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.

$\beta$-almost solitons on almost co-k\"{a}hler manifolds

Abstract: The object of the present paper is to study $\beta$-almost Yamabe solitons and $\beta$-almost Ricci solitons on almost co-Kahler manifolds. In this paper, we prove that if an almost co-Kahler manifold $M$ with the Reeb vector field $\xi$ admits a $\beta$-almost Yamabe solitons with the potential vector field $\xi$ or $b\xi$, where $b$ is a smooth function then manifold is $K$-almost co-Kahler manifold or the soliton is trivial, respectively. Also, we show if a closed $(\kappa,\mu)$-almost co-Kahler manifold with $n>1$ and $\kappa<0$ admits a $\beta$-almost Yamabe soliton then the soliton is trivial and expanding. Then we study an almost co-Kahler manifold admits a $\beta$-almost Yamabe soliton or $\beta$-almost Ricci soliton with $V$ as the potential vector field, $V$ is a special geometric vector field.

Clairaut anti-invariant submersions from normal almost contact metric manifolds

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.

Sasaki–Ricci Flow and Deformations of Contact Action–Angle Coordinates on Spaces T1,1 and Yp,q

Abstract: In this paper, we are concerned with completely integrable Hamiltonian systems and generalized action–angle coordinates in the setting of contact geometry We investigate the deformations of the Sasaki–Einstein structures, keeping the Reeb vector field fixed, but changing the contact form We examine the modifications of the action–angle coordinates by the Sasaki–Ricci flow We then pass to the particular cases of the contact structures of the five-dimensional Sasaki–Einstein manifolds T1,1 and Yp,q

Ricci collineations on 3-dimensional paracontact metric manifolds

Abstract: We classify three-dimensional paracontact metric manifold whose Ricci operator Q is invariant along Reeb vector field, that is, $${\mathcal {L}} _{\xi }Q=0$$ .

Minimality of invariant submanifolds in Metric Contact Pair Geometry

Abstract: We study invariant submanifolds of manifolds endowed with a normal or complex metric contact pair with decomposable endomorphism field $\phi$. For the normal case, we prove that a $\phi$-invariant submanifold tangent to a Reeb vector field and orthogonal to the other one is minimal. For a $\phi$-invariant submanifold $N$ everywhere transverse to both the Reeb vector fields but not orthogonal to them, we prove that it is minimal if and only if the angle between the tangential component $\xi$ (with respect to $N$) of a Reeb vector field and the Reeb vector field itself is constant along the integral curves of $\xi$. For the complex case (when just one of the two natural almost complex structures is supposed to be integrable), we prove that a complex submanifold is minimal if and only if it is tangent to both the Reeb vector fields.