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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, it was shown that an almost cosymplectic manifold admits a quasi-Einstein structure and is locally isomorphic to a Lie group if and only if the potential vector field V is collinear with the Reeb vector field.
Abstract: In this article, we study almost cosymplectic manifolds admitting quasi-Einstein structures $$(g, V, m, \lambda )$$. First we prove that an almost cosymplectic $$(\kappa ,\mu )$$-manifold is locally isomorphic to a Lie group if $$(g, V, m, \lambda )$$ is closed and on a compact almost $$(\kappa ,\mu )$$-cosymplectic manifold there do not exist quasi-Einstein structures $$(g, V, m, \lambda )$$, in which the potential vector field V is collinear with the Reeb vector field $$\xi $$. Next we consider an almost $$\alpha $$-cosymplectic manifold admitting a quasi-Einstein structure and obtain some results. Finally, for a K-cosymplectic manifold with a closed, non-steady quasi-Einstein structure, we prove that it is $$\eta $$-Einstein. If $$(g, V, m, \lambda )$$ is non-steady and V is a conformal vector field, we obtain the same conclusion.
8 citations
20 Jul 2021
TL;DR: In this article, a new type of soliton with a potential Reeb vector field on Riemannian manifolds with an almost paracontact structure was introduced and studied.
Abstract: We introduce and study a new type of soliton with a potential Reeb vector field on Riemannian manifolds with an almost paracontact structure corresponding to an almost paracomplex structure. The special cases of para-Einstein-like, para-Sasaki-like and having a torse-forming Reeb vector field were considered. It was proved a necessary and sufficient condition for the manifold to admit a para-Ricci-like soliton, which is the structure that is para-Einstein-like. Explicit examples are provided in support of the proven statements.
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 paper, it was shown that a curve is extremal of the total curvature energy if and only if it lies into either the rectifying plane or the osculating plane along that curve.
7 citations
TL;DR: In this article, the Ricci-like solitons on Sasaki-like almost contact B-metric manifolds are studied, where the potential of the soliton is the Reeb vector field or pointwise collinear to it.
Abstract: Ricci-like solitons on Sasaki-like almost contact B-metric manifolds are the object of study. Cases, where the potential of the Ricci-like soliton is the Reeb vector field or pointwise collinear to it, are considered. In the former case, the properties for a parallel or recurrent Ricci-tensor are studied. In the latter case, it is shown that the potential of the considered Ricci-like soliton has a constant length and the manifold is $\eta$-Einstein. Other curvature conditions are also found, which imply that the main metric is Einstein. After that, some results are obtained for a parallel symmetric second-order covariant tensor on the manifolds under study. Finally, an explicit example of dimension 5 is given and some of the results are illustrated.
7 citations