Is the development map in Hyperbolic geometry related to development in Cartan geometry?

Question

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?

Answer 1

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

Answer 2

Here's a sketch of a proof I've thought up: Since $M$ is flat, the connection is complete and parallel transport is homotopy invariant (rel endpoints). Thus, the development map $\mathrm{dev}$ takes homotopy classes of paths (rel. endpoints) $[c]: [0, 1] \to M$ from $x$ to $y$ in $M$ to paths starting at $O(x)$ in $S_x$ (or starting at $eP$ in $G/P$ if we pick a frame). By the usual construction, the universal cover $\widetilde{M}$ is given by homotopy classes of paths (rel. endpoints). Thus, by taking the endpoint of the developed paths on $S_x$, the development map lifts to a map $\widetilde{M} \to S_x$. Choosing a frame gives a map $\widetilde{M} \to G/P$.

Since hyperbolic manifolds can be realized as flat Cartan geometries modeled on $\mathbb{H}^n \cong SO(n,1)/SO(n)$ by model mutation of the usual Riemannian structure, this recovers the usual hyperbolic development map.