Showing papers on "Reeb vector field published in 2020"

On the existence of supporting broken book decompositions for contact forms in dimension 3

Abstract: We prove that in dimension 3 every nondegenerate contact form is carried by a broken book decomposition. As an application we get that if M is a closed irreducible oriented 3-manifold that is not a graph manifold, for example a hyperbolic manifold, then every nondegenerate Reeb vector field on M has positive topological entropy. Moreover, we obtain that on a closed 3-manifold, every nondegenerate Reeb vector field has either two or infinitely many periodic orbits, and two periodic orbits are possible only on the sphere or on a lens space.

Contact 3-manifolds and $*$-Ricci soliton

Abstract: Let $(M,\phi,\xi,\eta,g)$ be a three-dimensional Kenmotsu manifold. In this paper, we prove that the triple $(g,V,\lambda)$ on $M$ is a $*$-Ricci soliton if and only if $M$ is locally isometric to the hyperbolic 3-space $\mathbf{H}^3(-1)$ and $\lambda=0$. Moreover, if $g$ is a gradient $*$-Ricci soliton, then the potential vector field coincides with the Reeb vector field. We also show that the metric of a coKahler 3-manifold is a $*$-Ricci soliton if and only if it is a Ricci soliton.

Almost $$*$$∗ -Ricci soliton on paraKenmotsu manifolds

Abstract: We consider almost $$*$$ -Ricci solitons in the context of paracontact geometry, precisely, on a paraKenmotsu manifold. First, we prove that if the metric g of $$\eta $$ -Einstein paraKenmotsu manifold is $$*$$ Ricci soliton, then M is Einstein. Next, we show that if $$\eta $$ -Einstein paraKenmotsu manifold admits a gradient almost $$*$$ -Ricci soliton, then either M is Einstein or the potential vector field collinear with Reeb vector field $$\xi $$ . Finally, for three-dimensional case we show that paraKenmotsu manifold is of constant curvature $$-1$$ . An illustrative example is given to support the obtained results.

Hypersurfaces of a Sasakian Manifold

Abstract: We extend the study of orientable hypersurfaces in a Sasakian manifold initiated by Watanabe. The Reeb vector field ξ of the Sasakian manifold induces a vector field ξ T on the hypersurface, namely the tangential component of ξ to hypersurface, and it also gives a smooth function ρ on the hypersurface, which is the projection of the Reeb vector field on the unit normal. First, we find volume estimates for a compact orientable hypersurface and then we use them to find an upper bound of the first nonzero eigenvalue of the Laplace operator on the hypersurface, showing that if the equality holds then the hypersurface is isometric to a certain sphere. Also, we use a bound on the energy of the vector field ∇ ρ on a compact orientable hypersurface in a Sasakian manifold in order to find another geometric condition (in terms of mean curvature and integral curves of ξ T ) under which the hypersurface is isometric to a sphere. Finally, we study compact orientable hypersurfaces with constant mean curvature in a Sasakian manifold and find a sharp upper bound on the first nonzero eigenvalue of the Laplace operator on the hypersurface. In particular, we show that this upper bound is attained if and only if the hypersurface is isometric to a sphere, provided that the Ricci curvature of the hypersurface along ∇ ρ has a certain lower bound.

Stochastic processes on surfaces in three-dimensional contact sub-Riemannian manifolds

Abstract: We are concerned with stochastic processes on surfaces in three-dimensional contact sub-Riemannian manifolds. Employing the Riemannian approximations to the sub-Riemannian manifold which make use of the Reeb vector field, we obtain a second order partial differential operator on the surface arising as the limit of Laplace-Beltrami operators. The stochastic process associated with the limiting operator moves along the characteristic foliation induced on the surface by the contact distribution. We show that for this stochastic process elliptic characteristic points are inaccessible, while hyperbolic characteristic points are accessible from the separatrices. We illustrate the results with examples and we identify canonical surfaces in the Heisenberg group, and in ${\rm SU}(2)$ and ${\rm SL}(2,\mathbb{R})$ equipped with the standard sub-Riemannian contact structures as model cases for this setting. Our techniques further allow us to derive an expression for an intrinsic Gaussian curvature of a surface in a general three-dimensional contact sub-Riemannian manifold.

A note on quasi-Yamabe solitons on contact metric manifolds

Abstract: We prove that a contact metric manifold does not admit a proper quasi-Yamabe soliton ($$M,\, g,\,\xi ,\,\lambda ,\,\mu $$). Next we prove that if a contact metric manifold admits a quasi-Yamabe soliton ($$M,\, g,\, V,\, \lambda ,\, \mu $$) whose soliton field is pointwise collinear with the Reeb vector field, then the scalar curvature is constant, and the quasi-Yamabe soliton reduces to Yamabe soliton. Finally, it is shown that if a contact metric manifold admits a quasi-Yamabe soliton whose soliton field is the V-Ric vector field, then the Ricci operator Q commutes with the (1, 1) tensor $$\phi $$. As a consequence of the main result we obtain several corollaries.

What does a vector field know about volume

Abstract: This note provides an affirmative answer to a question of Viterbo concerning the existence of nondiffeomorphic contact forms that share the same Reeb vector field. Starting from an observation by Croke-Kleiner and Abbondandolo that such contact forms define the same total volume, we discuss various related issues for the wider class of geodesible vector fields. In particular, we define an Euler class of a geodesible vector field in the associated basic cohomology and give a topological characterisation of vector fields with vanishing Euler class. We prove the theorems of Gauss-Bonnet and Poincare-Hopf for closed, oriented 2-dimensional orbifolds using global surfaces of section and the volume determined by a geodesible vector field. This volume is computed for Seifert fibred 3-manifolds and for some transversely holomorphic flows.

The singular Weinstein conjecture

Abstract: In this article, we investigate Reeb dynamics on $b^m$-contact manifolds, previously introduced in [MiO], which are contact away from a hypersurface $Z$ but satisfy certain transversality conditions on $Z$. The study of these contact structures is motivated by that of contact manifolds with boundary. The search of periodic Reeb orbits on those manifolds thereby starts with a generalization of the well-known Weinstein conjecture. Contrary to the initial expectations, examples of compact $b^m$-contact manifolds without periodic Reeb orbits outside $Z$ are provided. Furthermore, we prove that in dimension $3$, there are always infinitely many periodic orbits on the critical set if it is compact. We prove that traps for the $b^m$-Reeb flow exist in any dimension. This investigation goes hand-in-hand with the Weinstein conjecture on non-compact manifolds having compact ends of convex type. In particular, we extend Hofer's arguments to open overtwisted contact manifolds that are $\mathbb R^+$-invariant in the open ends, obtaining as a corollary the existence of periodic $b^m$-Reeb orbits away from the critical set. The study of $b^m$-Reeb dynamics is motivated by well-known problems in fluid dynamics and celestial mechanics, where those geometric structures naturally appear. In particular, we prove that the dynamics on positive energy level-sets in the restricted planar circular three-body problem are described by the Reeb vector field of a $b^3$-contact form that admits an infinite number of periodic orbits at the critical set.

Quasi-Einstein structures and almost cosymplectic manifolds

Abstract: In this article, we study almost cosymplectic manifolds admitting quasi-Einstein structures $$(g, V, m, \lambda )$$. First we prove that an almost cosymplectic $$(\kappa ,\mu )$$-manifold is locally isomorphic to a Lie group if $$(g, V, m, \lambda )$$ is closed and on a compact almost $$(\kappa ,\mu )$$-cosymplectic manifold there do not exist quasi-Einstein structures $$(g, V, m, \lambda )$$, in which the potential vector field V is collinear with the Reeb vector field $$\xi $$. Next we consider an almost $$\alpha $$-cosymplectic manifold admitting a quasi-Einstein structure and obtain some results. Finally, for a K-cosymplectic manifold with a closed, non-steady quasi-Einstein structure, we prove that it is $$\eta $$-Einstein. If $$(g, V, m, \lambda )$$ is non-steady and V is a conformal vector field, we obtain the same conclusion.

Ricci-like solitons with vertical potential on Sasaki-like almost contact B-metric manifolds

Abstract: Ricci-like solitons on Sasaki-like almost contact B-metric manifolds are the object of study. Cases, where the potential of the Ricci-like soliton is the Reeb vector field or pointwise collinear to it, are considered. In the former case, the properties for a parallel or recurrent Ricci-tensor are studied. In the latter case, it is shown that the potential of the considered Ricci-like soliton has a constant length and the manifold is $\eta$-Einstein. Other curvature conditions are also found, which imply that the main metric is Einstein. After that, some results are obtained for a parallel symmetric second-order covariant tensor on the manifolds under study. Finally, an explicit example of dimension 5 is given and some of the results are illustrated.

Ricci-Like Solitons with Vertical Potential on Sasaki-Like Almost Contact B-Metric Manifolds

Abstract: Ricci-like solitons on Sasaki-like almost contact B-metric manifolds are the object of study Cases, where the potential of the Ricci-like soliton is the Reeb vector field or pointwise collinear to it, are considered In the former case, the properties for a parallel or recurrent Ricci-tensor are studied In the latter case, it is shown that the potential of the considered Ricci-like soliton has a constant length and the manifold is $$\eta $$ -Einstein Other curvature conditions are also found, which imply that the main metric is Einstein After that, some results are obtained for a parallel symmetric second-order covariant tensor on the manifolds under study Finally, an explicit example of dimension 5 is given and some of the results are illustrated

Almost Kenmotsu metric as quasi Yamabe soliton

Abstract: The object of the present paper is to characterize two classes of almost Kenmotsu manifolds admitting Ricci-Yamabe soliton. It is shown that a $(k,\mu)'$-almost Kenmotsu manifold admitting a Ricci-Yamabe soliton or gradient Ricci-Yamabe soliton is locally isometric to the Riemannian product $\mathbb{H}^{n+1}(-4) \times \mathbb{R}^n$. For the later case, the potential vector field is pointwise collinear with the Reeb vector field. Also, a $(k,\mu)$-almost Kenmotsu manifold admitting certain Ricci-Yamabe soliton with the curvature property $Q \cdot P = 0$ is locally isometric to the hyperbolic space $\mathbb{H}^{2n+1}(-1)$ and the non-existense of the curvature property $Q \cdot R = 0$ is proved.

On non-gradient $(m,\rho)$-quasi-Einstein contact metric manifolds.

Abstract: Many authors have studied Ricci solitons and their analogs within the framework of (almost) contact geometry. In this article, we thoroughly study the $(m,\rho)$-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 $(m,\rho)$-quasi-Einstein structure, then it is an Einstein manifold of constant scalar curvature $2n(2n+1)$, and for the particular case -- a non-Sasakian $(k,\mu)$-contact structure -- it is locally isometric to the product of a Euclidean space $\RR^{n+1}$ and a sphere $S^n$ of constant curvature $4$. Next, we prove that if a compact contact or $H$-contact metric manifold admits an $(m,\rho)$-quasi-Einstein structure, whose potential vector field $V$ is collinear to the Reeb vector field, then it is a $K$-contact $\eta$-Einstein manifold.

Geometric solitons in a $D$-homothetically deformed Kenmotsu manifold

Abstract: We consider almost Riemann and almost Ricci solitons in a $D$-homothetically deformed Kenmotsu manifold having as potential vector field a gradient vector field, a solenoidal vector field or the Reeb vector field of the deformed structure, and explicitly obtain the Ricci and scalar curvatures for some cases. We also provide a lower bound for the Ricci curvature of the initial Kenmotsu manifold when the deformed manifold admits a gradient almost Riemann or almost Ricci soliton.

$\beta$-almost solitons on almost co-k\"{a}hler manifolds

Abstract: The object of the present paper is to study $\beta$-almost Yamabe solitons and $\beta$-almost Ricci solitons on almost co-Kahler manifolds. In this paper, we prove that if an almost co-Kahler manifold $M$ with the Reeb vector field $\xi$ admits a $\beta$-almost Yamabe solitons with the potential vector field $\xi$ or $b\xi$, where $b$ is a smooth function then manifold is $K$-almost co-Kahler manifold or the soliton is trivial, respectively. Also, we show if a closed $(\kappa,\mu)$-almost co-Kahler manifold with $n>1$ and $\kappa<0$ admits a $\beta$-almost Yamabe soliton then the soliton is trivial and expanding. Then we study an almost co-Kahler manifold admits a $\beta$-almost Yamabe soliton or $\beta$-almost Ricci soliton with $V$ as the potential vector field, $V$ is a special geometric vector field.

Sasakian geometry of the second order ODEs and Hamiltonian dynamical systems

Abstract: We show that the existence of a Sasakian structure on a manifold corresponding to a second order ordinary differential equation $d^2y/dx^2=f(x,y')$ is equivalent to the existence of the Poisson structure determined by the one-form $dy-pdx$. We consider Hamiltonian dynamical system associated with this Poisson structure and show that the compatibility condition for the bi-Hamiltonian structure of the Reeb vector field manifests that the structure equations for the coframe encoding a certain family of second order ODEs are the Maurer-Cartan equations for the Heisenberg group and the independent variable plays the role of a Hamiltonian function. As a final remark we discuss some global aspects of the study and we show that the first Chern class of the normal bundle of the bi-Hamiltonian Reeb vector field vanishes iff $f_x+ff_p = \psi (x)$.

Almost Ricci-like solitons with torse-forming vertical potential on almost contact B-metric manifolds

Abstract: Almost Ricci-like solitons on almost contact B-metric manifolds with torse-forming potential have been studied. The case in which this potential is further vertical is considered, i.e. the potential is pointwise collinear to the Reeb vector field. The conditions under which the investigated manifolds with almost Ricci-like solitons are equivalent to almost Einstein-like manifolds have been established. In this case, necessary and sufficient conditions have been found for a number of properties of the curvature tensor and its Ricci tensor. Then some results are obtained for a parallel symmetric second-order covariant tensor on the manifolds under study. Finally, an explicit example of an arbitrary dimension is given and some of the results are illustrated.

Real hypersurfaces in the complex projective plane satisfying an equality involving $\delta(2)$

Abstract: It was proved in Chen's paper \cite{chen} that every real hypersurface in the complex projective plane of constant holomorphic sectional curvature $4$ satisfies $$ \delta(2)\leq \frac{9}{4}H^2+5,$$ where $H$ is the mean curvature and $\delta(2)$ is a $\delta$-invariant introduced by him. In this paper, we study non-Hopf real hypersurfaces satisfying the equality case of the inequality under the condition that the mean curvature is constant along each integral curve of the Reeb vector field. We describe how to obtain all such hypersurfaces.

Almost $\eta$-Ricci solitons on Kenmotsu manifolds

Abstract: In this paper we characterize the Einstein metrics in such broader classes of metrics as almost $\eta$-Ricci solitons and $\eta$-Ricci solitons on Kenmotsu manifolds, and generalize some results of other authors. First, we prove that a Kenmotsu metric as an $\eta$-Ricci soliton is Einstein metric if either it is $\eta$-Einstein or the potential vector field $V$ is an infinitesimal contact transformation or $V$ is collinear to the Reeb vector field. Further, we prove that if a Kenmotsu manifold admits a gradient almost $\eta$-Ricci soliton with a Reeb vector field leaving the scalar curvature invariant, then it is an Einstein manifold. Finally, we present new examples of $\eta$-Ricci solitons and gradient $\eta$-Ricci solitons, which illustrate our results.

Sasakian structure associated with a second order ODE and Hamiltonian dynamical systems

Abstract: We define a contact metric structure on the manifold corresponding to a second order ordinary differential equation $d^2y/dx^2=f(x,y,y')$ and show that the contact metric structure is Sasakian if and only if the 1-form $\frac{1}{2}(dp-fdx)$ defines a Poisson structure. We consider a Hamiltonian dynamical system defined by this Poisson structure and show that the Hamiltonian vector field, which is a multiple of the Reeb vector field, admits a compatible bi-Hamiltonian structure for which $f$ can be regarded as a Hamiltonian function. As a particular case, we give a compatible bi-Hamiltonian structure of the Reeb vector field such that the structure equations correspond to the Maurer-Cartan equations of an invariant coframe on the Heisenberg group and the independent variable plays the role of a Hamiltonian function. We also show that the first Chern class of the normal bundle of an integral curve of a multiple of the Reeb vector field vanishes iff $f_x+ff_p = \Psi (x)$ for some $\Psi$.

Sasaki-Ricci flow on five-dimensional Sasaki-Einstein space T1,1

Abstract: Within the scope of Sasaki-Ricci flow we investigate the transverse Kahler structure of the Sasaki-Einstein space T1,1. We consider deformations of the Sasaki structure modifying the contact form but preserving the Reeb vector field. We produce new families of metrics as solutions of the Sasaki-Ricci flow equation on the five-dimensional Sasaki-Einstein space T1,1.Within the scope of Sasaki-Ricci flow we investigate the transverse Kahler structure of the Sasaki-Einstein space T1,1. We consider deformations of the Sasaki structure modifying the contact form but preserving the Reeb vector field. We produce new families of metrics as solutions of the Sasaki-Ricci flow equation on the five-dimensional Sasaki-Einstein space T1,1.

$\eta$-normality, CR-structures, para-CR structures on almost contact metric and almost paracontact metric manifolds

Abstract: For almost contact metric or almost paracontact metric manifolds there is natural notion of $\eta$-normality. Manifold is called $\eta$-normal if is normal along kernel distribution of characteristic form. In the paper it is proved that $\eta$-normal manifolds are in one-one correspondence with Cauchy-Riemann almost contact metric manifolds or para Cauchy-Riemann in case of almost paracontact metric manifolds. There is provided characterization of $\eta$-normal manifolds in terms of Levi-Civita covariant derivative of structure tensor. It is established existence a Tanaka-like connection on $\eta$-normal manifold with autoparallel Reeb vector field. In particular case contact metric CR-manifold it is usual Tanaka connection. Similar results are obtained for almost paracontact metric manifolds. For manifold with closed fundamental form we shall state uniqueness of this connection. In the last part is studied bi-Legendrian structure of almost paracontact metric manifold with contact characteristic form. It is established that such manifold is bi-Legendrian flat if and only if is normal. There are characterized semi-flat bi-Legendrian manifolds.

Conformal semi-invariant $\xi ^{\perp }$-submersions from almost contact metric manifolds

Abstract: The purpose of the present paper is to introduce the conformal semi-invariant $\xi ^{\perp }$-Riemannian submersions from almost contact metric manifolds onto Riemannian manifolds as a generalization of anti-invariant submersions, semi-invariant submersions, anti-invariant $\xi ^{\perp }$-submersions, semi-invariant $\xi ^{\perp }$% -submersions, conformal anti-invariant $\xi ^{\perp }$-submersions, conformal semi-invariant submersions. We obtain characterizations and investigate the integrability of distributions which are arisen from the definition of conformal semi-invariant $\xi ^{\perp }$-Riemannian submersions. We find out the necessary and sufficient conditions for a conformal semi-invariant $\xi ^{\perp }$-Riemannian submersions to be totally geodesic and harmonic. Finally, examples are also given for conformal semi-invariant submersions with horizontal Reeb vector field.

Almost Kenmotsu manifolds admitting certain vector fields

Abstract: The object of the present paper is to characterize almost Kenmotsu manifolds admitting holomorphically planar conformal vector (in short, HPCV) fields. It is shown that an almost Kenmotsu manifold $M^{2n+1}$ admitting a non-zero HPCV field $V$ such that $V$ is pointwise collinear with the Reeb vector field $xi$ is locally a warped product of an almost Kaehler manifold and an open interval. Further, if an almost Kenmotsu manifold with constant $xi$-sectional curvature admits a non-zero HPCV field $V$, then $M^{2n+1}$ is locally a warped product of an almost Kaehler manifold and an open interval. Moreover, a $(k,mu)'$-almost Kenmotsu manifold admitting a HPCV field $V$ such that $phi V neq 0$ is either locally isometric to $mathbb{H}^{n+1}(-4)$ $times$ $mathbb{R}^n$ or $V$ is an eigenvector of $h'$.

Toric Sasaki-Einstein metrics with conical singularities

Abstract: We show that any toric Kahler cone with smooth compact cross-section admits a family of Calabi-Yau cone metrics with conical singularities along its toric divisors. The family is parametrized by the Reeb cone and the angles are given explicitly in terms of the Reeb vector field. The result is optimal, in the sense that any toric Calabi-Yau cone metric with conical singularities along the toric divisor (and smooth elsewhere) belongs to this family. We also provide examples and interpret our results in terms of Sasaki-Einstein metrics.

Cotton Solitons on Almost Kenmotsu 3-$h$-Manifolds

Abstract: In this paper, we consider the notion of Cotton soliton within the framework of almost Kenmotsu 3-$h$-manifolds First we consider that the potential vector field is pointwise collinear with the Reeb vector field and prove a non-existence of such Cotton soliton Next we assume that the potential vector field is orthogonal to the Reeb vector field It is proved that such a Cotton soliton on a non-Kenmotsu almost Kenmotsu 3-$h$-manifold such that the Reeb vector field is an eigen vector of the Ricci operator is steady and the manifold is locally isometric to $\mathbb{H}^2(-4) \times \mathbb{R}$