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 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. 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. 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.
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. 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.
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.
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.
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 …
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…
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.
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 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.
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.
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.
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.
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.
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.
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 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
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.
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.
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. 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.
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…
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.
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.
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.
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…
We develop a new geometric framework suitable for dealing with Hamiltonian field theories with dissipation. To this end we define the notions of k-contact structure and k-contact Hamiltonian system. This is a generalization of both the contact Hamiltonian systems in mechanics and the k-symplectic Hamiltonian syst…
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.
Survey of three geometric frameworks for action-dependent field theories.
problem Understanding action-dependent field theories through geometric structures.
method Introduction and analysis of three geometric frameworks: k-contact, k-cocontact, and multicontact.
result Analysis of relationships among these geometric structures and comparison with other definitions.
Sasakian structures found on tangent sphere bundles of certain symmetric spaces.
problem Existence of Sasakian structures on tangent sphere bundles of compact rank-one symmetric spaces.
method Proof of existence using K-contact structures and induced structures from almost Hermitian structures.
result Tangent sphere bundles of compact rank-one symmetric spaces admit unique K-contact structures that are Sasakian.
We construct the first example of a 5-dimensional simply connected compact manifold that admits a K-contact structure but does not admit a semi-regular Sasakian structure. For this, we need two ingredients: (a) to construct a suitable simply connected symplectic 4-manifold with disjoint symplectic surfaces spanning the…
The paper connects complex spacetimes to twistor spaces and finds an almost contact structure.
problem Understanding the geometry of complex spacetimes and their relation to twistor spaces.
method Using a correspondence between 5D complex spacetimes and 4D twistor spaces, the paper constructs almost contact structures.
result A 5-dimensional K-contact manifold can be derived from the Ren-Wang twistor space.
New geometric framework for non-conservative field theories with time-dependent terms.
problem Describing non-conservative field theories with explicit space-time dependence.
method Combining k-cosymplectic and k-contact formulations to develop Hamiltonian and Lagrangian formalisms.
result Illustrated with the nonlinear damped wave equation, demonstrating the new formalism's applicability.
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.
We provide some examples of harmonic unit vector fields as normalized gradients of isoparametric functions from a K-contact geometry setting.
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.
Geometrically describes pseudo-gauge freedom in relativistic hydrodynamics.
problem Pseudo-gauge ambiguity in relativistic hydrodynamics.
method Develops k-contact geometry to describe thermodynamic states and conservation laws. result Identifies pseudo-gauge freedom as the non-uniqueness of thermodynamic equilibrium redefinition.
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 …
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…
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
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 collect our recent results ([5] and [8]) and we get the equivalence of the three notions of the title under some conditions. We then use this equivalence in order to prove some consequences about Sasakian manifolds, complex almost contact structures and complex k-contact structures.
Regularisation method studies Lie algebroids via foliated structures.
problem Geometric structures on Lie algebroids.
method Regularisation procedure applied to various Lie algebroids.
result Proof of Weinstein conjecture for specific Lie algebroids.