arXiv subjects

Dibakar Dey

Publications and source records attributed to Dibakar Dey.

7 recordsLinked to original sources

arXiv2021

$\ast$-Ricci-Yamabe Soliton and Contact Geometry

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 $\ast$-Ricci-Yamabe soliton is completely classified. Further, it is proved that if a Sasakian 3-manifold $M$ admits $\ast$-Ricci-Yamabe soliton $(g,V,λ,α,β)$ under certain conditions on the soliton vector field $V$, then $M$ is $\ast$-Ricci flat, positive Sasakian and the transverse geometry of $M$ is Fano. In addition, the Sasakian 3-metric $g$ is homothetic to a Berger sphere and the soliton is steady. Also, the potential vector field $V$ is an infinitesimal automorphism of the contact metric structure.

math.DG↗

arXiv2021

Almost Kenmotsu Manifolds Admitting Certain Critical Metric

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.

math.DG↗

arXiv2020

Cotton Solitons on Almost Kenmotsu 3-$h$-Manifolds

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 $\mathbb{H}^2(-4) \times \mathbb{R}$.

math.DG↗

arXiv2020

Almost Kenmotsu metric as Ricci-Yamabe soliton

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.

math.DG↗

arXiv2020

Almost Kenmotsu metric as quasi Yamabe soliton

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.

math.DG↗

arXiv2020

$N(k)$-contact metric as $\ast$-conformal Ricci soliton

The aim of this paper is characterize a class of contact metric manifolds admitting $\ast$-conformal Ricci soliton. It is shown that if a $(2n + 1)$-dimensional $N(k)$-contact metric manifold $M$ admits $\ast$-conformal Ricci soliton or $\ast$-conformal gradient Ricci soliton, then the manifold M is $\ast$-Ricci at and locally isometric to the Riemannian of a flat $(n + 1)$-dimensional manifold and an $n$-dimensional manifold of constant curvature 4 for $n > 1$ and flat for $n = 1$. Further, for the first case, the soliton vector field is conformal and for the $\ast$-gradient case, the potential function $f$ is either harmonic or satisfy a Poisson equation. Finally, an example is presented to support the results.

math.DG↗

arXiv2020

Almost Kenmotsu manifolds admitting certain vector fields

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 with positive constant $ξ$-sectional curvature admits a non-zero HPCV field $V$, then either $M^{2n+1}$ is locally a warped product of an almost Kaehler manifold and an open interval or isometric to a sphere. Moreover, a $(k,μ)'$-almost Kenmotsu manifold admitting a HPCV field $V$ such that $ϕV = 0$ is either locally isometric to $\mathbb{H}^{n+1}(-4) \times \mathbb{R}^n$ or $V$ is an eigenvector of $h'$. Finally, an example is presented.

math.DG↗