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| In [[mathematics]], a '''complex geodesic''' is a generalization of the notion of [[geodesic]] to [[complex number|complex]] spaces.
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| ==Definition==
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| Let (''X'', || ||) be a complex [[Banach space]] and let ''B'' be the [[open set|open]] [[unit ball]] in ''X''. Let Δ denote the open unit disc in the [[Complex plane#Other meanings of "complex plane"|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
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| :<math>\rho (a, b) = \tanh^{-1} \frac{| a - b |}{|1 - \bar{a} b |}</math>
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| 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
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| :<math>d(f(w), f(z)) = \rho (w, z) \,</math>
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| for all points ''w'' and ''z'' in Δ.
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| ==Properties and examples of complex geodesics==
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| * Given ''u'' ∈ ''X'' with ||''u''|| = 1, the map ''f'' : Δ → ''B'' given by ''f''(''z'') = ''zu'' is a complex geodesic.
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| * 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'' <small>o</small> ''g'' is also a complex geodesic. In fact, any complex geodesic ''f''<sub>1</sub> with the same image as ''f'' (i.e., ''f''<sub>1</sub>(Δ) = ''f''(Δ)) arises as such a reparametrization of ''f''.
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| * If
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| ::<math>d(f(0), f(z)) = \rho (0, z)</math> | |
| :for some ''z'' ≠ 0, then ''f'' is a complex geodesic.
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| *If
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| ::<math>\alpha (f(0), f'(0)) = 1,</math>
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| :where ''α'' denotes the Caratheodory length of a tangent vector, then ''f'' is a complex geodesic.
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| ==References== | |
| * {{cite book
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| | author = Earle, Clifford J. and Harris, Lawrence A. and Hubbard, John H. and Mitra, Sudeb
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| | chapter = Schwarz's lemma and the Kobayashi and Carathéodory pseudometrics on complex Banach manifolds
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| | title = Kleinian groups and hyperbolic 3-manifolds (Warwick, 2001)
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| | editor = Komori, Y., Markovic, V. and Series, C. (eds)
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| | series = London Math. Soc. Lecture Note Ser. 299
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| | pages = 363–384
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| |publisher = Cambridge Univ. Press
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| | location = Cambridge
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| | year = 2003
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| }}
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| [[Category:Hyperbolic geometry]]
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| [[Category:Metric geometry]]
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