Holomorphic maps acting as Kobayashi isometries on a family of geodesics

Consider a holomorphic map \(F: D \rightarrow G\) between two domains in \({{\mathbb {C}}}^N\). Let \({\mathscr {F}}\) denote a family of geodesics for the Kobayashi distance, such that F acts as an isometry on each element of \({\mathscr {F}}\). 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 \({\mathscr {F}}\) 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 \({\mathscr {F}}\) 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.