 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 2dimensional real/1dimensional 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.
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 biholomorphic automorphism of the disc Δ, then f o g is also a complex geodesic. In fact, any complex geodesic f_{1} with the same image as f (i.e., f_{1}(Δ) = f(Δ)) arises as such a reparametrization of f.
 If

 d(f(0),f(z)) = ρ(0,z)
 for some z ≠ 0, then f is a complex geodesic.
 If

 α(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. and Series, C. (eds). Kleinian groups and hyperbolic 3manifolds (Warwick, 2001). London Math. Soc. Lecture Note Ser. 299. Cambridge: Cambridge Univ. Press. pp. 363–384.
Categories: Hyperbolic geometry
 Metric geometry
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