Showing papers on "Reeb vector field published in 2017"
TL;DR: In this paper, it was shown that the Reeb vector field of an ideal fluid on a closed 3-manifold can be rescaled to a stable Hamiltonian structure if the velocity field of such a solution has no zeroes and real analytic Bernoulli function.
Abstract: This note concerns stationary solutions of the Euler equations for an ideal fluid on a closed 3-manifold. We prove that if the velocity field of such a solution has no zeroes and real analytic Bernoulli function, then it can be rescaled to the Reeb vector field of a stable Hamiltonian structure. In particular, such a vector field has a periodic orbit unless the 3-manifold is a torus bundle over the circle. We provide a counterexample showing that the correspondence breaks down without the real analyticity hypothesis.
24 citations
TL;DR: In this paper, it was shown that a pseudoholomorphic curve can be asymptotic at each of its nonremovable punctures to a single periodic orbit of the Reeb vector field and that the convergence is exponential.
Abstract: It’s known from (Ann Inst H Poincare Anal Non Lineaire 13(3):337–379 [11], Properties of Pseudoholomorphic Curves in Symplectisation. IV. Asymptotics with Degeneracies. In: Contact and symplectic geometry (Cambridge, 1994), vol 8 Publ. Newton Inst., pp 78–117. Cambridge Univ. Press, Cambridge [10], A Morse-Bott Approach to Contact Homology. PhD thesis, Stanford University [2]) that in a contact manifold equipped with either a nondegenerate or Morse-Bott contact form, a finite-energy pseudoholomorphic curve will be asymptotic at each of its nonremovable punctures to a single periodic orbit of the Reeb vector field and that the convergence is exponential. We provide examples here to show that this need not be the case if the contact form is degenerate. More specifically, we show that on any contact manifold \((M, \xi )\) with cooriented contact structure one can choose a contact form \(\lambda \) with \(\ker \lambda =\xi \) and a compatible complex structure J on \(\xi \) so that for the associated \(\mathbb {R}\)-invariant almost complex structure \(\tilde{J}\) on \(\mathbb {R}\times M\) there exist families of embedded finite-energy \(\tilde{J}\)-holomorphic cylinders and planes having embedded tori as limit sets.
19 citations
TL;DR: In this paper, it was shown that there is no Clairaut anti-invariant submersion admitting vertical Reeb vector field in case of the total manifold is Sasakian.
Abstract: We investigate new Clairaut conditions for anti-invariant submersions from Sasakian and Kenmotsu manifolds onto Riemannian manifolds. We prove that there do not exist Clairaut anti-invariant submersion admitting vertical Reeb vector field in case of the total manifold is Sasakian. Several illustrative examples are also included.
16 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 article, the horizontal distribution of a Lagrangian submersion from a Sasakian manifold onto a Riemannian manifold admitting vertical Reeb vector field is integrable.
Abstract: We study Lagrangian submersions from Sasakian and Kenmotsu manifolds onto Riemannian manifolds. We prove that the horizontal distribution of a Lagrangian submersion from a Sasakian manifold onto a Riemannian manifold admitting vertical Reeb vector field is integrable, but the one admitting horizontal Reeb vector field is not. We also show that the horizontal distribution of a such submersion is integrable when the total manifold is Kenmotsu. Moreover, we give some applications of these results.
12 citations
TL;DR: In this paper, a necessary and sufficient condition for an almost Kenmotsu 3-manifold to be conformally flat is given. But this condition is not applicable to the case of the Riemannian product.
Abstract: In this paper, by virtue of a system of partial differential equations, we give a necessary and sufficient condition for an almost Kenmotsu 3-manifold to be conformally flat. As an application, we obtain that an almost Kenmotsu 3-H-manifold with scalar curvature invariant along the Reeb vector field is conformally flat if and only if it is locally isometric to either the hyperbolic space $$\mathbb {H}^3(-1)$$
or the Riemannian product $$\mathbb {H}^{2}(-4)\times \mathbb {R}$$
. Some concrete examples verifying main results are presented.
11 citations
TL;DR: In this paper, the Reeb principal curvature of Hopf hypersurfaces in complex Grassmannians of rank two was shown to be constant along integral curves of Reeb vector field.
Abstract: In this paper, we study real hypersurfaces in complex Grassmannians of rank two. First, the nonexistence of mixed foliate real hypersurfaces is proven. With this result, we show that for Hopf hypersurfaces in complex Grassmannians of rank two, the Reeb principal curvature is constant along integral curves of the Reeb vector field. As a result the classification of contact real hypersurfaces is obtained. We also introduce the notion of q-umbilical real hypersurfaces in complex Grassmannians of rank two and obtain a classification of such real hypersurfaces.
8 citations
Journal Article•
TL;DR: In this article, it was shown that a Ricci soliton admits a Reeb vector field if and only if the manifold is a paraSasaki-Einstein manifold, and an illustrative example is constructed.
Abstract: The aim of this paper is to characterize $3$-dimensional $N(k)$-paracontact metric manifolds satisfying certain curvature conditions. We prove that a $3$-dimensional $N(k)$-paracontact metric manifold $M$ admits a Ricci soliton whose potential vector field is the Reeb vector field $xi$ if and only if the manifold is a paraSasaki-Einstein manifold. Several consequences of this result are discussed. Finally, an illustrative example is constructed.
8 citations
22 Jun 2017
TL;DR: In this paper, the existence and non-existence of periodic Reeb orbits on contact manifolds is surveyed. But the authors place these statements in the context of Finsler geometry by including a proof of the folklore theorem that the finsler geodesic flow can be interpreted as a Reeb flow.
Abstract: We survey some results on the existence (and non-existence) of periodic Reeb orbits on contact manifolds, both in the open and closed case. We place these statements in the context of Finsler geometry by including a proof of the folklore theorem that the Finsler geodesic flow can be interpreted as a Reeb flow. As a mild extension of previous results we present existence statements on periodic Reeb orbits on contact manifolds with suitable supporting open books.
8 citations
TL;DR: In this article, the authors studied second order symmetric parallel tensors in generalized contact metric manifolds and its applications to Ricci solitons, and proved that a generalized contact manifold admits a Ricci-soliton whose potential vector field is the Reeb vector field if and only if the manifold is a Sasaki-Einstein manifold.
Abstract: The object of the present paper is to study second order symmetric parallel tensors in generalized \((k,\,\mu )\)-contact metric manifolds and its applications to Ricci solitons. Next, we prove that a generalized \((k,\,\mu )\)-contact metric manifold M admits a Ricci soliton whose potential vector field is the Reeb vector field \(\xi \) if and only if M is a Sasaki–Einstein manifold. Finally, we give some examples of generalized \((k,\,\mu )\)-contact metric manifold.
5 citations
Journal Article•
TL;DR: In this article, the Ricci tensor tensor is used to classify a 3D almost co-Kahler manifold M3 under some additional conditions related to Ricci's tensor.
Abstract: Let M3 be a three-dimensional almost co-Kahler manifold whose Reeb vector field is harmonic. We obtain some local classification results of M3 under some additional conditions related to the Ricci tensor.
TL;DR: In this paper, the authors studied slant null curves with respect to the original parameter on 3-dimensional normal almost contact B-metric manifolds with parallel Reeb vector field and proved that for non-geodesic such curves there exists a unique Frenet frame for which the original parameters are distinguished.
Abstract: In this paper we study slant null curves with respect to the original parameter on 3-dimensional normal almost contact B-metric manifolds with parallel Reeb vector field. We prove that for non-geodesic such curves there exists a unique Frenet frame for which the original parameter is distinguished. Moreover, we obtain a necessary condition this Frenet frame to be a Cartan Frenet frame with respect to the original parameter. Examples of the considered curves are constructed.
TL;DR: In this article, the authors use the Duistermaat-Heckman localization formula and an extension of it to give rational and explicit expressions of the volume, the total transversal scalar curvature and the Einstein-Hilbert functional, seen as functionals on the Sasaki cone (Reeb cone).
Abstract: Building on an idea laid out by Martelli--Sparks--Yau, we use the Duistermaat-Heckman localization formula and an extension of it to give rational and explicit expressions of the volume, the total transversal scalar curvature and the Einstein--Hilbert functional, seen as functionals on the Sasaki cone (Reeb cone). Studying the leading terms we prove they are all proper. Among consequences we get that the Einstein-Hilbert functional attains its minimal value and each Sasaki cone possess at least one Reeb vector field with vanishing transverse Futaki invariant.
TL;DR: In this article, it was shown that an almost coKahler 3-manifold is semi-symmetric if and only if it is co-kahler and the vertical Ricci curvatures are invariant along the Reeb vector field.
Abstract: Let M be an almost coKahler 3-manifold such that the vertical Ricci curvatures are invariant along the Reeb vector field. In this paper, we prove that M is semi-symmetric if and only if it is coKahler.
Posted Content•
TL;DR: In this paper, it was shown that there is no Clairaut anti-invariant submersion admitting vertical Reeb vector field when the total manifold is Sasakian.
Abstract: We investigate new Clairaut conditions for anti-invariant submersions from normal almost contact metric manifolds onto Riemannian manifolds. We prove that there is no Clairaut anti-invariant submersion admitting vertical Reeb vector field when the total manifold is Sasakian. Several illustrative examples are also included.
TL;DR: In this paper, an analogue of Kirwan surjectivity in the setting of equivariant basic cohomology of K-contact manifolds was shown, where the Reeb vector field induces a free $$S^1$$¯¯ -action, and the quasi-quotient is a symplectic manifold.
Abstract: We prove an analogue of Kirwan surjectivity in the setting of equivariant basic cohomology of K-contact manifolds. If the Reeb vector field induces a free $$S^1$$
-action, the $$S^1$$
-quotient is a symplectic manifold, and our result reproduces Kirwan’s surjectivity for these symplectic manifolds. We further prove a Tolman–Weitsman type description of the kernel of the basic Kirwan map for $$S^1$$
-actions and show that torus actions on a K-contact manifold that preserve the contact form and admit 0 as a regular value of the contact moment map are equivariantly formal in the basic setting.
23 Sep 2017
TL;DR: In this article, it was shown that the Ricci almost soliton is not present in a metric manifold with potential Reeb vector field (RVF) whose potential vector field is a product of a flat (n+1) manifold and an n-dimensional manifold of negative constant curvature.
Abstract: The purpose of this paper is to study Ricci almost soliton and gradient Ricci almost soliton in $(k,\mu)$-paracontact metric manifolds. We prove the non-existence of Ricci almost soliton in a $(k,\mu)$-paracontact metric manifold $M$ with $k -1$ and whose potential vector field is the Reeb vector field $\xi$. Further, if the metric $g$ of a $(k,\mu)$-paracontact metric manifold $M^{2n+1}$ with $k
eq-1$ is a gradient Ricci almost soliton, then we prove either the manifold is locally isometric to a product of a flat $(n+1)$-dimensional manifold and an $n$-dimensional manifold of negative constant curvature equal to $-4$, or, $M^{2n+1}$ is an Einstein manifold.
TL;DR: In this article, isometric immersions into an almost contact metric manifold which falls in the Chinea-Gonzalez class were studied under the hypothesis that the Reeb vector field of the ambient space is normal to the considered submanifolds.
Abstract: We studied isometric immersions into an almost contact metric manifold which falls in the Chinea–Gonzalez class $$C_5\oplus C_{12}$$
, under the hypothesis that the Reeb vector field of the ambient space is normal to the considered submanifolds. Particular attention to the case of a slant immersion is paid. We relate immersions into a Kahler manifold to suitable submanifolds of a $$C_5\oplus C_{12}$$
-manifold. More generally, in the framework of Gray–Hervella, we specify the type of the almost Hermitian structure induced on a non anti-invariant slant submanifold. The cases of totally umbilical or austere submanifolds are discussed.
TL;DR: In this paper, it was shown that an odd-dimensional submanifold of a metric contact pair with orthogonal characteristic foliations inherits a contact form with an almost contact metric structure.
Abstract: We show that \(\phi \)-invariant submanifolds of metric contact pairs with orthogonal characteristic foliations make constant angles with the Reeb vector fields. Our main result is that for the normal case such submanifolds of dimension at least 2 are all minimal. We prove that an odd-dimensional \(\phi \)-invariant submanifold of a metric contact pair with orthogonal characteristic foliations inherits a contact form with an almost contact metric structure, and this induced structure is contact metric if and only if the submanifold is tangent to one Reeb vector field and orthogonal to the other one. Furthermore, we show that the leaves of the two characteristic foliations of the differentials of the contact pair are minimal. We also prove that when one Reeb vector field is Killing and spans one characteristic foliation, the metric contact pair is locally the product of a contact metric manifold with \({\mathbb {R}}\).
TL;DR: In this paper, a necessary and sufficient condition for the Reeb vector field of a 3D non-Kenmotsu almost k-meansu manifold to be minimal is obtained.
Abstract: A necessary and sufficient condition for the Reeb vector field of a three dimensional non-Kenmotsu almost Kenmotsu manifold to be minimal is obtained. Using this result, we obtain some classifications of some types of (k, μ, v)-almost Kenmotsu manifolds. Also, we give some characterizations of the minimality of the Reeb vector fields of (k, μ, v)-almost Kenmotsu manifolds. In addition, we prove that the Reeb vector field of an almost Kenmotsu manifold with conformal Reeb foliation is minimal.
TL;DR: On a real hypersurface in a complex two-plane Grassmannian, the Lie derivation and a differential operator of order one associated with the generalized Tanaka-Webster connection were given in this paper.
Abstract: On a real hypersurface in a complex two-plane Grassmannian we have the Lie derivation and a differential operator of order one associated with the generalized Tanaka–Webster connection . We give a classification of real hypersurfaces on satisfying , where is the Reeb vector field on and the Ricci tensor of .