Cotton solitons on almost cokähler 3-manifolds

Abstract

Let (M ³, g) be a three dimensional almost coKähler manifold such that the Reeb vector field ξ is an eigenvector field of the Ricci operator Q , i.e. Qξ = ρξ, where ρ is a smooth function on M. In this article, we prove that if g represents a Cotton soliton with potential vector field being collinear with ξ, or a gradient Cotton soliton, then M is coKähler or locally conformally flat. Furthermore, when g represents a nontrivial Cotton soliton with potential vector field being orthogonal to ξ, we prove that M is coKähler or locally isometric to one of the following Lie groups: E(2) or E(1, 1) if ρ is constant along ξ. Finally, for a (κ, µ, ν)-almost coKähler manifold, we also consider that g is a nontrivial Cotton soliton with potential vector field being orthogonal to ξ.