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

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

In this paper, by virtue of a system of partial differential equations, we give a necessary and sufficient condition for an almost Kenmotsu 3-manifold to be conformally flat. As an application, we obtain that an almost Kenmotsu 3-H-manifold with scalar curvature invariant along the Reeb vector field is conformally flat if and only if it is locally isometric to either the hyperbolic space $$\mathbb {H}^3(-1)$$
 or the Riemannian product $$\mathbb {H}^{2}(-4)\times \mathbb {R}$$
 . Some concrete examples verifying main results are presented.

Conformally Flat Almost Kenmotsu 3-Manifolds

In this paper, we prove that the Ricci tensor of an almost Kenmotsu 3-h-manifold is cyclic-parallel if and only if it is parallel and hence, the manifold is locally isometric to either the hyperbolic space H3(−1) or the Riemannian product H2(−4)× R. c ©2016 All rights reserved.

/pdf/almost-kenmotsu-3-h-manifolds-with-cyclic-parallel-ricci-5ejgufe753.pdf

Almost Kenmotsu 3−h -manifolds with cyclic-parallel Ricci tensor

A note on almost contact Riemannian 3-manifolds

Abstract Let M3 be a three-dimensional almost Kenmotsu manifold satisfying ▽ξh = 0. In this paper, we prove that the curvature tensor of M3 is harmonic if and only if M3 is locally isometric to either the hyperbolic space ℍ3(-1) or the Riemannian product ℍ2(−4) × ℝ. This generalizes a recent result obtained by [Wang Y., Three-dimensional locally symmetric almost Kenmotsu manifolds, Ann. Polon. Math., 2016, 116, 79-86] and [Cho J.T., Local symmetry on almost Kenmotsu three-manifolds, Hokkaido Math. J., 2016, 45, 435-442].

Yaning Wang

Papers

Ricci solitons on almost Kenmotsu 3-manifolds

A class of 3-dimensional almost Kenmotsu manifolds with harmonic curvature tensors