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 nontrivial Ricci soliton if and only if it is flat. Thus we show that rank one symmetric spaces of compact as well as noncompact type are stable under a Ricci soliton. In particular, we obtain a strengthening of Theorems 1 and 2 in [R. H. Bamler, “Stability of symmetric spaces of noncompact type under Ricci flow,” Geom. Funct. Anal. 25 (2), 342–416 (2015)].