Kobayashi metric

In mathematics, the original Kobayashi metric is a pseudometric (or pseudodistance) on complex manifolds introduced by Kobayashi (1967). It can be viewed as the dual of the Carathéodory metric, and has been extended to complex analytic spaces and almost complex manifolds. 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 Kobayashi (1977) and then generalized to (normal) projective connections. Essentially the same construction has been applied to (normal, pseudo-Riemannian) conformal connections and, more recently, to general (regular) parabolic geometries.


If X is a complex manifold, the Kobayashi pseudometric d may be characterized as the largest pseudometric on X such that


for all holomorphic maps f from the unit disk D to X (where denotes distance in the Poincaré metric on D).


Further reading

This article is issued from Wikipedia - version of the 11/3/2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.