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.

Curvature of K -contact Semi-Riemannian Manifolds

Abstract: In this paper we characterize -contact semi-Riemannian manifolds and Sasakian semi-Riemannian manifolds in terms of curvature. Moreover, we show that any conformally flat -contact semi-Riemannian manifold is Sasakian and of constant sectional curvature , where denotes the causal character of the Reeb vector field. Finally, we give some results about the curvature of a -contact Lorentzian manifold.

Almost $$*$$∗ -Ricci soliton on paraKenmotsu manifolds

Abstract: We consider almost $$*$$ -Ricci solitons in the context of paracontact geometry, precisely, on a paraKenmotsu manifold. First, we prove that if the metric g of $$\eta $$ -Einstein paraKenmotsu manifold is $$*$$ Ricci soliton, then M is Einstein. Next, we show that if $$\eta $$ -Einstein paraKenmotsu manifold admits a gradient almost $$*$$ -Ricci soliton, then either M is Einstein or the potential vector field collinear with Reeb vector field $$\xi $$ . Finally, for three-dimensional case we show that paraKenmotsu manifold is of constant curvature $$-1$$ . An illustrative example is given to support the obtained results.

Lagrangian submersions from normal almost contact manifolds

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.

m-quasi-Einstein metric and contact geometry

Abstract: We study m-quasi-Einstein metric in the framework of contact metric manifolds. The existence of such metric has been confirmed on the class of $$ \eta $$ -Einstein K-contact manifold, in which the potential vector field V is a constant multiple of the Reeb vector field $$\xi $$ . Next, we consider closed m-quasi-Einstein metric on a complete K-contact manifold and prove that it is Sasakian and Einstein provided $$m e 1$$ . We also proved that, if a K-contact manifold M admits an m-quasi-Einstein metric such that the potential vector field V is conformal, then V becomes Killing and M is $$ \eta $$ -Einstein. Finally, we obtain a couple of results on a contact metric manifold M admitting an m-quasi-Einstein metric, whose potential vector field is a point wise collinear with the Reeb vector field.

Conformal slant submersions from cosymplectic manifolds

Abstract: Akyol [Conformal anti-invariant submersions from cosymplectic manifolds, Hacettepe Journal of Mathematics and Statistics 2017; 462: 177-192] defined and studied conformal antiinvariant submersions from cosymplectic manifolds. The aim of the present paper is to define and study the notion of conformal slant submersions (it means the Reeb vector field $\xi$ is a vertical vector field) from cosymplectic manifolds onto Riemannian manifolds as a generalization of Riemannian submersions, horizontally conformal submersions, slant submersions, and conformal antiinvariant submersions. More precisely, we mention many examples and obtain the geometries of the leaves of vertical distribution and horizontal distribution, including the integrability of the distributions, the geometry of foliations, some conditions related to total geodesicness, and harmonicity of the submersions. Finally, we consider a decomposition theorem on the total space of the new submersion.