Characterisation of geodesic preserving functions

Let Ω₁, Ω₂ be two domains in Cⁿ with Kobayashi metrics kΩᵢ and consider f ∈ O(Ω₁,Ω₂) a holomorphic mapping. Let F₁ and F₂ be a family of geodesics defined on Ω₁ and Ω₂ respectively, where a geodesic between z and w in Ωᵢ is the length minimizing curve between the two points for the metric kΩᵢ. We say that a holomorphic function preserves geodesics if for any geodesic γ₁ in F₁ its image is a subset of a geodesic γ₂ in F₂ (f(γ₁)⊂ γ₂). We aim to characterise the family of such functions between families of Kobayashi geodesics passing through a point in the unit disc D and in the unit ball Bⁿ. Some additional results in the complex plane C and Cⁿ.