V. Venkatesha, H. Aruna Kumara, Devaraja Mallesha Naik
The aim of this article is to study the Riemann soliton and gradient almost Riemann soliton on certain class of almost Kenmotsu manifolds. Also some suitable examples of Kenmotsu and $(κ,μ)'$-almost Kenmotsu manifolds are constructed to justify our results.
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The object of the present paper is to study some properties of Kenmotsu manifold whose metric is conformal $η$-Einstein soliton. We have studied some certain properties of Kenmotsu manifold admitting conformal $η$-Einstein soliton. We have also constructed a 3-dimensional Kenmotsu manifold satisfying conformal $η$-Einstein soliton.
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 $λ=2$.
Characterizations of $\ast$-Ricci-Bourguignon solitons on Kenmotsu manifolds
本文研究了Kenmotsu流形上* -Ricci-Bourguignon溶iton的特征，探讨了其压缩、平衡和扩大的条件，以及与tors-forming向量场的关系，并给出了一个5维例子。
Solitonic aspects of submanifolds in Kenmotsu statistical manifolds
本文研究Kenmotsu统计流形及其子流形上的统计孤子、Yamabe孤子及几乎拟Yamabe孤子，探讨其统计曲率性质，并给出5维示例。主要贡献在于系统分析这些孤子在共圆与共点向量场下的行为。
The Eisenhart problem of finding parallel tensors is solved for the symmetric case in the regular $f$-Kenmotsu framework. On this way, the Olszack-Rosca example of Einstein manifolds provided by $f$-Kenmotsu manifolds via locally symmetric Ricci tensors is recovered as well as a case of Killing vector fields. Some other classes of Einstein-Kenmotsu manifolds are presented. Our result is interpreted in terms of Ricci solitons and special quadratic first integrals.
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 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…
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.
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 $\mathbb{H}^2(-4) \times \mathbb{R}$.
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…
Geometry of $*$-$k$-Ricci-Yamabe soliton and gradient $*$-$k$-Ricci-Yamabe soliton on Kenmotsu manifolds
该文在Kenmotsu流形框架下研究$*$-$k$-Ricci-Yamabe孤立子的几何性质，推导标量曲率、向量场刻画及相关应用，结合梯度$*$-$η$-Einstein孤立子分析黎曼曲率张量，并构造5维实例验证结论。
Existence, Stability, and Geometric Implications of $\ast$-$η$-Schouten Solitons on Kenmotsu Manifolds
本文研究Kenmotsu流形上当势向量场为torsion-forming时$\ast$-$\eta$-Schouten孤子的刻画，推导标量曲率，讨论向量场的性质，并考察无穷小CL变换和Schouten-Van Kampen联络，最后构造三维例子验证结果。
Geometric solitons in a $D$-homothetically deformed Kenmotsu manifold
该文研究$D$-同伦形变Kenmotsu流形中三类势向量场对应的近Riemann与近Ricci孤立子，推导部分情形的曲率表达式，并给出形变流形含梯度孤立子时初始流形Ricci曲率的下界。
$η$-Ricci Solitons on Kenmotsu 3-Manifolds
本文研究 Kenmotsu 3-流形上的 η-Ricci 孤子，涵盖 Codazzi 型 Ricci 张量、循环平行 Ricci 张量、ϕ-Ricci 对称性及曲率条件 R.R=Q(S,R)，并通过构造例子证明适当 η-Ricci 孤子的存在性。
A note on gradient Solitons on two classes of almost Kenmotsu Manifolds
该研究在两类近Kenmotsu流形框架下刻画梯度$(m,ρ)$-拟爱因斯坦孤立子，并借助实例验证相关结果，丰富了近Kenmotsu流形上梯度孤立子的研究内容。
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…
Kenmotsu Contact Geometry Through the Lens of $\ast-\boldsymbolκ$-Ricci-Bourguignon Almost Solitons
研究 Kenmotsu 流形上 $\ast-\boldsymbol{\kappa}$-Ricci-Bourguignon 几乎孤子的性质，推导标量曲率，构造向量场，并探索扭形向量场的应用，给出五维实例。
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…
$\ast$-$η$-Ricci solitons and Einstein metrics on a weak $β$-Kenmotsu manifold
本文在弱$\beta$-Kenmotsu流形上引入$\ast$-Ricci张量，研究$\ast$-$\eta$-Ricci孤立子与结构（$\beta$为常数）的相互作用，得到Einstein度量的新特征。
Bi-slant Riemannian maps to Kenmotsu manifolds and some optimal inequalities
本文提出到 Kenmotsu 流形的双斜黎曼映射，统一推广多类已知黎曼映射，并建立 Chen-Ricci、DDVV 及 Casorati 曲率相关的最优不等式。
$η$-Ricci solitons on para-Kenmotsu manifolds
本文在仿切触几何背景下研究满足特定曲率条件的para-Kenmotsu流形上的$η$-Ricci孤立子，证明其存在蕴含流形拟爱因斯坦，满足特定曲率条件时为爱因斯坦流形，并给出其存在的充分条件。
$\ast$-$η$-Ricci solitons on weak Kenmotsu $f$-manifolds
本文将 *-Ricci 张量引入弱度量 f-流形，研究 *-η-Ricci 孤立子与弱 βf-Kenmotsu 结构的相互作用，并给出 η-Einstein 度量的新特征。
$η$-Ricci solitons and $η$-Einstein metrics on weak $β$-Kenmotsu $f$-manifolds
本文研究弱$β$-Kenmotsu $f$-流形，证明其局部为扭曲积，并得出满足条件的$η$-Ricci孤立子结构使流形成为常数量曲率的$η$-Einstein流形。
The purpose of this paper is to study anti-invariant Riemannian submersions from Kenmotsu manifolds onto Riemannian manifolds. Several fundamental results in this respect are proved. The integrability of the distributions and the geometry of foliations are investigated. We proved that there do not exist (anti-invariant) Riemannian submersions from Kenmotsu manifolds onto Riemannian manifolds such that characteristic vector field ξ is a vertical vector field. We gave a method to get horizontally conformal submersion examples from warped product manifolds onto Riemannian manifolds. Furthermore, …
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…
Kenmotsu LIe groups
本文研究了切触黎曼流形的重要子类 Kenmotsu 流形，完全确定了具有左不变 Kenmotsu 结构的连通且单连通李群，并证明这些李群是 Einstein 黎曼流形。
In 1972, K. Kenmotsu studied a class of almost contact Riemannian manifolds. Later, such a manifold was called a Kenmotsu manifold. This paper, we studied Kenmotsu manifolds with $(2n+s)$-dimensional $s-$contact metric manifold and this manifold, we have called generalized Kenmotsu manifolds. Necessary and sufficient condition is given for an almost $s-$contact metric manifold to be a generalized Kenmotsu manifold.We show that a generalized Kenmotsu manifold is a locally warped product space. In addition, we study some curvature properties of generalized Kenmotsu manifolds. Moreover, we show t…
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.
We study the Riemann curvature tensor of (κ,μ,ν)-spaces when they have almost cosymplectic and almost Kenmotsu structures, giving its writing explicitly. This leads to the definition and study of a natural generalisation of the contact metric (κ,μ,ν)-spaces. We present examples or obstruction results of these spaces in all possible cases.
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.
H-almost Ricci-Yamabe solitons in paracontact geometry
本文在仿切触几何框架下分类了 h-几乎 Ricci-Yamabe 孤立子，系统刻画了 para-Kenmotsu、para-Sasakian 和 para-cosymplectic 流形上的相关结构，并给出两个实例加以说明。
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…
Kenmotsu manifolds and spin-c Killing spinors
该研究拓展了第一作者的结果，证明奇数维黎曼流形存在特定spin-c Killing旋量等价于其为精确$\mu$-Kenmotsu流形，还通过两种理论结合给出刻画证明并描述了该流形的整体结构。
Generalized curvature tensor and the hypersurfaces of the Hermitian manifold for the class of Kenmotsu type
本文研究Kenmotsu型流形，确定其广义曲率张量分量，引入GΦSH-曲率张量及Φ-广义半对称概念，建立其与η-Einstein流形的关系，并证明该类可作为Hermitian流形超曲面存在。
Conformal vector fields on almost Kenmotsu manifolds
本文研究几乎Kenmotsu流形上的共形向量场，推导两类共形向量场情形下流形的结构与扭曲积形式，证明带Killing向量场的$(k,μ)'$-几乎Kenmotsu流形的两种结构，还研究了其上的η-Ricci-Yamabe孤立子并构造实例。
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…
In this note, we completely classify left-invariant Riemann solitons on three-dimensional Lorentzian Lie groups.
In this paper, we study Riemannian submersions whose total manifolds admitting a Ricci soliton. Here, we characterize any fiber of such a submersion is Ricci soliton or almost Ricci soliton. Indeed, we obtain necessary conditions for which the target manifold of Riemannian submersion is a Ricci soliton. Moreover, we study the harmonicity of Riemannian submersion from Ricci soliton and give a characterization for such a submersion to be harmonic.
The main purpose of the paper is to prove that if a compact Riemannian manifold admits a gradient $ρ$-Einstein soliton such that the gradient Einstein potential is a non-trivial conformal vector field, then the manifold is isometric to the Euclidean sphere. We have showed that a Riemannian manifold satisfying gradient $ρ$-Einstein soliton with convex Einstein potential possesses non-negative scalar curvature. We have also deduced a sufficient condition for a Riemannian manifold to be compact which satisfies almost $η$-Ricci soliton (see Theorem 2).
Rigidity results for Riemann solitons
该文研究黎曼孤立子的刚性问题，给出平凡孤立子的条件，证明非紧梯度黎曼孤立子的无穷远性态可使其成为爱因斯坦流形，并得到完备收缩/稳定梯度黎曼孤立子的标量曲率估计。
A note on Almost Riemann Soliton and gradient almost Riemann soliton
本文研究三维非余辛法向近切触度量流形中的近Riemann孤立子与梯度近Riemann孤立子，推导了相关曲率与流形类型的等价条件，并构造了对应实例。
Gradient almost para-Ricci-like solitons on para-Sasaki-like Riemannian $Π$-manifolds are studied. It is proved that these objects have constant soliton coefficients. For the soliton under study is shown that the corresponding scalar curvatures of the considered both metrics are equal and constant and its Ricci tensor is a constant multiple of the vertical component. Explicit example of a 3-dimensional para-Sasaki-like Riemannian $Π$-manifold is provided in support of the proved assertions.
Remarks on Riemann and Ricci solitons in $(α,β)$-contact metric manifolds
该研究聚焦满足Ricci对称条件的$(α,β)$-切触度量流形，分析其中近Riemann与近Ricci孤立子在势向量场与结构向量场逐点共线时的性质，补充了相关几何结构的研究内容。
In this paper we have proved that a compact Riemannian manifold does not admit a metric with positive scalar curvature if there exists a real valued function in this manifold which is strictly positive along a geodesic ray satisfying expanding or steady Yamabe soliton. We have also deduced a relation between scalar curvature and surface area of a geodesic ball in a Riemannian manifold with a pole satisfying steady Yamabe soliton.
In this paper, we recall some general properties and theorems about Translating Solitons in Semi Riemannian Manifolds. Moreover, we investigate those which are invariant by the action of a Lie group of isometries of the ambient space, by paying attention to the behaviour close to the singular orbit (if any) and at infinity. Then, we provide some related examples.
Object of study are para-Ricci-like solitons on para-Sasaki-like almost paracontact almost paracomplex Riemannian manifolds, briefly, Riemannian $Π$-manifolds. Different cases when the potential of the soliton is the Reeb vector field or pointwise collinear to it are considered. Some additional geometric properties of the constructed objects are proven. Results for a parallel symmetric second-order covariant tensor on the considered manifolds are obtained. Explicit example of dimension 5 in support of the given assertions is provided.
Para-Ricci-like solitons with arbitrary potential on para-Sasaki-like Riemannian $Π$-manifolds are introduced and studied. For the studied soliton, it is proved that its Ricci tensor is a constant multiple of the vertical component of both metrics. Thus, the corresponding scalar curvatures of both considered metrics are equal and constant. An explicit example of the Lie group as the manifold under study is presented.
Remarks on almost Riemann solitons with gradient or torse-forming vector field
本文研究黎曼流形中近Riemann孤立子与近Ricci孤立子的关联，利用Bochner公式给出梯度型近Riemann孤立子的λ函数显式表达式，并推导螺管式、扭形势向量场情形下的曲率相关性质。
We recall the notion of (vertical) translating solitons in a product of a semi-Riemannian manifold $(M,g)$ and the real line. Mainly, we restrict our attention to those which are the graph of a smooth function. When dealing with submersions, we show a criteria to lift (or project) translating solitons from the base manifold to the total space (or viceversa). In particular, manifolds foliated by codimension 1 orbits of a Lie group action give rise to such solitons, up to solving a first-order ordinary differential equation. This gives us explicit criteria under which the graph of a function is …
Riemannian maps whose base manifolds admit a Ricci soliton
本文研究底流形容许 Ricci soliton 的 Riemannian 映射，推导了其曲率张量及叶为 Ricci soliton 或 Einstein 的必要条件，并给出了映射为调和与双调和的充要条件。
Killing Mean Curvature Solitons from Riemannian Submersions
针对容许非零Killing向量场的流形，利用黎曼淹没技术将平均曲率孤立子构造问题从PDE简化为ODE，得到双曲空间中旋转子的新实例。
Almost Riemann Solitons with Vertical Potential on Conformal Cosymplectic Contact Complex Riemannian Manifolds
本文在由余辛流形经接触共形变换得到的近接触B-度量流形上引入并研究具有垂直势（与Reeb向量场共线）的近黎曼孤立子，推导曲率性质，并构造五维及七维显式例子。
Considering pseudo-Riemannian $g$-natural metrics on tangent bundles, we prove that the condition of being Ricci soliton is hereditary in the sense that a Ricci soliton structure on the tangent bundle gives rise to a Ricci soliton structure on the base manifold. Restricting ourselves to some class of pseudo-Riemannian $g$-natural metrics, we show that the tangent bundle is a Ricci soliton if and only if the base manifold is flat and the potential vector field is a complete lift of a conformal vector field. We give then a classification of conformal vector fields on a flat Riemannian manifold. …
Ricci--Bourguignon Almost Solitons with Special Potential on Sasaki-like Almost Contact Complex Riemannian Manifolds
该论文在Sasaki型近切触复黎曼流形上引入$\eta$-Ricci-Bourguignon-like近孤子，刻画其几何性质，并构造两类例子验证理论结果。
Pairs of associated Yamabe almost solitons with vertical potential on almost contact complex Riemannian manifolds
该论文研究几乎接触复 Riemann 流形上相伴 Yamabe 几乎孤子对，在 Sasaki-like 和 torse-forming 势两种情形下刻画孤子，并给出三维显式例子验证理论结果。
Investigations on a Riemannian manifold with a semi-symmetric non-metric connection and gradient solitons
本文研究具有半对称非度量联络的三维黎曼流形，构造存在性例子，并证明梯度 Ricci 孤立子、梯度 Yamabe 孤立子、Einstein 孤立子及 m-拟 Einstein 孤立子条件下的曲率或平凡性结论。
We characterize Ricci almost solitons on semi-Riemannian warped products, considering the potential function to depend on the fiber or not. We show that the fiber is necessarily an Einstein manifold. As a consequence of our characterization we prove that when the potential function depends on the fiber, if the gradient of the warping function does not act by translations then the base and the warped product are also Einstein manifolds. Moreover, we show the existence of conformal vector fields on the base, the fiber and on the warped product. Assuming completeness of the warped product we prov…
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