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In mathematics, the original '''Kobayashi metric''' is a [[pseudometric space|pseudometric]] (or pseudodistance) on [[complex manifold]]s introduced by {{harvs|txt|authorlink=Shoshichi Kobayashi|last=Kobayashi|year=1967}}. It can be viewed as the [[dual (mathematics)|dual]] of the [[Carathéodory metric]], and has been extended to [[complex analytic space]]s and [[almost complex manifold]]s. On [[Teichmüller space]] the Kobayashi metric coincides with the [[Teichmüller metric]]; on the unit ball, it coincides with the [[Bergman metric]].

An analogous pseudodistance was constructed for flat affine and projective structures in {{harvs|txt|authorlink=Shoshichi Kobayashi|last=Kobayashi|year=1977}} and then generalized to (normal) [[projective connection]]s. Essentially the same construction has been applied to (normal, pseudo-Riemannian) [[conformal connection]]s and, more recently, to general (regular) parabolic geometries.

==Definition==

If ''X'' is a complex manifold, the Kobayashi pseudometric ''d'' may be characterized as the largest pseudometric on ''X'' such that
:<math>d(f(x),f(y)) \le \rho(x,y)</math>,
for all holomorphic maps ''f''  from the unit disk ''D'' to ''X'' (where <math> \rho(x,y)</math> denotes distance in the [[Poincaré metric]] on ''D'').

==References==

*{{Cite journal|last=Kobayashi|first=Shoshichi|title=Intrinsic Metrics on Complex Manifolds|journal=Bull. Amer. Math. Soc.|year=1967|volume=73|pages=347–349|url=http://www.ams.org/journals/bull/1967-73-03/S0002-9904-1967-11745-2/S0002-9904-1967-11745-2.pdf}}

*{{Citation | last1=Kobayashi | first1=Shoshichi | title=Hyperbolic manifolds and holomorphic mappings | url=http://books.google.com/books?id=rleQdMhML6kC | publisher=Marcel Dekker Inc. | location=New York | series=Pure and Applied Mathematics | isbn=978-0-8247-1380-5 | mr=0277770 | year=1970 | volume=2}}

*{{cite journal|first=Shoshichi|last=Kobayashi|title=Intrinsic distances associated with flat affine or projective structures|journal=J. Fac. Sci. Univ. Tokyo|year=1977|volume=24|pages=129–135| mr=445016 }}

==Further reading==
* {{cite journal | first=Serge | last=Lang | authorlink=Serge Lang | title=Hyperbolic and Diophantine anaysis | journal=[[Bulletin of the American Mathematical Society]] | volume=14 | number=2 | year=1986 | pages=159–205 | zbl=0602.14019 | url=http://www.ams.org/journals/bull/1986-14-02/S0273-0979-1986-15426-1/S0273-0979-1986-15426-1.pdf }} 

[[Category:Complex manifolds]]