Integral formulas in Riemannian geometry

In the present paper, we initiate the study of $$*$$
 -
 $$\eta $$
 -Ricci soliton within the framework of Kenmotsu manifolds as a characterization of Einstein metrics. Here we display that a Kenmotsu metric as a $$*$$
 -
 $$\eta $$
 -Ricci soliton is Einstein metric if the soliton vector field is contact. Further, we have developed the characterization of the Kenmotsu manifold or the nature of the potential vector field when the manifold satisfies gradient almost $$*$$
 -
 $$\eta $$
 -Ricci soliton. Next, we deliberate $$*$$
 -
 $$\eta $$
 -Ricci soliton admitting $$(\kappa ,\mu )^\prime $$
 -almost Kenmotsu manifold and proved that the manifold is Ricci flat and is locally isometric to $${\mathbb {H}}^{n+1}(-4)\times {\mathbb {R}}^n$$
 . Finally we present some examples to decorate the existence of $$*$$
 -
 $$\eta $$
 -Ricci soliton, gradient almost $$*$$
 -
 $$\eta $$
 -Ricci soliton on Kenmotsu manifold.

$$*$$ ∗ - $$\eta $$ η -Ricci soliton and contact geometry

Let $$(M^{2n+1},\phi ,\xi ,\eta ,g)$$
 be a non-Kenmotsu almost Kenmotsu $$(k,\mu )'$$
 -manifold. If the metric g represents a Yamabe soliton, then either $$M^{2n+1}$$
 is locally isometric to the product space $$\mathbb {H}^{n+1}(-4)\times \mathbb {R}^n$$
 or $$\eta $$
 is a strict infinitesimal contact transformation. The later case can not occur if a Yamabe soliton is replaced by a gradient Yamabe soliton. Some corollaries of this theorem are given and an example illustrating this theorem is constructed.

Almost Kenmotsu $$(k,\mu )'$$ ( k , μ ) ′ -manifolds with Yamabe solitons

In this short note, we prove a non-existence result for $$*$$
 -Ricci solitons on non-cosymplectic $$(\kappa ,\mu )$$
 -almost cosymplectic manifolds.

Non-existence of $$*$$ ∗ -Ricci solitons on $$(\kappa ,\mu )$$ ( κ , μ ) -almost cosymplectic manifolds

In this paper, we study an almost coKahler manifold admitting certain metrics such as $$*$$
 -Ricci solitons, satisfying the critical point equation (CPE) or Bach flat. First, we consider a coKahler 3-manifold (M, g) admitting a $$*$$
 -Ricci soliton (g, X) and we show in this case that either M is locally flat or X is an infinitesimal contact transformation. Next, we study non-coKahler $$(\kappa ,\mu )$$
 -almost coKahler metrics as CPE metrics and prove that such a g cannot be a solution of CPE with non-trivial function f. Finally, we prove that a $$(\kappa , \mu )$$
 -almost coKahler manifold (M, g) is coKahler if either M admits a divergence free Cotton tensor or the metric g is Bach flat. In contrast to this, we show by a suitable example that there are Bach flat almost coKahler manifolds which are non-coKahler.

Certain types of metrics on almost 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

Yan Zhao

Papers

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