We consider a natural condition determining a large class of almost contact metric structures. We study their geometry, emphasizing that this class shares several properties with contact metric manifolds. We then give a complete classification of left-invariant examples on three-dimensional Lie groups, and show that any simply connected homogeneous Riemannian three-manifold $(M, g)$ admits a natural almost contact structure having $g$ as a compatible metric. Moreover, we investigate left-invariant CR structures corresponding to natural almost contact metric structures.
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