The object of the present paper is to characterize two classes of almost Kenmotsu manifolds admitting Ricci-Yamabe soliton. It is shown that a $(k,μ)'$-almost Kenmotsu manifold admitting a Ricci-Yamabe soliton or gradient Ricci-Yamabe soliton is locally isometric to the Riemannian product $\mathbb{H}^{n+1}(-4) \times \mathbb{R}^n$. For the later case, the potential vector field is pointwise collinear with the Reeb vector field. Also, a $(k,μ)$-almost Kenmotsu manifold admitting certain Ricci-Yamabe soliton with the curvature property $Q \cdot P = 0$ is locally isometric to the hyperbolic space $\mathbb{H}^{2n+1}(-1)$ and the non-existense of the curvature property $Q \cdot R = 0$ is proved.
In the present paper, we characterize a class of almost Kenmotsu manifolds admitting quasi Yamabe soliton. It is shown that if a $(k,μ)'$-almost Kenmotsu manifold admits a quasi Yamabe soliton $(g,V,λ,α)$ with $V$ pointwise collinear with $ξ$, then (1) $V$ is a constant multiple of $ξ$, (2) $V$ is a strict infinitesimal contact transformation and (3) $(£_V h')X = 0$ for any vector field $X$. Finally an illustrative example is presented to support the result.
This manuscript 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 $η$-Einstein is established. We also show that an ARYS on Kenmotsu manifold becomes Ricci-Yamabe soliton under certain restrictions. In this series, it is proven that a $(2n+1)$-dimensional $(κ, μ)'$-AKM equipped with a gradient ARYS is either locally isometric to $\mathbb{H}^{n+1}(-4)\times\mathbb{R}^n$ or the Reeb vector field and the soliton vector field are codirectional. The properties of $3$-dimensional non-K…
The goal of our present paper is to deliberate $*$-conformal $η$-Ricci soliton within the framework of Kenmotsu manifolds. Here we have shown that a Kenmotsu metric as a $*$-conformal $η$-Ricci soliton is Einstein metric if the soliton vector field is contact. Further, we have evolved the characterization of the Kenmotsu manifold or the nature of the potential vector field when the manifold satisfies gradient almost $*$-conformal $η$-Ricci soliton. Next, we have contrived $*$-conformal $η$-Ricci soliton admitting $(κ,μ)^\prime$-almost Kenmotsu manifold and proved that the manifold is Ricci fla…
In this short note it is established that there does not exist Ricci soliton with the Reeb potential vector field in an almost Kenmotsu manifold (briefly, $\mathcal{AKM}$).
In this paper the notion of Ricci $ρ$-soliton as a generalization of Ricci soliton is defined. We are motivated by the Ricci-Bourguignon flow to define this concept. We show that if a 3- dimensional almost Kenmotsu Einstein manifold $M$ be a $ρ$-soliton, then $M$ is a Kenmotsu manifold of constant sectional curvature $-1$ and the $ρ$-soliton is expanding, with $λ=2$.
In this paper we characterize the Einstein metrics in such broader classes of metrics as almost $η$-Ricci solitons and $η$-Ricci solitons on Kenmotsu manifolds, and generalize some results of other authors. First, we prove that a Kenmotsu metric as an $η$-Ricci soliton is Einstein metric if either it is $η$-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 $η$-Ricci soliton with a Reeb vector field leaving the scalar curvature invariant, then …
The objective of the present paper is to study the $η$-Ricci solitons on Kenmotsu manifold with generalized symmetric metric connection of type $(α,β)$. There are discussed Ricci and $η$-Ricci solitons with generalized symmetric metric connection of type $(α,β)$ satisfying the conditions $\bar{R}.\bar{S}=0$, $\bar{S}.\bar{R}=0$, $\bar{W_{2}}.\bar{S}=0$ and $\bar{S}.\bar{W_{2}}=0.$. Finally, we construct an example of Kenmotsu manifold with generalized symmetric metric connection of type $(α,β)$ admitting $η$-Ricci solitons.
In this paper, we consider $*$-Ricci soliton in the frame-work of Kenmotsu manifolds. First, we prove that if the metric of a Kenmotsu manifold $M$ is a $*$-Ricci soliton, then soliton constant $λ$ is zero. For 3-dimensional case, if $M$ admits a $*$-Ricci soliton, then we show that $M$ is of constant sectional curvature -1. Next, we show that if $M$ admits a $*$-Ricci soliton whose potential vector field is collinear with the characteristic vector field $ξ$, then $M$ is Einstein and soliton vector field is equal to $ξ$. Finally, we prove that if $g$ is a gradient almost $*$-Ricci soliton, the…
In this paper, a systematic study of Kenmotsu pseudo-metric manifolds are introduced. After studying the properties of this manifolds, we provide necessary and sufficient condition for Kenmotsu pseudo-metric manifold to have constant $φ$-sectional curvature, and prove the structure theorem for $ξ$-conformally flat and $φ$-conformally flat Kenmotsu pseudo-metric manifolds. Next, we consider Ricci solitons on this manifolds. In particular, we prove that an $η$-Einstein Kenmotsu pseudo-metric manifold of dimension higher than 3 admitting a Ricci soliton is Einstein, and a Kenmotsu pseudo-metric 3…
The aim of this paper is to study generalized recurrent, generalized Ricci-recurrent, weakly symmetric and weakly Ricci-symmetric Kenmotsu manifolds with respect to the semi-symmetric non-metric connection.
Kenmotsu geometry is a valuable part of contact geometry with nice applications in other fields such as theoretical physics. In this article, we study the statistical counterpart of a Kenmotsu manifold, that is, Kenmotsu statistical manifold with some related examples. We investigate some statistical curvature properties of Kenmotsu statistical manifolds. We prove that a Kenmotsu statistical manifold is not a Ricci-flat statistical manifold with an example. Finally, we prove a very well-known Chen-Ricci inequality for statistical submanifolds in Kenmotsu statistical manifolds of constant $ϕ-$s…
The purpose of the paper is to study Yamabe solitons on three-dimensional para-Sasakian, paracosymplectic and para-Kenmotsu manifolds. Mainly, we proved that *If the semi-Riemannian metric of a three-dimensional para-Sasakian manifold is a Yamabe soliton, then it is of constant scalar curvature, and the flow vector field V is Killing. In the next step, we proved that either manifold has constant curvature -1 and reduces to an Einstein manifold, or V is an infinitesimal automorphism of the paracontact metric structure on the manifold. *If the semi-Riemannian metric of a three-dimensional paraco…
In this paper we study para-Kenmotsu manifolds. We characterize this manifolds by tensor equations and study their properties. We are devoted to a study of $η-$Einstein manifolds. We show that a conformally flat para-Kenmotsu manifold is a space of constant negative curvature $-1$ and we prove that if a para-Kenmotsu manifold is a space of constant $φ-$para-holomorphic sectional curvature $H$, then it is a space of constant curvature and $H=-1$. Finally the object of the present paper is to study a 3-dimensional para-Kenmotsu manifold, satisfying certain curvature conditions. Among other, it i…
The object of the present paper is to characterize the class of Kenmotsu manifolds which admits conformal $η$-Ricci soliton. Here, we have investigated the nature of the conformal $η$-Ricci soliton within the framework of Kenmotsu manifolds. It is shown that an $η$-Einstein Kenmotsu manifold admitting conformal $η$-Ricci soliton is an Einstein one. Moving further, we have considered gradient conformal $η$-Ricci soliton on Kenmotsu manifold and established a relation between the potential vector field and the Reeb vector field. Next, it is proved that under certain condition, a conformal $η$-Ri…
In this research article, we establish the geometrical bearing on Riemannian submersions in terms of $η$-Ricci-Yamabe Soliton with the potential field and giving the classification of any fiber of Riemannian submersion is an $η$-Ricci-Yamabe soliton, $η$-Ricci soliton and $η$-Yamabe soliton. We also discuss the various conditions for which the target manifold of Riemannian submersion is an $η$-Ricci-Yamabe soliton, $η$-Ricci soliton, $η$-Yamabe soliton and quasi-Yamabe soliton. In a particular case when the potential filed $V$ of the $η$-Ricci-Yamabe soliton is of gradient type, we derive a La…
This paper is concerned with the study of generalized gradient Ricci-Yamabe solitons. We characterize the compact generalized gradient Ricci-Yamabe soliton and find certain conditions under which the scalar curvature becomes constant. The estimation of Ricci curvature is deduced and also an isometry theorem is found in gradient Ricci-Yamabe soliton satisfying a finite weighted Dirichlet integral. Further, it is proved that a Ricci-Yamabe soliton reduces to an Einstein manifold when the potential vector field becomes concircular. Moreover, the eigenvalue and the corresponding eigenspace of the …
For an asymptotically Poincare-Einstein manifold with a lower Ricci curvature bound, we establish a sharp inequality relating the type II Yamabe invariant of the interior and the Yamabe invariant of its conformal infinity
The present paper is to deliberate the geometric composition of a perfect fluid spacetime with torse-forming vector field ξ in connection with conformal Ricci-Yamabe metric and conformal η-Ricci-Yamabe metric. Here we have delineated the conditions for conformal Ricci-Yamabe soliton to be expanding, steady, or shrinking. Later, we have acquired Laplace equation from conformal η-Ricci-Yamabe soliton equation when the potential vector field ξ of the soliton is of gradient type. Lastly, we have designated perfect fluid with Robertson-Walker spacetime and some applications of physics and gravity.
In this paper, we consider the notion of Cotton soliton within the framework of almost Kenmotsu 3-$h$-manifolds. First we consider that the potential vector field is pointwise collinear with the Reeb vector field and prove a non-existence of such Cotton soliton. Next we assume that the potential vector field is orthogonal to the Reeb vector field. It is proved that such a Cotton soliton on a non-Kenmotsu almost Kenmotsu 3-$h$-manifold such that the Reeb vector field is an eigen vector of the Ricci operator is steady and the manifold is locally isometric to $\mathbb{H}^2(-4) \times \mathbb{R}$.
In the present paper, we introduce the notion of $\ast$-Miao-Tam critical equation on almost contact metric manifolds and studied on a class of almost Kenmotsu manifold. It is shown that if the metric of a $(2n + 1)$-dimensional $(k,μ)'$-almost Kenmotsu manifold $(M,g)$ satisfies the $\ast$-Miao-Tam critical equation, then the manifold $(M,g)$ is $\ast$-Ricci flat and locally isometric to the Riemannian product of a $(n + 1)$-dimensional manifold of constant sectional curvature $-4$ and a flat $n$-dimensional manifold. Finally, an illustrative example is presented to support the main theorem.
The present study initially identified the generalized symmetric connections $(α,β)$ typed, which can be regarded as more generalised forms of quarter and semi-symmetric connections. The quarter and semi-symmetric connections are obtained respectively particularly when $(α,β)=(1,0)$ and $(α,β)=(0,1)$ are taken into consideration. Taking that into account, a new generalized symmetric metric connection was attained upon Kenmotsu manifolds. In compliance with the new connection, some results were provided through calculation of tensors belonging to Kenmotsu manifold involving curvature tensor, ri…
The aim of this short note is to produce new examples of geometrical flows associated to a given Riemannian flow $g(t)$. The considered flow in covariant symmetric $2$-tensor fields will be called Ricci-Yamabe map since it involves a scalar combination of Ricci tensor and scalar curvature of $g(t)$. Due to the signs of considered scalars the Ricci-Yamabe flow can be also a Riemannian or semi-Riemannian or singular Riemannian flow. We study the associated function of volume variation as well as the volume entropy. Finally, since the two-dimensional case was the most handled situation we express…
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 $η$-Einstein or the potential vector field $V$ is collinear to the Reeb vector field or $V$ is an infinitesimal contact transformation.
In this paper, we show that steady or shrinking complete gradient Yamabe solitons with finite total scalar curvature and non-positive Ricci curvature are Ricci flat. Moreover, under certain pinching condition for Ricci curvature, we show that steady or shrinking complete gradient Yamabe solitons with finite total scalar curvature and non-positive scalar curvature have zero scalar curvature.
We consider almost Ricci-Yamabe soliton in the context of certain contact metric manifolds. Firstly, we prove that if the metric $g$ admits an almost $(α,β)$-Ricci-Yamabe soliton with $α\neq 0$ and potential vector field collinear with the Reeb vector field $ξ$ on a complete contact metric manifold with the Reeb vector field $ξ$ as an eigenvector of the Ricci operator, then the manifold is compact Einstein Sasakian and the potential vector field is a constant multiple of the Reeb vector field $ξ$. Next, if complete $K$-contact manifold admits gradient Ricci-Yamabe soliton with $α\neq 0$, then …
It is well known that a unit sphere admits Sasakian 3-structure. Also, Sasakian manifolds are locally isometric to a unit sphere under several curvature and critical conditions. So, a natural question is: Does there exist any curvature or critical condition under which a Sasakian 3-manifold represents a geometrical object other than the unit sphere? In this regard, as an extension of the $\ast$-Ricci soliton, the notion of $\ast$-Ricci-Yamabe soliton is introduced and studied on two classes contact metric manifolds. A $(2n + 1)$-dimensional non-Sasakian $N(k)$-contact metric manifold admitting…
The goal of the present paper is to deliberate certain types of metric such as $*$-$η$-Ricci-Yamabe soliton on $α$-Cosymplectic manifolds with respect to quarter-symmetric metric connection. Further, we have proved some curvature properties of $α$-Cosymplectic manifolds admitting quarter-symmetric metric connection. Here, we have shown the characteristics of the soliton when the manifold satisfies quarter-symmetric metric connection on $α$-Cosymplectic manifolds. Later, we have acquired Laplace equation from $*$-$η$-Ricci-Yamabe soliton equation when the potential vector field $ξ$ of the solit…
Let $W$ be a manifold with boundary $M$ given together with a conformal class $\bar C$ which restricts to a conformal class $C$ on $M$. Then the relative Yamabe constant $Y_{\bar C}(W,M;C)$ is well-defined. We study the short-time behavior of the relative Yamabe constant $Y_{[\bar g_t]}(W,M;C)$ under the Ricci flow $\bar g_t$ on $W$ with boundary conditions that mean curvature $H_{\bar g_t}\equiv 0$ and $\bar{g}_t|_M\in C = [\bar{g}_0]$. In particular, we show that if the initial metric $\bar{g}_0$ is a Yamabe metric, then, under some natural assumptions, $\left.\frac{d}{dt}\right|_{t=0}Y_{[\b…
本文证明了：若单位体积黎曼流形$(M^n, g)$的里奇曲率由r>0下有界，则$(M^n, g)$的共形类的Yamabe常数至少为$n.r$ .这个不等式已由S. Ilias（Constantes explicites pour les inegalites de Sobolev sur les varietyriemannienes prontes，Ann. Inst. Fourier 33，151-165）.如果度量是爱因斯坦（具有里奇曲率r），则实现相等。这意味着，例如，如果$h$是$CP^2$上的Fubini-Study指标，而$g$是$Ricci(g) \geq Ricci(h)$上的任何其他指标，则$Vol(CP^2, g) \leq Vol(CP^2, h)$。
On a smooth closed Riemannian manifold, we show short time existence of smooth solutions to the $(α,β)$-Ricci-Yamabe flow, which is a natural generalization of the Ricci flow and the Yamabe flow. We also establish some long time existence theorems for the closed $(α,β)$-Ricci-Yamabe flow by estimating its curvatures.
In this paper, we classify 3-dimensional complete gradient Yamabe solitons with divergence-free Cotton tensor. We also give some classifications of complete gradient Yamabe solitons with nonpositively curved Ricci curvature in the direction of the gradient of the potential function.
In this paper, we completely classify almost Yamabe solitons on hypersurfaces in Euclidean spaces arisen from the position vector field. Some results of almost Yamabe solitons with a concurrent vector field and almost Yamabe solitons on submanifolds in Riemannian manifolds equipped with a concurrent vector field are also presented. Moreover, we classify complete Ricci solitons on minimal submanifolds in non-positively curved space forms. For almost Yamabe solitons, all of results in this paper can be applied to Yamabe solitons.
In this paper, we study Ricci flow on compact manifolds with a continuous initial metric. It was known from Simon that the Ricci flow exists for a short time. We prove that the scalar curvature lower bound is preserved along the Ricci flow if the initial metric has a scalar curvature lower bound in distributional sense provided that the initial metric is $W^{1,p}$ for some $n<p\le \infty$. As an application, we use this result to study the relation between Yamabe invariant and Ricci flat metrics. We prove that if the Yamabe invariant is nonpositive and the scalar curvature is nonnegative in di…
In this paper we investigate the structure of certain solutions of the fully nonlinear Yamabe flow, which we call quotient almost Yamabe solitons because they extend quite naturally those called quotient Yamabe solitons. We then present sufficient conditions for a compact quotient almost Yamabe soliton to be either trivial or isometric with an Euclidean sphere. We also characterize noncompact quotient gradient almost Yamabe solitons satisfying certain conditions on both its Ricci tensor and potential function.
We consider a closed cohomogeneity one Riemannian manifold $(M^n,g) $ of dimension $n\geq 3$. If the Ricci curvature of $M$ is positive, we prove the existence of infinite nodal solutions for equations of the form $-Δ_g u + λu = λu^q$ with $λ>0$, $q>1$. In particular for a positive Einstein manifold which is of cohomogeneity one or fibers over a cohomogeniety one Einstein manifold we prove the existence of infinite nodal solutions for the Yamabe equation, with a prescribed number of connected components of its nodal domain.
