Nature of Some Solitons on Almost coK\"{a}hler Manifolds and Asymptotically Harmonic Manifolds
In this research, we study the nature of $\eta$-Einstein and gradient $\eta$-Einstein soliton in the framework of almost coK\"{a}hler manifolds and $(\kappa, \mu)$-almost coK\"{a}hler manifolds. We find some expressions for scalar curvature of the almost coK\"{a}hler manifold admitting $\eta$-Einstein soliton in various cases. We also prove that if a $(\kappa, \mu)$-almost coK\"{a}hler manifold admits a gradient $\eta$-Einstein soliton, then either the manifold is coK\"{a}hler, or $N(\kappa)$-almost coKahler, or the soliton is trivial. We present an example which validates our results. Finally, we investigate the asymptotically harmonic manifolds admitting non-trivial Ricci solitons and show that they exhibit rigid behavior if for example, scalar curvature attains maximum.