Characterisation of Geodesic-Preserving Functions

Abstract

Abstract Let $$\Omega _1$$ Ω 1 , $$\Omega _2$$ Ω 2 be two domains in $$\mathbb {C}^n$$ C n with Kobayashi metrics $$k_{\Omega _i}$$ k Ω i and consider a holomorphic mapping $$f \in \mathcal {O}(\Omega _1,\Omega _2)$$ f ∈ O ( Ω 1 , Ω 2 ) . Let $$\mathfrak {F}_1$$ F 1 and $$\mathfrak {F}_2$$ F 2 be families of geodesics defined in $$\Omega _1$$ Ω 1 and $$\Omega _2$$ Ω 2 , respectively, where a geodesic between z and w in $$\Omega _i$$ Ω i is a length-minimizing curve for the metric $$k_{\Omega _i}$$ k Ω i . We say that a holomorphic mapping preserves geodesics if for any geodesic $$\gamma _1$$ γ 1 in $$\mathfrak {F}_1$$ F 1 its image is a subset of a geodesic $$\gamma _2$$ γ 2 in $$\mathfrak {F}_2$$ F 2 ( $$f(\gamma _1)\subset \gamma _2$$