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

Abstract

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