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:
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)
“No rules exist, and examples are simply life-savers answering the appeals of rules making vain attempts to exist.”
—André Breton (1896–1966)
“When distant and unfamiliar and complex things are communicated to great masses of people, the truth suffers a considerable and often a radical distortion. The complex is made over into the simple, the hypothetical into the dogmatic, and the relative into an absolute.”
—Walter Lippmann (1889–1974)