Bergman metric

In differential geometry, the Bergman metric is a Hermitian metric that can be defined on certain types of complex manifold.

It is so called because it is derived from the Bergman kernel, both of which are named after Stefan Bergman.

be the Bergman kernel on G. We define a Hermitian metric on the tangent bundle

Then the length of a tangent vector

is given by This metric is called the Bergman metric on G. The length of a (piecewise) C1 curve

is then defined as The distance dG is called the Bergman distance.

The Bergman metric is in fact a positive definite matrix at each point if G is a bounded domain.

More importantly, the distance dG is invariant under biholomorphic mappings of G to another domain

This article incorporates material from Bergman metric on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

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