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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 paper, the authors investigated timelike and null vector flows on closed Riemannian 3-manifolds and their relationship to Ricci curvature, and they showed that positive curvature tends to yield contact forms, namely, 1-forms metrically equivalent to unit vector fields with geodesic flow.
Abstract: We investigate timelike and null vector flows on closed Lorentzian manifolds and their relationship to Ricci curvature. The guiding observation, first observed for closed Riemannian 3-manifolds in Harris & Paternain (2013), is that positive Ricci curvature tends to yield contact forms, namely, 1-forms metrically equivalent to unit vector fields with geodesic flow. We carry this line of thought over to the Lorentzian setting. First, we observe that the same is true on a closed Lorentzian 3-manifold: if ${\boldsymbol k}$ is a global timelike unit vector field with geodesic flow satisfying $\text{Ric}({\boldsymbol k},{\boldsymbol k}) > 0$, then $g({\boldsymbol k},\cdot)$ is a contact form with Reeb vector field ${\boldsymbol k}$, at least one of whose integral curves is closed. Second, we show that on a closed Lorentzian 4-manifold, if ${\boldsymbol k}$ is a global null vector field satisfying $
abla_{{\boldsymbol k}}{\boldsymbol k} = {\boldsymbol k}$ and $\text{Ric}({\boldsymbol k},{\boldsymbol k}) > \text{div}\,{\boldsymbol k}-1$, then $dg({\boldsymbol k},\cdot)$ is a symplectic form and ${\boldsymbol k}$ is a Liouville vector field.
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TL;DR: In this paper, a simple proof of the Poincar\'e conjecture by using the contact Ricci flow associated with the Reeb vector field was given, and the proof was later extended to prove the existence of Poincare conjecture.
Abstract: We give a simple proof of the Poincar\'e conjecture by using the contact Ricci flow associated with the Reeb vector field.
TL;DR: In this article, the periodic orbits of the Reeb vector field created by the bypass attachment are described in terms of Reeb chords of the attachment arc. And the contact homology of a product neighbourhood of a convex surface after a bypass attachment is computed.
Abstract: On a three-dimensional contact manifold with boundary, a bypass attachment is an elementary change of the contact structure consisting in the attachment of a thickened half-disc with a prescribed contact structure along an arc on the boundary. We give a model bypass attachment in which we describe the periodic orbits of the Reeb vector field created by the bypass attachment in terms of Reeb chords of the attachment arc. As an application, we compute the contact homology of a product neighbourhood of a convex surface after a bypass attachment, and the contact homology of some contact structures on solid tori.
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TL;DR: In this paper, the authors investigated the undecidability of different dynamical properties of Turing complete Euler flows, and showed that a stationary Euler flow of the Beltrami type exhibits undecidable particle paths.
Abstract: The dynamics of an inviscid and incompressible fluid flow on a Riemannian manifold is governed by the Euler equations. In recent papers [5, 6, 7, 8] several unknown facets of the Euler flows have been discovered, including universality properties of the stationary solutions to the Euler equations. The study of these universality features was suggested by Tao as a novel way to address the problem of global existence for Euler and Navier-Stokes [28]. Universality of the Euler equations was proved in [7] for stationary solutions using a contact mirror which reflects a Beltrami flow as a Reeb vector field. This contact mirror permits the use of advanced geometric techniques in fluid dynamics. On the other hand, motivated by Tao's approach relating Turing machines to Navier-Stokes equations, a Turing complete stationary Euler solution on a Riemannian $3$-dimensional sphere was constructed in [8]. Since the Turing completeness of a vector field can be characterized in terms of the halting problem, which is known to be undecidable [30], a striking consequence of this fact is that a Turing complete Euler flow exhibits undecidable particle paths [8]. In this article, we give a panoramic overview of this fascinating subject, and go one step further in investigating the undecidability of different dynamical properties of Turing complete flows. In particular, we show that variations of [8] allow us to construct a stationary Euler flow of Beltrami type (and, via the contact mirror, a Reeb vector field) for which it is undecidable to determine whether its orbits through an explicit set of points are periodic.
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TL;DR: In this article, the authors considered the notion of Cotton soliton within the framework of almost Kenmotsu 3-$h$-manifolds and proved a non-existence of such a soliton.
Abstract: In this paper, we consider the notion of Cotton soliton within the framework of almost Kenmotsu 3-$h$-manifolds First we consider that the potential vector field is pointwise collinear with the Reeb vector field and prove a non-existence of such Cotton soliton Next we assume that the potential vector field is orthogonal to the Reeb vector field It is proved that such a Cotton soliton on a non-Kenmotsu almost Kenmotsu 3-$h$-manifold such that the Reeb vector field is an eigen vector of the Ricci operator is steady and the manifold is locally isometric to $\mathbb{H}^2(-4) \times \mathbb{R}$