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)].