Characterisation of geodesic preserving functions

Let ${\mathrm{\Omega }}_1$, ${\mathrm{\Omega }}_2$ be two domains in ${\mathbb{C}}^n$ with Kobayashi metrics $k_{{\mathrm{\Omega }}_i}$ and consider $f\in \mathcal{O}({\mathrm{\Omega }}_1,{\mathrm{\Omega }}_2)$ a holomorphic mapping. Let ${\mathfrak{F}}_1$ and ${\mathfrak{F}}_2$ be a family of geodesics defined on ${\mathrm{\Omega }}_1$ and ${\mathrm{\Omega }}_2$ respectively, where a geodesic between $z$ and $w$ in ${\mathrm{\Omega }}_i$ is the length minimizing curve between the two points for the metric $k_{{\mathrm{\Omega }}_i}$. We say that a holomorphic function $\mathit{\text{preserves}}\mathit{\text{geodesics}}$ if for any geodesic ${\gamma }_1$ in ${\mathfrak{F}}_1$ its image is a subset of a geodesic ${\gamma }_2$ in ${\mathfrak{F}}_2$ ($f({\gamma }_1)\subset {\gamma }_2$). We aim to characterise the family of such functions between families of Kobayashi geodesics passing through a point in the unit disc $\mathbb{D}$ and in the unit ball ${\mathbb{B}}^n$. Some additional results in the complex plane $\mathbb{C}$ and ${\mathbb{C}}^n$.