Compact K-contact Research Articles

How to construct all metric f-K-contact manifolds

Abstract 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.

The $k$-almost Yamabe solitons and contact metric manifolds

We introduce the concept of a k-almost Yamabe soliton which extends naturally from Yamabe solitons. Our aim is to study the k-almost Yamabe soliton (g,V,k,λ) on a contact metric manifold M2n+1. Firstly, for a general contact metric manifold, it is proved that V is Killing if the potential vector field V is a contact vector field and that M is K-contact if V is collinear with Reeb vector field. Secondly, we prove that a compact K-contact manifold, admitting a k-almost Yamabe gradient soliton, is isometric to a standard unit sphere. Moreover, for a complete Sasakian manifold admitting a k-almost Yamabe soliton, we show that it is isometric to a standard unit sphere 𝕊2n+1(1) when n>1 and for n=1, M is also isometric to a standard unit sphere if it admits a closed k-almost Yamabe soliton. Finally, we consider a contact metric (κ,μ)-manifold with a nontrivial k-almost Yamabe gradient soliton and show that it is flat in dimension 3 and in higher dimension M is locally isometric to En+1×𝕊n(4). In the end, we construct two examples of contact metric manifolds with a k-almost Yamabe soliton.

Riemann soliton within the framework of contact geometry

In this paper, we study contact metric manifold whose metric is a Riemann soliton. First, we consider Riemann soliton (g, V ) with V as contact vector field on a Sasakian manifold (M, g) and in this case we prove that M is either of constant curvature +1 (and V is Killing) or D-homothetically fixed η-Einstein manifold (and V leaves the structure tensor φ invariant). Next, we prove that if a compact K-contact manifold whose metric g is a gradient almost Riemann soliton, then it is Sasakian and isometric to a unit sphere S 2n+1. Further, we study H-contact manifold admitting a Riemann soliton (g, V ) where V is pointwise collinear with ξ.

Weyl–Einstein structures on K-contact manifolds

We show that a compact K-contact manifold $(M,g,\xi)$ has a closed Weyl-Einstein connection compatible with the conformal structure $[g]$ if and only if it is Sasaki-Einstein.

Basic Cohomology Group Decomposition of K-contact 5-manifolds

In this paper, we consider decompositions of basic degree 2 cohomology for a compact K-contact 5-manifold $(M,\xi,\eta,\Phi,g)$, and conclude the pureness and fullness of $\Phi$-invariant and $\Phi$-anti-invariant cohomology groups. Moreover, we discuss the decomposition of the complexified basic degree 2 cohomology group. This is an analogue problem when Draghici, Li and Zhang \cite{DLZ1} considered the $C^{\infty}$ pureness and fullness of $J$-invariant and $J$-anti-invariant subgroups of the degree 2 real cohomology group $H^2(M,\mathbb{R})$ of any compact almost complex manifold $(M, J)$.

Basic Cohomology Group Decomposition of K-Contact 5-Manifolds

In this paper, we consider decompositions of basic degree 2 cohomology for a compact K-contact 5-manifold \({(M,\xi,\eta,\Phi,g)}\), and conclude the pureness and fullness of \({\Phi}\)-invariant and \({\Phi}\)-anti-invariant cohomology groups. Moreover, we discuss the decomposition of the complexified basic degree 2 cohomology group. This is an analogue problem when Draghici et al. (Int Math Res Not IMRN 2010(1):1–17, 2010) considered the \({C^{\infty}}\) pureness and fullness of J-invariant and J-anti-invariant subgroups of the degree 2 real cohomology group \({H^2(M,{\mathbb{R}})}\) of any compact almost complex manifold (M, J).

Examples of compact K-contact manifolds with no Sasakian metric

Using the hard Lefschetz theorem for Sasakian manifolds, we find two examples of compact K-contact nilmanifolds with no compatible Sasakian metric in dimensions 5 and 7, respectively.

Certain Contact Metrics as Ricci Almost Solitons

We show that if a compact K-contact metric is a gradient Ricci almost soliton, then it is isometric to a unit sphere S2n+1. Next, we prove that if the metric of a non-Sasakian (κ, μ)-contact metric is a gradient Ricci almost soliton, then in dimension 3 it is flat and in higher dimensions it is locally isometric to En+1 × Sn (4). Finally, a couple of results on contact metric manifolds whose metric is a Ricci almost soliton and the potential vector field is point wise collinear with the Reeb vector field of the contact metric structure were obtained.

Closed Einstein–Weyl structures on compact Sasakian and cosymplectic manifolds

AbstractWe study closed Einstein–Weyl structures on compact K-contact, Sasakian and cosymplectic manifolds. In particular we prove that compact Sasakian and cosymplectic manifolds endowed with a closed Einstein–Weyl structure are η-Einstein.

Stability of the Reeb vector field of H-contact manifolds

It is well known that a Hopf vector field on the unit sphere S 2n+1 is the Reeb vector field of a natural Sasakian structure on S 2n+1. A contact metric manifold whose Reeb vector field ξ is a harmonic vector field is called an H-contact manifold. Sasakian and K-contact manifolds, generalized (k, μ)-spaces and contact metric three-manifolds with ξ strongly normal, are H-contact manifolds. In this paper we study, in dimension three, the stability with respect to the energy of the Reeb vector field ξ for such special classes of H-contact manifolds (and with respect to the volume when ξ is also minimal) in terms of Webster scalar curvature. Finally, we extend for the Reeb vector field of a compact K-contact (2n+1)-manifold the obtained results for the Hopf vector fields to minimize the energy functional with mean curvature correction.

Stability numbers in K-contact manifolds

The aim of this paper is to study the stability of the characteristic vector field of a compact K-contact manifold with respect to the energy and volume functionals when we consider on the manifold a two-parameter variation of the metric. First of all, we multiply the metric in the direction of the characteristic vector field by a constant and then we change the metric by homotheties. We will study to what extent the results obtained in [V. Borrelli, Stability of the characteristic vector field of a Sasakian manifold, Soochow J. Math. 30 (2004) 283–292. Erratum on the article: Stability of the characteristic vector field of a Sasakian manifold, Soochow J. Math. 32 (2006) 179–180] for Sasakian manifolds are valid for a general K-contact manifold. Finally, as an example, we will study the stability of Hopf vector fields on Berger spheres when we consider homotheties of Berger metrics.