Showing papers on "Reeb vector field published in 2013"
TL;DR: In this paper, the authors considered supersymmetric gauge theories on Riemannian three-manifolds with the topology of a three-sphere and showed that the partition function depends only on this vector field, giving an explicit expression in terms of the double sine function.
Abstract: We consider supersymmetric gauge theories on Riemannian three-manifolds with the topology of a three-sphere. The three-manifold is always equipped with a contact structure and an associated Reeb vector field. We show that the partition function depends only on this vector field, giving an explicit expression in terms of the double sine function. In the large N limit our formula agrees with a recently discovered two-parameter family of dual supergravity solutions. We also explain how our results may be applied to prove vortex-antivortex factorization. Finally, we comment on the extension of our results to three-manifolds with non-trivial fundamental group.
139 citations
TL;DR: In this article, the authors studied the contact homology of contact 3-manifolds in the framework of open book decompositions and verified the Weinstein conjecture for an open book with pseudo-Anosov monodromy.
Abstract: We determine parts of the contact homology of certain contact 3-manifolds in the framework of open book decompositions, due to Giroux. We study two cases: when the monodromy map of the compatible open book is periodic and when it is pseudo-Anosov. For an open book with periodic monodromy, we verify the Weinstein conjecture. In the case of an open book with pseudo-Anosov monodromy, suppose the boundary of a page of the open book is connected and the fractional Dehn twist coefficient $c={k\over n}$, where $n$ is the number of prongs along the boundary. If $k\geq 2$, then there is a well-defined linearized contact homology group. If $k\geq 3$, then the linearized contact homology is exponentially growing with respect to the action, and every Reeb vector field of the corresponding contact structure admits an infinite number of simple periodic orbits.
60 citations
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TL;DR: In this article, the authors introduce the notion of paracontact Ricci eigenvectors and prove that they are characterized by the condition that the Reeb vector field is a Ricci Eigenvector.
Abstract: We introduce and study $H$-paracontact metric manifolds, that is, paracontact metric manifolds whose Reeb vector field $\xi$ is harmonic. We prove that they are characterized by the condition that $\xi$ is a Ricci eigenvector. We then investigate how harmonicity of the Reeb vector field $\xi$ of a paracontact metric manifold is related to some other relevant geometric properties, like infinitesimal harmonic transformations and paracontact Ricci solitons.
47 citations
TL;DR: In this paper, a general scheme which associates conjugacy classes of tori in the contactomorphism group to transverse almost complex structures on a compact contact manifold is described, and for tori of Reeb type whose Lie algebra contains a Reeb vector field one can associate a Sasaki cone.
Abstract: I describe a general scheme which associates conjugacy classes of tori in the contactomorphism group to transverse almost complex structures on a compact contact manifold. Moreover, to tori of Reeb type whose Lie algebra contains a Reeb vector field one can associate a Sasaki cone. Thus, for contact structures D of K-contact type one obtains a configuration of Sasaki cones called a bouquet such that each Sasaki cone is associated with a conjugacy class of tori of Reeb type.
31 citations
TL;DR: In this article, the authors considered supersymmetric gauge theories on Riemannian three-manifolds with the topology of a three-sphere and showed that the partition function depends only on this vector field, giving an explicit expression in terms of the double sine function.
Abstract: We consider supersymmetric gauge theories on Riemannian three-manifolds with the topology of a three-sphere. The three-manifold is always equipped with an almost contact structure and an associated Reeb vector field. We show that the partition function depends only on this vector field, giving an explicit expression in terms of the double sine function. In the large N limit our formula agrees with a recently discovered two-parameter family of dual supergravity solutions. We also explain how our results may be applied to prove vortex-antivortex factorization. Finally, we comment on the extension of our results to three-manifolds with non-trivial fundamental group.
30 citations
TL;DR: In this paper, a Ricci soliton with the Reeb potential vector field or a transversal vector field is of constant sectional curvature 0.1 and a cosymplectic 3-manifold admits Ricci s solitons of constant curvature -1.
Abstract: 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.
25 citations
TL;DR: In this paper, the authors studied slant curves of three-dimensional normal almost contact manifolds as natural generalization of Legendre curves, which is characterized by the scalar product between the normal vector field and the Reeb vector field.
Abstract: 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.
23 citations
TL;DR: In this article, it was shown that the Reeb vector field of an almost cosymplectic three-manifold is minimal if and only if it is an eigenvector of the Ricci operator.
Abstract: We show that the Reeb vector field of an almost cosymplectic three-manifold is minimal if and only if it is an eigenvector of the Ricci operator. Then, we show that Reeb vector field ξ of an almost cosymplectic three-manifold M is minimal if and only if M is (κ, μ, ν)-space on an open dense subset. After, using the notion of strongly normal unit vector field introduced in [8], we study the minimality of ξ for an almost cosymplectic (2n + 1)-manifold. Finally, we classify a special class of almost cosymplectic three-manifold whose Reeb vector field is minimal.
20 citations
TL;DR: In this article, the authors characterized contact semi-Riemannian manifolds with Ricci eigenvectors by the condition that the Reeb vector field is an infinitesimal harmonic transformation.
19 citations
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
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TL;DR: In this article, a closed orientable Riemannian 3-manifold and a vector field with unit norm whose integral curves are geodesics of the flow were considered.
Abstract: We consider a closed orientable Riemannian 3-manifold $(M,g)$ and a vector field $X$ with unit norm whose integral curves are geodesics of $g$. Any such vector field determines naturally a 2-plane bundle contained in the kernel of the contact form of the geodesic flow of $g$. We study when this 2-plane bundle remains invariant under two natural almost complex structures. We also provide a geometric condition that ensures that $X$ is the Reeb vector field of the 1-form $\lambda$ obtained by contracting $g$ with $X$. We apply these results to the case of great circle flows on the 3-sphere with two objectives in mind: one is to recover the result in \cite{GG} that a volume preserving great circle flow must be Hopf and the other is to characterize in a similar fashion great circle flows that are conformal relative to the almost complex structure in the kernel of $\lambda$ given by rotation by $\pi/2$ according to the orientation of $M$.
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TL;DR: In this paper, the authors introduce the notion of D-homothetic warping, give a few rudimentary properties and a couple of applications, and show that this idea can be used for generating further results and examples of various structures.
Abstract: The goal of this lecture will be to introduce the notion ofD-homothetic warping, give a few rudimentary properties and a couple of applications. As with the usual warped product it is hoped that this idea will prove useful for generating further results and examples of various structures. Details of the proofs will appear in [3]. For this purpose we must first review the geometry of contact metric and almost contact metric manifolds. By a contact manifold we mean a C manifold M2n+1 together with a 1-form η such that η ∧ (dη) 6= 0. It is well known that given η there exists a unique vector field ξ such that dη(ξ,X) = 0 and η(ξ) = 1. The vector field ξ is known as the characteristic vector field or Reeb vector field of the contact structure η. Denote by D the contact subbundle defined by {X ∈ TmM : η(X) = 0}. A Riemannian metric g is an associated metric for a contact form η if, first of all, η(X) = g(X, ξ) and secondly, there exists a field of endomorphisms, φ, such that φ2 = −I + η ⊗ ξ, dη(X,Y ) = g(X,φY ). We refer to (φ, ξ, η, g) as a contact metric structure and to M2n+1 with such a structure as a contact metric manifold. By an almost contact manifold we mean a C manifold M2n+1 together with a field of endomorphisms φ, a 1-form η and a vector field ξ such that
TL;DR: In this paper, the authors proved the non-existence of Hopf hypersurfaces in G 2 (G 2 ) with normal Jacobi operator semi-parallel, if the principal curvature of the Reeb vector field is non-vanishing.
Abstract: It is proved the non-existence of Hopf hypersurfaces in \(G_{2}({\mathbb{C }}^{m+2}), m\ge 3\), whose normal Jacobi operator is semi-parallel, if the principal curvature of the Reeb vector field is non-vanishing and the component of the Reeb vector field in the maximal quaternionic subbundle \(\mathfrak{D }\) or its orthogonal complement \(\mathfrak{D } ^{\bot }\) is invariant by the shape operator.
TL;DR: In this paper, the authors studied generalized Sasakian space form M(f1, f2, f3) when the Reeb vector field of the almost contact metric structure is Killing and the Ricci tensor satisfies Einstein-like conditions.
Abstract: 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.
TL;DR: In this article, the Riemannian geometry of contact manifolds with respect to a fixed admissible metric was studied and the Reeb vector field was shown to be unitary and orthogonal to the contact distribution.
Abstract: 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.
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07 May 2013
TL;DR: In this article, a study of three-dimensional paracontact metric manifolds whose Reeb vector field is harmonic is presented, and the curvature properties of such manifolds are investigated.
Abstract: This paper is a study of three-dimensional paracontact metric (\k{appa},{\mu},{
u})-manifolds. Three dimensional paracontact metric manifolds whose Reeb vector field {\xi} is harmonic are characterized. We focus on some curvature properties by considering the class of paracontact metric (\k{appa},{\mu},{
u})-manifolds under a condition which is given at Definition 3.1. We study properties of such manifolds according to the cases \k{appa}>-1, \k{appa}=-1, \k{appa}<-1 and construct new examples of such manifolds for each case.
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TL;DR: In this paper, the authors give a characterization of real hypersurfaces in non-compact complex two-plane Grassmannian with Reeb vector field for a horosphere whose center at infinity is singular or another exceptional case.
Abstract: In this paper we give a characterization of real hypersurfaces in noncompact complex two-plane Grassmannian $SU_{2,m}/S(U_2 U_m)$, $m \geq 2$ with Reeb vector field $\xi$ belonging to the maximal quaternionic subbundle $\mathcal Q$. Then it becomes a tube over a totally real totally geodesic ${\mathbb H}H^n$, $m=2n$, in noncompact complex two-plane Grassmannian $SU_{2,m}/S(U_2 U_m)$, a horosphere whose center at the infinity is singular or another exceptional case.
TL;DR: In this article, it was shown that the CR-Yamabe equation has infinitely many changing-sign solutions and that the associated functional does not satisfy the Palais-Smale compactness condition.
Abstract: In this note we will prove that the CR-Yamabe equation has infinitely many changing-sign solutions. The problem is variational but the associated functional does not satisfy the Palais-Smale compactness condition; by mean of a suitable group action we will define a subspace on which we can apply the minimax argument of Ambrosetti-Rabinowitz. The result solves a question left open from the classification results of positive solutions by Jerison-Lee in the '80s.