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.
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.
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.
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.
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. Unified framework for classifying Sasakian, K-contact, and (κ, μ)-manifolds.
problem Classifying and understanding different types of contact metric manifolds.
method Investigating metric structures on symplectizations and proving the existence of a unique metric symplectization.
result Unified framework for classifying Sasakian, K-contact, and (κ, μ)-manifolds.
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.
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.
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. 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.
In this paper, we prove that evry 3-dimensional manifold M is a ?- recurrent N(k)-contact metric manifold if and only if it is flat. Then we classify the ?-recurrent contact metric manifolds of constant curvature. This implies that there exists no ?-recurrent N(k)-contact metric manifold, which is neither symmetric nor…
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. 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…
In this article, we study an almost contact metric structure on a G2-manifold constructed by Arikan, Cho and Salur in via the classification of almost contact metric structures given by Chinea and Gonzalez. In particular, we characterize when this almost contact metric structure is cosymplectic and narrow down the p…
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. In this paper the notion of the intrinsic geometry of an almost contact metric manifold is introduced. Description of some classes of spaces with almost contact metric structures in terms of the intrinsic geometry is given. A new type of almost contact metric spaces, more precisely, Hermitian almost contact metric spac…
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.
The paper classifies special types of contact metric manifolds with curvature conditions.
problem Classifying N(κ)-contact metric manifolds with specific curvature tensors. method Examining flatness conditions on T-curvature tensor and analyzing specific curvature tensors. result A classification of N(κ)-contact metric manifolds under various curvature conditions. In this paper the geometry of normal metric contact pair manifolds is studied under the flatness of conformal, concircular and quasi-conformal curvature tensors. It is proved that a conformal flat normal metric contact pair manifold is an Einstein manifold with a negative scalar curvature and has positive sectional cur…
The paper classifies contact 3-manifolds with critical metrics and connects entropy to optimization.
problem Classifying contact 3-manifolds with critical metrics and understanding their entropy.
method Critical metrics optimization and entropy analysis.
result Anosov contact metrics' optimization is linked to Reeb dynamics and entropy.
The study defines and analyzes semi-invariant submanifolds in complex contact metric manifolds.
problem Characterizing semi-invariant submanifolds in complex contact metric manifolds.
method Definition and derivation of relations, integrability conditions of distributions.
result Obtained useful relations and integrability conditions for semi-invariant submanifolds.
Families of linear connections are constructed on almost contact manifolds with Norden metric. An analogous connection to the symmetric Yano connection is obtained on a normal almost contact manifold with Norden metric and closed structural 1-form. The curvature properties of this connection are studied on two basic cl…
It is introduced a differentiable manifold with almost contact 3-structure which consists of an almost contact metric structure and two almost contact B-metric structures. The product of this manifold and a real line is an almost hypercomplex manifold with Hermitian-Norden metrics. It is proven that the introduced mani…
Study new warped product metric in contact geometry.
problem Define a new type warped product metric.
method Introduced a new warped product metric on product manifolds.
result Showed existence of α-Sasakian structure. The paper defines η-normality for contact and paracontact manifolds and explores their properties.
problem Defining and characterizing η-normality for contact and paracontact manifolds. method Using the Levi-Civita covariant derivative and Tanaka-like connections.
result Existence and uniqueness of connections on η-normal manifolds. In this paper, we write down Seiberg-Witten equations on contact metric manifolds of dimension 5. Any contact metric manifold has a spin^c structure. For Dirac equation we use Dirac type operators associated to the generalized Tanaka-Webster connection on spin^c spinor bundle of a contact metric manifold. For curvature…
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. Study on harmonicity of maps between different types of almost contact metric manifolds.
problem Understanding harmonicity of maps between various almost contact metric manifolds.
method Analyzing and deriving new results for different subclasses of almost contact metric manifolds.
result Obtained new results and recovered, generalized, and corrected known results.
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.…
The φ-sectional curvature of statistical structures on almost contact metric manifolds is always non-positive.
problem Analyzing the φ-sectional curvature of statistical structures on almost contact metric manifolds.
method Investigating the φ-sectional curvature induced by a statistical structure and deriving sufficient conditions.
result The φ-sectional curvature is always non-positive.
Almost contact manifolds with B-metric are considered. A special linear connection is introduced, which preserves the almost contact B-metric structure on these manifolds. This connection is investigated on some classes of the considered manifolds.
Almost contact manifolds with B-metric are considered. Of special interest are the so-called vertical classes of the almost contact B-metric manifolds. Curvature properties of these manifolds are studied. An example of 5-dimensional manifolds is constructed and characterized.
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.
The paper characterizes contact metric manifolds with specific solitons.
problem Characterizing contact metric manifolds with ∗-conformal Ricci solitons. method Analyzing properties of (2n+1)-dimensional N(k)-contact metric manifolds. result The manifold is locally isometric to a flat (n+1)-dimensional manifold and an n-dimensional manifold of constant curvature 4. Yamabe solitons defined on specific Sasaki-like manifolds.
problem Defining Yamabe solitons on Sasaki-like almost contact B-metric manifolds.
method Contact conformal transformation of manifold components to define Yamabe solitons.
result Explicit 5-dimensional Lie group example of a Yamabe soliton.
Determining the associated metrics we get a local classification of contact metric three manifolds.
We observe that the class of metric f-K-contact manifolds, which naturally contains that of K-contact manifolds, is closed under forming mapping tori of automorphisms of the structure. We show that the de Rham cohomology of compact metric f-K-contact manifolds naturally splits off an exterior algebra, and rel…
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.
We study 5-dimensional Riemannian manifolds that admit an almost contact metric structure. We classify these structures by their intrinsic torsion and review the literature in terms of this scheme. Moreover, we determine necessary and sufficient conditions for the existence of metric connections with vectorial, totally…
The paper defines a new structure on tangent sphere bundles and characterizes their properties.
problem Characterizing properties of tangent sphere bundles with contact pseudo-metric structures.
method Introduced a contact pseudo-metric structure on TεM and proved manifold properties based on constant sectional curvature. result The tangent sphere bundle TεM is (κ,μ)-contact pseudo-metric manifold if and only if the manifold M has constant sectional curvature. We study harmonic almost contact structures in the context of contact metric manifolds, and an analysis is carried out when such a manifold fibres over an almost Hermitian manifold, as exemplified by the Boothby-Wang fibration. Two types of almost contact metric warped products are also studied, relating their harmonic…
New HyperKahler structure found for 3-contact distributions on Sasakian manifolds.
problem Finding a HyperKahler structure for 3-contact distributions on Sasakian manifolds.
method Defined a special metric connection and proved curvature properties.
result 3-Sasakian manifolds with constant φα-sectional curvatures have constant holomorphic sectional curvatures in their HyperKahler contact distribution. The paper explores conditions for manifolds to have specific geometric structures.
problem Conditions for manifolds to admit almost contact structures and related structures.
method Using dual connections and statistical manifolds, the paper defines conditions for specific geometric structures.
result The paper provides conditions for a manifold to admit almost contact structures and related structures.
The paper studies special contact metric manifolds and their properties.
problem Investigating properties of contact metric manifolds with a specific equation.
method Analyzing K-contact and (κ,μ)-contact manifolds with a smooth function f satisfying a given equation. result Complete and simply connected K-contact manifolds admitting such a function are isometric to the unit sphere. We disprove the generalized Chern-Hamilton conjecture on the existence of critical compatible metrics on contact 3-manifolds. More precisely, we show that a contact 3-manifold (M,α) admits a critical compatible metric for the Chern-Hamilton energy functional if and only if it is Sasakian or its associated Reeb fl…
We study the geometry of almost contact pseudo-metric manifolds in terms of tensor fields h:=21£ξφ and ℓ:=R(⋅,ξ)ξ, emphasizing analogies and differences with respect to the contact metric case. Certain identities involving ξ-sectional curvatures are obtained. We establish necessary and su…
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.
On connected manifolds of dimension higher than three, the non-existence of 132 Chinea and González-Dávila types of almost contact metric structures is proved. This is a consequence of some interrelations among components of the intrinsic torsion of an almost contact metric structure. Such interrelations allow to des…