We consider the Lie algebra of all vector fields on a contact manifold as a module over the Lie subalgebra of contact vector fields. This module is split into a direct sum of two submodules: the contact algebra itself and the space of tangent vector fields. We study the geometric nature of these two modules.
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Contact Lie systemsmath-ph
Contact Lie systems analyze integral curves of Hamiltonian vector fields.
problem Analyzing integral curves of Hamiltonian vector fields.
method Define and study contact Lie systems, including conservative systems.
result Develop Liouville theorems, contact reductions, and Gromov non-squeezing theorems.
Geodesic vector fields on flat 3-manifolds are related to contact structures.
problem Understanding geodesic vector fields on flat 3-manifolds.
method Analyzing geodesic and Reeb vector fields on flat 3-manifolds.
result Geodesic vector fields on closed flat 3-manifolds are Reeb vector fields of contact forms.
The paper simplifies proofs and characterizes contact structures in 3D.
problem Contact structures induced by geodesic vector fields in 3D.
method New proofs and characterizations of contact structures.
result Contact structures in 3D are universally tight under certain conditions.
The paper studies Ricci solitons on contact pseudo-metric manifolds and their properties.
problem Characterizing properties of contact pseudo-metric manifolds with Ricci solitons.
method Analyzing specific types of Ricci solitons on Sasakian and contact pseudo-metric manifolds.
result Properties of Ricci solitons on contact pseudo-metric manifolds, leading to Einstein manifolds under certain conditions.
Study on Yamabe solitons on specific complex manifolds, focusing on torse-forming vector fields.
problem Exploring Yamabe solitons on a specific class of complex manifolds.
method Analyzing Yamabe solitons on almost contact complex Riemannian manifolds with a vertical torse-forming vector field.
result Explicit examples of 5-dimensional Lie groups characterized by the study.
The paper characterizes Kenmotsu metrics as almost -Ricci solitons.
problem Characterizing Kenmotsu metrics as almost -Ricci solitons.
method Analyzing the geometry of almost contact metrics through -Ricci solitons.
result Kenmotsu metrics are characterized as almost -Ricci solitons under specific conditions.
The study classifies contact metric manifolds based on Ricci-Yamabe solitons.
problem Classifying contact metric manifolds based on Ricci-Yamabe solitons.
method Analyzing specific types of solitons in contact metric manifolds.
result Contact metric manifolds are classified based on the properties of Ricci-Yamabe solitons.
We provide some examples of harmonic unit vector fields as normalized gradients of isoparametric functions from a K-contact geometry setting.
Study shows how to realize Ricci curvature as Reeb vector field for contact 3-manifolds.
problem When can a function be realized as Ricci curvature of a Reeb vector field?
method Topological tools to show realization, resolving singularities depend on contact topology.
result Every admissible function can be realized as Ricci curvature for a singular metric away from a measure zero set.
The aim of this article is to study the k-almost Ricci soliton and k-almost gradient Ricci soliton on contact metric manifold. First, we prove that if a compact K-contact metric is a k-almost gradient Ricci soliton then it is isometric to a unit sphere S2n+1. Next, we extend this result on a compact k-almost Ricci soli…
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.
The abstract proves that certain Reeb vector fields on 3-manifolds have Birkhoff sections.
problem Existence of Birkhoff sections for Reeb vector fields on 3-manifolds.
method Showed existence of Birkhoff sections for Reeb vector fields satisfying Kupka-Smale condition.
result Reeb vector fields on closed 3-manifolds with Kupka-Smale condition admit Birkhoff sections.
We show that -invariant submanifolds of metric contact pairs with orthogonal characteristic foliations make constant angles with the Reeb vector fields. Our main result is that for the normal case such submanifolds of dimension at least are all minimal. We prove that an odd-dimensional -invariant submanifold …
Decomposition of symmetric tensor fields in the presence of a flat contact projective structuremath.DG
Let be an odd-dimensional Euclidean space endowed with a contact 1-form . We investigate the space of symmetric contravariant tensor fields on as a module over the Lie algebra of contact vector fields, i.e. over the Lie subalgebra made up by those vector fields that preserve the contact structure. If we cons…
Study of solitons in a specific type of contact metric manifold.
problem Characterizing solitons in -contact metric manifolds.
method Analyzing almost Riemann and Ricci solitons under Ricci symmetry conditions.
result Characterization of solitons in -contact metric manifolds.
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.
We consider a contact manifold with a pseudo-Riemannian metric and define a contact vector field intrinsically associated to this pair of structures. We call this new differential invariant the contact Riemannian curl. On a Riemannian manifold, Killing vector fields are those that annihilate the metric; a Killing -f…
The study explores -quasi-Einstein structures on contact metric manifolds.
problem Exploring -quasi-Einstein structures in contact geometry.
method Proving properties of -quasi-Einstein structures on contact metric manifolds.
result Compact contact or -contact metric manifolds with -quasi-Einstein structures have specific properties.
The paper studies regular contact manifolds and their products.
problem Characterizing regular contact manifolds and their products.
method Elementary proof for compact manifolds, topological tools for general manifolds.
result Regular contact manifolds are principal bundles with or structure group.
Characterizes invariant contact metric structures on tangent sphere bundles of compact symmetric spaces.
problem Identifying invariant contact metric structures on tangent sphere bundles of compact symmetric spaces.
method Characterization through invariant Riemannian metrics and Killing vector fields.
result A family of invariant contact metric structures is obtained on tangent sphere bundles of compact symmetric spaces with rank greater than or equal to two.
3D contact forms have supporting decompositions, leading to entropy results.
problem Existence of supporting decompositions for contact forms in 3D.
method Proving existence of broken book decompositions for nondegenerate contact forms.
result Nondegenerate Reeb vector fields on 3-manifolds have positive entropy or infinitely many periodic orbits.
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.
The study of quasi Yamabe solitons on 3D contact metric manifolds with specific curvature condition.
problem Investigating quasi Yamabe solitons on 3D contact metric manifolds with a specific curvature condition.
method Analyzing the properties of quasi Yamabe solitons on 3D contact metric manifolds with Qφ = φQ and proving the conditions under which the soliton vector field is constant, the scalar curvature is constant, and the manifold is Sasakian.
result If a 3D contact metric manifold M with Qφ = φQ admits a quasi Yamabe soliton with a non-zero soliton vector field V collinear with the Reeb vector field ξ, then V is a constant multiple of ξ, the scalar curvature is constant, and the manifold is Sasakian.
The study explores rigid constraints on almost Ricci-Bourguignon solitons on contact metric three-manifolds.
problem Investigating constraints on almost Ricci-Bourguignon solitons on contact metric three-manifolds.
method Using a local orthonormal \(\varphi\)-basis, derived the full component form of the almost Ricci-Bourguignon soliton equation.
result For contact metric three-manifolds satisfying \(Qξ=σξ\), a collinear potential field must vanish on the non-Sasakian region whenever \(ξ(σ)=0\).
We study vector fields of the plane preserving the form of Liouville. We present their local models up to the natural equivalence relation, and describe local bifurcations of low codimension. To achieve that, a classification of univariate functions is given, according to a relation stricter than contact equivalence. W…
Study lightlike hypersurfaces in a specific type of manifold.
problem Characterize lightlike hypersurfaces in indefinite almost contact metric-manifolds.
method Analyze the position of the structure vector field and classify hypersurfaces into two types.
result Prove there are only two types of lightlike hypersurfaces: ascreen and inascreen.
Develops theory of contact systems with nonholonomic constraints.
problem Nonholonomic constraints in contact systems.
method Variational principle and projection of Hamiltonian vector field.
result Nonholonomic dynamics as projection of unconstrained dynamics.
Biharmonic or polyharmonic curves and surfaces in 3-dimensional contact manifolds are investigated.
Geometric study of non-constant vector fields making hyperbolic space Ricci-Bourguignon solitonsmath.DG
Study vector fields on hyperbolic spaces to create Ricci-Bourguignon solitons.
problem Characterize vector fields on hyperbolic spaces that transform them into Ricci-Bourguignon solitons.
method Detailed geometric study of vector fields in dimensions and , focusing on dual forms in odd dimensions.
result Dual forms of these vectors are contact forms in odd dimensions.
New contact structures extend supergravity solutions.
problem Extend supergravity solutions using new contact structures.
method Introduce and investigate -contact metric structures, focusing on null contact structures.
result Appropriate direct products of -Einstein structures produce solutions of six-dimensional minimal supergravity.
Contact metric three manifolds and Lorentzian geometry with torsion in six-dimensional supergravitymath.DG
Novel contact metric structures lead to supergravity solutions.
problem Developing new contact metric structures for supergravity.
method Introducing and studying -Einstein structures.
result Constructed families of six-dimensional supergravity solutions.
The paper recovers contact forms from boundary data using vector fields and Lyapunov functions.
problem Recovering contact forms from boundary data.
method Using vector fields and Lyapunov functions, the paper describes boundary data and proves reconstruction of up to diffeomorphism.
result Boundary data allow for the reconstruction of up to a diffeomorphism of .
New theorem generalizes contact manifolds with symplectic properties.
problem Generalizing contact manifolds with symplectic structures.
method Introducing regular contact manifolds and proving properties of fibrations.
result Existence of unique symplectic form and prequantization.
In this paper, we study normal complex contact metric manifolds and we get some general results on them. Moreover, we obtained the general expression of the curvature tensor field for arbitrary vector fields. Furthermore, we show that the necessary and succient conditions to be normal a complex contact metric manifold.…
Contact Ricci flowmath.DG
We introduce the notion of contact Ricci flow associated with the Reeb vector field. Using it, we give a simple proof of the Poincare conjecture.
The paper is devoted to the complete classification of all real Lie algebras of contact vector fields on the first jet space of one-dimensional submanifolds in the plane. This completes Sophus Lie's classification of all possible Lie algebras of contact symmetries for ordinary differential equations. As a main tool we …
We employ the relationship between contact structures and Beltrami fields derived in part I of this series to construct steady nonsingular solutions to the Euler equations on a Riemannian whose flowlines trace out closed curves of all possible knot and link types simultaneously. Using careful contact-topological …
A contact metric manifold is said to be -contact, if the characteristic vector field is harmonic. We prove that the unit tangent bundle of a Riemannian manifold equipped with the standard contact metric structure is -contact if and only if is -stein.
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.
We study invariant submanifolds of manifolds endowed with a normal or complex metric contact pair with decomposable endomorphism field . For the normal case, we prove that a -invariant submanifold tangent to a Reeb vector field and orthogonal to the other one is minimal. For a -invariant submanifold everyw…
Study weak -K-contact manifolds, finding Einstein-type metrics and solitons.
problem Characterize and study geometric properties of weak -K-contact manifolds.
method Analyzing weak metric -structures, using Killing vector fields, and Jacobi operators.
result Einstein weak -K-contact manifolds are Ricci flat.
The paper connects a second order ODE to Sasakian structures and bi-Hamiltonian systems.
problem Defining and analyzing Sasakian structures associated with second order ODEs.
method Defining contact metric structures and Poisson structures, showing compatibility with bi-Hamiltonian systems.
result A compatible bi-Hamiltonian structure for the Reeb vector field is found, and conditions for the vanishing of the first Chern class are derived.
Study on Ricci-like solitons on specific geometric manifolds.
problem Characterizing Ricci-like solitons on almost contact B-metric manifolds.
method Introduced and analyzed Ricci-like solitons with Reeb vector fields on these manifolds, considering special cases and providing examples.
result Ricci-like solitons on these manifolds coincide with Einstein-like structures.
In this paper we proof that the Holomorphic angle for compact minimal surfaces in the sphere with constant Contact angle and with a parallel normal vector field must be constant.
On contact p-spheresmath.GT
We study invariant contact p-spheres on principal circle-bundles and solve the corresponding existence problem in dimension 3. Moreover, we show that contact p-spheres can only exist on (4n-1)-dimensional manifolds and we construct examples of contact p-spheres on such manifolds. We also consider relations between taut…
In -contact metric manifolds and/or -manifolds, gradient Ricci solitons, compact Ricci solitons and Ricci solitons with pointwise collinear with the structure vector field are studied.
The paper classifies geometric structures of δ-almost Yamabe solitons on paracontact metric manifolds.
problem Characterizing δ-almost Yamabe solitons on paracontact metric manifolds.
method Investigation of geometric structures under specific assumptions, including quarter-symmetric non-metric connections.
result Conditions for δ-almost Yamabe solitons to be expanding, steady, or shrinking.