This is a reference request on what are surely well known facts.

Let $M$ be a compact hyperbolic surface and $S(M)$ its unit tangent bundle. It follows from facts about Möebius tranformations in the upper half-plane of $\mathbb{C}$ that $S(M)$ admits a Thurston geometry given by the universal covering of the group $\text{PSL}(2,\mathbb{R})$. This is surprising to me since this group has sections of positive curvature (see "Curvatures of left invariant metrics on Lie groups." by Milnor) and it's not apparent from looking at the hyperbolic plane where this curvature would come from.

One (very questionable) explanation is that the positive curvature comes from restricting oneself to vectors of norm $1$. Hence, I was wondering if the Sasaki metric on the tangent $TM$ also has sections of positive curvature.

References discussing the particular case of $S(M)$ and $TM$ for hyperbolic manifolds in detail would be very appreciated. Thank you.