/pdf/ricci-solitons-and-certain-related-metrics-on-almost-co-246jt351lr.pdf

Ricci Solitons and Certain Related Metrics on Almost Co-Kaehler Manifolds

Let $$(M^{2n+1},\phi ,\xi ,\eta ,g)$$
 be a non-Kenmotsu almost Kenmotsu $$(k,\mu )'$$
 -manifold. If the metric g represents a Yamabe soliton, then either $$M^{2n+1}$$
 is locally isometric to the product space $$\mathbb {H}^{n+1}(-4)\times \mathbb {R}^n$$
 or $$\eta $$
 is a strict infinitesimal contact transformation. The later case can not occur if a Yamabe soliton is replaced by a gradient Yamabe soliton. Some corollaries of this theorem are given and an example illustrating this theorem is constructed.

Almost Kenmotsu $$(k,\mu )'$$ ( k , μ ) ′ -manifolds with Yamabe solitons

The purpose of the present paper is to investigate the almost quasi- Yamabe soliton and gradient almost quasi-Yamabe solitons under the framework of three-dimensional normal almost paracontact metr...

Almost quasi-Yamabe solitons and gradient almost quasi-yamabe solitons in paracontact geometry

Many authors have studied Ricci solitons and their analogs within the framework of (almost) contact geometry. In this article, we thoroughly study the $(m,\rho)$-quasi-Einstein structure on a contact metric manifold. First, we prove that if a $K$-contact or Sasakian manifold $M^{2n+1}$ admits a closed $(m,\rho)$-quasi-Einstein structure, then it is an Einstein manifold of constant scalar curvature $2n(2n+1)$, and for the particular case -- a non-Sasakian $(k,\mu)$-contact structure -- it is locally isometric to the product of a Euclidean space $\RR^{n+1}$ and a sphere $S^n$ of constant curvature $4$. Next, we prove that if a compact contact or $H$-contact metric manifold admits an $(m,\rho)$-quasi-Einstein structure, whose potential vector field $V$ is collinear to the Reeb vector field, then it is a $K$-contact $\eta$-Einstein manifold.

On non-gradient $(m,\rho)$-quasi-Einstein contact metric manifolds.

The goal of the current paper is to characterize paracontact metric manifolds conceding $$\delta $$
 -almost Yamabe solitons. A few fascinating results of such solitons are established. Specifically, we classify $$\delta $$
 -almost Yamabe solitons on $$(k,\mu )$$
 and N(k)-paracontact metric manifolds.

$$\delta $$ δ -Almost Yamabe Solitons in Paracontact Metric Manifolds

In this paper, we study almost cosymplectic manifolds admitting almost quasi-Yamabe solitons (g,V,m,λ). First, we prove that an almost cosymplectic (κ,μ)-manifold is locally isomorphic to a Lie gro...

Almost quasi-Yamabe solitons on almost cosymplectic manifolds

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.

Quasi-Einstein structures and almost cosymplectic manifolds

In this article we study an almost $f$-cosymplectic manifold admitting a Ricci soliton. We first prove that there do not exist Ricci solitons on an almost cosymplectic $(\kappa,\mu)$-manifold. Further, we consider an almost $f$-cosymplectic manifold admitting a Ricci soliton whose potential vector field is the Reeb vector field and show that a three dimensional almost $f$-cosymplectic is a cosymplectic manifold. Finally we classify a three dimensional $\eta$-Einstein almost $f$-cosymplectic manifold admitting a Ricci soliton..

Ricci solitons in almost $f$-cosymplectic manifolds

In this article, we study Einstein-Weyl structures on almost cosymplectic manifolds. First we prove that an almost cosymplectic $(\kappa,\mu)$-manifold is Einstein or cosymplectic if it admits a closed Einstein-Weyl structure or two Einstein-Weyl structures. Next for a three dimensional compact almost $\alpha$-cosymplectic manifold admitting closed Einstein-Weyl structures, we prove that it is Ricc-flat. Further, we show that an almost $\alpha$-cosymplectic admitting two Einstein-Weyl structures is either Einstein or $\alpha$-cosymplectic, provided that its Ricci tensor is commuting. Finally, we prove that a compact $K$-cosymplectic manifold with a closed Einstein-Weyl structure or two special Einstein-Weyl structures is cosymplectic.

Einstein-Weyl structures on almost cosymplectic manifolds

In this article, we study Einstein–Weyl structures on almost cosymplectic manifolds. First we prove that an almost cosymplectic $$(\kappa ,\mu )$$
 -manifold is Einstein or cosymplectic if it admits a closed Einstein–Weyl structure or two Einstein–Weyl structures. Next for a three dimensional compact almost $$\alpha $$
 -cosymplectic manifold admitting closed Einstein–Weyl structures, we prove that it is Ricc-flat. Further, we show that an almost $$\alpha $$
 -cosymplectic admitting two Einstein–Weyl structures is either Einstein or $$\alpha $$
 -cosymplectic, provided that its Ricci tensor is commuting. Finally, we prove that a compact K-cosymplectic manifold with a closed Einstein–Weyl structure or two special Einstein–Weyl structures is cosymplectic.

/pdf/einstein-weyl-structures-on-almost-cosymplectic-manifolds-3zlqrkjych.pdf

Xiaomin Chen

Papers

Almost quasi-Yamabe solitons on almost cosymplectic manifolds

Quasi-Einstein structures and almost cosymplectic manifolds

Ricci solitons in almost $f$-cosymplectic manifolds

Einstein-Weyl structures on almost cosymplectic manifolds

Einstein–Weyl structures on almost cosymplectic manifolds