Bergman metric

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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 for Stefan Bergman.

Definition[edit]

Let be a domain and let be the Bergman kernel on G. We define a Hermitian metric on the tangent bundle by

for . 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 computed as

The distance of two points 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 . That is if f is a biholomorphism of G and , then .

References[edit]

  • Steven G. Krantz. Function Theory of Several Complex Variables, AMS Chelsea Publishing, Providence, Rhode Island, 1992.

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