Complex geodesic

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

ρ ( a , b ) = tanh − 1 ⁡ | a − b | | 1 − 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 ) ) = ρ ( 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

for some z ≠ 0, then f is a complex geodesic.

If

where α denotes the Caratheodory length of a tangent vector, then f is a complex geodesic.