We introduce the notion of contact pair structure and the corresponding associated metrics, in the same spirit of the geometry of almost contact structures. We prove that, with respect to these metrics, the integral curves of the Reeb vector fields are geodesics and that the leaves of the Reeb action are totally geodesic. Mreover, we show that, in the case of a metric contact pair with decomposable endomorphism, the characteristic foliations are orthogonal and their leaves carry induced contact metric structures.
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.
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.
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 locally ?-symmetric.
We study almost contact metric structures induced by 2-fold vector cross products on manifolds with $G_2$ structures. We get some results on possible classes of almost contact metric structures. Finally we give examples.
The study of harmonicity for almost contact metric structures was initiated by Vergara-Díaz and Wood and continued by González-Dávila and the present author. By using the intrinsic torsion and some restriction on the type of almost contact metric structure, González-Dávila and the present author have characterised harmonic structures by showing conditions relating harmonicity and classes of almost contact metric structures. Here we do this in a more general context. Moreover, the harmonicity of almost contact metric structures as a map is also done in such a general context. Finally, some rema…
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 of a metric contact pair with orthogonal characteristic foliations inherits a contact form with an almost contact metric structure, and this induced structure is contact metric if and only if the submanifold is tangent to one Reeb vector field and orthogonal to the other one. Fur…
In this paper, we construct complex metric structures on complex hypersurfaces in hyperkahler manifolds. This construction is that in contact geometry.
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.
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.
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.
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 describe the exterior derivatives of some relevant forms in the context of almost contact metric geometry.
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 harmonicity to that of the almost Hermitian structure on the base or fibre.
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.
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.
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.
Given a manifold $M$ endowed with a contact 1-form $α$, a bi-invariant pseudo-metric $\varrho_α$ is introduced on $Cont(M,α)$, the compactly supported identity component of the group of all strict contactomorphisms of $(M,α)$. For $M$ open $\varrho_α$ is a 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.
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.
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.
Contact metric $(κ,μ)$-spaces are generalizations of Sasakian spaces. We introduce a weak $(κ,μ)$ condition as a generalization of the K-contact one and show that many of the known results from generalized Sasakian geometry hold in the weaker generalized K-contact geometry setting. In particular, we prove existence of K-contact and $(κ,μ=2)$-structures under some conditions on the Boeckx invariant.
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.
There are introduced and studied a pair of associated Schouten-van Kampen affine connections adapted to the contact distribution and an almost contact B-metric structure generated by the pair of associated B-metrics and their Levi-Civita connections. By means of the constructed non-symmetric connections, the basic classes of almost contact B-metric manifolds are characterized. Curvature properties of the considered connections are obtained.
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.
We study the Schouten-van Kampen connection associated to an almost contact or paracontact metric structure. With the help of such a connection, some classes of almost (para) contact metric manifolds are characterized. Certain curvature properties of this connection are found.
We present a classification of the complete, simply connected, contact metric $(κ,μ)$-spaces as homogeneous contact metric manifolds, by studying the base space of their canonical fibration. According to the value of the Boeckx invariant, it turns out that the base is a complexification or a para-complexification of a sphere or of a hyperbolic space. In particular, we obtain a new homogeneous representation of the contact metric $(κ,μ)$-spaces with Boeckx invariant less than $-1$.
On a main class of the almost contact manifolds with B-metric, it is described the family of the linear connections preserving the manifold's structures by 4 parameters. In this family there are determined the canonical-type connection and the connection with zero parameters.
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.
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…
A contact pair on a manifold always admits an associated metric for which the two characteristic contact foliations are orthogonal. We show that all these metrics have the same volume element. We also prove that the leaves of the characteristic foliations are minimal with respect to these metrics. We give an example where these leaves are not totally geodesic submanifolds.
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.
In this paper, we obtain Chen's inequalities for submanifolds in $(κ,μ)$-contact space form with two kinds of generalized semi-symmetric non-metric connections.
On a manifold with an almost contact metric structure $(φ,\vecξ,η,g,X,D)$ the notions of the interior and the $N$-prolonged connections are introduced. Using the $N$-prolonged connection, a new almost contact metric structure is defined on the distribution $D$. The properties of this structure are studied.
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 curvature. It is also shown that a concircular flat normal metric contact pair manifold an Einstein manifold. Finally it is obtained that a quasi-conformal flat normal metric contact pair manifold is an Einstein manifold with a positive scalar curvature and, is a space of constant c…
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.
We introduce a class of half lightlike submanifolds of almost contact B-metric manifolds and prove that such submanifolds are semi-Riemannian with respect to the associated B-metric. Object of investigations are also minimal of the considered submanifolds and a non-trivial example for them is given.
We study non-paraSasakian paracontact metric $(κ,μ)$-spaces with $κ=-1$ (equivalent to $h^2=0$ but $h\neq0$). These manifolds, which do not have a contact geometry counterpart, will be classified locally in terms of the rank of $h$. We will also give explicit examples of every possible constant rank of $h$.
We prove that any contact metric $(κ,μ)$-space $(M,ξ,ϕ,η,g)$ admits a canonical paracontact metric structure which is compatible with the contact form $η$. We study such canonical paracontact structure, proving that it verifies a nullity condition and induces on the underlying contact manifold $(M,η)$ a sequence of compatible contact and paracontact metric structures verifying nullity conditions. The behavior of that sequence, related to the Boeckx invariant $I_M$ and to the bi-Legendrian structure of $(M,ξ,ϕ,η,g)$, is then studied. Finally we are able to define a canonical Sasakian structure …
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…
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.
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.
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.
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.
We study the geometry of almost contact pseudo-metric manifolds in terms of tensor fields $h:=\frac{1}{2}£_ξφ$ and $\ell := R(\cdot,ξ)ξ$, emphasizing analogies and differences with respect to the contact metric case. Certain identities involving $ξ$-sectional curvatures are obtained. We establish necessary and sufficient condition for a nondegenerate almost $CR$ structure $(\mathcal{H}(M), J, θ)$ corresponding to almost contact pseudo-metric manifold $M$ to be $CR$ manifold. Finally, we prove that a contact pseudo-metric manifold $(M,φ,ξ,η,g)$ is Sasakian if and only if the corresponding nonde…
The notions of a twistor space of a contact manifold and a contact connection on such a manifold have been introduced by L. Vezzoni as extensions of the corresponding notions in the case of a symplectic manifold. Given a contact connection on a contact manifold one can define an almost $CR$-structure on its twistor space and Vezzoni has found the integrability condition for this structure. In the present paper it is observed that the $CR$-structure is induced by an almost contact metric structure. The main goal of the paper is to obtain necessary and sufficient conditions for normality of this…
Contact variation of almost contact pseudo-metric structures
Institute of Mathematics, Jagiellonian University in Kraków（克拉科夫雅盖隆大学数学研究所）；Department of Applied Mathematics, University of Agriculture in Krakow（克拉科夫农业大学应用数学系）
In this paper, we present the invalidities of the results in [3], because of their definition of a semi-invariant submanifold of an almost complex contact metric manifold is not true .
The object of the present study is to study 3-dimensional generalized ($κ,μ$)-contact metric manifolds with $\tilde{W}\cdot R=0$ and $\tilde{W}\cdot H=0$ to cover all the eight equivalent classes given in \cite{Shaikh2}.
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.
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…
We consider manifolds endowed with metric contact pairs for which the two characteristic foliations are orthogonal. We give some properties of the curvature tensor and in particular a formula for the Ricci curvature in the direction of the sum of the two Reeb vector fields. This shows that metrics associated to normal contact pairs cannot be flat. Therefore flat non-Kähler Vaisman manifolds do not exist. Furthermore we give a local classification of metric contact pair manifolds whose curvature vanishes on the vertical subbundle. As a corollary we have that flat associated metrics can only exi…
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.
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…
