Let M be an odd-dimensional Euclidean space endowed with a contact 1-form α. We investigate the space of symmetric contravariant tensor fields on M 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…
Paper shows how a Lie superalgebra can be realized using matrices.
problem Realizing the Lie superalgebra of contact projective vector fields.
method Using embedding techniques from projectively equivariant quantizations, the paper constructs a matrix realization.
result The Lie superalgebra spo(2l+2∣n) is realized as the intersection of pgl(2l+2∣n) and K(2l+1∣n). 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 1-f…
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.
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.
We study manifolds endowed with mixed metric 3--contact structures, proving that the distribution spanned by the Reeb vector fields is integrable, with totally geodesic integral manifolds, of constant sectional curvature k=±1. We also prove a result of projectability of such structures onto paraquaternionic Kähleri…
Lie systems method simplifies Riccati hierarchy study.
problem Simplifying study of Riccati hierarchy equations.
method Lie systems approach to projective Riccati equations.
result Characterization of Riccati chain equations geometrically.
Study contact structures on projective spaces, proving infinite non-isotopic structures.
problem Classify contact structures on projective spaces with specific surgery numbers.
method Detailed analysis of Gompf's Γ-invariant and d_3-invariant of tangential 2-plane fields.
result Infinitely many non-isotopic contact structures on real projective 3-space.
New Bol operators identified on superstrings.
problem Classifying Bol operators on superstrings.
method Invariant classification of Bol operators on supermanifolds.
result Many new Bol operators discovered.
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.
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 η−Ricci solitons on contact pseudo-metric manifolds and their properties.
problem Characterizing properties of contact pseudo-metric manifolds with η−Ricci solitons. method Analyzing specific types of η−Ricci solitons on Sasakian and K−contact pseudo-metric manifolds. result Properties of η−Ricci solitons on contact pseudo-metric manifolds, leading to η−Einstein manifolds under certain conditions. Study new Hopf real hypersurfaces in indefinite complex projective space.
problem Problems posed by H.~Anciaux and K.~Panagiotidou on non-degenerate real hypersurfaces.
method Changed point of view, constructed new families, obtained rigidity results.
result Classified η-umbilical real hypersurfaces and characterized Killing Reeb vector field. 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.
Study vector fields preserving Liouville form, classifying them and their bifurcations.
problem Classifying vector fields preserving Liouville form and understanding their bifurcations.
method Classification of univariate functions, local models, contact equivalence.
result Local models and bifurcations of vector fields preserved by Liouville form.
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.
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. Develops a universal Hermitian projective calculus for complex hyperbolic two-space
problem Complex hyperbolic geometry
method Algebraic invariant calculus
result Denominator-cleared identities for various geometric quantities
Study on H-contact structures on Riemannian manifolds.
problem Characterizing H-contact unit tangent bundles of Riemannian manifolds. method Analyzing the conditions for a Riemannian manifold to have a H-contact unit tangent bundle. result The unit tangent bundle of a Riemannian manifold is H-contact if and only if the manifold is 2-stein. 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.
We provide some examples of harmonic unit vector fields as normalized gradients of isoparametric functions from a K-contact geometry setting.
Study k-almost Ricci solitons on contact metric manifolds.
problem Characterize k-almost Ricci solitons on contact metric manifolds.
method Prove isometric properties and extend results for k-almost gradient Ricci solitons and k-almost Ricci solitons.
result Compact K-contact metric manifolds that are k-almost gradient Ricci solitons are isometric to a unit sphere.
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 consider Courant and Courant-Jacobi brackets on the stable tangent bundle $TM\times\mathds{R}^h$ of a differentiable manifold and corresponding Dirac, Dirac-Jacobi and generalized complex structures. We prove that Dirac and Dirac-Jacobi structures on $TM\times\mathds{R}^h$ can be prolonged to $TM\times\mathds{R}^k$,…
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 φ-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 2 are all minimal. We prove that an odd-dimensional φ-invariant submanifold …
Study of solitons in a specific type of contact metric manifold.
problem Characterizing solitons in (α,β)-contact metric manifolds. method Analyzing almost Riemann and Ricci solitons under Ricci symmetry conditions.
result Characterization of solitons in (α,β)-contact metric manifolds. Survey reviews Hamilton-Jacobi theory in various geometric settings, focusing on Jacobi and Leibniz identities.
problem Analyzing Hamilton-Jacobi theory across different geometric backgrounds.
method Geometric review of Hamilton-Jacobi theory, focusing on Jacobi and Leibniz identities.
result Novel Hamilton-Jacobi equation for conformal Hamiltonian vector fields.
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.
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 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 S1 or R 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.
Outer billiards maps on foliated surfaces with specific vector fields.
problem Characterizing vector fields that induce outer billiards maps on foliated surfaces.
method Analyzing necessary and sufficient conditions for foliation of the exterior of a hypersurface.
result Explicit periodic and unbounded orbits in a specific outer billiard map.
New method finds vector fields with maximal Jacobi operator rank in manifolds.
problem Existence of non-isotropic vector fields with maximal rank Jacobi operator.
method Effective algorithmic procedure using a quadratic parallel differential form.
result Existence of non-isotropic vector field with maximal rank Jacobi operator implies manifold is locally non-reducible.
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 \(ξ(σ)=0\).
Study projective symmetries in Finsler spaces, showing reductions and constant flag curvature.
problem Exploring projective symmetries in Finsler spaces and their properties.
method Analyzing algebraic sub-algebras and curvature invariants of projective vector fields.
result Closed Finsler spaces with negative Ricci curvature reduce to Killing vector fields.
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.
3D manifolds with certain projective fields are projectively flat.
problem Geodesic rigidity of Levi-Civita connections with essential projective vector fields.
method Proved projective flatness for specific manifolds with projective vector fields.
result Connected 3D Riemannian and closed semi-Riemannian manifolds with non-linearizable projective singularities are projectively flat.
Study vector fields on hyperbolic spaces to create Ricci-Bourguignon solitons.
problem Characterize vector fields on hyperbolic spaces Hn that transform them into Ricci-Bourguignon solitons. method Detailed geometric study of vector fields in dimensions n=2,3 and n≥3, focusing on dual forms in odd dimensions. result Dual forms of these vectors are contact forms in odd dimensions.
Biharmonic or polyharmonic curves and surfaces in 3-dimensional contact manifolds are investigated.
Novel contact metric structures lead to supergravity solutions.
problem Developing new contact metric structures for supergravity.
method Introducing and studying εη-Einstein structures. result Constructed families of six-dimensional supergravity solutions.
New contact structures extend supergravity solutions.
problem Extend supergravity solutions using new contact structures.
method Introduce and investigate ε-contact metric structures, focusing on null contact structures. result Appropriate direct products of ε-Einstein structures produce solutions of six-dimensional minimal supergravity. 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 (X,β) up to diffeomorphism. result Boundary data allow for the reconstruction of (X,β) up to a diffeomorphism of X.