Study on isometries in contact sub-pseudo-Riemannian geometry.
problem Understanding isometries in a specific geometric structure.
method Analyzing upper bounds and maximal dimensions of isometry groups.
result Maximal dimension of isometry groups is achieved for Heisenberg group structures.
Invariants from contact structures lead to Einstein-Weyl geometry.
problem Understanding local geometry of contact sub-pseudo-Riemannian structures.
method Constructing invariants and showing conditions for Einstein-Weyl geometry.
result Contact structures on contact manifolds lead to Einstein-Weyl geometries in dimension 3.
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 demonstrate how the novel approach to the local geometry of structures of nonholonomic nature, originated by Andrei Agrachev, works in the following two situations: rank 2 distributions of maximal class in R^n with non-zero generalized Wilczynski invariants and rank 2 distributions of maximal class in R^n with addit…
The paper studies geometric structures on SL(n,R) induced by the Killing form.
problem Understanding geometric structures on SL(n,R) induced by the Killing form.
method Constructing manifolds, studying Poisson-commutation relations, and solving Hamiltonian systems.
result Explicit solutions of Hamiltonian systems for n=2.
New examples show contact structures can be isotopic but not contact-isotopic.
problem Identifying contact structures that are isotopic but not contact-isotopic.
method Constructing specific contactomorphisms in all dimensions.
result Infinitely many examples of contactomorphisms that are isotopic but not contact-isotopic.
The study explores embedding closed contact manifolds in higher dimensions.
problem Iso-contact embeddings of closed contact manifolds in co-dimension 2.
method Analyzes conditions for iso-contact embeddings and applies results to specific cases.
result Closed contact 3-manifolds without 2-torsion in their cohomology iso-contact embed in the standard contact 5-sphere.
Study on contact Hamiltonian functions for singular contact structures.
problem Understanding infinitesimal contact transformations on singular contact structures.
method Showed injectivity and provided an explicit local formula for the inverse map.
result Explicit local formula for the inverse map when contact structure has singularities of the first type.
Contact round surgery proves existence of contact structures on 3-manifolds.
problem Existence of contact structures on 3-manifolds.
method Contact round surgery and Lutz twist/Giroux torsion operations.
result Any contact structure on a closed orientable 3-manifold is constructed from the standard contact structure on the 3-sphere via contact round surgeries.
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.
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.
Simplified construction of contact triad connection for analysis.
problem Analyzing contact instanton equation on contact triads.
method Characterization via almost contact structure and construction using almost contact moving frame.
result Simpler and more canonical construction of contact triad connection.
The paper examines conditions for contact surgeries on rational homology 3-spheres.
problem Conditions for contact surgeries on rational homology 3-spheres.
method Analyzes sufficient conditions for contact surgeries using Legendrian knots and links.
result Provides sufficient conditions for surgeries to have vanishing contact invariants or to be overtwisted.
We study the contact equivalence problem for toric contact structures on S3-bundles over S2. That is, given two toric contact structures, one can ask the question: when are they equivalent as contact structures while inequivalent as toric contact structures? In general this appears to be a difficult problem. To f…
Contact surgeries transform manifolds, and all admit symplectic caps.
problem Transforming contact manifolds using surgeries.
method Contact surgeries on isotropic and coisotropic spheres.
result All closed oriented contact manifolds admit symplectic caps.
The paper studies how to embed contact 3-manifolds into the standard contact 5-sphere using braidings.
problem Embedding contact 3-manifolds into the standard contact 5-sphere.
method Using braidings of one manifold about another.
result Many contact 3-manifolds can be embedded into the standard contact 5-sphere.
Introduces contact dual pairs using line bundles.
problem Understanding contact geometry and Jacobi structures.
method Line bundle approach to contact and Jacobi geometry.
result Characteristic Leaf Correspondence Theorem for contact dual pairs.
New contact structures on folded sums of contact mapping tori are tight under certain conditions.
problem Understanding tight contact structures on folded sums of contact mapping tori.
method Alternative bundle-theoretical construction and gluing process near the fold.
result Folded contact structures on folded sums of contact mapping tori are tight under specific conditions.
The study characterizes surgeries on fibred knots that preserve tight contact structures.
problem Understanding when surgeries on fibred knots in contact 3-manifolds result in tight contact structures.
method Leveraging information about the contact structure supported by the mirror knot, we derive characterizations and corollaries.
result Negative surgeries on fibred knots can result in contact structures with non-vanishing Heegaard Floer contact classes.
Abstract: Survey on contact submanifolds.
problem Understanding contact submanifolds.
method None specified, survey of existing knowledge.
result Discussion of various contact submanifolds.
Embeds all contact 3-manifolds into specific 5-manifolds.
problem Embedding all contact 3-manifolds into fixed contact 5-manifolds.
method Spun embeddings and Lefschetz fibrations.
result Embeds all contact 3-manifolds into a Stein fillable contact structure on the twisted S3-bundle over S2 and a unique overtwisted contact structure on S3imesS2. 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. 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.
Local flatness theorem for paraquaternionic contact structures.
problem Local flatness of paraquaternionic contact manifolds.
method Defined paraquaternionic contact conformal curvature tensor and showed local flatness condition.
result Paraquaternionic contact conformal curvature vanishing implies local flatness.
New examples show contact invariants can be non-zero even with half Giroux torsion.
problem Understanding contact invariants and their obstructions in 3-manifolds.
method Use bordered contact invariants and innermost contact structures.
result Found closed contact 3-manifolds with non-vanishing contact invariants.
Study of cosmetic contact surgeries on knots, excluding some cases.
problem Determining cosmetic contact surgeries on transverse knots.
method Analyzing contact surgeries and their outcomes on transverse knots.
result Exclude non-trivial cosmetic contact surgeries for certain transverse knots.
We consider certain type of fiber bundles with odd dimensional compact contact base, exact symplectic fibers, and the structure group contained in the group of exact symplectomorphisms of the fiber. We call such fibrations "contact symplectic fibrations". By a result of Hajduk-Walczak, some of these admit contact struc…
The paper finds minimal contact triangulations for 3-manifolds, proving a linear bound on vertex count.
problem Finding the minimum number of vertices in contact triangulations for 3-manifolds.
method Explicit examples and linear growth bound analysis for overtwisted contact structures.
result The number of vertices for minimal contact triangulations grows at most linearly with respect to the d3 invariant. Survey of contact homology for contact manifolds.
problem Understanding contact manifolds through homology.
method Holomorphic curves invariants.
result Useful for students and young mathematicians.
New result on contact surgeries and overtwistedness.
problem When contact (+1/n)-surgery is overtwisted.
method Analyzes contact surgeries and provides counterexamples.
result Provides conditions for overtwisted contact surgeries.
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.
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.
Study contact structures on specific manifolds, proving infinite non-isotopic structures and tight/overtwisted classifications.
problem Classifying contact structures with specific contact surgery numbers on Brieskorn spheres and lens spaces.
method Algorithm for computing Euler class from rational contact surgery, Legendrian knot classification, and analysis of contact structures.
result Infinitely many non-isotopic contact structures on Brieskorn spheres and lens spaces cannot be obtained by a single rational contact surgery.
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). 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.
We consider contact elements in the sutured Floer homology of solid tori with longitudinal sutures, as part of the (1+1)-dimensional topological quantum field theory defined by Honda--Kazez--Matić in \cite{HKM08}. The Z2 SFH of these solid tori forms a "categorification of Pascal's triangle", and contact structur…
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 connects contact structures to twistor spaces and almost contact metric structures.
problem Integrability and normality conditions for almost contact metric structures on twistor spaces.
method Definition of contact connection and almost CR-structure on twistor spaces, and conditions for normality in terms of curvature. result Necessary and sufficient conditions for normality of the almost contact metric structure are derived in terms of the curvature of the contact connection.
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.
Contact forms on 3-manifolds can have very large systolic ratios.
problem Understanding the systolic ratios of contact forms on 3-manifolds.
method Analyzing the systolic ratio of contact forms defined by co-orientable contact structures.
result Contact forms on any closed 3-manifold can have arbitrarily large systolic ratios.
A contact projective structure is a contact path geometry the paths of which are among the geodesics of some affine connection. In the manner of T.Y. Thomas there is associated to each contact projective structure an ambient affine connection on a symplectic manifold with one-dimensional fibers over the contact manifol…
Generalizes Tulczyjew triples for contact manifolds in Hamiltonian and Lagrangian formalisms.
problem Tackles the need for a geometric tool in contact manifolds.
method Introduces a generalized Tulczyjew triple for contact manifolds.
result Contact Hamiltonians and Lagrangians as sections of line bundles determine dynamics on contact phase space.
The study finds tight contact structures without fillings in high dimensions.
problem Finding tight contact structures that cannot be filled by symplectic forms.
method Construction of specific contact structures on manifolds of various dimensions.
result Existence of tight contact structures without fillings in all dimensions n≥3 and for n=2 under certain conditions. The paper introduces new contact structure modifications via round surgery.
problem Studying higher-dimensional contact structures and their modifications.
method Contact round surgery to introduce three types of modifications: overtwisted, weakly symplectically non-fillable, and strongly symplectically non-fillable.
result The introduced modifications can be realized by sequences of contact round surgeries and contribute to the Euler classes of contact structures.
This thesis improves protein contact prediction using unsupervised and supervised methods.
problem Improving accuracy of protein contact prediction.
method Unsupervised and supervised deep learning methods.
result A scoring system called diversity score for measuring contact novelty.
Study contact surgeries on Legendrian links, proving invariant vanishing and overtwistedness.
problem Understanding contact surgeries on Legendrian links and their properties.
method Algebraic methods using Heegaard Floer homology and contact-geometric arguments.
result Vanishing of contact Ozsváth-Szabó invariant and overtwistedness of contact 3-manifolds.
We construct an open book decomposition compatible with a contact structure given by a rational contact surgery on a Legendrian link in the standard contact S3. As an application we show that some rational contact surgeries on certain Legendrian knots induce overtwisted contact structures.
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.