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The $\rho $ -Einstein soliton is a self-similar solution of the Ricci–Bourguignon flow, which includes or relates to some famous geometric solitons, for example, the Ricci soliton and the Yamabe soliton, and so on. This paper deals with the study of $\rho $ -Einstein solitons on Sasakian manifolds. First, we prove that if a Sasakian manifold M admits a nontrivial $\rho $ -Einstein soliton $(M,g,V,\lambda )$ , then M is $\mathcal {D}$ -homothetically fixed null $\eta $ -Einstein and the soliton vector field V is Jacobi field along trajectories of the Reeb vector field $\xi $ , nonstrict infinitesimal contact transformation and leaves $\varphi $ invariant. Next, we find two sufficient conditions for a compact $\rho $ -Einstein almost soliton to be trivial (Einstein) under the assumption that the soliton vector field is an infinitesimal contact transformation or is parallel to the Reeb vector field $\xi $ .

The purpose of this article is to investigate almost Ricci solitons on para- contact manifolds. We demonstrate that a gradient almost Ricci soliton whose metric is para-sasakian turns into Einstein with a constant scalar curvature -2n(2n + 1). Next, we get a few relationships between almost Ricci solitons and Ricci solitons on a K-paracontact manifold and its generalizations. Finally, some ndings on para-contact manifolds and H-paracontact manifolds admitting an almost Ricci soliton with a potential vector field that is a pointwise collinear with the Reeb vector field are discovered.&nbsp

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 $${\mathbb {R}}^{n+1}$$ 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.

In the present paper, we give some characterizations by considering $*$-Ricci soliton as a Kenmotsu metric. We prove that if a Kenmotsu manifold represents an almost $*$-Ricci soliton with the potential vector field $V$ is a Jacobi along the Reeb vector field, then it is a steady $*$-Ricci soliton. Next, we show that a Kenmotsu matric endowed an almost $*$-Ricci soliton is Einstein metric if it is $\eta$-Einstein or the potential vector field $V$ is collinear to the Reeb vector field or $V$ is an infinitesimal contact transformation., Comment: 12 pages. arXiv admin note: text overlap with arXiv:2008.12497

In this paper, we prove two fundamental results of Ricci almost soliton on K-contact manifold. First we prove that if a complete K-contact metric represents a gradient Ricci almost soliton, then it is compact Einstein Sasakian and isometric to a unit sphere. Next we prove that if a K-contact metric represents a Ricci almost soliton whose potential vector field V is contact and the Ricci operator commutes with the structure tensor $$\varphi $$ , then it is Einstein with Einstein constant 2n.

In this paper we characterize the Einstein metrics in such broader classes of metrics as almost $\eta$-Ricci solitons and $\eta$-Ricci solitons on Kenmotsu manifolds, and generalize some results of other authors. First, we prove that a Kenmotsu metric as an $\eta$-Ricci soliton is Einstein metric if either it is $\eta$-Einstein or the potential vector field $V$ is an infinitesimal contact transformation or $V$ is collinear to the Reeb vector field. Further, we prove that if a Kenmotsu manifold admits a gradient almost $\eta$-Ricci soliton with a Reeb vector field leaving the scalar curvature invariant, then it is an Einstein manifold. Finally, we present new examples of $\eta$-Ricci solitons and gradient $\eta$-Ricci solitons, which illustrate our results., Comment: 10 pages

In this paper, we consider an Einstein-type equation on contact metric manifolds. First, we prove that any K-contact manifold admitting an Einstein-type metric is isometric to a unit sphere. Similar conclusion also holds for a contact metric manifold admitting an Einstein type metric with zero radial Weyl curvature and satisfying a commutativity condition. Finally, we study an Einstein-type equation on ( k , μ ) -contact manifold.

In this article, first we prove that if a metric of a para-Sasakian manifold is a Ricci soliton, then either it is an Einstein (and V Killing) or a $$\eta $$-Einstein (and V leaves $$\varphi $$ invariant) manifold. Next, we prove that if a K-paracontact metric g is a gradient Ricci soliton, then it becomes a expanding soliton which is Einstein with constant scalar curvature. Further, we study the Ricci soliton where the potential vector field V is point wise collinear with the Reeb vector field on paracontact manifold. Finally, we consider the gradient Ricci soliton on $$(\kappa ,\mu )$$-paracontact manifold.

In this paper, we have studied the critical point equation (shortly, CPE) within the frame-work of Kenmotsu and almost Kenmotsu manifold satisfying certain nullity conditions. First, we prove that a complete Kenmotsu metric satisfies the CPE is Einstein and locally isometric to the hyperbolic space H2n+1. In case of Kenmotsu manifolds, it is possible to determine the potential function explicitly (locally). We also provide some examples of Kenmotsu and almost Kenmotsu manifolds that satisfies the CPE., Comment: arXiv admin note: text overlap with arXiv:1701.04996

In this paper we characterize certain class of almost Kenmotsu metrics satisfying the Miao-Tam equation.

In this paper, we consider the CPE conjecture in the frame-work of $K$-contact and $(\kappa, \mu)$-contact manifolds. First, we prove that if a complete $K$-contact metric satisfies the CPE is Einstein and is isometric to a unit sphere $S^{2n+1}$. Next, we prove that if a non-Sasakian $ (\kappa, \mu) $-contact metric satisfies the CPE, then $ M^{3} $ is flat and for $ n > 1 $, $ M^{2n+1} $ is locally isometric to $ E^{n+1}\times S^{n}(4)$., Comment: In the published version there was a sign error in Eq. (1.3). We have fixed it here

We prove that if a Sasakian metric is a ∗-Ricci Soliton, then it is either positive Sasakian, or null-Sasakian. Next, we prove that if a complete Sasakian metric is an almost gradient ∗-Ricci Soliton, then it is positive-Sasakian and isometric to a unit sphere [Formula: see text]. Finally, we classify nontrivial ∗-Ricci Solitons on non-Sasakian [Formula: see text]-contact manifolds.