Yamabe Solitons Research Articles

Solitonic View of Generic Contact CR-Submanifolds of Sasakian Manifolds with Concurrent Vector Fields

This paper mainly devotes to the study of some solitons such as Ricci and Yamabe solitons and also their combination called Ricci-Yamabe solitons. In the geometry of solitons, a fundamental question is to identify the conditions under which these solitons can be trivial. Firstly, in this paper we study some extensive results on generic contact CR-submanifolds of Sasakian manifolds endowed with concurrent vector fields. Then some applications of solitons such as Ricci and Ricci-Yamabe solitons on such submanifolds with concurrent vector fields in the same ambient manifold have been discussed.

A study of conformal Riemannian maps whose total manifold admits a 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.

Characterization of a paraSasakian manifold admitting Bach tensor

In the present article, our aim is to characterize Bach flat paraSasakian manifolds. It is established that a Bach flat paraSasakian manifold of dimension greater than three is of constant scalar curvature. Next, we prove that if the metric of a Bach flat paraSasakian manifold is a Yamabe soliton, then the soliton field becomes a Killing vector field. Finally, it is shown that a 3-dimensional Bach flat paraSasakian manifold is locally isometric to the hyperbolic space $H^{2n+1}(1)$.

A Note on Yamabe Solitons on 3-dimensional Almost Kenmotsu Manifolds with $\: \textbf{Q}\phi=\phi \textbf{Q}$

In the present paper, we prove that if the metric of a three dimensional almost Kenmotsu manifold with $\textbf{Q}\phi=\phi \textbf{Q}$ whose scalar curvature remains invariant under the chracterstic vector field $\zeta$, admits a non-trivial Yamabe solitons, then the manifold is of constant sectional curvature or the manifold is Ricci simple.

On the problem of prescribing weighted scalar curvature and the weighted Yamabe flow

Abstract The weighted Yamabe problem introduced by Case is the generalization of the Gagliardo-Nirenberg inequalities to smooth metric measure spaces. More precisely, given a smooth metric measure space ( M , g , e − ϕ d V g , m ) \left(M,g,{e}^{-\phi }{\rm{d}}{V}_{g},m) , the weighted Yamabe problem consists on finding another smooth metric measure space conformal to ( M , g , e − ϕ d V g , m ) \left(M,g,{e}^{-\phi }{\rm{d}}{V}_{g},m) such that its weighted scalar curvature is equal to λ + μ e − ϕ ∕ m \lambda +\mu {e}^{-\phi /m} for some constants μ \mu and λ \lambda , satisfying a certain condition. In this article, we consider the problem of prescribing the weighted scalar curvature. We first prove some uniqueness and nonuniqueness results and then some existence result about prescribing the weighted scalar curvature. We also estimate the first nonzero eigenvalue of the weighted Laplacian of ( M , g , e − ϕ d V g , m ) \left(M,g,{e}^{-\phi }{\rm{d}}{V}_{g},m) . On the other hand, we prove a version of the conformal Schwarz lemma on ( M , g , e − ϕ d V g , m ) \left(M,g,{e}^{-\phi }{\rm{d}}{V}_{g},m) . All these results are achieved by using geometric flows related to the weighted Yamabe flow. We also prove the backward uniqueness of the weighted Yamabe flow. Finally, we consider weighted Yamabe solitons, which are the self-similar solutions of the weighted Yamabe flow.

Applications of some types of solitons within the framework of Kählerian spacetime manifolds

In this paper, we study applications of some certain types of solitons such as conformal Ricci soliton, conformal [Formula: see text]-Ricci–Yamabe soliton and [Formula: see text]-Ricci soliton on Kählerian spacetime manifolds. Further, we have developed the characteristics of conformal Ricci soliton and conformal [Formula: see text]-Ricci–Yamabe soliton on almost pseudo-symmetric Kählerian spacetime manifolds. Here, we have signalized the nature of solitons in terms of shrinking, steady or expanding and we have also presented the relationship between [Formula: see text] and [Formula: see text] in terms of conformal [Formula: see text]-Ricci–Yamabe soliton. Finally, we have embellished the classification of the potential function with respect to gradient [Formula: see text]-Ricci soliton on Kählerian spacetime manifolds.

Geometry of almost contact metrics as a ∗-conformal Ricci–Yamabe solitons and related results

The goal of this paper is to study certain types of metric such as ∗-conformal Ricci–Yamabe soliton (RYS), whose potential vector field is torse-forming on Kenmotsu manifold. Here, we establish the conditions for solitons to be expanding, shrinking or steady and find the scalar curvature when the manifold admits ∗-conformal RYS on Kenmotsu manifold. Next, we developed the nature of the vector field when the manifold satisfies ∗-conformal RYS. Also, we have adorned some applications of torse-forming vector field in terms of ∗-conformal RYS on Kenmotsu manifold. We have also studied infinitesimal CL-transformation and Schouten–van Kampen connection on Kenmotsu manifold, whose metric is ∗-conformal RYS. We present an example of ∗-conformal RYS on three-dimensional Kenmotsu manifold, and verify some of our findings.

Almost Ricci–Yamabe solitons on almost Kenmotsu manifolds

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 [Formula: see text]-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 [Formula: see text]-dimensional [Formula: see text]-AKM equipped with a gradient ARYS is either locally isometric to [Formula: see text] 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.

Yamabe Solitons and τ-Quasi Yamabe Gradient Solitons on Riemannian Manifolds Admitting Concurrent-Recurrent Vector Fields

Abstract We consider a Riemannian manifold (M,g) admitting a concurrent-recurrent vector field for which the metric g is a Yamabe soliton or a τ-quasi Yamabe gradient soliton. We show that if the metric of a Riemannian three-manifold (M, g) admitting a concurrent-recurrent vector field is a Yamabe soliton, then M is of constant negative curvature –α 2. In this case, we see that the potential vector field is Killing. Next, we show that if the metric of a Riemannian manifold M admitting concurrent-recurrent vector field is a non-trivial r-quasi Yamabe gradient soliton with potential function f, then M has constant scalar curvature and is equal to –n(n – 1)α 2. Finally, an illustrative example is presented.

Solitonic Aspect of Relativistic Magneto-Fluid Spacetime with Some Specific Vector Fields

The target of the current research article is to investigate the solitonic attributes of relativistic magneto-fluid spacetime (MFST) if its metrics are Ricci–Yamabe soliton (RY-soliton) and gradient Ricci–Yamabe soliton (GRY-soliton). We exhibit that a magneto-fluid spacetime filled with a magneto-fluid density ρ, magnetic field strength H, and magnetic permeability μ obeys the Einstein field equation without the cosmic constant being a generalized quasi-Einstein spacetime manifold (GQE). In such a spacetime, we obtain an EoS with a constant scalar curvature R in terms of the magnetic field strength H and magnetic permeability μ. Next, we achieve some cauterization of the magneto-fluid spacetime in terms of Ricci–Yamabe solitons with a time-like torse-forming vector field ξ and a φ(Ric) vector field. We establish the existence of a black hole in the relativistic magneto-fluid spacetime by demonstrating that it admits a shrinking Ricci–Yamabe soliton and satisfies the time-like energy convergence criteria. In addition, we examine the magneto-fluid spacetime with a gradient Ricci–Yamabe soliton and deduce some conditions for an equation of state (EoS) ω=−15 with a Killing vector field. Furthermore, we demonstrate that the EoS ω=−15 of the magneto-fluid spacetime under some constraints represents a star model and a static, spherically symmetric perfect fluid spacetime. Finally, we prove that a gradient Ricci–Yamabe soliton with the conditions μ=0 or H=2; μ≠0, H>2 and obeying the equation of state ω=−15 is conceded in a magneto-fluid spacetime, and a naked singularity with a Cauchy horizon subsequently emerges, respectively.

Yamabe solitons with boundary

Yamabe solitons with boundary

Geometry of paracontact metric as an almost Yamabe solitons

In this offering exposition, we intend to study paracontact metric manifold [Formula: see text] admitting almost Yamabe solitons. First, for a general paracontact metric manifold, it is proved that [Formula: see text] is Killing if the vector field [Formula: see text] is an infinitesimal contact transformation and that [Formula: see text] is [Formula: see text]-paracontact if [Formula: see text] is collinear with Reeb vector field. Second, we proved that a [Formula: see text]-paracontact manifold admitting a Yamabe gradient soliton is of constant curvature [Formula: see text] when [Formula: see text] and for [Formula: see text], the soliton is trivial and the manifold has constant scalar curvature. Moreover, for a paraSasakian manifold admitting a Yamabe soliton, we show that it has constant scalar curvature and [Formula: see text] is Killing when [Formula: see text]. Finally, we consider a paracontact metric [Formula: see text]-manifold with a non-trivial almost Yamabe gradient soliton. In the end, we construct two examples of paracontact metric manifolds with an almost Yamabe soliton.

Perfect fluid spacetimes and $k$-almost Yamabe solitons

In this article, we presumed that a perfect fluid is the source of the gravitational field while analyzing the solutions to the Einstein field equations. With this new and creative approach, here we study $k$-almost Yamabe solitons and gradient $k$-almost Yamabe solitons. First, two examples are constructed to ensure the existence of gradient $k$-almost Yamabe solitons. Then we show that if a perfect fluid spacetime admits a $k$-almost Yamabe soliton, then its potential vector field is Killing if and only if the divergence of the potential vector field vanishes. Besides, we prove that if a perfect fluid spacetime permits a $k$-almost Yamabe soliton ($g,k,\rho,\lambda$), then the integral curves of the vector field $\rho$ are geodesics, the spacetime becomes stationary and the isotopic pressure and energy density remain invariant under the velocity vector field $\rho$. Also, we establish that if the potential vector field is pointwise collinear with the velocity vector field and $\rho(a)=0$ where a is a scalar, then either the perfect fluid spacetime represents a phantom era, or the potential function $\Phi$ is invariant under the velocity vector field $\rho$. Finally, we prove that if a perfect fluid spacetime permits a gradient $k$-almost Yamabe soliton ($g,k,D\Phi,\lambda$) and $R, \lambda, k$ are invariant under $\rho$, then the vorticity of the fluid vanishes.

Yamabe Solitons on Conformal Almost-Contact Complex Riemannian Manifolds with Vertical Torse-Forming Vector Field

A Yamabe soliton is considered on an almost-contact complex Riemannian manifold (also known as an almost-contact B-metric manifold), which is obtained by a contact conformal transformation of the Reeb vector field, its dual contact 1-form, the B-metric, and its associated B-metric. A case in which the potential is a torse-forming vector field of constant length on the vertical distribution determined by the Reeb vector field is studied. In this way, manifolds from one of the main classes of the studied manifolds are obtained. The same class contains the conformally equivalent manifolds of cosymplectic manifolds by the usual conformal transformation of the given B-metric. An explicit five-dimensional example of a Lie group is given, which is characterized in relation to the obtained results.

Conformal vector fields and geometric solitons on the tangent bundle with the ciconia metric

Let (X2k, F, g) be an almost anti-paraHermitian manifold with an almost paracomplex structure F and a Riemannian metric g and let TX be its tangent bundle with the ciconia metric g?. The purpose of this paper is divided into two folds. The first one is to examine the curvature properties of the tangent bundle TX with the ciconia metric g?. The second one is to study conformal vector fields and almost Ricci and Yamabe solitons on the tangent bundle TX according to the ciconia metric g?.

ζ-Conformally Flat LP-Kenmotsu Manifolds and Ricci–Yamabe Solitons

In the present paper, we characterize m-dimensional ζ-conformally flat LP-Kenmotsu manifolds (briefly, (LPK)m) equipped with the Ricci–Yamabe solitons (RYS) and gradient Ricci–Yamabe solitons (GRYS). It is proven that the scalar curvature r of an (LPK)m admitting an RYS satisfies the Poisson equation Δr=4(m−1)δ{β(m−1)+ρ}+2(m−3)r−4m(m−1)(m−2), where ρ,δ(≠0)∈R. In this sequel, the condition for which the scalar curvature of an (LPK)m admitting an RYS holds the Laplace equation is established. We also give an affirmative answer for the existence of a GRYS on an (LPK)m. Finally, a non-trivial example of an LP-Kenmotsu manifold (LPK) of dimension four is constructed to verify some of our results.

Some characterizations of quasi Yamabe solitons

In this article, we have proved some results in connection with the potential vector field having finite global norm in quasi Yamabe soliton. We have derived some criteria for the potential vector field on the non-positive Ricci curvature of the quasi Yamabe soliton. Also, a necessary condition for a compact quasi Yamabe soliton has been formulated. We further showed that if the potential vector field has a finite global norm in a complete non-trivial, non-compact quasi Yamabe soliton with finite volume, then the scalar curvature becomes constant and the soliton reduces to a Yamabe soliton.

Results related to the transverse Yamabe problem

Let [Formula: see text] be a Riemannian minimal foliation. The transverse Yamabe problem is to find a metric [Formula: see text] in the basic conformal class of [Formula: see text] such that the transverse scalar curvature of [Formula: see text] is constant. We first study the uniqueness of the solutions of the transverse Yamabe problem. As a generalization of the transverse Yamabe problem, we study the problem of prescribing transverse scalar curvature by using geometric flow. We then prove a version of conformal Schwarz lemma on [Formula: see text]. Finally, we consider the transverse Yamabe soliton, which is the self-similar solution of the transverse Yamabe flow.

Isometries on almost Ricci–Yamabe solitons

The purpose of the present paper is to examine the isometries of almost Ricci–Yamabe solitons. Firstly, the conditions under which a compact gradient almost Ricci–Yamabe soliton is isometric to Euclidean sphere S^n(r) are obtained. Moreover, we have shown that the potential f of a compact gradient almost Ricci–Yamabe soliton agrees with the Hodge–de Rham potential h. Next, we studied complete gradient almost Ricci–Yamabe soliton with alpha ne 0 and non-trivial conformal vector field with non-negative scalar curvature and proved that it is either isometric to Euclidean space E^n or Euclidean sphere S^n. Also, solenoidal and torse-forming vector fields are considered. Lastly, some non-trivial examples are constructed to verify the obtained results.

Riemannian 3-manifolds and Ricci–Yamabe solitons

In this paper, we characterize three-dimensional Riemannian manifolds [Formula: see text] admitting Ricci–Yamabe solitons (RYSs). It is proved that if an [Formula: see text] endowed with a semi-symmetric metric [Formula: see text]-connection admits an RYS, then the scalar curvature of [Formula: see text] satisfies the Poisson equation [Formula: see text] where [Formula: see text] and [Formula: see text] We also discuss the existence of gradient RYS in Riemannian setting. Finally, we construct a nontrivial example of three-dimensional Riemannian 3-manifolds admitting RYS to prove some of our results.