Nature of Some Solitons on Almost coKähler Manifolds and Asymptotically Harmonic Manifolds

Abstract

In this research, we study the nature of $\eta$-Einstein and gradient $\eta$-Einstein soliton in the framework of almost coKähler manifolds and $(\kappa ,\mu )$-almost coKähler manifolds. We find some expressions for scalar curvature of the almost coKähler manifold admitting $\eta$-Einstein soliton in various cases. We also prove that if a $(\kappa ,\mu )$-almost coKähler manifold admits a gradient $\eta$-Einstein soliton, then either the manifold is coKähler, 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.