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

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

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

${\displaystyle d(f(w),f(z))=\rho (w,z)\,}$

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

${\displaystyle d(f(0),f(z))=\rho (0,z)}$

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

• If

${\displaystyle \alpha (f(0),f'(0))=1,}$

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

References

• Earle, Clifford J. and Harris, Lawrence A. and Hubbard, John H. and Mitra, Sudeb (2003). "Schwarz's lemma and the Kobayashi and Carathéodory pseudometrics on complex Banach manifolds". In Komori, Y.; Markovic, V.; Series, C. Kleinian groups and hyperbolic 3-manifolds (Warwick, 2001). London Math. Soc. Lecture Note Ser. 299. Cambridge: Cambridge Univ. Press. pp. 363–384.