If you consider hyperbolic $3$-space $H^3$, modeled by the open unit ball in $\mathbb{R}^3$, then given any two distinct points $x_1$, $x_2$ in $H^3$, there is a natural way of identifying the unit tangent spheres $S_{x_1}$ and $S_{x_2}$ at $x_1$ and $x_2$ respectively. Start at $x_1$. Given a unit tangent vector $v$ at $x_1$, draw the geodesic ray starting at $x_1$ with initial velocity $v$, and define $f_1(v)$ to be the ideal point which is the limiting point of that geodesic ray. Then $f_1: S_{x_1} \to S_\infty$ is a diffeomorphism from $S_{x_1}$ onto the sphere at infinity.

Similarly, one may define the diffeomorphism $f_2: S_{x_2} \to S_\infty$. Then the composition $f_2^{-1} \circ f_1$ is a naturally defined diffeomorphism from $S_{x_1}$ onto $S_{x_2}$.

Now if we think of each unit tangent sphere as "rolling" onto $H^3$, then if we roll the sphere along two different paths joining $x_1$ to $x_2$, then at the end we will end up with the same "sphere configuration" at $x_2$, so to speak. Thus, if I am not mistaken, this defines a flat Cartan geometry, where the principal fiber bundle is a sphere bundle (in this case topologically trivial).

I am a bit rusty on Cartan geometry (which I never studied sufficiently carefully), but am I right that this is an example of a flat Cartan geometry? Could someone please write an answer with some more details perhaps?