In Peter Scott's "The geometry of 3-manifolds" he shows that $\widetilde{SL(2,\mathbb{R})}$ is distinct from $\mathbb{H}^2\times\mathbb{R}$ by showing that the horizontal plane field induced from $T\mathbb{H}^2$ is non integrable for $\widetilde{SL(2,\mathbb{R})}$ but the horizontal plane field is integrable for $\mathbb{H}^2\times\mathbb{R}$. I'm wondering how I can show that this is the case.
I understand his argument for $\widetilde{SL(2,\mathbb{R})}$ but I dont see why it is integrable for $\mathbb{H}^2\times \mathbb{R}$. He defines horizontalI by: Let $x\in\mathbb{H}^2$, and $v\in\mathbb T_x\mathbb{H}^2$ and consider all of the geodesic paths through $x$ and lift them to horizontal paths (paths which transport any tangent vector parallel) in $T\mathbb{H}^2$ through $v$ this collection forms a surface in $T\mathbb{H}^2$. The tangent plane to this surface at $v$ is called horizontal. To obtain a plane field on $T\mathbb{H}^2$ we can consider all such tangent planes at every point in $T\mathbb{H}^2$.
This is horizontal plane field that then can be induced onto $\widetilde{SL(2,\mathbb{R})}$. Is there a more intuitive way to define it for $\mathbb{H}^2\times\mathbb{R}$? If not how can I see that this same horizontal plane field is integrable for $\mathbb{H}^2\times\mathbb{R}$?