Homogeneous Structures on ${\mathbb{S}}^2\times \mathbb{R}$ and ${\mathbb{H}}^2\times \mathbb{R}$

We determine all the homogeneous structure tensors on ${\mathbb{S}}^2\times \mathbb{R}$ and ${\mathbb{H}}^2\times \mathbb{R}$. This work together with previous articles yields a complete classification of all the homogeneous structure tensors on three-dimensional homogeneous Riemannian manifolds.