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 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
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- for some z ≠ 0, then f is a complex geodesic.
- If
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- where α denotes the Caratheodory length of a tangent vector, then f is a complex geodesic.
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