Classification of homogeneous almost α-coKähler three-manifolds

Abstract An orientable Riemannian three-manifold ( M , g ) admits an almost α -coKahler 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 -coKahler three-manifold is either a Riemannian product of type R × S 2 ( k 2 ) , equipped with its standard coKahler structure, or it is a semidirect product Lie group G = R 2 ⋊ A R equipped with a left invariant almost α -coKahler structure. Moreover, we distinguish the several spaces of this classification by using the Gaussian curvature K G of the canonical foliation. In particular, R × S 2 ( k 2 ) is the only simply connected homogeneous almost α -coKahler three-manifolds whose canonical foliation has Gaussian curvature K G > 0 .