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.
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.
The purpose of this article is to study co-dimension 2 iso-contact embeddings of closed contact manifolds. We first show that a closed contact manifold (M2n−1,ξM) iso-contact embeds in a contact manifold (N2n+1,ξN), provided M contact embeds in (N,ξN) with a trivial normal bundle and the contact s…
Study of higher-dimensional contact manifolds and their properties.
problem Understanding contact manifolds in higher dimensions.
method Topological and symplectic methods, including open books.
result Many contact manifolds can be realized as iterated planar contact manifolds.
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.
We give a characterization of a contact metric manifold as a special almost contact metric manifold and discuss an almost contact metric manifold which is {a} natural generalization of the contact metric manifolds introduced by Y. Tashiro.
In this paper we study embeddings of contact manifolds using braidings of one manifold about another. In particular we show how to embed many contact 3-manifolds into the standard contact 5-sphere. We also show how to obstruct braidings of one manifold about another using contact geometry.
Study on generalized quasi-Einstein structures in contact geometry.
problem Characterizing and understanding generalized quasi-Einstein structures in contact geometry.
method Investigation of properties, existence, and characterizations of generalized quasi-Einstein normal metric contact pair manifolds.
result Normal metric contact pair manifolds with generalized quasi-constant curvature are generalized quasi-Einstein manifolds.
Contact surgeries yield algebraically overtwisted manifolds.
problem Understanding algebraically overtwisted contact manifolds through surgeries.
method Contact (+1)-surgeries on Legendrian spheres in flexibly fillable contact manifolds. result Yielding algebraically overtwisted manifolds when the Legendrian's homology class is not annihilated.
The paper defines and studies contact surgery numbers for contact 3-manifolds.
problem Understanding the minimal number of components of a surgery link describing a contact 3-manifold.
method Defined and studied various versions of contact surgery numbers, relating them to other invariants and computing specific cases.
result There exist infinitely many non-isotopic contact structures on certain manifolds that cannot be obtained by a single rational contact surgery from the standard tight contact 3-sphere.
Survey of contact homology for contact manifolds.
problem Understanding contact manifolds through homology.
method Holomorphic curves invariants.
result Useful for students and young mathematicians.
Contact round surgery of contact 3-manifolds is introduced in this paper. By using this method, an alternative proof of the existence of a contact structure on any closed orientable 3-manifold is given. It is also proved that any contact structure on any closed orientable 3-manifold is constructed from the standard con…
Contact manifolds' momentum polytopes are convex.
problem Understanding the structure of contact manifolds.
method Using isomorphism to toric varieties.
result Momentum polytopes of contact manifolds are convex.
Graphs describe contact surgery on 3-manifolds.
problem Understanding contact surgery on 3-manifolds.
method Defined contact surgery graphs to analyze their properties.
result Analyzed basic properties and interesting subgraphs of contact surgery graphs.
3D contact manifolds have optimal higher systolic ratios.
problem Optimizing higher systolic ratios in 3D contact manifolds.
method Proving Besse contact forms maximize certain ratios.
result Besse contact forms are local maximizers of higher systolic ratios.
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. Abstract: Survey on contact submanifolds.
problem Understanding contact submanifolds.
method None specified, survey of existing knowledge.
result Discussion of various contact submanifolds.
We use contact handle decompositions and a stabilization process to compute the cylindrical contact homology of a subcritical Stein-fillable contact manifold with vanishing first Chern class, and show that it is completely determined by the homology of a subcritical Stein-filling of the contact manifold.
The study explores (m,ρ)-quasi-Einstein structures on contact metric manifolds.
problem Exploring (m,ρ)-quasi-Einstein structures in contact geometry. method Proving properties of (m,ρ)-quasi-Einstein structures on contact metric manifolds. result Compact contact or H-contact metric manifolds with (m,ρ)-quasi-Einstein structures have specific properties. The paper connects complex contact structures to specific types of almost contact 3-structures.
problem Understanding the relationship between complex contact structures and almost contact 3-structures.
method Proving that every complex contact structure gives rise to a distinguished almost contact metric 3-structure.
result The paper provides new examples of manifolds with specific contact and almost contact structures.
We introduce the cutting construction of possibly non-compact symplectic toric manifolds, in particular, toric symplectic cones that correspond to a weakly convex good cone. Since the symplectization of a toric contact manifold is a toric symplectic cone, we can also construct toric contact manifolds that correspond to…
Study on null hypersurfaces in complex contact manifolds.
problem Characterizing null hypersurfaces in indefinite complex contact manifolds.
method Proved classification results for various null hypersurfaces and characterized the ambient space.
result The ambient complex contact manifold must have constant GH-sectional curvature of −3 for certain null hypersurfaces. 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.
The paper introduces a statistical version of contact CR-product for Sasakian statistical manifolds.
problem Characterizing geometric properties of contact CR-submanifolds in Sasakian statistical manifolds.
method Characterization of integrability of invariant and anti-invariant distributions, development of results on specific types of contact CR submanifolds, introduction of statistical contact CR-product.
result Introduction of a statistical version of contact CR-product for Sasakian statistical manifolds.
Contact connected sums do not increase support genus.
problem Understanding how support genus changes under contact connected sums.
method Analyzing the support genus of contact connected sums of 3-manifolds.
result The support genus of contact connected sums is at most the maximum of the summands' support genera.
The paper studies contact pseudo-metric manifolds with a specific curvature condition.
problem Investigating properties of contact pseudo-metric manifolds under a nullity condition.
method Introducing and analyzing (κ,μ)-contact pseudo-metric manifolds and generalized (κ,μ)-contact pseudo-metric manifolds. result The curvature of these manifolds is constant if the φ-sectional curvature is independent of the φ-section. 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.
We give a complete classification of foliations on open contact manifolds whose leaves are contact submanifolds of the ambient manifold. The results are analogues of Haefliger's classification of foliations on open manifold.
In this note we observe that one can contact embed all contact 3-manifolds into a Stein fillable contact structure on the twisted S3-bundle over S2 and also into a unique overtwisted contact structure on S3×S2. These results are proven using "spun embeddings" and Lefschetz fibrations.
Monopole invariant studies contact structures on 3-manifolds.
problem Topology of contact structures on 3-manifolds.
method Generalized Kronheimer--Mrowka--Ozsváth--Szabó contact invariant for families of contact structures.
result Obstructs the existence of sections and detects exotic loops of contact structures.
We give the parallelism between locally conformal symplectic manifolds and contact manifolds. We also give the generalization of exact contact manifolds.
No contact Anosov diffeomorphisms exist on odd-dimensional manifolds.
problem Existence of contact Anosov diffeomorphisms on odd-dimensional manifolds.
method Analysis of Anosov diffeomorphisms and invariant contact structures.
result No invariant contact structure exists for Anosov diffeomorphisms on odd-dimensional manifolds.
Stabilization operation for high-dimensional contact manifolds, proving many links are non-simple.
problem Understanding the structure and properties of high-dimensional contact manifolds.
method Definition and proof of stabilization operation for codimension 2 contact submanifolds in dim≥5 contact manifolds. result Many transverse links are non-simple.
The abstract discusses embedding manifolds in open books and contact structures.
problem Embedding manifolds in open books and contact structures.
method Using open book embeddings and contact structures, the abstract proves embedding theorems for various manifolds.
result Closed manifolds can be embedded in open books and contact structures.
Using contact surgery we define families of contact structures on certain Seifert fibered three-manifolds. We prove that all these contact structures are tight using contact Ozsath-Szabo invariants. We use these examples to show that, given a natural number n, there exists a Seifert fibered three-manifold carrying at l…
New contactomorphisms found via Dehn twists on 3-manifold sums.
problem Detecting exotic contactomorphisms in 3-manifold sums.
method Combining Kronheimer-Mrowka invariant and h-principle for convex spheres.
result First examples of infinite-order exotic contactomorphisms.
Proposes a contact dynamics framework using generalized geometries.
problem Contact dynamics and related geometries.
method Generalizes symplectic and Morse families to contact framework.
result Establishes contact Hamiltonian and Lagrangian Dynamics as Legendrian submanifolds.
Contact round surgeries on (S3,ξst) help in constructing and understanding contact 3-manifolds.
problem Constructing contact 3-manifolds using Legendrian surgeries.
method Introducing contact round surgeries of indices 1 and 2, and associating them with surgery diagrams.
result Every closed connected contact 3-manifold can be obtained by a sequence of contact round surgeries on Legendrian knots in (S3,ξst). We extract a nonnegative integer-valued invariant, which we call the "order of algebraic torsion", from the Symplectic Field Theory of a closed contact manifold, and show that its finiteness gives obstructions to the existence of symplectic fillings and exact symplectic cobordisms. A contact manifold has algebraic tors…
A contact metric manifold is said to be H-contact, if the characteristic vector field is harmonic. We prove that the unit tangent bundle of a Riemannian manifold M equipped with the standard contact metric structure is H-contact if and only if M is 2-stein.
According to Giroux, contact manifolds can be described as open books whose pages are Stein manifolds. For 5-dimensional contact manifolds the pages are Stein surfaces, which permit a description via Kirby diagrams. We introduce handle moves on such diagrams that do not change the corresponding contact manifold. As an …
Odd-dimensional manifolds have contact maps of non-zero degree.
problem Contact domination in odd-dimensional manifolds.
method Proving existence of maps from tight contact manifolds.
result Existence of non-zero degree maps from Liouville-fillable but not Weinstein-fillable contact manifolds.
Defines contact surgery distance and shows it's bounded by topological surgery distance by 5.
problem Comparing contact structures on 3-manifolds.
method Defining contact surgery distance and proving upper bound on its value.
result Contact surgery distance is at most 5 larger than topological surgery distance.
Proves existence of strongly overtwisted contact structures on 3-manifolds.
problem Existence of overtwisted contact structures on 3-manifolds.
method Complete h-principle for overtwisted disks and Legendrians.
result Homotopy type of overtwisted disk embeddings is determined.
The author is planning if possible classify all three-dimensional (κ,μ)-manifolds wether contact metric, almost cosymplectic, para-contact metric, almost para-cosymplectic. Of course classification in contact or almost cosymplectic cases already is provdied. Up to authors knowledge there is no classification for para…
The paper defines quasi-isometry for almost contact metric manifolds and explores its properties.
problem Understanding quasi-isometry in almost contact metric manifolds.
method Definition and study of quasi-isometry for almost contact metric manifolds.
result Established a relation between scalar curvature and quasi-isometric constants.
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.
Recently Francisco Presas Mata constructed the first examples of closed contact manifolds of dimension larger than 3 that contain a plastikstufe, and hence are non-fillable. Using contact surgery on his examples we create on every sphere S^{2n-1}, n>1, an exotic contact structure ξ_- that also contains a plastikstufe. …