isometric Reeb flow.
problem Characterizing real hypersurfaces with isometric Reeb flow.
method Investigation of structure and classification in Kaehler manifolds.
result Classification of real hypersurfaces in irreducible Hermitian symmetric spaces.
Characterizes smooth functions on manifolds with simple Reeb spaces.
problem Understanding the structure of Reeb spaces for smooth functions.
method Analyzes smooth functions on closed manifolds to determine their Reeb spaces' structure.
result Characterizes smooth functions whose Reeb spaces are finite graphs.
The paper constructs contact-hyperbolic manifolds with large automorphism groups.
problem Finding contact-hyperbolic manifolds with large automorphism groups.
method Holomorphic contact structures, pseudometrics, and symplectic quotients.
result Explicit examples of contact-hyperbolic contact manifolds are constructed.
Explains how Reeb dynamics connects to topology.
problem Understanding the relationship between Reeb dynamics and topology.
method Contact cuts, lifting group actions, and other construction methods.
result Methods for constructing Reeb flows with finite periodic orbits.
The paper generalizes Reeb spaces for special generic maps and lifts smooth functions.
problem Constructing lifts of smooth maps, especially Morse functions.
method Defining and generalizing quotient maps onto Reeb spaces of special generic maps and constructing lifts.
result Lifts of Morse functions can be constructed using the generalized maps.
A new stable edit distance for Reeb graphs is shown to be universal.
problem Stability and comparability of Reeb graphs under function similarity.
method Defined and proved stability and universality of Reeb graph edit distance.
result Reeb graph edit distance is the most stable and universal among distances.
Paper connects dynamics of mechanical systems to Reeb dynamics.
problem Understanding dynamics in mechanical systems with Poisson structures.
method Using Jacobi bundle metrics and linear Poisson structures.
result Extends classical results on Reeb dynamics to mechanical systems.
Study on characteristic classes for codimension-one foliations, showing non-triviality under specific conditions.
problem Characterizing non-trivial characteristic classes for codimension-one foliations.
method Using Gelfand-Fuchs cohomology and dynamical properties of holonomy groups.
result Non-triviality of certain characteristic classes for Reeb foliations, contradicting classical theorems.
Study on knots and dynamics on three-sphere, linking bounds, and upper action bounds.
problem Existence and properties of Reeb orbits on three-sphere with torus knots.
method Knot filtered embedded contact homology and obstructions to exact symplectic cobordisms.
result Upper bound on mean action of periodic orbits and linking bounds for Reeb orbits.
New method computes contact invariants for non-simply connected Gorenstein toric contact manifolds.
problem Computing contact invariants for non-simply connected Gorenstein toric contact manifolds.
method Using moment map data and Conley-Zehnder index to compute invariants for non-contractible periodic Reeb orbits.
result Explicit computation of contact invariants for specific examples of Gorenstein contact lens spaces.
Isomorphic cosymplectomorphism groups imply diffeomorphic manifolds.
problem Determining when cosymplectic manifolds are diffeomorphic based on their cosymplectomorphism groups.
method Characterized Reeb flow, used to descend isomorphism to symplectic base manifolds, preserved monodromy class ensuring bundle equivalence.
result Isomorphic cosymplectomorphism groups imply diffeomorphic manifolds.
Study obstructions to conformally Anosov Reeb flows on 3-manifolds.
problem Existence of conformally Anosov Reeb flows on 3-manifolds.
method Geometric and topological obstructions, Riemannian conditions.
result Obtained obstructions and conditions for the existence of conformally Anosov Reeb flows.
The paper characterizes Einstein metrics in Kenmotsu manifolds using specific soliton types.
problem Characterizing Einstein metrics in Kenmotsu manifolds using specific soliton types.
method Proving properties of Kenmotsu metrics as η-Ricci solitons and gradient η-Ricci solitons. result Kenmotsu metrics as η-Ricci solitons are Einstein if certain conditions are met. Let X be a simplicial complex with a piecewise linear function f:X→R. The Reeb graph Reeb(f,X) is the quotient of X, where we collapse each connected component of f−1(t) to a single point. Let the nodes of Reeb(f,X) be all homologically critical points where any homology of the corresponding c…
Researchers prove any graph can be realized as Reeb graph, linking it to manifold properties.
problem Realizing graphs as Reeb graphs on manifolds.
method Proving graphs can be realized as Reeb graphs under natural conditions, linking Reeb number to fundamental group corank.
result Reeb number equals corank of fundamental group, extending previous results.
Uniform foliations with Reeb components on 3-manifolds.
problem Characterizing uniform foliations with Reeb components.
method Study of foliations on 3-manifolds with infinite fundamental group.
result Examples and results on behavior of foliations.
Assume that we are given a closed chord-generic Legendrian submanifold Λ⊂P×R of the contactisation of a Liouville manifold, where Λ moreover admits an exact Lagrangian filling LΛ⊂R×P×R inside the symplectisation. Under the further assumptions that this …
New method constructs smooth functions with specific Reeb graphs and preimages on 3D manifolds.
problem Construct smooth functions with prescribed Reeb graphs and preimages on 3D closed manifolds.
method Develops a new approach to realize graphs as Reeb graphs of smooth functions on 3D closed manifolds.
result Provides a best possible solution for functions on 3D closed manifolds.
The paper explores how vector fields relate to volume in geometric contexts.
problem Existence of nondiffeomorphic contact forms with identical Reeb vector fields.
method Analyzes geodesible vector fields and their associated Euler classes, applying topological and geometric theorems.
result Proves the Gauss-Bonnet and Poincaré-Hopf theorems for 2D orbifolds using geodesible vector fields.
Geodesics spiral around Reeb orbits in 3D contact manifolds.
problem Understanding geodesics in sub-Riemannian geometry.
method Normal form along Reeb orbits due to Melrose.
result Sub-Riemannian geodesics spiral around Reeb orbits in both phase and configuration spaces.
New findings on how graphs can be realized as Reeb graphs of smooth functions.
problem Realizing a given graph as the Reeb graph of a Morse function on a manifold.
method Investigates the conditions under which a graph can be realized as the Reeb graph of a Morse function on a manifold.
result For any graph with a good orientation, there exists an n-manifold and a Morse function whose Reeb graph is isomorphic to the given graph.
Study on periodic orbits in contact geometry and Finsler geodesics.
problem Existence and non-existence of periodic Reeb orbits on contact manifolds.
method Interpretation of Finsler geodesics as Reeb flows and use of open books.
result Existence of periodic Reeb orbits on contact manifolds with suitable open books.