New criteria for Sasakian manifolds and classification of nearly cosymplectic manifolds.
problem Characterizing Sasakian and cosymplectic manifolds.
method Analyzing nearly Sasakian and nearly cosymplectic manifolds of dimensions greater than five.
result Every nearly Sasakian manifold of dimension greater than five is Sasakian.
The study examines conditions for weak nearly cosymplectic manifolds to split into products.
problem Understanding the curvature and topology of weak nearly cosymplectic manifolds.
method Analyzes the conditions for splitting and characterizes specific manifolds.
result Conditions for weak nearly cosymplectic manifolds to become Riemannian products are identified.
Recently, we have shown that there do not exist the warped product semi-slant submanifolds of cosymplectic manifolds [10]. As nearly cosymplectic structure generalizes cosymplectic ones same as nearly Kaehler generalizes Kaehler structure in almost Hermitian setting. It is interesting that the warped product semi-slant…
Study characterizes geometric properties of QGCR-null submanifolds in cosymplectic manifolds.
problem Characterizing geometric properties of QGCR-null submanifolds in cosymplectic manifolds.
method Characterization through totally umbilical and irrotational properties, and discussion of geodesity conditions.
result Characterizes geometric properties of ascreen QGCR-null submanifolds.
Almost contact structures can be identified with sections of a twistor bundle and this allows to define their harmonicity, as sections or maps. We consider the class of nearly cosymplectic almost contact structures on a Riemannian manifold and prove curvature identities which imply the harmonicity of their parametrizin…
We carry on a systematic study of nearly Sasakian manifolds. We prove that any nearly Sasakian manifold admits two types of integrable distributions with totally geodesic leaves which are, respectively, Sasakian or 5-dimensional nearly Sasakian manifolds. As a consequence, any nearly Sasakian manifold is a contact ma…
The paper derives inequalities for contact CR-warped product submanifolds in cosymplectic space forms.
problem Establishing inequalities for contact CR-warped product submanifolds in cosymplectic space forms.
method Using the Gauss equation and hypotheses for cosymplectic and nearly cosymplectic manifolds, the paper derives inequalities for the norm of the second fundamental form and the shape operator.
result The contact warped product submanifolds in cosymplectic manifolds exhibit a geometric property called D1-minimality, leading to an optimal general inequality. Only the 6-sphere has constant curvature hypersurfaces in nearly Kähler manifolds.
problem Characterizing hypersurfaces with constant curvature in six-dimensional nearly Kähler manifolds.
method Proving hypersurfaces are η-quasi umbilical in specific spaces, then using non-existence results. result Only the 6-sphere has constant curvature hypersurfaces in nearly Kähler manifolds.
The study examines Ricci solitons in specific types of manifolds.
problem Investigating Ricci solitons in almost f-cosymplectic manifolds. method Analyzing properties of almost f-cosymplectic manifolds and proving theorems about the existence and classification of Ricci solitons. result Classification of three-dimensional η-Einstein almost f-cosymplectic manifolds admitting Ricci solitons. The study explores Einstein-Weyl structures on specific types of manifolds.
problem Investigating properties of Einstein-Weyl structures on almost cosymplectic manifolds.
method Analyzing conditions for Einstein-Weyl structures on (κ,μ)-manifolds, three-dimensional compact manifolds, and K-cosymplectic manifolds. result Conditions for manifolds to be Einstein, cosymplectic, or Ricc-flat.
Paper defines cosymplectic conformal connections and shows their curvature implications.
problem Understanding conformal connections in cosymplectic manifolds.
method Introduced a cosymplectic analogue of conformal connections and proved curvature properties.
result If a cosymplectic manifold admits a cosymplectic conformal connection of zero curvature, its Bochner curvature tensor vanishes.
The study examines almost cosymplectic statistical manifolds and their properties.
problem Characterizing and understanding almost cosymplectic statistical manifolds.
method Analyzing basic properties, proving a characterization theorem, studying curvature, and constructing examples.
result Characterization theorem and corollary for almost cosymplectic statistical manifolds with Kaehler leaves.
Extends integrability to cosymplectic manifolds.
problem Integrability of Hamiltonian systems on cosymplectic manifolds.
method Extended Arnold-Liouville and noncommutative integrability to cosymplectic manifolds, proved a variant of non-commutative integrability for specific fields, constructed action-angle variables.
result Variant of non-commutative integrability for evaluation and Reeb vector fields on cosymplectic manifolds.
In this paper we study K-cosymplectic manifolds, i.e., smooth cosymplectic manifolds for which the Reeb field is Killing with respect to some Riemannian metric. These structures generalize coKähler structures, in the same way as K-contact structures generalize Sasakian structures. In analogy to the contact case, we dis…
Study Ricci solitons on f-cosymplectic manifolds.
problem Characterize Ricci solitons on f-cosymplectic manifolds. method Analyze two classes of Ricci solitons: contact and gradient. Provide local classifications and properties.
result Local classifications of f-cosymplectic manifolds with gradient Ricci solitons. The study examines quasi-Einstein structures on specific types of manifolds.
problem Characterizing quasi-Einstein structures on almost cosymplectic manifolds.
method Analyzes properties of quasi-Einstein structures on almost cosymplectic manifolds, Lie groups, and K-cosymplectic manifolds.
result Closed quasi-Einstein structures on compact almost cosymplectic manifolds do not exist.
Toric actions in cosymplectic geometry linked to symplectomorphisms.
problem Understanding toric actions in cosymplectic geometry.
method Showed that compact toric cosymplectic manifolds are mapping tori of equivariant symplectomorphisms of toric symplectic manifolds.
result Compact toric cosymplectic manifolds are mapping tori of equivariant symplectomorphisms of toric symplectic manifolds.
The paper classifies cosymplectic manifolds with critical metrics in dimension 3.
problem Classifying cosymplectic manifolds with critical metrics.
method Study of Chern-Hamilton energy functional on compact cosymplectic manifolds.
result Classification of manifolds admitting critical compatible metrics in dimension 3.
The paper derives Chen inequalities for statistical submanifolds in cosymplectic manifolds.
problem Deriving Chen inequalities for statistical submanifolds in cosymplectic manifolds.
method Analyzing statistical cosymplectic manifolds and Legendrian submanifolds to derive Chen inequalities.
result Chen inequalities for statistical submanifolds in cosymplectic manifolds and Legendrian submanifolds are derived.
We introduce cosymplectic circles and cosymplectic spheres, which are the analogues in the cosymplectic setting of contact circles and contact spheres. We provide a complete classification of compact 3-manifolds that admit a cosymplectic circle. The properties of tautness and roundness for a cosymplectic p-sphere are…
We continue the program of Chinea, De Leon and Marrero who studied the topology of cosymplectic manifolds. We study 3-cosymplectic manifolds which are the closest odd-dimensional analogue of hyper-Kaehler structures. We show that there is an action of the Lie algebra so(4,1) on the basic cohomology spaces of a compact …
The author aims to classify 3D (κ,μ)-manifolds, focusing on para-contact and almost para-cosymplectic cases.
problem Classifying 3D (κ,μ)-manifolds, especially para-contact and almost para-cosymplectic. method Analyzing and conjecturing about structures that provide classification.
result These 3D manifolds are fundamental building blocks for higher-dimensional manifolds.
Abstract: Study Hamiltonian systems on almost cosymplectic manifolds, extending contact Hamiltonian systems.
problem Extend Hamiltonian systems to almost cosymplectic manifolds.
method Determine Hamiltonian vector field on odd-dimensional almost cosymplectic manifolds.
result Extend equations of motion to generalized transitive almost cosymplectic structures.
Researchers tackle the globalization problem of locally cosymplectic Hamiltonian dynamics.
problem Globalization problem of locally cosymplectic Hamiltonian dynamics.
method Investigate the geometry of locally conformally cosymplectic manifolds and provide a geometric Hamilton-Jacobi theory.
result Provide a geometric Hamilton-Jacobi theory on locally conformally cosymplectic manifolds.
Contact manifolds have a fixed dimension of 5 under certain curvature conditions.
problem Identifying the dimension of contact manifolds with specific curvature properties.
method Analyzing a class of contact manifolds with an almost contact metric structure and constant sectional curvature.
result All manifolds in the class considered have dimension 5 under the given curvature assumption.
Study cosymplectic diffeomorphisms and their properties.
problem Understanding cosymplectic manifolds and their diffeomorphisms.
method Analyzes the group of cosymplectic and almost cosymplectic diffeomorphisms, proving theorems and deriving norms.
result Proves the identity component of cosymplectic diffeomorphisms is C0-closed, and gives conditions for almost cosymplectic diffeomorphisms to be cosymplectic. In the paper we develop a framework for the alternative way of the study of a local geometry of almost cosymplectic manifolds with Kahlerian leaves. The main idea is to apply the concept of a geometry and analysis of CR manifolds. Locally the almost cosymplectic manifold is modeled on the 'mixed' space RxCn. There is g…
The paper studies ∗-η-Ricci-Yamabe solitons on α-Cosymplectic manifolds.
problem Exploring ∗-η-Ricci-Yamabe solitons on α-Cosymplectic manifolds. method Analyzing curvature properties and developing soliton characteristics with respect to quarter-symmetric metric connection.
result Characteristics and nature of ∗-η-Ricci-Yamabe solitons on α-Cosymplectic manifolds. In this paper we introduce slant Riemannian submersions from cosymplectic manifolds onto Riemannian manifolds. We obtain some results on slant Riemannian submersions of a cosymplectic manifolds. We also give examples and inequalities between the scalar curvature and squared mean curvature of fibres of such slant submer…
Recently, M. Atcken studied Contact CR-warped prod- uct submanifolds in cosymplectic space forms and established gen- eral sharp inequalities for CR-warped products in a cosymplectic manifold [1]. In the present paper, we obtain an inequality for the squared norm of the second fundamental form in terms of constant ?φ…
Generalizes symplectic reduction to cosymplectic groupoid actions.
problem Symplectic reduction for cosymplectic groupoid actions.
method Introduced cosymplectic groupoid actions and proved a theorem.
result Proved a theorem analogous to Mikami-Weinstein theorem.
This paper extends symplectic reduction to cosymplectic groupoids and explores their properties.
problem Cosymplectic groupoids and their reductions.
method Analogous to symplectic reduction, the authors extend the Marsden-Weinstein-Meyer reduction to cosymplectic groupoids.
result Integration commutes with reduction for algebroids associated with cosymplectic groupoids.
We obtain a generalization of the Kodaira-Morrow stability theorem for cosymplectic structures. We investigate cosymplectic geometry on Lie groups and on their compact quotients by uniform discrete subgroups. In this way we show that a compact solvmanifold admits a cosymplectic structure if and only if it is a finite q…
Main interest of the present paper is to investigate the almost α-cosymplectic manifolds for which the characteristic vector field of the almost α-cosymplectic structure satisfies a specific (κ,μ,ν)-nullity condition. This condition is invariant under D-homothetic deformation of the almost cosymplectic (κ,μ,ν)-spaces i…
Study calculates Poisson cohomology for symplectic-like manifolds.
problem Computing cohomology of symplectic-like manifolds.
method Analyzes Poisson structures on manifolds with symplectic regions outside a set of hypersurfaces.
result Poisson cohomology computed for a specific class of manifolds.
In the paper there are described new examples of conformally flat three dimensional almost cosymplectic manifolds. All these manifolds form a class which was completely characterized.
We review topological properties of Kähler and symplectic manifolds, and of their odd-dimensional counterparts, coKähler and cosymplectic manifolds. We focus on formality, Lefschetz property and parity of Betti numbers, also distinguishing the simply-connected case (in the Kähler/symplectic situation) and the b1=1 c…
We present a compared analysis of some properties of 3-Sasakian and 3-cosymplectic manifolds. We construct a canonical connection on an almost 3-contact metric manifold which generalises the Tanaka-Webster connection of a contact metric manifold and we use this connection to show that a 3-Sasakian manifold does not adm…
We introduce anti-invariant Riemannian submersions from cosymplectic manifolds onto Riemannian manifolds. We survey main results of anti-invariant Riemannian submersions defined on cosymplectic manifolds. We investigate necessary and sufficient condition for an anti-invariant Riemannian submersion to be totally geodesi…
New method for studying t-dependent Hamilton equations on cosymplectic manifolds.
problem Existence and stability of solutions of t-dependent Hamilton equations. method Develops a cosymplectic energy-momentum method for Hamilton equations with more types of symmetries.
result Provides a more general framework for studying t-dependent Hamilton equations. New metric structures generalize Sasakian and cosymplectic structures, proving rigidity and finding conditions.
problem Generalizing Sasakian and cosymplectic structures to new metric structures.
method Introducing weak structures and proving rigidity of Sasakian structures.
result Any weak Sasakian structure is homothetically equivalent to a Sasakian structure.
Study Lie algebras of symmetries for odd-dimensional structures.
problem Understanding Lie algebras of infinitesimal symmetries in almost-cosymplectic-contact structures.
method Analyzing Lie subalgebras of a Lie algebra defined by pairs of 1-forms and functions.
result Described Lie subalgebras generating infinitesimal symmetries of basic tensor fields.
In this paper we study the hemi-slant submanifolds of cosymplectic manifolds. Necessary and sufficient conditions for distributions to be integrable are worked out. Some important results are obtained in this direction.
In this note we briefly review some recent results of the authors on the topological and geometrical properties of 3-cosymplectic manifolds.
New geometric structures on 3-manifolds discovered and proven for all closed orientable ones.
problem Existence of specific geometric structures on 3-manifolds.
method Generalized geometry and surgery techniques.
result Any closed orientable 3-manifold admits a B3-generalized complex structure. The paper extends Vlasov kinetic theory to time-dependent dynamics using cosymplectic and cocontact manifolds.
problem Extending Vlasov kinetic theory to time-dependent dynamics.
method Introducing geometric kinetic theories within cosymplectic and cocontact manifolds.
result Alternative realizations of cosymplectic and cocontact kinetic theories linked via Poisson/momentum maps.
Examines Lie algebras for symmetries in odd-dimensional manifolds.
problem Symmetry analysis in odd-dimensional manifolds.
method Analyzes local Lie algebras of pairs of functions.
result Identifies infinitesimal symmetries of almost-cosymplectic-contact structures.
We give an up-to-date overview of geometric and topological properties of cosymplectic and coKaehler manifolds. We also mention some of their applications to time-dependent mechanics.