Some Solitons on Homogeneous Almost $\alpha$-Cosymplectic $3$-Manifolds and Harmonic Manifolds

In this paper, we investigate the nature of Einstein solitons, whether it is steady, shrinking or expanding on almost $\alpha$-cosymplectic $3$-manifolds. We also prove that a simply connected homogeneous almost $\alpha$-cosymplectic $3$-manifold, admitting a contact Einstein soliton, is an unimodular semidirect product Lie group. Finally, we show that a harmonic manifold admits a Ricci soliton if and only if it is flat.