Showing papers on "Reeb vector field published in 2019"

Nonstandard Hamiltonian structures of the Liénard equation and contact geometry

Abstract: The construction of nonstandard Lagrangians and Hamiltonian structures for Lienard equations satisfying Chiellini condition is presented and their connection to time-dependent Hamiltonian formalism...

Ricci soliton and Ricci almost soliton within the framework of Kenmotsu manifold

Abstract: First, we prove that if the Reeb vector field $\xi$ of a Kenmotsu manifold $M$ leaves the Ricci operator $Q$ invariant, then $M$ is Einstein. Next, we study Kenmotsu manifold whose metric represents a Ricci soliton and prove that it is expanding. Moreover, the soliton is trivial (Einstein) if either (i) $V$ is a contact vector field, or (ii) the Reeb vector field $\xi$ leaves the scalar curvature invariant. Finally, it is shown that if the metric of a Kenmotsu manifold represents a gradient Ricci almost soliton, then it is $\eta$-Einstein and the soliton is expanding. We also exhibited some examples of Kenmotsu manifold that admit Ricci almost solitons.

Yamabe solitons on 3-dimensional contact metric manifolds with Qφ = φQ

Abstract: If M is a 3-dimensional contact metric manifold such that Qφ = φQ which admits a Yamabe soliton (g,V ) with the flow vector field V pointwise collinear with the Reeb vector field ξ, then we show th...

Gradient $$\rho $$ ρ -Einstein soliton on almost Kenmotsu manifolds

Abstract: In this paper, we prove that if the metric of an almost Kenmotsu manifold with conformal Reeb foliation admits a gradient $$\rho $$ -Einstein soliton, then either $$M^{2n+1}$$ is Einstein or the potential function is pointwise collinear with the Reeb vector field $$\xi $$ on an open set $${\mathcal {O}}$$ of $$M^{2n+1}$$ . Moreover, we prove that if the metric of a $$(\kappa ,-2)'$$ -almost Kenmotsu manifold with $$h' e 0$$ admits a gradient $$\rho $$ -Einstein soliton, then the manifold is locally isometric to the Riemannian product $${\mathbb {H}}^{n+1}(-4)\times {\mathbb {R}}^n$$ and potential vector field is tangential to the Euclidean factor $${\mathbb {R}}^n$$ . We show that there does not exist gradient $$\rho $$ -Einstein soliton on generalized $$(\kappa ,\mu )$$ -almost Kenmotsu manifold of constant scalar curvature. Finally, we construct an example for gradient $$\rho $$ -Einstein soliton.

m-quasi-Einstein metric and contact geometry

Abstract: We study m-quasi-Einstein metric in the framework of contact metric manifolds. The existence of such metric has been confirmed on the class of $$ \eta $$ -Einstein K-contact manifold, in which the potential vector field V is a constant multiple of the Reeb vector field $$\xi $$ . Next, we consider closed m-quasi-Einstein metric on a complete K-contact manifold and prove that it is Sasakian and Einstein provided $$m e 1$$ . We also proved that, if a K-contact manifold M admits an m-quasi-Einstein metric such that the potential vector field V is conformal, then V becomes Killing and M is $$ \eta $$ -Einstein. Finally, we obtain a couple of results on a contact metric manifold M admitting an m-quasi-Einstein metric, whose potential vector field is a point wise collinear with the Reeb vector field.

Ricci Solitons and Paracontact Geometry

Abstract: In this article, first we prove that if a metric of a para-Sasakian manifold is a Ricci soliton, then either it is an Einstein (and V Killing) or a $$\eta $$-Einstein (and V leaves $$\varphi $$ invariant) manifold. Next, we prove that if a K-paracontact metric g is a gradient Ricci soliton, then it becomes a expanding soliton which is Einstein with constant scalar curvature. Further, we study the Ricci soliton where the potential vector field V is point wise collinear with the Reeb vector field on paracontact manifold. Finally, we consider the gradient Ricci soliton on $$(\kappa ,\mu )$$-paracontact manifold.

Ricci Almost Solitons on Three-Dimensional Quasi-Sasakian Manifolds

Abstract: In this paper it is shown that a three-dimensional non-cosymplectic quasi-Sasakian manifold admitting Ricci almost soliton is locally $$\phi $$-symmetric. It is proved that a Ricci almost soliton on a three-dimensional quasi-Sasakian manifold reduces to a Ricci soliton. It is also proved that if a three-dimensional non-cosymplectic quasi-Sasakian manifold admits gradient Ricci soliton, then the potential function is invariant in the orthogonal distribution of the Reeb vector field $$\xi$$. We also improve some previous results regarding gradient Ricci soliton on three-dimensional quasi-Sasakian manifolds. An illustrative example is given to support the obtained results.

Transverse Kähler-Ricci flow and deformations of the metric on the Sasaki space $T^{1,1}$

Abstract: In this paper we investigate the possibility to obtain locally new Sasaki-Einstein metrics on the space $T^{1,1}$ considering a deformation of the standard metric tensor field. We show that from the geometric point of view this deformation leaves transverse and the leafwise metric intact, but changes the orthogonal complement of the Reeb vector field using a particular basic function. In particular, the family of metric obtained using this method can be regarded as solutions of the equation associated to the Sasaki-Ricci flow on the underlying manifold.

Ricci Solitons in Kenmotsu Manifold under Generalized D-Conformal Deformation

Abstract: In this paper we study Ricci solitons in generalized D-conformally deformed Kenmotsu manifold and we analyzed the nature of Ricci solitons when associated vector field is orthagonal to Reeb vector field.

Ricci Curvature, Reeb Flows and Contact 3-Manifolds

Abstract: Given a contact 3-manifold we consider the problem of when a given function can be realized as the Ricci curvature of a Reeb vector field for the contact structure. We will use topological tools to show that every admissible function can be realized as such Ricci curvature for a singular metric which is an honest compatible metric away from a measure zero set. However, we will see that resolving such singularities depends on contact topological data and is yet to be fully understood.

On cosymplectic dynamics

Abstract: Cosymplectic geometry can be viewed as an odd dimensional counterpart of symplectic geometry. Likely in the symplectic case, a related property which is preservation of closed forms $\omega$ and $\eta$, refers to the theoretical possibility of further understanding a cosymplectic manifold $(M, \omega, \eta)$ from its group of diffeomorphisms. In this paper we study the structures of the group of cosymplectic diffeomorphisms and the group of almost cosymplectic diffeomorphisms of a cosymplectic manifold $(M, \omega, \eta)$ in threefold:first of all, we study cosymplectics counterpart of the Moser isotopy method, a proof of a cosymplectic version of Darboux theorem follows, and we present the features of the space of almost cosymplectic vector fields, this set forms a Lie group whose Lie algebra is the group of all almost cosymplectic diffeomorphisms; $(II)$ we prove by a direct method that the identity component in the group of all cosymplectic diffeomorphisms is $C^0-$closed in the group $Diff^\infty(M)$, while in the almost cosymplectic case, we prove that the Reeb vector field determines the almost cosymplectic nature of the $C^0-$limit $\phi$ of a sequence of almost cosymplectic diffeomorphisms (a rigidity result). A sufficient condition (based on Reeb's vector field) which guarantees that $\phi$ is a cosymplectic diffeomorphism is given (a flexibility condition), and also an attempt to the study cosymplectic counterpart of flux geometry follows: this gives rise to a group homomorphism whose kernel is path connected; and $(III)$ we study the almost cosymplectic analogues of Hofer geometry and Hofer-like geometry: the group of almost co-Hamiltonian diffeomorphisms carries two bi-invariant norms, the cosymplectic analogues of the usual symplectic capacity-inequality are derived and the cosymplectic analogues of a result that was proved by Hofer-Zehnder follow.

On the Sign of the Curvature of a Contact Metric Manifold

Abstract: In this expository article, we discuss the author’s conjecture that an associated metric for a given contact form on a contact manifold of dimension ≥5 must have some positive curvature. In dimension 3, the standard contact structure on the 3-torus admits a flat associated metric; we also discuss a local example, due to Krouglov, where there exists a neighborhood of negative curvature on a particular 3-dimensional contact metric manifold. In the last section, we review some results on contact metric manifolds with negative sectional curvature for sections containing the Reeb vector field.

Disk Filling Methods and Applications

Abstract: Let (M, λ) be a closed three dimensional contact manifold with overtwisted contact structure \(\ker \lambda \). Then there exists a contractible periodic orbit for the Reeb vector field Xλ.

Hopf Real Hypersurfaces in the Indefinite Complex Projective Space

Abstract: We wish to attack the problems that Anciaux and Panagiotidou posed in (Differ Geom Appl 42:1–14, https://doi.org/10.1016/j.difgeo.2015.05.004 , 2015), for non-degenerate real hypersurfaces in indefinite complex projective space. We will slightly change these authors’ point of view, obtaining cleaner equations for the almost-contact metric structure. To make the theory meaningful, we construct new families of non-degenerate Hopf real hypersurfaces whose shape operator is diagonalisable, and one Hopf example with degenerate metric and non-diagonalisable shape operator. Next, we obtain a rigidity result. We classify those real hypersurfaces which are $$\eta $$ -umbilical. As a consequence, we characterize some of our new examples as those whose Reeb vector field $$\xi $$ is Killing.

On the Riemannian Curvature Tensor of a Real Hypersurface in Complex Two-Plane Grassmannians

Abstract: We prove the nonexistence of Hopf real hypersurfaces in complex two-plane Grassmannians such that the covariant derivatives with respect to Levi-Civita and kth generalized Tanaka–Webster connections in the direction of the Reeb vector field applied to the Riemannian curvature tensor coincide when the shape operator and the structure operator commute on the \(\mathcal Q\)-component of the Reeb vector field.

Finite Energy Planes and Periodic Orbits

Abstract: In this chapter we will prove the main result on finite energy planes due to H. Hofer [64] (see also [2]). Namely, given any manifold M equipped with a contact form λ, denote by ξ → M the associated contact structure and by Xλ the associated Reeb vector field.