Integral formulas in Riemannian geometry

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

We characterize almost co-Kahler manifolds with gradient Yamabe, gradient Einstein and quasi-Yamabe solitons. It is proved that if the metric of a (κ,μ)-almost co-Kahler manifold M2n+1 is a gradien...

A note on almost co-Kähler manifolds

In this paper, we characterize the gradient Yamabe and the gradient m-quasi Einstein solitons within the framework of three-dimensional cosymplectic manifolds.

Gradient Yamabe and Gradient m-Quasi Einstein Metrics on Three-dimensional Cosymplectic Manifolds

In this paper, we prove that the Ricci tensor of a threedimensional almost Kenmotsu manifold satisfying ∇ξh = 0, h 6= 0, is η-parallel if and only if the manifold is locally isometric to either the Riemannian product H(−4) × R or a non-unimodular Lie group equipped with a left invariant non-Kenmotsu almost Kenmotsu structure.

http://koreascience.or.kr:80/article/JAKO201717234703018.pdf

THREE-DIMENSIONAL ALMOST KENMOTSU MANIFOLDS WITH η-PARALLEL RICCI TENSOR

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

Let M3 be a three-dimensional almost coKahler manifold such that the Ricci curvature of the Reeb vector field is invariant along the Reeb vector field. In this paper, we obtain some classification results of M3 for which the Ricci tensor is η-parallel or the Riemannian curvature tensor is harmonic.

/pdf/ricci-tensors-on-three-dimensional-almost-cokahler-manifolds-r6prtxn8t5.pdf

Yaning Wang

Papers

Contact 3-manifolds and $*$-Ricci soliton

Ricci tensors on three-dimensional almost coKähler manifolds