The study explores new metric structures on manifolds, linking them to Einstein metrics.
problem Characterizing and understanding weak K-contact manifolds and their properties.
method Analyzing weak K-contact manifolds and their properties, including the parallel Ricci tensor and generalized Ricci soliton structures.
result Sufficient conditions for weak K-contact manifolds with specific properties to be Einstein manifolds.
Study weak f-K-contact manifolds, finding Einstein-type metrics and solitons.
problem Characterize and study geometric properties of weak f-K-contact manifolds. method Analyzing weak metric f-structures, using Killing vector fields, and Jacobi operators. result Einstein weak f-K-contact manifolds are Ricci flat. Study weak quasi contact metric manifolds to generalize K-contact and Sasakian manifolds criteria.
problem Generalize K-contact and Sasakian manifolds criteria using weak quasi contact metric manifolds.
method Study weak quasi contact metric manifolds and generalize theorems for K-contact and Sasakian manifolds.
result Provide new criterions for K-contact and Sasakian manifolds in terms of curvature tensor and geometric objects.
Study explores weak generalized K-contact structures in contact metric spaces.
problem Exploring weak generalized K-contact structures in contact metric spaces.
method Introducing a weak (κ,μ) condition and proving existence of K-contact and (κ,μ=2)-structures. result Existence of K-contact and (κ,μ=2)-structures under certain conditions on the Boeckx invariant. Compact metric f-K-contact manifolds constructed via specific transformations.
problem Constructing all metric f-K-contact manifolds.
method Iteration of constructions of mapping tori, rotations, and type II deformations.
result Compact metric f-K-contact manifolds are derived from compact K-contact manifolds.
New Lie systems defined on k-contact manifolds, with applications.
problem Understanding Lie systems on k-contact manifolds. method Distributional approach to k-contact manifolds and Hamiltonian vector fields. result Lie systems can be understood as Hamiltonian relative to a k-contact manifold. Develops k-contact geometry theory for field theories.
problem Analyse field theories using k-contact geometry.
method Distributions maximally non-integrable with k commuting Lie symmetries.
result Established k-contact distributions and their relationships.
The study constructs K-contact manifolds with minimal closed Reeb orbits and provides conditions for their homeomorphism to spheres.
problem Understanding K-contact manifolds with minimal closed Reeb orbits and their homeomorphism properties.
method Using Boothby-Wang fibration and Hamiltonian torus actions, the study constructs and analyzes K-contact manifolds.
result The existence of K-contact manifolds with minimal closed Reeb orbits that are not homeomorphic to spheres and have unique cohomology rings.
First example of a 5D manifold with K-contact but no Sasakian structure.
problem Long-standing question of Boyer and Galicki about existence of Sasakian structures.
method Construction of a specific 5-manifold.
result First example of a simply connected compact 5-manifold with K-contact but no Sasakian structure.
Holonomy groups of K-contact sub-Riemannian manifolds are studied.
problem Understanding the holonomy groups of K-contact sub-Riemannian manifolds.
method Analyzing the horizontal holonomy group and comparing it to the holonomy group of a Riemannian manifold.
result The horizontal holonomy group either coincides with the holonomy group of a Riemannian manifold or is a codimension-one subgroup.
We observe that the class of metric f-K-contact manifolds, which naturally contains that of K-contact manifolds, is closed under forming mapping tori of automorphisms of the structure. We show that the de Rham cohomology of compact metric f-K-contact manifolds naturally splits off an exterior algebra, and rel…
Clarifies construction of K-contact non-Sasakian Smale-Barden manifolds.
problem Determining which Smale-Barden manifolds admit K-contact but not Sasakian structures.
method Explicitly determines the number of symplectic surfaces needed for isotropy loci, refines constructions of symplectic surfaces.
result Determines the number N of symplectic surfaces needed for K-contact but non-Sasakian manifolds.
Study new k-contact distributions and Lie systems.
problem Characterizing and understanding k-contact distributions. method Analyzing Goursat distributions and Lie systems.
result Characterized new types of k-contact distributions. The paper classifies K-contact forms on 3-manifolds and connects their orbits to spectral invariants.
problem Classifying K-contact forms with specific properties on 3-manifolds.
method Analyzing the Reeb vector field and its orbits, proving diffeomorphism results, and relating to spectral invariants.
result Compact 3-manifolds carrying such K-contact forms are diffeomorphic to lens spaces with specific orbit properties.
Developed Gompf connected sum for orbifolds, constructing symplectic and K-contact manifolds.
problem Constructing symplectic and K-contact manifolds with specific properties.
method Developed Gompf fiber connected sum operation for symplectic orbifolds and used it to construct the required manifolds.
result Constructed a K-contact Smale-Barden manifold with specified 2-homology and sharper estimates.
Kollár has found subtle obstructions to the existence of Sasakian structures on 5-dimensional manifolds. In the present article we develop methods of using these obstructions to distinguish K-contact manifolds from Sasakian ones. In particular, we find the first example of a closed 5-manifold M with H1(M,Z)=0 which …
Holonomy groups of K-contact sub-Riemannian manifolds are isomorphic.
problem Holonomy groups of K-contact sub-Riemannian manifolds.
method Proved isomorphism of holonomy groups under specific conditions.
result Holonomy groups of K-contact sub-Riemannian manifolds are isomorphic.
The paper studies special solitons on specific contact metric manifolds.
problem Characterizing solitons on N(k)-contact metric manifolds.
method Analyzing ∗-conformal Einstein solitons and gradient solitons on N(k)-contact metric manifolds. result Conditions for solitons to be expanding, steady, or shrinking are determined.
New constructions of Sasakian and K-contact structures on Smale-Barden manifolds.
problem Deciding when a Smale-Barden manifold admits a Sasakian or K-contact structure.
method Developing quasi-regular Seifert fibrations and applying them to constructions.
result Determined all Smale-Barden manifolds admitting null Sasakian structures and provided counterexamples to conjectures.
We investigate the equivariant cohomology of the natural torus action on a K-contact manifold and its relation to the topology of the Reeb flow. Using the contact moment map, we show that the equivariant cohomology of this action is Cohen-Macaulay, which is a generalization of equivariant formality for torus actions wi…
We show that a compact K-contact manifold (M,g,ξ) has a closed Weyl-Einstein connection compatible with the conformal structure [g] if and only if it is Sasaki-Einstein.
Using the Hard Lefschetz Theorem for Sasakian manifolds, we find two examples of compact K-contact nilmanifolds with no compatible Sasakian metric in dimensions five and seven, respectively
This paper constructs symplectic surfaces in 4-manifolds with transversal intersections.
problem Constructing symplectic surfaces in symplectic 4-manifolds with transversal intersections.
method Revision and generalization of existing constructions.
result Produces the ramification locus of Seifert bundles over symplectic 4-orbifolds.
Defines generalized Sasakian structures in contact geometry.
problem Understanding k-contact manifolds and CR manifolds.
method Introduced generalized Sasakian structures and proved their equivalence to Sasakian structures.
result k-contact manifolds are generalized Sasakian if and only if they are classically Sasakian.
We find a family of five dimensional completely solvable compact manifolds that constitute the first examples of K-contact manifolds which satisfy the Hard Lefschetz Theorem and have a model of Tievsky type just as Sasakian manifolds but do not admit any Sasakian structure.
Eta-Einstein and (κ,μ)-structures studied in dimension 3.
problem Characterizing and understanding Eta-Einstein and (κ,μ)-structures in 3D. method Analyzing closed manifolds and constructing examples.
result Almost regular Eta-Einstein structures not D-homothetic to Einstein structures exist.
This paper classifies holonomy groups of K-contact sub-pseudo-Riemannian manifolds.
problem The problem of subspace degeneracy in indefinite signature metrics.
method Adapted for metrics of indefinite signature, bypassing subspace degeneracy.
result Horizontal holonomy group either coincides with the adapted holonomy group or acts as its normal subgroup of codimension one.
We prove an analogue of Kirwan surjectivity in the setting of equivariant basic cohomology of K-contact manifolds. If the Reeb vector field induces a free S1-action, the S1-quotient is a symplectic manifold and our result reproduces Kirwan's surjectivity for these symplectic manifolds. We further prove a Tolman-W…
In this paper, we prove that evry 3-dimensional manifold M is a ?- recurrent N(k)-contact metric manifold if and only if it is flat. Then we classify the ?-recurrent contact metric manifolds of constant curvature. This implies that there exists no ?-recurrent N(k)-contact metric manifold, which is neither symmetric nor…
We construct a compact simply-connected 7-dimensional manifold admitting a K-contact structure but not a Sasakian structure. We also study rational homotopy properties of such manifolds, proving in particular that a simply-connected 7-dimensional Sasakian manifold has vanishing cup-product on the second cohomology and …
Study examines tangential real hypersurfaces on Hermite-like manifolds.
problem Characterizing real hypersurfaces on Hermite-like manifolds.
method Introduced tangential real hypersurfaces and derived main identities.
result Discussed contact metric structures in K-contact and cosymplectic cases.
Jet bundles as higher-order polarised k-contact manifolds
problem Recognizing finite-order jet bundles as polarised Nπr-contact manifolds method Introducing new classes of polarisations for k-contact distributions result Main recognition theorem
We solve the problem posed by Boyer and Galicki about the existence of K-contact simply connected manifolds with no Sasakian structure. Although the result lies in the framework of metric contact geometry, our methods come from contact and symplectic geometry and are based on the method of fat bundles developed by Ster…
Unified framework for classifying Sasakian, K-contact, and (κ, μ)-manifolds.
problem Classifying and understanding different types of contact metric manifolds.
method Investigating metric structures on symplectizations and proving the existence of a unique metric symplectization.
result Unified framework for classifying Sasakian, K-contact, and (κ, μ)-manifolds.
Extending a result of He to the non-integrable case of K-contact manifolds, it is shown that transverse Hermitian scalar curvature may be interpreted as a moment map for the strict contactomorphism group. As a consequence, we may generalize the Sasaki-Futaki invariant to K-contact geometry and establish a number of ele…
The paper characterizes contact metric manifolds with specific solitons.
problem Characterizing contact metric manifolds with ∗-conformal Ricci solitons. method Analyzing properties of (2n+1)-dimensional N(k)-contact metric manifolds. result The manifold is locally isometric to a flat (n+1)-dimensional manifold and an n-dimensional manifold of constant curvature 4. Geometric quantization for specific symplectic structures proved.
problem Quantization of specific symplectic structures.
method Geometric quantization for constant rank presymplectic structures with Riemannian null foliation.
result Quantization-commutes-with-reduction theorem proved in this context.
We show that every K-contact Einstein manifold is Sasakian-Einstein and discuss several corollaries of this result.
Develops Marsden-Meyer-Weinstein reduction for k-contact field theories.
problem None explicitly stated, but related to field theories and contact geometry.
method Marsden-Meyer-Weinstein reduction techniques applied to k-contact field theories. result Clarifies and corrects previous contact reduction literature.
We prove an analogue of the Atiyah-Bott-Berline-Vergne localization formula in the setting of equivariant basic cohomology of K-contact manifolds. As a consequence, we deduce analogues of Witten's nonabelian localization and the Jeffrey-Kirwan residue formula, which relate equivariant basic integrals on a contact man…
Develops Hamilton-Jacobi theory for non-conservative field theories in k-contact geometry.
problem Analyzes non-conservative field theories, especially dissipative systems.
method Introduces evolution k-contact k-vector fields and develops two Hamilton-Jacobi theories.
result Recover ordinary contact Hamilton-Jacobi theory as k=1, and enlarges application range.
In this paper, we consider decompositions of basic degree 2 cohomology for a compact K-contact 5-manifold (M,ξ,η,Φ,g), and conclude the pureness and fullness of Φ-invariant and Φ-anti-invariant cohomology groups. Moreover, we discuss the decomposition of the complexified basic degree 2 cohomology group. This is a…
Study on curvature properties of N(κ)-contact metric manifolds with generalized Tanaka-Webster connection.
problem Curvature properties of N(κ)-contact metric manifolds.
method Analysis using generalized Tanaka-Webster connection.
result If a N(κ)-contact metric manifold with generalized Tanaka-Webster connection is K-contact, it is a generalized Sasakian space form.
Develops geometric framework for dissipative field equations.
problem Dissipative field equations and their geometric analysis.
method Canonical k-contact manifolds, k-contactifications, splitting results, regularity conditions, criteria for PDEs. result Explicit Hamiltonian descriptions for various nonlinear PDEs.
We compute the first and second homotopy groups of a class of contact toric manifolds in terms of the images of the associated moment map.
In N(k)-contact metric manifolds and/or (k,μ)-manifolds, gradient Ricci solitons, compact Ricci solitons and Ricci solitons with V pointwise collinear with the structure vector field ξ are studied.
We study the Lie algebra of infinitesimal isometries on compact Sasakian and K--contact manifolds. On a Sasakian manifold which is not a space form or 3--Sasakian, every Killing vector field is an infinitesimal automorphism of the Sasakian structure. For a manifold with K--contact structure, we prove that there exists …
With a view toward sub-Riemannian geometry, we introduce and study H-type foliations. These structures are natural generalizations of K-contact geometries which encompass as special cases K-contact manifolds, twistor spaces, 3K contact manifolds and H-type groups. Under an horizontal Ricci curvature lower bound, we pro…