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

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

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 ≠ 0, then f is a complex geodesic.

• If

$\alpha (f(0),f^{\prime }(0))=1,$

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

References

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