Abstract Let (M, g) be a non-Kenmotsu (κ, μ)′-almost Kenmotsu manifold of dimension 2n + 1. In this paper, we prove that if the metric g of M is a *-Ricci soliton, then either M is locally isometric to the product ℍn+1(−4)×ℝn or the potential vector field is strict infinitesimal contact transformation. Moreover, two concrete examples of (κ, μ)′-almost Kenmotsu 3-manifolds admitting a Killing vector field and strict infinitesimal contact transformation are given.

https://www.degruyter.com/document/doi/10.1515/math-2019-0056/pdf

*-Ricci soliton on (κ, μ)′-almost Kenmotsu manifolds

Let $(M,\phi,\xi,\eta,g)$ be a three-dimensional Kenmotsu manifold. In this paper, we prove that the triple $(g,V,\lambda)$ on $M$ is a $*$-Ricci soliton if and only if $M$ is locally isometric to the hyperbolic 3-space $\mathbf{H}^3(-1)$ and $\lambda=0$. Moreover, if $g$ is a gradient $*$-Ricci soliton, then the potential vector field coincides with the Reeb vector field. We also show that the metric of a coKahler 3-manifold is a $*$-Ricci soliton if and only if it is a Ricci soliton.

Contact 3-manifolds and $*$-Ricci soliton

In this paper, we prove that if the metric of an almost Kenmotsu manifold with conformal Reeb foliation admits a gradient $$\rho $$
 -Einstein soliton, then either $$M^{2n+1}$$
 is Einstein or the potential function is pointwise collinear with the Reeb vector field $$\xi $$
 on an open set $${\mathcal {O}}$$
 of $$M^{2n+1}$$
 . Moreover, we prove that if the metric of a $$(\kappa ,-2)'$$
 -almost Kenmotsu manifold with $$h'\ne 0$$
 admits a gradient $$\rho $$
 -Einstein soliton, then the manifold is locally isometric to the Riemannian product $${\mathbb {H}}^{n+1}(-4)\times {\mathbb {R}}^n$$
 and potential vector field is tangential to the Euclidean factor $${\mathbb {R}}^n$$
 . We show that there does not exist gradient $$\rho $$
 -Einstein soliton on generalized $$(\kappa ,\mu )$$
 -almost Kenmotsu manifold of constant scalar curvature. Finally, we construct an example for gradient $$\rho $$
 -Einstein soliton.

Gradient $$\rho $$ ρ -Einstein soliton on almost Kenmotsu manifolds

Abstract Let (M3, g) be an almost Kenmotsu 3-manifold such that the Reeb vector field is an eigenvector field of the Ricci operator. In this paper, we prove that if g represents a Ricci soliton whose potential vector field is orthogonal to the Reeb vector field, then M3 is locally isometric to either the hyperbolic space ℍ3(−1) or a non-unimodular Lie group equipped with a left invariant non-Kenmotsu almost Kenmotsu structure. In particular, when g represents a gradient Ricci soliton whose potential vector field is orthogonal to the Reeb vector field, then M3 is locally isometric to either ℍ3(−1) or ℍ2(−4) × ℝ.

Ricci solitons on almost Kenmotsu 3-manifolds

We establish a link between a connection symmetry, called conformal collineation, and almost Ricci soliton (in particular Ricci soliton) in reducible Ricci symmetric semi-Riemannian manifolds. As a physical application, by investigating the kinematic and dynamic properties of almost Ricci soliton manifolds, we present a physical model of imperfect fluid spacetimes. This model gives a general relation between the physical quantities (u, μ, p, α, η, σij ) of the matter tensor of the field equations and does not provide any exact solution. Therefore, we propose further study on finding exact solutions of our viscous fluid physical model for which it is required that the fluid velocity vector u be tilted. We also suggest two open problems.

https://downloads.hindawi.com/archive/2016/4903520.pdf

Yaning Wang

Papers

Gradient ricci almost solitons on two classes of almost kenmotsu manifolds