Narrow Results

First, we want to give a complete classification of Yamabe solitons and gradient Yamabe solitons for real hypersurfaces in the complex hyperbolic two-plane Grassmannians $G_2^{*}({ℂ}^{m+2})$. Next, as an application we also give a complete classification of quasi-Yamabe and gradient quasi-Yamabe solitons on real hypersurfaces in the complex hyperbolic two-plane Grassmannians $G_2^{*}({ℂ}^{m+2})$.

If a 3-dimensional Sasakian metric on a complete manifold (M, g) is a Yamabe soliton, then we show that g has constant scalar curvature, and the flow vector field V is Killing. We further show that, either M has constant curvature 1, or V is an infinitesimal automorphism of the contact metric structure on M.

In this paper, we first introduce the notion of almost quasi-Yamabe solitons and get some interesting formulas for them. Then, we explore conditions under which an almost quasi-Yamabe soliton is trivial and give some characterization results for it. Finally, we give a necessary and sufficient condition under which an arbitrary compact almost Yamabe soliton is necessarily gradiant.

Here, the scalar curvature of an Einstein-type manifold or equivalently an almost Yamabe quasi-soliton is explicitly determined in terms of its soliton function and some rigidity theorems are obtained. Among the others its shown; if the soliton function is negative, then every compact conformal Yamabe quasi-soliton is isometric to the standard Euclidean sphere.

If $M$ is a 3-dimensional contact metric manifold such that $Q\phi =\phi Q$ which admits a Yamabe soliton $(g,V)$ with the flow vector field $V$ pointwise collinear with the Reeb vector field $\xi$, then we show that the scalar curvature is constant and the manifold is Sasakian. Moreover, we prove that if $M$ is endowed with a Yamabe soliton $(g,V)$, then either $M$ is flat or it has constant scalar curvature and the flow vector field $V$ is Killing. Furthermore, we show that if $M$ is non-flat, then either $M$ is a Sasakian manifold of constant curvature $1$ or $V$ is an infinitesimal automorphism of the contact metric structure on $M$.

In this paper, we use less topological restrictions and more geometric and analytic conditions to obtain some sufficient conditions on Yamabe solitons such that their metrics are Yamabe metrics, that is, metrics of constant scalar curvature. More precisely, we use properties of conformal vector fields to find several sufficient conditions on the soliton vector fields of Yamabe solitons under which their metrics are Yamabe metrics.

We characterize almost co-Kähler manifolds with gradient Yamabe, gradient Einstein and quasi-Yamabe solitons. It is proved that if the metric of a $(\kappa ,\mu )$-almost co-Kähler manifold $M^{2n+1}$ is a gradient Yamabe soliton, then $M^{2n+1}$ is either $K$-almost co-Kähler or $N(\kappa )$-almost co-Kähler or the metric of $M^{2n+1}$ is a trivial gradient Yamabe soliton. A $(\kappa ,\mu )$-almost co-Kähler manifold with gradient Einstein soliton is $K$-almost co-Kähler. Finally, it is shown that an almost co-Kähler manifold admitting a quasi-Yamabe soliton, whose soliton vector is pointwise collinear with the Reeb vector field of the manifold, is $K$-almost co-Kähler. Consequently, some results of almost co-Kähler manifolds are deduced.

If a three-dimensional $N(k)$-contact metric manifold $M$ admits a Yamabe soliton of type $(M,g,V)$, then the manifold has a constant scalar curvature and the flow vector field $V$ is Killing. Furthermore, either $M$ has a constant curvature $k$ or the flow vector field $V$ is a strict contact infinitesimal transformation. Also, we prove that if the metric of a three-dimensional $N(k)$-contact metric manifold $M$ admits a gradient Yamabe soliton, then either the manifold is flat or the scalar curvature is constant. Moreover, either the potential function is constant or the manifold is of constant sectional curvature $k$. Finally, we have given an example to verify our result.

This paper is concerned with the study of $LCS$-manifolds and Ricci solitons. It is shown that in a $CS$-spacetime, the fluid has vanishing vorticity and vanishing shear. It is found that in an $LCS$-manifold, $\mathrm{\text{grad}}\alpha$ is an irrotational vector field, where $\alpha$ is a non-zero smooth scalar function. It is proved that in a $CS$-spacetime with generator vector field $\xi$ obeying Einstein equation, $T(\xi ,\xi )>0$ or $<0$ according to $\rho >{\alpha }^2$ or $\rho <{\alpha }^2$, where $\rho$ is a scalar function and $T$ is the energy momentum tensor. Also, it is shown that if $X$ is a non-null spacelike (respectively, timelike) vector field on a $CS$-spacetime with scalar curvature $r$ and cosmological constant $\mathrm{\Lambda }$, then $T(X,X)>0$ if and only if $r>2\mathrm{\Lambda }$ (respectively, $r<2\mathrm{\Lambda }$), and $T(X,X)<0$ if and only if $r<2\mathrm{\Lambda }$ (respectively, $r>2\mathrm{\Lambda }$), and further $T(X,X)=0$ if and only if $r=2\mathrm{\Lambda }$. The nature of the scalar curvature of an $LCS$-manifold admitting Yamabe soliton is obtained. Also, it is proved that an $LCS$-manifold admitting $\eta$-Ricci soliton is $\eta$-Einstein and its scalar curvature is constant if and only if $\alpha$ is constant. Further, it is shown that if $\beta$ is a scalar function with $\beta =-(\xi \rho )$ and $2\alpha \rho -\beta$ vanishes, then the gradients of $\alpha$, $\beta$, $\rho$ are co-directional with the generator $\xi$. In a perfect fluid $CS$-spacetime admitting $\eta$-Ricci soliton, it is proved that the pressure density $p$ and energy density $\sigma$ are constants, and if it agrees Einstein field equation, then we obtain a necessary and sufficient condition for the scalar curvature to be constant. If such a spacetime possesses Ricci collineation, then it must admit an almost $\eta$-Yamabe soliton and the converse holds when the Ricci operator is of constant norm. Also, in a perfect fluid $CS$-spacetime satisfying Einstein equation, it is shown that if Ricci collineation is admitted with respect to the generator $\xi$, then the matter content cannot be perfect fluid, and further $\kappa (p-\sigma )\ne 2\mathrm{\Lambda }$ with gravitational constant $\kappa$ implies that $\xi$ is a Killing vector field. Finally, in an $LCS$-manifold, it is proved that if the $CL$-curvature tensor is conservative, then scalar potential and the generator vector field are co-directional, and if the manifold possesses pseudosymmetry due to the $CL$-curvature tensor, then it is an $\eta$-Einstein manifold.

The objective of this paper is to deal with Kenmotsu manifolds admitting $\eta$-Ricci-Yamabe solitons. First, it is proved that if a Kenmotsu manifold $M$ which admits an $\eta$-Ricci-Yamabe soliton, then the manifold $M$ is Einstein and is of constant scalar curvature. Then, some important characterizations, which classify Kenmotsu manifolds admitting such solitons, are obtained and an example given which supports our results.

In this paper, we give a complete classification of Yamabe solitons and gradient Yamabe solitons on real hypersurfaces in the complex quadric $Q^m=S{O}_{m+2}/S{O}_{2}S{O}_m$. In the following, as an application, we show a complete classification of quasi-Yamabe and gradient quasi-Yamabe solitons on Hopf real hypersurfaces in the complex quadric $Q^m$.

The aim of this paper is to study certain types of metrics such as conformal $\eta$-Ricci soliton and Yamabe soliton in general relativistic spacetime. Here, we have shown the nature of the soliton when the spacetime with semisymmetric energy–momentum tensor admits conformal $\eta$-Ricci soliton, whose potential vector field is torse-forming. We have studied certain curvature conditions on the spacetime that admits conformal $\eta$-Ricci soliton. Also, we have enriched the importance of the Laplace equation on the spacetime admitting conformal $\eta$-Ricci soliton. Next, we have given some applications of physical connection of dust fluid, dark fluid and radiation era on general relativistic spacetime admitting conformal $\eta$-Ricci soliton and Yamabe soliton.

We study conformal Riemannian maps between Riemannian manifolds and derive expression of scalar curvature for its total manifold. Later, we study conformal Riemannian maps whose total manifold admits a Yamabe soliton and obtain conditions for fiber and range space of such maps to be Yamabe soliton. We also present a characterization theorem for a Yamabe soliton to be an almost Yamabe soliton for conformal Riemannian maps. Finally, we derive a nontrivial example of a conformal Riemannian map whose total manifold admits a Yamabe soliton.

This paper examines almost Kenmotsu manifolds (briefly, AKMs) endowed with the almost Ricci–Yamabe solitons (ARYSs) and gradient ARYSs. The condition for an AKM with ARYS to be $\eta$-Einstein is established. We also show that an ARYS on Kenmotsu manifold becomes a Ricci–Yamabe soliton under certain restrictions. In this series, it is proven that a $(2n+1)$-dimensional ${(\kappa ,\mu )}^{\prime }$-AKM equipped with a gradient ARYS is either locally isometric to ${ℍ}^{n+1}(-4)\times {ℝ}^n$ or the Reeb vector field and the soliton vector field are codirectional. The properties of three-dimensional non-Kenmotsu AKMs endowed with a gradient ARYS are studied.