Reeb Vector Field Research Articles

On the Curvature of Metric Contact Pairs

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 exist if the leaves of the characteristic foliations are at most three-dimensional.

Geodesic Ricci solitons on unit tangent sphere bundles

In the paper, (Abbassi and Kowalski, Ann Glob Anal Geom, 38: 11–20, 2010) the authors study Einstein Riemannian \(g\) natural metrics on unit tangent sphere bundles. In this study, we equip the unit tangent sphere bundle \(T_1 M\) of a Riemannian manifold \((M,g)\) with an arbitrary Riemannian \(g\) natural metric \(\tilde{G}\) and we show that if the geodesic flow \(\tilde{\xi }\) is the potential vector field of a Ricci soliton\((\tilde{G},\tilde{\xi },\lambda )\) on \(T_1M,\) then \((T_1M,\tilde{G})\) is Einstein. Moreover, we show that the Reeb vector field of a contact metric manifold is an infinitesimal harmonic transformation if and only if it is Killing. Thus, we consider a natural contact metric structure \((\tilde{G}, \tilde{\eta }, \tilde{\varphi }, \tilde{\xi })\) over \(T_1 M\) and we show that the geodesic flow \(\tilde{\xi }\) is an infinitesimal harmonic transformation if and only if the structure \((\tilde{G}, \tilde{\eta }, \tilde{\varphi },\tilde{\xi })\) is Sasaki \(\eta \)-Einstein. Consequently, we get that \((\tilde{G},\tilde{\xi }, \lambda )\) is a Ricci soliton if and only if the structure \((\tilde{G}, \tilde{\eta }, \tilde{\varphi }, \tilde{\xi })\) is Sasaki-Einstein with \(\lambda = 2(n-1) >0.\) This last result gives new examples of Sasaki–Einstein structures.

ALMOST CONTACT 3-MANIFOLDS AND RICCI SOLITONS

A Kenmotsu 3-manifold M admitting a Ricci soliton (g, w) with a transversal potential vector field w (orthogonal to the Reeb vector field) is of constant sectional curvature -1. A cosymplectic 3-manifold admitting a Ricci soliton with the Reeb potential vector field or a transversal vector field is of constant sectional curvature 0.

Levi-parallel contact Riemannian manifolds

We study the Riemannian geometry of contact manifolds with respect to a fixed admissible metric, making the Reeb vector field unitary and orthogonal to the contact distribution, under the assumption that the Levi–Tanaka form is parallel with respect to a canonical connection with torsion.

Nullity conditions in paracontact geometry

The paper is a complete study of paracontact metric manifolds for which the Reeb vector field of the underlying contact structure satisfies a nullity condition (the condition (1.2), for some real numbers κ˜ and μ˜). This class of pseudo-Riemannian manifolds, which includes para-Sasakian manifolds, was recently defined in Cappelletti Montano (2010) [13]. In this paper we show in fact that there is a kind of duality between those manifolds and contact metric (κ,μ)-spaces. In particular, we prove that, under some natural assumption, any such paracontact metric manifold admits a compatible contact metric (κ,μ)-structure (eventually Sasakian). Moreover, we prove that the nullity condition is invariant under D-homothetic deformations and determines the whole curvature tensor field completely. Finally non-trivial examples in any dimension are presented and the many differences with the contact metric case, due to the non-positive definiteness of the metric, are discussed.

$$\mathfrak D $$ -parallelism of normal and structure Jacobi operators for hypersurfaces in complex two-plane Grassmannians

In this paper, we give non-existence theorems for Hopf hypersurfaces in complex two-plane Grassmannians $$G_2(\mathbb{C }^{m+2})$$ with $$\mathfrak D $$ -parallel normal Jacobi operator $${\bar{R}}_N$$ and $$\mathfrak D $$ -parallel structure Jacobi operator $$R_{\xi }$$ if the distribution $$\mathfrak D $$ or $$\mathfrak D ^{\bot }$$ component of the Reeb vector field is invariant by the shape operator, respectively.

Killing Vector Fields and Twistor Forms on Generalized Sasakian Space Forms

We study generalized Sasakian space form M(f1, f2, f3) when (i) the Reeb vector field of the almost contact metric structure is Killing, (ii) the Ricci tensor satisfies Einstein-like conditions and (iii) the fundamental 2-form of the almost contact metric structure is a twistor form.

Slant Curves in 3-dimensional Normal Almost Contact Geometry

The aim of this paper is to study slant curves of three-dimensional normal almost contact manifolds as natural generalization of Legendre curves. Such a curve is characterized by means of the scalar product between its normal vector field and the Reeb vector field of the ambient space. In the particular case of a helix we show that it has a proper (non-harmonic) mean curvature vector field. The general expressions of the curvature and torsion of these curves and the associated Lancret invariant are computed as well as the corresponding variants for some particular cases, namely β-Sasakian and cosymplectic. A class of examples is discussed for a normal not quasi-Sasakian 3-manifold.

Slant curves in three-dimensional [formula omitted]-Kenmotsu manifolds

The aim of this paper is to study slant curves of three-dimensional f-Kenmotsu manifolds. These curves are characterized through the scalar product between the normal at the curve and the Reeb vector field. The classification of slant curves in the hyperbolic three-dimensional space is provided as well as some remarkable cases. Slant curves with proper mean curvature vector field are characterized.

Almost contact metric manifolds whose Reeb vector field is a harmonic section

We investigate almost contact metric manifolds whose Reeb vector field is a harmonic unit vector field, equivalently a harmonic section. We first consider an arbitrary Riemannian manifold and characterize the harmonicity of a unit vector field ξ, when ∇ξ is symmetric, in terms of Ricci curvature. Then, we show that for the class of locally conformal almost cosymplectic manifolds whose Reeb vector field ξ is geodesic, ξ is a harmonic section if and only if it is an eigenvector of the Ricci operator. Moreover, we build a large class of locally conformal almost cosymplectic manifolds whose Reeb vector field is a harmonic section. Finally, we exhibit several classes of almost contact metric manifolds where the associated almost contact metric structures σ are harmonic sections, in the sense of Vergara-Diaz and Wood [25], and in some cases they are also harmonic maps.

The harmonicity of the Reeb vector field with respect to Riemannian g-natural metrics

In this paper we show that a 3-dimensional non-Sasakian contact metric manifold [M,(η,ξ,ϕ,g)] is a (κ,μ,ν)-contact metric manifold with ν=const., if and only if there exists a Riemannian g-natural metric G˜ on T1M for which ξ:(M,g)↦(T1M,G˜) is a harmonic map. Furthermore, we give examples of 3-dimensional non-Sasakian contact metric manifolds [M,(η,ξ,ϕ,g)] such that the corresponding Reeb vector fields ξ:(M,g)↦(T1M,G˜) are harmonic maps, for suitable Riemannian g-natural metrics G˜ on T1M which are not of Kaluza–Klein type. Finally, we prove that if (M,g) is an Einstein manifold and (η˜,ξ˜,ϕ˜,G˜) a g-natural contact metric structure on T1M, then the contact metric manifold [T1M,(η˜,ξ˜,ϕ˜,G˜)] is H-contact if and only if (M,g) is 2-stein.

Generalized geometry in AdS/CFT and volume minimization

We study the general structure of the AdS5/CFT4 correspondence in type IIB string theory from the perspective of generalized geometry. We begin by defining a notion of “generalized Sasakian geometry”, which consists of a contact structure together with a differential system for three symplectic forms on the four-dimensional transverse space to the Reeb vector field. A generalized Sasakian manifold which satisfies an additional “Einstein” condition provides a general supersymmetric AdS5 solution of type IIB supergravity with fluxes. We then show that the supergravity action restricted to a space of generalized Sasakian structures is simply the contact volume, and that its minimization determines the Reeb vector field for such a solution. We conjecture that this contact volume is equal to the inverse of the trial central charge whose maximization determines the R-symmetry of any four-dimensional N=1 superconformal field theory. This variational procedure allows us to compute the contact volumes for a predicted infinite family of solutions, and we find perfect agreement with the central charges and R-charges of BPS operators in the dual mass-deformed generalized conifold theories.

ENTROPY FORM AND THE CONTACT GEOMETRY OF THE MATERIAL POINT MODEL

In this work we investigate a material point model (MP-model) and exploit the geometrical meaning of the "entropy form" introduced by Coleman and Owen. We show that a modification of the thermodynamical phase space (studied and exploited in numerous works) is an appropriate setting for the development of the MP-model in different physical situations. This approach allows to formulate the MP-model and the corresponding entropy form in terms similar to those of homogeneous thermodynamics. Closeness condition of the entropy form is reformulated as the requirement that admissible processes curves belong to the (extended) constitutive surfaces foliating the extended thermodynamical phase space [Formula: see text] of the model over the space X of basic variables. Extended constitutive surfaces ΣS,κ are described as the Legendre submanifolds ΣS of the space [Formula: see text]shifted by the flow of Reeb vector field. This shift is controlled by the entropy production function κ. We determine which contact Hamiltonian dynamical systems ξK are tangent to the surfaces ΣS,κ, introduce conformally Hamiltonian systems μξK where conformal factor μ characterizes the increase of entropy along the trajectories. These considerations are then illustrated by applying them to the Coleman–Owen model of thermoelastic point.

Classification of homogeneous almost cosymplectic three-manifolds

The purpose of this paper is to classify all simply connected homogeneous almost cosymplectic three-manifolds. We show that each such three-manifold is either a Lie group G equipped with a left invariant almost cosymplectic structure or a Riemannian product of type R × N , where N is a Kähler surface of constant curvature. Moreover, we find that the Reeb vector field of any homogeneous almost cosymplectic three-manifold, except one case, defines a harmonic map.

Closed trajectories on symmetric convex Hamiltonian energy surfaces

In this article, let $\Sigma\subset\mathbf{R}^{2n}$ be a compactconvex Hamiltonian energy surface which is symmetric with respect tothe origin, where $n\ge 2$. We prove that there exist at least twogeometrically distinct symmetric closed trajectories of the Reebvector field on $\Sigma$.

Existence and non-uniqueness of constant scalar curvature toric Sasaki metrics

AbstractWe study compatible toric Sasaki metrics with constant scalar curvature on co-oriented compact toric contact manifolds of Reeb type of dimension at least five. These metrics come in rays of transversal homothety due to the possible rescaling of the Reeb vector fields. We prove that there exist Reeb vector fields for which the transversal Futaki invariant (restricted to the Lie algebra of the torus) vanishes. Using an existence result of E. Legendre [Toric geometry of convex quadrilaterals, J. Symplectic Geom. 9 (2011), 343–385], we show that a co-oriented compact toric contact 5-manifold whose moment cone has four facets admits a finite number of rays of transversal homothetic compatible toric Sasaki metrics with constant scalar curvature. We point out a family of well-known toric contact structures on S2×S3 admitting two non-isometric and non-transversally homothetic compatible toric Sasaki metrics with constant scalar curvature.

K-contact Lie groups of dimension five or greater

We prove that a K-contact Lie group of dimension five or greater is the central extension of a symplectic Lie group by complexifying the Lie algebra and applying a result from complex contact geometry, namely, that, if the adjoint action of the complex Reeb vector field on a complex contact Lie algebra is diagonalizable, then it is trivial.

Attaching maps in the standard geodesics problem on $S^2$

Unstable manifolds of critical points at infinity in the variational problems relating to periodic orbits of Reeb vector-fields in Contact Form Geometry are viewed in this paper as part of the attaching maps along which these variational problems attach themselves to natural generalizations that they have. The specific periodic orbit problem for the Reeb vector-field $\xi_0$ of the standard contact structure/form of $S^3$ is studied; the extended variational problem is the closed geodesics problem on $S^2$. The attaching maps are studied for low-dimensional (at most $4$) cells. Some circle and ''loop" actions on the loop space of $S^3$, that are lifts (via Hopf-fibration map) of the standard $S^1$-action on the free loop space of $S^2$, are also defined. ''Conjugacy" relations relating these actions are established.

Contact homology of orbit complements and implied existence

For Reeb vector fields on closed 3-manifolds, cylindrical contact homology is used to show that the existence of a set of closed Reeb orbit with certain knotting/linking properties implies the existence of other Reeb orbits with other knotting/linking properties relative to the original set. We work out a few examples on the 3-sphere to illustrate the theory, and describe an application to closed geodesics on $S^2$ (a version of a result due to Angenent).

Real hypersurfaces in complex two-plane Grassmannians with [formula omitted]-invariant Ricci tensor

In this paper, first we introduce the full expression of the Ricci tensor of a real hypersurface M in complex two-plane Grassmannians G 2 ( C m + 2 ) from the equation of Gauss. Next, we give a new characterization of real hypersurfaces of type ( A ) in complex two-plane Grassamnnians with a vanishing Lie derivative of the Ricci tensor S in the direction of the Reeb vector field ξ , that is, an ξ -invariant Ricci tensor.