Complex geodesic

In mathematics, a complex geodesic is a generalization of the notion of geodesic to complex spaces.

Definition
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


 * $$\rho (a, b) = \tanh^{-1} \frac{| a - b |}{|1 - \bar{a} b |}$$

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


 * $$d(f(w), f(z)) = \rho (w, z) \,$$

for all points w and z in Δ.

Properties and examples of complex geodesics

 * Given u ∈ X with ||u|| = 1, the map f : Δ → B given by f(z) = zu is a complex geodesic.
 * Geodesics can be reparametrized: if f is a complex geodesic and g ∈ Aut(Δ) is a bi-holomorphic automorphism of the disc Δ, then f o g is also a complex geodesic. In fact, any complex geodesic f1 with the same image as f (i.e., f1(Δ) = f(Δ)) arises as such a reparametrization of f.
 * If
 * $$d(f(0), f(z)) = \rho (0, z)$$
 * for some z &ne; 0, then f is a complex geodesic.


 * If
 * $$\alpha (f(0), f'(0)) = 1,$$
 * where &alpha; denotes the Caratheodory length of a tangent vector, then f is a complex geodesic.