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Reeb vector field
About: Reeb vector field is a research topic. Over the lifetime, 254 publications have been published within this topic receiving 4118 citations.
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TL;DR: In this article, the authors investigated almost contact metric manifolds whose Reeb vector field is a harmonic unit vector field, equivalently a harmonic section, and constructed a large class of locally conformal almost cosymplectic manifolds where the associated almost contact metrics σ are harmonic sections, in the sense of Vergara-Diaz and Wood [25] and in some cases they are also harmonic maps.
Abstract: 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.
13 citations
TL;DR: In this paper, the Ricci curvature of the Reeb vector field is invariant to the Riemannian curvature tensor in a 3D almost co-Kahler manifold.
Abstract: Let M3 be a three-dimensional almost coKahler manifold such that the Ricci curvature of the Reeb vector field is invariant along the Reeb vector field. In this paper, we obtain some classification results of M3 for which the Ricci tensor is η-parallel or the Riemannian curvature tensor is harmonic.
13 citations
TL;DR: In this paper, a study of three-dimensional paracontact metric (κ,μ,ν)-manifolds whose Reeb vector field ξ is harmonic is presented.
Abstract: This paper is a study of three-dimensional paracontact metric (κ,μ,ν)-manifolds. Three-dimensional paracontact metric manifolds whose Reeb vector field ξ is harmonic are characterized. We focus on some curvature properties by considering the class of paracontact metric (κ,μ,ν)-manifolds under a condition which is given at Definition 3.1. We study properties of such manifolds according to the cases κ > −1, κ = −1, κ < −1 and construct new examples of such manifolds for each case. We also show the existence of paracontact metric (−1,μ≠0,ν≠0) spaces with dimension greater than 3, such that h2 = 0 but h≠0.
13 citations
TL;DR: In this paper, it was shown that the Reeb vector field ξ of a K-contact manifold defines a harmonic map for any Riemannian natural metric and is a special metric of the Kaluza-Klein type.
Abstract: Let (M, g) be a Riemannian manifold and T1 M its unit tangent sphere bundle. Minimality and harmonicity of unit vector fields have been extensively studied by considering on T1M the Sasaki metric . This metric, and other well-known Riemannian metrics on T1 M, are particular examples of Riemannian natural metrics. In this paper we equip T1 M with a Riemannian natural metric and in particular with a natural contact metric structure. Then, we study the minimality for Reeb vector fields of contact metric manifolds and of quasi-umbilical hypersurfaces of a Kahler manifold. Several explicit examples are given. In particular, the Reeb vector field ξ of a K-contact manifold is minimal for any that belongs to a family depending on two parameters of metrics of the Kaluza–Klein type. Next, we show that the Reeb vector field ξ of a K-contact manifold defines a harmonic map for any Riemannian natural metric . Besides this, if the Reeb vector ξ of an almost contact metric manifold is a CR map then the induced almost CR structure on M is strictly pseudoconvex and ξ is a pseudo-Hermitian map; if in addition ξ is geodesic then is a harmonic map. Moreover, the Reeb vector field ξ of a contact metric manifold is a CR map iff ξ is Killing and is a special metric of the Kaluza–Klein type. Finally, in the final section, we obtain that there is a family of strictly pseudoconvex CR structures on T1S2n+1 depending on one parameter, for which a Hopf vector field ξ determines a pseudo-harmonic map (in the sense of Barletta–Dragomir–Urakawa [8]) from S2n+1 to T1S2n+1.
13 citations
TL;DR: In this article, it was shown that if an integrable contact pseudo-metric manifold of dimension 2n + 1, n ≥ 2, has constant sectional curvature, then the structure is Sasakian and \({\kappa=\varepsilon=g(\xi,\xi)}) is the Reeb vector field.
Abstract: In this paper, we show that if an integrable contact pseudo-metric manifold of dimension 2n + 1, n ≥ 2, has constant sectional curvature \({\kappa}\), then the structure is Sasakian and \({\kappa=\varepsilon=g(\xi,\xi)}\), where \({\xi}\) is the Reeb vector field. We note that the notion of contact pseudo-metric structure is equivalent to the notion of non-degenerate almost CR manifold, then an equivalent statement of this result holds in terms of CR geometry. Moreover, we study the pseudohermitian torsion \({\tau}\) of a non-degenerate almost CR manifold.
13 citations