Classification of homogeneous almost α-coKähler three-manifolds

An orientable Riemannian three-manifold (M,g) admits an almost α-coKähler structure with g as a compatible metric if and only if M admits a foliation, defined by a unit closed 1-form, of constant mean curvature. Then, we show that a simply connected homogeneous almost α-coKähler three-manifold is either a Riemannian product of type R×S2(k2), equipped with its standard coKähler structure, or it is a semidirect product Lie group G=R2⋊AR equipped with a left invariant almost α-coKähler structure. Moreover, we distinguish the several spaces of this classification by using the Gaussian curvature KG of the canonical foliation. In particular, R×S2(k2) is the only simply connected homogeneous almost α-coKähler three-manifolds whose canonical foliation has Gaussian curvature KG>0.