The geometry of contact metric three manifolds

Generalized Quasi-Einstein Manifolds in Contact Geometry

In this study, we investigate generalized quasi-Einstein structure for normal metric contact pair manifolds. Firstly, we deal with elementary properties and examine, existence, and characterizations of generalized quasi-Einstein normal metric contact pair manifold. Secondly, the generalized quasi-constant curvature of normal metric contact pair manifolds are studied and it is proven that a normal metric contact pair manifold with generalized quasi-constant curvature is a generalized quasi-Einstein manifold. Normal metric contact pair manifolds satisfying cyclic Ricci tensor and the Codazzi typ…

$H$-contact unit tangent sphere bundles of Riemannian manifolds

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.

On phi-Recurrent (k, m)-Contact Metric Manifolds

In this paper, we show that there is no phi-recurrent Sasakian manifold. Then we prove that the only flat 3-dimensional manifolds are phi-recurrent (k, m)-contact metric manifolds.

How to construct all metric $f$-$K$-contact manifolds

We show that any compact metric $f$-$K$-contact, respectively $S$-manifold is obtained from a compact $K$-contact, respectively Sasakian manifold by an iteration of constructions of mapping tori, rotations, and type II deformations.

Seiberg-Witten Like equations on 5-dimensional contact metric 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 equation we need to self-duality concept. Self-duality concept is significant on odd dimensional manifolds, particularly, on 5-dimensional contact manifolds. Finally, we give a global solution to these equations on strictly pseudoconvex CR manifolds.

Ricci-generalized pseudosymmetric $(κ, μ)$-contact metric manifolds

In this paper we classify Ricci-generalized pseudosymmetric $(κ, μ)$-contact metric manifolds in the sense of Deszcz .

Sasaki-like almost contact complex Riemannian manifolds

A Sasaki-like almost contact complex Riemannian manifold is defined as an almost contact complex Riemannian manifold which complex cone is a holomorphic complex Riemannian manifold. Explicit compact and non-compact examples are given. A canonical construction producing a Sasaki-like almost contact complex Riemannian manifold from a holomorphic complex Riemannian manifold is presented and called an $S^1$-solvable extension.

A class of 3-dimensional contact metric manifolds

We classify the contact metric 3-manifolds that satisfy ||gradλ||=1 and \nabla_{ξ}τ=2aτϕ.

Almost contact curves in trans-Sasakian 3-manifolds

This paper is devoted to the study of curvature and torsion of almost contact curves in trans-Sasakian 3-Manifolds. The conditions for the frenet curves to be almost contact curves in trans-Sasakian 3-manifolds have been obtained.

Curvature properties on some classes of almost contact manifolds with B-metric

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.

Complex almost contact metric structures on complex hypersurfaces in hyperkahler manifolds

In this paper, we construct complex metric structures on complex hypersurfaces in hyperkahler manifolds. This construction is that in contact geometry.

The topology of quaternionic contact manifolds

We explore the consequences of curvature and torsion on the topology of quaternionic contact manifolds with integrable vertical distribution. We prove a general Myers theorem and establish a Cartan-Hadamard result for almost qc-Einstein manifolds.

Curvature Properties of Normal Complex Contact metric Manifolds

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. Also, we give some new equailities for Ricci curvature of normal complex contact metric manifolds.

A natural connection on some classes of almost contact manifolds with B-metric

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.

Mixed metric 3-contact manifolds and paraquaternionic Kähler manifolds

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 $k=\pm1$. We also prove a result of projectability of such structures onto paraquaternionic Kählerian structures.

The Intrinsic Geometry of Almost Contact Metric Manifolds

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 spaces, is introduced.

Quasi-isometry between two almost contact metric manifolds

In this paper the notion of quasi-isometry between two Riemannian manifolds has been introduced. This idea is also imposed to study quasi-isometry between two almost contact metric manifolds. Moving further, some curvature properties of two quasi-isometrically embedded almost contact metric manifolds, $N(k)-$contact metric manifolds and Sasakian manifolds are investigated. Next, an illustrative example of a quasi-isometry between two Sasakian structures is constructed. Finally, a relation between the scalar curvature and the quasi-isometric constants for two quasi-isometric Riemannian manifold…

Contact hypersurfaces in Kaehler manifolds

A contact hypersurface in a Kaehler manifold is a real hypersurface for which the induced almost contact metric structure determines a contact structure. We carry out a systematic study of contact hypersurfaces in Kaehler manifolds. We then apply these general results to obtain classifications of contact hypersurfaces with constant mean curvature in the complex quadric SO(n+2)/SO(n)SO(2) and its noncompact dual space SO(n,2)/SO(n)SO(2) for n > 2.

On the twistor space of a quaternionic contact manifold

In this note, we prove that the CR manifold which is induced from the canonical parabolic geometry of a quaternionic contact (qc) manifold via a Fefferman-type construction is equivalent to the CR twistor space of the qc manifold defined by O. Biquard.

弱 $K$-接触流形上的广义 Ricci 孤立子与 Einstein 度量

Generalized Ricci solitons and Einstein metrics on weak $K$-contact manifolds

本文研究推广经典理论的弱接触度量结构与弱 K-结构，刻画了弱 K-接触流形，并找出了具有平行 Ricci 张量或广义 Ricci 孤立子结构的弱 K-接触流形成为 Einstein 流形的充分条件。

Submanifolds with harmonic mean curvature vector field in contact 3-manifolds

Biharmonic or polyharmonic curves and surfaces in 3-dimensional contact manifolds are investigated.

Manifolds with almost contact 3-structure and metrics of Hermitian-Norden type

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 manifold of cosymplectic type is at. Some examples of the studied manifolds are given.

On basic curvature identities for almost (para)contact metric manifolds

In monograph of D. E. Blair "Riemannian geometry of contact and symplectic manifolds" and in the paper of S. Zamkovoy "Canonical connections on paracontact manifolds", the curvature identities respectively for contact and paracontact metric manifold are proved. We obtain the curvature identity in the wider class of manifolds, which generalizes results presented in above mentioned publications. Moreover, we present some properties of almost (para)hermitian structure on a special semiproduct of R+ and an almost (para)contact metric manifold. This semiproduct plays an auxiliary role in proving ma…

On contact pseudo-metric manifolds satisfying a nullity condition

In this paper, we aim to introduce and study $(κ, μ)$-contact pseudo-metric manifold and prove that if the $φ$-sectional curvature of any point of $M$ is independent of the choice of $φ$-section at the point, then it is constant on $M$ and accordingly the curvature tensor. Also, we introduce generalized $(κ, μ)$-contact pseudo-metric manifold and prove for $n>1$, that a non-Sasakian generalized $(κ, μ)$-contact pseudo-metric manifold is a $(κ, μ)$-contact pseudo-metric manifold.

Clairaut anti-invariant submersions from normal almost contact metric manifolds

We investigate new Clairaut conditions for anti-invariant submersions from normal almost contact metric manifolds onto Riemannian manifolds. We prove that there is no Clairaut anti-invariant submersion admitting vertical Reeb vector field when the total manifold is Sasakian. Several illustrative examples are also included.

Curvature identities on contact metric manifolds and their applications

We study curvature identities on contact metric manifolds on the geometry of the corresponding almost Käehler cones, and we provide applications of the derived curvature identities.

Bishop and Laplacian Comparison Theorems on Three Dimensional Contact Subriemannian Manifolds with Symmetry

We prove a Bishop volume comparison theorem and a Laplacian comparison theorem for three dimensional contact subriemannian manifolds with symmetry.

关于Sasakian统计流形的切触CR-积

On Contact CR-Product of Sasakian statistical manifold

本文研究Sasakian统计流形中切触CR-子流形的几何性质，刻画其分布可积性，建立关于对偶联络的全测地与脐性结果，并引入切触CR-积的统计版本。

Lie groups as 3-dimensional almost contact B-metric manifolds in the main vertical classes

Almost contact B-metric manifolds of dimension 3 are constructed by a two-parametric family of Lie groups. The class of these manifolds in a known classification of almost contact B-metric manifolds is determined as the direct sum of the main vertical classes. The type of the corresponding Lie algebras in the Bianchi classification is given. Some geometric characteristics and properties of the considered manifolds are obtained.

On non-gradient $(m,ρ)$-quasi-Einstein contact metric manifolds

Many authors have studied Ricci solitons and their analogs within the framework of (almost) contact geometry. In this article, we thoroughly study the $(m,ρ)$-quasi-Einstein structure on a contact metric manifold. First, we prove that if a $K$-contact or Sasakian manifold $M^{2n+1}$ admits a closed $(m,ρ)$-quasi-Einstein structure, then it is an Einstein manifold of constant scalar curvature $2n(2n+1)$, and for the particular case -- a non-Sasakian $(k,μ)$-contact structure -- it is locally isometric to the product of a Euclidean space $\RR^{n+1}$ and a sphere $S^n$ of constant curvature $4$. …

On the geometry of null hypersurfaces of indefinite complex contact manifolds

We study the geometry of null hypersurfaces in indefinite complex contact manifolds. We prove several classification results for a variety of well-known null hypersurfaces, including the totally umbilic, totally screen umbilic, and the screen conformal ones. Furthermore, a characterization of the ambient space is given in case the underlying null hypersurface is totally contact umbilic, totally contact screen umbilic or contact screen conformal, i.e. we have proved that the ambient complex contact manifold must be a space of constant $GH$-sectional curvature of $-3$.

Subclasses of the Conformal Almost Contact Metric Manifolds

A classification scheme of the conformal almost contact metric manifolds with respect to the covariant derivative of the Lee form is given. The subclasses of one basic class and their exact characterizations by the maximal subgroups of the contact conformal group preserving itself are found.

准四元数切触流形的扭量与反射空间

Twistor and Reflector spaces for paraquaternionic contact manifolds

本文研究准四元数切触流形上的扭量空间与反射空间纤维丛，证明它们具有一种总是可积的内在几何结构。

Matrix Lie groups as 3-dimensional almost contact B-metric manifolds

The object of investigation are Lie groups considered as almost contact B-metric manifolds of the lowest dimension three. It is established a correspondence of all basic-class-manifolds of the Ganchev-Mihova-Gribachev classification of the studied manifolds and the explicit matrix representation of Lie groups. Some known Lie groups are equiped with almost contact B-metric structure of different types.

Semi-invariant Submanifolds of Normal Complex Contact Metric Manifolds

In this paper, we study on semi-invariant submanifolds of normal complex contact metric manifolds. We give the definition of such submanifolds and we obtain useful relations. Moreover, we give the integrability conditions of distributions.

Contact 3-manifolds and Ricci solitons

A contact 3-manifold $M$ admitting a transversal Ricci soliton $(g,v,λ)$ is either Sasakian or locally isometric to one of the Lie groups SU(2), $SL(2,R)$, E(2), E(1,1) with a left invariant metric.

Geometric connections and geometric Dirac operators on contact manifolds

We construct several natural connections and Dirac type operators on a general metric contact manifold which are more sensitive to the geometric background. In the special case of CR manifolds these connections are also compatible with the CR structure and include among them the Webster connection. We also describe several Weitzenbock type formulae. Our method is based on work of P. Gauduchon applied to two almost complex manifolds naturally associated to a given metric contact manifold.

对称接触流形的解析挠率

Analytic Torsion for Symmetric Contact Manifolds

本文回顾Rumin复形与解析接触挠率，利用表示论及Boothby-Wang构造，将对称接触流形的接触挠率计算化为调和形式与Dolbeault上同调，并推广至等变情形以确定孤立不动点处的等变接触挠率。

Linear Connections on Normal Almost Contact Manifolds with Norden Metric

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 classes of normal almost contact manifolds with Norden metric.

Gray Curvature Identities for Almost Contact Metric Manifolds

The aim of this research is the study of Gray curvature identities, introduced by Alfred Gray in \cite{kn:Gra76} for the class of almost hermitian manifolds. As known till now, there is no equivalent for the class of almost contact manifolds. For this purpose we use the Boohby-Wang fibration and the warped manifolds construction in order to establish which identities could be satisfied by an almost contact manifold.

Almost contact metric 5-manifolds and connections with torsion

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 skew-symmetric or traceless cyclic torsion that are compatible with the almost contact metric structure. Finally, we examine explicit examples of almost contact metric 5-manifolds from this perspective.

Quaternionic contact Einstein manifolds

The main result is that the qc-scalar curvature of a seven dimensional quaternionic contact Einstein manifold is a constant. In addition, we characterize qc-Einstein structures with certain flat vertical connection and develop their local structure equations. Finally, regular qc-Ricci flat structures are shown to fibre over hyper-Kahler manifolds.

On Harmonicity Of Holomorphic Maps Between Various Types Of Almost Contact Metric Manifolds

Harmonicity of holomorphic maps between various subclasses of almost contact metric manifolds is discussed. Consequently, some new results are obtained. Also some known results are recovered, some of them are generalized and some of them are corrected.

$η-$Ricci solitons on contact pseudo-metric manifolds

In this paper, we prove that a Sasakian pseudo-metric manifold which admits an $η-$Ricci soliton is an $η-$Einstein manifold, and if the potential vector field of the $η-$Ricci soliton is not a Killing vector field then the manifold is $\mathcal{D}-$homothetically fixed, and the vector field leaves the structure tensor field invariant. Next, we prove that a $K-$contact pseudo-metric manifold with a gradient $η-$Ricci soliton metric is $η-$Einstein. Moreover, we study contact pseudo-metric manifolds admitting an $η-$Ricci soliton with a potential vector field point-wise colinear with the Reeb v…

A dimensional restriction for a class of contact manifolds

In this work we consider a class of contact manifolds $(M,η)$ with an associated almost contact metric structure $(ϕ, ξ, η,g)$. This class contains, for example, nearly cosymplectic manifolds and the manifolds in the class $C_9\oplus C_{10}$ defined by Chinea and Gonzalez. All manifolds in the class considered turn out to have dimension $4n+1$. Under the assumption that the sectional curvature of the horizontal $2$-planes is constant at one point, we obtain that these manifolds must have dimension $5$.

Invariants and Infinitesimal Transformations for Contact Sub-Lorentzian Structures on 3-Dimensional Manifolds

In this article we develop some elementary aspects of a theory of symmetry in sub-Lorentzian geometry. First of all we construct invariants characterizing isometric classes of sub-Lorentzian contact 3 manifolds. Next we characterize vector fields which generate isometric and conformal symmetries in general sub-Lorentzian manifolds. We then focus attention back to the case where the underlying manifold is a contact 3 manifold and more specifically when the manifold is also a Lie group and the structure is left-invariant.

Natural operations on differential forms on contact manifolds

We characterize all natural linear operations between spaces of differential forms on contact manifolds. Our main theorem says roughly that such operations are built from some algebraic operators which we introduce and the exterior derivative.

On simply connected K-contact non-Sasakian manifolds

We solve the problem posed by Boyer and Galicki about the existence of K-contact simply connected manifolds with no Sasakian structure. Although the result lies in the framework of metric contact geometry, our methods come from contact and symplectic geometry and are based on the method of fat bundles developed by Sternberg, Weinstein and Lerman.

The space of Cauchy-Riemann structures on 3-D compact contact manifolds

We study the action of the group of contact diffeomorphisms on CR deformations of compact three-dimensional CR manifolds. Using anisotropic function spaces and an anisotropic structure on the space of contact diffeomorphisms, we establish the existence of local transverse slices to the action of the contact diffeomorphism group in the neighbourhood of a fixed embeddable strongly pseudoconvex CR structure.

On canonical-type connections on almost contact complex Riemannian manifolds

We consider a pair of smooth manifolds, which are the counterparts in the even-dimensional and odd-dimensional cases. They are separately an almost complex manifold with Norden metric and an almost contact manifolds with B-metric, respectively. They can be combined as the so-called almost contact complex Riemannian manifold. This paper is a survey with additions of results on differential geometry of canonical-type connections (i.e. metric connections with torsion satisfying a certain algebraic identity) on the considered manifolds.

On some Lie groups as 5-dimensional almost contact B-metric manifolds with three natural connections

Almost contact manifolds with B-metric are considered. There are studied three natural connections (i.e. linear connections preserving the structure tensors) determined by conditions for their torsions. These connections are investigated on a family of Lie groups considered as 5-dimensional almost contact B-metric manifolds.

Contact metric $(κ,μ)$-spaces as bi-Legendrian manifolds

We regard a contact metric manifold whose Reeb vector field belongs to the $(κ,μ)$-nullity distribution as a bi-Legendrian manifold and we study its canonical bi-Legendrian structure. Then we characterize contact metric $(κ,μ)$-spaces in terms of a canonical connection which can be naturally defined on them.

Almost contact B-metric manifolds as extensions of a 2-dimensional space-form

The object of investigations are almost contact B-metric manifolds which are derived as a product of a real line and a 2-dimensional manifold equipped with a complex structure and a Norden metric. There are used two different methods for generation of the B-metric on the product manifold. The constructed manifolds are characterised with respect to the Ganchev-Mihova-Gribachev classification and their basic curvature properties.

On generalized $ϕ$-recurrent generalized $(k, μ)$-contact metric manifolds

The present paper deals with the study of generalized $ϕ$-recurrent generalized $(k,μ)$-contact metric manifolds with the existence of such notion by a proper example.