Locally homogeneous non-gradient quasi-Einstein 3-manifolds

In this paper, we classify the compact locally homogeneous non-gradient m-quasi Einstein 3- manifolds. Along the way, we also prove that given a compact quotient of a Lie group of any dimension that is m-quasi Einstein, the potential vector field X must be left invariant and Killing. We also classify the nontrivial m-quasi Einstein metrics that are a compact quotient of the product of two Einstein metrics. We also show that S1 is the only compact manifold of any dimension which admits a metric which is nontrivially m-quasi Einstein and Einstein.

Imperfect Fluid Generalized Robertson Walker Spacetime Admitting Ricci-Yamabe Metric

In the present paper, we investigate the nature of Ricci-Yamabe soliton on an imperfect fluid generalized Robertson-Walker spacetime with a torse-forming vector field ξ . Furthermore, if the potential vector field ξ of the Ricci-Yamabe soliton is of the gradient type, the Laplace-Poisson equation is derived. Also, we explore the harmonic aspects of η -Ricci-Yamabe soliton on an imperfect fluid GRW spacetime with a harmonic potential function ψ . Finally, we examine necessary and sufficient conditions for a 1 -form η , which is the g -dual of the vector field ξ on imperfect fluid GRW spacetime to be a solution of the Schrödinger-Ricci equation.

A Note on LP -Sasakian Manifolds with Almost Quasi-Yamabe Solitons

We categorize almost quasi-Yamabe solitons on LP -Sasakian manifolds and their CR -submanifolds whose potential vector field is torse-forming, admitting a generalized symmetric metric connection of type α , β . Finally, a nontrivial example is provided to confirm some of our results.

Geometry of α-Cosymplectic Metric as ∗-Conformal η-Ricci–Yamabe Solitons Admitting Quarter-Symmetric Metric Connection

The outline of this research article is to initiate the development of a ∗-conformal η-Ricci–Yamabe soliton in α-Cosymplectic manifolds according to the quarter-symmetric metric connection. Here, we have established some curvature properties of α-Cosymplectic manifolds in regard to the quarter-symmetric metric connection. Further, the attributes of the soliton when the manifold gratifies a quarter-symmetric metric connection have been displayed in this article. Later, we picked up the Laplace equation from ∗-conformal η-Ricci–Yamabe soliton equation when the potential vector field ξ of the soliton is of gradient type, admitting quarter-symmetric metric connection. Next, we evolved the nature of the soliton when the vector field’s conformal killing reveals a quarter-symmetric metric connection. We show an example of a 5-dimensional α-cosymplectic metric as a ∗-conformal η-Ricci–Yamabe soliton acknowledges quarter-symmetric metric connection to prove our results.

Almost Cosymplectic $$(k,\mu )$$-metrics as $$\eta$$-Ricci Solitons

In this paper, we study $$\eta$$ η -Ricci solitons on almost cosymplectic $$(k,\mu )$$ ( k , μ ) -manifolds. As an application, it is proved that if an almost cosymplectic $$(k,\mu )$$ ( k , μ ) -metric with $$k<0$$ k < 0 represents a Ricci soliton, then the potential vector field of the Ricci soliton is a strict infinitesimal contact transformation, and the corresponding almost cosymplectic manifold is locally isometric to a Lie group whose local structure is determined completely by $$k<0$$ k < 0 . In addition, a concrete example is constructed to illustrate the above result.

The k-almost Yamabe solitons and almost coKähler manifolds

In this paper, we study almost coKähler manifolds admitting [Formula: see text]-almost Yamabe solitons [Formula: see text]. First, we obtain a classification of almost coKähler [Formula: see text]-manifolds admitting nontrivial closed [Formula: see text]-almost Yamabe solitons. Next, we consider an almost [Formula: see text]-coKähler manifold admitting a nontrivial [Formula: see text]-almost Yamabe soliton and prove that it is locally the Riemannian product of an almost Kähler manifold with the real line if the potential vector field [Formula: see text] is collinear with the Reeb vector field. For the potential vector field [Formula: see text] being orthogonal to the Reeb vector field, we also obtain two results.

A Note on Solitons with Generalized Geodesic Vector Field

We consider a general notion of an almost Ricci soliton and establish some curvature properties for the case in which the potential vector field of the soliton is a generalized geodesic or a 2-Killing vector field. In this vein, we characterize trivial generalized Ricci solitons.

N(k)-paracontact three metric as a Eta-Ricci soliton

In this paper, we study Eta-Ricci soliton (η-Ricci soliton) on three dimensional N(k)-paracontact metric manifolds. We prove that the scalar curvature of an N(k)-paracontact metric manifold admitting η-Ricci solitons is constant and the manifold is of constant curvature k. Also, we prove that such manifolds are Einstein. Moreover, we show the condition of that the η-Ricci soliton to be expanding, steady or shrinking. In such a case we prove that the potential vector field is Killing vector field. Also, we show that the potential vector field is an infinitesimal automorphism or it leaves the structure tensor in the direction perpendicular to the Reeb vector field ξ. Finally, we illustrate an example of a three dimensional N(k)-paracontact metric manifold admitting an η-Ricci soliton