Holomorphic maps acting as Kobayashi isometries on a family of geodesics

Consider a holomorphic map $F:D\to G$ between two domains in ${\mathbb{C}}^N$. Let $\mathcal{F}$ denote a family of geodesics for the Kobayashi distance, such that $F$ acts as an isometry on each element of $\mathcal{F}$. This paper is dedicated to characterizing the scenarios in which the aforementioned condition implies that $F$ is a biholomorphism. Specifically, we establish this when $D$ is a complete hyperbolic domain, and $\mathcal{F}$ comprises all geodesic segments originating from a specific point. Another case is when $D$ and $G$ are $C^{2+\alpha }$-smooth bounded pseudoconvex domains, and $\mathcal{F}$ consists of all geodesic rays converging at a designated boundary point of $D$. Furthermore, we provide examples to demonstrate that these assumptions are essentially optimal.