Integral formulas in Riemannian geometry

In this paper, we prove that if an almost co-Kahler manifold of dimension greater than three satisfying of rigid motions of the Minkowski 2-space.

Ricci Solitons on Almost Co-Kähler Manifolds

Let M be a compact almost coKahler manifold. If the metric g of M is a Ricci soliton and the potential vector field is pointwise collinear with the Reeb vector field, then we prove that M is Ricci-flat and coKahler and the soliton g is steady. This generalizes a Goldberg-like conjecture for coKahler manifolds obtained by Cappelletti-Montano and Pastore, namely any compact Einstein K-almost coKahler manifold is coKahler. Without the assumption of compactness, Ricci solitons with the potential vector fields pointwise collinear with the Reeb vector fields on K-almost coKahler manifolds are also studied. Moreover, we prove that there exist no gradient Ricci solitons on proper ${(\kappa, \mu)}$-almost coKahler manifolds.

A Generalization of the Goldberg Conjecture for CoKähler Manifolds

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 In this paper, we study Yamabe soliton and quasi Yamabe soliton on Kenmotsu manifold. First, we prove that if a Kenmotsu metric is a Yamabe soliton, then it has constant scalar curvature. Examples has been provided on a larger class of almost Kenmotsu manifolds, known as β-Kenmotsu manifold. Next, we study quasi Yamabe soliton on a complete Kenmotsu manifold M and proved that it has warped product structure with constant scalar curvature in a region Σ where ∣Df∣ ≠ 0.

Yamabe soliton and quasi Yamabe soliton on Kenmotsu manifold

Abstract In this paper, we prove that if a 3-dimensional cosymplectic manifold M3 admits a Ricci soliton, then either M3 is locally flat or the potential vector field is an infinitesimal contact transformation.

Ricci solitons on 3-dimensional cosymplectic manifolds

Abstract Kaimakamis and Panagiotidou in [Taiwanese J. Math. 18(6) (2014), 1991–1998] proposed an open question: are there real hypersurfaces in nonflat complex space forms whose ∗-Ricci tensor satisfies the condition of 𝔻-parallelism? In this short note, we present an affirmative answer and prove that a three-dimensional real hypersurface in a nonflat complex space form has 𝔻-parallel ∗-Ricci tensor if and only if it is locally congruent to either a geodesic hypersphere of radius r in ℂ H2(c) with tanh⁡(|c|2r)=12 $\\begin{array}{} \\displaystyle \\tanh(\\frac{\\sqrt{|c|}}{2}r) = \\frac{1}{2} \\end{array}$ or a ruled real hypersurface.

Yaning Wang

Papers

Ricci solitons on 3-dimensional cosymplectic manifolds

-recurrent ∗-Ricci tensor on three-dimensional real hypersurfaces in nonflat complex space forms