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 aim of the present paper is to study the properties of Kenmotsu manifolds equipped with a non-symmetric non-metric connection. We also establish some curvature properties of Kenmotsu manifolds. It is proved that a Kenmotsu manifold endowed with a non-symmetric non-metric is irregular.
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.
In the present paper, we characterize almost Kenmotsu manifolds admitting holomorphically planar conformal vector (HPCV) fields. We have shown that if an almost Kenmotsu manifold $M^{2n+1}$ admits a non-zero HPCV field $V$ such that $ϕV = 0$, then $M^{2n+1}$ is locally a warped product of an almost Kaehler manifold and an open interval. As a corollary of this we obtain few classifications of an almost Kenmotsu manifold to be a Kenmotsu manifold and also prove that the integral manifolds of D are totally umbilical submanifolds of $M^{2n+1}$. Further, we prove that if an almost Kenmotsu manifold…
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 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 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 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.
In this paper, we have studied the critical point equation (shortly, CPE) within the frame-work of Kenmotsu and almost Kenmotsu manifold satisfying certain nullity conditions. First, we prove that a complete Kenmotsu metric satisfies the CPE is Einstein and locally isometric to the hyperbolic space H2n+1. In case of Kenmotsu manifolds, it is possible to determine the potential function explicitly (locally). We also provide some examples of Kenmotsu and almost Kenmotsu manifolds that satisfies the CPE.
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 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…
We study $\mathcal D$-homothetic deformations of almost $α$-Kenmotsu structures. We characterize almost contact metric manifolds which are $CR$-integrable almost $α$-Kenmotsu manifolds, through the existence of a canonical linear connection, invariant under $\mathcal D$-homothetic deformations. If the canonical connection associated to the structure $(φ,ξ,η,g)$ has parallel torsion and curvature, then the local geometry is completely determined by the dimension of the manifold and the spectrum of the operator $h'$ defined by $2αh'=({\mathcal L}_ξφ)\circφ$. In particular, the manifold is locall…
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 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 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 $\mathbb{H}^2(-4) \times \mathbb{R}$.
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 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.
For a finite family of 3-dimensional almost contact metric manifolds with closed the structure form $η$ is described a construction of an almost contact metric manifold, where the members of the family are building blocks - cells. Obtained manifold share many properties of cells. One of the more important are nullity conditions. If cells satisfy nullity conditions - then - in the case of almost cosymplectic or almost $α$-Kenmotsu manifolds - "sewed cells" also satisfies nullity condition - but generally with different constants. It is important that even in the case of the generalized nullity …
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…
We investigate 3-dimensional almost Kenmotsu manifolds satisfying special types of nullity conditions depending on two smooth functions $κ,μ$. When either $κ<-1$ and $μ=0$ or $h=0$, such conditions coincide with the $κ$-nullity condition which we show to be equivalent to the $η$-Einstein one. As an application of this result, we obtain examples of $N(κ)$-quasi Einstein manifolds. Moreover, for the aforementioned manifolds, some complete local descriptions of their structure are established, building local "models" for each of them.
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…
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 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.
As a generalization of slant submanifolds and semi-slant submanifolds, we introduce the notions of pointwise slant submanifolds and pointwise semi-slant sunmanifolds of an almost contact metric manifold. We obtain a characterization at each notion, investigate the topological properties of pointwise slant submanifolds, and give some examples of them. We also consider some distributions on cosymplectic, Sasakian, Kenmotsu manifolds and deal with some properties of warped product pointwise semi-slant submanifolds. Finally, we give some inequalities for the squared norm of the second fundamental …
A structure on an almost contact metric manifold is defined as a generalization of well-known cases: Sasakian, quasi-Sasakian, Kenmotsu and cosymplectic. Then we consider a semi-invariant $ξ^{\bot}$-submanifold of a manifold endowed with such a structure and two topics are studied: the integrability of distributions defined by this submanifold and characterizations for the totally umbilical case. In particular we recover results of Kenmotsu, Eum and Papaghiuc.
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…
This paper is a complete study of almost α-paracosmplectic manifolds. We characterize almost α-paracosmplectic manifolds which have para Kaehler leaves. Main curvature identities which are fulfilled by any almost α-paracosmplectic manifold are found. We also proved that ξ is a harmonic vector field if and only if it is an eigen vector field of the Ricci operator. We locally classify three dimensional almost α-para-Kenmotsu manifolds satisfying a certain nullity condition. We show that this condition is invariant under D_{γ,β}-homothetic deformation. Furthermore, we construct examples of almost…
It is provided an overview of existed results concerning classification of contact metric, almost cosymplectic and almost Kenmotsu $(κ,μ)$-manifolds. In the case of dimension three it is described in full details structure of contact metric or almost cosymplectic $(κ,μ)$-spaces. The second part of the paper addresses three-dimensional paracontact metric and almost paracosymplectic $(κ,μ)$-spaces. There is obtained local classification of paracontact metric $(κ,μ)$-spaces, and almost paracosymplectic $(κ,μ)$-spaces, for every possible value of $κ$.
This paper has two goals; the first is to generalize results for the existence and nonexistence of warped product submanifolds of almost contact manifolds, accordingly a self-contained reference of such submanifolds is offered to save efforts of potential research. Most of the results of this paper are general and decisive enough to generalize both discovered and not discovered results. Moreover, a discrete example of contact CR-warped product submanifold in Kenmotsu manifold is constructed. For further research direction, we addressed a couple of open problems arose from the results of this p…
The vector space of the tensors $\mathcal F$ of type (0,3) having the same symmetries as the covariant derivative of the fundamental form of an almost contact metric manifold is considered. A scheme of decomposition of $\mathcal F$ into orthogonal components which are invariant under the action of $U(n)\times 1$ is given. Using this decomposition there are found 12 natural basic classes of almost contact metric manifolds. The classes of cosymplectic, $α$-Sasakian, $α$-Kenmotsu, etc. manifolds fit nicely to these considerations. On the other hand, many new interesting classes of almost contact …
This paper is a continuation of our previous work, where eleven basic classes of almost paracontact metric manifolds with respect to the covariant derivative of the structure tensor field were obtained. First we decompose one of the eleven classes into two classes and the basic classes of the considered manifolds become twelve. Also, we determine the classes of $α$-para-Sasakian, $α$-para-Kenmotsu, normal, paracontact metric, para-Sasakian, K-paracontact and quasi-para-Sasakian manifolds. Moreover, we study 3-dimensional almost paracontact metric manifolds and show that they belong to four bas…
We study locally conformally balanced metrics on almost abelian Lie algebras, namely solvable Lie algebras admitting an abelian ideal of codimension one, providing characterizations in every dimension. Moreover, we classify six-dimensional almost abelian Lie algebras admitting locally conformally balanced metrics and study some compatibility results between different types of special Hermitian metrics on almost abelian Lie groups and their compact quotients. We end by classifying almost abelian Lie algebras admitting locally conformally hyperkähler structures.
In this article we study an almost $f$-cosymplectic manifold admitting a Ricci soliton. We first prove that there do not exist Ricci solitons on an almost cosymplectic $(κ,μ)$-manifold. Further, we consider an almost $f$-cosymplectic manifold admitting a Ricci soliton whose potential vector field is the Reeb vector field and show that a three dimesional almost $f$-cosymplectic is a cosymplectic manifold. Finally we classify a three dimensional $η$-Einstein almost $f$-cosymplectic manifold admitting a Ricci soliton.
In this article, we study Einstein-Weyl structures on almost cosymplectic manifolds. First we prove that an almost cosymplectic $(κ,μ)$-manifold is Einstein or cosymplectic if it admits a closed Einstein-Weyl structure or two Einstein-Weyl structures. Next for a three dimensional compact almost $α$-cosymplectic manifold admitting closed Einstein-Weyl structures, we prove that it is Ricc-flat. Further, we show that an almost $α$-cosymplectic admitting two Einstein-Weyl structures is either Einstein or $α$-cosymplectic, provided that its Ricci tensor is commuting. Finally, we prove that a compac…
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.
In this article, we study almost cosymplectic manifolds admitting quasi-Einstein structures $(g, V, m, λ)$. First we prove that an almost cosymplectic $(κ,μ)$-manifold is locally isomorphic to a Lie group if $(g, V, m, λ)$ is closed and on a compact almost $(κ,μ)$-cosymplectic manifold there do not exist quasi-Einstein structures $(g, V, m, λ)$, in which the potential vector field $V$ is collinear with the Reeb vector filed $ξ$. Next we consider an almost $α$-cosymplectic manifold admitting a quasi-Einstein structure and obtain some results. Finally, for a $K$-cosymplectic manifold with a clos…
We prove the rigidity and vanishing of several indices of "geometrically natural" twisted Dirac operators on almost even-Clifford Hermitian manifolds admitting circle actions by automorphisms.
We investigate new Clairaut conditions for anti-invariant submersions from normal almost contact metric manifolds onto Riemannian manifolds. We prove that there is no Clairaut anti-invariant submersion admitting vertical Reeb vector field when the total manifold is Sasakian. Several illustrative examples are also included.
We study manifolds endowed with an (almost) even Clifford (hermitian) structure and admitting a large automorphism group. We classify them when they are simply connected and the dimension of the automorphism group is maximal, and also prove a gap theorem for the dimension of the automorphism group.
In order to look for a well-behaved counterpart to Dolbeault cohomology in D-complex geometry, we study the de Rham cohomology of an almost D-complex manifold and its subgroups made up of the classes admitting invariant, respectively anti-invariant, representatives with respect to the almost D-complex structure, miming the theory introduced by T.-J. Li and W. Zhang in [T.-J. Li, W. Zhang, Comparing tamed and compatible symplectic cones and cohomological properties of almost complex manifolds, Comm. Anal. Geom. 17 (2009), no. 4, 651-684] for almost complex manifolds. In particular, we prove tha…
