Showing papers on "Reeb vector field published in 2009"
TL;DR: In this article, it was shown that a real hypersurface in a non-flat complex space form does not admit a Ricci soliton whose potential vector field is the Reeb vector field.
Abstract: We prove that a real hypersurface in a non-flat complex space form does not admit a Ricci soliton whose potential vector field is the Reeb vector field. Moreover, we classify a real hypersurface admitting so-called “$\eta$-Ricci soliton” in a non-flat complex space form.
154 citations
TL;DR: In this article, the equivalence between embedded contact homology and Seiberg-Witten Floer homology was used to obtain the following improvements on the Weinstein conjecture: if Y is a closed oriented connected 3-manifold with a stable Hamiltonian structure, then R denotes the associated Reeb vector field on Y.
Abstract: We use the equivalence between embedded contact homology and Seiberg–Witten Floer homology to obtain the following improvements on the Weinstein conjecture. Let Y be a closed oriented connected 3–manifold with a stable Hamiltonian structure, and let R denote the associated Reeb vector field on Y . We prove that if Y is not a T 2 –bundle over S , then R has a closed orbit. Along the way we prove that if Y is a closed oriented connected 3–manifold with a contact form such that all Reeb orbits are nondegenerate and elliptic, then Y is a lens space. Related arguments show that if Y is a closed oriented 3–manifold with a contact form such that all Reeb orbits are nondegenerate, and if Y is not a lens space, then there exist at least three distinct embedded Reeb orbits.
85 citations
TL;DR: In this paper, the authors describe other classes in H 1.M IZ/ that are represented by formal, positively weighted sums of closed integral curves of v. To set the stage for a precise statement of what is proved here, digress for a moment to introduce the set, SM, of SpinC structures on M, this a principle homogeneous space for the group H 1.
Abstract: The purpose of this article is to describe other classes in H1.M IZ/ that are represented by formal, positively weighted sums of closed integral curves of v . To set the stage for a precise statement of what is proved here, digress for a moment to introduce the set, SM , of SpinC structures on M , this a principle homogeneous space for the group H .M IZ/. Each element in S has a canonically associated Z–module, a version
50 citations
TL;DR: The existence of a "Plastikstufe" for a contact structure implies the Weinstein con- jecture for all supporting contact forms as mentioned in this paper, which is the case for all contact structures.
Abstract: The existence of a "Plastikstufe" for a contact structure implies the Weinstein con- jecture for all supporting contact forms.
46 citations
TL;DR: In this article, the authors considered almost Kenmotsu manifolds (M 2 n + 1, φ, ξ, η, g ) with a parallel tensor φ = h ○ φ, 2h being the Lie derivative of φ with respect to the Reeb vector field.
Abstract: We consider almost Kenmotsu manifolds ( M 2 n + 1 , φ , ξ , η , g ) with η-parallel tensor h ′ = h ○ φ , 2h being the Lie derivative of the structure tensor φ with respect to the Reeb vector field ξ. We describe the Riemannian geometry of an integral submanifold of the distribution orthogonal to ξ, characterizing the CR-integrability of the structure. Under the additional condition ∇ ξ h ′ = 0 , the almost Kenmotsu manifold is locally a warped product. Finally, some lightlike structures on M 2 n + 1 are introduced and studied.
42 citations
01 Apr 2009
TL;DR: In this paper, the authors introduce the notion of contact pair structure and corresponding associated metrics, in the same spirit of the geometry of almost contact structures, and prove that, with respect to these metrics, the integral curves of the Reeb vector field s are geodesics and that the leaves of the reeb action are totally geodesic.
Abstract: We introduce the notion of contact pair structure and the corresponding associated metrics, in the same spirit of the geometry of almost contact structures. We prove that, with respect to these metrics, the integral curves of the Reeb vector field s are geodesics and that the leaves of the Reeb action are totally geodesic. Moreover, we show that, in the case of a metric contact pair with decomposable endomorphism, the characteristic foliations are orthogonal and their leaves carry induced contact metric structures.
16 citations
TL;DR: A summary of some of the author's work on Sasaki-Einstein geometry can be found in this paper, where a rather general conjecture in string theory known as the AdS/CFT correspondence relates Sasaki einstein geometry, in low dimensions, to superconformal field theory.
Abstract: This article is a summary of some of the author’s work on Sasaki–Einstein geometry. A rather general conjecture in string theory known as the AdS/CFT correspondence relates Sasaki–Einstein geometry, in low dimensions, to superconformal field theory; properties of the latter are therefore reflected in the former, and vice versa. Despite this physical motivation, many recent results are of independent geometrical interest and are described here in purely mathematical terms: explicit constructions of infinite families of both quasi-regular and irregular Sasaki–Einstein metrics; toric Sasakian geometry; an extremal problem that determines the Reeb vector field for, and hence also the volume of, a Sasaki–Einstein manifold; and finally, obstructions to the existence of Sasaki–Einstein metrics. Some of these results also provide new insights into Kahler geometry, and in particular new obstructions to the existence of Kahler–Einstein metrics on Fano orbifolds.
15 citations
TL;DR: In this paper, it was shown that a complete K-contact manifold admitting both the Einstein-Weyl structures W± = (g, ±ω) is Sasakian.
Abstract: In this paper we study Einstein-Weyl structures in the framework of contact metric manifolds. First, we prove that a complete K-contact manifold admitting both the Einstein-Weyl structures W± = (g, ±ω) is Sasakian. Next, we show that a compact contact metric manifold admitting an Einstein-Weyl structure is either K-contact or the dual field of ω is orthogonal to the Reeb vector field, provided the Reeb vector field is an eigenvector of the Ricci operator. We also prove that a contact metric manifold admitting both the Einstein-Weyl structures and satisfying \({Q\varphi = \varphi Q}\) is either K-contact or Einstein. Finally, a couple of results on contact metric manifold admitting an Einstein-Weyl structure W = (g, fη) are presented.
14 citations
TL;DR: In this article, the stability of the Reeb vector field with respect to the energy functional with mean curvature correction was studied for H-contact manifolds in terms of the Webster scalar curvature.
Abstract: 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.
11 citations
TL;DR: In this article, it was shown that a nonsingular Morse-Smale vector field in the kernel of a contact form α, with Reeb vector field defined on M3, satisfies the Palais-smale condition.
Abstract: Let v be a nonsingular Morse–Smale vector field in the kernel of a contact form α, with Reeb vector field \(\xi\), defined on M3. We establish that the associated variational problem at infinity defined by the action functional on the stratified space \(\bigcup \Gamma_{2k}\) of curves made of \(\xi\)-pieces of orbits alternating with \(\pm v\)-pieces of orbits satisfies the Palais–Smale condition. This result takes a more special form for the standard contact structure of S3.
4 citations
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TL;DR: In this article, it was shown that if the integral curves of a real hypersurface are real analytic, then there exists an open neighborhood M0 ⊂ M of V and a solution u ∈ C k (M0) = 0 on M0 which is a defining equation for V. The result is obtaine d solving a Cauchy problem for infinitesimal symmetries of CR distribu tions of codimension.
Abstract: Let V be a real hypersurface of class C k , k ≥ 3, in a com- plex manifold M of complex dimension n + 1, HT(V) the holomorphic tangent bundle to V giving the induced CR structure on V. Let θ be a contact form for (V, HT(V)), ξ0 the Reeb vector field determined by θ and assume that ξ0 is of class C k . In this paper we prove the following theorem (cf. Theorem 4.1): if the integral curves of ξ0 are real analytic then there exist an open neighbourhood M0 ⊂ M of V and a solution u ∈ C k (M0) of the complex Monge-Ampere equation (dd c u) n+1 = 0 on M0 which is a defining equation for V. Moreover, the Monge-Ampere foliation associated to u induces on V that one associated to the Reeb vector field. The converse is also true. The result is obtaine d solving a Cauchy problem for infinitesimal symmetries of CR distribu tions of codimension one which is of independent interest (cf. Theorem 3.1 be- low).
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TL;DR: In this paper, it was shown that there exist at least two geometrically distinct symmetric closed trajectories of the Reeb vector field on a convex Hamiltonian energy surface which is symmetric with respect to the origin.
Abstract: In this article, let $\Sigma\subset\R^{2n}$ be a compact convex Hamiltonian energy surface which is symmetric with respect to the origin. where $n\ge 2$. We prove that there exist at least two geometrically distinct symmetric closed trajectories of the Reeb vector field on $\Sg$.