I am more familiar with Cartan geometry, and in this setting we have a notion of development of curves. As described in Cap & Slovak 1.5.17, on a Cartan geometry $(\mathcal{P} \to M, \omega)$ modelled on $G/P$, we have a "Cartan space" which is an associated bundle $S :=\mathcal{P} \times^P G/P \to M$ that comes equipped with a canonical section $O: x \mapsto [u, eP]$ as well as a connection induced by the Cartan connection. Then there is a development map which takes germs of curves in the base $M$ to germs of curves in the model $G/P$ (technically the fibre of the Cartan space until we pick a frame in the $\mathcal{P}$). Explicitly, for a (germ of a) curve $c$ in $M$ with $c(0)=x$, we get (a germ of a) curve $\mathrm{dev}\_c$ in the fibre $S\_x$ of the Cartan space given by $$\mathrm{dev}\_c(t) := \mathrm{Pt}\_{c\_t}((O(c(t)),-t)$$
where $c\_t(s) := c(t+s)$. After picking a frame $u \in S\_x$ we can identify $S\_x$ with the model $G/P$.


Since n-dimensional (oriented) Hyperbolic geometry can be viewed as a Cartan geometry modelled on $\mathrm{SO}(n,1)/\mathrm{SO}(n)$, we also have this notion of development of curves in hyperbolic geometry.


But in the hyperbolic geometry literature (which I am mostly unfamiliar with) there is another notion of developing/development: For $(M,g)$ a connected hyperbolic $n$-manifold, there is a developing/development map on the universal cover $\mathrm{dev}: \widetilde{M} \to \mathbb{H}^n$ which is an isometry if $M$ is complete (if I remember correctly).


Is there some nice way to related these two notions of development? Or is the naming mostly coincidence?



Is the development map in Hyperbolic geometry related to development in Cartan geometry?
Of course it is the same. Any submanifold on which the curvature of a Cartan geometry vanishes has a developing map from its universal covering space. I don't know any great reference, but I have used this in many of my papers, for example: <https://arxiv.org/abs/1005.1472>


