Complex geodesic

In mathematics, a complex geodesic is a generalization of the notion of geodesic to complex spaces. In mathematics, a complex geodesic is a generalization of the notion of geodesic to complex spaces. Let (X, || ||) be a complex Banach space and let B be the open unit ball in X. Let Δ denote the open unit disc in the complex plane C, thought of as the Poincaré disc model for 2-dimensional real/1-dimensional complex hyperbolic geometry. Let the Poincaré metric ρ on Δ be given by and denote the corresponding Carathéodory metric on B by d. Then a holomorphic function f : Δ → B is said to be a complex geodesic if for all points w and z in Δ.