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

TL;DR

This paper studies Einstein and Ricci solitons on almost $eta$-cosymplectic 3-manifolds, establishing conditions for their existence and characterizing harmonic manifolds admitting Ricci solitons as flat.

Contribution

It characterizes Einstein solitons on homogeneous almost $eta$-cosymplectic 3-manifolds and proves that harmonic manifolds admit Ricci solitons only if they are flat.

Findings

Simply connected homogeneous almost $eta$-cosymplectic 3-manifolds with contact Einstein solitons are unimodular semidirect product Lie groups.

Harmonic manifolds admit Ricci solitons if and only if they are flat.

Abstract

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.