Gradient Yamabe Solitons Research Articles

Structure of generalized Yamabe solitons and its applications

AbstractWe consider the broadest concept of the gradient Yamabe soliton, the conformal gradient soliton. In this paper, we elucidate the structure of complete gradient conformal solitons under some assumption, and provide some applications to gradient Yamabe solitons. These results enhance the understanding gained from previous research. Furthermore, we give an affirmative partial answer to the Yamabe soliton version of Perelman’s conjecture.

Investigations on a Riemannian Manifold with a Semi- Symmetric Non-Metric Connection and Gradient Solitons

This article carries out the investigation of a three-dimensional Riemannian manifold N3 endowed with a semi-symmetric type non-metric connection. Firstly, we construct a non-trivial example to prove the existence of a semi-symmetric type non-metric connection on N3. It is established that a N3 with the semi-symmetric type non-metric connection, whose metric is a gradient Ricci soliton, is a manifold of constant sectional curvature with respect to the semi-symmetric type non-metric connection. Moreover, we prove that if the Riemannian metric of N3 with the semi-symmetric type non-metric connection is a gradient Yamabe soliton, then either N3 is a manifold of constant scalar curvature or the gradient Yamabe soliton is trivial with respect to the semi-symmetric type non-metric connection. We also characterize the manifold N3 with a semi-symmetric type non-metric connection whose metrics are Einstein solitons and m-quasi Einstein solitons of gradient type, respectively.

Kenmotsu 3-manifolds and gradient solitons

The aim of this article is to characterize a Kenmotsu 3-manifold whose metric is either a gradient Yamabe soliton or gradient Einstein soliton. It is proven that in both cases this manifold is reduced to the manifold of constant sectional curvature. Finally, we verify the obtained results by an example.

3-Dimensional f-Kenmotsu manifolds and solitons

In this paper, Yamabe solitons, gradient Yamabe solitons and gradient [Formula: see text]-Einstein solitons have been investigated on 3-dimensional [Formula: see text]-Kenmotsu manifolds. Moreover, we characterize 3-dimensional [Formula: see text]-Kenmotsu manifolds admitting *-Ricci solitons and gradient *-Ricci solitons, respectively.

Gradient solitons on twisted product manifolds and their applications in general relativity

In this paper, first, we find the necessary and sufficient condition for a Riemannian manifold to be the locally warped product. Then we investigate the existence of different types of gradient solitons, such as gradient (almost) Yamabe soliton, conformal soliton and gradient Ricci soliton on the twisted product manifolds. We also study the concircular flatness condition on a twisted product and examine the Einstein-type relations on its base and fiber manifold. Moreover, we introduce the notions of twisted generalized Robertson–Walker spacetime and twisted standard static spacetime. We get an ordinary differential equation (ODE) that determines the twisting function of the former and the exact form of the twisting function for the latter one.

Yamabe and gradient Yamabe solitons in the complex hyperbolic two-plane Grassmannians

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 [Formula: see text]. 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 [Formula: see text].

Perfect Fluid Spacetimes and Gradient Solitons

In this article, we investigate perfect fluid spacetimes equipped with concircular vector field. At first, in a perfect fluid spacetime admitting concircular vector field, we prove that the velocity vector field annihilates the conformal curvature tensor. In addition, in dimension 4, we show that a perfect fluid spacetime is a generalized Robertson–Walker spacetime with Einstein fibre. It is proved that if a perfect fluid spacetime furnished with concircular vector field admits a second order symmetric parallel tensor P, then either the equation of state of the perfect fluid spacetime is characterized by p=frac{3-n}{n-1} sigma , or the tensor P is a constant multiple of the metric tensor. Finally, The perfect fluid spacetimes with concircular vector field whose Lorentzian metrics are Ricci soliton, gradient Ricci soliton, gradient Yamabe solitons, and gradient m -quasi Einstein solitons, are characterized.

A note on four-dimensional gradient Yamabe solitons

In this paper, we prove that four-dimensional gradient Yamabe solitons must have a Yamabe metric, provided that an asymptotic condition holds. The [Formula: see text]-dimensional gradient Yamabe solitons are also considered.

Remarks on scalar curvature of gradient Yamabe solitons with non-positive Ricci curvature

The aim of this short note is the study of the scalar curvature of a complete gradient Yamabe solitons. In particular we show an integral inequality for a gradient Yamabe soliton and as a consequence we proved that under a linear growth of the potential function f the gradient Yamabe soliton has constant scalar curvature. Also, with natural conditions and non-positive Ricci curvature, any complete Yamabe soliton has constant scalar curvature, namely, it is a Yamabe metric and become rotationally symmetric.

Yamabe and gradient Yamabe solitons on real hypersurfaces in the complex quadric

In this paper, we give a complete classification of Yamabe solitons and gradient Yamabe solitons on real hypersurfaces in the complex quadric [Formula: see text]. 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 [Formula: see text].

Some characterizations of gradient Yamabe solitons

In this article we have proved that a gradient Yamabe soliton satisfying some additional conditions must be of constant scalar curvature. Later, we have showed that in a gradient expanding Yamabe soliton with non-negative scalar curvature if the potential function satisfies an integral condition then it is isometric to the Euclidean space Rn, and for steady case with the same condition the potential function becomes harmonic. Also we have proved that, in a compact gradient Yamabe soliton, the potential function agrees with the Hodge-de Rham potential up to a constant.

Study on Twisted Product Almost Gradient Yamabe Solitons

In this paper, we first study gradient Yamabe solitons on the twisted product spaces. Then, we classify and characterize the warped product and twisted product spaces with almost gradient Yamabe solitons. We also study the construction of almost gradient Yamabe solitons in the Riemannian product spaces.

A Note on Gradient Solitons in Three-Dimensional Riemannian Manifolds

We charecterize three-dimensional Riemannian manifolds endowed with a special type of vector field if the Riemannian metrices are gradient Yamabe solitons and gradient Einstein solitons respectively.

Perfect fluid spacetimes and Yamabe solitons

This paper deals with the study of perfect fluid spacetimes. It is proven that a perfect fluid spacetime is Ricci recurrent if and only if the velocity vector field of perfect fluid spacetime is parallel and α = β. In addition, in a stiff matter perfect fluid Yang pure space with p + σ ≠ 0, the integral curves generated by the velocity vector field are geodesics. Moreover, it is shown that in a generalized Robertson–Walker perfect fluid spacetime, the Weyl tensor is divergence-free and the gradient of the potential function of the concircular vector field is pointwise collinear with the velocity vector field of perfect fluid spacetime. We also characterize the perfect fluid spacetimes whose Lorentzian metrics are Yamabe and gradient Yamabe solitons, respectively.

Certain solitons on generalized $(\kappa, \mu)$ contact metric manifolds

The aim of the present paper is to study some solitons on three dimensional generalized $(\kappa, \mu)$-contact metric manifolds. We study gradient Yamabe solitons on three dimensional generalized $(\kappa, \mu)$-contact metric manifolds. It is proved that if the metric of a three dimensional generalized $(\kappa, \mu)$-contact metric manifold is gradient Einstein soliton then $\mu = \frac{2\kappa}{\kappa - 2}.$ It is shown that if the metric of a three dimensional generalized $(\kappa, \mu)$-contact metric manifold is closed m-quasi Einstein metric then $\kappa = \frac{\lambda}{m + 2}$ and $\mu = 0.$ We also study conformal gradient Ricci solitons on three dimensional generalized $(\kappa, \mu)$-contact metric manifolds.

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

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.

Yamabe solitons and gradient Yamabe solitons on three-dimensional N(k)-contact manifolds

If a three-dimensional [Formula: see text]-contact metric manifold [Formula: see text] admits a Yamabe soliton of type [Formula: see text], then the manifold has a constant scalar curvature and the flow vector field [Formula: see text] is Killing. Furthermore, either [Formula: see text] has a constant curvature [Formula: see text] or the flow vector field [Formula: see text] is a strict contact infinitesimal transformation. Also, we prove that if the metric of a three-dimensional [Formula: see text]-contact metric manifold [Formula: see text] 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 [Formula: see text]. Finally, we have given an example to verify our result.

A note on almost co-Kähler manifolds

We characterize almost co-Kähler manifolds with gradient Yamabe, gradient Einstein and quasi-Yamabe solitons. It is proved that if the metric of a [Formula: see text]-almost co-Kähler manifold [Formula: see text] is a gradient Yamabe soliton, then [Formula: see text] is either [Formula: see text]-almost co-Kähler or [Formula: see text]-almost co-Kähler or the metric of [Formula: see text] is a trivial gradient Yamabe soliton. A [Formula: see text]-almost co-Kähler manifold with gradient Einstein soliton is [Formula: see text]-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 [Formula: see text]-almost co-Kähler. Consequently, some results of almost co-Kähler manifolds are deduced.

Correction to: Three-dimensional complete gradient Yamabe solitons with divergence-free Cotton tensor

In the original article the formula chapter 4, just above Remark 4.2 in the paragraph starting with Case (2) the formula should read.

Solutions to the affine quasi-Einstein equation for homogeneous surfaces

Abstract We examine the space of solutions to the affine quasi–Einstein equation in the context of homogeneous surfaces. As these spaces can be used to create gradient Yamabe solitons, conformally Einstein metrics, and warped product Einstein manifolds using the modified Riemannian extension, we provide very explicit descriptions of these solution spaces. We use the dimension of the space of affine Killing vector fields to structure our discussion as this provides a convenient organizational framework.