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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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Book ChapterDOI
01 Jan 2019
TL;DR: In this article, a contractible periodic orbit for the Reeb vector field Xλ has been proposed, where Xλ is a closed three dimensional contact manifold with overtwisted contact structure.
Abstract: Let (M, λ) be a closed three dimensional contact manifold with overtwisted contact structure \(\ker \lambda \). Then there exists a contractible periodic orbit for the Reeb vector field Xλ.
Journal ArticleDOI
Tuna Bayrakdar1
TL;DR: In this article, the authors define a contact metric structure on the manifold corresponding to a second order ordinary differential equation, and show that the contact metric is Sasakian if and only if the 1-form of the first order differential equation defines a Poisson structure.
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 $\Psi$.
Posted Content
TL;DR: In this paper, the authors examined conditions under which smooth coordinates may be defined in a tubular neighbourhood of the orbit of the Reeb vector field such that the smooth coordinates resemble a holomorphic curve, invoking comparison with the topological linking of plane complex algebroid curves near a singularity.
Abstract: Let $M$ be a three-dimensional contact manifold and $\psi:D\setminus\{0\}\to M\times{\Bbb R}$ a finite-energy pseudoholomorphic map from a punctured disc in ${\Bbb C}$, that is asymptotic to a periodic orbit of the Reeb vector field. This article examines conditions under which smooth coordinates may be defined in a tubular neighbourhood of the orbit such that $\psi$ resembles a holomorphic curve, invoking comparison with the theory of topological linking of plane complex algebroid curves near a singularity. Examples of this behaviour which are studied in some detail include pseudoholomorphic maps into ${\Bbb E}_{p,q}\times{\Bbb R}$, where ${\Bbb E}_{p,q}$ denotes a rational ellipsoid with contact structure induced by the complex structure of the ambient ${\Bbb C}^{2}$. Contact structures arising from non-standard circle-fibrations of the three-sphere are also examined.
Journal ArticleDOI
TL;DR: In this article, a necessary and sufficient condition for a compact orientable hypersurface to be totally umbilical was given, with the assumption that the vector field u is an eigenvector of the Laplace operator.
Abstract: We study orientable hypersurfaces in a Sasakian manifold. The structure vector field ξ of a Sasakian manifold determines a vector field v on a hypersurface that is the component of the Reeb vector field ξ tangential to the hypersurface, and it also gives rise to a smooth function σ on the hypersurface, namely the projection of ξ on the unit normal vector field N. Moreover, we have a second vector field tangent to the hypersurface, given by $\mathbf{u}=-\varphi (N)$ . In this paper, we first find a necessary and sufficient condition for a compact orientable hypersurface to be totally umbilical. Then, with the assumption that the vector field u is an eigenvector of the Laplace operator, we find a necessary condition for a compact orientable hypersurface to be isometric to a sphere. It is shown that the converse of this result holds, provided that the Sasakian manifold is the odd dimensional sphere $\mathbf{S}^{2n+1}$ . Similar results are obtained for the vector field v under the hypothesis that this is an eigenvector of the Laplace operator. Also, we use a bound on the integral of the Ricci curvature $Ric ( \mathbf{u},\mathbf{u} ) $ of the compact hypersurface to find a necessary condition for the hypersurface to be isometric to a sphere. We show that this condition is also sufficient if the Sasakian manifold is $\mathbf{S}^{2n+1}$ .
Journal ArticleDOI
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.
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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20221
202126
202028
201918
201813
201721