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
-
- 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.
Read more about this topic: Complex Geodesic
Famous quotes containing the words properties, examples and/or complex:
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (18031882)
“There are many examples of women that have excelled in learning, and even in war, but this is no reason we should bring em all up to Latin and Greek or else military discipline, instead of needle-work and housewifry.”
—Bernard Mandeville (16701733)
“By object is meant some element in the complex whole that is defined in abstraction from the whole of which it is a distinction.”
—John Dewey (18591952)