Three-dimensional homogeneous almost contact metric structures

We consider left-invariant almost contact metric structures on three-dimensional Lie groups, satisfying a quite natural and mild condition. We prove that any three-dimensional Riemannian Lie group admits one of such structures. Moreover, our study leads to the complete classification of three-dimensional left-invariant normal almost contact metric structures, as well as all cases where the one-form η is contact. We then study almost contact metric properties of these examples and harmonicity properties of their characteristic vector fields.