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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 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

Journal ArticleDOI
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

DOI
01 Jan 2001
TL;DR: In this paper, the authors consider a 3-manifold M equipped with the contact form λ and consider smooth maps u : C → R × M solving the Cauchy-Riemann equations Tu i = J(u) Tu for a distinguished class of almost complex structures J on R ×M which are R-invariant and related to λ.
Abstract: Given a compact 3-manifold M equipped with the contact form λ we consider smooth maps u : C → R × M solving the Cauchy-Riemann equations Tu i = J(u) Tu, for a distinguished class of almost complex structures J on R ×M which are R-invariant and related to λ. If the map is non constant and of finite energy, the projection into M necessarily approaches as |z| → ∞ a periodic solution of the Reeb vector field associated with the contact form. Assuming the periodic solution to be non degenerate we shall describe the asymptotic behavior of the map u. The paper is a revised version of [5] and includes also [6].

11 citations

Journal ArticleDOI
TL;DR: In this paper, the authors introduced the notion of e η -Einstein e -contact metric three-manifolds, which allows for the Reeb vector field to be null.

10 citations

01 Jan 2012
TL;DR: In this article, a scalar product between the normal at the curve and the Reeb vector field is used to characterize slant curves of three-dimensional f-Kenmotsu manifolds.
Abstract: The aim of this paper is to study slant curves of three-dimensional f-Kenmotsu manifolds. These curves are characterized through the scalar product between the normal at the curve and the Reeb vector field. The classification of slant curves in the hyperbolic 3-dimensional space is provided as well as some remarkable cases. Slant curves with proper mean curvature vector field are characterized.

10 citations

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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20221
202126
202028
201918
201813
201721